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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Query-Efficient Fixpoints of ℓ_𝑝-Contractions</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.16089v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS1" title="In 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS2" title="In 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Proof Techniques</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3" title="In 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Discussion</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3.SSS0.Px1" title="In 1.3 Discussion ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Comparison to Results for <math alttext="p=2" class="ltx_Math" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation-xml encoding="MathML-Content"><apply><eq></eq><ci>𝑝</ci><cn type="integer">2</cn></apply></annotation-xml><annotation encoding="application/x-tex">p=2</annotation><annotation encoding="application/x-llamapun">italic_p = 2</annotation></semantics></math>.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3.SSS0.Px2" title="In 1.3 Discussion ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Comparison to Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref">5</span>]</cite> on <math alttext="p=\infty" class="ltx_Math" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mi mathvariant="normal">∞</mi></mrow><annotation-xml encoding="MathML-Content"><apply><eq></eq><ci>𝑝</ci><infinity></infinity></apply></annotation-xml><annotation encoding="application/x-tex">p=\infty</annotation><annotation encoding="application/x-llamapun">italic_p = ∞</annotation></semantics></math>.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3.SSS0.Px3" title="In 1.3 Discussion ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><math alttext="\mathsf{CLS}" class="ltx_Math" display="inline"><semantics><mi>𝖢𝖫𝖲</mi><annotation-xml encoding="MathML-Content"><ci>𝖢𝖫𝖲</ci></annotation-xml><annotation encoding="application/x-tex">\mathsf{CLS}</annotation><annotation encoding="application/x-llamapun">sansserif_CLS</annotation></semantics></math>-Completeness of General Banach.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3.SSS0.Px4" title="In 1.3 Discussion ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Finding Centerpoints.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS3.SSS0.Px5" title="In 1.3 Discussion ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Proof of Centerpoint Theorem.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS4" title="In 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.4 </span>Further Related Work</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS4.SSS0.Px1" title="In 1.4 Further Related Work ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Generalized Centerpoint Theorems.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS4.SSS0.Px2" title="In 1.4 Further Related Work ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title">Fixpoint Theorems.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.SS5" title="In 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.5 </span>Overview and Organization</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S2" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces and <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoints</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS1" title="In 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS2" title="In 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Properties of <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS3" title="In 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoints of Mass Distributions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS4" title="In 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Tightness of Centerpoint Theorems</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Finding Fixpoints of <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Contraction Maps</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS1" title="In 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Solving <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps">-ContractionFixpoint</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS2" title="In 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Rounding to the Grid in the <math alttext="\ell_{1}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mn>1</mn></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><cn type="integer">1</cn></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{1}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-Case</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS3" title="In 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Total Search Version</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>More on <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS1" title="In Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.1 </span>Fundamentals of <math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS2" title="In Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.2 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline"><semantics><msub><mi mathvariant="normal">ℓ</mi><mi>p</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℓ</ci><ci>𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex">\ell_{p}</annotation><annotation encoding="application/x-llamapun">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces and Mass Distributions</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A2" title="In Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Proof of <span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Query-Efficient Fixpoints of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="id1.m1.1"><semantics id="id1.m1.1b"><msub id="id1.m1.1.1" xref="id1.m1.1.1.cmml"><mi id="id1.m1.1.1.2" mathvariant="normal" xref="id1.m1.1.1.2.cmml">ℓ</mi><mi id="id1.m1.1.1.3" xref="id1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id1.m1.1c"><apply id="id1.m1.1.1.cmml" xref="id1.m1.1.1"><csymbol cd="ambiguous" id="id1.m1.1.1.1.cmml" xref="id1.m1.1.1">subscript</csymbol><ci id="id1.m1.1.1.2.cmml" xref="id1.m1.1.1.2">ℓ</ci><ci id="id1.m1.1.1.3.cmml" xref="id1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="id1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Contractions</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Sebastian Haslebacher </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Computer Science, ETH Zürich, Switzerland </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Jonas Lill </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Computer Science, ETH Zürich, Switzerland </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Patrick Schnider </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Computer Science, ETH Zürich, Switzerland </span> <span class="ltx_contact ltx_role_affiliation">Department of Mathematics and Computer Science, University of Basel, Switzerland </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Simon Weber </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Computer Science, ETH Zürich, Switzerland </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id15.14">We prove that an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="id2.1.m1.1"><semantics id="id2.1.m1.1a"><mi id="id2.1.m1.1.1" xref="id2.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="id2.1.m1.1b"><ci id="id2.1.m1.1.1.cmml" xref="id2.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="id2.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="id2.1.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint of a map <math alttext="f:[0,1]^{d}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="id3.2.m2.4"><semantics id="id3.2.m2.4a"><mrow id="id3.2.m2.4.5" xref="id3.2.m2.4.5.cmml"><mi id="id3.2.m2.4.5.2" xref="id3.2.m2.4.5.2.cmml">f</mi><mo id="id3.2.m2.4.5.1" lspace="0.278em" rspace="0.278em" xref="id3.2.m2.4.5.1.cmml">:</mo><mrow id="id3.2.m2.4.5.3" xref="id3.2.m2.4.5.3.cmml"><msup id="id3.2.m2.4.5.3.2" xref="id3.2.m2.4.5.3.2.cmml"><mrow id="id3.2.m2.4.5.3.2.2.2" xref="id3.2.m2.4.5.3.2.2.1.cmml"><mo id="id3.2.m2.4.5.3.2.2.2.1" stretchy="false" xref="id3.2.m2.4.5.3.2.2.1.cmml">[</mo><mn id="id3.2.m2.1.1" xref="id3.2.m2.1.1.cmml">0</mn><mo id="id3.2.m2.4.5.3.2.2.2.2" xref="id3.2.m2.4.5.3.2.2.1.cmml">,</mo><mn id="id3.2.m2.2.2" xref="id3.2.m2.2.2.cmml">1</mn><mo id="id3.2.m2.4.5.3.2.2.2.3" stretchy="false" xref="id3.2.m2.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="id3.2.m2.4.5.3.2.3" xref="id3.2.m2.4.5.3.2.3.cmml">d</mi></msup><mo id="id3.2.m2.4.5.3.1" stretchy="false" xref="id3.2.m2.4.5.3.1.cmml">→</mo><msup id="id3.2.m2.4.5.3.3" xref="id3.2.m2.4.5.3.3.cmml"><mrow id="id3.2.m2.4.5.3.3.2.2" xref="id3.2.m2.4.5.3.3.2.1.cmml"><mo id="id3.2.m2.4.5.3.3.2.2.1" stretchy="false" xref="id3.2.m2.4.5.3.3.2.1.cmml">[</mo><mn id="id3.2.m2.3.3" xref="id3.2.m2.3.3.cmml">0</mn><mo id="id3.2.m2.4.5.3.3.2.2.2" xref="id3.2.m2.4.5.3.3.2.1.cmml">,</mo><mn id="id3.2.m2.4.4" xref="id3.2.m2.4.4.cmml">1</mn><mo id="id3.2.m2.4.5.3.3.2.2.3" stretchy="false" xref="id3.2.m2.4.5.3.3.2.1.cmml">]</mo></mrow><mi id="id3.2.m2.4.5.3.3.3" xref="id3.2.m2.4.5.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="id3.2.m2.4b"><apply id="id3.2.m2.4.5.cmml" xref="id3.2.m2.4.5"><ci id="id3.2.m2.4.5.1.cmml" xref="id3.2.m2.4.5.1">:</ci><ci id="id3.2.m2.4.5.2.cmml" xref="id3.2.m2.4.5.2">𝑓</ci><apply id="id3.2.m2.4.5.3.cmml" xref="id3.2.m2.4.5.3"><ci id="id3.2.m2.4.5.3.1.cmml" xref="id3.2.m2.4.5.3.1">→</ci><apply id="id3.2.m2.4.5.3.2.cmml" xref="id3.2.m2.4.5.3.2"><csymbol cd="ambiguous" id="id3.2.m2.4.5.3.2.1.cmml" xref="id3.2.m2.4.5.3.2">superscript</csymbol><interval closure="closed" id="id3.2.m2.4.5.3.2.2.1.cmml" xref="id3.2.m2.4.5.3.2.2.2"><cn id="id3.2.m2.1.1.cmml" type="integer" xref="id3.2.m2.1.1">0</cn><cn id="id3.2.m2.2.2.cmml" type="integer" xref="id3.2.m2.2.2">1</cn></interval><ci id="id3.2.m2.4.5.3.2.3.cmml" xref="id3.2.m2.4.5.3.2.3">𝑑</ci></apply><apply id="id3.2.m2.4.5.3.3.cmml" xref="id3.2.m2.4.5.3.3"><csymbol cd="ambiguous" id="id3.2.m2.4.5.3.3.1.cmml" xref="id3.2.m2.4.5.3.3">superscript</csymbol><interval closure="closed" id="id3.2.m2.4.5.3.3.2.1.cmml" xref="id3.2.m2.4.5.3.3.2.2"><cn id="id3.2.m2.3.3.cmml" type="integer" xref="id3.2.m2.3.3">0</cn><cn id="id3.2.m2.4.4.cmml" type="integer" xref="id3.2.m2.4.4">1</cn></interval><ci id="id3.2.m2.4.5.3.3.3.cmml" xref="id3.2.m2.4.5.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.2.m2.4c">f:[0,1]^{d}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="id3.2.m2.4d">italic_f : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> can be found with <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))" class="ltx_Math" display="inline" id="id4.3.m3.1"><semantics id="id4.3.m3.1a"><mrow id="id4.3.m3.1.1" xref="id4.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="id4.3.m3.1.1.3" xref="id4.3.m3.1.1.3.cmml">𝒪</mi><mo id="id4.3.m3.1.1.2" xref="id4.3.m3.1.1.2.cmml">⁢</mo><mrow 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id="id4.3.m3.1.1.1.1.1.1.1.1.2.2" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2.cmml"><mn id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.2" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2.2.cmml">1</mn><mi id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.3" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2.3.cmml">ε</mi></mfrac></mrow><mo id="id4.3.m3.1.1.1.1.1.1.1.1.1" xref="id4.3.m3.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="id4.3.m3.1.1.1.1.1.1.1.1.3" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.cmml"><mi id="id4.3.m3.1.1.1.1.1.1.1.1.3.1" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="id4.3.m3.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mfrac id="id4.3.m3.1.1.1.1.1.1.1.1.3.2" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.2" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mrow id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.cmml"><mn id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.2" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.2.cmml">1</mn><mo id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.1" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.1.cmml">−</mo><mi id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.3" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mrow></mrow><mo id="id4.3.m3.1.1.1.1.1.1.1.3" stretchy="false" xref="id4.3.m3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="id4.3.m3.1.1.1.1.3" stretchy="false" xref="id4.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id4.3.m3.1b"><apply id="id4.3.m3.1.1.cmml" xref="id4.3.m3.1.1"><times id="id4.3.m3.1.1.2.cmml" xref="id4.3.m3.1.1.2"></times><ci id="id4.3.m3.1.1.3.cmml" xref="id4.3.m3.1.1.3">𝒪</ci><apply id="id4.3.m3.1.1.1.1.1.cmml" xref="id4.3.m3.1.1.1.1"><times id="id4.3.m3.1.1.1.1.1.2.cmml" xref="id4.3.m3.1.1.1.1.1.2"></times><apply id="id4.3.m3.1.1.1.1.1.3.cmml" xref="id4.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="id4.3.m3.1.1.1.1.1.3.1.cmml" xref="id4.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="id4.3.m3.1.1.1.1.1.3.2.cmml" xref="id4.3.m3.1.1.1.1.1.3.2">𝑑</ci><cn id="id4.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="id4.3.m3.1.1.1.1.1.3.3">2</cn></apply><apply id="id4.3.m3.1.1.1.1.1.1.1.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1"><plus id="id4.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.1"></plus><apply id="id4.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.2"><log id="id4.3.m3.1.1.1.1.1.1.1.1.2.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.1"></log><apply id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2"><divide id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2"></divide><cn id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.2.cmml" type="integer" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2.2">1</cn><ci id="id4.3.m3.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="id4.3.m3.1.1.1.1.1.1.1.1.3.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3"><log id="id4.3.m3.1.1.1.1.1.1.1.1.3.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.1"></log><apply id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2"><divide id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2"></divide><cn id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3"><minus id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="id4.3.m3.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.3.m3.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="id4.3.m3.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> queries to <math alttext="f" class="ltx_Math" display="inline" id="id5.4.m4.1"><semantics id="id5.4.m4.1a"><mi id="id5.4.m4.1.1" xref="id5.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="id5.4.m4.1b"><ci id="id5.4.m4.1.1.cmml" xref="id5.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="id5.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="id5.4.m4.1d">italic_f</annotation></semantics></math> if <math alttext="f" class="ltx_Math" display="inline" id="id6.5.m5.1"><semantics id="id6.5.m5.1a"><mi id="id6.5.m5.1.1" xref="id6.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="id6.5.m5.1b"><ci id="id6.5.m5.1.1.cmml" xref="id6.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="id6.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="id6.5.m5.1d">italic_f</annotation></semantics></math> is <math alttext="\lambda" class="ltx_Math" display="inline" id="id7.6.m6.1"><semantics id="id7.6.m6.1a"><mi id="id7.6.m6.1.1" xref="id7.6.m6.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="id7.6.m6.1b"><ci id="id7.6.m6.1.1.cmml" xref="id7.6.m6.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.6.m6.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="id7.6.m6.1d">italic_λ</annotation></semantics></math>-contracting with respect to an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="id8.7.m7.1"><semantics id="id8.7.m7.1a"><msub id="id8.7.m7.1.1" xref="id8.7.m7.1.1.cmml"><mi id="id8.7.m7.1.1.2" mathvariant="normal" xref="id8.7.m7.1.1.2.cmml">ℓ</mi><mi id="id8.7.m7.1.1.3" xref="id8.7.m7.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id8.7.m7.1b"><apply id="id8.7.m7.1.1.cmml" xref="id8.7.m7.1.1"><csymbol cd="ambiguous" id="id8.7.m7.1.1.1.cmml" xref="id8.7.m7.1.1">subscript</csymbol><ci id="id8.7.m7.1.1.2.cmml" xref="id8.7.m7.1.1.2">ℓ</ci><ci id="id8.7.m7.1.1.3.cmml" xref="id8.7.m7.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.7.m7.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="id8.7.m7.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metric for some <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="id9.8.m8.3"><semantics id="id9.8.m8.3a"><mrow id="id9.8.m8.3.4" xref="id9.8.m8.3.4.cmml"><mi id="id9.8.m8.3.4.2" xref="id9.8.m8.3.4.2.cmml">p</mi><mo id="id9.8.m8.3.4.1" xref="id9.8.m8.3.4.1.cmml">∈</mo><mrow id="id9.8.m8.3.4.3" xref="id9.8.m8.3.4.3.cmml"><mrow id="id9.8.m8.3.4.3.2.2" xref="id9.8.m8.3.4.3.2.1.cmml"><mo id="id9.8.m8.3.4.3.2.2.1" stretchy="false" xref="id9.8.m8.3.4.3.2.1.cmml">[</mo><mn id="id9.8.m8.1.1" xref="id9.8.m8.1.1.cmml">1</mn><mo id="id9.8.m8.3.4.3.2.2.2" xref="id9.8.m8.3.4.3.2.1.cmml">,</mo><mi id="id9.8.m8.2.2" mathvariant="normal" xref="id9.8.m8.2.2.cmml">∞</mi><mo id="id9.8.m8.3.4.3.2.2.3" stretchy="false" xref="id9.8.m8.3.4.3.2.1.cmml">)</mo></mrow><mo id="id9.8.m8.3.4.3.1" xref="id9.8.m8.3.4.3.1.cmml">∪</mo><mrow id="id9.8.m8.3.4.3.3.2" xref="id9.8.m8.3.4.3.3.1.cmml"><mo id="id9.8.m8.3.4.3.3.2.1" stretchy="false" xref="id9.8.m8.3.4.3.3.1.cmml">{</mo><mi id="id9.8.m8.3.3" mathvariant="normal" xref="id9.8.m8.3.3.cmml">∞</mi><mo id="id9.8.m8.3.4.3.3.2.2" stretchy="false" xref="id9.8.m8.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id9.8.m8.3b"><apply id="id9.8.m8.3.4.cmml" xref="id9.8.m8.3.4"><in id="id9.8.m8.3.4.1.cmml" xref="id9.8.m8.3.4.1"></in><ci id="id9.8.m8.3.4.2.cmml" xref="id9.8.m8.3.4.2">𝑝</ci><apply id="id9.8.m8.3.4.3.cmml" xref="id9.8.m8.3.4.3"><union id="id9.8.m8.3.4.3.1.cmml" xref="id9.8.m8.3.4.3.1"></union><interval closure="closed-open" id="id9.8.m8.3.4.3.2.1.cmml" xref="id9.8.m8.3.4.3.2.2"><cn id="id9.8.m8.1.1.cmml" type="integer" xref="id9.8.m8.1.1">1</cn><infinity id="id9.8.m8.2.2.cmml" xref="id9.8.m8.2.2"></infinity></interval><set id="id9.8.m8.3.4.3.3.1.cmml" xref="id9.8.m8.3.4.3.3.2"><infinity id="id9.8.m8.3.3.cmml" xref="id9.8.m8.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.8.m8.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="id9.8.m8.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. This generalizes a recent result of Chen, Li, and Yannakakis [STOC’24] from the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="id10.9.m9.1"><semantics id="id10.9.m9.1a"><msub id="id10.9.m9.1.1" xref="id10.9.m9.1.1.cmml"><mi id="id10.9.m9.1.1.2" mathvariant="normal" xref="id10.9.m9.1.1.2.cmml">ℓ</mi><mi id="id10.9.m9.1.1.3" mathvariant="normal" xref="id10.9.m9.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="id10.9.m9.1b"><apply id="id10.9.m9.1.1.cmml" xref="id10.9.m9.1.1"><csymbol cd="ambiguous" id="id10.9.m9.1.1.1.cmml" xref="id10.9.m9.1.1">subscript</csymbol><ci id="id10.9.m9.1.1.2.cmml" xref="id10.9.m9.1.1.2">ℓ</ci><infinity id="id10.9.m9.1.1.3.cmml" xref="id10.9.m9.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="id10.9.m9.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="id10.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case to all <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="id11.10.m10.1"><semantics id="id11.10.m10.1a"><msub id="id11.10.m10.1.1" xref="id11.10.m10.1.1.cmml"><mi id="id11.10.m10.1.1.2" mathvariant="normal" xref="id11.10.m10.1.1.2.cmml">ℓ</mi><mi id="id11.10.m10.1.1.3" xref="id11.10.m10.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id11.10.m10.1b"><apply id="id11.10.m10.1.1.cmml" xref="id11.10.m10.1.1"><csymbol cd="ambiguous" id="id11.10.m10.1.1.1.cmml" xref="id11.10.m10.1.1">subscript</csymbol><ci id="id11.10.m10.1.1.2.cmml" xref="id11.10.m10.1.1.2">ℓ</ci><ci id="id11.10.m10.1.1.3.cmml" xref="id11.10.m10.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.10.m10.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="id11.10.m10.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics. Previously, all query upper bounds for <math alttext="p\in[1,\infty)\setminus\{2\}" class="ltx_Math" display="inline" id="id12.11.m11.3"><semantics id="id12.11.m11.3a"><mrow id="id12.11.m11.3.4" xref="id12.11.m11.3.4.cmml"><mi id="id12.11.m11.3.4.2" xref="id12.11.m11.3.4.2.cmml">p</mi><mo id="id12.11.m11.3.4.1" xref="id12.11.m11.3.4.1.cmml">∈</mo><mrow id="id12.11.m11.3.4.3" xref="id12.11.m11.3.4.3.cmml"><mrow id="id12.11.m11.3.4.3.2.2" xref="id12.11.m11.3.4.3.2.1.cmml"><mo id="id12.11.m11.3.4.3.2.2.1" stretchy="false" xref="id12.11.m11.3.4.3.2.1.cmml">[</mo><mn id="id12.11.m11.1.1" xref="id12.11.m11.1.1.cmml">1</mn><mo id="id12.11.m11.3.4.3.2.2.2" xref="id12.11.m11.3.4.3.2.1.cmml">,</mo><mi id="id12.11.m11.2.2" mathvariant="normal" xref="id12.11.m11.2.2.cmml">∞</mi><mo id="id12.11.m11.3.4.3.2.2.3" stretchy="false" xref="id12.11.m11.3.4.3.2.1.cmml">)</mo></mrow><mo id="id12.11.m11.3.4.3.1" xref="id12.11.m11.3.4.3.1.cmml">∖</mo><mrow id="id12.11.m11.3.4.3.3.2" xref="id12.11.m11.3.4.3.3.1.cmml"><mo id="id12.11.m11.3.4.3.3.2.1" stretchy="false" xref="id12.11.m11.3.4.3.3.1.cmml">{</mo><mn id="id12.11.m11.3.3" xref="id12.11.m11.3.3.cmml">2</mn><mo id="id12.11.m11.3.4.3.3.2.2" stretchy="false" xref="id12.11.m11.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id12.11.m11.3b"><apply id="id12.11.m11.3.4.cmml" xref="id12.11.m11.3.4"><in id="id12.11.m11.3.4.1.cmml" xref="id12.11.m11.3.4.1"></in><ci id="id12.11.m11.3.4.2.cmml" xref="id12.11.m11.3.4.2">𝑝</ci><apply id="id12.11.m11.3.4.3.cmml" xref="id12.11.m11.3.4.3"><setdiff id="id12.11.m11.3.4.3.1.cmml" xref="id12.11.m11.3.4.3.1"></setdiff><interval closure="closed-open" id="id12.11.m11.3.4.3.2.1.cmml" xref="id12.11.m11.3.4.3.2.2"><cn id="id12.11.m11.1.1.cmml" type="integer" xref="id12.11.m11.1.1">1</cn><infinity id="id12.11.m11.2.2.cmml" xref="id12.11.m11.2.2"></infinity></interval><set id="id12.11.m11.3.4.3.3.1.cmml" xref="id12.11.m11.3.4.3.3.2"><cn id="id12.11.m11.3.3.cmml" type="integer" xref="id12.11.m11.3.3">2</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id12.11.m11.3c">p\in[1,\infty)\setminus\{2\}</annotation><annotation encoding="application/x-llamapun" id="id12.11.m11.3d">italic_p ∈ [ 1 , ∞ ) ∖ { 2 }</annotation></semantics></math> were either exponential in <math alttext="d" class="ltx_Math" display="inline" id="id13.12.m12.1"><semantics id="id13.12.m12.1a"><mi id="id13.12.m12.1.1" xref="id13.12.m12.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="id13.12.m12.1b"><ci id="id13.12.m12.1.1.cmml" xref="id13.12.m12.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="id13.12.m12.1c">d</annotation><annotation encoding="application/x-llamapun" id="id13.12.m12.1d">italic_d</annotation></semantics></math>, <math alttext="\log\frac{1}{\varepsilon}" class="ltx_Math" display="inline" id="id14.13.m13.1"><semantics id="id14.13.m13.1a"><mrow id="id14.13.m13.1.1" xref="id14.13.m13.1.1.cmml"><mi id="id14.13.m13.1.1.1" xref="id14.13.m13.1.1.1.cmml">log</mi><mo id="id14.13.m13.1.1a" lspace="0.167em" xref="id14.13.m13.1.1.cmml">⁡</mo><mfrac id="id14.13.m13.1.1.2" xref="id14.13.m13.1.1.2.cmml"><mn id="id14.13.m13.1.1.2.2" xref="id14.13.m13.1.1.2.2.cmml">1</mn><mi id="id14.13.m13.1.1.2.3" xref="id14.13.m13.1.1.2.3.cmml">ε</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="id14.13.m13.1b"><apply id="id14.13.m13.1.1.cmml" xref="id14.13.m13.1.1"><log id="id14.13.m13.1.1.1.cmml" xref="id14.13.m13.1.1.1"></log><apply id="id14.13.m13.1.1.2.cmml" xref="id14.13.m13.1.1.2"><divide id="id14.13.m13.1.1.2.1.cmml" xref="id14.13.m13.1.1.2"></divide><cn id="id14.13.m13.1.1.2.2.cmml" type="integer" xref="id14.13.m13.1.1.2.2">1</cn><ci id="id14.13.m13.1.1.2.3.cmml" xref="id14.13.m13.1.1.2.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id14.13.m13.1c">\log\frac{1}{\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="id14.13.m13.1d">roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG</annotation></semantics></math>, or <math alttext="\log\frac{1}{1-\lambda}" class="ltx_Math" display="inline" id="id15.14.m14.1"><semantics id="id15.14.m14.1a"><mrow id="id15.14.m14.1.1" xref="id15.14.m14.1.1.cmml"><mi id="id15.14.m14.1.1.1" xref="id15.14.m14.1.1.1.cmml">log</mi><mo id="id15.14.m14.1.1a" lspace="0.167em" xref="id15.14.m14.1.1.cmml">⁡</mo><mfrac id="id15.14.m14.1.1.2" xref="id15.14.m14.1.1.2.cmml"><mn id="id15.14.m14.1.1.2.2" xref="id15.14.m14.1.1.2.2.cmml">1</mn><mrow id="id15.14.m14.1.1.2.3" xref="id15.14.m14.1.1.2.3.cmml"><mn id="id15.14.m14.1.1.2.3.2" xref="id15.14.m14.1.1.2.3.2.cmml">1</mn><mo id="id15.14.m14.1.1.2.3.1" xref="id15.14.m14.1.1.2.3.1.cmml">−</mo><mi id="id15.14.m14.1.1.2.3.3" xref="id15.14.m14.1.1.2.3.3.cmml">λ</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="id15.14.m14.1b"><apply id="id15.14.m14.1.1.cmml" xref="id15.14.m14.1.1"><log id="id15.14.m14.1.1.1.cmml" xref="id15.14.m14.1.1.1"></log><apply id="id15.14.m14.1.1.2.cmml" xref="id15.14.m14.1.1.2"><divide id="id15.14.m14.1.1.2.1.cmml" xref="id15.14.m14.1.1.2"></divide><cn id="id15.14.m14.1.1.2.2.cmml" type="integer" xref="id15.14.m14.1.1.2.2">1</cn><apply id="id15.14.m14.1.1.2.3.cmml" xref="id15.14.m14.1.1.2.3"><minus id="id15.14.m14.1.1.2.3.1.cmml" xref="id15.14.m14.1.1.2.3.1"></minus><cn id="id15.14.m14.1.1.2.3.2.cmml" type="integer" xref="id15.14.m14.1.1.2.3.2">1</cn><ci id="id15.14.m14.1.1.2.3.3.cmml" xref="id15.14.m14.1.1.2.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id15.14.m14.1c">\log\frac{1}{1-\lambda}</annotation><annotation encoding="application/x-llamapun" id="id15.14.m14.1d">roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG</annotation></semantics></math>.</p> <p class="ltx_p" id="id20.19">Chen, Li, and Yannakakis also show how to ensure that all queries to <math alttext="f" class="ltx_Math" display="inline" id="id16.15.m1.1"><semantics id="id16.15.m1.1a"><mi id="id16.15.m1.1.1" xref="id16.15.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="id16.15.m1.1b"><ci id="id16.15.m1.1.1.cmml" xref="id16.15.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="id16.15.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="id16.15.m1.1d">italic_f</annotation></semantics></math> lie on a discrete grid of limited granularity in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="id17.16.m2.1"><semantics id="id17.16.m2.1a"><msub id="id17.16.m2.1.1" xref="id17.16.m2.1.1.cmml"><mi id="id17.16.m2.1.1.2" mathvariant="normal" xref="id17.16.m2.1.1.2.cmml">ℓ</mi><mi id="id17.16.m2.1.1.3" mathvariant="normal" xref="id17.16.m2.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="id17.16.m2.1b"><apply id="id17.16.m2.1.1.cmml" xref="id17.16.m2.1.1"><csymbol cd="ambiguous" id="id17.16.m2.1.1.1.cmml" xref="id17.16.m2.1.1">subscript</csymbol><ci id="id17.16.m2.1.1.2.cmml" xref="id17.16.m2.1.1.2">ℓ</ci><infinity id="id17.16.m2.1.1.3.cmml" xref="id17.16.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="id17.16.m2.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="id17.16.m2.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case. We provide such a rounding for the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="id18.17.m3.1"><semantics id="id18.17.m3.1a"><msub id="id18.17.m3.1.1" xref="id18.17.m3.1.1.cmml"><mi id="id18.17.m3.1.1.2" mathvariant="normal" xref="id18.17.m3.1.1.2.cmml">ℓ</mi><mn id="id18.17.m3.1.1.3" xref="id18.17.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="id18.17.m3.1b"><apply id="id18.17.m3.1.1.cmml" xref="id18.17.m3.1.1"><csymbol cd="ambiguous" id="id18.17.m3.1.1.1.cmml" xref="id18.17.m3.1.1">subscript</csymbol><ci id="id18.17.m3.1.1.2.cmml" xref="id18.17.m3.1.1.2">ℓ</ci><cn id="id18.17.m3.1.1.3.cmml" type="integer" xref="id18.17.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id18.17.m3.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="id18.17.m3.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, placing an appropriately defined version of the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="id19.18.m4.1"><semantics id="id19.18.m4.1a"><msub id="id19.18.m4.1.1" xref="id19.18.m4.1.1.cmml"><mi id="id19.18.m4.1.1.2" mathvariant="normal" xref="id19.18.m4.1.1.2.cmml">ℓ</mi><mn id="id19.18.m4.1.1.3" xref="id19.18.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="id19.18.m4.1b"><apply id="id19.18.m4.1.1.cmml" xref="id19.18.m4.1.1"><csymbol cd="ambiguous" id="id19.18.m4.1.1.1.cmml" xref="id19.18.m4.1.1">subscript</csymbol><ci id="id19.18.m4.1.1.2.cmml" xref="id19.18.m4.1.1.2">ℓ</ci><cn id="id19.18.m4.1.1.3.cmml" type="integer" xref="id19.18.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id19.18.m4.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="id19.18.m4.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="id20.19.m5.1"><semantics id="id20.19.m5.1a"><msup id="id20.19.m5.1.1" xref="id20.19.m5.1.1.cmml"><mi id="id20.19.m5.1.1.2" xref="id20.19.m5.1.1.2.cmml">𝖥𝖯</mi><mtext id="id20.19.m5.1.1.3" xref="id20.19.m5.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="id20.19.m5.1b"><apply id="id20.19.m5.1.1.cmml" xref="id20.19.m5.1.1"><csymbol cd="ambiguous" id="id20.19.m5.1.1.1.cmml" xref="id20.19.m5.1.1">superscript</csymbol><ci id="id20.19.m5.1.1.2.cmml" xref="id20.19.m5.1.1.2">𝖥𝖯</ci><ci id="id20.19.m5.1.1.3a.cmml" xref="id20.19.m5.1.1.3"><mtext id="id20.19.m5.1.1.3.cmml" mathsize="70%" xref="id20.19.m5.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id20.19.m5.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="id20.19.m5.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> <p class="ltx_p" id="id26.25">To prove our results, we introduce the notion of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="id21.20.m1.1"><semantics id="id21.20.m1.1a"><msub id="id21.20.m1.1.1" xref="id21.20.m1.1.1.cmml"><mi id="id21.20.m1.1.1.2" mathvariant="normal" xref="id21.20.m1.1.1.2.cmml">ℓ</mi><mi id="id21.20.m1.1.1.3" xref="id21.20.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id21.20.m1.1b"><apply id="id21.20.m1.1.1.cmml" xref="id21.20.m1.1.1"><csymbol cd="ambiguous" id="id21.20.m1.1.1.1.cmml" xref="id21.20.m1.1.1">subscript</csymbol><ci id="id21.20.m1.1.1.2.cmml" xref="id21.20.m1.1.1.2">ℓ</ci><ci id="id21.20.m1.1.1.3.cmml" xref="id21.20.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id21.20.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="id21.20.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="id22.21.m2.3"><semantics id="id22.21.m2.3a"><mrow id="id22.21.m2.3.4" xref="id22.21.m2.3.4.cmml"><mi id="id22.21.m2.3.4.2" xref="id22.21.m2.3.4.2.cmml">p</mi><mo id="id22.21.m2.3.4.1" xref="id22.21.m2.3.4.1.cmml">∈</mo><mrow id="id22.21.m2.3.4.3" xref="id22.21.m2.3.4.3.cmml"><mrow id="id22.21.m2.3.4.3.2.2" xref="id22.21.m2.3.4.3.2.1.cmml"><mo id="id22.21.m2.3.4.3.2.2.1" stretchy="false" xref="id22.21.m2.3.4.3.2.1.cmml">[</mo><mn id="id22.21.m2.1.1" xref="id22.21.m2.1.1.cmml">1</mn><mo id="id22.21.m2.3.4.3.2.2.2" xref="id22.21.m2.3.4.3.2.1.cmml">,</mo><mi id="id22.21.m2.2.2" mathvariant="normal" xref="id22.21.m2.2.2.cmml">∞</mi><mo id="id22.21.m2.3.4.3.2.2.3" stretchy="false" xref="id22.21.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="id22.21.m2.3.4.3.1" xref="id22.21.m2.3.4.3.1.cmml">∪</mo><mrow id="id22.21.m2.3.4.3.3.2" xref="id22.21.m2.3.4.3.3.1.cmml"><mo id="id22.21.m2.3.4.3.3.2.1" stretchy="false" xref="id22.21.m2.3.4.3.3.1.cmml">{</mo><mi id="id22.21.m2.3.3" mathvariant="normal" xref="id22.21.m2.3.3.cmml">∞</mi><mo id="id22.21.m2.3.4.3.3.2.2" stretchy="false" xref="id22.21.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id22.21.m2.3b"><apply id="id22.21.m2.3.4.cmml" xref="id22.21.m2.3.4"><in id="id22.21.m2.3.4.1.cmml" xref="id22.21.m2.3.4.1"></in><ci id="id22.21.m2.3.4.2.cmml" xref="id22.21.m2.3.4.2">𝑝</ci><apply id="id22.21.m2.3.4.3.cmml" xref="id22.21.m2.3.4.3"><union id="id22.21.m2.3.4.3.1.cmml" xref="id22.21.m2.3.4.3.1"></union><interval closure="closed-open" id="id22.21.m2.3.4.3.2.1.cmml" xref="id22.21.m2.3.4.3.2.2"><cn id="id22.21.m2.1.1.cmml" type="integer" xref="id22.21.m2.1.1">1</cn><infinity id="id22.21.m2.2.2.cmml" xref="id22.21.m2.2.2"></infinity></interval><set id="id22.21.m2.3.4.3.3.1.cmml" xref="id22.21.m2.3.4.3.3.2"><infinity id="id22.21.m2.3.3.cmml" xref="id22.21.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id22.21.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="id22.21.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and any mass distribution (or point set), we prove that there exists a centerpoint <math alttext="c" class="ltx_Math" display="inline" id="id23.22.m3.1"><semantics id="id23.22.m3.1a"><mi id="id23.22.m3.1.1" xref="id23.22.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="id23.22.m3.1b"><ci id="id23.22.m3.1.1.cmml" xref="id23.22.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="id23.22.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="id23.22.m3.1d">italic_c</annotation></semantics></math> such that every <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="id24.23.m4.1"><semantics id="id24.23.m4.1a"><msub id="id24.23.m4.1.1" xref="id24.23.m4.1.1.cmml"><mi id="id24.23.m4.1.1.2" mathvariant="normal" xref="id24.23.m4.1.1.2.cmml">ℓ</mi><mi id="id24.23.m4.1.1.3" xref="id24.23.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id24.23.m4.1b"><apply id="id24.23.m4.1.1.cmml" xref="id24.23.m4.1.1"><csymbol cd="ambiguous" id="id24.23.m4.1.1.1.cmml" xref="id24.23.m4.1.1">subscript</csymbol><ci id="id24.23.m4.1.1.2.cmml" xref="id24.23.m4.1.1.2">ℓ</ci><ci id="id24.23.m4.1.1.3.cmml" xref="id24.23.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id24.23.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="id24.23.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace defined by <math alttext="c" class="ltx_Math" display="inline" id="id25.24.m5.1"><semantics id="id25.24.m5.1a"><mi id="id25.24.m5.1.1" xref="id25.24.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="id25.24.m5.1b"><ci id="id25.24.m5.1.1.cmml" xref="id25.24.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="id25.24.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="id25.24.m5.1d">italic_c</annotation></semantics></math> and a normal vector contains at least a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="id26.25.m6.1"><semantics id="id26.25.m6.1a"><mfrac id="id26.25.m6.1.1" xref="id26.25.m6.1.1.cmml"><mn id="id26.25.m6.1.1.2" xref="id26.25.m6.1.1.2.cmml">1</mn><mrow id="id26.25.m6.1.1.3" xref="id26.25.m6.1.1.3.cmml"><mi id="id26.25.m6.1.1.3.2" xref="id26.25.m6.1.1.3.2.cmml">d</mi><mo id="id26.25.m6.1.1.3.1" xref="id26.25.m6.1.1.3.1.cmml">+</mo><mn id="id26.25.m6.1.1.3.3" xref="id26.25.m6.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="id26.25.m6.1b"><apply id="id26.25.m6.1.1.cmml" xref="id26.25.m6.1.1"><divide id="id26.25.m6.1.1.1.cmml" xref="id26.25.m6.1.1"></divide><cn id="id26.25.m6.1.1.2.cmml" type="integer" xref="id26.25.m6.1.1.2">1</cn><apply id="id26.25.m6.1.1.3.cmml" xref="id26.25.m6.1.1.3"><plus id="id26.25.m6.1.1.3.1.cmml" xref="id26.25.m6.1.1.3.1"></plus><ci id="id26.25.m6.1.1.3.2.cmml" xref="id26.25.m6.1.1.3.2">𝑑</ci><cn id="id26.25.m6.1.1.3.3.cmml" type="integer" xref="id26.25.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id26.25.m6.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="id26.25.m6.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of the mass (or points).</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.7">A <em class="ltx_emph ltx_font_italic" id="S1.p1.7.1">contraction map</em> is a function that maps any two points in such a way that their two images lie closer to each other than the original points. Formally, a function <math alttext="f:X\rightarrow X" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mi id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">f</mi><mo id="S1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml"><mi id="S1.p1.1.m1.1.1.3.2" xref="S1.p1.1.m1.1.1.3.2.cmml">X</mi><mo id="S1.p1.1.m1.1.1.3.1" stretchy="false" xref="S1.p1.1.m1.1.1.3.1.cmml">→</mo><mi id="S1.p1.1.m1.1.1.3.3" xref="S1.p1.1.m1.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><apply id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1"><ci id="S1.p1.1.m1.1.1.1.cmml" xref="S1.p1.1.m1.1.1.1">:</ci><ci id="S1.p1.1.m1.1.1.2.cmml" xref="S1.p1.1.m1.1.1.2">𝑓</ci><apply id="S1.p1.1.m1.1.1.3.cmml" xref="S1.p1.1.m1.1.1.3"><ci id="S1.p1.1.m1.1.1.3.1.cmml" xref="S1.p1.1.m1.1.1.3.1">→</ci><ci id="S1.p1.1.m1.1.1.3.2.cmml" xref="S1.p1.1.m1.1.1.3.2">𝑋</ci><ci id="S1.p1.1.m1.1.1.3.3.cmml" xref="S1.p1.1.m1.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">f:X\rightarrow X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_f : italic_X → italic_X</annotation></semantics></math> on a metric space <math alttext="(X,d_{X})" class="ltx_Math" display="inline" id="S1.p1.2.m2.2"><semantics id="S1.p1.2.m2.2a"><mrow id="S1.p1.2.m2.2.2.1" xref="S1.p1.2.m2.2.2.2.cmml"><mo id="S1.p1.2.m2.2.2.1.2" stretchy="false" xref="S1.p1.2.m2.2.2.2.cmml">(</mo><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">X</mi><mo id="S1.p1.2.m2.2.2.1.3" xref="S1.p1.2.m2.2.2.2.cmml">,</mo><msub id="S1.p1.2.m2.2.2.1.1" xref="S1.p1.2.m2.2.2.1.1.cmml"><mi id="S1.p1.2.m2.2.2.1.1.2" xref="S1.p1.2.m2.2.2.1.1.2.cmml">d</mi><mi id="S1.p1.2.m2.2.2.1.1.3" xref="S1.p1.2.m2.2.2.1.1.3.cmml">X</mi></msub><mo id="S1.p1.2.m2.2.2.1.4" stretchy="false" xref="S1.p1.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.2b"><interval closure="open" id="S1.p1.2.m2.2.2.2.cmml" xref="S1.p1.2.m2.2.2.1"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝑋</ci><apply id="S1.p1.2.m2.2.2.1.1.cmml" xref="S1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S1.p1.2.m2.2.2.1.1.1.cmml" xref="S1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S1.p1.2.m2.2.2.1.1.2.cmml" xref="S1.p1.2.m2.2.2.1.1.2">𝑑</ci><ci id="S1.p1.2.m2.2.2.1.1.3.cmml" xref="S1.p1.2.m2.2.2.1.1.3">𝑋</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.2c">(X,d_{X})</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.2d">( italic_X , italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT )</annotation></semantics></math> is a contraction map if there exists some <math alttext="\lambda<1" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mrow id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">λ</mi><mo id="S1.p1.3.m3.1.1.1" xref="S1.p1.3.m3.1.1.1.cmml"><</mo><mn id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><lt id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1.1"></lt><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">𝜆</ci><cn id="S1.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\lambda<1</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_λ < 1</annotation></semantics></math>, such that any two points <math alttext="x,y\in X" class="ltx_Math" display="inline" id="S1.p1.4.m4.2"><semantics id="S1.p1.4.m4.2a"><mrow id="S1.p1.4.m4.2.3" xref="S1.p1.4.m4.2.3.cmml"><mrow id="S1.p1.4.m4.2.3.2.2" xref="S1.p1.4.m4.2.3.2.1.cmml"><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">x</mi><mo id="S1.p1.4.m4.2.3.2.2.1" xref="S1.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S1.p1.4.m4.2.2" xref="S1.p1.4.m4.2.2.cmml">y</mi></mrow><mo id="S1.p1.4.m4.2.3.1" xref="S1.p1.4.m4.2.3.1.cmml">∈</mo><mi id="S1.p1.4.m4.2.3.3" xref="S1.p1.4.m4.2.3.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.2b"><apply id="S1.p1.4.m4.2.3.cmml" xref="S1.p1.4.m4.2.3"><in id="S1.p1.4.m4.2.3.1.cmml" xref="S1.p1.4.m4.2.3.1"></in><list id="S1.p1.4.m4.2.3.2.1.cmml" xref="S1.p1.4.m4.2.3.2.2"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝑥</ci><ci id="S1.p1.4.m4.2.2.cmml" xref="S1.p1.4.m4.2.2">𝑦</ci></list><ci id="S1.p1.4.m4.2.3.3.cmml" xref="S1.p1.4.m4.2.3.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.2c">x,y\in X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.2d">italic_x , italic_y ∈ italic_X</annotation></semantics></math> satisfy <math alttext="d_{X}(f(x),f(y))\leq\lambda\cdot d_{X}(x,y)" class="ltx_Math" display="inline" id="S1.p1.5.m5.6"><semantics id="S1.p1.5.m5.6a"><mrow id="S1.p1.5.m5.6.6" xref="S1.p1.5.m5.6.6.cmml"><mrow id="S1.p1.5.m5.6.6.2" xref="S1.p1.5.m5.6.6.2.cmml"><msub id="S1.p1.5.m5.6.6.2.4" xref="S1.p1.5.m5.6.6.2.4.cmml"><mi id="S1.p1.5.m5.6.6.2.4.2" xref="S1.p1.5.m5.6.6.2.4.2.cmml">d</mi><mi id="S1.p1.5.m5.6.6.2.4.3" xref="S1.p1.5.m5.6.6.2.4.3.cmml">X</mi></msub><mo id="S1.p1.5.m5.6.6.2.3" xref="S1.p1.5.m5.6.6.2.3.cmml">⁢</mo><mrow id="S1.p1.5.m5.6.6.2.2.2" xref="S1.p1.5.m5.6.6.2.2.3.cmml"><mo id="S1.p1.5.m5.6.6.2.2.2.3" stretchy="false" xref="S1.p1.5.m5.6.6.2.2.3.cmml">(</mo><mrow id="S1.p1.5.m5.5.5.1.1.1.1" xref="S1.p1.5.m5.5.5.1.1.1.1.cmml"><mi id="S1.p1.5.m5.5.5.1.1.1.1.2" xref="S1.p1.5.m5.5.5.1.1.1.1.2.cmml">f</mi><mo id="S1.p1.5.m5.5.5.1.1.1.1.1" xref="S1.p1.5.m5.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p1.5.m5.5.5.1.1.1.1.3.2" xref="S1.p1.5.m5.5.5.1.1.1.1.cmml"><mo 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id="S1.p1.5.m5.6.6.3" xref="S1.p1.5.m5.6.6.3.cmml">≤</mo><mrow id="S1.p1.5.m5.6.6.4" xref="S1.p1.5.m5.6.6.4.cmml"><mrow id="S1.p1.5.m5.6.6.4.2" xref="S1.p1.5.m5.6.6.4.2.cmml"><mi id="S1.p1.5.m5.6.6.4.2.2" xref="S1.p1.5.m5.6.6.4.2.2.cmml">λ</mi><mo id="S1.p1.5.m5.6.6.4.2.1" lspace="0.222em" rspace="0.222em" xref="S1.p1.5.m5.6.6.4.2.1.cmml">⋅</mo><msub id="S1.p1.5.m5.6.6.4.2.3" xref="S1.p1.5.m5.6.6.4.2.3.cmml"><mi id="S1.p1.5.m5.6.6.4.2.3.2" xref="S1.p1.5.m5.6.6.4.2.3.2.cmml">d</mi><mi id="S1.p1.5.m5.6.6.4.2.3.3" xref="S1.p1.5.m5.6.6.4.2.3.3.cmml">X</mi></msub></mrow><mo id="S1.p1.5.m5.6.6.4.1" xref="S1.p1.5.m5.6.6.4.1.cmml">⁢</mo><mrow id="S1.p1.5.m5.6.6.4.3.2" xref="S1.p1.5.m5.6.6.4.3.1.cmml"><mo id="S1.p1.5.m5.6.6.4.3.2.1" stretchy="false" xref="S1.p1.5.m5.6.6.4.3.1.cmml">(</mo><mi id="S1.p1.5.m5.3.3" xref="S1.p1.5.m5.3.3.cmml">x</mi><mo id="S1.p1.5.m5.6.6.4.3.2.2" xref="S1.p1.5.m5.6.6.4.3.1.cmml">,</mo><mi id="S1.p1.5.m5.4.4" xref="S1.p1.5.m5.4.4.cmml">y</mi><mo id="S1.p1.5.m5.6.6.4.3.2.3" stretchy="false" xref="S1.p1.5.m5.6.6.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.6b"><apply id="S1.p1.5.m5.6.6.cmml" xref="S1.p1.5.m5.6.6"><leq id="S1.p1.5.m5.6.6.3.cmml" xref="S1.p1.5.m5.6.6.3"></leq><apply id="S1.p1.5.m5.6.6.2.cmml" xref="S1.p1.5.m5.6.6.2"><times id="S1.p1.5.m5.6.6.2.3.cmml" xref="S1.p1.5.m5.6.6.2.3"></times><apply id="S1.p1.5.m5.6.6.2.4.cmml" xref="S1.p1.5.m5.6.6.2.4"><csymbol cd="ambiguous" id="S1.p1.5.m5.6.6.2.4.1.cmml" xref="S1.p1.5.m5.6.6.2.4">subscript</csymbol><ci id="S1.p1.5.m5.6.6.2.4.2.cmml" xref="S1.p1.5.m5.6.6.2.4.2">𝑑</ci><ci id="S1.p1.5.m5.6.6.2.4.3.cmml" xref="S1.p1.5.m5.6.6.2.4.3">𝑋</ci></apply><interval closure="open" id="S1.p1.5.m5.6.6.2.2.3.cmml" xref="S1.p1.5.m5.6.6.2.2.2"><apply id="S1.p1.5.m5.5.5.1.1.1.1.cmml" xref="S1.p1.5.m5.5.5.1.1.1.1"><times id="S1.p1.5.m5.5.5.1.1.1.1.1.cmml" xref="S1.p1.5.m5.5.5.1.1.1.1.1"></times><ci id="S1.p1.5.m5.5.5.1.1.1.1.2.cmml" xref="S1.p1.5.m5.5.5.1.1.1.1.2">𝑓</ci><ci id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1">𝑥</ci></apply><apply id="S1.p1.5.m5.6.6.2.2.2.2.cmml" xref="S1.p1.5.m5.6.6.2.2.2.2"><times id="S1.p1.5.m5.6.6.2.2.2.2.1.cmml" xref="S1.p1.5.m5.6.6.2.2.2.2.1"></times><ci id="S1.p1.5.m5.6.6.2.2.2.2.2.cmml" xref="S1.p1.5.m5.6.6.2.2.2.2.2">𝑓</ci><ci id="S1.p1.5.m5.2.2.cmml" xref="S1.p1.5.m5.2.2">𝑦</ci></apply></interval></apply><apply id="S1.p1.5.m5.6.6.4.cmml" xref="S1.p1.5.m5.6.6.4"><times id="S1.p1.5.m5.6.6.4.1.cmml" xref="S1.p1.5.m5.6.6.4.1"></times><apply id="S1.p1.5.m5.6.6.4.2.cmml" xref="S1.p1.5.m5.6.6.4.2"><ci id="S1.p1.5.m5.6.6.4.2.1.cmml" xref="S1.p1.5.m5.6.6.4.2.1">⋅</ci><ci id="S1.p1.5.m5.6.6.4.2.2.cmml" xref="S1.p1.5.m5.6.6.4.2.2">𝜆</ci><apply id="S1.p1.5.m5.6.6.4.2.3.cmml" xref="S1.p1.5.m5.6.6.4.2.3"><csymbol cd="ambiguous" id="S1.p1.5.m5.6.6.4.2.3.1.cmml" xref="S1.p1.5.m5.6.6.4.2.3">subscript</csymbol><ci id="S1.p1.5.m5.6.6.4.2.3.2.cmml" xref="S1.p1.5.m5.6.6.4.2.3.2">𝑑</ci><ci id="S1.p1.5.m5.6.6.4.2.3.3.cmml" xref="S1.p1.5.m5.6.6.4.2.3.3">𝑋</ci></apply></apply><interval closure="open" id="S1.p1.5.m5.6.6.4.3.1.cmml" xref="S1.p1.5.m5.6.6.4.3.2"><ci id="S1.p1.5.m5.3.3.cmml" xref="S1.p1.5.m5.3.3">𝑥</ci><ci id="S1.p1.5.m5.4.4.cmml" xref="S1.p1.5.m5.4.4">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.6c">d_{X}(f(x),f(y))\leq\lambda\cdot d_{X}(x,y)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.6d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_f ( italic_x ) , italic_f ( italic_y ) ) ≤ italic_λ ⋅ italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_x , italic_y )</annotation></semantics></math>. Banach’s fixpoint theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib1" title="">1</a>]</cite> famously states that a contraction map must have a unique fixpoint, i.e., a point <math alttext="x\in X" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mrow id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml"><mi id="S1.p1.6.m6.1.1.2" xref="S1.p1.6.m6.1.1.2.cmml">x</mi><mo id="S1.p1.6.m6.1.1.1" xref="S1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S1.p1.6.m6.1.1.3" xref="S1.p1.6.m6.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><apply id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1"><in id="S1.p1.6.m6.1.1.1.cmml" xref="S1.p1.6.m6.1.1.1"></in><ci id="S1.p1.6.m6.1.1.2.cmml" xref="S1.p1.6.m6.1.1.2">𝑥</ci><ci id="S1.p1.6.m6.1.1.3.cmml" xref="S1.p1.6.m6.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_x ∈ italic_X</annotation></semantics></math> with <math alttext="f(x)=x" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mrow id="S1.p1.7.m7.1.2" xref="S1.p1.7.m7.1.2.cmml"><mrow id="S1.p1.7.m7.1.2.2" xref="S1.p1.7.m7.1.2.2.cmml"><mi id="S1.p1.7.m7.1.2.2.2" xref="S1.p1.7.m7.1.2.2.2.cmml">f</mi><mo id="S1.p1.7.m7.1.2.2.1" xref="S1.p1.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S1.p1.7.m7.1.2.2.3.2" xref="S1.p1.7.m7.1.2.2.cmml"><mo id="S1.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="S1.p1.7.m7.1.2.2.cmml">(</mo><mi id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml">x</mi><mo id="S1.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="S1.p1.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p1.7.m7.1.2.1" xref="S1.p1.7.m7.1.2.1.cmml">=</mo><mi id="S1.p1.7.m7.1.2.3" xref="S1.p1.7.m7.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><apply id="S1.p1.7.m7.1.2.cmml" xref="S1.p1.7.m7.1.2"><eq id="S1.p1.7.m7.1.2.1.cmml" xref="S1.p1.7.m7.1.2.1"></eq><apply id="S1.p1.7.m7.1.2.2.cmml" xref="S1.p1.7.m7.1.2.2"><times id="S1.p1.7.m7.1.2.2.1.cmml" xref="S1.p1.7.m7.1.2.2.1"></times><ci id="S1.p1.7.m7.1.2.2.2.cmml" xref="S1.p1.7.m7.1.2.2.2">𝑓</ci><ci id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1">𝑥</ci></apply><ci id="S1.p1.7.m7.1.2.3.cmml" xref="S1.p1.7.m7.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">f(x)=x</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_f ( italic_x ) = italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.8">In this paper, we consider the problem of finding <em class="ltx_emph ltx_font_italic" id="S1.p2.8.1">approximate</em> fixpoints based on the <em class="ltx_emph ltx_font_italic" id="S1.p2.8.2">residual error criterion</em>: we wish to find any point with <math alttext="d_{X}(f(x),x)\leq\varepsilon" class="ltx_Math" display="inline" id="S1.p2.1.m1.3"><semantics id="S1.p2.1.m1.3a"><mrow id="S1.p2.1.m1.3.3" xref="S1.p2.1.m1.3.3.cmml"><mrow id="S1.p2.1.m1.3.3.1" xref="S1.p2.1.m1.3.3.1.cmml"><msub id="S1.p2.1.m1.3.3.1.3" xref="S1.p2.1.m1.3.3.1.3.cmml"><mi id="S1.p2.1.m1.3.3.1.3.2" xref="S1.p2.1.m1.3.3.1.3.2.cmml">d</mi><mi id="S1.p2.1.m1.3.3.1.3.3" xref="S1.p2.1.m1.3.3.1.3.3.cmml">X</mi></msub><mo id="S1.p2.1.m1.3.3.1.2" xref="S1.p2.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S1.p2.1.m1.3.3.1.1.1" xref="S1.p2.1.m1.3.3.1.1.2.cmml"><mo id="S1.p2.1.m1.3.3.1.1.1.2" stretchy="false" xref="S1.p2.1.m1.3.3.1.1.2.cmml">(</mo><mrow id="S1.p2.1.m1.3.3.1.1.1.1" xref="S1.p2.1.m1.3.3.1.1.1.1.cmml"><mi id="S1.p2.1.m1.3.3.1.1.1.1.2" xref="S1.p2.1.m1.3.3.1.1.1.1.2.cmml">f</mi><mo id="S1.p2.1.m1.3.3.1.1.1.1.1" xref="S1.p2.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p2.1.m1.3.3.1.1.1.1.3.2" xref="S1.p2.1.m1.3.3.1.1.1.1.cmml"><mo id="S1.p2.1.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S1.p2.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">x</mi><mo id="S1.p2.1.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S1.p2.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p2.1.m1.3.3.1.1.1.3" xref="S1.p2.1.m1.3.3.1.1.2.cmml">,</mo><mi id="S1.p2.1.m1.2.2" xref="S1.p2.1.m1.2.2.cmml">x</mi><mo id="S1.p2.1.m1.3.3.1.1.1.4" stretchy="false" xref="S1.p2.1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.p2.1.m1.3.3.2" xref="S1.p2.1.m1.3.3.2.cmml">≤</mo><mi id="S1.p2.1.m1.3.3.3" xref="S1.p2.1.m1.3.3.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.3b"><apply id="S1.p2.1.m1.3.3.cmml" xref="S1.p2.1.m1.3.3"><leq id="S1.p2.1.m1.3.3.2.cmml" xref="S1.p2.1.m1.3.3.2"></leq><apply id="S1.p2.1.m1.3.3.1.cmml" xref="S1.p2.1.m1.3.3.1"><times id="S1.p2.1.m1.3.3.1.2.cmml" xref="S1.p2.1.m1.3.3.1.2"></times><apply id="S1.p2.1.m1.3.3.1.3.cmml" xref="S1.p2.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S1.p2.1.m1.3.3.1.3.1.cmml" xref="S1.p2.1.m1.3.3.1.3">subscript</csymbol><ci id="S1.p2.1.m1.3.3.1.3.2.cmml" xref="S1.p2.1.m1.3.3.1.3.2">𝑑</ci><ci id="S1.p2.1.m1.3.3.1.3.3.cmml" xref="S1.p2.1.m1.3.3.1.3.3">𝑋</ci></apply><interval closure="open" id="S1.p2.1.m1.3.3.1.1.2.cmml" xref="S1.p2.1.m1.3.3.1.1.1"><apply id="S1.p2.1.m1.3.3.1.1.1.1.cmml" xref="S1.p2.1.m1.3.3.1.1.1.1"><times id="S1.p2.1.m1.3.3.1.1.1.1.1.cmml" xref="S1.p2.1.m1.3.3.1.1.1.1.1"></times><ci id="S1.p2.1.m1.3.3.1.1.1.1.2.cmml" xref="S1.p2.1.m1.3.3.1.1.1.1.2">𝑓</ci><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝑥</ci></apply><ci id="S1.p2.1.m1.2.2.cmml" xref="S1.p2.1.m1.2.2">𝑥</ci></interval></apply><ci id="S1.p2.1.m1.3.3.3.cmml" xref="S1.p2.1.m1.3.3.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.3c">d_{X}(f(x),x)\leq\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.3d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_f ( italic_x ) , italic_x ) ≤ italic_ε</annotation></semantics></math>, called an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. Banach’s proof actually provides a simple algorithm to find such an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_ε</annotation></semantics></math>-approximate fixpoint: start at any point <math alttext="x\in X" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mrow id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml"><mi id="S1.p2.4.m4.1.1.2" xref="S1.p2.4.m4.1.1.2.cmml">x</mi><mo id="S1.p2.4.m4.1.1.1" xref="S1.p2.4.m4.1.1.1.cmml">∈</mo><mi id="S1.p2.4.m4.1.1.3" xref="S1.p2.4.m4.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><apply id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1"><in id="S1.p2.4.m4.1.1.1.cmml" xref="S1.p2.4.m4.1.1.1"></in><ci id="S1.p2.4.m4.1.1.2.cmml" xref="S1.p2.4.m4.1.1.2">𝑥</ci><ci id="S1.p2.4.m4.1.1.3.cmml" xref="S1.p2.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_x ∈ italic_X</annotation></semantics></math> and iteratively apply <math alttext="f" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_f</annotation></semantics></math> until <math alttext="d_{X}(f^{k+1}(x),f^{k}(x))\leq\varepsilon" class="ltx_Math" display="inline" id="S1.p2.6.m6.4"><semantics id="S1.p2.6.m6.4a"><mrow id="S1.p2.6.m6.4.4" xref="S1.p2.6.m6.4.4.cmml"><mrow id="S1.p2.6.m6.4.4.2" xref="S1.p2.6.m6.4.4.2.cmml"><msub id="S1.p2.6.m6.4.4.2.4" xref="S1.p2.6.m6.4.4.2.4.cmml"><mi id="S1.p2.6.m6.4.4.2.4.2" xref="S1.p2.6.m6.4.4.2.4.2.cmml">d</mi><mi id="S1.p2.6.m6.4.4.2.4.3" xref="S1.p2.6.m6.4.4.2.4.3.cmml">X</mi></msub><mo id="S1.p2.6.m6.4.4.2.3" xref="S1.p2.6.m6.4.4.2.3.cmml">⁢</mo><mrow id="S1.p2.6.m6.4.4.2.2.2" xref="S1.p2.6.m6.4.4.2.2.3.cmml"><mo id="S1.p2.6.m6.4.4.2.2.2.3" stretchy="false" xref="S1.p2.6.m6.4.4.2.2.3.cmml">(</mo><mrow id="S1.p2.6.m6.3.3.1.1.1.1" xref="S1.p2.6.m6.3.3.1.1.1.1.cmml"><msup id="S1.p2.6.m6.3.3.1.1.1.1.2" xref="S1.p2.6.m6.3.3.1.1.1.1.2.cmml"><mi id="S1.p2.6.m6.3.3.1.1.1.1.2.2" xref="S1.p2.6.m6.3.3.1.1.1.1.2.2.cmml">f</mi><mrow id="S1.p2.6.m6.3.3.1.1.1.1.2.3" xref="S1.p2.6.m6.3.3.1.1.1.1.2.3.cmml"><mi 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id="S1.p2.6.m6.4.4.2.2.2.2.2.3" xref="S1.p2.6.m6.4.4.2.2.2.2.2.3.cmml">k</mi></msup><mo id="S1.p2.6.m6.4.4.2.2.2.2.1" xref="S1.p2.6.m6.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S1.p2.6.m6.4.4.2.2.2.2.3.2" xref="S1.p2.6.m6.4.4.2.2.2.2.cmml"><mo id="S1.p2.6.m6.4.4.2.2.2.2.3.2.1" stretchy="false" xref="S1.p2.6.m6.4.4.2.2.2.2.cmml">(</mo><mi id="S1.p2.6.m6.2.2" xref="S1.p2.6.m6.2.2.cmml">x</mi><mo id="S1.p2.6.m6.4.4.2.2.2.2.3.2.2" stretchy="false" xref="S1.p2.6.m6.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p2.6.m6.4.4.2.2.2.5" stretchy="false" xref="S1.p2.6.m6.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S1.p2.6.m6.4.4.3" xref="S1.p2.6.m6.4.4.3.cmml">≤</mo><mi id="S1.p2.6.m6.4.4.4" xref="S1.p2.6.m6.4.4.4.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.4b"><apply id="S1.p2.6.m6.4.4.cmml" xref="S1.p2.6.m6.4.4"><leq id="S1.p2.6.m6.4.4.3.cmml" xref="S1.p2.6.m6.4.4.3"></leq><apply id="S1.p2.6.m6.4.4.2.cmml" xref="S1.p2.6.m6.4.4.2"><times id="S1.p2.6.m6.4.4.2.3.cmml" xref="S1.p2.6.m6.4.4.2.3"></times><apply id="S1.p2.6.m6.4.4.2.4.cmml" xref="S1.p2.6.m6.4.4.2.4"><csymbol cd="ambiguous" id="S1.p2.6.m6.4.4.2.4.1.cmml" xref="S1.p2.6.m6.4.4.2.4">subscript</csymbol><ci id="S1.p2.6.m6.4.4.2.4.2.cmml" xref="S1.p2.6.m6.4.4.2.4.2">𝑑</ci><ci id="S1.p2.6.m6.4.4.2.4.3.cmml" xref="S1.p2.6.m6.4.4.2.4.3">𝑋</ci></apply><interval closure="open" id="S1.p2.6.m6.4.4.2.2.3.cmml" xref="S1.p2.6.m6.4.4.2.2.2"><apply id="S1.p2.6.m6.3.3.1.1.1.1.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1"><times id="S1.p2.6.m6.3.3.1.1.1.1.1.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.1"></times><apply id="S1.p2.6.m6.3.3.1.1.1.1.2.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.p2.6.m6.3.3.1.1.1.1.2.1.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2">superscript</csymbol><ci id="S1.p2.6.m6.3.3.1.1.1.1.2.2.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2.2">𝑓</ci><apply id="S1.p2.6.m6.3.3.1.1.1.1.2.3.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2.3"><plus id="S1.p2.6.m6.3.3.1.1.1.1.2.3.1.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2.3.1"></plus><ci id="S1.p2.6.m6.3.3.1.1.1.1.2.3.2.cmml" xref="S1.p2.6.m6.3.3.1.1.1.1.2.3.2">𝑘</ci><cn id="S1.p2.6.m6.3.3.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.p2.6.m6.3.3.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝑥</ci></apply><apply id="S1.p2.6.m6.4.4.2.2.2.2.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2"><times id="S1.p2.6.m6.4.4.2.2.2.2.1.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2.1"></times><apply id="S1.p2.6.m6.4.4.2.2.2.2.2.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.p2.6.m6.4.4.2.2.2.2.2.1.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2.2">superscript</csymbol><ci id="S1.p2.6.m6.4.4.2.2.2.2.2.2.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2.2.2">𝑓</ci><ci id="S1.p2.6.m6.4.4.2.2.2.2.2.3.cmml" xref="S1.p2.6.m6.4.4.2.2.2.2.2.3">𝑘</ci></apply><ci id="S1.p2.6.m6.2.2.cmml" xref="S1.p2.6.m6.2.2">𝑥</ci></apply></interval></apply><ci id="S1.p2.6.m6.4.4.4.cmml" xref="S1.p2.6.m6.4.4.4">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.4c">d_{X}(f^{k+1}(x),f^{k}(x))\leq\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.4d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_f start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT ( italic_x ) , italic_f start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_x ) ) ≤ italic_ε</annotation></semantics></math>. This algorithm requires at most <math alttext="\mathcal{O}\left(\frac{\log(\frac{1}{\varepsilon})}{\log(\frac{1}{\lambda})}\right)" class="ltx_Math" display="inline" id="S1.p2.7.m7.4"><semantics id="S1.p2.7.m7.4a"><mrow id="S1.p2.7.m7.4.5" xref="S1.p2.7.m7.4.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.7.m7.4.5.2" xref="S1.p2.7.m7.4.5.2.cmml">𝒪</mi><mo id="S1.p2.7.m7.4.5.1" xref="S1.p2.7.m7.4.5.1.cmml">⁢</mo><mrow id="S1.p2.7.m7.4.5.3.2" xref="S1.p2.7.m7.4.4.cmml"><mo id="S1.p2.7.m7.4.5.3.2.1" xref="S1.p2.7.m7.4.4.cmml">(</mo><mfrac id="S1.p2.7.m7.4.4" xref="S1.p2.7.m7.4.4.cmml"><mrow id="S1.p2.7.m7.2.2.2.4" xref="S1.p2.7.m7.2.2.2.3.cmml"><mi id="S1.p2.7.m7.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.cmml">log</mi><mo id="S1.p2.7.m7.2.2.2.4a" xref="S1.p2.7.m7.2.2.2.3.cmml">⁡</mo><mrow id="S1.p2.7.m7.2.2.2.4.1" xref="S1.p2.7.m7.2.2.2.3.cmml"><mo id="S1.p2.7.m7.2.2.2.4.1.1" stretchy="false" xref="S1.p2.7.m7.2.2.2.3.cmml">(</mo><mfrac id="S1.p2.7.m7.2.2.2.2" xref="S1.p2.7.m7.2.2.2.2.cmml"><mn id="S1.p2.7.m7.2.2.2.2.2" xref="S1.p2.7.m7.2.2.2.2.2.cmml">1</mn><mi id="S1.p2.7.m7.2.2.2.2.3" xref="S1.p2.7.m7.2.2.2.2.3.cmml">ε</mi></mfrac><mo id="S1.p2.7.m7.2.2.2.4.1.2" stretchy="false" xref="S1.p2.7.m7.2.2.2.3.cmml">)</mo></mrow></mrow><mrow id="S1.p2.7.m7.4.4.4.4" xref="S1.p2.7.m7.4.4.4.3.cmml"><mi id="S1.p2.7.m7.3.3.3.1" xref="S1.p2.7.m7.3.3.3.1.cmml">log</mi><mo id="S1.p2.7.m7.4.4.4.4a" xref="S1.p2.7.m7.4.4.4.3.cmml">⁡</mo><mrow id="S1.p2.7.m7.4.4.4.4.1" xref="S1.p2.7.m7.4.4.4.3.cmml"><mo id="S1.p2.7.m7.4.4.4.4.1.1" stretchy="false" xref="S1.p2.7.m7.4.4.4.3.cmml">(</mo><mfrac id="S1.p2.7.m7.4.4.4.2" xref="S1.p2.7.m7.4.4.4.2.cmml"><mn id="S1.p2.7.m7.4.4.4.2.2" xref="S1.p2.7.m7.4.4.4.2.2.cmml">1</mn><mi id="S1.p2.7.m7.4.4.4.2.3" xref="S1.p2.7.m7.4.4.4.2.3.cmml">λ</mi></mfrac><mo id="S1.p2.7.m7.4.4.4.4.1.2" stretchy="false" xref="S1.p2.7.m7.4.4.4.3.cmml">)</mo></mrow></mrow></mfrac><mo id="S1.p2.7.m7.4.5.3.2.2" xref="S1.p2.7.m7.4.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.4b"><apply id="S1.p2.7.m7.4.5.cmml" xref="S1.p2.7.m7.4.5"><times id="S1.p2.7.m7.4.5.1.cmml" xref="S1.p2.7.m7.4.5.1"></times><ci id="S1.p2.7.m7.4.5.2.cmml" xref="S1.p2.7.m7.4.5.2">𝒪</ci><apply id="S1.p2.7.m7.4.4.cmml" xref="S1.p2.7.m7.4.5.3.2"><divide id="S1.p2.7.m7.4.4.5.cmml" xref="S1.p2.7.m7.4.5.3.2"></divide><apply id="S1.p2.7.m7.2.2.2.3.cmml" xref="S1.p2.7.m7.2.2.2.4"><log id="S1.p2.7.m7.1.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1.1"></log><apply id="S1.p2.7.m7.2.2.2.2.cmml" xref="S1.p2.7.m7.2.2.2.2"><divide id="S1.p2.7.m7.2.2.2.2.1.cmml" xref="S1.p2.7.m7.2.2.2.2"></divide><cn id="S1.p2.7.m7.2.2.2.2.2.cmml" type="integer" xref="S1.p2.7.m7.2.2.2.2.2">1</cn><ci id="S1.p2.7.m7.2.2.2.2.3.cmml" xref="S1.p2.7.m7.2.2.2.2.3">𝜀</ci></apply></apply><apply id="S1.p2.7.m7.4.4.4.3.cmml" xref="S1.p2.7.m7.4.4.4.4"><log id="S1.p2.7.m7.3.3.3.1.cmml" xref="S1.p2.7.m7.3.3.3.1"></log><apply id="S1.p2.7.m7.4.4.4.2.cmml" xref="S1.p2.7.m7.4.4.4.2"><divide id="S1.p2.7.m7.4.4.4.2.1.cmml" xref="S1.p2.7.m7.4.4.4.2"></divide><cn id="S1.p2.7.m7.4.4.4.2.2.cmml" type="integer" xref="S1.p2.7.m7.4.4.4.2.2">1</cn><ci id="S1.p2.7.m7.4.4.4.2.3.cmml" xref="S1.p2.7.m7.4.4.4.2.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.4c">\mathcal{O}\left(\frac{\log(\frac{1}{\varepsilon})}{\log(\frac{1}{\lambda})}\right)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.4d">caligraphic_O ( divide start_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) end_ARG start_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG italic_λ end_ARG ) end_ARG )</annotation></semantics></math> queries to <math alttext="f" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><mi id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><ci id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.1d">italic_f</annotation></semantics></math> on metric spaces of constant diameter.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.18">Unfortunately, Banach’s iterative algorithm is quite slow when <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_λ</annotation></semantics></math> is close to <math alttext="1" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mn id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><cn id="S1.p3.2.m2.1.1.cmml" type="integer" xref="S1.p3.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">1</annotation></semantics></math>, which is often the regime that is useful in practice <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib36" title="">36</a>]</cite>. For example, Simple Stochastic Games (SSG) reduce to approximating the fixpoint of a contraction map on the metric space induced on the unit cube <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S1.p3.3.m3.2"><semantics id="S1.p3.3.m3.2a"><msup id="S1.p3.3.m3.2.3" xref="S1.p3.3.m3.2.3.cmml"><mrow id="S1.p3.3.m3.2.3.2.2" xref="S1.p3.3.m3.2.3.2.1.cmml"><mo id="S1.p3.3.m3.2.3.2.2.1" stretchy="false" xref="S1.p3.3.m3.2.3.2.1.cmml">[</mo><mn id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">0</mn><mo id="S1.p3.3.m3.2.3.2.2.2" xref="S1.p3.3.m3.2.3.2.1.cmml">,</mo><mn id="S1.p3.3.m3.2.2" xref="S1.p3.3.m3.2.2.cmml">1</mn><mo id="S1.p3.3.m3.2.3.2.2.3" stretchy="false" xref="S1.p3.3.m3.2.3.2.1.cmml">]</mo></mrow><mi id="S1.p3.3.m3.2.3.3" xref="S1.p3.3.m3.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.2b"><apply id="S1.p3.3.m3.2.3.cmml" xref="S1.p3.3.m3.2.3"><csymbol cd="ambiguous" id="S1.p3.3.m3.2.3.1.cmml" xref="S1.p3.3.m3.2.3">superscript</csymbol><interval closure="closed" id="S1.p3.3.m3.2.3.2.1.cmml" xref="S1.p3.3.m3.2.3.2.2"><cn id="S1.p3.3.m3.1.1.cmml" type="integer" xref="S1.p3.3.m3.1.1">0</cn><cn id="S1.p3.3.m3.2.2.cmml" type="integer" xref="S1.p3.3.m3.2.2">1</cn></interval><ci id="S1.p3.3.m3.2.3.3.cmml" xref="S1.p3.3.m3.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> by the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><msub id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mi id="S1.p3.4.m4.1.1.2" mathvariant="normal" xref="S1.p3.4.m4.1.1.2.cmml">ℓ</mi><mi id="S1.p3.4.m4.1.1.3" mathvariant="normal" xref="S1.p3.4.m4.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1">subscript</csymbol><ci id="S1.p3.4.m4.1.1.2.cmml" xref="S1.p3.4.m4.1.1.2">ℓ</ci><infinity id="S1.p3.4.m4.1.1.3.cmml" xref="S1.p3.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-norm, with <math alttext="1-\lambda" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mrow id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><mn id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml">1</mn><mo id="S1.p3.5.m5.1.1.1" xref="S1.p3.5.m5.1.1.1.cmml">−</mo><mi id="S1.p3.5.m5.1.1.3" xref="S1.p3.5.m5.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><apply id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1"><minus id="S1.p3.5.m5.1.1.1.cmml" xref="S1.p3.5.m5.1.1.1"></minus><cn id="S1.p3.5.m5.1.1.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.2">1</cn><ci id="S1.p3.5.m5.1.1.3.cmml" xref="S1.p3.5.m5.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">1-\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">1 - italic_λ</annotation></semantics></math> and <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mi id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><ci id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_ε</annotation></semantics></math> exponentially small <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib7" title="">7</a>]</cite>. Similarly, the ARRIVAL problem reduces to approximating the fixpoint of a contraction map on the unit cube <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S1.p3.7.m7.2"><semantics id="S1.p3.7.m7.2a"><msup id="S1.p3.7.m7.2.3" xref="S1.p3.7.m7.2.3.cmml"><mrow id="S1.p3.7.m7.2.3.2.2" xref="S1.p3.7.m7.2.3.2.1.cmml"><mo id="S1.p3.7.m7.2.3.2.2.1" stretchy="false" xref="S1.p3.7.m7.2.3.2.1.cmml">[</mo><mn id="S1.p3.7.m7.1.1" xref="S1.p3.7.m7.1.1.cmml">0</mn><mo id="S1.p3.7.m7.2.3.2.2.2" xref="S1.p3.7.m7.2.3.2.1.cmml">,</mo><mn id="S1.p3.7.m7.2.2" xref="S1.p3.7.m7.2.2.cmml">1</mn><mo id="S1.p3.7.m7.2.3.2.2.3" stretchy="false" xref="S1.p3.7.m7.2.3.2.1.cmml">]</mo></mrow><mi id="S1.p3.7.m7.2.3.3" xref="S1.p3.7.m7.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.7.m7.2b"><apply id="S1.p3.7.m7.2.3.cmml" xref="S1.p3.7.m7.2.3"><csymbol cd="ambiguous" id="S1.p3.7.m7.2.3.1.cmml" xref="S1.p3.7.m7.2.3">superscript</csymbol><interval closure="closed" id="S1.p3.7.m7.2.3.2.1.cmml" xref="S1.p3.7.m7.2.3.2.2"><cn id="S1.p3.7.m7.1.1.cmml" type="integer" xref="S1.p3.7.m7.1.1">0</cn><cn id="S1.p3.7.m7.2.2.cmml" type="integer" xref="S1.p3.7.m7.2.2">1</cn></interval><ci id="S1.p3.7.m7.2.3.3.cmml" xref="S1.p3.7.m7.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m7.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m7.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, again with <math alttext="1-\lambda" class="ltx_Math" display="inline" id="S1.p3.8.m8.1"><semantics id="S1.p3.8.m8.1a"><mrow id="S1.p3.8.m8.1.1" xref="S1.p3.8.m8.1.1.cmml"><mn id="S1.p3.8.m8.1.1.2" xref="S1.p3.8.m8.1.1.2.cmml">1</mn><mo id="S1.p3.8.m8.1.1.1" xref="S1.p3.8.m8.1.1.1.cmml">−</mo><mi id="S1.p3.8.m8.1.1.3" xref="S1.p3.8.m8.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.8.m8.1b"><apply id="S1.p3.8.m8.1.1.cmml" xref="S1.p3.8.m8.1.1"><minus id="S1.p3.8.m8.1.1.1.cmml" xref="S1.p3.8.m8.1.1.1"></minus><cn id="S1.p3.8.m8.1.1.2.cmml" type="integer" xref="S1.p3.8.m8.1.1.2">1</cn><ci id="S1.p3.8.m8.1.1.3.cmml" xref="S1.p3.8.m8.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m8.1c">1-\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m8.1d">1 - italic_λ</annotation></semantics></math> and <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p3.9.m9.1"><semantics id="S1.p3.9.m9.1a"><mi id="S1.p3.9.m9.1.1" xref="S1.p3.9.m9.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p3.9.m9.1b"><ci id="S1.p3.9.m9.1.1.cmml" xref="S1.p3.9.m9.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m9.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m9.1d">italic_ε</annotation></semantics></math> exponentially small, but here the metric is induced by the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.p3.10.m10.1"><semantics id="S1.p3.10.m10.1a"><msub id="S1.p3.10.m10.1.1" xref="S1.p3.10.m10.1.1.cmml"><mi id="S1.p3.10.m10.1.1.2" mathvariant="normal" xref="S1.p3.10.m10.1.1.2.cmml">ℓ</mi><mn id="S1.p3.10.m10.1.1.3" xref="S1.p3.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p3.10.m10.1b"><apply id="S1.p3.10.m10.1.1.cmml" xref="S1.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S1.p3.10.m10.1.1.1.cmml" xref="S1.p3.10.m10.1.1">subscript</csymbol><ci id="S1.p3.10.m10.1.1.2.cmml" xref="S1.p3.10.m10.1.1.2">ℓ</ci><cn id="S1.p3.10.m10.1.1.3.cmml" type="integer" xref="S1.p3.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.10.m10.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.10.m10.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-norm instead <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib17" title="">17</a>]</cite>. Note that no polynomial-time algorithms are known for SSG and ARRIVAL, despite both of them being contained in <math alttext="\mathsf{NP}\cap\mathsf{CoNP}" class="ltx_Math" display="inline" id="S1.p3.11.m11.1"><semantics id="S1.p3.11.m11.1a"><mrow id="S1.p3.11.m11.1.1" xref="S1.p3.11.m11.1.1.cmml"><mi id="S1.p3.11.m11.1.1.2" xref="S1.p3.11.m11.1.1.2.cmml">𝖭𝖯</mi><mo id="S1.p3.11.m11.1.1.1" xref="S1.p3.11.m11.1.1.1.cmml">∩</mo><mi id="S1.p3.11.m11.1.1.3" xref="S1.p3.11.m11.1.1.3.cmml">𝖢𝗈𝖭𝖯</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.11.m11.1b"><apply id="S1.p3.11.m11.1.1.cmml" xref="S1.p3.11.m11.1.1"><intersect id="S1.p3.11.m11.1.1.1.cmml" xref="S1.p3.11.m11.1.1.1"></intersect><ci id="S1.p3.11.m11.1.1.2.cmml" xref="S1.p3.11.m11.1.1.2">𝖭𝖯</ci><ci id="S1.p3.11.m11.1.1.3.cmml" xref="S1.p3.11.m11.1.1.3">𝖢𝗈𝖭𝖯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.11.m11.1c">\mathsf{NP}\cap\mathsf{CoNP}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.11.m11.1d">sansserif_NP ∩ sansserif_CoNP</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib10" title="">10</a>]</cite>. If we could approximate the fixpoint of a <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.p3.12.m12.1"><semantics id="S1.p3.12.m12.1a"><mi id="S1.p3.12.m12.1.1" xref="S1.p3.12.m12.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.12.m12.1b"><ci id="S1.p3.12.m12.1.1.cmml" xref="S1.p3.12.m12.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.12.m12.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p3.12.m12.1d">italic_λ</annotation></semantics></math>-contraction map on <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S1.p3.13.m13.2"><semantics id="S1.p3.13.m13.2a"><msup id="S1.p3.13.m13.2.3" xref="S1.p3.13.m13.2.3.cmml"><mrow id="S1.p3.13.m13.2.3.2.2" xref="S1.p3.13.m13.2.3.2.1.cmml"><mo id="S1.p3.13.m13.2.3.2.2.1" stretchy="false" xref="S1.p3.13.m13.2.3.2.1.cmml">[</mo><mn id="S1.p3.13.m13.1.1" xref="S1.p3.13.m13.1.1.cmml">0</mn><mo id="S1.p3.13.m13.2.3.2.2.2" xref="S1.p3.13.m13.2.3.2.1.cmml">,</mo><mn id="S1.p3.13.m13.2.2" xref="S1.p3.13.m13.2.2.cmml">1</mn><mo id="S1.p3.13.m13.2.3.2.2.3" stretchy="false" xref="S1.p3.13.m13.2.3.2.1.cmml">]</mo></mrow><mi id="S1.p3.13.m13.2.3.3" xref="S1.p3.13.m13.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.13.m13.2b"><apply id="S1.p3.13.m13.2.3.cmml" xref="S1.p3.13.m13.2.3"><csymbol cd="ambiguous" id="S1.p3.13.m13.2.3.1.cmml" xref="S1.p3.13.m13.2.3">superscript</csymbol><interval closure="closed" id="S1.p3.13.m13.2.3.2.1.cmml" xref="S1.p3.13.m13.2.3.2.2"><cn id="S1.p3.13.m13.1.1.cmml" type="integer" xref="S1.p3.13.m13.1.1">0</cn><cn id="S1.p3.13.m13.2.2.cmml" type="integer" xref="S1.p3.13.m13.2.2">1</cn></interval><ci id="S1.p3.13.m13.2.3.3.cmml" xref="S1.p3.13.m13.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.13.m13.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.13.m13.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with respect to an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.p3.14.m14.1"><semantics id="S1.p3.14.m14.1a"><msub id="S1.p3.14.m14.1.1" xref="S1.p3.14.m14.1.1.cmml"><mi id="S1.p3.14.m14.1.1.2" mathvariant="normal" xref="S1.p3.14.m14.1.1.2.cmml">ℓ</mi><mi id="S1.p3.14.m14.1.1.3" xref="S1.p3.14.m14.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p3.14.m14.1b"><apply id="S1.p3.14.m14.1.1.cmml" xref="S1.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S1.p3.14.m14.1.1.1.cmml" xref="S1.p3.14.m14.1.1">subscript</csymbol><ci id="S1.p3.14.m14.1.1.2.cmml" xref="S1.p3.14.m14.1.1.2">ℓ</ci><ci id="S1.p3.14.m14.1.1.3.cmml" xref="S1.p3.14.m14.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.14.m14.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.14.m14.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm in time polynomial in <math alttext="d" class="ltx_Math" display="inline" id="S1.p3.15.m15.1"><semantics id="S1.p3.15.m15.1a"><mi id="S1.p3.15.m15.1.1" xref="S1.p3.15.m15.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.p3.15.m15.1b"><ci id="S1.p3.15.m15.1.1.cmml" xref="S1.p3.15.m15.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.15.m15.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.p3.15.m15.1d">italic_d</annotation></semantics></math>, <math alttext="\log(\frac{1}{\varepsilon})" class="ltx_Math" display="inline" id="S1.p3.16.m16.2"><semantics id="S1.p3.16.m16.2a"><mrow id="S1.p3.16.m16.2.3.2" xref="S1.p3.16.m16.2.3.1.cmml"><mi id="S1.p3.16.m16.1.1" xref="S1.p3.16.m16.1.1.cmml">log</mi><mo id="S1.p3.16.m16.2.3.2a" xref="S1.p3.16.m16.2.3.1.cmml">⁡</mo><mrow id="S1.p3.16.m16.2.3.2.1" xref="S1.p3.16.m16.2.3.1.cmml"><mo id="S1.p3.16.m16.2.3.2.1.1" stretchy="false" xref="S1.p3.16.m16.2.3.1.cmml">(</mo><mfrac id="S1.p3.16.m16.2.2" xref="S1.p3.16.m16.2.2.cmml"><mn id="S1.p3.16.m16.2.2.2" xref="S1.p3.16.m16.2.2.2.cmml">1</mn><mi id="S1.p3.16.m16.2.2.3" xref="S1.p3.16.m16.2.2.3.cmml">ε</mi></mfrac><mo id="S1.p3.16.m16.2.3.2.1.2" stretchy="false" xref="S1.p3.16.m16.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.16.m16.2b"><apply id="S1.p3.16.m16.2.3.1.cmml" xref="S1.p3.16.m16.2.3.2"><log id="S1.p3.16.m16.1.1.cmml" xref="S1.p3.16.m16.1.1"></log><apply id="S1.p3.16.m16.2.2.cmml" xref="S1.p3.16.m16.2.2"><divide id="S1.p3.16.m16.2.2.1.cmml" xref="S1.p3.16.m16.2.2"></divide><cn id="S1.p3.16.m16.2.2.2.cmml" type="integer" xref="S1.p3.16.m16.2.2.2">1</cn><ci id="S1.p3.16.m16.2.2.3.cmml" xref="S1.p3.16.m16.2.2.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.16.m16.2c">\log(\frac{1}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S1.p3.16.m16.2d">roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math>, and <math alttext="\log(\frac{1}{1-\lambda})" class="ltx_Math" display="inline" id="S1.p3.17.m17.2"><semantics id="S1.p3.17.m17.2a"><mrow id="S1.p3.17.m17.2.3.2" xref="S1.p3.17.m17.2.3.1.cmml"><mi id="S1.p3.17.m17.1.1" xref="S1.p3.17.m17.1.1.cmml">log</mi><mo id="S1.p3.17.m17.2.3.2a" xref="S1.p3.17.m17.2.3.1.cmml">⁡</mo><mrow id="S1.p3.17.m17.2.3.2.1" xref="S1.p3.17.m17.2.3.1.cmml"><mo id="S1.p3.17.m17.2.3.2.1.1" stretchy="false" xref="S1.p3.17.m17.2.3.1.cmml">(</mo><mfrac id="S1.p3.17.m17.2.2" xref="S1.p3.17.m17.2.2.cmml"><mn id="S1.p3.17.m17.2.2.2" xref="S1.p3.17.m17.2.2.2.cmml">1</mn><mrow id="S1.p3.17.m17.2.2.3" xref="S1.p3.17.m17.2.2.3.cmml"><mn id="S1.p3.17.m17.2.2.3.2" xref="S1.p3.17.m17.2.2.3.2.cmml">1</mn><mo id="S1.p3.17.m17.2.2.3.1" xref="S1.p3.17.m17.2.2.3.1.cmml">−</mo><mi id="S1.p3.17.m17.2.2.3.3" xref="S1.p3.17.m17.2.2.3.3.cmml">λ</mi></mrow></mfrac><mo id="S1.p3.17.m17.2.3.2.1.2" stretchy="false" xref="S1.p3.17.m17.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.17.m17.2b"><apply id="S1.p3.17.m17.2.3.1.cmml" xref="S1.p3.17.m17.2.3.2"><log id="S1.p3.17.m17.1.1.cmml" xref="S1.p3.17.m17.1.1"></log><apply id="S1.p3.17.m17.2.2.cmml" xref="S1.p3.17.m17.2.2"><divide id="S1.p3.17.m17.2.2.1.cmml" xref="S1.p3.17.m17.2.2"></divide><cn id="S1.p3.17.m17.2.2.2.cmml" type="integer" xref="S1.p3.17.m17.2.2.2">1</cn><apply id="S1.p3.17.m17.2.2.3.cmml" xref="S1.p3.17.m17.2.2.3"><minus id="S1.p3.17.m17.2.2.3.1.cmml" xref="S1.p3.17.m17.2.2.3.1"></minus><cn id="S1.p3.17.m17.2.2.3.2.cmml" type="integer" xref="S1.p3.17.m17.2.2.3.2">1</cn><ci id="S1.p3.17.m17.2.2.3.3.cmml" xref="S1.p3.17.m17.2.2.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.17.m17.2c">\log(\frac{1}{1-\lambda})</annotation><annotation encoding="application/x-llamapun" id="S1.p3.17.m17.2d">roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG )</annotation></semantics></math>, then this would put both ARRIVAL and SSG in <math alttext="\mathsf{P}" class="ltx_Math" display="inline" id="S1.p3.18.m18.1"><semantics id="S1.p3.18.m18.1a"><mi id="S1.p3.18.m18.1.1" xref="S1.p3.18.m18.1.1.cmml">𝖯</mi><annotation-xml encoding="MathML-Content" id="S1.p3.18.m18.1b"><ci id="S1.p3.18.m18.1.1.cmml" xref="S1.p3.18.m18.1.1">𝖯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.18.m18.1c">\mathsf{P}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.18.m18.1d">sansserif_P</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.4">In the Euclidean case (<math alttext="p=2" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">p</mi><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">=</mo><mn id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><eq id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1"></eq><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝑝</ci><cn id="S1.p4.1.m1.1.1.3.cmml" type="integer" xref="S1.p4.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">p=2</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_p = 2</annotation></semantics></math>), such efficient algorithms have been known for quite some time <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib36" title="">36</a>]</cite>. Concretely, the Inscribed Ellipsoid algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib19" title="">19</a>]</cite> requires <math alttext="\mathcal{O}(d\log(\frac{1}{\varepsilon}))" class="ltx_Math" display="inline" id="S1.p4.2.m2.3"><semantics id="S1.p4.2.m2.3a"><mrow id="S1.p4.2.m2.3.3" xref="S1.p4.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.2.m2.3.3.3" xref="S1.p4.2.m2.3.3.3.cmml">𝒪</mi><mo id="S1.p4.2.m2.3.3.2" xref="S1.p4.2.m2.3.3.2.cmml">⁢</mo><mrow id="S1.p4.2.m2.3.3.1.1" xref="S1.p4.2.m2.3.3.1.1.1.cmml"><mo id="S1.p4.2.m2.3.3.1.1.2" stretchy="false" xref="S1.p4.2.m2.3.3.1.1.1.cmml">(</mo><mrow id="S1.p4.2.m2.3.3.1.1.1" xref="S1.p4.2.m2.3.3.1.1.1.cmml"><mi id="S1.p4.2.m2.3.3.1.1.1.2" xref="S1.p4.2.m2.3.3.1.1.1.2.cmml">d</mi><mo id="S1.p4.2.m2.3.3.1.1.1.1" lspace="0.167em" xref="S1.p4.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S1.p4.2.m2.3.3.1.1.1.3.2" xref="S1.p4.2.m2.3.3.1.1.1.3.1.cmml"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">log</mi><mo id="S1.p4.2.m2.3.3.1.1.1.3.2a" xref="S1.p4.2.m2.3.3.1.1.1.3.1.cmml">⁡</mo><mrow id="S1.p4.2.m2.3.3.1.1.1.3.2.1" xref="S1.p4.2.m2.3.3.1.1.1.3.1.cmml"><mo id="S1.p4.2.m2.3.3.1.1.1.3.2.1.1" stretchy="false" xref="S1.p4.2.m2.3.3.1.1.1.3.1.cmml">(</mo><mfrac id="S1.p4.2.m2.2.2" xref="S1.p4.2.m2.2.2.cmml"><mn id="S1.p4.2.m2.2.2.2" xref="S1.p4.2.m2.2.2.2.cmml">1</mn><mi id="S1.p4.2.m2.2.2.3" xref="S1.p4.2.m2.2.2.3.cmml">ε</mi></mfrac><mo id="S1.p4.2.m2.3.3.1.1.1.3.2.1.2" stretchy="false" xref="S1.p4.2.m2.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.p4.2.m2.3.3.1.1.3" stretchy="false" xref="S1.p4.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.3b"><apply id="S1.p4.2.m2.3.3.cmml" xref="S1.p4.2.m2.3.3"><times id="S1.p4.2.m2.3.3.2.cmml" xref="S1.p4.2.m2.3.3.2"></times><ci id="S1.p4.2.m2.3.3.3.cmml" xref="S1.p4.2.m2.3.3.3">𝒪</ci><apply id="S1.p4.2.m2.3.3.1.1.1.cmml" xref="S1.p4.2.m2.3.3.1.1"><times id="S1.p4.2.m2.3.3.1.1.1.1.cmml" xref="S1.p4.2.m2.3.3.1.1.1.1"></times><ci id="S1.p4.2.m2.3.3.1.1.1.2.cmml" xref="S1.p4.2.m2.3.3.1.1.1.2">𝑑</ci><apply id="S1.p4.2.m2.3.3.1.1.1.3.1.cmml" xref="S1.p4.2.m2.3.3.1.1.1.3.2"><log id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"></log><apply id="S1.p4.2.m2.2.2.cmml" xref="S1.p4.2.m2.2.2"><divide id="S1.p4.2.m2.2.2.1.cmml" xref="S1.p4.2.m2.2.2"></divide><cn id="S1.p4.2.m2.2.2.2.cmml" type="integer" xref="S1.p4.2.m2.2.2.2">1</cn><ci id="S1.p4.2.m2.2.2.3.cmml" xref="S1.p4.2.m2.2.2.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.3c">\mathcal{O}(d\log(\frac{1}{\varepsilon}))</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.3d">caligraphic_O ( italic_d roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) )</annotation></semantics></math> queries (independent of <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_λ</annotation></semantics></math>) to find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. Moreover, each query point can be found efficiently, implying an overall time-efficient (and not just query-efficient) algorithm.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.8">In contrast, only little is known for <math alttext="p\neq 2" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">p</mi><mo id="S1.p5.1.m1.1.1.1" xref="S1.p5.1.m1.1.1.1.cmml">≠</mo><mn id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><neq id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1"></neq><ci id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">𝑝</ci><cn id="S1.p5.1.m1.1.1.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">p\neq 2</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_p ≠ 2</annotation></semantics></math>. The problem is known to be contained in <math alttext="\mathsf{CLS}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mi id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">𝖢𝖫𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><ci id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1">𝖢𝖫𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\mathsf{CLS}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">sansserif_CLS</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib8" title="">8</a>]</cite> and was explicitely mentioned as an open problem by Fearnley, Goldberg, Hollender, and Savani <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib12" title="">12</a>]</cite> when they proved <math alttext="\mathsf{CLS}=\mathsf{PPAD}\cap\mathsf{EOPL}" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mi id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml">𝖢𝖫𝖲</mi><mo id="S1.p5.3.m3.1.1.1" xref="S1.p5.3.m3.1.1.1.cmml">=</mo><mrow id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml"><mi id="S1.p5.3.m3.1.1.3.2" xref="S1.p5.3.m3.1.1.3.2.cmml">𝖯𝖯𝖠𝖣</mi><mo id="S1.p5.3.m3.1.1.3.1" xref="S1.p5.3.m3.1.1.3.1.cmml">∩</mo><mi id="S1.p5.3.m3.1.1.3.3" xref="S1.p5.3.m3.1.1.3.3.cmml">𝖤𝖮𝖯𝖫</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><eq id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1.1"></eq><ci id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2">𝖢𝖫𝖲</ci><apply id="S1.p5.3.m3.1.1.3.cmml" xref="S1.p5.3.m3.1.1.3"><intersect id="S1.p5.3.m3.1.1.3.1.cmml" xref="S1.p5.3.m3.1.1.3.1"></intersect><ci id="S1.p5.3.m3.1.1.3.2.cmml" xref="S1.p5.3.m3.1.1.3.2">𝖯𝖯𝖠𝖣</ci><ci id="S1.p5.3.m3.1.1.3.3.cmml" xref="S1.p5.3.m3.1.1.3.3">𝖤𝖮𝖯𝖫</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">\mathsf{CLS}=\mathsf{PPAD}\cap\mathsf{EOPL}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">sansserif_CLS = sansserif_PPAD ∩ sansserif_EOPL</annotation></semantics></math>. For <math alttext="p\in\mathbb{N}\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mrow id="S1.p5.4.m4.1.2" xref="S1.p5.4.m4.1.2.cmml"><mi id="S1.p5.4.m4.1.2.2" xref="S1.p5.4.m4.1.2.2.cmml">p</mi><mo id="S1.p5.4.m4.1.2.1" xref="S1.p5.4.m4.1.2.1.cmml">∈</mo><mrow id="S1.p5.4.m4.1.2.3" xref="S1.p5.4.m4.1.2.3.cmml"><mi id="S1.p5.4.m4.1.2.3.2" xref="S1.p5.4.m4.1.2.3.2.cmml">ℕ</mi><mo id="S1.p5.4.m4.1.2.3.1" xref="S1.p5.4.m4.1.2.3.1.cmml">∪</mo><mrow id="S1.p5.4.m4.1.2.3.3.2" xref="S1.p5.4.m4.1.2.3.3.1.cmml"><mo id="S1.p5.4.m4.1.2.3.3.2.1" stretchy="false" xref="S1.p5.4.m4.1.2.3.3.1.cmml">{</mo><mi id="S1.p5.4.m4.1.1" mathvariant="normal" xref="S1.p5.4.m4.1.1.cmml">∞</mi><mo id="S1.p5.4.m4.1.2.3.3.2.2" stretchy="false" xref="S1.p5.4.m4.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><apply id="S1.p5.4.m4.1.2.cmml" xref="S1.p5.4.m4.1.2"><in id="S1.p5.4.m4.1.2.1.cmml" xref="S1.p5.4.m4.1.2.1"></in><ci id="S1.p5.4.m4.1.2.2.cmml" xref="S1.p5.4.m4.1.2.2">𝑝</ci><apply id="S1.p5.4.m4.1.2.3.cmml" xref="S1.p5.4.m4.1.2.3"><union id="S1.p5.4.m4.1.2.3.1.cmml" xref="S1.p5.4.m4.1.2.3.1"></union><ci id="S1.p5.4.m4.1.2.3.2.cmml" xref="S1.p5.4.m4.1.2.3.2">ℕ</ci><set id="S1.p5.4.m4.1.2.3.3.1.cmml" xref="S1.p5.4.m4.1.2.3.3.2"><infinity id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">p\in\mathbb{N}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">italic_p ∈ blackboard_N ∪ { ∞ }</annotation></semantics></math>, Fearnley, Gordon, Mehta, and Savani proved containment in <math alttext="\mathsf{UEOPL}" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mi id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml">𝖴𝖤𝖮𝖯𝖫</mi><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><ci id="S1.p5.5.m5.1.1.cmml" xref="S1.p5.5.m5.1.1">𝖴𝖤𝖮𝖯𝖫</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">\mathsf{UEOPL}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">sansserif_UEOPL</annotation></semantics></math> by reduction to One-Permutation Discrete Contraction (OPDC) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib13" title="">13</a>]</cite>. Moreover, their algorithm for OPDC implies an algorithm for finding an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mi id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><ci id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">italic_ε</annotation></semantics></math>-approximate fixpoint in <math alttext="\mathcal{O}(\log^{d}(\frac{1}{\varepsilon}))" class="ltx_Math" display="inline" id="S1.p5.7.m7.2"><semantics id="S1.p5.7.m7.2a"><mrow id="S1.p5.7.m7.2.2" xref="S1.p5.7.m7.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p5.7.m7.2.2.3" xref="S1.p5.7.m7.2.2.3.cmml">𝒪</mi><mo id="S1.p5.7.m7.2.2.2" xref="S1.p5.7.m7.2.2.2.cmml">⁢</mo><mrow id="S1.p5.7.m7.2.2.1.1" xref="S1.p5.7.m7.2.2.cmml"><mo id="S1.p5.7.m7.2.2.1.1.2" stretchy="false" xref="S1.p5.7.m7.2.2.cmml">(</mo><mrow id="S1.p5.7.m7.2.2.1.1.1.1" xref="S1.p5.7.m7.2.2.1.1.1.2.cmml"><msup id="S1.p5.7.m7.2.2.1.1.1.1.1" xref="S1.p5.7.m7.2.2.1.1.1.1.1.cmml"><mi id="S1.p5.7.m7.2.2.1.1.1.1.1.2" xref="S1.p5.7.m7.2.2.1.1.1.1.1.2.cmml">log</mi><mi id="S1.p5.7.m7.2.2.1.1.1.1.1.3" xref="S1.p5.7.m7.2.2.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S1.p5.7.m7.2.2.1.1.1.1a" xref="S1.p5.7.m7.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S1.p5.7.m7.2.2.1.1.1.1.2" xref="S1.p5.7.m7.2.2.1.1.1.2.cmml"><mo id="S1.p5.7.m7.2.2.1.1.1.1.2.1" stretchy="false" xref="S1.p5.7.m7.2.2.1.1.1.2.cmml">(</mo><mfrac id="S1.p5.7.m7.1.1" xref="S1.p5.7.m7.1.1.cmml"><mn id="S1.p5.7.m7.1.1.2" xref="S1.p5.7.m7.1.1.2.cmml">1</mn><mi id="S1.p5.7.m7.1.1.3" xref="S1.p5.7.m7.1.1.3.cmml">ε</mi></mfrac><mo id="S1.p5.7.m7.2.2.1.1.1.1.2.2" stretchy="false" xref="S1.p5.7.m7.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.p5.7.m7.2.2.1.1.3" stretchy="false" xref="S1.p5.7.m7.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.7.m7.2b"><apply id="S1.p5.7.m7.2.2.cmml" xref="S1.p5.7.m7.2.2"><times id="S1.p5.7.m7.2.2.2.cmml" xref="S1.p5.7.m7.2.2.2"></times><ci id="S1.p5.7.m7.2.2.3.cmml" xref="S1.p5.7.m7.2.2.3">𝒪</ci><apply id="S1.p5.7.m7.2.2.1.1.1.2.cmml" xref="S1.p5.7.m7.2.2.1.1.1.1"><apply id="S1.p5.7.m7.2.2.1.1.1.1.1.cmml" xref="S1.p5.7.m7.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.7.m7.2.2.1.1.1.1.1.1.cmml" xref="S1.p5.7.m7.2.2.1.1.1.1.1">superscript</csymbol><log id="S1.p5.7.m7.2.2.1.1.1.1.1.2.cmml" xref="S1.p5.7.m7.2.2.1.1.1.1.1.2"></log><ci id="S1.p5.7.m7.2.2.1.1.1.1.1.3.cmml" xref="S1.p5.7.m7.2.2.1.1.1.1.1.3">𝑑</ci></apply><apply id="S1.p5.7.m7.1.1.cmml" xref="S1.p5.7.m7.1.1"><divide id="S1.p5.7.m7.1.1.1.cmml" xref="S1.p5.7.m7.1.1"></divide><cn id="S1.p5.7.m7.1.1.2.cmml" type="integer" xref="S1.p5.7.m7.1.1.2">1</cn><ci id="S1.p5.7.m7.1.1.3.cmml" xref="S1.p5.7.m7.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m7.2c">\mathcal{O}(\log^{d}(\frac{1}{\varepsilon}))</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m7.2d">caligraphic_O ( roman_log start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) )</annotation></semantics></math> queries. Unfortunately, this is still exponential in <math alttext="d" class="ltx_Math" display="inline" id="S1.p5.8.m8.1"><semantics id="S1.p5.8.m8.1a"><mi id="S1.p5.8.m8.1.1" xref="S1.p5.8.m8.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.p5.8.m8.1b"><ci id="S1.p5.8.m8.1.1.cmml" xref="S1.p5.8.m8.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m8.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m8.1d">italic_d</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.10">Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> recently (STOC’24) gave the first query-efficient algorithm in the special case of the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><msub id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml"><mi id="S1.p6.1.m1.1.1.2" mathvariant="normal" xref="S1.p6.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.p6.1.m1.1.1.3" mathvariant="normal" xref="S1.p6.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p6.1.m1.1.1.1.cmml" xref="S1.p6.1.m1.1.1">subscript</csymbol><ci id="S1.p6.1.m1.1.1.2.cmml" xref="S1.p6.1.m1.1.1.2">ℓ</ci><infinity id="S1.p6.1.m1.1.1.3.cmml" xref="S1.p6.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-norm. Concretely, their algorithm can find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mi id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><ci id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixed point with <math alttext="\mathcal{O}(d^{2}\log\frac{1}{\varepsilon})" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><mrow id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p6.3.m3.1.1.3" xref="S1.p6.3.m3.1.1.3.cmml">𝒪</mi><mo id="S1.p6.3.m3.1.1.2" xref="S1.p6.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.p6.3.m3.1.1.1.1" xref="S1.p6.3.m3.1.1.1.1.1.cmml"><mo id="S1.p6.3.m3.1.1.1.1.2" stretchy="false" xref="S1.p6.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.p6.3.m3.1.1.1.1.1" xref="S1.p6.3.m3.1.1.1.1.1.cmml"><msup id="S1.p6.3.m3.1.1.1.1.1.2" xref="S1.p6.3.m3.1.1.1.1.1.2.cmml"><mi id="S1.p6.3.m3.1.1.1.1.1.2.2" xref="S1.p6.3.m3.1.1.1.1.1.2.2.cmml">d</mi><mn id="S1.p6.3.m3.1.1.1.1.1.2.3" xref="S1.p6.3.m3.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S1.p6.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S1.p6.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p6.3.m3.1.1.1.1.1.3" xref="S1.p6.3.m3.1.1.1.1.1.3.cmml"><mi id="S1.p6.3.m3.1.1.1.1.1.3.1" xref="S1.p6.3.m3.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.p6.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S1.p6.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mfrac id="S1.p6.3.m3.1.1.1.1.1.3.2" xref="S1.p6.3.m3.1.1.1.1.1.3.2.cmml"><mn id="S1.p6.3.m3.1.1.1.1.1.3.2.2" xref="S1.p6.3.m3.1.1.1.1.1.3.2.2.cmml">1</mn><mi id="S1.p6.3.m3.1.1.1.1.1.3.2.3" xref="S1.p6.3.m3.1.1.1.1.1.3.2.3.cmml">ε</mi></mfrac></mrow></mrow><mo id="S1.p6.3.m3.1.1.1.1.3" stretchy="false" xref="S1.p6.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1"><times id="S1.p6.3.m3.1.1.2.cmml" xref="S1.p6.3.m3.1.1.2"></times><ci id="S1.p6.3.m3.1.1.3.cmml" xref="S1.p6.3.m3.1.1.3">𝒪</ci><apply id="S1.p6.3.m3.1.1.1.1.1.cmml" xref="S1.p6.3.m3.1.1.1.1"><times id="S1.p6.3.m3.1.1.1.1.1.1.cmml" xref="S1.p6.3.m3.1.1.1.1.1.1"></times><apply id="S1.p6.3.m3.1.1.1.1.1.2.cmml" xref="S1.p6.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.p6.3.m3.1.1.1.1.1.2.1.cmml" xref="S1.p6.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S1.p6.3.m3.1.1.1.1.1.2.2.cmml" xref="S1.p6.3.m3.1.1.1.1.1.2.2">𝑑</ci><cn id="S1.p6.3.m3.1.1.1.1.1.2.3.cmml" type="integer" xref="S1.p6.3.m3.1.1.1.1.1.2.3">2</cn></apply><apply id="S1.p6.3.m3.1.1.1.1.1.3.cmml" xref="S1.p6.3.m3.1.1.1.1.1.3"><log id="S1.p6.3.m3.1.1.1.1.1.3.1.cmml" xref="S1.p6.3.m3.1.1.1.1.1.3.1"></log><apply id="S1.p6.3.m3.1.1.1.1.1.3.2.cmml" xref="S1.p6.3.m3.1.1.1.1.1.3.2"><divide id="S1.p6.3.m3.1.1.1.1.1.3.2.1.cmml" xref="S1.p6.3.m3.1.1.1.1.1.3.2"></divide><cn id="S1.p6.3.m3.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S1.p6.3.m3.1.1.1.1.1.3.2.2">1</cn><ci id="S1.p6.3.m3.1.1.1.1.1.3.2.3.cmml" xref="S1.p6.3.m3.1.1.1.1.1.3.2.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">\mathcal{O}(d^{2}\log\frac{1}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> (independent of <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.p6.4.m4.1"><semantics id="S1.p6.4.m4.1a"><mi id="S1.p6.4.m4.1.1" xref="S1.p6.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.p6.4.m4.1b"><ci id="S1.p6.4.m4.1.1.cmml" xref="S1.p6.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p6.4.m4.1d">italic_λ</annotation></semantics></math>) queries to <math alttext="f" class="ltx_Math" display="inline" id="S1.p6.5.m5.1"><semantics id="S1.p6.5.m5.1a"><mi id="S1.p6.5.m5.1.1" xref="S1.p6.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.p6.5.m5.1b"><ci id="S1.p6.5.m5.1.1.cmml" xref="S1.p6.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.p6.5.m5.1d">italic_f</annotation></semantics></math>. One of the main questions that they ask is whether similar query upper bounds can be shown for <math alttext="p\neq\{2,\infty\}" class="ltx_Math" display="inline" id="S1.p6.6.m6.2"><semantics id="S1.p6.6.m6.2a"><mrow id="S1.p6.6.m6.2.3" xref="S1.p6.6.m6.2.3.cmml"><mi id="S1.p6.6.m6.2.3.2" xref="S1.p6.6.m6.2.3.2.cmml">p</mi><mo id="S1.p6.6.m6.2.3.1" xref="S1.p6.6.m6.2.3.1.cmml">≠</mo><mrow id="S1.p6.6.m6.2.3.3.2" xref="S1.p6.6.m6.2.3.3.1.cmml"><mo id="S1.p6.6.m6.2.3.3.2.1" stretchy="false" xref="S1.p6.6.m6.2.3.3.1.cmml">{</mo><mn id="S1.p6.6.m6.1.1" xref="S1.p6.6.m6.1.1.cmml">2</mn><mo id="S1.p6.6.m6.2.3.3.2.2" xref="S1.p6.6.m6.2.3.3.1.cmml">,</mo><mi id="S1.p6.6.m6.2.2" mathvariant="normal" xref="S1.p6.6.m6.2.2.cmml">∞</mi><mo id="S1.p6.6.m6.2.3.3.2.3" stretchy="false" xref="S1.p6.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.6.m6.2b"><apply id="S1.p6.6.m6.2.3.cmml" xref="S1.p6.6.m6.2.3"><neq id="S1.p6.6.m6.2.3.1.cmml" xref="S1.p6.6.m6.2.3.1"></neq><ci id="S1.p6.6.m6.2.3.2.cmml" xref="S1.p6.6.m6.2.3.2">𝑝</ci><set id="S1.p6.6.m6.2.3.3.1.cmml" xref="S1.p6.6.m6.2.3.3.2"><cn id="S1.p6.6.m6.1.1.cmml" type="integer" xref="S1.p6.6.m6.1.1">2</cn><infinity id="S1.p6.6.m6.2.2.cmml" xref="S1.p6.6.m6.2.2"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.6.m6.2c">p\neq\{2,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.6.m6.2d">italic_p ≠ { 2 , ∞ }</annotation></semantics></math> as well. We answer this question affirmatively by achieving a query upper bound of <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))" class="ltx_Math" display="inline" id="S1.p6.7.m7.1"><semantics id="S1.p6.7.m7.1a"><mrow id="S1.p6.7.m7.1.1" xref="S1.p6.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p6.7.m7.1.1.3" xref="S1.p6.7.m7.1.1.3.cmml">𝒪</mi><mo id="S1.p6.7.m7.1.1.2" xref="S1.p6.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.p6.7.m7.1.1.1.1" xref="S1.p6.7.m7.1.1.1.1.1.cmml"><mo id="S1.p6.7.m7.1.1.1.1.2" stretchy="false" xref="S1.p6.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S1.p6.7.m7.1.1.1.1.1" xref="S1.p6.7.m7.1.1.1.1.1.cmml"><msup id="S1.p6.7.m7.1.1.1.1.1.3" xref="S1.p6.7.m7.1.1.1.1.1.3.cmml"><mi id="S1.p6.7.m7.1.1.1.1.1.3.2" xref="S1.p6.7.m7.1.1.1.1.1.3.2.cmml">d</mi><mn id="S1.p6.7.m7.1.1.1.1.1.3.3" xref="S1.p6.7.m7.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S1.p6.7.m7.1.1.1.1.1.2" xref="S1.p6.7.m7.1.1.1.1.1.2.cmml">⁢</mo><mrow 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xref="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3"><minus id="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S1.p6.7.m7.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.7.m7.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S1.p6.7.m7.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.p6.8.m8.3"><semantics id="S1.p6.8.m8.3a"><mrow id="S1.p6.8.m8.3.4" xref="S1.p6.8.m8.3.4.cmml"><mi id="S1.p6.8.m8.3.4.2" xref="S1.p6.8.m8.3.4.2.cmml">p</mi><mo id="S1.p6.8.m8.3.4.1" xref="S1.p6.8.m8.3.4.1.cmml">∈</mo><mrow id="S1.p6.8.m8.3.4.3" xref="S1.p6.8.m8.3.4.3.cmml"><mrow id="S1.p6.8.m8.3.4.3.2.2" xref="S1.p6.8.m8.3.4.3.2.1.cmml"><mo id="S1.p6.8.m8.3.4.3.2.2.1" stretchy="false" xref="S1.p6.8.m8.3.4.3.2.1.cmml">[</mo><mn id="S1.p6.8.m8.1.1" xref="S1.p6.8.m8.1.1.cmml">1</mn><mo id="S1.p6.8.m8.3.4.3.2.2.2" xref="S1.p6.8.m8.3.4.3.2.1.cmml">,</mo><mi id="S1.p6.8.m8.2.2" mathvariant="normal" xref="S1.p6.8.m8.2.2.cmml">∞</mi><mo id="S1.p6.8.m8.3.4.3.2.2.3" stretchy="false" xref="S1.p6.8.m8.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.p6.8.m8.3.4.3.1" xref="S1.p6.8.m8.3.4.3.1.cmml">∪</mo><mrow id="S1.p6.8.m8.3.4.3.3.2" xref="S1.p6.8.m8.3.4.3.3.1.cmml"><mo id="S1.p6.8.m8.3.4.3.3.2.1" stretchy="false" xref="S1.p6.8.m8.3.4.3.3.1.cmml">{</mo><mi id="S1.p6.8.m8.3.3" mathvariant="normal" xref="S1.p6.8.m8.3.3.cmml">∞</mi><mo id="S1.p6.8.m8.3.4.3.3.2.2" stretchy="false" xref="S1.p6.8.m8.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.8.m8.3b"><apply id="S1.p6.8.m8.3.4.cmml" xref="S1.p6.8.m8.3.4"><in id="S1.p6.8.m8.3.4.1.cmml" xref="S1.p6.8.m8.3.4.1"></in><ci id="S1.p6.8.m8.3.4.2.cmml" xref="S1.p6.8.m8.3.4.2">𝑝</ci><apply id="S1.p6.8.m8.3.4.3.cmml" xref="S1.p6.8.m8.3.4.3"><union id="S1.p6.8.m8.3.4.3.1.cmml" xref="S1.p6.8.m8.3.4.3.1"></union><interval closure="closed-open" id="S1.p6.8.m8.3.4.3.2.1.cmml" xref="S1.p6.8.m8.3.4.3.2.2"><cn id="S1.p6.8.m8.1.1.cmml" type="integer" xref="S1.p6.8.m8.1.1">1</cn><infinity id="S1.p6.8.m8.2.2.cmml" xref="S1.p6.8.m8.2.2"></infinity></interval><set id="S1.p6.8.m8.3.4.3.3.1.cmml" xref="S1.p6.8.m8.3.4.3.3.2"><infinity id="S1.p6.8.m8.3.3.cmml" xref="S1.p6.8.m8.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.8.m8.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.8.m8.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. Note that using an observation of Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>, we can also get rid of the factor <math alttext="\log\frac{1}{1-\lambda}" class="ltx_Math" display="inline" id="S1.p6.9.m9.1"><semantics id="S1.p6.9.m9.1a"><mrow id="S1.p6.9.m9.1.1" xref="S1.p6.9.m9.1.1.cmml"><mi id="S1.p6.9.m9.1.1.1" xref="S1.p6.9.m9.1.1.1.cmml">log</mi><mo id="S1.p6.9.m9.1.1a" lspace="0.167em" xref="S1.p6.9.m9.1.1.cmml">⁡</mo><mfrac id="S1.p6.9.m9.1.1.2" xref="S1.p6.9.m9.1.1.2.cmml"><mn id="S1.p6.9.m9.1.1.2.2" xref="S1.p6.9.m9.1.1.2.2.cmml">1</mn><mrow id="S1.p6.9.m9.1.1.2.3" xref="S1.p6.9.m9.1.1.2.3.cmml"><mn id="S1.p6.9.m9.1.1.2.3.2" xref="S1.p6.9.m9.1.1.2.3.2.cmml">1</mn><mo id="S1.p6.9.m9.1.1.2.3.1" xref="S1.p6.9.m9.1.1.2.3.1.cmml">−</mo><mi id="S1.p6.9.m9.1.1.2.3.3" xref="S1.p6.9.m9.1.1.2.3.3.cmml">λ</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.9.m9.1b"><apply id="S1.p6.9.m9.1.1.cmml" xref="S1.p6.9.m9.1.1"><log id="S1.p6.9.m9.1.1.1.cmml" xref="S1.p6.9.m9.1.1.1"></log><apply id="S1.p6.9.m9.1.1.2.cmml" xref="S1.p6.9.m9.1.1.2"><divide id="S1.p6.9.m9.1.1.2.1.cmml" xref="S1.p6.9.m9.1.1.2"></divide><cn id="S1.p6.9.m9.1.1.2.2.cmml" type="integer" xref="S1.p6.9.m9.1.1.2.2">1</cn><apply id="S1.p6.9.m9.1.1.2.3.cmml" xref="S1.p6.9.m9.1.1.2.3"><minus id="S1.p6.9.m9.1.1.2.3.1.cmml" xref="S1.p6.9.m9.1.1.2.3.1"></minus><cn id="S1.p6.9.m9.1.1.2.3.2.cmml" type="integer" xref="S1.p6.9.m9.1.1.2.3.2">1</cn><ci id="S1.p6.9.m9.1.1.2.3.3.cmml" xref="S1.p6.9.m9.1.1.2.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.9.m9.1c">\log\frac{1}{1-\lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.9.m9.1d">roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG</annotation></semantics></math> for <math alttext="p=\infty" class="ltx_Math" display="inline" id="S1.p6.10.m10.1"><semantics id="S1.p6.10.m10.1a"><mrow id="S1.p6.10.m10.1.1" xref="S1.p6.10.m10.1.1.cmml"><mi id="S1.p6.10.m10.1.1.2" xref="S1.p6.10.m10.1.1.2.cmml">p</mi><mo id="S1.p6.10.m10.1.1.1" xref="S1.p6.10.m10.1.1.1.cmml">=</mo><mi id="S1.p6.10.m10.1.1.3" mathvariant="normal" xref="S1.p6.10.m10.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.10.m10.1b"><apply id="S1.p6.10.m10.1.1.cmml" xref="S1.p6.10.m10.1.1"><eq id="S1.p6.10.m10.1.1.1.cmml" xref="S1.p6.10.m10.1.1.1"></eq><ci id="S1.p6.10.m10.1.1.2.cmml" xref="S1.p6.10.m10.1.1.2">𝑝</ci><infinity id="S1.p6.10.m10.1.1.3.cmml" xref="S1.p6.10.m10.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.10.m10.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="S1.p6.10.m10.1d">italic_p = ∞</annotation></semantics></math>, hence matching their bound.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">Neither our algorithm nor the algorithm of Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> is time-efficient, i.e., in both cases it is unclear how to algorithmically find the query points efficiently. However, Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> ensure that all queries made by their algorithm lie on a discrete grid of limited granularity, which is certainly a prerequisite for any time-efficient algorithms. We provide a similar rounding of queries to a grid in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><msub id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.2" mathvariant="normal" xref="S1.p7.1.m1.1.1.2.cmml">ℓ</mi><mn id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p7.1.m1.1.1.1.cmml" xref="S1.p7.1.m1.1.1">subscript</csymbol><ci id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2">ℓ</ci><cn id="S1.p7.1.m1.1.1.3.cmml" type="integer" xref="S1.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.2">Note that since our algorithm is only query-efficient, it does not imply polynomial-time algorithms for SSG, ARRIVAL, or other applications. However, as pointed out by Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>, the existence of query-efficient algorithms could be viewed as evidence for the existence of time-efficient algorithms for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><msub id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" mathvariant="normal" xref="S1.p8.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p8.1.m1.1.1.1.cmml" xref="S1.p8.1.m1.1.1">subscript</csymbol><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">ℓ</ci><ci id="S1.p8.1.m1.1.1.3.cmml" xref="S1.p8.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-contraction (at least for <math alttext="p\in\{1,\infty\}" class="ltx_Math" display="inline" id="S1.p8.2.m2.2"><semantics id="S1.p8.2.m2.2a"><mrow id="S1.p8.2.m2.2.3" xref="S1.p8.2.m2.2.3.cmml"><mi id="S1.p8.2.m2.2.3.2" xref="S1.p8.2.m2.2.3.2.cmml">p</mi><mo id="S1.p8.2.m2.2.3.1" xref="S1.p8.2.m2.2.3.1.cmml">∈</mo><mrow id="S1.p8.2.m2.2.3.3.2" xref="S1.p8.2.m2.2.3.3.1.cmml"><mo id="S1.p8.2.m2.2.3.3.2.1" stretchy="false" xref="S1.p8.2.m2.2.3.3.1.cmml">{</mo><mn id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml">1</mn><mo id="S1.p8.2.m2.2.3.3.2.2" xref="S1.p8.2.m2.2.3.3.1.cmml">,</mo><mi id="S1.p8.2.m2.2.2" mathvariant="normal" xref="S1.p8.2.m2.2.2.cmml">∞</mi><mo id="S1.p8.2.m2.2.3.3.2.3" stretchy="false" xref="S1.p8.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.2b"><apply id="S1.p8.2.m2.2.3.cmml" xref="S1.p8.2.m2.2.3"><in id="S1.p8.2.m2.2.3.1.cmml" xref="S1.p8.2.m2.2.3.1"></in><ci id="S1.p8.2.m2.2.3.2.cmml" xref="S1.p8.2.m2.2.3.2">𝑝</ci><set id="S1.p8.2.m2.2.3.3.1.cmml" xref="S1.p8.2.m2.2.3.3.2"><cn id="S1.p8.2.m2.1.1.cmml" type="integer" xref="S1.p8.2.m2.1.1">1</cn><infinity id="S1.p8.2.m2.2.2.cmml" xref="S1.p8.2.m2.2.2"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.2c">p\in\{1,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.2d">italic_p ∈ { 1 , ∞ }</annotation></semantics></math>).</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Results</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.16">For any <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><msub id="S1.SS1.p1.1.m1.1.1" xref="S1.SS1.p1.1.m1.1.1.cmml"><mi id="S1.SS1.p1.1.m1.1.1.2" mathvariant="normal" xref="S1.SS1.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p1.1.m1.1.1.3" xref="S1.SS1.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><apply id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.1.m1.1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS1.p1.1.m1.1.1.2.cmml" xref="S1.SS1.p1.1.m1.1.1.2">ℓ</ci><ci id="S1.SS1.p1.1.m1.1.1.3.cmml" xref="S1.SS1.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm with <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.3"><semantics id="S1.SS1.p1.2.m2.3a"><mrow id="S1.SS1.p1.2.m2.3.4" xref="S1.SS1.p1.2.m2.3.4.cmml"><mi id="S1.SS1.p1.2.m2.3.4.2" xref="S1.SS1.p1.2.m2.3.4.2.cmml">p</mi><mo id="S1.SS1.p1.2.m2.3.4.1" xref="S1.SS1.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S1.SS1.p1.2.m2.3.4.3" xref="S1.SS1.p1.2.m2.3.4.3.cmml"><mrow id="S1.SS1.p1.2.m2.3.4.3.2.2" xref="S1.SS1.p1.2.m2.3.4.3.2.1.cmml"><mo id="S1.SS1.p1.2.m2.3.4.3.2.2.1" stretchy="false" xref="S1.SS1.p1.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">1</mn><mo id="S1.SS1.p1.2.m2.3.4.3.2.2.2" xref="S1.SS1.p1.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S1.SS1.p1.2.m2.2.2" mathvariant="normal" xref="S1.SS1.p1.2.m2.2.2.cmml">∞</mi><mo id="S1.SS1.p1.2.m2.3.4.3.2.2.3" stretchy="false" xref="S1.SS1.p1.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.SS1.p1.2.m2.3.4.3.1" xref="S1.SS1.p1.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S1.SS1.p1.2.m2.3.4.3.3.2" xref="S1.SS1.p1.2.m2.3.4.3.3.1.cmml"><mo id="S1.SS1.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="S1.SS1.p1.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S1.SS1.p1.2.m2.3.3" mathvariant="normal" xref="S1.SS1.p1.2.m2.3.3.cmml">∞</mi><mo id="S1.SS1.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="S1.SS1.p1.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.3b"><apply id="S1.SS1.p1.2.m2.3.4.cmml" xref="S1.SS1.p1.2.m2.3.4"><in id="S1.SS1.p1.2.m2.3.4.1.cmml" xref="S1.SS1.p1.2.m2.3.4.1"></in><ci id="S1.SS1.p1.2.m2.3.4.2.cmml" xref="S1.SS1.p1.2.m2.3.4.2">𝑝</ci><apply id="S1.SS1.p1.2.m2.3.4.3.cmml" xref="S1.SS1.p1.2.m2.3.4.3"><union id="S1.SS1.p1.2.m2.3.4.3.1.cmml" xref="S1.SS1.p1.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S1.SS1.p1.2.m2.3.4.3.2.1.cmml" xref="S1.SS1.p1.2.m2.3.4.3.2.2"><cn id="S1.SS1.p1.2.m2.1.1.cmml" type="integer" xref="S1.SS1.p1.2.m2.1.1">1</cn><infinity id="S1.SS1.p1.2.m2.2.2.cmml" xref="S1.SS1.p1.2.m2.2.2"></infinity></interval><set id="S1.SS1.p1.2.m2.3.4.3.3.1.cmml" xref="S1.SS1.p1.2.m2.3.4.3.3.2"><infinity id="S1.SS1.p1.2.m2.3.3.cmml" xref="S1.SS1.p1.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, we present an algorithm that can find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.1"><semantics id="S1.SS1.p1.3.m3.1a"><mi id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.1b"><ci id="S1.SS1.p1.3.m3.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.1d">italic_ε</annotation></semantics></math>-approximate 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xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.cmml"><mn id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.2" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.2.cmml">1</mn><mi id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.3" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml">ε</mi></mfrac></mrow><mo id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.1" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.cmml"><mi id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.1" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mfrac id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.2" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mrow id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml"><mn id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.2" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml">1</mn><mo id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.1" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml">−</mo><mi id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.3" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mrow></mrow><mo id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S1.SS1.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.6.m6.1b"><apply id="S1.SS1.p1.6.m6.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1"><times id="S1.SS1.p1.6.m6.1.1.2.cmml" xref="S1.SS1.p1.6.m6.1.1.2"></times><ci id="S1.SS1.p1.6.m6.1.1.3.cmml" xref="S1.SS1.p1.6.m6.1.1.3">𝒪</ci><apply id="S1.SS1.p1.6.m6.1.1.1.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1"><times id="S1.SS1.p1.6.m6.1.1.1.1.1.2.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.2"></times><apply id="S1.SS1.p1.6.m6.1.1.1.1.1.3.cmml" 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xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.2">1</cn><ci id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3"><log id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.1.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.1"></log><apply id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2"><divide id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2"></divide><cn id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3"><minus id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S1.SS1.p1.6.m6.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.6.m6.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.6.m6.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> queries to <math alttext="f" class="ltx_Math" display="inline" id="S1.SS1.p1.7.m7.1"><semantics id="S1.SS1.p1.7.m7.1a"><mi id="S1.SS1.p1.7.m7.1.1" xref="S1.SS1.p1.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.7.m7.1b"><ci id="S1.SS1.p1.7.m7.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.7.m7.1d">italic_f</annotation></semantics></math>. In the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS1.p1.8.m8.1"><semantics id="S1.SS1.p1.8.m8.1a"><msub id="S1.SS1.p1.8.m8.1.1" xref="S1.SS1.p1.8.m8.1.1.cmml"><mi id="S1.SS1.p1.8.m8.1.1.2" mathvariant="normal" xref="S1.SS1.p1.8.m8.1.1.2.cmml">ℓ</mi><mn id="S1.SS1.p1.8.m8.1.1.3" xref="S1.SS1.p1.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.8.m8.1b"><apply id="S1.SS1.p1.8.m8.1.1.cmml" xref="S1.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.8.m8.1.1.1.cmml" xref="S1.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S1.SS1.p1.8.m8.1.1.2.cmml" xref="S1.SS1.p1.8.m8.1.1.2">ℓ</ci><cn id="S1.SS1.p1.8.m8.1.1.3.cmml" type="integer" xref="S1.SS1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.8.m8.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.8.m8.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, we show how to ensure that all queries are made on points of a given discrete grid, as long as the distance between two neighboring grid points is at most <math alttext="\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}" class="ltx_Math" display="inline" id="S1.SS1.p1.9.m9.1"><semantics id="S1.SS1.p1.9.m9.1a"><mrow id="S1.SS1.p1.9.m9.1.1" xref="S1.SS1.p1.9.m9.1.1.cmml"><mfrac id="S1.SS1.p1.9.m9.1.1.2" xref="S1.SS1.p1.9.m9.1.1.2.cmml"><mrow id="S1.SS1.p1.9.m9.1.1.2.2" xref="S1.SS1.p1.9.m9.1.1.2.2.cmml"><mn id="S1.SS1.p1.9.m9.1.1.2.2.2" xref="S1.SS1.p1.9.m9.1.1.2.2.2.cmml">2</mn><mo id="S1.SS1.p1.9.m9.1.1.2.2.1" xref="S1.SS1.p1.9.m9.1.1.2.2.1.cmml">⁢</mo><mi id="S1.SS1.p1.9.m9.1.1.2.2.3" xref="S1.SS1.p1.9.m9.1.1.2.2.3.cmml">d</mi></mrow><mi id="S1.SS1.p1.9.m9.1.1.2.3" xref="S1.SS1.p1.9.m9.1.1.2.3.cmml">ε</mi></mfrac><mo id="S1.SS1.p1.9.m9.1.1.1" xref="S1.SS1.p1.9.m9.1.1.1.cmml">⁢</mo><mfrac id="S1.SS1.p1.9.m9.1.1.3" xref="S1.SS1.p1.9.m9.1.1.3.cmml"><mrow id="S1.SS1.p1.9.m9.1.1.3.2" xref="S1.SS1.p1.9.m9.1.1.3.2.cmml"><mn id="S1.SS1.p1.9.m9.1.1.3.2.2" xref="S1.SS1.p1.9.m9.1.1.3.2.2.cmml">1</mn><mo id="S1.SS1.p1.9.m9.1.1.3.2.1" xref="S1.SS1.p1.9.m9.1.1.3.2.1.cmml">+</mo><mi id="S1.SS1.p1.9.m9.1.1.3.2.3" xref="S1.SS1.p1.9.m9.1.1.3.2.3.cmml">λ</mi></mrow><mrow id="S1.SS1.p1.9.m9.1.1.3.3" xref="S1.SS1.p1.9.m9.1.1.3.3.cmml"><mn id="S1.SS1.p1.9.m9.1.1.3.3.2" xref="S1.SS1.p1.9.m9.1.1.3.3.2.cmml">1</mn><mo id="S1.SS1.p1.9.m9.1.1.3.3.1" xref="S1.SS1.p1.9.m9.1.1.3.3.1.cmml">−</mo><mi id="S1.SS1.p1.9.m9.1.1.3.3.3" xref="S1.SS1.p1.9.m9.1.1.3.3.3.cmml">λ</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.9.m9.1b"><apply id="S1.SS1.p1.9.m9.1.1.cmml" xref="S1.SS1.p1.9.m9.1.1"><times id="S1.SS1.p1.9.m9.1.1.1.cmml" xref="S1.SS1.p1.9.m9.1.1.1"></times><apply id="S1.SS1.p1.9.m9.1.1.2.cmml" xref="S1.SS1.p1.9.m9.1.1.2"><divide id="S1.SS1.p1.9.m9.1.1.2.1.cmml" xref="S1.SS1.p1.9.m9.1.1.2"></divide><apply id="S1.SS1.p1.9.m9.1.1.2.2.cmml" xref="S1.SS1.p1.9.m9.1.1.2.2"><times id="S1.SS1.p1.9.m9.1.1.2.2.1.cmml" xref="S1.SS1.p1.9.m9.1.1.2.2.1"></times><cn id="S1.SS1.p1.9.m9.1.1.2.2.2.cmml" type="integer" xref="S1.SS1.p1.9.m9.1.1.2.2.2">2</cn><ci id="S1.SS1.p1.9.m9.1.1.2.2.3.cmml" xref="S1.SS1.p1.9.m9.1.1.2.2.3">𝑑</ci></apply><ci id="S1.SS1.p1.9.m9.1.1.2.3.cmml" xref="S1.SS1.p1.9.m9.1.1.2.3">𝜀</ci></apply><apply id="S1.SS1.p1.9.m9.1.1.3.cmml" xref="S1.SS1.p1.9.m9.1.1.3"><divide id="S1.SS1.p1.9.m9.1.1.3.1.cmml" xref="S1.SS1.p1.9.m9.1.1.3"></divide><apply id="S1.SS1.p1.9.m9.1.1.3.2.cmml" xref="S1.SS1.p1.9.m9.1.1.3.2"><plus id="S1.SS1.p1.9.m9.1.1.3.2.1.cmml" xref="S1.SS1.p1.9.m9.1.1.3.2.1"></plus><cn id="S1.SS1.p1.9.m9.1.1.3.2.2.cmml" type="integer" xref="S1.SS1.p1.9.m9.1.1.3.2.2">1</cn><ci id="S1.SS1.p1.9.m9.1.1.3.2.3.cmml" xref="S1.SS1.p1.9.m9.1.1.3.2.3">𝜆</ci></apply><apply id="S1.SS1.p1.9.m9.1.1.3.3.cmml" xref="S1.SS1.p1.9.m9.1.1.3.3"><minus id="S1.SS1.p1.9.m9.1.1.3.3.1.cmml" xref="S1.SS1.p1.9.m9.1.1.3.3.1"></minus><cn id="S1.SS1.p1.9.m9.1.1.3.3.2.cmml" type="integer" xref="S1.SS1.p1.9.m9.1.1.3.3.2">1</cn><ci id="S1.SS1.p1.9.m9.1.1.3.3.3.cmml" xref="S1.SS1.p1.9.m9.1.1.3.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.9.m9.1c">\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.9.m9.1d">divide start_ARG 2 italic_d end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG</annotation></semantics></math>. A similar rounding was provided by Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS1.p1.10.m10.1"><semantics id="S1.SS1.p1.10.m10.1a"><msub id="S1.SS1.p1.10.m10.1.1" xref="S1.SS1.p1.10.m10.1.1.cmml"><mi id="S1.SS1.p1.10.m10.1.1.2" mathvariant="normal" xref="S1.SS1.p1.10.m10.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p1.10.m10.1.1.3" mathvariant="normal" xref="S1.SS1.p1.10.m10.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.10.m10.1b"><apply id="S1.SS1.p1.10.m10.1.1.cmml" xref="S1.SS1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.10.m10.1.1.1.cmml" xref="S1.SS1.p1.10.m10.1.1">subscript</csymbol><ci id="S1.SS1.p1.10.m10.1.1.2.cmml" xref="S1.SS1.p1.10.m10.1.1.2">ℓ</ci><infinity id="S1.SS1.p1.10.m10.1.1.3.cmml" xref="S1.SS1.p1.10.m10.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.10.m10.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.10.m10.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case, and their rounding can also be used in combination with our algorithm. If the function <math alttext="f" class="ltx_Math" display="inline" id="S1.SS1.p1.11.m11.1"><semantics id="S1.SS1.p1.11.m11.1a"><mi id="S1.SS1.p1.11.m11.1.1" xref="S1.SS1.p1.11.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.11.m11.1b"><ci id="S1.SS1.p1.11.m11.1.1.cmml" xref="S1.SS1.p1.11.m11.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.11.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.11.m11.1d">italic_f</annotation></semantics></math> produces only <math alttext="\operatorname{poly}(k)" class="ltx_Math" display="inline" id="S1.SS1.p1.12.m12.2"><semantics id="S1.SS1.p1.12.m12.2a"><mrow id="S1.SS1.p1.12.m12.2.3.2" xref="S1.SS1.p1.12.m12.2.3.1.cmml"><mi id="S1.SS1.p1.12.m12.1.1" xref="S1.SS1.p1.12.m12.1.1.cmml">poly</mi><mo id="S1.SS1.p1.12.m12.2.3.2a" xref="S1.SS1.p1.12.m12.2.3.1.cmml">⁡</mo><mrow id="S1.SS1.p1.12.m12.2.3.2.1" xref="S1.SS1.p1.12.m12.2.3.1.cmml"><mo id="S1.SS1.p1.12.m12.2.3.2.1.1" stretchy="false" xref="S1.SS1.p1.12.m12.2.3.1.cmml">(</mo><mi id="S1.SS1.p1.12.m12.2.2" xref="S1.SS1.p1.12.m12.2.2.cmml">k</mi><mo id="S1.SS1.p1.12.m12.2.3.2.1.2" stretchy="false" xref="S1.SS1.p1.12.m12.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.12.m12.2b"><apply id="S1.SS1.p1.12.m12.2.3.1.cmml" xref="S1.SS1.p1.12.m12.2.3.2"><ci id="S1.SS1.p1.12.m12.1.1.cmml" xref="S1.SS1.p1.12.m12.1.1">poly</ci><ci id="S1.SS1.p1.12.m12.2.2.cmml" xref="S1.SS1.p1.12.m12.2.2">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.12.m12.2c">\operatorname{poly}(k)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.12.m12.2d">roman_poly ( italic_k )</annotation></semantics></math> bits of output when given <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.p1.13.m13.1"><semantics id="S1.SS1.p1.13.m13.1a"><mi id="S1.SS1.p1.13.m13.1.1" xref="S1.SS1.p1.13.m13.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.13.m13.1b"><ci id="S1.SS1.p1.13.m13.1.1.cmml" xref="S1.SS1.p1.13.m13.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.13.m13.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.13.m13.1d">italic_k</annotation></semantics></math> bits of input (as it is the case with many applications of the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS1.p1.14.m14.1"><semantics id="S1.SS1.p1.14.m14.1a"><msub id="S1.SS1.p1.14.m14.1.1" xref="S1.SS1.p1.14.m14.1.1.cmml"><mi id="S1.SS1.p1.14.m14.1.1.2" mathvariant="normal" xref="S1.SS1.p1.14.m14.1.1.2.cmml">ℓ</mi><mn id="S1.SS1.p1.14.m14.1.1.3" xref="S1.SS1.p1.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.14.m14.1b"><apply id="S1.SS1.p1.14.m14.1.1.cmml" xref="S1.SS1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.14.m14.1.1.1.cmml" xref="S1.SS1.p1.14.m14.1.1">subscript</csymbol><ci id="S1.SS1.p1.14.m14.1.1.2.cmml" xref="S1.SS1.p1.14.m14.1.1.2">ℓ</ci><cn id="S1.SS1.p1.14.m14.1.1.3.cmml" type="integer" xref="S1.SS1.p1.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.14.m14.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.14.m14.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case and the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS1.p1.15.m15.1"><semantics id="S1.SS1.p1.15.m15.1a"><msub id="S1.SS1.p1.15.m15.1.1" xref="S1.SS1.p1.15.m15.1.1.cmml"><mi id="S1.SS1.p1.15.m15.1.1.2" mathvariant="normal" xref="S1.SS1.p1.15.m15.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p1.15.m15.1.1.3" mathvariant="normal" xref="S1.SS1.p1.15.m15.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.15.m15.1b"><apply id="S1.SS1.p1.15.m15.1.1.cmml" xref="S1.SS1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.15.m15.1.1.1.cmml" xref="S1.SS1.p1.15.m15.1.1">subscript</csymbol><ci id="S1.SS1.p1.15.m15.1.1.2.cmml" xref="S1.SS1.p1.15.m15.1.1.2">ℓ</ci><infinity id="S1.SS1.p1.15.m15.1.1.3.cmml" xref="S1.SS1.p1.15.m15.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.15.m15.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.15.m15.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case), our algorithm only needs to input and output polynomially many bits of information into the function <math alttext="f" class="ltx_Math" display="inline" id="S1.SS1.p1.16.m16.1"><semantics id="S1.SS1.p1.16.m16.1a"><mi id="S1.SS1.p1.16.m16.1.1" xref="S1.SS1.p1.16.m16.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.16.m16.1b"><ci id="S1.SS1.p1.16.m16.1.1.cmml" xref="S1.SS1.p1.16.m16.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.16.m16.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.16.m16.1d">italic_f</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.2">As a main technical tool, we consider a generalization of the notion of halfspaces from Euclidean geometry into <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.1"><semantics id="S1.SS1.p2.1.m1.1a"><msub id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml"><mi id="S1.SS1.p2.1.m1.1.1.2" mathvariant="normal" xref="S1.SS1.p2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p2.1.m1.1.1.3" xref="S1.SS1.p2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.1b"><apply id="S1.SS1.p2.1.m1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p2.1.m1.1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S1.SS1.p2.1.m1.1.1.2.cmml" xref="S1.SS1.p2.1.m1.1.1.2">ℓ</ci><ci id="S1.SS1.p2.1.m1.1.1.3.cmml" xref="S1.SS1.p2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-geometry. In particular, we prove a generalization of the classical <em class="ltx_emph ltx_font_italic" id="S1.SS1.p2.2.1">centerpoint theorem</em> from discrete and computational geometry that works for any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.3"><semantics id="S1.SS1.p2.2.m2.3a"><mrow id="S1.SS1.p2.2.m2.3.4" xref="S1.SS1.p2.2.m2.3.4.cmml"><mi id="S1.SS1.p2.2.m2.3.4.2" xref="S1.SS1.p2.2.m2.3.4.2.cmml">p</mi><mo id="S1.SS1.p2.2.m2.3.4.1" xref="S1.SS1.p2.2.m2.3.4.1.cmml">∈</mo><mrow id="S1.SS1.p2.2.m2.3.4.3" xref="S1.SS1.p2.2.m2.3.4.3.cmml"><mrow id="S1.SS1.p2.2.m2.3.4.3.2.2" xref="S1.SS1.p2.2.m2.3.4.3.2.1.cmml"><mo id="S1.SS1.p2.2.m2.3.4.3.2.2.1" stretchy="false" xref="S1.SS1.p2.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml">1</mn><mo id="S1.SS1.p2.2.m2.3.4.3.2.2.2" xref="S1.SS1.p2.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S1.SS1.p2.2.m2.2.2" mathvariant="normal" xref="S1.SS1.p2.2.m2.2.2.cmml">∞</mi><mo id="S1.SS1.p2.2.m2.3.4.3.2.2.3" stretchy="false" xref="S1.SS1.p2.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.SS1.p2.2.m2.3.4.3.1" xref="S1.SS1.p2.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S1.SS1.p2.2.m2.3.4.3.3.2" xref="S1.SS1.p2.2.m2.3.4.3.3.1.cmml"><mo id="S1.SS1.p2.2.m2.3.4.3.3.2.1" stretchy="false" xref="S1.SS1.p2.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S1.SS1.p2.2.m2.3.3" mathvariant="normal" xref="S1.SS1.p2.2.m2.3.3.cmml">∞</mi><mo id="S1.SS1.p2.2.m2.3.4.3.3.2.2" stretchy="false" xref="S1.SS1.p2.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.3b"><apply id="S1.SS1.p2.2.m2.3.4.cmml" xref="S1.SS1.p2.2.m2.3.4"><in id="S1.SS1.p2.2.m2.3.4.1.cmml" xref="S1.SS1.p2.2.m2.3.4.1"></in><ci id="S1.SS1.p2.2.m2.3.4.2.cmml" xref="S1.SS1.p2.2.m2.3.4.2">𝑝</ci><apply id="S1.SS1.p2.2.m2.3.4.3.cmml" xref="S1.SS1.p2.2.m2.3.4.3"><union id="S1.SS1.p2.2.m2.3.4.3.1.cmml" xref="S1.SS1.p2.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S1.SS1.p2.2.m2.3.4.3.2.1.cmml" xref="S1.SS1.p2.2.m2.3.4.3.2.2"><cn id="S1.SS1.p2.2.m2.1.1.cmml" type="integer" xref="S1.SS1.p2.2.m2.1.1">1</cn><infinity id="S1.SS1.p2.2.m2.2.2.cmml" xref="S1.SS1.p2.2.m2.2.2"></infinity></interval><set id="S1.SS1.p2.2.m2.3.4.3.3.1.cmml" xref="S1.SS1.p2.2.m2.3.4.3.3.2"><infinity id="S1.SS1.p2.2.m2.3.3.cmml" xref="S1.SS1.p2.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. We think that this might be of independent interest. Our generalized centerpoint theorem works for both mass distributions and point sets and is tight in both cases.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.7">Our results also imply that a properly defined total search problem version of the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.1.m1.1"><semantics id="S1.SS1.p3.1.m1.1a"><msub id="S1.SS1.p3.1.m1.1.1" xref="S1.SS1.p3.1.m1.1.1.cmml"><mi id="S1.SS1.p3.1.m1.1.1.2" mathvariant="normal" xref="S1.SS1.p3.1.m1.1.1.2.cmml">ℓ</mi><mn id="S1.SS1.p3.1.m1.1.1.3" xref="S1.SS1.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.1.m1.1b"><apply id="S1.SS1.p3.1.m1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.1.m1.1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S1.SS1.p3.1.m1.1.1.2.cmml" xref="S1.SS1.p3.1.m1.1.1.2">ℓ</ci><cn id="S1.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.SS1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case lies in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.1"><semantics id="S1.SS1.p3.2.m2.1a"><msup id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml"><mi id="S1.SS1.p3.2.m2.1.1.2" xref="S1.SS1.p3.2.m2.1.1.2.cmml">𝖥𝖯</mi><mtext id="S1.SS1.p3.2.m2.1.1.3" xref="S1.SS1.p3.2.m2.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.1b"><apply id="S1.SS1.p3.2.m2.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.2.m2.1.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1">superscript</csymbol><ci id="S1.SS1.p3.2.m2.1.1.2.cmml" xref="S1.SS1.p3.2.m2.1.1.2">𝖥𝖯</ci><ci id="S1.SS1.p3.2.m2.1.1.3a.cmml" xref="S1.SS1.p3.2.m2.1.1.3"><mtext id="S1.SS1.p3.2.m2.1.1.3.cmml" mathsize="70%" xref="S1.SS1.p3.2.m2.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>, the class of black-box total search problems that can be solved efficiently by decision trees <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib15" title="">15</a>]</cite>. Again, the analogous observation for the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS1.p3.3.m3.1"><semantics id="S1.SS1.p3.3.m3.1a"><msub id="S1.SS1.p3.3.m3.1.1" xref="S1.SS1.p3.3.m3.1.1.cmml"><mi id="S1.SS1.p3.3.m3.1.1.2" mathvariant="normal" xref="S1.SS1.p3.3.m3.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p3.3.m3.1.1.3" mathvariant="normal" xref="S1.SS1.p3.3.m3.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.3.m3.1b"><apply id="S1.SS1.p3.3.m3.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.2.cmml" xref="S1.SS1.p3.3.m3.1.1.2">ℓ</ci><infinity id="S1.SS1.p3.3.m3.1.1.3.cmml" xref="S1.SS1.p3.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.3.m3.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case was made by Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> already. In a total search problem, the algorithm has to either return a solution (an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS1.p3.4.m4.1"><semantics id="S1.SS1.p3.4.m4.1a"><mi id="S1.SS1.p3.4.m4.1.1" xref="S1.SS1.p3.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.4.m4.1b"><ci id="S1.SS1.p3.4.m4.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.4.m4.1d">italic_ε</annotation></semantics></math>-approximate fixpoint) or a proof that the given instance does not fulfill the promise (i.e., that the given function on the grid does not extend to a contraction map on all of <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S1.SS1.p3.5.m5.2"><semantics id="S1.SS1.p3.5.m5.2a"><msup id="S1.SS1.p3.5.m5.2.3" xref="S1.SS1.p3.5.m5.2.3.cmml"><mrow id="S1.SS1.p3.5.m5.2.3.2.2" xref="S1.SS1.p3.5.m5.2.3.2.1.cmml"><mo id="S1.SS1.p3.5.m5.2.3.2.2.1" stretchy="false" xref="S1.SS1.p3.5.m5.2.3.2.1.cmml">[</mo><mn id="S1.SS1.p3.5.m5.1.1" xref="S1.SS1.p3.5.m5.1.1.cmml">0</mn><mo id="S1.SS1.p3.5.m5.2.3.2.2.2" xref="S1.SS1.p3.5.m5.2.3.2.1.cmml">,</mo><mn id="S1.SS1.p3.5.m5.2.2" xref="S1.SS1.p3.5.m5.2.2.cmml">1</mn><mo id="S1.SS1.p3.5.m5.2.3.2.2.3" stretchy="false" xref="S1.SS1.p3.5.m5.2.3.2.1.cmml">]</mo></mrow><mi id="S1.SS1.p3.5.m5.2.3.3" xref="S1.SS1.p3.5.m5.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.5.m5.2b"><apply id="S1.SS1.p3.5.m5.2.3.cmml" xref="S1.SS1.p3.5.m5.2.3"><csymbol cd="ambiguous" id="S1.SS1.p3.5.m5.2.3.1.cmml" xref="S1.SS1.p3.5.m5.2.3">superscript</csymbol><interval closure="closed" id="S1.SS1.p3.5.m5.2.3.2.1.cmml" xref="S1.SS1.p3.5.m5.2.3.2.2"><cn id="S1.SS1.p3.5.m5.1.1.cmml" type="integer" xref="S1.SS1.p3.5.m5.1.1">0</cn><cn id="S1.SS1.p3.5.m5.2.2.cmml" type="integer" xref="S1.SS1.p3.5.m5.2.2">1</cn></interval><ci id="S1.SS1.p3.5.m5.2.3.3.cmml" xref="S1.SS1.p3.5.m5.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.5.m5.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.5.m5.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>). In the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS1.p3.6.m6.1"><semantics id="S1.SS1.p3.6.m6.1a"><msub id="S1.SS1.p3.6.m6.1.1" xref="S1.SS1.p3.6.m6.1.1.cmml"><mi id="S1.SS1.p3.6.m6.1.1.2" mathvariant="normal" xref="S1.SS1.p3.6.m6.1.1.2.cmml">ℓ</mi><mi id="S1.SS1.p3.6.m6.1.1.3" mathvariant="normal" xref="S1.SS1.p3.6.m6.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.6.m6.1b"><apply id="S1.SS1.p3.6.m6.1.1.cmml" xref="S1.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.6.m6.1.1.1.cmml" xref="S1.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S1.SS1.p3.6.m6.1.1.2.cmml" xref="S1.SS1.p3.6.m6.1.1.2">ℓ</ci><infinity id="S1.SS1.p3.6.m6.1.1.3.cmml" xref="S1.SS1.p3.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.6.m6.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case, it suffices to consider pairs of grid points violating the contraction property as proof for the violation of the promise <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>. For the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.7.m7.1"><semantics id="S1.SS1.p3.7.m7.1a"><msub id="S1.SS1.p3.7.m7.1.1" xref="S1.SS1.p3.7.m7.1.1.cmml"><mi id="S1.SS1.p3.7.m7.1.1.2" mathvariant="normal" xref="S1.SS1.p3.7.m7.1.1.2.cmml">ℓ</mi><mn id="S1.SS1.p3.7.m7.1.1.3" xref="S1.SS1.p3.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.7.m7.1b"><apply id="S1.SS1.p3.7.m7.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.7.m7.1.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S1.SS1.p3.7.m7.1.1.2.cmml" xref="S1.SS1.p3.7.m7.1.1.2">ℓ</ci><cn id="S1.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S1.SS1.p3.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.7.m7.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.7.m7.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, this does not seem to suffice, which means that we need to consider a slightly more complicated proof of violation to properly define a total search version of the problem.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Proof Techniques</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.11">Our algorithm is based on a simple observation that has also been used in previous work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib35" title="">35</a>]</cite>: due to the contraction property, given any query point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p1.1.m1.1"><semantics id="S1.SS2.p1.1.m1.1a"><mi id="S1.SS2.p1.1.m1.1.1" xref="S1.SS2.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.1.m1.1b"><ci id="S1.SS2.p1.1.m1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.1.m1.1d">italic_x</annotation></semantics></math>, the fixpoint <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S1.SS2.p1.2.m2.1"><semantics id="S1.SS2.p1.2.m2.1a"><msup id="S1.SS2.p1.2.m2.1.1" xref="S1.SS2.p1.2.m2.1.1.cmml"><mi id="S1.SS2.p1.2.m2.1.1.2" xref="S1.SS2.p1.2.m2.1.1.2.cmml">x</mi><mo id="S1.SS2.p1.2.m2.1.1.3" xref="S1.SS2.p1.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.2.m2.1b"><apply id="S1.SS2.p1.2.m2.1.1.cmml" xref="S1.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.2.m2.1.1.1.cmml" xref="S1.SS2.p1.2.m2.1.1">superscript</csymbol><ci id="S1.SS2.p1.2.m2.1.1.2.cmml" xref="S1.SS2.p1.2.m2.1.1.2">𝑥</ci><ci id="S1.SS2.p1.2.m2.1.1.3.cmml" xref="S1.SS2.p1.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.2.m2.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.2.m2.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> must be closer to the query’s response <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS2.p1.3.m3.1"><semantics id="S1.SS2.p1.3.m3.1a"><mrow id="S1.SS2.p1.3.m3.1.2" xref="S1.SS2.p1.3.m3.1.2.cmml"><mi id="S1.SS2.p1.3.m3.1.2.2" xref="S1.SS2.p1.3.m3.1.2.2.cmml">f</mi><mo id="S1.SS2.p1.3.m3.1.2.1" xref="S1.SS2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p1.3.m3.1.2.3.2" xref="S1.SS2.p1.3.m3.1.2.cmml"><mo id="S1.SS2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S1.SS2.p1.3.m3.1.2.cmml">(</mo><mi id="S1.SS2.p1.3.m3.1.1" xref="S1.SS2.p1.3.m3.1.1.cmml">x</mi><mo id="S1.SS2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S1.SS2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.3.m3.1b"><apply id="S1.SS2.p1.3.m3.1.2.cmml" xref="S1.SS2.p1.3.m3.1.2"><times id="S1.SS2.p1.3.m3.1.2.1.cmml" xref="S1.SS2.p1.3.m3.1.2.1"></times><ci id="S1.SS2.p1.3.m3.1.2.2.cmml" xref="S1.SS2.p1.3.m3.1.2.2">𝑓</ci><ci id="S1.SS2.p1.3.m3.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.3.m3.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.3.m3.1d">italic_f ( italic_x )</annotation></semantics></math> than to the query <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p1.4.m4.1"><semantics id="S1.SS2.p1.4.m4.1a"><mi id="S1.SS2.p1.4.m4.1.1" xref="S1.SS2.p1.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.4.m4.1b"><ci id="S1.SS2.p1.4.m4.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.4.m4.1d">italic_x</annotation></semantics></math> itself. To quickly hone in on the fixpoint, we therefore wish to query a point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p1.5.m5.1"><semantics id="S1.SS2.p1.5.m5.1a"><mi id="S1.SS2.p1.5.m5.1.1" xref="S1.SS2.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.5.m5.1b"><ci id="S1.SS2.p1.5.m5.1.1.cmml" xref="S1.SS2.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.5.m5.1d">italic_x</annotation></semantics></math> such that for every possible response <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS2.p1.6.m6.1"><semantics id="S1.SS2.p1.6.m6.1a"><mrow id="S1.SS2.p1.6.m6.1.2" xref="S1.SS2.p1.6.m6.1.2.cmml"><mi id="S1.SS2.p1.6.m6.1.2.2" xref="S1.SS2.p1.6.m6.1.2.2.cmml">f</mi><mo id="S1.SS2.p1.6.m6.1.2.1" xref="S1.SS2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p1.6.m6.1.2.3.2" xref="S1.SS2.p1.6.m6.1.2.cmml"><mo id="S1.SS2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S1.SS2.p1.6.m6.1.2.cmml">(</mo><mi id="S1.SS2.p1.6.m6.1.1" xref="S1.SS2.p1.6.m6.1.1.cmml">x</mi><mo id="S1.SS2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S1.SS2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.6.m6.1b"><apply id="S1.SS2.p1.6.m6.1.2.cmml" xref="S1.SS2.p1.6.m6.1.2"><times id="S1.SS2.p1.6.m6.1.2.1.cmml" xref="S1.SS2.p1.6.m6.1.2.1"></times><ci id="S1.SS2.p1.6.m6.1.2.2.cmml" xref="S1.SS2.p1.6.m6.1.2.2">𝑓</ci><ci id="S1.SS2.p1.6.m6.1.1.cmml" xref="S1.SS2.p1.6.m6.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.6.m6.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.6.m6.1d">italic_f ( italic_x )</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S1.SS2.p1.7.m7.1"><semantics id="S1.SS2.p1.7.m7.1a"><mi id="S1.SS2.p1.7.m7.1.1" xref="S1.SS2.p1.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.7.m7.1b"><ci id="S1.SS2.p1.7.m7.1.1.cmml" xref="S1.SS2.p1.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.7.m7.1d">italic_f</annotation></semantics></math>, a significantly large part of the remaining search space lies at least as close to <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p1.8.m8.1"><semantics id="S1.SS2.p1.8.m8.1a"><mi id="S1.SS2.p1.8.m8.1.1" xref="S1.SS2.p1.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.8.m8.1b"><ci id="S1.SS2.p1.8.m8.1.1.cmml" xref="S1.SS2.p1.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.8.m8.1d">italic_x</annotation></semantics></math> as to <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS2.p1.9.m9.1"><semantics id="S1.SS2.p1.9.m9.1a"><mrow id="S1.SS2.p1.9.m9.1.2" xref="S1.SS2.p1.9.m9.1.2.cmml"><mi id="S1.SS2.p1.9.m9.1.2.2" xref="S1.SS2.p1.9.m9.1.2.2.cmml">f</mi><mo id="S1.SS2.p1.9.m9.1.2.1" xref="S1.SS2.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p1.9.m9.1.2.3.2" xref="S1.SS2.p1.9.m9.1.2.cmml"><mo id="S1.SS2.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S1.SS2.p1.9.m9.1.2.cmml">(</mo><mi id="S1.SS2.p1.9.m9.1.1" xref="S1.SS2.p1.9.m9.1.1.cmml">x</mi><mo id="S1.SS2.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S1.SS2.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.9.m9.1b"><apply id="S1.SS2.p1.9.m9.1.2.cmml" xref="S1.SS2.p1.9.m9.1.2"><times id="S1.SS2.p1.9.m9.1.2.1.cmml" xref="S1.SS2.p1.9.m9.1.2.1"></times><ci id="S1.SS2.p1.9.m9.1.2.2.cmml" xref="S1.SS2.p1.9.m9.1.2.2">𝑓</ci><ci id="S1.SS2.p1.9.m9.1.1.cmml" xref="S1.SS2.p1.9.m9.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.9.m9.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.9.m9.1d">italic_f ( italic_x )</annotation></semantics></math>. Querying <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p1.10.m10.1"><semantics id="S1.SS2.p1.10.m10.1a"><mi id="S1.SS2.p1.10.m10.1.1" xref="S1.SS2.p1.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.10.m10.1b"><ci id="S1.SS2.p1.10.m10.1.1.cmml" xref="S1.SS2.p1.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.10.m10.1d">italic_x</annotation></semantics></math> then allows us to considerably shrink the remaining search space no matter the response <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS2.p1.11.m11.1"><semantics id="S1.SS2.p1.11.m11.1a"><mrow id="S1.SS2.p1.11.m11.1.2" xref="S1.SS2.p1.11.m11.1.2.cmml"><mi id="S1.SS2.p1.11.m11.1.2.2" xref="S1.SS2.p1.11.m11.1.2.2.cmml">f</mi><mo id="S1.SS2.p1.11.m11.1.2.1" xref="S1.SS2.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p1.11.m11.1.2.3.2" xref="S1.SS2.p1.11.m11.1.2.cmml"><mo id="S1.SS2.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S1.SS2.p1.11.m11.1.2.cmml">(</mo><mi id="S1.SS2.p1.11.m11.1.1" xref="S1.SS2.p1.11.m11.1.1.cmml">x</mi><mo id="S1.SS2.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S1.SS2.p1.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.11.m11.1b"><apply id="S1.SS2.p1.11.m11.1.2.cmml" xref="S1.SS2.p1.11.m11.1.2"><times id="S1.SS2.p1.11.m11.1.2.1.cmml" xref="S1.SS2.p1.11.m11.1.2.1"></times><ci id="S1.SS2.p1.11.m11.1.2.2.cmml" xref="S1.SS2.p1.11.m11.1.2.2">𝑓</ci><ci id="S1.SS2.p1.11.m11.1.1.cmml" xref="S1.SS2.p1.11.m11.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.11.m11.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.11.m11.1d">italic_f ( italic_x )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS2.p2"> <p class="ltx_p" id="S1.SS2.p2.13">In the Euclidean case (<math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S1.SS2.p2.1.m1.1"><semantics id="S1.SS2.p2.1.m1.1a"><msub id="S1.SS2.p2.1.m1.1.1" xref="S1.SS2.p2.1.m1.1.1.cmml"><mi id="S1.SS2.p2.1.m1.1.1.2" mathvariant="normal" xref="S1.SS2.p2.1.m1.1.1.2.cmml">ℓ</mi><mn id="S1.SS2.p2.1.m1.1.1.3" xref="S1.SS2.p2.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.1.m1.1b"><apply id="S1.SS2.p2.1.m1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p2.1.m1.1.1.2.cmml" xref="S1.SS2.p2.1.m1.1.1.2">ℓ</ci><cn id="S1.SS2.p2.1.m1.1.1.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.1.m1.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-metric), a <em class="ltx_emph ltx_font_italic" id="S1.SS2.p2.13.1">centerpoint</em> of the remaining search space makes for a good query: a point <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p2.2.m2.1"><semantics id="S1.SS2.p2.2.m2.1a"><mi id="S1.SS2.p2.2.m2.1.1" xref="S1.SS2.p2.2.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.2.m2.1b"><ci id="S1.SS2.p2.2.m2.1.1.cmml" xref="S1.SS2.p2.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.2.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.2.m2.1d">italic_c</annotation></semantics></math> is a <math alttext="\rho" class="ltx_Math" display="inline" id="S1.SS2.p2.3.m3.1"><semantics id="S1.SS2.p2.3.m3.1a"><mi id="S1.SS2.p2.3.m3.1.1" xref="S1.SS2.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.3.m3.1b"><ci id="S1.SS2.p2.3.m3.1.1.cmml" xref="S1.SS2.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.3.m3.1d">italic_ρ</annotation></semantics></math>-centerpoint of a mass distribution <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS2.p2.4.m4.1"><semantics id="S1.SS2.p2.4.m4.1a"><mi id="S1.SS2.p2.4.m4.1.1" xref="S1.SS2.p2.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.4.m4.1b"><ci id="S1.SS2.p2.4.m4.1.1.cmml" xref="S1.SS2.p2.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.4.m4.1d">italic_μ</annotation></semantics></math> if any halfspace that contains <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p2.5.m5.1"><semantics id="S1.SS2.p2.5.m5.1a"><mi id="S1.SS2.p2.5.m5.1.1" xref="S1.SS2.p2.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.5.m5.1b"><ci id="S1.SS2.p2.5.m5.1.1.cmml" xref="S1.SS2.p2.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.5.m5.1d">italic_c</annotation></semantics></math> also contains at least a <math alttext="\rho" class="ltx_Math" display="inline" id="S1.SS2.p2.6.m6.1"><semantics id="S1.SS2.p2.6.m6.1a"><mi id="S1.SS2.p2.6.m6.1.1" xref="S1.SS2.p2.6.m6.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.6.m6.1b"><ci id="S1.SS2.p2.6.m6.1.1.cmml" xref="S1.SS2.p2.6.m6.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.6.m6.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.6.m6.1d">italic_ρ</annotation></semantics></math>-fraction of the mass of <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS2.p2.7.m7.1"><semantics id="S1.SS2.p2.7.m7.1a"><mi id="S1.SS2.p2.7.m7.1.1" xref="S1.SS2.p2.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.7.m7.1b"><ci id="S1.SS2.p2.7.m7.1.1.cmml" xref="S1.SS2.p2.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.7.m7.1d">italic_μ</annotation></semantics></math>. The celebrated centerpoint theorem originally due to Rado <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib31" title="">31</a>]</cite> guarantees that a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS2.p2.8.m8.1"><semantics id="S1.SS2.p2.8.m8.1a"><mfrac id="S1.SS2.p2.8.m8.1.1" xref="S1.SS2.p2.8.m8.1.1.cmml"><mn id="S1.SS2.p2.8.m8.1.1.2" xref="S1.SS2.p2.8.m8.1.1.2.cmml">1</mn><mrow id="S1.SS2.p2.8.m8.1.1.3" xref="S1.SS2.p2.8.m8.1.1.3.cmml"><mi id="S1.SS2.p2.8.m8.1.1.3.2" xref="S1.SS2.p2.8.m8.1.1.3.2.cmml">d</mi><mo id="S1.SS2.p2.8.m8.1.1.3.1" xref="S1.SS2.p2.8.m8.1.1.3.1.cmml">+</mo><mn id="S1.SS2.p2.8.m8.1.1.3.3" xref="S1.SS2.p2.8.m8.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.8.m8.1b"><apply id="S1.SS2.p2.8.m8.1.1.cmml" xref="S1.SS2.p2.8.m8.1.1"><divide id="S1.SS2.p2.8.m8.1.1.1.cmml" xref="S1.SS2.p2.8.m8.1.1"></divide><cn id="S1.SS2.p2.8.m8.1.1.2.cmml" type="integer" xref="S1.SS2.p2.8.m8.1.1.2">1</cn><apply id="S1.SS2.p2.8.m8.1.1.3.cmml" xref="S1.SS2.p2.8.m8.1.1.3"><plus id="S1.SS2.p2.8.m8.1.1.3.1.cmml" xref="S1.SS2.p2.8.m8.1.1.3.1"></plus><ci id="S1.SS2.p2.8.m8.1.1.3.2.cmml" xref="S1.SS2.p2.8.m8.1.1.3.2">𝑑</ci><cn id="S1.SS2.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S1.SS2.p2.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.8.m8.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.8.m8.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-centerpoint (commonly just called a centerpoint) always exists. Since the set of points that are at least as close to <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p2.9.m9.1"><semantics id="S1.SS2.p2.9.m9.1a"><mi id="S1.SS2.p2.9.m9.1.1" xref="S1.SS2.p2.9.m9.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.9.m9.1b"><ci id="S1.SS2.p2.9.m9.1.1.cmml" xref="S1.SS2.p2.9.m9.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.9.m9.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.9.m9.1d">italic_c</annotation></semantics></math> as to <math alttext="f(c)" class="ltx_Math" display="inline" id="S1.SS2.p2.10.m10.1"><semantics id="S1.SS2.p2.10.m10.1a"><mrow id="S1.SS2.p2.10.m10.1.2" xref="S1.SS2.p2.10.m10.1.2.cmml"><mi id="S1.SS2.p2.10.m10.1.2.2" xref="S1.SS2.p2.10.m10.1.2.2.cmml">f</mi><mo id="S1.SS2.p2.10.m10.1.2.1" xref="S1.SS2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p2.10.m10.1.2.3.2" xref="S1.SS2.p2.10.m10.1.2.cmml"><mo id="S1.SS2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S1.SS2.p2.10.m10.1.2.cmml">(</mo><mi id="S1.SS2.p2.10.m10.1.1" xref="S1.SS2.p2.10.m10.1.1.cmml">c</mi><mo id="S1.SS2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S1.SS2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.10.m10.1b"><apply id="S1.SS2.p2.10.m10.1.2.cmml" xref="S1.SS2.p2.10.m10.1.2"><times id="S1.SS2.p2.10.m10.1.2.1.cmml" xref="S1.SS2.p2.10.m10.1.2.1"></times><ci id="S1.SS2.p2.10.m10.1.2.2.cmml" xref="S1.SS2.p2.10.m10.1.2.2">𝑓</ci><ci id="S1.SS2.p2.10.m10.1.1.cmml" xref="S1.SS2.p2.10.m10.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.10.m10.1c">f(c)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.10.m10.1d">italic_f ( italic_c )</annotation></semantics></math> is a halfspace containing <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p2.11.m11.1"><semantics id="S1.SS2.p2.11.m11.1a"><mi id="S1.SS2.p2.11.m11.1.1" xref="S1.SS2.p2.11.m11.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.11.m11.1b"><ci id="S1.SS2.p2.11.m11.1.1.cmml" xref="S1.SS2.p2.11.m11.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.11.m11.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.11.m11.1d">italic_c</annotation></semantics></math>, querying a centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p2.12.m12.1"><semantics id="S1.SS2.p2.12.m12.1a"><mi id="S1.SS2.p2.12.m12.1.1" xref="S1.SS2.p2.12.m12.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.12.m12.1b"><ci id="S1.SS2.p2.12.m12.1.1.cmml" xref="S1.SS2.p2.12.m12.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.12.m12.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.12.m12.1d">italic_c</annotation></semantics></math> of the remaining search space allows us to discard at least a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS2.p2.13.m13.1"><semantics id="S1.SS2.p2.13.m13.1a"><mfrac id="S1.SS2.p2.13.m13.1.1" xref="S1.SS2.p2.13.m13.1.1.cmml"><mn id="S1.SS2.p2.13.m13.1.1.2" xref="S1.SS2.p2.13.m13.1.1.2.cmml">1</mn><mrow id="S1.SS2.p2.13.m13.1.1.3" xref="S1.SS2.p2.13.m13.1.1.3.cmml"><mi id="S1.SS2.p2.13.m13.1.1.3.2" xref="S1.SS2.p2.13.m13.1.1.3.2.cmml">d</mi><mo id="S1.SS2.p2.13.m13.1.1.3.1" xref="S1.SS2.p2.13.m13.1.1.3.1.cmml">+</mo><mn id="S1.SS2.p2.13.m13.1.1.3.3" xref="S1.SS2.p2.13.m13.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.13.m13.1b"><apply id="S1.SS2.p2.13.m13.1.1.cmml" xref="S1.SS2.p2.13.m13.1.1"><divide id="S1.SS2.p2.13.m13.1.1.1.cmml" xref="S1.SS2.p2.13.m13.1.1"></divide><cn id="S1.SS2.p2.13.m13.1.1.2.cmml" type="integer" xref="S1.SS2.p2.13.m13.1.1.2">1</cn><apply id="S1.SS2.p2.13.m13.1.1.3.cmml" xref="S1.SS2.p2.13.m13.1.1.3"><plus id="S1.SS2.p2.13.m13.1.1.3.1.cmml" xref="S1.SS2.p2.13.m13.1.1.3.1"></plus><ci id="S1.SS2.p2.13.m13.1.1.3.2.cmml" xref="S1.SS2.p2.13.m13.1.1.3.2">𝑑</ci><cn id="S1.SS2.p2.13.m13.1.1.3.3.cmml" type="integer" xref="S1.SS2.p2.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.13.m13.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.13.m13.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of the search space.</p> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.17">We generalize these concepts from Euclidean geometry to arbitrary <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS2.p3.1.m1.1"><semantics id="S1.SS2.p3.1.m1.1a"><msub id="S1.SS2.p3.1.m1.1.1" xref="S1.SS2.p3.1.m1.1.1.cmml"><mi id="S1.SS2.p3.1.m1.1.1.2" mathvariant="normal" xref="S1.SS2.p3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p3.1.m1.1.1.3" xref="S1.SS2.p3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.1.m1.1b"><apply id="S1.SS2.p3.1.m1.1.1.cmml" xref="S1.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.1.m1.1.1.1.cmml" xref="S1.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p3.1.m1.1.1.2.cmml" xref="S1.SS2.p3.1.m1.1.1.2">ℓ</ci><ci id="S1.SS2.p3.1.m1.1.1.3.cmml" xref="S1.SS2.p3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics. We first define <em class="ltx_emph ltx_font_italic" id="S1.SS2.p3.2.1"><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS2.p3.2.1.m1.1"><semantics id="S1.SS2.p3.2.1.m1.1a"><msub id="S1.SS2.p3.2.1.m1.1.1" xref="S1.SS2.p3.2.1.m1.1.1.cmml"><mi id="S1.SS2.p3.2.1.m1.1.1.2" mathvariant="normal" xref="S1.SS2.p3.2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p3.2.1.m1.1.1.3" xref="S1.SS2.p3.2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.2.1.m1.1b"><apply id="S1.SS2.p3.2.1.m1.1.1.cmml" xref="S1.SS2.p3.2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.2.1.m1.1.1.1.cmml" xref="S1.SS2.p3.2.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p3.2.1.m1.1.1.2.cmml" xref="S1.SS2.p3.2.1.m1.1.1.2">ℓ</ci><ci id="S1.SS2.p3.2.1.m1.1.1.3.cmml" xref="S1.SS2.p3.2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.2.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces</em> as a generalization of halfspaces. Informally speaking, an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS2.p3.3.m2.1"><semantics id="S1.SS2.p3.3.m2.1a"><msub id="S1.SS2.p3.3.m2.1.1" xref="S1.SS2.p3.3.m2.1.1.cmml"><mi id="S1.SS2.p3.3.m2.1.1.2" mathvariant="normal" xref="S1.SS2.p3.3.m2.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p3.3.m2.1.1.3" xref="S1.SS2.p3.3.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.3.m2.1b"><apply id="S1.SS2.p3.3.m2.1.1.cmml" xref="S1.SS2.p3.3.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.3.m2.1.1.1.cmml" xref="S1.SS2.p3.3.m2.1.1">subscript</csymbol><ci id="S1.SS2.p3.3.m2.1.1.2.cmml" xref="S1.SS2.p3.3.m2.1.1.2">ℓ</ci><ci id="S1.SS2.p3.3.m2.1.1.3.cmml" xref="S1.SS2.p3.3.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.3.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.3.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S1.SS2.p3.4.m3.2"><semantics id="S1.SS2.p3.4.m3.2a"><msubsup id="S1.SS2.p3.4.m3.2.3" xref="S1.SS2.p3.4.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.p3.4.m3.2.3.2.2" xref="S1.SS2.p3.4.m3.2.3.2.2.cmml">ℋ</mi><mrow id="S1.SS2.p3.4.m3.2.2.2.4" xref="S1.SS2.p3.4.m3.2.2.2.3.cmml"><mi id="S1.SS2.p3.4.m3.1.1.1.1" xref="S1.SS2.p3.4.m3.1.1.1.1.cmml">x</mi><mo id="S1.SS2.p3.4.m3.2.2.2.4.1" xref="S1.SS2.p3.4.m3.2.2.2.3.cmml">,</mo><mi id="S1.SS2.p3.4.m3.2.2.2.2" xref="S1.SS2.p3.4.m3.2.2.2.2.cmml">v</mi></mrow><mi id="S1.SS2.p3.4.m3.2.3.2.3" xref="S1.SS2.p3.4.m3.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.4.m3.2b"><apply id="S1.SS2.p3.4.m3.2.3.cmml" xref="S1.SS2.p3.4.m3.2.3"><csymbol cd="ambiguous" id="S1.SS2.p3.4.m3.2.3.1.cmml" xref="S1.SS2.p3.4.m3.2.3">subscript</csymbol><apply id="S1.SS2.p3.4.m3.2.3.2.cmml" xref="S1.SS2.p3.4.m3.2.3"><csymbol cd="ambiguous" id="S1.SS2.p3.4.m3.2.3.2.1.cmml" xref="S1.SS2.p3.4.m3.2.3">superscript</csymbol><ci id="S1.SS2.p3.4.m3.2.3.2.2.cmml" xref="S1.SS2.p3.4.m3.2.3.2.2">ℋ</ci><ci id="S1.SS2.p3.4.m3.2.3.2.3.cmml" xref="S1.SS2.p3.4.m3.2.3.2.3">𝑝</ci></apply><list id="S1.SS2.p3.4.m3.2.2.2.3.cmml" xref="S1.SS2.p3.4.m3.2.2.2.4"><ci id="S1.SS2.p3.4.m3.1.1.1.1.cmml" xref="S1.SS2.p3.4.m3.1.1.1.1">𝑥</ci><ci id="S1.SS2.p3.4.m3.2.2.2.2.cmml" xref="S1.SS2.p3.4.m3.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.4.m3.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.4.m3.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is defined by a point <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S1.SS2.p3.5.m4.1"><semantics id="S1.SS2.p3.5.m4.1a"><mrow id="S1.SS2.p3.5.m4.1.1" xref="S1.SS2.p3.5.m4.1.1.cmml"><mi id="S1.SS2.p3.5.m4.1.1.2" xref="S1.SS2.p3.5.m4.1.1.2.cmml">x</mi><mo id="S1.SS2.p3.5.m4.1.1.1" xref="S1.SS2.p3.5.m4.1.1.1.cmml">∈</mo><msup id="S1.SS2.p3.5.m4.1.1.3" xref="S1.SS2.p3.5.m4.1.1.3.cmml"><mi id="S1.SS2.p3.5.m4.1.1.3.2" xref="S1.SS2.p3.5.m4.1.1.3.2.cmml">ℝ</mi><mi id="S1.SS2.p3.5.m4.1.1.3.3" xref="S1.SS2.p3.5.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.5.m4.1b"><apply id="S1.SS2.p3.5.m4.1.1.cmml" xref="S1.SS2.p3.5.m4.1.1"><in id="S1.SS2.p3.5.m4.1.1.1.cmml" xref="S1.SS2.p3.5.m4.1.1.1"></in><ci id="S1.SS2.p3.5.m4.1.1.2.cmml" xref="S1.SS2.p3.5.m4.1.1.2">𝑥</ci><apply id="S1.SS2.p3.5.m4.1.1.3.cmml" xref="S1.SS2.p3.5.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p3.5.m4.1.1.3.1.cmml" xref="S1.SS2.p3.5.m4.1.1.3">superscript</csymbol><ci id="S1.SS2.p3.5.m4.1.1.3.2.cmml" xref="S1.SS2.p3.5.m4.1.1.3.2">ℝ</ci><ci id="S1.SS2.p3.5.m4.1.1.3.3.cmml" xref="S1.SS2.p3.5.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.5.m4.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.5.m4.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and a direction <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S1.SS2.p3.6.m5.1"><semantics id="S1.SS2.p3.6.m5.1a"><mrow id="S1.SS2.p3.6.m5.1.1" xref="S1.SS2.p3.6.m5.1.1.cmml"><mi id="S1.SS2.p3.6.m5.1.1.2" xref="S1.SS2.p3.6.m5.1.1.2.cmml">v</mi><mo id="S1.SS2.p3.6.m5.1.1.1" xref="S1.SS2.p3.6.m5.1.1.1.cmml">∈</mo><msup id="S1.SS2.p3.6.m5.1.1.3" xref="S1.SS2.p3.6.m5.1.1.3.cmml"><mi id="S1.SS2.p3.6.m5.1.1.3.2" xref="S1.SS2.p3.6.m5.1.1.3.2.cmml">S</mi><mrow id="S1.SS2.p3.6.m5.1.1.3.3" xref="S1.SS2.p3.6.m5.1.1.3.3.cmml"><mi id="S1.SS2.p3.6.m5.1.1.3.3.2" xref="S1.SS2.p3.6.m5.1.1.3.3.2.cmml">d</mi><mo id="S1.SS2.p3.6.m5.1.1.3.3.1" xref="S1.SS2.p3.6.m5.1.1.3.3.1.cmml">−</mo><mn id="S1.SS2.p3.6.m5.1.1.3.3.3" xref="S1.SS2.p3.6.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.6.m5.1b"><apply id="S1.SS2.p3.6.m5.1.1.cmml" xref="S1.SS2.p3.6.m5.1.1"><in id="S1.SS2.p3.6.m5.1.1.1.cmml" xref="S1.SS2.p3.6.m5.1.1.1"></in><ci id="S1.SS2.p3.6.m5.1.1.2.cmml" xref="S1.SS2.p3.6.m5.1.1.2">𝑣</ci><apply id="S1.SS2.p3.6.m5.1.1.3.cmml" xref="S1.SS2.p3.6.m5.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p3.6.m5.1.1.3.1.cmml" xref="S1.SS2.p3.6.m5.1.1.3">superscript</csymbol><ci id="S1.SS2.p3.6.m5.1.1.3.2.cmml" xref="S1.SS2.p3.6.m5.1.1.3.2">𝑆</ci><apply id="S1.SS2.p3.6.m5.1.1.3.3.cmml" xref="S1.SS2.p3.6.m5.1.1.3.3"><minus id="S1.SS2.p3.6.m5.1.1.3.3.1.cmml" xref="S1.SS2.p3.6.m5.1.1.3.3.1"></minus><ci id="S1.SS2.p3.6.m5.1.1.3.3.2.cmml" xref="S1.SS2.p3.6.m5.1.1.3.3.2">𝑑</ci><cn id="S1.SS2.p3.6.m5.1.1.3.3.3.cmml" type="integer" xref="S1.SS2.p3.6.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.6.m5.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.6.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and contains all points that are at least as close (with respect to the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS2.p3.7.m6.1"><semantics id="S1.SS2.p3.7.m6.1a"><msub id="S1.SS2.p3.7.m6.1.1" xref="S1.SS2.p3.7.m6.1.1.cmml"><mi id="S1.SS2.p3.7.m6.1.1.2" mathvariant="normal" xref="S1.SS2.p3.7.m6.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p3.7.m6.1.1.3" xref="S1.SS2.p3.7.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.7.m6.1b"><apply id="S1.SS2.p3.7.m6.1.1.cmml" xref="S1.SS2.p3.7.m6.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.7.m6.1.1.1.cmml" xref="S1.SS2.p3.7.m6.1.1">subscript</csymbol><ci id="S1.SS2.p3.7.m6.1.1.2.cmml" xref="S1.SS2.p3.7.m6.1.1.2">ℓ</ci><ci id="S1.SS2.p3.7.m6.1.1.3.cmml" xref="S1.SS2.p3.7.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.7.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.7.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metric) to <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p3.8.m7.1"><semantics id="S1.SS2.p3.8.m7.1a"><mi id="S1.SS2.p3.8.m7.1.1" xref="S1.SS2.p3.8.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.8.m7.1b"><ci id="S1.SS2.p3.8.m7.1.1.cmml" xref="S1.SS2.p3.8.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.8.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.8.m7.1d">italic_x</annotation></semantics></math> as to <math alttext="x-\varepsilon v" class="ltx_Math" display="inline" id="S1.SS2.p3.9.m8.1"><semantics id="S1.SS2.p3.9.m8.1a"><mrow id="S1.SS2.p3.9.m8.1.1" xref="S1.SS2.p3.9.m8.1.1.cmml"><mi id="S1.SS2.p3.9.m8.1.1.2" xref="S1.SS2.p3.9.m8.1.1.2.cmml">x</mi><mo id="S1.SS2.p3.9.m8.1.1.1" xref="S1.SS2.p3.9.m8.1.1.1.cmml">−</mo><mrow id="S1.SS2.p3.9.m8.1.1.3" xref="S1.SS2.p3.9.m8.1.1.3.cmml"><mi id="S1.SS2.p3.9.m8.1.1.3.2" xref="S1.SS2.p3.9.m8.1.1.3.2.cmml">ε</mi><mo id="S1.SS2.p3.9.m8.1.1.3.1" xref="S1.SS2.p3.9.m8.1.1.3.1.cmml">⁢</mo><mi id="S1.SS2.p3.9.m8.1.1.3.3" xref="S1.SS2.p3.9.m8.1.1.3.3.cmml">v</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.9.m8.1b"><apply id="S1.SS2.p3.9.m8.1.1.cmml" xref="S1.SS2.p3.9.m8.1.1"><minus id="S1.SS2.p3.9.m8.1.1.1.cmml" xref="S1.SS2.p3.9.m8.1.1.1"></minus><ci id="S1.SS2.p3.9.m8.1.1.2.cmml" xref="S1.SS2.p3.9.m8.1.1.2">𝑥</ci><apply id="S1.SS2.p3.9.m8.1.1.3.cmml" xref="S1.SS2.p3.9.m8.1.1.3"><times id="S1.SS2.p3.9.m8.1.1.3.1.cmml" xref="S1.SS2.p3.9.m8.1.1.3.1"></times><ci id="S1.SS2.p3.9.m8.1.1.3.2.cmml" xref="S1.SS2.p3.9.m8.1.1.3.2">𝜀</ci><ci id="S1.SS2.p3.9.m8.1.1.3.3.cmml" xref="S1.SS2.p3.9.m8.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.9.m8.1c">x-\varepsilon v</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.9.m8.1d">italic_x - italic_ε italic_v</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S1.SS2.p3.10.m9.1"><semantics id="S1.SS2.p3.10.m9.1a"><mrow id="S1.SS2.p3.10.m9.1.1" xref="S1.SS2.p3.10.m9.1.1.cmml"><mi id="S1.SS2.p3.10.m9.1.1.2" xref="S1.SS2.p3.10.m9.1.1.2.cmml">ε</mi><mo id="S1.SS2.p3.10.m9.1.1.1" xref="S1.SS2.p3.10.m9.1.1.1.cmml">></mo><mn id="S1.SS2.p3.10.m9.1.1.3" xref="S1.SS2.p3.10.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.10.m9.1b"><apply id="S1.SS2.p3.10.m9.1.1.cmml" xref="S1.SS2.p3.10.m9.1.1"><gt id="S1.SS2.p3.10.m9.1.1.1.cmml" xref="S1.SS2.p3.10.m9.1.1.1"></gt><ci id="S1.SS2.p3.10.m9.1.1.2.cmml" xref="S1.SS2.p3.10.m9.1.1.2">𝜀</ci><cn id="S1.SS2.p3.10.m9.1.1.3.cmml" type="integer" xref="S1.SS2.p3.10.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.10.m9.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.10.m9.1d">italic_ε > 0</annotation></semantics></math> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S1.F1" title="In 1.2 Proof Techniques ‣ 1 Introduction ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>). We then prove a generalized centerpoint theorem for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS2.p3.11.m10.1"><semantics id="S1.SS2.p3.11.m10.1a"><msub id="S1.SS2.p3.11.m10.1.1" xref="S1.SS2.p3.11.m10.1.1.cmml"><mi id="S1.SS2.p3.11.m10.1.1.2" mathvariant="normal" xref="S1.SS2.p3.11.m10.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p3.11.m10.1.1.3" xref="S1.SS2.p3.11.m10.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.11.m10.1b"><apply id="S1.SS2.p3.11.m10.1.1.cmml" xref="S1.SS2.p3.11.m10.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.11.m10.1.1.1.cmml" xref="S1.SS2.p3.11.m10.1.1">subscript</csymbol><ci id="S1.SS2.p3.11.m10.1.1.2.cmml" xref="S1.SS2.p3.11.m10.1.1.2">ℓ</ci><ci id="S1.SS2.p3.11.m10.1.1.3.cmml" xref="S1.SS2.p3.11.m10.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.11.m10.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.11.m10.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics, which says that for every mass distribution <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS2.p3.12.m11.1"><semantics id="S1.SS2.p3.12.m11.1a"><mi id="S1.SS2.p3.12.m11.1.1" xref="S1.SS2.p3.12.m11.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.12.m11.1b"><ci id="S1.SS2.p3.12.m11.1.1.cmml" xref="S1.SS2.p3.12.m11.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.12.m11.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.12.m11.1d">italic_μ</annotation></semantics></math> on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S1.SS2.p3.13.m12.1"><semantics id="S1.SS2.p3.13.m12.1a"><msup id="S1.SS2.p3.13.m12.1.1" xref="S1.SS2.p3.13.m12.1.1.cmml"><mi id="S1.SS2.p3.13.m12.1.1.2" xref="S1.SS2.p3.13.m12.1.1.2.cmml">ℝ</mi><mi id="S1.SS2.p3.13.m12.1.1.3" xref="S1.SS2.p3.13.m12.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.13.m12.1b"><apply id="S1.SS2.p3.13.m12.1.1.cmml" xref="S1.SS2.p3.13.m12.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.13.m12.1.1.1.cmml" xref="S1.SS2.p3.13.m12.1.1">superscript</csymbol><ci id="S1.SS2.p3.13.m12.1.1.2.cmml" xref="S1.SS2.p3.13.m12.1.1.2">ℝ</ci><ci id="S1.SS2.p3.13.m12.1.1.3.cmml" xref="S1.SS2.p3.13.m12.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.13.m12.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.13.m12.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists a point <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p3.14.m13.1"><semantics id="S1.SS2.p3.14.m13.1a"><mi id="S1.SS2.p3.14.m13.1.1" xref="S1.SS2.p3.14.m13.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.14.m13.1b"><ci id="S1.SS2.p3.14.m13.1.1.cmml" xref="S1.SS2.p3.14.m13.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.14.m13.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.14.m13.1d">italic_c</annotation></semantics></math> satisfying <math alttext="\mu(\mathcal{H}^{p}_{c,v})\geq\frac{1}{d+1}\mu(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S1.SS2.p3.15.m14.4"><semantics id="S1.SS2.p3.15.m14.4a"><mrow id="S1.SS2.p3.15.m14.4.4" xref="S1.SS2.p3.15.m14.4.4.cmml"><mrow id="S1.SS2.p3.15.m14.3.3.1" xref="S1.SS2.p3.15.m14.3.3.1.cmml"><mi id="S1.SS2.p3.15.m14.3.3.1.3" xref="S1.SS2.p3.15.m14.3.3.1.3.cmml">μ</mi><mo id="S1.SS2.p3.15.m14.3.3.1.2" xref="S1.SS2.p3.15.m14.3.3.1.2.cmml">⁢</mo><mrow id="S1.SS2.p3.15.m14.3.3.1.1.1" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.cmml"><mo id="S1.SS2.p3.15.m14.3.3.1.1.1.2" stretchy="false" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.cmml">(</mo><msubsup id="S1.SS2.p3.15.m14.3.3.1.1.1.1" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.p3.15.m14.3.3.1.1.1.1.2.2" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S1.SS2.p3.15.m14.2.2.2.4" xref="S1.SS2.p3.15.m14.2.2.2.3.cmml"><mi id="S1.SS2.p3.15.m14.1.1.1.1" xref="S1.SS2.p3.15.m14.1.1.1.1.cmml">c</mi><mo id="S1.SS2.p3.15.m14.2.2.2.4.1" xref="S1.SS2.p3.15.m14.2.2.2.3.cmml">,</mo><mi id="S1.SS2.p3.15.m14.2.2.2.2" xref="S1.SS2.p3.15.m14.2.2.2.2.cmml">v</mi></mrow><mi id="S1.SS2.p3.15.m14.3.3.1.1.1.1.2.3" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S1.SS2.p3.15.m14.3.3.1.1.1.3" stretchy="false" xref="S1.SS2.p3.15.m14.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS2.p3.15.m14.4.4.3" xref="S1.SS2.p3.15.m14.4.4.3.cmml">≥</mo><mrow id="S1.SS2.p3.15.m14.4.4.2" xref="S1.SS2.p3.15.m14.4.4.2.cmml"><mfrac id="S1.SS2.p3.15.m14.4.4.2.3" xref="S1.SS2.p3.15.m14.4.4.2.3.cmml"><mn id="S1.SS2.p3.15.m14.4.4.2.3.2" xref="S1.SS2.p3.15.m14.4.4.2.3.2.cmml">1</mn><mrow id="S1.SS2.p3.15.m14.4.4.2.3.3" xref="S1.SS2.p3.15.m14.4.4.2.3.3.cmml"><mi id="S1.SS2.p3.15.m14.4.4.2.3.3.2" xref="S1.SS2.p3.15.m14.4.4.2.3.3.2.cmml">d</mi><mo id="S1.SS2.p3.15.m14.4.4.2.3.3.1" xref="S1.SS2.p3.15.m14.4.4.2.3.3.1.cmml">+</mo><mn id="S1.SS2.p3.15.m14.4.4.2.3.3.3" xref="S1.SS2.p3.15.m14.4.4.2.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S1.SS2.p3.15.m14.4.4.2.2" xref="S1.SS2.p3.15.m14.4.4.2.2.cmml">⁢</mo><mi id="S1.SS2.p3.15.m14.4.4.2.4" xref="S1.SS2.p3.15.m14.4.4.2.4.cmml">μ</mi><mo id="S1.SS2.p3.15.m14.4.4.2.2a" xref="S1.SS2.p3.15.m14.4.4.2.2.cmml">⁢</mo><mrow id="S1.SS2.p3.15.m14.4.4.2.1.1" xref="S1.SS2.p3.15.m14.4.4.2.1.1.1.cmml"><mo id="S1.SS2.p3.15.m14.4.4.2.1.1.2" stretchy="false" xref="S1.SS2.p3.15.m14.4.4.2.1.1.1.cmml">(</mo><msup id="S1.SS2.p3.15.m14.4.4.2.1.1.1" xref="S1.SS2.p3.15.m14.4.4.2.1.1.1.cmml"><mi id="S1.SS2.p3.15.m14.4.4.2.1.1.1.2" xref="S1.SS2.p3.15.m14.4.4.2.1.1.1.2.cmml">ℝ</mi><mi 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( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ) ≥ divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S1.SS2.p3.16.m15.1"><semantics id="S1.SS2.p3.16.m15.1a"><mrow id="S1.SS2.p3.16.m15.1.1" xref="S1.SS2.p3.16.m15.1.1.cmml"><mi id="S1.SS2.p3.16.m15.1.1.2" xref="S1.SS2.p3.16.m15.1.1.2.cmml">v</mi><mo id="S1.SS2.p3.16.m15.1.1.1" xref="S1.SS2.p3.16.m15.1.1.1.cmml">∈</mo><msup id="S1.SS2.p3.16.m15.1.1.3" xref="S1.SS2.p3.16.m15.1.1.3.cmml"><mi id="S1.SS2.p3.16.m15.1.1.3.2" xref="S1.SS2.p3.16.m15.1.1.3.2.cmml">S</mi><mrow id="S1.SS2.p3.16.m15.1.1.3.3" xref="S1.SS2.p3.16.m15.1.1.3.3.cmml"><mi id="S1.SS2.p3.16.m15.1.1.3.3.2" xref="S1.SS2.p3.16.m15.1.1.3.3.2.cmml">d</mi><mo id="S1.SS2.p3.16.m15.1.1.3.3.1" xref="S1.SS2.p3.16.m15.1.1.3.3.1.cmml">−</mo><mn id="S1.SS2.p3.16.m15.1.1.3.3.3" xref="S1.SS2.p3.16.m15.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.16.m15.1b"><apply id="S1.SS2.p3.16.m15.1.1.cmml" xref="S1.SS2.p3.16.m15.1.1"><in id="S1.SS2.p3.16.m15.1.1.1.cmml" xref="S1.SS2.p3.16.m15.1.1.1"></in><ci id="S1.SS2.p3.16.m15.1.1.2.cmml" xref="S1.SS2.p3.16.m15.1.1.2">𝑣</ci><apply id="S1.SS2.p3.16.m15.1.1.3.cmml" xref="S1.SS2.p3.16.m15.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p3.16.m15.1.1.3.1.cmml" xref="S1.SS2.p3.16.m15.1.1.3">superscript</csymbol><ci id="S1.SS2.p3.16.m15.1.1.3.2.cmml" xref="S1.SS2.p3.16.m15.1.1.3.2">𝑆</ci><apply id="S1.SS2.p3.16.m15.1.1.3.3.cmml" xref="S1.SS2.p3.16.m15.1.1.3.3"><minus id="S1.SS2.p3.16.m15.1.1.3.3.1.cmml" xref="S1.SS2.p3.16.m15.1.1.3.3.1"></minus><ci id="S1.SS2.p3.16.m15.1.1.3.3.2.cmml" xref="S1.SS2.p3.16.m15.1.1.3.3.2">𝑑</ci><cn id="S1.SS2.p3.16.m15.1.1.3.3.3.cmml" type="integer" xref="S1.SS2.p3.16.m15.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.16.m15.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.16.m15.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. This theorem implies existence of good query points for our algorithm, allowing us to discard a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS2.p3.17.m16.1"><semantics id="S1.SS2.p3.17.m16.1a"><mfrac id="S1.SS2.p3.17.m16.1.1" xref="S1.SS2.p3.17.m16.1.1.cmml"><mn id="S1.SS2.p3.17.m16.1.1.2" xref="S1.SS2.p3.17.m16.1.1.2.cmml">1</mn><mrow id="S1.SS2.p3.17.m16.1.1.3" xref="S1.SS2.p3.17.m16.1.1.3.cmml"><mi id="S1.SS2.p3.17.m16.1.1.3.2" xref="S1.SS2.p3.17.m16.1.1.3.2.cmml">d</mi><mo id="S1.SS2.p3.17.m16.1.1.3.1" xref="S1.SS2.p3.17.m16.1.1.3.1.cmml">+</mo><mn id="S1.SS2.p3.17.m16.1.1.3.3" xref="S1.SS2.p3.17.m16.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.17.m16.1b"><apply id="S1.SS2.p3.17.m16.1.1.cmml" xref="S1.SS2.p3.17.m16.1.1"><divide id="S1.SS2.p3.17.m16.1.1.1.cmml" xref="S1.SS2.p3.17.m16.1.1"></divide><cn id="S1.SS2.p3.17.m16.1.1.2.cmml" type="integer" xref="S1.SS2.p3.17.m16.1.1.2">1</cn><apply id="S1.SS2.p3.17.m16.1.1.3.cmml" xref="S1.SS2.p3.17.m16.1.1.3"><plus id="S1.SS2.p3.17.m16.1.1.3.1.cmml" xref="S1.SS2.p3.17.m16.1.1.3.1"></plus><ci id="S1.SS2.p3.17.m16.1.1.3.2.cmml" xref="S1.SS2.p3.17.m16.1.1.3.2">𝑑</ci><cn id="S1.SS2.p3.17.m16.1.1.3.3.cmml" type="integer" xref="S1.SS2.p3.17.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.17.m16.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.17.m16.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of the remaining search space with each query.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="359" id="S1.F1.g1" src="extracted/6296433/figs/halfspace.png" width="359"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S1.F1.10.5.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S1.F1.8.4" style="font-size:90%;">The <math alttext="\ell_{5}" class="ltx_Math" display="inline" id="S1.F1.5.1.m1.1"><semantics id="S1.F1.5.1.m1.1b"><msub id="S1.F1.5.1.m1.1.1" xref="S1.F1.5.1.m1.1.1.cmml"><mi id="S1.F1.5.1.m1.1.1.2" mathvariant="normal" xref="S1.F1.5.1.m1.1.1.2.cmml">ℓ</mi><mn id="S1.F1.5.1.m1.1.1.3" xref="S1.F1.5.1.m1.1.1.3.cmml">5</mn></msub><annotation-xml encoding="MathML-Content" id="S1.F1.5.1.m1.1c"><apply id="S1.F1.5.1.m1.1.1.cmml" xref="S1.F1.5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.F1.5.1.m1.1.1.1.cmml" xref="S1.F1.5.1.m1.1.1">subscript</csymbol><ci id="S1.F1.5.1.m1.1.1.2.cmml" xref="S1.F1.5.1.m1.1.1.2">ℓ</ci><cn id="S1.F1.5.1.m1.1.1.3.cmml" type="integer" xref="S1.F1.5.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.5.1.m1.1d">\ell_{5}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.5.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{5}_{\mathbf{0},v}" class="ltx_Math" display="inline" id="S1.F1.6.2.m2.2"><semantics id="S1.F1.6.2.m2.2b"><msubsup id="S1.F1.6.2.m2.2.3" xref="S1.F1.6.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.F1.6.2.m2.2.3.2.2" xref="S1.F1.6.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S1.F1.6.2.m2.2.2.2.4" xref="S1.F1.6.2.m2.2.2.2.3.cmml"><mn id="S1.F1.6.2.m2.1.1.1.1" xref="S1.F1.6.2.m2.1.1.1.1.cmml">𝟎</mn><mo id="S1.F1.6.2.m2.2.2.2.4.1" xref="S1.F1.6.2.m2.2.2.2.3.cmml">,</mo><mi id="S1.F1.6.2.m2.2.2.2.2" xref="S1.F1.6.2.m2.2.2.2.2.cmml">v</mi></mrow><mn id="S1.F1.6.2.m2.2.3.2.3" xref="S1.F1.6.2.m2.2.3.2.3.cmml">5</mn></msubsup><annotation-xml encoding="MathML-Content" id="S1.F1.6.2.m2.2c"><apply id="S1.F1.6.2.m2.2.3.cmml" xref="S1.F1.6.2.m2.2.3"><csymbol cd="ambiguous" id="S1.F1.6.2.m2.2.3.1.cmml" xref="S1.F1.6.2.m2.2.3">subscript</csymbol><apply id="S1.F1.6.2.m2.2.3.2.cmml" xref="S1.F1.6.2.m2.2.3"><csymbol cd="ambiguous" id="S1.F1.6.2.m2.2.3.2.1.cmml" xref="S1.F1.6.2.m2.2.3">superscript</csymbol><ci id="S1.F1.6.2.m2.2.3.2.2.cmml" xref="S1.F1.6.2.m2.2.3.2.2">ℋ</ci><cn id="S1.F1.6.2.m2.2.3.2.3.cmml" type="integer" xref="S1.F1.6.2.m2.2.3.2.3">5</cn></apply><list id="S1.F1.6.2.m2.2.2.2.3.cmml" xref="S1.F1.6.2.m2.2.2.2.4"><cn id="S1.F1.6.2.m2.1.1.1.1.cmml" type="integer" xref="S1.F1.6.2.m2.1.1.1.1">0</cn><ci id="S1.F1.6.2.m2.2.2.2.2.cmml" xref="S1.F1.6.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.6.2.m2.2d">\mathcal{H}^{5}_{\mathbf{0},v}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.6.2.m2.2e">caligraphic_H start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_0 , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="v=(-2,-0.5,-0.6)" class="ltx_Math" display="inline" id="S1.F1.7.3.m3.3"><semantics id="S1.F1.7.3.m3.3b"><mrow id="S1.F1.7.3.m3.3.3" xref="S1.F1.7.3.m3.3.3.cmml"><mi id="S1.F1.7.3.m3.3.3.5" xref="S1.F1.7.3.m3.3.3.5.cmml">v</mi><mo id="S1.F1.7.3.m3.3.3.4" xref="S1.F1.7.3.m3.3.3.4.cmml">=</mo><mrow id="S1.F1.7.3.m3.3.3.3.3" xref="S1.F1.7.3.m3.3.3.3.4.cmml"><mo id="S1.F1.7.3.m3.3.3.3.3.4" stretchy="false" xref="S1.F1.7.3.m3.3.3.3.4.cmml">(</mo><mrow id="S1.F1.7.3.m3.1.1.1.1.1" xref="S1.F1.7.3.m3.1.1.1.1.1.cmml"><mo id="S1.F1.7.3.m3.1.1.1.1.1b" xref="S1.F1.7.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S1.F1.7.3.m3.1.1.1.1.1.2" xref="S1.F1.7.3.m3.1.1.1.1.1.2.cmml">2</mn></mrow><mo id="S1.F1.7.3.m3.3.3.3.3.5" xref="S1.F1.7.3.m3.3.3.3.4.cmml">,</mo><mrow id="S1.F1.7.3.m3.2.2.2.2.2" xref="S1.F1.7.3.m3.2.2.2.2.2.cmml"><mo id="S1.F1.7.3.m3.2.2.2.2.2b" xref="S1.F1.7.3.m3.2.2.2.2.2.cmml">−</mo><mn id="S1.F1.7.3.m3.2.2.2.2.2.2" xref="S1.F1.7.3.m3.2.2.2.2.2.2.cmml">0.5</mn></mrow><mo id="S1.F1.7.3.m3.3.3.3.3.6" xref="S1.F1.7.3.m3.3.3.3.4.cmml">,</mo><mrow id="S1.F1.7.3.m3.3.3.3.3.3" xref="S1.F1.7.3.m3.3.3.3.3.3.cmml"><mo id="S1.F1.7.3.m3.3.3.3.3.3b" xref="S1.F1.7.3.m3.3.3.3.3.3.cmml">−</mo><mn id="S1.F1.7.3.m3.3.3.3.3.3.2" xref="S1.F1.7.3.m3.3.3.3.3.3.2.cmml">0.6</mn></mrow><mo id="S1.F1.7.3.m3.3.3.3.3.7" stretchy="false" xref="S1.F1.7.3.m3.3.3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.7.3.m3.3c"><apply id="S1.F1.7.3.m3.3.3.cmml" xref="S1.F1.7.3.m3.3.3"><eq id="S1.F1.7.3.m3.3.3.4.cmml" xref="S1.F1.7.3.m3.3.3.4"></eq><ci id="S1.F1.7.3.m3.3.3.5.cmml" xref="S1.F1.7.3.m3.3.3.5">𝑣</ci><vector id="S1.F1.7.3.m3.3.3.3.4.cmml" xref="S1.F1.7.3.m3.3.3.3.3"><apply id="S1.F1.7.3.m3.1.1.1.1.1.cmml" xref="S1.F1.7.3.m3.1.1.1.1.1"><minus id="S1.F1.7.3.m3.1.1.1.1.1.1.cmml" xref="S1.F1.7.3.m3.1.1.1.1.1"></minus><cn id="S1.F1.7.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S1.F1.7.3.m3.1.1.1.1.1.2">2</cn></apply><apply id="S1.F1.7.3.m3.2.2.2.2.2.cmml" xref="S1.F1.7.3.m3.2.2.2.2.2"><minus id="S1.F1.7.3.m3.2.2.2.2.2.1.cmml" xref="S1.F1.7.3.m3.2.2.2.2.2"></minus><cn id="S1.F1.7.3.m3.2.2.2.2.2.2.cmml" type="float" xref="S1.F1.7.3.m3.2.2.2.2.2.2">0.5</cn></apply><apply id="S1.F1.7.3.m3.3.3.3.3.3.cmml" xref="S1.F1.7.3.m3.3.3.3.3.3"><minus id="S1.F1.7.3.m3.3.3.3.3.3.1.cmml" xref="S1.F1.7.3.m3.3.3.3.3.3"></minus><cn id="S1.F1.7.3.m3.3.3.3.3.3.2.cmml" type="float" xref="S1.F1.7.3.m3.3.3.3.3.3.2">0.6</cn></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.7.3.m3.3d">v=(-2,-0.5,-0.6)</annotation><annotation encoding="application/x-llamapun" id="S1.F1.7.3.m3.3e">italic_v = ( - 2 , - 0.5 , - 0.6 )</annotation></semantics></math> is drawn in red. The vector shown in the image is <math alttext="-v" class="ltx_Math" display="inline" id="S1.F1.8.4.m4.1"><semantics id="S1.F1.8.4.m4.1b"><mrow id="S1.F1.8.4.m4.1.1" xref="S1.F1.8.4.m4.1.1.cmml"><mo id="S1.F1.8.4.m4.1.1b" xref="S1.F1.8.4.m4.1.1.cmml">−</mo><mi id="S1.F1.8.4.m4.1.1.2" xref="S1.F1.8.4.m4.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.8.4.m4.1c"><apply id="S1.F1.8.4.m4.1.1.cmml" xref="S1.F1.8.4.m4.1.1"><minus id="S1.F1.8.4.m4.1.1.1.cmml" xref="S1.F1.8.4.m4.1.1"></minus><ci id="S1.F1.8.4.m4.1.1.2.cmml" xref="S1.F1.8.4.m4.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.8.4.m4.1d">-v</annotation><annotation encoding="application/x-llamapun" id="S1.F1.8.4.m4.1e">- italic_v</annotation></semantics></math>. Image created with the Desmos 3D calculator.</span></figcaption> </figure> <div class="ltx_para" id="S1.SS2.p4"> <p class="ltx_p" id="S1.SS2.p4.6">To prove our generalized centerpoint theorem, we use Brouwer’s fixpoint theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib3" title="">3</a>]</cite>. Concretely, given a mass distribution <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS2.p4.1.m1.1"><semantics id="S1.SS2.p4.1.m1.1a"><mi id="S1.SS2.p4.1.m1.1.1" xref="S1.SS2.p4.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.1.m1.1b"><ci id="S1.SS2.p4.1.m1.1.1.cmml" xref="S1.SS2.p4.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S1.SS2.p4.2.m2.1"><semantics id="S1.SS2.p4.2.m2.1a"><msup id="S1.SS2.p4.2.m2.1.1" xref="S1.SS2.p4.2.m2.1.1.cmml"><mi id="S1.SS2.p4.2.m2.1.1.2" xref="S1.SS2.p4.2.m2.1.1.2.cmml">ℝ</mi><mi id="S1.SS2.p4.2.m2.1.1.3" xref="S1.SS2.p4.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.2.m2.1b"><apply id="S1.SS2.p4.2.m2.1.1.cmml" xref="S1.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p4.2.m2.1.1.1.cmml" xref="S1.SS2.p4.2.m2.1.1">superscript</csymbol><ci id="S1.SS2.p4.2.m2.1.1.2.cmml" xref="S1.SS2.p4.2.m2.1.1.2">ℝ</ci><ci id="S1.SS2.p4.2.m2.1.1.3.cmml" xref="S1.SS2.p4.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, we consider a function <math alttext="F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S1.SS2.p4.3.m3.1"><semantics id="S1.SS2.p4.3.m3.1a"><mrow id="S1.SS2.p4.3.m3.1.1" xref="S1.SS2.p4.3.m3.1.1.cmml"><mi id="S1.SS2.p4.3.m3.1.1.2" xref="S1.SS2.p4.3.m3.1.1.2.cmml">F</mi><mo id="S1.SS2.p4.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.SS2.p4.3.m3.1.1.1.cmml">:</mo><mrow id="S1.SS2.p4.3.m3.1.1.3" xref="S1.SS2.p4.3.m3.1.1.3.cmml"><msup id="S1.SS2.p4.3.m3.1.1.3.2" xref="S1.SS2.p4.3.m3.1.1.3.2.cmml"><mi id="S1.SS2.p4.3.m3.1.1.3.2.2" xref="S1.SS2.p4.3.m3.1.1.3.2.2.cmml">ℝ</mi><mi id="S1.SS2.p4.3.m3.1.1.3.2.3" xref="S1.SS2.p4.3.m3.1.1.3.2.3.cmml">d</mi></msup><mo id="S1.SS2.p4.3.m3.1.1.3.1" stretchy="false" xref="S1.SS2.p4.3.m3.1.1.3.1.cmml">→</mo><msup id="S1.SS2.p4.3.m3.1.1.3.3" xref="S1.SS2.p4.3.m3.1.1.3.3.cmml"><mi id="S1.SS2.p4.3.m3.1.1.3.3.2" xref="S1.SS2.p4.3.m3.1.1.3.3.2.cmml">ℝ</mi><mi id="S1.SS2.p4.3.m3.1.1.3.3.3" xref="S1.SS2.p4.3.m3.1.1.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.3.m3.1b"><apply id="S1.SS2.p4.3.m3.1.1.cmml" xref="S1.SS2.p4.3.m3.1.1"><ci id="S1.SS2.p4.3.m3.1.1.1.cmml" xref="S1.SS2.p4.3.m3.1.1.1">:</ci><ci id="S1.SS2.p4.3.m3.1.1.2.cmml" xref="S1.SS2.p4.3.m3.1.1.2">𝐹</ci><apply id="S1.SS2.p4.3.m3.1.1.3.cmml" xref="S1.SS2.p4.3.m3.1.1.3"><ci id="S1.SS2.p4.3.m3.1.1.3.1.cmml" xref="S1.SS2.p4.3.m3.1.1.3.1">→</ci><apply id="S1.SS2.p4.3.m3.1.1.3.2.cmml" xref="S1.SS2.p4.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p4.3.m3.1.1.3.2.1.cmml" xref="S1.SS2.p4.3.m3.1.1.3.2">superscript</csymbol><ci id="S1.SS2.p4.3.m3.1.1.3.2.2.cmml" xref="S1.SS2.p4.3.m3.1.1.3.2.2">ℝ</ci><ci id="S1.SS2.p4.3.m3.1.1.3.2.3.cmml" xref="S1.SS2.p4.3.m3.1.1.3.2.3">𝑑</ci></apply><apply id="S1.SS2.p4.3.m3.1.1.3.3.cmml" xref="S1.SS2.p4.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS2.p4.3.m3.1.1.3.3.1.cmml" xref="S1.SS2.p4.3.m3.1.1.3.3">superscript</csymbol><ci id="S1.SS2.p4.3.m3.1.1.3.3.2.cmml" xref="S1.SS2.p4.3.m3.1.1.3.3.2">ℝ</ci><ci id="S1.SS2.p4.3.m3.1.1.3.3.3.cmml" xref="S1.SS2.p4.3.m3.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.3.m3.1c">F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.3.m3.1d">italic_F : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> which intuitively maps any point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p4.4.m4.1"><semantics id="S1.SS2.p4.4.m4.1a"><mi id="S1.SS2.p4.4.m4.1.1" xref="S1.SS2.p4.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.4.m4.1b"><ci id="S1.SS2.p4.4.m4.1.1.cmml" xref="S1.SS2.p4.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.4.m4.1d">italic_x</annotation></semantics></math> by pushing it in all directions <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S1.SS2.p4.5.m5.1"><semantics id="S1.SS2.p4.5.m5.1a"><mrow id="S1.SS2.p4.5.m5.1.1" xref="S1.SS2.p4.5.m5.1.1.cmml"><mi id="S1.SS2.p4.5.m5.1.1.2" xref="S1.SS2.p4.5.m5.1.1.2.cmml">v</mi><mo id="S1.SS2.p4.5.m5.1.1.1" xref="S1.SS2.p4.5.m5.1.1.1.cmml">∈</mo><msup id="S1.SS2.p4.5.m5.1.1.3" xref="S1.SS2.p4.5.m5.1.1.3.cmml"><mi id="S1.SS2.p4.5.m5.1.1.3.2" xref="S1.SS2.p4.5.m5.1.1.3.2.cmml">S</mi><mrow id="S1.SS2.p4.5.m5.1.1.3.3" xref="S1.SS2.p4.5.m5.1.1.3.3.cmml"><mi id="S1.SS2.p4.5.m5.1.1.3.3.2" xref="S1.SS2.p4.5.m5.1.1.3.3.2.cmml">d</mi><mo id="S1.SS2.p4.5.m5.1.1.3.3.1" xref="S1.SS2.p4.5.m5.1.1.3.3.1.cmml">−</mo><mn id="S1.SS2.p4.5.m5.1.1.3.3.3" xref="S1.SS2.p4.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.5.m5.1b"><apply id="S1.SS2.p4.5.m5.1.1.cmml" xref="S1.SS2.p4.5.m5.1.1"><in id="S1.SS2.p4.5.m5.1.1.1.cmml" xref="S1.SS2.p4.5.m5.1.1.1"></in><ci id="S1.SS2.p4.5.m5.1.1.2.cmml" xref="S1.SS2.p4.5.m5.1.1.2">𝑣</ci><apply id="S1.SS2.p4.5.m5.1.1.3.cmml" xref="S1.SS2.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p4.5.m5.1.1.3.1.cmml" xref="S1.SS2.p4.5.m5.1.1.3">superscript</csymbol><ci id="S1.SS2.p4.5.m5.1.1.3.2.cmml" xref="S1.SS2.p4.5.m5.1.1.3.2">𝑆</ci><apply id="S1.SS2.p4.5.m5.1.1.3.3.cmml" xref="S1.SS2.p4.5.m5.1.1.3.3"><minus id="S1.SS2.p4.5.m5.1.1.3.3.1.cmml" xref="S1.SS2.p4.5.m5.1.1.3.3.1"></minus><ci id="S1.SS2.p4.5.m5.1.1.3.3.2.cmml" xref="S1.SS2.p4.5.m5.1.1.3.3.2">𝑑</ci><cn id="S1.SS2.p4.5.m5.1.1.3.3.3.cmml" type="integer" xref="S1.SS2.p4.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.5.m5.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.5.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> for which <math alttext="\mu(\mathcal{H}^{p}_{x,-v})" class="ltx_Math" display="inline" id="S1.SS2.p4.6.m6.3"><semantics id="S1.SS2.p4.6.m6.3a"><mrow id="S1.SS2.p4.6.m6.3.3" xref="S1.SS2.p4.6.m6.3.3.cmml"><mi id="S1.SS2.p4.6.m6.3.3.3" xref="S1.SS2.p4.6.m6.3.3.3.cmml">μ</mi><mo id="S1.SS2.p4.6.m6.3.3.2" xref="S1.SS2.p4.6.m6.3.3.2.cmml">⁢</mo><mrow id="S1.SS2.p4.6.m6.3.3.1.1" xref="S1.SS2.p4.6.m6.3.3.1.1.1.cmml"><mo id="S1.SS2.p4.6.m6.3.3.1.1.2" stretchy="false" xref="S1.SS2.p4.6.m6.3.3.1.1.1.cmml">(</mo><msubsup id="S1.SS2.p4.6.m6.3.3.1.1.1" xref="S1.SS2.p4.6.m6.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS2.p4.6.m6.3.3.1.1.1.2.2" xref="S1.SS2.p4.6.m6.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="S1.SS2.p4.6.m6.2.2.2.2" xref="S1.SS2.p4.6.m6.2.2.2.3.cmml"><mi id="S1.SS2.p4.6.m6.1.1.1.1" xref="S1.SS2.p4.6.m6.1.1.1.1.cmml">x</mi><mo id="S1.SS2.p4.6.m6.2.2.2.2.2" xref="S1.SS2.p4.6.m6.2.2.2.3.cmml">,</mo><mrow id="S1.SS2.p4.6.m6.2.2.2.2.1" xref="S1.SS2.p4.6.m6.2.2.2.2.1.cmml"><mo id="S1.SS2.p4.6.m6.2.2.2.2.1a" xref="S1.SS2.p4.6.m6.2.2.2.2.1.cmml">−</mo><mi id="S1.SS2.p4.6.m6.2.2.2.2.1.2" xref="S1.SS2.p4.6.m6.2.2.2.2.1.2.cmml">v</mi></mrow></mrow><mi id="S1.SS2.p4.6.m6.3.3.1.1.1.2.3" xref="S1.SS2.p4.6.m6.3.3.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S1.SS2.p4.6.m6.3.3.1.1.3" stretchy="false" xref="S1.SS2.p4.6.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.6.m6.3b"><apply id="S1.SS2.p4.6.m6.3.3.cmml" xref="S1.SS2.p4.6.m6.3.3"><times id="S1.SS2.p4.6.m6.3.3.2.cmml" xref="S1.SS2.p4.6.m6.3.3.2"></times><ci id="S1.SS2.p4.6.m6.3.3.3.cmml" xref="S1.SS2.p4.6.m6.3.3.3">𝜇</ci><apply id="S1.SS2.p4.6.m6.3.3.1.1.1.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p4.6.m6.3.3.1.1.1.1.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1">subscript</csymbol><apply id="S1.SS2.p4.6.m6.3.3.1.1.1.2.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p4.6.m6.3.3.1.1.1.2.1.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1">superscript</csymbol><ci id="S1.SS2.p4.6.m6.3.3.1.1.1.2.2.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1.1.2.2">ℋ</ci><ci id="S1.SS2.p4.6.m6.3.3.1.1.1.2.3.cmml" xref="S1.SS2.p4.6.m6.3.3.1.1.1.2.3">𝑝</ci></apply><list id="S1.SS2.p4.6.m6.2.2.2.3.cmml" xref="S1.SS2.p4.6.m6.2.2.2.2"><ci id="S1.SS2.p4.6.m6.1.1.1.1.cmml" xref="S1.SS2.p4.6.m6.1.1.1.1">𝑥</ci><apply id="S1.SS2.p4.6.m6.2.2.2.2.1.cmml" xref="S1.SS2.p4.6.m6.2.2.2.2.1"><minus id="S1.SS2.p4.6.m6.2.2.2.2.1.1.cmml" xref="S1.SS2.p4.6.m6.2.2.2.2.1"></minus><ci id="S1.SS2.p4.6.m6.2.2.2.2.1.2.cmml" xref="S1.SS2.p4.6.m6.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.6.m6.3c">\mu(\mathcal{H}^{p}_{x,-v})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.6.m6.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> is not yet large enough. Formally, we define it using the following integral</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{i}(x)\coloneqq x_{i}+\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}-\mu(% \mathcal{H}^{p}_{x,-v}),0\right)dv" class="ltx_Math" display="block" id="S1.Ex1.m1.6"><semantics id="S1.Ex1.m1.6a"><mrow id="S1.Ex1.m1.6.6" xref="S1.Ex1.m1.6.6.cmml"><mrow id="S1.Ex1.m1.6.6.3" xref="S1.Ex1.m1.6.6.3.cmml"><msub id="S1.Ex1.m1.6.6.3.2" xref="S1.Ex1.m1.6.6.3.2.cmml"><mi id="S1.Ex1.m1.6.6.3.2.2" xref="S1.Ex1.m1.6.6.3.2.2.cmml">F</mi><mi id="S1.Ex1.m1.6.6.3.2.3" xref="S1.Ex1.m1.6.6.3.2.3.cmml">i</mi></msub><mo id="S1.Ex1.m1.6.6.3.1" xref="S1.Ex1.m1.6.6.3.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.6.6.3.3.2" xref="S1.Ex1.m1.6.6.3.cmml"><mo id="S1.Ex1.m1.6.6.3.3.2.1" stretchy="false" xref="S1.Ex1.m1.6.6.3.cmml">(</mo><mi id="S1.Ex1.m1.3.3" xref="S1.Ex1.m1.3.3.cmml">x</mi><mo id="S1.Ex1.m1.6.6.3.3.2.2" stretchy="false" xref="S1.Ex1.m1.6.6.3.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.6.6.2" xref="S1.Ex1.m1.6.6.2.cmml">≔</mo><mrow id="S1.Ex1.m1.6.6.1" xref="S1.Ex1.m1.6.6.1.cmml"><msub id="S1.Ex1.m1.6.6.1.3" xref="S1.Ex1.m1.6.6.1.3.cmml"><mi id="S1.Ex1.m1.6.6.1.3.2" xref="S1.Ex1.m1.6.6.1.3.2.cmml">x</mi><mi id="S1.Ex1.m1.6.6.1.3.3" xref="S1.Ex1.m1.6.6.1.3.3.cmml">i</mi></msub><mo id="S1.Ex1.m1.6.6.1.2" rspace="0.055em" xref="S1.Ex1.m1.6.6.1.2.cmml">+</mo><mrow id="S1.Ex1.m1.6.6.1.1" xref="S1.Ex1.m1.6.6.1.1.cmml"><msub id="S1.Ex1.m1.6.6.1.1.2" xref="S1.Ex1.m1.6.6.1.1.2.cmml"><mo id="S1.Ex1.m1.6.6.1.1.2.2" xref="S1.Ex1.m1.6.6.1.1.2.2.cmml">∫</mo><msup id="S1.Ex1.m1.6.6.1.1.2.3" xref="S1.Ex1.m1.6.6.1.1.2.3.cmml"><mi id="S1.Ex1.m1.6.6.1.1.2.3.2" xref="S1.Ex1.m1.6.6.1.1.2.3.2.cmml">S</mi><mrow id="S1.Ex1.m1.6.6.1.1.2.3.3" xref="S1.Ex1.m1.6.6.1.1.2.3.3.cmml"><mi id="S1.Ex1.m1.6.6.1.1.2.3.3.2" xref="S1.Ex1.m1.6.6.1.1.2.3.3.2.cmml">d</mi><mo id="S1.Ex1.m1.6.6.1.1.2.3.3.1" xref="S1.Ex1.m1.6.6.1.1.2.3.3.1.cmml">−</mo><mn id="S1.Ex1.m1.6.6.1.1.2.3.3.3" xref="S1.Ex1.m1.6.6.1.1.2.3.3.3.cmml">1</mn></mrow></msup></msub><mrow id="S1.Ex1.m1.6.6.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.cmml"><msub id="S1.Ex1.m1.6.6.1.1.1.3" xref="S1.Ex1.m1.6.6.1.1.1.3.cmml"><mi id="S1.Ex1.m1.6.6.1.1.1.3.2" xref="S1.Ex1.m1.6.6.1.1.1.3.2.cmml">v</mi><mi id="S1.Ex1.m1.6.6.1.1.1.3.3" xref="S1.Ex1.m1.6.6.1.1.1.3.3.cmml">i</mi></msub><mo id="S1.Ex1.m1.6.6.1.1.1.2" lspace="0.167em" xref="S1.Ex1.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.2.cmml"><mi id="S1.Ex1.m1.4.4" xref="S1.Ex1.m1.4.4.cmml">max</mi><mo id="S1.Ex1.m1.6.6.1.1.1.1.1a" xref="S1.Ex1.m1.6.6.1.1.1.1.2.cmml">⁡</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.2.cmml"><mo id="S1.Ex1.m1.6.6.1.1.1.1.1.1.2" xref="S1.Ex1.m1.6.6.1.1.1.1.2.cmml">(</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.cmml"><mfrac id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.cmml"><mn id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.2" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.2.cmml">1</mn><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.cmml"><mi id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.2" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.2.cmml">d</mi><mo id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.1" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.3" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.3.cmml">μ</mi><mo id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1" 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id="S1.Ex1.m1.6c">F_{i}(x)\coloneqq x_{i}+\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}-\mu(% \mathcal{H}^{p}_{x,-v}),0\right)dv</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.6d">italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ≔ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∫ start_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT ) , 0 ) italic_d italic_v</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS2.p4.8">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S1.SS2.p4.7.m1.1"><semantics id="S1.SS2.p4.7.m1.1a"><mrow id="S1.SS2.p4.7.m1.1.2" xref="S1.SS2.p4.7.m1.1.2.cmml"><mi id="S1.SS2.p4.7.m1.1.2.2" xref="S1.SS2.p4.7.m1.1.2.2.cmml">i</mi><mo id="S1.SS2.p4.7.m1.1.2.1" xref="S1.SS2.p4.7.m1.1.2.1.cmml">∈</mo><mrow id="S1.SS2.p4.7.m1.1.2.3.2" xref="S1.SS2.p4.7.m1.1.2.3.1.cmml"><mo id="S1.SS2.p4.7.m1.1.2.3.2.1" stretchy="false" xref="S1.SS2.p4.7.m1.1.2.3.1.1.cmml">[</mo><mi id="S1.SS2.p4.7.m1.1.1" xref="S1.SS2.p4.7.m1.1.1.cmml">d</mi><mo id="S1.SS2.p4.7.m1.1.2.3.2.2" stretchy="false" xref="S1.SS2.p4.7.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.7.m1.1b"><apply id="S1.SS2.p4.7.m1.1.2.cmml" xref="S1.SS2.p4.7.m1.1.2"><in id="S1.SS2.p4.7.m1.1.2.1.cmml" xref="S1.SS2.p4.7.m1.1.2.1"></in><ci id="S1.SS2.p4.7.m1.1.2.2.cmml" xref="S1.SS2.p4.7.m1.1.2.2">𝑖</ci><apply id="S1.SS2.p4.7.m1.1.2.3.1.cmml" xref="S1.SS2.p4.7.m1.1.2.3.2"><csymbol cd="latexml" id="S1.SS2.p4.7.m1.1.2.3.1.1.cmml" 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We show that <math alttext="F" class="ltx_Math" display="inline" id="S1.SS2.p4.8.m2.1"><semantics id="S1.SS2.p4.8.m2.1a"><mi id="S1.SS2.p4.8.m2.1.1" xref="S1.SS2.p4.8.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.8.m2.1b"><ci id="S1.SS2.p4.8.m2.1.1.cmml" xref="S1.SS2.p4.8.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.8.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.8.m2.1d">italic_F</annotation></semantics></math> is continuous, which means that if we restrict it to a compact domain, Brouwer’s fixpoint theorem implies existence of a fixpoint. We then prove that this fixpoint must be a centerpoint.</p> </div> <div class="ltx_para" id="S1.SS2.p5"> <p class="ltx_p" id="S1.SS2.p5.9">After proving that we can reduce our remaining search space by a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS2.p5.1.m1.1"><semantics id="S1.SS2.p5.1.m1.1a"><mfrac id="S1.SS2.p5.1.m1.1.1" xref="S1.SS2.p5.1.m1.1.1.cmml"><mn id="S1.SS2.p5.1.m1.1.1.2" xref="S1.SS2.p5.1.m1.1.1.2.cmml">1</mn><mrow id="S1.SS2.p5.1.m1.1.1.3" xref="S1.SS2.p5.1.m1.1.1.3.cmml"><mi id="S1.SS2.p5.1.m1.1.1.3.2" xref="S1.SS2.p5.1.m1.1.1.3.2.cmml">d</mi><mo id="S1.SS2.p5.1.m1.1.1.3.1" xref="S1.SS2.p5.1.m1.1.1.3.1.cmml">+</mo><mn id="S1.SS2.p5.1.m1.1.1.3.3" xref="S1.SS2.p5.1.m1.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.1.m1.1b"><apply id="S1.SS2.p5.1.m1.1.1.cmml" xref="S1.SS2.p5.1.m1.1.1"><divide id="S1.SS2.p5.1.m1.1.1.1.cmml" xref="S1.SS2.p5.1.m1.1.1"></divide><cn id="S1.SS2.p5.1.m1.1.1.2.cmml" type="integer" xref="S1.SS2.p5.1.m1.1.1.2">1</cn><apply id="S1.SS2.p5.1.m1.1.1.3.cmml" xref="S1.SS2.p5.1.m1.1.1.3"><plus id="S1.SS2.p5.1.m1.1.1.3.1.cmml" xref="S1.SS2.p5.1.m1.1.1.3.1"></plus><ci id="S1.SS2.p5.1.m1.1.1.3.2.cmml" xref="S1.SS2.p5.1.m1.1.1.3.2">𝑑</ci><cn id="S1.SS2.p5.1.m1.1.1.3.3.cmml" type="integer" xref="S1.SS2.p5.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.1.m1.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.1.m1.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction with every query, we still need a termination condition for our algorithm. To this end, we show that as long as a queried point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p5.2.m2.1"><semantics id="S1.SS2.p5.2.m2.1a"><mi id="S1.SS2.p5.2.m2.1.1" xref="S1.SS2.p5.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.2.m2.1b"><ci id="S1.SS2.p5.2.m2.1.1.cmml" xref="S1.SS2.p5.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.2.m2.1d">italic_x</annotation></semantics></math> is not an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS2.p5.3.m3.1"><semantics id="S1.SS2.p5.3.m3.1a"><mi id="S1.SS2.p5.3.m3.1.1" xref="S1.SS2.p5.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.3.m3.1b"><ci id="S1.SS2.p5.3.m3.1.1.cmml" xref="S1.SS2.p5.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.3.m3.1d">italic_ε</annotation></semantics></math>-approximate fixpoint, any point sufficiently close to the fixpoint <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S1.SS2.p5.4.m4.1"><semantics id="S1.SS2.p5.4.m4.1a"><msup id="S1.SS2.p5.4.m4.1.1" xref="S1.SS2.p5.4.m4.1.1.cmml"><mi id="S1.SS2.p5.4.m4.1.1.2" xref="S1.SS2.p5.4.m4.1.1.2.cmml">x</mi><mo id="S1.SS2.p5.4.m4.1.1.3" xref="S1.SS2.p5.4.m4.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.4.m4.1b"><apply id="S1.SS2.p5.4.m4.1.1.cmml" xref="S1.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS2.p5.4.m4.1.1.1.cmml" xref="S1.SS2.p5.4.m4.1.1">superscript</csymbol><ci id="S1.SS2.p5.4.m4.1.1.2.cmml" xref="S1.SS2.p5.4.m4.1.1.2">𝑥</ci><ci id="S1.SS2.p5.4.m4.1.1.3.cmml" xref="S1.SS2.p5.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.4.m4.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.4.m4.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> cannot lie closer to <math alttext="x" class="ltx_Math" display="inline" id="S1.SS2.p5.5.m5.1"><semantics id="S1.SS2.p5.5.m5.1a"><mi id="S1.SS2.p5.5.m5.1.1" xref="S1.SS2.p5.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.5.m5.1b"><ci id="S1.SS2.p5.5.m5.1.1.cmml" xref="S1.SS2.p5.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.5.m5.1d">italic_x</annotation></semantics></math> than to <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS2.p5.6.m6.1"><semantics id="S1.SS2.p5.6.m6.1a"><mrow id="S1.SS2.p5.6.m6.1.2" xref="S1.SS2.p5.6.m6.1.2.cmml"><mi id="S1.SS2.p5.6.m6.1.2.2" xref="S1.SS2.p5.6.m6.1.2.2.cmml">f</mi><mo id="S1.SS2.p5.6.m6.1.2.1" xref="S1.SS2.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.SS2.p5.6.m6.1.2.3.2" xref="S1.SS2.p5.6.m6.1.2.cmml"><mo id="S1.SS2.p5.6.m6.1.2.3.2.1" stretchy="false" xref="S1.SS2.p5.6.m6.1.2.cmml">(</mo><mi id="S1.SS2.p5.6.m6.1.1" xref="S1.SS2.p5.6.m6.1.1.cmml">x</mi><mo id="S1.SS2.p5.6.m6.1.2.3.2.2" stretchy="false" xref="S1.SS2.p5.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.6.m6.1b"><apply id="S1.SS2.p5.6.m6.1.2.cmml" xref="S1.SS2.p5.6.m6.1.2"><times id="S1.SS2.p5.6.m6.1.2.1.cmml" xref="S1.SS2.p5.6.m6.1.2.1"></times><ci id="S1.SS2.p5.6.m6.1.2.2.cmml" xref="S1.SS2.p5.6.m6.1.2.2">𝑓</ci><ci id="S1.SS2.p5.6.m6.1.1.cmml" xref="S1.SS2.p5.6.m6.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.6.m6.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.6.m6.1d">italic_f ( italic_x )</annotation></semantics></math>. Thus, there exists a small ball around <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S1.SS2.p5.7.m7.1"><semantics id="S1.SS2.p5.7.m7.1a"><msup id="S1.SS2.p5.7.m7.1.1" xref="S1.SS2.p5.7.m7.1.1.cmml"><mi id="S1.SS2.p5.7.m7.1.1.2" xref="S1.SS2.p5.7.m7.1.1.2.cmml">x</mi><mo id="S1.SS2.p5.7.m7.1.1.3" xref="S1.SS2.p5.7.m7.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.7.m7.1b"><apply id="S1.SS2.p5.7.m7.1.1.cmml" xref="S1.SS2.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS2.p5.7.m7.1.1.1.cmml" xref="S1.SS2.p5.7.m7.1.1">superscript</csymbol><ci id="S1.SS2.p5.7.m7.1.1.2.cmml" xref="S1.SS2.p5.7.m7.1.1.2">𝑥</ci><ci id="S1.SS2.p5.7.m7.1.1.3.cmml" xref="S1.SS2.p5.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.7.m7.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.7.m7.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> that cannot be discarded as long as we have not queried an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS2.p5.8.m8.1"><semantics id="S1.SS2.p5.8.m8.1a"><mi id="S1.SS2.p5.8.m8.1.1" xref="S1.SS2.p5.8.m8.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.8.m8.1b"><ci id="S1.SS2.p5.8.m8.1.1.cmml" xref="S1.SS2.p5.8.m8.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.8.m8.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.8.m8.1d">italic_ε</annotation></semantics></math>-approximate fixpoint yet. This in turn implies a lower bound <math alttext="V_{\varepsilon,\lambda}" class="ltx_Math" display="inline" id="S1.SS2.p5.9.m9.2"><semantics id="S1.SS2.p5.9.m9.2a"><msub id="S1.SS2.p5.9.m9.2.3" xref="S1.SS2.p5.9.m9.2.3.cmml"><mi id="S1.SS2.p5.9.m9.2.3.2" xref="S1.SS2.p5.9.m9.2.3.2.cmml">V</mi><mrow id="S1.SS2.p5.9.m9.2.2.2.4" xref="S1.SS2.p5.9.m9.2.2.2.3.cmml"><mi id="S1.SS2.p5.9.m9.1.1.1.1" xref="S1.SS2.p5.9.m9.1.1.1.1.cmml">ε</mi><mo id="S1.SS2.p5.9.m9.2.2.2.4.1" xref="S1.SS2.p5.9.m9.2.2.2.3.cmml">,</mo><mi id="S1.SS2.p5.9.m9.2.2.2.2" xref="S1.SS2.p5.9.m9.2.2.2.2.cmml">λ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.9.m9.2b"><apply id="S1.SS2.p5.9.m9.2.3.cmml" xref="S1.SS2.p5.9.m9.2.3"><csymbol cd="ambiguous" id="S1.SS2.p5.9.m9.2.3.1.cmml" xref="S1.SS2.p5.9.m9.2.3">subscript</csymbol><ci id="S1.SS2.p5.9.m9.2.3.2.cmml" xref="S1.SS2.p5.9.m9.2.3.2">𝑉</ci><list id="S1.SS2.p5.9.m9.2.2.2.3.cmml" xref="S1.SS2.p5.9.m9.2.2.2.4"><ci id="S1.SS2.p5.9.m9.1.1.1.1.cmml" xref="S1.SS2.p5.9.m9.1.1.1.1">𝜀</ci><ci id="S1.SS2.p5.9.m9.2.2.2.2.cmml" xref="S1.SS2.p5.9.m9.2.2.2.2">𝜆</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.9.m9.2c">V_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.9.m9.2d">italic_V start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math> for the volume of the remaining search space that holds as long as the algorithm does not terminate. From this, we can then conclude the upper bound</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\log_{\frac{d}{d+1}}\left(\frac{1}{V_{\varepsilon,\lambda}}\right)=\mathcal{O}% \left(d^{2}\left(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d% \right)\right)" class="ltx_Math" display="block" id="S1.Ex2.m1.4"><semantics id="S1.Ex2.m1.4a"><mrow id="S1.Ex2.m1.4.4" xref="S1.Ex2.m1.4.4.cmml"><mrow id="S1.Ex2.m1.3.3.1.1" xref="S1.Ex2.m1.3.3.1.2.cmml"><msub id="S1.Ex2.m1.3.3.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.cmml"><mi id="S1.Ex2.m1.3.3.1.1.1.2" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml">log</mi><mfrac id="S1.Ex2.m1.3.3.1.1.1.3" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml"><mi id="S1.Ex2.m1.3.3.1.1.1.3.2" xref="S1.Ex2.m1.3.3.1.1.1.3.2.cmml">d</mi><mrow id="S1.Ex2.m1.3.3.1.1.1.3.3" xref="S1.Ex2.m1.3.3.1.1.1.3.3.cmml"><mi id="S1.Ex2.m1.3.3.1.1.1.3.3.2" xref="S1.Ex2.m1.3.3.1.1.1.3.3.2.cmml">d</mi><mo id="S1.Ex2.m1.3.3.1.1.1.3.3.1" xref="S1.Ex2.m1.3.3.1.1.1.3.3.1.cmml">+</mo><mn id="S1.Ex2.m1.3.3.1.1.1.3.3.3" xref="S1.Ex2.m1.3.3.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></msub><mo id="S1.Ex2.m1.3.3.1.1a" xref="S1.Ex2.m1.3.3.1.2.cmml">⁡</mo><mrow id="S1.Ex2.m1.3.3.1.1.2" xref="S1.Ex2.m1.3.3.1.2.cmml"><mo id="S1.Ex2.m1.3.3.1.1.2.1" xref="S1.Ex2.m1.3.3.1.2.cmml">(</mo><mfrac id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml"><mn id="S1.Ex2.m1.2.2.4" xref="S1.Ex2.m1.2.2.4.cmml">1</mn><msub id="S1.Ex2.m1.2.2.2" xref="S1.Ex2.m1.2.2.2.cmml"><mi id="S1.Ex2.m1.2.2.2.4" xref="S1.Ex2.m1.2.2.2.4.cmml">V</mi><mrow id="S1.Ex2.m1.2.2.2.2.2.4" xref="S1.Ex2.m1.2.2.2.2.2.3.cmml"><mi id="S1.Ex2.m1.1.1.1.1.1.1" xref="S1.Ex2.m1.1.1.1.1.1.1.cmml">ε</mi><mo id="S1.Ex2.m1.2.2.2.2.2.4.1" xref="S1.Ex2.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S1.Ex2.m1.2.2.2.2.2.2" xref="S1.Ex2.m1.2.2.2.2.2.2.cmml">λ</mi></mrow></msub></mfrac><mo 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\left(d^{2}\left(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d% \right)\right)</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.4d">roman_log start_POSTSUBSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG italic_V start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT end_ARG ) = caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG + roman_log italic_d ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS2.p5.14">on the number of queries needed to find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS2.p5.10.m1.1"><semantics id="S1.SS2.p5.10.m1.1a"><mi id="S1.SS2.p5.10.m1.1.1" xref="S1.SS2.p5.10.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.10.m1.1b"><ci id="S1.SS2.p5.10.m1.1.1.cmml" xref="S1.SS2.p5.10.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.10.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.10.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. To get rid of the <math alttext="\log d" class="ltx_Math" display="inline" id="S1.SS2.p5.11.m2.1"><semantics id="S1.SS2.p5.11.m2.1a"><mrow id="S1.SS2.p5.11.m2.1.1" xref="S1.SS2.p5.11.m2.1.1.cmml"><mi id="S1.SS2.p5.11.m2.1.1.1" xref="S1.SS2.p5.11.m2.1.1.1.cmml">log</mi><mo id="S1.SS2.p5.11.m2.1.1a" lspace="0.167em" xref="S1.SS2.p5.11.m2.1.1.cmml">⁡</mo><mi id="S1.SS2.p5.11.m2.1.1.2" xref="S1.SS2.p5.11.m2.1.1.2.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.11.m2.1b"><apply id="S1.SS2.p5.11.m2.1.1.cmml" xref="S1.SS2.p5.11.m2.1.1"><log id="S1.SS2.p5.11.m2.1.1.1.cmml" xref="S1.SS2.p5.11.m2.1.1.1"></log><ci id="S1.SS2.p5.11.m2.1.1.2.cmml" xref="S1.SS2.p5.11.m2.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.11.m2.1c">\log d</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.11.m2.1d">roman_log italic_d</annotation></semantics></math> term, we additionally observe that a simple iterative algorithm is faster if <math alttext="d" class="ltx_Math" display="inline" id="S1.SS2.p5.12.m3.1"><semantics id="S1.SS2.p5.12.m3.1a"><mi id="S1.SS2.p5.12.m3.1.1" xref="S1.SS2.p5.12.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.12.m3.1b"><ci id="S1.SS2.p5.12.m3.1.1.cmml" xref="S1.SS2.p5.12.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.12.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.12.m3.1d">italic_d</annotation></semantics></math> dominates <math alttext="\frac{1}{\varepsilon}" class="ltx_Math" display="inline" id="S1.SS2.p5.13.m4.1"><semantics id="S1.SS2.p5.13.m4.1a"><mfrac id="S1.SS2.p5.13.m4.1.1" xref="S1.SS2.p5.13.m4.1.1.cmml"><mn id="S1.SS2.p5.13.m4.1.1.2" xref="S1.SS2.p5.13.m4.1.1.2.cmml">1</mn><mi id="S1.SS2.p5.13.m4.1.1.3" xref="S1.SS2.p5.13.m4.1.1.3.cmml">ε</mi></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.13.m4.1b"><apply id="S1.SS2.p5.13.m4.1.1.cmml" xref="S1.SS2.p5.13.m4.1.1"><divide id="S1.SS2.p5.13.m4.1.1.1.cmml" xref="S1.SS2.p5.13.m4.1.1"></divide><cn id="S1.SS2.p5.13.m4.1.1.2.cmml" type="integer" xref="S1.SS2.p5.13.m4.1.1.2">1</cn><ci id="S1.SS2.p5.13.m4.1.1.3.cmml" xref="S1.SS2.p5.13.m4.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.13.m4.1c">\frac{1}{\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.13.m4.1d">divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG</annotation></semantics></math> and <math alttext="\frac{1}{1-\lambda}" class="ltx_Math" display="inline" id="S1.SS2.p5.14.m5.1"><semantics id="S1.SS2.p5.14.m5.1a"><mfrac id="S1.SS2.p5.14.m5.1.1" xref="S1.SS2.p5.14.m5.1.1.cmml"><mn id="S1.SS2.p5.14.m5.1.1.2" xref="S1.SS2.p5.14.m5.1.1.2.cmml">1</mn><mrow id="S1.SS2.p5.14.m5.1.1.3" xref="S1.SS2.p5.14.m5.1.1.3.cmml"><mn id="S1.SS2.p5.14.m5.1.1.3.2" xref="S1.SS2.p5.14.m5.1.1.3.2.cmml">1</mn><mo id="S1.SS2.p5.14.m5.1.1.3.1" xref="S1.SS2.p5.14.m5.1.1.3.1.cmml">−</mo><mi id="S1.SS2.p5.14.m5.1.1.3.3" xref="S1.SS2.p5.14.m5.1.1.3.3.cmml">λ</mi></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.14.m5.1b"><apply id="S1.SS2.p5.14.m5.1.1.cmml" xref="S1.SS2.p5.14.m5.1.1"><divide id="S1.SS2.p5.14.m5.1.1.1.cmml" xref="S1.SS2.p5.14.m5.1.1"></divide><cn id="S1.SS2.p5.14.m5.1.1.2.cmml" type="integer" xref="S1.SS2.p5.14.m5.1.1.2">1</cn><apply id="S1.SS2.p5.14.m5.1.1.3.cmml" xref="S1.SS2.p5.14.m5.1.1.3"><minus id="S1.SS2.p5.14.m5.1.1.3.1.cmml" xref="S1.SS2.p5.14.m5.1.1.3.1"></minus><cn id="S1.SS2.p5.14.m5.1.1.3.2.cmml" type="integer" xref="S1.SS2.p5.14.m5.1.1.3.2">1</cn><ci id="S1.SS2.p5.14.m5.1.1.3.3.cmml" xref="S1.SS2.p5.14.m5.1.1.3.3">𝜆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.14.m5.1c">\frac{1}{1-\lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.14.m5.1d">divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS2.p6"> <p class="ltx_p" id="S1.SS2.p6.7">As mentioned before, Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> provide a rounding scheme in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS2.p6.1.m1.1"><semantics id="S1.SS2.p6.1.m1.1a"><msub id="S1.SS2.p6.1.m1.1.1" xref="S1.SS2.p6.1.m1.1.1.cmml"><mi id="S1.SS2.p6.1.m1.1.1.2" mathvariant="normal" xref="S1.SS2.p6.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p6.1.m1.1.1.3" mathvariant="normal" xref="S1.SS2.p6.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.1.m1.1b"><apply id="S1.SS2.p6.1.m1.1.1.cmml" xref="S1.SS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p6.1.m1.1.1.1.cmml" xref="S1.SS2.p6.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p6.1.m1.1.1.2.cmml" xref="S1.SS2.p6.1.m1.1.1.2">ℓ</ci><infinity id="S1.SS2.p6.1.m1.1.1.3.cmml" xref="S1.SS2.p6.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.1.m1.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case that allows them to ensure that all queries lie on a discrete grid of limited granularity. Their rounding scheme can also be used in combination with our algorithm, but is specific to the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS2.p6.2.m2.1"><semantics id="S1.SS2.p6.2.m2.1a"><msub id="S1.SS2.p6.2.m2.1.1" xref="S1.SS2.p6.2.m2.1.1.cmml"><mi id="S1.SS2.p6.2.m2.1.1.2" mathvariant="normal" xref="S1.SS2.p6.2.m2.1.1.2.cmml">ℓ</mi><mi id="S1.SS2.p6.2.m2.1.1.3" mathvariant="normal" xref="S1.SS2.p6.2.m2.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.2.m2.1b"><apply id="S1.SS2.p6.2.m2.1.1.cmml" xref="S1.SS2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p6.2.m2.1.1.1.cmml" xref="S1.SS2.p6.2.m2.1.1">subscript</csymbol><ci id="S1.SS2.p6.2.m2.1.1.2.cmml" xref="S1.SS2.p6.2.m2.1.1.2">ℓ</ci><infinity id="S1.SS2.p6.2.m2.1.1.3.cmml" xref="S1.SS2.p6.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.2.m2.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case. For the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS2.p6.3.m3.1"><semantics id="S1.SS2.p6.3.m3.1a"><msub id="S1.SS2.p6.3.m3.1.1" xref="S1.SS2.p6.3.m3.1.1.cmml"><mi id="S1.SS2.p6.3.m3.1.1.2" mathvariant="normal" xref="S1.SS2.p6.3.m3.1.1.2.cmml">ℓ</mi><mn id="S1.SS2.p6.3.m3.1.1.3" xref="S1.SS2.p6.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.3.m3.1b"><apply id="S1.SS2.p6.3.m3.1.1.cmml" xref="S1.SS2.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p6.3.m3.1.1.1.cmml" xref="S1.SS2.p6.3.m3.1.1">subscript</csymbol><ci id="S1.SS2.p6.3.m3.1.1.2.cmml" xref="S1.SS2.p6.3.m3.1.1.2">ℓ</ci><cn id="S1.SS2.p6.3.m3.1.1.3.cmml" type="integer" xref="S1.SS2.p6.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.3.m3.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, which is also motivated by applications, we therefore provide our own rounding scheme. Concretely, we show that for <math alttext="p=1" class="ltx_Math" display="inline" id="S1.SS2.p6.4.m4.1"><semantics id="S1.SS2.p6.4.m4.1a"><mrow id="S1.SS2.p6.4.m4.1.1" xref="S1.SS2.p6.4.m4.1.1.cmml"><mi id="S1.SS2.p6.4.m4.1.1.2" xref="S1.SS2.p6.4.m4.1.1.2.cmml">p</mi><mo id="S1.SS2.p6.4.m4.1.1.1" xref="S1.SS2.p6.4.m4.1.1.1.cmml">=</mo><mn id="S1.SS2.p6.4.m4.1.1.3" xref="S1.SS2.p6.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.4.m4.1b"><apply id="S1.SS2.p6.4.m4.1.1.cmml" xref="S1.SS2.p6.4.m4.1.1"><eq id="S1.SS2.p6.4.m4.1.1.1.cmml" xref="S1.SS2.p6.4.m4.1.1.1"></eq><ci id="S1.SS2.p6.4.m4.1.1.2.cmml" xref="S1.SS2.p6.4.m4.1.1.2">𝑝</ci><cn id="S1.SS2.p6.4.m4.1.1.3.cmml" type="integer" xref="S1.SS2.p6.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.4.m4.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.4.m4.1d">italic_p = 1</annotation></semantics></math>, each centerpoint query <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p6.5.m5.1"><semantics id="S1.SS2.p6.5.m5.1a"><mi id="S1.SS2.p6.5.m5.1.1" xref="S1.SS2.p6.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.5.m5.1b"><ci id="S1.SS2.p6.5.m5.1.1.cmml" xref="S1.SS2.p6.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.5.m5.1d">italic_c</annotation></semantics></math> of the original algorithm can be replaced with a query to the closest grid point <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S1.SS2.p6.6.m6.1"><semantics id="S1.SS2.p6.6.m6.1a"><msup id="S1.SS2.p6.6.m6.1.1" xref="S1.SS2.p6.6.m6.1.1.cmml"><mi id="S1.SS2.p6.6.m6.1.1.2" xref="S1.SS2.p6.6.m6.1.1.2.cmml">c</mi><mo id="S1.SS2.p6.6.m6.1.1.3" xref="S1.SS2.p6.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.6.m6.1b"><apply id="S1.SS2.p6.6.m6.1.1.cmml" xref="S1.SS2.p6.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS2.p6.6.m6.1.1.1.cmml" xref="S1.SS2.p6.6.m6.1.1">superscript</csymbol><ci id="S1.SS2.p6.6.m6.1.1.2.cmml" xref="S1.SS2.p6.6.m6.1.1.2">𝑐</ci><ci id="S1.SS2.p6.6.m6.1.1.3.cmml" xref="S1.SS2.p6.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.6.m6.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.6.m6.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="c" class="ltx_Math" display="inline" id="S1.SS2.p6.7.m7.1"><semantics id="S1.SS2.p6.7.m7.1a"><mi id="S1.SS2.p6.7.m7.1.1" xref="S1.SS2.p6.7.m7.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.7.m7.1b"><ci id="S1.SS2.p6.7.m7.1.1.cmml" xref="S1.SS2.p6.7.m7.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.7.m7.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.7.m7.1d">italic_c</annotation></semantics></math>. To make this work, we need to use a discrete variant of our centerpoint theorem to measure the size of the remaining search space in terms of remaining grid points.</p> </div> </section> <section class="ltx_subsection" id="S1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.3 </span>Discussion</h3> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Comparison to Results for <math alttext="p=2" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.1.m1.1"><semantics id="S1.SS3.SSS0.Px1.1.m1.1b"><mrow id="S1.SS3.SSS0.Px1.1.m1.1.1" xref="S1.SS3.SSS0.Px1.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.1.m1.1.1.2" xref="S1.SS3.SSS0.Px1.1.m1.1.1.2.cmml">p</mi><mo id="S1.SS3.SSS0.Px1.1.m1.1.1.1" xref="S1.SS3.SSS0.Px1.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS3.SSS0.Px1.1.m1.1.1.3" xref="S1.SS3.SSS0.Px1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.1.m1.1c"><apply id="S1.SS3.SSS0.Px1.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px1.1.m1.1.1"><eq id="S1.SS3.SSS0.Px1.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.1.m1.1.1.1"></eq><ci id="S1.SS3.SSS0.Px1.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.1.m1.1.1.2">𝑝</ci><cn id="S1.SS3.SSS0.Px1.1.m1.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.1.m1.1d">p=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.1.m1.1e">italic_p = 2</annotation></semantics></math>.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p1.13">The intersection of Euclidean halfspaces is a polyhedron, and Grünbaum’s theorem says that the centroid of a polyhedron is a <math alttext="\frac{1}{e}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S1.SS3.SSS0.Px1.p1.1.m1.1a"><mfrac id="S1.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mn id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">1</mn><mi id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml">e</mi></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1"><divide id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1"></divide><cn id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1.2">1</cn><ci id="S1.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.1.m1.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.1.m1.1c">\frac{1}{e}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.1.m1.1d">divide start_ARG 1 end_ARG start_ARG italic_e end_ARG</annotation></semantics></math>-centerpoint of the polyhedron <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib16" title="">16</a>]</cite>. Thus, there always exists a <math alttext="\frac{1}{e}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="S1.SS3.SSS0.Px1.p1.2.m2.1a"><mfrac id="S1.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1.cmml"><mn id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.2" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml">1</mn><mi id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.3" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml">e</mi></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.2.m2.1b"><apply id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1"><divide id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1"></divide><cn id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1.2">1</cn><ci id="S1.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.2.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.2.m2.1c">\frac{1}{e}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.2.m2.1d">divide start_ARG 1 end_ARG start_ARG italic_e end_ARG</annotation></semantics></math>-centerpoint of the remaining search space in the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.3.m3.1"><semantics id="S1.SS3.SSS0.Px1.p1.3.m3.1a"><msub id="S1.SS3.SSS0.Px1.p1.3.m3.1.1" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1.2.cmml">ℓ</mi><mn id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.3" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.3.m3.1b"><apply id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1.2">ℓ</ci><cn id="S1.SS3.SSS0.Px1.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.3.m3.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-case, rather than just a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.4.m4.1"><semantics id="S1.SS3.SSS0.Px1.p1.4.m4.1a"><mfrac id="S1.SS3.SSS0.Px1.p1.4.m4.1.1" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.cmml"><mn id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.2" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.2.cmml">1</mn><mrow id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.cmml"><mi id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.2" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.2.cmml">d</mi><mo id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.1" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.1.cmml">+</mo><mn id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.3" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.4.m4.1b"><apply id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1"><divide id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1"></divide><cn id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.2">1</cn><apply id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3"><plus id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.1"></plus><ci id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.2.cmml" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.2">𝑑</ci><cn id="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.4.m4.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.4.m4.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-centerpoint. Thanks to this, specialized algorithms such as the Inscribed Ellipsoid and the Centroid algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib36" title="">36</a>]</cite> achieve a query upper bound that depends only linearly on <math alttext="d" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.5.m5.1"><semantics id="S1.SS3.SSS0.Px1.p1.5.m5.1a"><mi id="S1.SS3.SSS0.Px1.p1.5.m5.1.1" xref="S1.SS3.SSS0.Px1.p1.5.m5.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.5.m5.1b"><ci id="S1.SS3.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.5.m5.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.5.m5.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.5.m5.1d">italic_d</annotation></semantics></math> (compared to our quadratic dependency). An interesting question would be whether something analogous to Grünbaum’s theorem exists for the intersection of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.6.m6.1"><semantics id="S1.SS3.SSS0.Px1.p1.6.m6.1a"><msub id="S1.SS3.SSS0.Px1.p1.6.m6.1.1" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.3" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.6.m6.1b"><apply id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.6.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces in general. Such a bound could then be plugged into our analysis to improve our upper bounds from <math alttext="\mathcal{O}(d^{2})" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.7.m7.1"><semantics id="S1.SS3.SSS0.Px1.p1.7.m7.1a"><mrow id="S1.SS3.SSS0.Px1.p1.7.m7.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.3" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.3.cmml">𝒪</mi><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.2" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml">(</mo><msup id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.2" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.2.cmml">d</mi><mn id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.3" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.7.m7.1b"><apply id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1"><times id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.2"></times><ci id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.3">𝒪</ci><apply id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1">superscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.2">𝑑</ci><cn id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.7.m7.1c">\mathcal{O}(d^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.7.m7.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> to <math alttext="\mathcal{O}(d\log d)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.8.m8.1"><semantics id="S1.SS3.SSS0.Px1.p1.8.m8.1a"><mrow id="S1.SS3.SSS0.Px1.p1.8.m8.1.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml">𝒪</mi><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.2" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml">d</mi><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml"><mi id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2.cmml">d</mi></mrow></mrow><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.8.m8.1b"><apply id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1"><times id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2"></times><ci id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3">𝒪</ci><apply id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1"><times id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.1"></times><ci id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.2">𝑑</ci><apply id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3"><log id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.1"></log><ci id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.8.m8.1c">\mathcal{O}(d\log d)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.8.m8.1d">caligraphic_O ( italic_d roman_log italic_d )</annotation></semantics></math> in terms of the dimension. Note that the <math alttext="\log d" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.9.m9.1"><semantics id="S1.SS3.SSS0.Px1.p1.9.m9.1a"><mrow id="S1.SS3.SSS0.Px1.p1.9.m9.1.1" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.9.m9.1.1.1" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.1.cmml">log</mi><mo id="S1.SS3.SSS0.Px1.p1.9.m9.1.1a" lspace="0.167em" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.cmml">⁡</mo><mi id="S1.SS3.SSS0.Px1.p1.9.m9.1.1.2" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.2.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.9.m9.1b"><apply id="S1.SS3.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1"><log id="S1.SS3.SSS0.Px1.p1.9.m9.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.1"></log><ci id="S1.SS3.SSS0.Px1.p1.9.m9.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.9.m9.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.9.m9.1c">\log d</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.9.m9.1d">roman_log italic_d</annotation></semantics></math> overhead (compared to aforementioned results with <math alttext="\mathcal{O}(d)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.10.m10.1"><semantics id="S1.SS3.SSS0.Px1.p1.10.m10.1a"><mrow id="S1.SS3.SSS0.Px1.p1.10.m10.1.2" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.2" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.2.cmml">𝒪</mi><mo id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.1" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.3.2" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.cmml"><mo id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.cmml">(</mo><mi id="S1.SS3.SSS0.Px1.p1.10.m10.1.1" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.1.cmml">d</mi><mo id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.10.m10.1b"><apply id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2"><times id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.1"></times><ci id="S1.SS3.SSS0.Px1.p1.10.m10.1.2.2.cmml" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.2.2">𝒪</ci><ci id="S1.SS3.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.10.m10.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.10.m10.1c">\mathcal{O}(d)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.10.m10.1d">caligraphic_O ( italic_d )</annotation></semantics></math> dependency for the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.11.m11.1"><semantics id="S1.SS3.SSS0.Px1.p1.11.m11.1a"><msub id="S1.SS3.SSS0.Px1.p1.11.m11.1.1" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1.2.cmml">ℓ</mi><mn id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.3" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.11.m11.1b"><apply id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1.2">ℓ</ci><cn id="S1.SS3.SSS0.Px1.p1.11.m11.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.11.m11.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.11.m11.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-case) would come from using a lower bound of <math 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id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.1" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.1.cmml">⁢</mo><msup id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.cmml"><mi id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.2" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.2.cmml">b</mi><mi id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.3" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.12.m12.4b"><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4"><geq id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.2"></geq><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1"><ci id="S1.SS3.SSS0.Px1.p1.12.m12.3.3.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.3.3">vol</ci><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1"><times id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.1"></times><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2">superscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.2">𝐵</ci><ci id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.3.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.2.3">𝑝</ci></apply><interval closure="open" id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.1.1.1.1.3.2"><cn id="S1.SS3.SSS0.Px1.p1.12.m12.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.12.m12.1.1">0</cn><ci id="S1.SS3.SSS0.Px1.p1.12.m12.2.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.2.2">𝑏</ci></interval></apply></apply><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3"><times 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id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.2.3.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.2.3.2">𝑑</ci></apply></apply><apply id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.1.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3">superscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.2.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.2">𝑏</ci><ci id="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.3.cmml" xref="S1.SS3.SSS0.Px1.p1.12.m12.4.4.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.12.m12.4c">\operatorname{vol}(B^{p}(0,b))\geq\frac{2^{d}}{d!}b^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.12.m12.4d">roman_vol ( italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( 0 , italic_b ) ) ≥ divide start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG italic_d ! end_ARG italic_b start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> for the volume of a general <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.13.m13.1"><semantics id="S1.SS3.SSS0.Px1.p1.13.m13.1a"><msub id="S1.SS3.SSS0.Px1.p1.13.m13.1.1" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.3" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.13.m13.1b"><apply id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px1.p1.13.m13.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.13.m13.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.13.m13.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.13.m13.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-ball in our analysis (see proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem2" title="Theorem 4.2. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.2</span></a> for more details).</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p2.1">As we have discussed earlier, the Inscribed Ellipsoid algorithm is not just query-efficient, but can actually be implemented in polynomial time. However, this is not straightforward. Even in the Euclidean case, computing the centroid of a polyhedron (given by its bounding hyperplanes) is <math alttext="\mathsf{\#P}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p2.1.m1.1"><semantics id="S1.SS3.SSS0.Px1.p2.1.m1.1a"><mrow id="S1.SS3.SSS0.Px1.p2.1.m1.1.1" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.2.cmml">#</mi><mo id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.1" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.3" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml">𝖯</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p2.1.m1.1b"><apply id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1"><times id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.1"></times><ci id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.2">#</ci><ci id="S1.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p2.1.m1.1.1.3">𝖯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p2.1.m1.1c">\mathsf{\#P}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p2.1.m1.1d"># sansserif_P</annotation></semantics></math>-hard <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib30" title="">30</a>]</cite>. One thus has to rely on approximate centerpoints that can be computed efficiently by taking the centroid of an ellipsoid that approximates the polyhedron (hence the name Inscribed Ellipsoid algorithm). In other words, using ellipsoids to approximate the remaining search space is the main tool to get from query-efficient algorithms to actual polynomial-time algorithms in the Euclidean case.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p3"> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p3.1">For more information regarding the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.1.m1.1"><semantics id="S1.SS3.SSS0.Px1.p3.1.m1.1a"><msub id="S1.SS3.SSS0.Px1.p3.1.m1.1.1" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.2.cmml">ℓ</mi><mn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.3" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.1.m1.1b"><apply id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.2">ℓ</ci><cn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.1.m1.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-case, we refer to the survey by Sikorksi <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib36" title="">36</a>]</cite>.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Comparison to Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> on <math alttext="p=\infty" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.1.m1.1"><semantics id="S1.SS3.SSS0.Px2.1.m1.1b"><mrow id="S1.SS3.SSS0.Px2.1.m1.1.1" xref="S1.SS3.SSS0.Px2.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.1.m1.1.1.2" xref="S1.SS3.SSS0.Px2.1.m1.1.1.2.cmml">p</mi><mo id="S1.SS3.SSS0.Px2.1.m1.1.1.1" xref="S1.SS3.SSS0.Px2.1.m1.1.1.1.cmml">=</mo><mi id="S1.SS3.SSS0.Px2.1.m1.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.1.m1.1c"><apply id="S1.SS3.SSS0.Px2.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px2.1.m1.1.1"><eq id="S1.SS3.SSS0.Px2.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.1.m1.1.1.1"></eq><ci id="S1.SS3.SSS0.Px2.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.1.m1.1.1.2">𝑝</ci><infinity id="S1.SS3.SSS0.Px2.1.m1.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.1.m1.1d">p=\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.1.m1.1e">italic_p = ∞</annotation></semantics></math>.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p1.12">Our approach is similar (but much more general) to the one of Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> for the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S1.SS3.SSS0.Px2.p1.1.m1.1a"><msub id="S1.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.1.m1.1b"><apply id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.1.m1.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case. They prove that there always exists a query point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.2.m2.1"><semantics id="S1.SS3.SSS0.Px2.p1.2.m2.1a"><mi id="S1.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S1.SS3.SSS0.Px2.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.2.m2.1b"><ci id="S1.SS3.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.2.m2.1d">italic_x</annotation></semantics></math>, such that any response <math alttext="f(x)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.3.m3.1"><semantics id="S1.SS3.SSS0.Px2.p1.3.m3.1a"><mrow id="S1.SS3.SSS0.Px2.p1.3.m3.1.2" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.cmml"><mi id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.2" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.2.cmml">f</mi><mo id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.1" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.3.2" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.cmml"><mo id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.cmml">(</mo><mi id="S1.SS3.SSS0.Px2.p1.3.m3.1.1" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.1.cmml">x</mi><mo id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.3.m3.1b"><apply id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.cmml" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2"><times id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.1.cmml" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.1"></times><ci id="S1.SS3.SSS0.Px2.p1.3.m3.1.2.2.cmml" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.2.2">𝑓</ci><ci id="S1.SS3.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.3.m3.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.3.m3.1d">italic_f ( italic_x )</annotation></semantics></math> will allow them to discard at least a <math alttext="\frac{1}{2d}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.4.m4.1"><semantics id="S1.SS3.SSS0.Px2.p1.4.m4.1a"><mfrac id="S1.SS3.SSS0.Px2.p1.4.m4.1.1" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.cmml"><mn id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.2" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.2.cmml">1</mn><mrow id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.cmml"><mn id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.2" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.2.cmml">2</mn><mo id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.1" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.3" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.3.cmml">d</mi></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.4.m4.1b"><apply id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1"><divide id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1"></divide><cn id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.2">1</cn><apply id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3"><times id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.1"></times><cn id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.2">2</cn><ci id="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.3.cmml" xref="S1.SS3.SSS0.Px2.p1.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.4.m4.1c">\frac{1}{2d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.4.m4.1d">divide start_ARG 1 end_ARG start_ARG 2 italic_d end_ARG</annotation></semantics></math>-fraction of the remaining search space. Concretely, the area that is discarded is one out of <math alttext="2d" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.5.m5.1"><semantics id="S1.SS3.SSS0.Px2.p1.5.m5.1a"><mrow id="S1.SS3.SSS0.Px2.p1.5.m5.1.1" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.cmml"><mn id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.2" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.2.cmml">2</mn><mo id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.1" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.3" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.5.m5.1b"><apply id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1"><times id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.1"></times><cn id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.2">2</cn><ci id="S1.SS3.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p1.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.5.m5.1c">2d</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.5.m5.1d">2 italic_d</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px2.p1.12.1">pyramids</em> (<math alttext="d" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.6.m6.1"><semantics id="S1.SS3.SSS0.Px2.p1.6.m6.1a"><mi id="S1.SS3.SSS0.Px2.p1.6.m6.1.1" xref="S1.SS3.SSS0.Px2.p1.6.m6.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.6.m6.1b"><ci id="S1.SS3.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.6.m6.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.6.m6.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.6.m6.1d">italic_d</annotation></semantics></math> pairs of antipodal pyramids) that touch at <math alttext="x" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.7.m7.1"><semantics id="S1.SS3.SSS0.Px2.p1.7.m7.1a"><mi id="S1.SS3.SSS0.Px2.p1.7.m7.1.1" xref="S1.SS3.SSS0.Px2.p1.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.7.m7.1b"><ci id="S1.SS3.SSS0.Px2.p1.7.m7.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.7.m7.1d">italic_x</annotation></semantics></math>. To prove existence of a good query <math alttext="x" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.8.m8.1"><semantics id="S1.SS3.SSS0.Px2.p1.8.m8.1a"><mi id="S1.SS3.SSS0.Px2.p1.8.m8.1.1" xref="S1.SS3.SSS0.Px2.p1.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.8.m8.1b"><ci id="S1.SS3.SSS0.Px2.p1.8.m8.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.8.m8.1d">italic_x</annotation></semantics></math>, they use Brouwer’s fixpoint theorem on a function that balances the mass among the two pyramids in each of the <math alttext="d" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.9.m9.1"><semantics id="S1.SS3.SSS0.Px2.p1.9.m9.1a"><mi id="S1.SS3.SSS0.Px2.p1.9.m9.1.1" xref="S1.SS3.SSS0.Px2.p1.9.m9.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.9.m9.1b"><ci id="S1.SS3.SSS0.Px2.p1.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.9.m9.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.9.m9.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.9.m9.1d">italic_d</annotation></semantics></math> pairs of antipodal pyramids simultaneously. In other words, their good query point <math alttext="x" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.10.m10.1"><semantics id="S1.SS3.SSS0.Px2.p1.10.m10.1a"><mi id="S1.SS3.SSS0.Px2.p1.10.m10.1.1" xref="S1.SS3.SSS0.Px2.p1.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.10.m10.1b"><ci id="S1.SS3.SSS0.Px2.p1.10.m10.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.10.m10.1d">italic_x</annotation></semantics></math> is essentially a <math alttext="\frac{1}{2d}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.11.m11.1"><semantics id="S1.SS3.SSS0.Px2.p1.11.m11.1a"><mfrac id="S1.SS3.SSS0.Px2.p1.11.m11.1.1" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.cmml"><mn id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.2" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.2.cmml">1</mn><mrow id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.cmml"><mn id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.2" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.2.cmml">2</mn><mo id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.1" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.3" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.3.cmml">d</mi></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.11.m11.1b"><apply id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1"><divide id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1"></divide><cn id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.2">1</cn><apply id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3"><times id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.1"></times><cn id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.2">2</cn><ci id="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.3.cmml" xref="S1.SS3.SSS0.Px2.p1.11.m11.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.11.m11.1c">\frac{1}{2d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.11.m11.1d">divide start_ARG 1 end_ARG start_ARG 2 italic_d end_ARG</annotation></semantics></math>-centerpoint, although they do not use this terminology. Their decomposition into pyramids is tailored to the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p1.12.m12.1"><semantics id="S1.SS3.SSS0.Px2.p1.12.m12.1a"><msub id="S1.SS3.SSS0.Px2.p1.12.m12.1.1" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p1.12.m12.1b"><apply id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px2.p1.12.m12.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p1.12.m12.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p1.12.m12.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p1.12.m12.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case and does not generalize.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p2.9">Our algorithm for arbitrary <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.1.m1.3"><semantics id="S1.SS3.SSS0.Px2.p2.1.m1.3a"><mrow id="S1.SS3.SSS0.Px2.p2.1.m1.3.4" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.cmml"><mi id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.2" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.2.cmml">p</mi><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.1" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.1.cmml">∈</mo><mrow id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.cmml"><mrow id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.2" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.1.cmml"><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.2.1" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S1.SS3.SSS0.Px2.p2.1.m1.1.1" xref="S1.SS3.SSS0.Px2.p2.1.m1.1.1.cmml">1</mn><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.2.2" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S1.SS3.SSS0.Px2.p2.1.m1.2.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p2.1.m1.2.2.cmml">∞</mi><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.2.3" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.1" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.2" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.1.cmml"><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S1.SS3.SSS0.Px2.p2.1.m1.3.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.3.cmml">∞</mi><mo id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.1.m1.3b"><apply id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4"><in id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.1.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.1"></in><ci id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.2.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.2">𝑝</ci><apply id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3"><union id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.1.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.2.2"><cn id="S1.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.1.m1.1.1">1</cn><infinity id="S1.SS3.SSS0.Px2.p2.1.m1.2.2.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.2.2"></infinity></interval><set id="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.4.3.3.2"><infinity id="S1.SS3.SSS0.Px2.p2.1.m1.3.3.cmml" xref="S1.SS3.SSS0.Px2.p2.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> uses <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.2.m2.1"><semantics id="S1.SS3.SSS0.Px2.p2.2.m2.1a"><mrow id="S1.SS3.SSS0.Px2.p2.2.m2.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml">𝒪</mi><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml"><msup id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.cmml"><mi id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.2" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.2.cmml">d</mi><mn id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.3" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.2" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow 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xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mrow></mrow><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.2.m2.1b"><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1"><times id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2"></times><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3">𝒪</ci><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1"><times id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.2"></times><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.3"><csymbol 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xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.2.2"></divide><cn id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.2.2.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.2.2.2">1</cn><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3"><log id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.1"></log><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2"><divide id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2"></divide><cn id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3"><minus id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.2.m2.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.2.m2.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> queries, essentially matching the bound in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>. Note that their upper bound does not include <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.3.m3.1"><semantics id="S1.SS3.SSS0.Px2.p2.3.m3.1a"><mi id="S1.SS3.SSS0.Px2.p2.3.m3.1.1" xref="S1.SS3.SSS0.Px2.p2.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.3.m3.1b"><ci id="S1.SS3.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.3.m3.1d">italic_λ</annotation></semantics></math> because they show how to get rid of the dependency on <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.4.m4.1"><semantics id="S1.SS3.SSS0.Px2.p2.4.m4.1a"><mi id="S1.SS3.SSS0.Px2.p2.4.m4.1.1" xref="S1.SS3.SSS0.Px2.p2.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.4.m4.1b"><ci id="S1.SS3.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.4.m4.1d">italic_λ</annotation></semantics></math> by applying a reduction that turns an instance with parameters <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.5.m5.1"><semantics id="S1.SS3.SSS0.Px2.p2.5.m5.1a"><mi id="S1.SS3.SSS0.Px2.p2.5.m5.1.1" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.5.m5.1b"><ci id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.5.m5.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.5.m5.1d">italic_ε</annotation></semantics></math> and <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.6.m6.1"><semantics id="S1.SS3.SSS0.Px2.p2.6.m6.1a"><mi id="S1.SS3.SSS0.Px2.p2.6.m6.1.1" xref="S1.SS3.SSS0.Px2.p2.6.m6.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.6.m6.1b"><ci id="S1.SS3.SSS0.Px2.p2.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.6.m6.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.6.m6.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.6.m6.1d">italic_λ</annotation></semantics></math> into an instance with <math alttext="\varepsilon^{\prime}=\varepsilon/2" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.7.m7.1"><semantics id="S1.SS3.SSS0.Px2.p2.7.m7.1a"><mrow id="S1.SS3.SSS0.Px2.p2.7.m7.1.1" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.cmml"><msup id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.cmml"><mi id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.2" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.2.cmml">ε</mi><mo id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.3" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.1" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.1.cmml">=</mo><mrow id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.cmml"><mi id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.2" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.2.cmml">ε</mi><mo id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.1" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.1.cmml">/</mo><mn id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.3" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.7.m7.1b"><apply id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1"><eq id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.1"></eq><apply id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.1.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.2.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.2">𝜀</ci><ci id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.3.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.2.3">′</ci></apply><apply id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3"><divide id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.1"></divide><ci id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.2.cmml" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.2">𝜀</ci><cn id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.7.m7.1c">\varepsilon^{\prime}=\varepsilon/2</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.7.m7.1d">italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_ε / 2</annotation></semantics></math> and <math alttext="\lambda^{\prime}=1-\varepsilon^{\prime}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.8.m8.1"><semantics id="S1.SS3.SSS0.Px2.p2.8.m8.1a"><mrow id="S1.SS3.SSS0.Px2.p2.8.m8.1.1" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.cmml"><msup id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml"><mi id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.2" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.2.cmml">λ</mi><mo id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.3" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.1" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.1.cmml">=</mo><mrow id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml"><mn id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.2" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.2.cmml">1</mn><mo id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.1" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.1.cmml">−</mo><msup id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.cmml"><mi id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.2" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.2.cmml">ε</mi><mo id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.3" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.8.m8.1b"><apply id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1"><eq id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.1"></eq><apply id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.1.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2">superscript</csymbol><ci id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.2.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.2">𝜆</ci><ci id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.3.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.2.3">′</ci></apply><apply id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3"><minus id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.1"></minus><cn id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.2.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.2">1</cn><apply id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.1.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3">superscript</csymbol><ci id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.2.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.2">𝜀</ci><ci id="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.3.cmml" xref="S1.SS3.SSS0.Px2.p2.8.m8.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.8.m8.1c">\lambda^{\prime}=1-\varepsilon^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.8.m8.1d">italic_λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1 - italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. This is again specific to the case <math alttext="p=\infty" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.9.m9.1"><semantics id="S1.SS3.SSS0.Px2.p2.9.m9.1a"><mrow id="S1.SS3.SSS0.Px2.p2.9.m9.1.1" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2.cmml">p</mi><mo id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml">=</mo><mi id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.9.m9.1b"><apply id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1"><eq id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1"></eq><ci id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2">𝑝</ci><infinity id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.9.m9.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.9.m9.1d">italic_p = ∞</annotation></semantics></math> and unlikely to work in general.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p3"> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p3.2">In order to prove the result for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p3.1.m1.3"><semantics id="S1.SS3.SSS0.Px2.p3.1.m1.3a"><mrow id="S1.SS3.SSS0.Px2.p3.1.m1.3.4" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.cmml"><mi id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.2" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.2.cmml">p</mi><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.1" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.1.cmml">∈</mo><mrow id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.cmml"><mrow id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.2" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.1.cmml"><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.2.1" stretchy="false" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S1.SS3.SSS0.Px2.p3.1.m1.1.1" xref="S1.SS3.SSS0.Px2.p3.1.m1.1.1.cmml">1</mn><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.2.2" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S1.SS3.SSS0.Px2.p3.1.m1.2.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p3.1.m1.2.2.cmml">∞</mi><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.2.3" stretchy="false" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.1" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.2" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.1.cmml"><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S1.SS3.SSS0.Px2.p3.1.m1.3.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.3.cmml">∞</mi><mo id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p3.1.m1.3b"><apply id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4"><in id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.1.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.1"></in><ci id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.2.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.2">𝑝</ci><apply id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3"><union id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.1.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.1.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.2.2"><cn id="S1.SS3.SSS0.Px2.p3.1.m1.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p3.1.m1.1.1">1</cn><infinity id="S1.SS3.SSS0.Px2.p3.1.m1.2.2.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.2.2"></infinity></interval><set id="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.1.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.4.3.3.2"><infinity id="S1.SS3.SSS0.Px2.p3.1.m1.3.3.cmml" xref="S1.SS3.SSS0.Px2.p3.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p3.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p3.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> simultaneously, we avoid a specific decomposition such as the pyramid decomposition in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p3.2.m2.1"><semantics id="S1.SS3.SSS0.Px2.p3.2.m2.1a"><msub id="S1.SS3.SSS0.Px2.p3.2.m2.1.1" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p3.2.m2.1b"><apply id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px2.p3.2.m2.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p3.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p3.2.m2.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p3.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case. We think that the generality of our approach also makes our arguments less technical than the ones used in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p4"> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p4.9">Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> show that their algorithm also places the following total version of the problem in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.1.m1.1"><semantics id="S1.SS3.SSS0.Px2.p4.1.m1.1a"><msup id="S1.SS3.SSS0.Px2.p4.1.m1.1.1" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.2" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.2.cmml">𝖥𝖯</mi><mtext id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.1.m1.1b"><apply id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1">superscript</csymbol><ci id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.2">𝖥𝖯</ci><ci id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3a.cmml" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3"><mtext id="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="S1.SS3.SSS0.Px2.p4.1.m1.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.1.m1.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.1.m1.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>: given query access to the function <math alttext="f" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.2.m2.1"><semantics id="S1.SS3.SSS0.Px2.p4.2.m2.1a"><mi id="S1.SS3.SSS0.Px2.p4.2.m2.1.1" xref="S1.SS3.SSS0.Px2.p4.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.2.m2.1b"><ci id="S1.SS3.SSS0.Px2.p4.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.2.m2.1d">italic_f</annotation></semantics></math> on a grid, either find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.3.m3.1"><semantics id="S1.SS3.SSS0.Px2.p4.3.m3.1a"><mi id="S1.SS3.SSS0.Px2.p4.3.m3.1.1" xref="S1.SS3.SSS0.Px2.p4.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.3.m3.1b"><ci id="S1.SS3.SSS0.Px2.p4.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.3.m3.1d">italic_ε</annotation></semantics></math>-approximate fixpoint, or two grid points violating the contraction property. Totality of this problem crucially relies on the fact that any <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.4.m4.1"><semantics id="S1.SS3.SSS0.Px2.p4.4.m4.1a"><msub id="S1.SS3.SSS0.Px2.p4.4.m4.1.1" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.4.m4.1b"><apply id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px2.p4.4.m4.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p4.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.4.m4.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-contracting function defined only on the grid can be extended to an <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.5.m5.1"><semantics id="S1.SS3.SSS0.Px2.p4.5.m5.1a"><msub id="S1.SS3.SSS0.Px2.p4.5.m5.1.1" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.5.m5.1b"><apply id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px2.p4.5.m5.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p4.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.5.m5.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-contracting map on the whole cube. Unfortunately, their extension construction, which is an implicit application of the extension theorem of McShane <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib25" title="">25</a>]</cite>, does not generalize to other values of <math alttext="p" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.6.m6.1"><semantics id="S1.SS3.SSS0.Px2.p4.6.m6.1a"><mi id="S1.SS3.SSS0.Px2.p4.6.m6.1.1" xref="S1.SS3.SSS0.Px2.p4.6.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.6.m6.1b"><ci id="S1.SS3.SSS0.Px2.p4.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.6.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.6.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.6.m6.1d">italic_p</annotation></semantics></math>. In fact, we do not expect a similar extension theorem to hold for general <math alttext="p" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.7.m7.1"><semantics id="S1.SS3.SSS0.Px2.p4.7.m7.1a"><mi id="S1.SS3.SSS0.Px2.p4.7.m7.1.1" xref="S1.SS3.SSS0.Px2.p4.7.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.7.m7.1b"><ci id="S1.SS3.SSS0.Px2.p4.7.m7.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.7.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.7.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.7.m7.1d">italic_p</annotation></semantics></math>. In order to define a total version of the problem in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.8.m8.1"><semantics id="S1.SS3.SSS0.Px2.p4.8.m8.1a"><msub id="S1.SS3.SSS0.Px2.p4.8.m8.1.1" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1.2.cmml">ℓ</mi><mn id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.3" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.8.m8.1b"><apply id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1.2">ℓ</ci><cn id="S1.SS3.SSS0.Px2.p4.8.m8.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p4.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.8.m8.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.8.m8.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, we therefore need to include a different certificate for violation of the contraction property. This certificate uses not just two, but polynomially many grid points, and exists whenever our algorithm would fail to find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p4.9.m9.1"><semantics id="S1.SS3.SSS0.Px2.p4.9.m9.1a"><mi id="S1.SS3.SSS0.Px2.p4.9.m9.1.1" xref="S1.SS3.SSS0.Px2.p4.9.m9.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p4.9.m9.1b"><ci id="S1.SS3.SSS0.Px2.p4.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px2.p4.9.m9.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p4.9.m9.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p4.9.m9.1d">italic_ε</annotation></semantics></math>-approximate fixpoint (on a non-contracting map).</p> </div> </section> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph"> <math alttext="\mathsf{CLS}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px3.1.m1.1"><semantics id="S1.SS3.SSS0.Px3.1.m1.1b"><mi id="S1.SS3.SSS0.Px3.1.m1.1.1" xref="S1.SS3.SSS0.Px3.1.m1.1.1.cmml">𝖢𝖫𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px3.1.m1.1c"><ci id="S1.SS3.SSS0.Px3.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px3.1.m1.1.1">𝖢𝖫𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px3.1.m1.1d">\mathsf{CLS}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px3.1.m1.1e">sansserif_CLS</annotation></semantics></math>-Completeness of General Banach.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px3.p1.6">Daskalakis, Tzamos, and Zampetakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib9" title="">9</a>]</cite> showed that the problem of computing the fixpoint of a contraction map on the metric space <math alttext="([0,1]^{3},d_{[0,1]^{3}})" class="ltx_Math" display="inline" 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This stands in stark contrast to our polynomial-query result for all <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px3.p1.4.m4.1"><semantics id="S1.SS3.SSS0.Px3.p1.4.m4.1a"><msub id="S1.SS3.SSS0.Px3.p1.4.m4.1.1" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.3" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px3.p1.4.m4.1b"><apply id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px3.p1.4.m4.1.1.3.cmml" xref="S1.SS3.SSS0.Px3.p1.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px3.p1.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px3.p1.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics, and highlights that the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px3.p1.5.m5.1"><semantics id="S1.SS3.SSS0.Px3.p1.5.m5.1a"><msub id="S1.SS3.SSS0.Px3.p1.5.m5.1.1" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1.cmml"><mi id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.3" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px3.p1.5.m5.1b"><apply id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.2.cmml" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px3.p1.5.m5.1.1.3.cmml" xref="S1.SS3.SSS0.Px3.p1.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px3.p1.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px3.p1.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics are quite special metrics. For example, as metrics induced by a norm, the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px3.p1.6.m6.1"><semantics id="S1.SS3.SSS0.Px3.p1.6.m6.1a"><msub id="S1.SS3.SSS0.Px3.p1.6.m6.1.1" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1.cmml"><mi id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.3" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px3.p1.6.m6.1b"><apply id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.1.cmml" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.2.cmml" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px3.p1.6.m6.1.1.3.cmml" xref="S1.SS3.SSS0.Px3.p1.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px3.p1.6.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px3.p1.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metrics are translation-invariant. In future work, it would be interesting to investigate whether super-polynomial query lower bounds can be proven for some fixed metric, or whether our results can be generalized to more general classes of metrics, for example all metrics induced by some norm.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Finding Centerpoints.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px4.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px4.p1.5">An important question left unanswered by our work is whether our algorithms can be implemented not only using polynomially many queries, but also in overall polynomial time. Such a result would have important implications. For example, it would imply the existence of a polynomial-time algorithm for SSGs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib7" title="">7</a>]</cite> in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p1.1.m1.1"><semantics id="S1.SS3.SSS0.Px4.p1.1.m1.1a"><msub id="S1.SS3.SSS0.Px4.p1.1.m1.1.1" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.3" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p1.1.m1.1b"><apply id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1.2">ℓ</ci><infinity id="S1.SS3.SSS0.Px4.p1.1.m1.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p1.1.m1.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case and a polynomial-time algorithm for ARRIVAL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib17" title="">17</a>]</cite> in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p1.2.m2.1"><semantics id="S1.SS3.SSS0.Px4.p1.2.m2.1a"><msub id="S1.SS3.SSS0.Px4.p1.2.m2.1.1" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1.2.cmml">ℓ</mi><mn id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.3" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p1.2.m2.1b"><apply id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1.2">ℓ</ci><cn id="S1.SS3.SSS0.Px4.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p1.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case. To be able to implement our algorithm in polynomial time, we would have to be able to find a centerpoint of an intersection of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p1.3.m3.1"><semantics id="S1.SS3.SSS0.Px4.p1.3.m3.1a"><msub id="S1.SS3.SSS0.Px4.p1.3.m3.1.1" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.3" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p1.3.m3.1b"><apply id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p1.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces (the remaining search space) efficiently. However, note that we would not necessarily need a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p1.4.m4.1"><semantics id="S1.SS3.SSS0.Px4.p1.4.m4.1a"><mfrac id="S1.SS3.SSS0.Px4.p1.4.m4.1.1" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.cmml"><mn id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.2" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.2.cmml">1</mn><mrow id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.cmml"><mi id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.2" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.2.cmml">d</mi><mo id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.1" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.1.cmml">+</mo><mn id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.3" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p1.4.m4.1b"><apply id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1"><divide id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1"></divide><cn id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.2">1</cn><apply id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3"><plus id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.1"></plus><ci id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.2.cmml" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.2">𝑑</ci><cn id="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p1.4.m4.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p1.4.m4.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-centerpoint; a <math alttext="\frac{1}{\operatorname{poly}(d)}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p1.5.m5.2"><semantics id="S1.SS3.SSS0.Px4.p1.5.m5.2a"><mfrac id="S1.SS3.SSS0.Px4.p1.5.m5.2.2" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.cmml"><mn id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.4" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.4.cmml">1</mn><mrow id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml"><mi id="S1.SS3.SSS0.Px4.p1.5.m5.1.1.1.1" xref="S1.SS3.SSS0.Px4.p1.5.m5.1.1.1.1.cmml">poly</mi><mo id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4a" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml">⁡</mo><mrow id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4.1" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml"><mo id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4.1.1" stretchy="false" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml">(</mo><mi id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.2" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.2.cmml">d</mi><mo id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4.1.2" stretchy="false" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p1.5.m5.2b"><apply id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.cmml" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2"><divide id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.3.cmml" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2"></divide><cn id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.4.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.4">1</cn><apply id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.3.cmml" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.4"><ci id="S1.SS3.SSS0.Px4.p1.5.m5.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p1.5.m5.1.1.1.1">poly</ci><ci id="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.2.cmml" xref="S1.SS3.SSS0.Px4.p1.5.m5.2.2.2.2">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p1.5.m5.2c">\frac{1}{\operatorname{poly}(d)}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p1.5.m5.2d">divide start_ARG 1 end_ARG start_ARG roman_poly ( italic_d ) end_ARG</annotation></semantics></math>-centerpoint would suffice.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px4.p2"> <p class="ltx_p" id="S1.SS3.SSS0.Px4.p2.10">In Euclidean geometry, such approximate centerpoints can be computed efficiently with a variety of approaches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib26" title="">26</a>]</cite>. One approach for a randomized algorithm that might potentially generalize to our setting is the one based on computing the centerpoint of a representative subset of a point set: given a point set <math alttext="X\subseteq[0,1]^{d}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.1.m1.2"><semantics id="S1.SS3.SSS0.Px4.p2.1.m1.2a"><mrow id="S1.SS3.SSS0.Px4.p2.1.m1.2.3" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.cmml"><mi id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.2" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.2.cmml">X</mi><mo id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.1" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.1.cmml">⊆</mo><msup id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.cmml"><mrow id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.2" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.1.cmml"><mo id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.2.1" stretchy="false" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.1.cmml">[</mo><mn id="S1.SS3.SSS0.Px4.p2.1.m1.1.1" xref="S1.SS3.SSS0.Px4.p2.1.m1.1.1.cmml">0</mn><mo id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.2.2" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.1.cmml">,</mo><mn id="S1.SS3.SSS0.Px4.p2.1.m1.2.2" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.2.cmml">1</mn><mo id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.2.3" stretchy="false" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.1.cmml">]</mo></mrow><mi id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.3" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.1.m1.2b"><apply id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3"><subset id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.1.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.1"></subset><ci id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.2.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.2">𝑋</ci><apply id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.1.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3">superscript</csymbol><interval closure="closed" id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.1.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.2.2"><cn id="S1.SS3.SSS0.Px4.p2.1.m1.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p2.1.m1.1.1">0</cn><cn id="S1.SS3.SSS0.Px4.p2.1.m1.2.2.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.2">1</cn></interval><ci id="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.3.cmml" xref="S1.SS3.SSS0.Px4.p2.1.m1.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.1.m1.2c">X\subseteq[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.1.m1.2d">italic_X ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, the set of all halfspaces induces a set system (or <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px4.p2.10.2">range space</em>) on <math alttext="X" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.2.m2.1"><semantics id="S1.SS3.SSS0.Px4.p2.2.m2.1a"><mi id="S1.SS3.SSS0.Px4.p2.2.m2.1.1" xref="S1.SS3.SSS0.Px4.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.2.m2.1b"><ci id="S1.SS3.SSS0.Px4.p2.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.2.m2.1d">italic_X</annotation></semantics></math>. This range space has a Vapnik–Chervonenkis dimension <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib37" title="">37</a>]</cite> of at most <math alttext="d+1" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.3.m3.1"><semantics id="S1.SS3.SSS0.Px4.p2.3.m3.1a"><mrow id="S1.SS3.SSS0.Px4.p2.3.m3.1.1" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.2" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.2.cmml">d</mi><mo id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.1" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.1.cmml">+</mo><mn id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.3" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.3.m3.1b"><apply id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1"><plus id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.1"></plus><ci id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.2">𝑑</ci><cn id="S1.SS3.SSS0.Px4.p2.3.m3.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px4.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.3.m3.1c">d+1</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.3.m3.1d">italic_d + 1</annotation></semantics></math>, as implied by Radon’s lemma <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib32" title="">32</a>]</cite>. For range spaces with bounded VC-dimension, a random sample of a constant number of points is a so-called <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px4.p2.4.1"><math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.4.1.m1.1"><semantics id="S1.SS3.SSS0.Px4.p2.4.1.m1.1a"><mi id="S1.SS3.SSS0.Px4.p2.4.1.m1.1.1" xref="S1.SS3.SSS0.Px4.p2.4.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.4.1.m1.1b"><ci id="S1.SS3.SSS0.Px4.p2.4.1.m1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.4.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.4.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.4.1.m1.1d">italic_ε</annotation></semantics></math>-approximation</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib27" title="">27</a>]</cite> with high probability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib27" title="">27</a>]</cite>. Furthermore, the centerpoint of such an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.5.m4.1"><semantics id="S1.SS3.SSS0.Px4.p2.5.m4.1a"><mi id="S1.SS3.SSS0.Px4.p2.5.m4.1.1" xref="S1.SS3.SSS0.Px4.p2.5.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.5.m4.1b"><ci id="S1.SS3.SSS0.Px4.p2.5.m4.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.5.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.5.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.5.m4.1d">italic_ε</annotation></semantics></math>-approximation of the range space of halfspaces is an approximate centerpoint of <math alttext="X" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.6.m5.1"><semantics id="S1.SS3.SSS0.Px4.p2.6.m5.1a"><mi id="S1.SS3.SSS0.Px4.p2.6.m5.1.1" xref="S1.SS3.SSS0.Px4.p2.6.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.6.m5.1b"><ci id="S1.SS3.SSS0.Px4.p2.6.m5.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.6.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.6.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.6.m5.1d">italic_X</annotation></semantics></math> itself <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib24" title="">24</a>]</cite>. More details about this series of results can be found in Chapter 47 of the <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px4.p2.10.3">Handbook of Discrete and Computational Geometry</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib27" title="">27</a>]</cite>. To generalize this approach to our setting, one would need to bound the VC-dimension of the range space induced by <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.7.m6.1"><semantics id="S1.SS3.SSS0.Px4.p2.7.m6.1a"><msub id="S1.SS3.SSS0.Px4.p2.7.m6.1.1" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.3" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.7.m6.1b"><apply id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px4.p2.7.m6.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p2.7.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.7.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.7.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, and to find an efficient way of sampling a random point from the intersection of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.8.m7.1"><semantics id="S1.SS3.SSS0.Px4.p2.8.m7.1a"><msub id="S1.SS3.SSS0.Px4.p2.8.m7.1.1" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.3" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.8.m7.1b"><apply id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px4.p2.8.m7.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p2.8.m7.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.8.m7.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.8.m7.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces. The first step towards such a sampling procedure may be to study the analogue of Linear Programming feasibility for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.9.m8.1"><semantics id="S1.SS3.SSS0.Px4.p2.9.m8.1a"><msub id="S1.SS3.SSS0.Px4.p2.9.m8.1.1" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.3" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.9.m8.1b"><apply id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px4.p2.9.m8.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p2.9.m8.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.9.m8.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.9.m8.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces: how quickly can we decide whether an intersection of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px4.p2.10.m9.1"><semantics id="S1.SS3.SSS0.Px4.p2.10.m9.1a"><msub id="S1.SS3.SSS0.Px4.p2.10.m9.1.1" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1.cmml"><mi id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.2" mathvariant="normal" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1.2.cmml">ℓ</mi><mi id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.3" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px4.p2.10.m9.1b"><apply id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1"><csymbol cd="ambiguous" id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.1.cmml" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1">subscript</csymbol><ci id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.2.cmml" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1.2">ℓ</ci><ci id="S1.SS3.SSS0.Px4.p2.10.m9.1.1.3.cmml" xref="S1.SS3.SSS0.Px4.p2.10.m9.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px4.p2.10.m9.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px4.p2.10.m9.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces is empty or not?</p> </div> </section> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Proof of Centerpoint Theorem.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px5.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px5.p1.1">Our proof of the generalized centerpoint theorem makes use of Brouwer’s fixpoint theorem. The classical centerpoint theorem is usually proven using Helly’s theorem, which can in turn be proven using Brouwer’s fixpoint theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib23" title="">23</a>]</cite>. A proof of the classical centerpoint theorem using Brouwer’s fixpoint theorem <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px5.p1.1.1">directly</em> in the style of our proof seems to be somewhat folklore, as this strategy is mentioned in Chapter 27 of the <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px5.p1.1.2">Handbook of Discrete and Computational Geometry</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib38" title="">38</a>]</cite>, but we were unable to find any reference containing the full proof.</p> </div> </section> </section> <section class="ltx_subsection" id="S1.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.4 </span>Further Related Work</h3> <section class="ltx_paragraph" id="S1.SS4.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Generalized Centerpoint Theorems.</h4> <div class="ltx_para" id="S1.SS4.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS4.SSS0.Px1.p1.1">The classical centerpoint theorem has been generalized in many different directions. For example, it has been generalized from single centerpoints (generalizing the concept of a median) to multiple centerpoints (generalizing the concept of quantiles) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib33" title="">33</a>]</cite>. It has also been generalized from centerpoints to centerdisks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib2" title="">2</a>]</cite>. The centerpoint theorem has further been generalized to projective spaces <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib21" title="">21</a>]</cite>, which is the only generalization to non-Euclidean geometries that we are aware of.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS4.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Fixpoint Theorems.</h4> <div class="ltx_para" id="S1.SS4.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS4.SSS0.Px2.p1.6">Many other fixpoint theorems have been studied from the lens of computational complexity. Finding a fixpoint of a continuous function from a hypercube to itself, as guaranteed by Brouwer’s fixpoint theorem, is famously <math alttext="\mathsf{PPAD}" class="ltx_Math" display="inline" id="S1.SS4.SSS0.Px2.p1.1.m1.1"><semantics id="S1.SS4.SSS0.Px2.p1.1.m1.1a"><mi id="S1.SS4.SSS0.Px2.p1.1.m1.1.1" xref="S1.SS4.SSS0.Px2.p1.1.m1.1.1.cmml">𝖯𝖯𝖠𝖣</mi><annotation-xml encoding="MathML-Content" id="S1.SS4.SSS0.Px2.p1.1.m1.1b"><ci id="S1.SS4.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S1.SS4.SSS0.Px2.p1.1.m1.1.1">𝖯𝖯𝖠𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.SSS0.Px2.p1.1.m1.1c">\mathsf{PPAD}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.SSS0.Px2.p1.1.m1.1d">sansserif_PPAD</annotation></semantics></math>-complete <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib28" title="">28</a>]</cite>. Finding a fixpoint of a monotone function from a lattice to itself, as guaranteed by Tarski’s fixpoint theorem, is known to lie in <math alttext="\mathsf{EOPL}" class="ltx_Math" display="inline" id="S1.SS4.SSS0.Px2.p1.2.m2.1"><semantics id="S1.SS4.SSS0.Px2.p1.2.m2.1a"><mi id="S1.SS4.SSS0.Px2.p1.2.m2.1.1" xref="S1.SS4.SSS0.Px2.p1.2.m2.1.1.cmml">𝖤𝖮𝖯𝖫</mi><annotation-xml encoding="MathML-Content" id="S1.SS4.SSS0.Px2.p1.2.m2.1b"><ci id="S1.SS4.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S1.SS4.SSS0.Px2.p1.2.m2.1.1">𝖤𝖮𝖯𝖫</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.SSS0.Px2.p1.2.m2.1c">\mathsf{EOPL}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.SSS0.Px2.p1.2.m2.1d">sansserif_EOPL</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib14" title="">14</a>]</cite>, but no hardness result is known. Chang and Lyuu <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib4" title="">4</a>]</cite> gave lower bounds on the query complexity of finding fixpoints on <em class="ltx_emph ltx_font_italic" id="S1.SS4.SSS0.Px2.p1.6.1">finite</em> metric spaces, as guaranteed by Banach’s and Caristi’s fixpoint theorems. 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The computational version of Brøndsted’s fixpoint theorem on the same metric space is known to lie in <math alttext="\mathsf{PPAD}" class="ltx_Math" display="inline" id="S1.SS4.SSS0.Px2.p1.5.m5.1"><semantics id="S1.SS4.SSS0.Px2.p1.5.m5.1a"><mi id="S1.SS4.SSS0.Px2.p1.5.m5.1.1" xref="S1.SS4.SSS0.Px2.p1.5.m5.1.1.cmml">𝖯𝖯𝖠𝖣</mi><annotation-xml encoding="MathML-Content" id="S1.SS4.SSS0.Px2.p1.5.m5.1b"><ci id="S1.SS4.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S1.SS4.SSS0.Px2.p1.5.m5.1.1">𝖯𝖯𝖠𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.SSS0.Px2.p1.5.m5.1c">\mathsf{PPAD}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.SSS0.Px2.p1.5.m5.1d">sansserif_PPAD</annotation></semantics></math> and to be <math alttext="\mathsf{CLS}" class="ltx_Math" display="inline" id="S1.SS4.SSS0.Px2.p1.6.m6.1"><semantics id="S1.SS4.SSS0.Px2.p1.6.m6.1a"><mi id="S1.SS4.SSS0.Px2.p1.6.m6.1.1" xref="S1.SS4.SSS0.Px2.p1.6.m6.1.1.cmml">𝖢𝖫𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.SS4.SSS0.Px2.p1.6.m6.1b"><ci id="S1.SS4.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S1.SS4.SSS0.Px2.p1.6.m6.1.1">𝖢𝖫𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.SSS0.Px2.p1.6.m6.1c">\mathsf{CLS}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.SSS0.Px2.p1.6.m6.1d">sansserif_CLS</annotation></semantics></math>-hard <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib20" title="">20</a>]</cite>, with the exact complexity still open.</p> </div> </section> </section> <section class="ltx_subsection" id="S1.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.5 </span>Overview and Organization</h3> <div class="ltx_para" id="S1.SS5.p1"> <p class="ltx_p" id="S1.SS5.p1.1">We start our exposition in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S2" title="2 Preliminaries ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2</span></a> by formally defining the problem <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS5.p1.1.m1.1"><semantics id="S1.SS5.p1.1.m1.1a"><msub id="S1.SS5.p1.1.m1.1.1" xref="S1.SS5.p1.1.m1.1.1.cmml"><mi id="S1.SS5.p1.1.m1.1.1.2" mathvariant="normal" xref="S1.SS5.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS5.p1.1.m1.1.1.3" xref="S1.SS5.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p1.1.m1.1b"><apply id="S1.SS5.p1.1.m1.1.1.cmml" xref="S1.SS5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS5.p1.1.m1.1.1.1.cmml" xref="S1.SS5.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS5.p1.1.m1.1.1.2.cmml" xref="S1.SS5.p1.1.m1.1.1.2">ℓ</ci><ci id="S1.SS5.p1.1.m1.1.1.3.cmml" xref="S1.SS5.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S1.SS5.p1.1.1">-ContractionFixpoint</span> that we are interested in. We also recall important notions and results from the literature that are needed to understand the rest of the paper.</p> </div> <div class="ltx_para" id="S1.SS5.p2"> <p class="ltx_p" id="S1.SS5.p2.1">We introduce the notion of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS5.p2.1.m1.1"><semantics id="S1.SS5.p2.1.m1.1a"><msub id="S1.SS5.p2.1.m1.1.1" xref="S1.SS5.p2.1.m1.1.1.cmml"><mi id="S1.SS5.p2.1.m1.1.1.2" mathvariant="normal" xref="S1.SS5.p2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS5.p2.1.m1.1.1.3" xref="S1.SS5.p2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p2.1.m1.1b"><apply id="S1.SS5.p2.1.m1.1.1.cmml" xref="S1.SS5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS5.p2.1.m1.1.1.1.cmml" xref="S1.SS5.p2.1.m1.1.1">subscript</csymbol><ci id="S1.SS5.p2.1.m1.1.1.2.cmml" xref="S1.SS5.p2.1.m1.1.1.2">ℓ</ci><ci id="S1.SS5.p2.1.m1.1.1.3.cmml" xref="S1.SS5.p2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p2.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS1" title="3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.1</span></a> and discuss some of their properties in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS2" title="3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.2</span></a>. Proofs of those properties are deferred to <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1" title="Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">A</span></a>. <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS3" title="3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.3</span></a> contains the proof of our generalized centerpoint theorem for mass distributions. We also provide a discrete variant for point sets instead of mass distributions, the proof of which can be found in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A2" title="Appendix B Proof of Theorem 3.18 ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">B</span></a>. Finally, we use <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS4" title="3.4 Tightness of Centerpoint Theorems ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.4</span></a> to briefly argue that both versions of our generalized centerpoint theorem are tight.</p> </div> <div class="ltx_para" id="S1.SS5.p3"> <p class="ltx_p" id="S1.SS5.p3.6">We use <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4" title="4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a> to describe our algorithms. In <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS1" title="4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>, we prove our query upper bound for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S1.SS5.p3.1.m1.1"><semantics id="S1.SS5.p3.1.m1.1a"><msub id="S1.SS5.p3.1.m1.1.1" xref="S1.SS5.p3.1.m1.1.1.cmml"><mi id="S1.SS5.p3.1.m1.1.1.2" mathvariant="normal" xref="S1.SS5.p3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S1.SS5.p3.1.m1.1.1.3" xref="S1.SS5.p3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.1.m1.1b"><apply id="S1.SS5.p3.1.m1.1.1.cmml" xref="S1.SS5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS5.p3.1.m1.1.1.1.cmml" xref="S1.SS5.p3.1.m1.1.1">subscript</csymbol><ci id="S1.SS5.p3.1.m1.1.1.2.cmml" xref="S1.SS5.p3.1.m1.1.1.2">ℓ</ci><ci id="S1.SS5.p3.1.m1.1.1.3.cmml" xref="S1.SS5.p3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S1.SS5.p3.6.1">-ContractionFixpoint</span> for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S1.SS5.p3.2.m2.3"><semantics id="S1.SS5.p3.2.m2.3a"><mrow id="S1.SS5.p3.2.m2.3.4" xref="S1.SS5.p3.2.m2.3.4.cmml"><mi id="S1.SS5.p3.2.m2.3.4.2" xref="S1.SS5.p3.2.m2.3.4.2.cmml">p</mi><mo id="S1.SS5.p3.2.m2.3.4.1" xref="S1.SS5.p3.2.m2.3.4.1.cmml">∈</mo><mrow id="S1.SS5.p3.2.m2.3.4.3" xref="S1.SS5.p3.2.m2.3.4.3.cmml"><mrow id="S1.SS5.p3.2.m2.3.4.3.2.2" xref="S1.SS5.p3.2.m2.3.4.3.2.1.cmml"><mo id="S1.SS5.p3.2.m2.3.4.3.2.2.1" stretchy="false" xref="S1.SS5.p3.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S1.SS5.p3.2.m2.1.1" xref="S1.SS5.p3.2.m2.1.1.cmml">1</mn><mo id="S1.SS5.p3.2.m2.3.4.3.2.2.2" xref="S1.SS5.p3.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S1.SS5.p3.2.m2.2.2" mathvariant="normal" xref="S1.SS5.p3.2.m2.2.2.cmml">∞</mi><mo id="S1.SS5.p3.2.m2.3.4.3.2.2.3" stretchy="false" xref="S1.SS5.p3.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S1.SS5.p3.2.m2.3.4.3.1" xref="S1.SS5.p3.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S1.SS5.p3.2.m2.3.4.3.3.2" xref="S1.SS5.p3.2.m2.3.4.3.3.1.cmml"><mo id="S1.SS5.p3.2.m2.3.4.3.3.2.1" stretchy="false" xref="S1.SS5.p3.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S1.SS5.p3.2.m2.3.3" mathvariant="normal" xref="S1.SS5.p3.2.m2.3.3.cmml">∞</mi><mo id="S1.SS5.p3.2.m2.3.4.3.3.2.2" stretchy="false" xref="S1.SS5.p3.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.2.m2.3b"><apply id="S1.SS5.p3.2.m2.3.4.cmml" xref="S1.SS5.p3.2.m2.3.4"><in id="S1.SS5.p3.2.m2.3.4.1.cmml" xref="S1.SS5.p3.2.m2.3.4.1"></in><ci id="S1.SS5.p3.2.m2.3.4.2.cmml" xref="S1.SS5.p3.2.m2.3.4.2">𝑝</ci><apply id="S1.SS5.p3.2.m2.3.4.3.cmml" xref="S1.SS5.p3.2.m2.3.4.3"><union id="S1.SS5.p3.2.m2.3.4.3.1.cmml" xref="S1.SS5.p3.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S1.SS5.p3.2.m2.3.4.3.2.1.cmml" xref="S1.SS5.p3.2.m2.3.4.3.2.2"><cn id="S1.SS5.p3.2.m2.1.1.cmml" type="integer" xref="S1.SS5.p3.2.m2.1.1">1</cn><infinity id="S1.SS5.p3.2.m2.2.2.cmml" xref="S1.SS5.p3.2.m2.2.2"></infinity></interval><set id="S1.SS5.p3.2.m2.3.4.3.3.1.cmml" xref="S1.SS5.p3.2.m2.3.4.3.3.2"><infinity id="S1.SS5.p3.2.m2.3.3.cmml" xref="S1.SS5.p3.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. In <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS2" title="4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, we then show how we can ensure that all queries lie on a discrete grid in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS5.p3.3.m3.1"><semantics id="S1.SS5.p3.3.m3.1a"><msub id="S1.SS5.p3.3.m3.1.1" xref="S1.SS5.p3.3.m3.1.1.cmml"><mi id="S1.SS5.p3.3.m3.1.1.2" mathvariant="normal" xref="S1.SS5.p3.3.m3.1.1.2.cmml">ℓ</mi><mn id="S1.SS5.p3.3.m3.1.1.3" xref="S1.SS5.p3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.3.m3.1b"><apply id="S1.SS5.p3.3.m3.1.1.cmml" xref="S1.SS5.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS5.p3.3.m3.1.1.1.cmml" xref="S1.SS5.p3.3.m3.1.1">subscript</csymbol><ci id="S1.SS5.p3.3.m3.1.1.2.cmml" xref="S1.SS5.p3.3.m3.1.1.2">ℓ</ci><cn id="S1.SS5.p3.3.m3.1.1.3.cmml" type="integer" xref="S1.SS5.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.3.m3.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, solving what we call the problem <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS5.p3.4.m4.1"><semantics id="S1.SS5.p3.4.m4.1a"><msub id="S1.SS5.p3.4.m4.1.1" xref="S1.SS5.p3.4.m4.1.1.cmml"><mi id="S1.SS5.p3.4.m4.1.1.2" mathvariant="normal" xref="S1.SS5.p3.4.m4.1.1.2.cmml">ℓ</mi><mn id="S1.SS5.p3.4.m4.1.1.3" xref="S1.SS5.p3.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.4.m4.1b"><apply id="S1.SS5.p3.4.m4.1.1.cmml" xref="S1.SS5.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS5.p3.4.m4.1.1.1.cmml" xref="S1.SS5.p3.4.m4.1.1">subscript</csymbol><ci id="S1.SS5.p3.4.m4.1.1.2.cmml" xref="S1.SS5.p3.4.m4.1.1.2">ℓ</ci><cn id="S1.SS5.p3.4.m4.1.1.3.cmml" type="integer" xref="S1.SS5.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.4.m4.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S1.SS5.p3.6.2">-GridContractionFixpoint</span>. Finally, we discuss membership in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S1.SS5.p3.5.m5.1"><semantics id="S1.SS5.p3.5.m5.1a"><msup id="S1.SS5.p3.5.m5.1.1" xref="S1.SS5.p3.5.m5.1.1.cmml"><mi id="S1.SS5.p3.5.m5.1.1.2" xref="S1.SS5.p3.5.m5.1.1.2.cmml">𝖥𝖯</mi><mtext id="S1.SS5.p3.5.m5.1.1.3" xref="S1.SS5.p3.5.m5.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.5.m5.1b"><apply id="S1.SS5.p3.5.m5.1.1.cmml" xref="S1.SS5.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS5.p3.5.m5.1.1.1.cmml" xref="S1.SS5.p3.5.m5.1.1">superscript</csymbol><ci id="S1.SS5.p3.5.m5.1.1.2.cmml" xref="S1.SS5.p3.5.m5.1.1.2">𝖥𝖯</ci><ci id="S1.SS5.p3.5.m5.1.1.3a.cmml" xref="S1.SS5.p3.5.m5.1.1.3"><mtext id="S1.SS5.p3.5.m5.1.1.3.cmml" mathsize="70%" xref="S1.SS5.p3.5.m5.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.5.m5.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.5.m5.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math> of a total search version <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S1.SS5.p3.6.m6.1"><semantics id="S1.SS5.p3.6.m6.1a"><msub id="S1.SS5.p3.6.m6.1.1" xref="S1.SS5.p3.6.m6.1.1.cmml"><mi id="S1.SS5.p3.6.m6.1.1.2" mathvariant="normal" xref="S1.SS5.p3.6.m6.1.1.2.cmml">ℓ</mi><mn id="S1.SS5.p3.6.m6.1.1.3" xref="S1.SS5.p3.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS5.p3.6.m6.1b"><apply id="S1.SS5.p3.6.m6.1.1.cmml" xref="S1.SS5.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS5.p3.6.m6.1.1.1.cmml" xref="S1.SS5.p3.6.m6.1.1">subscript</csymbol><ci id="S1.SS5.p3.6.m6.1.1.2.cmml" xref="S1.SS5.p3.6.m6.1.1.2">ℓ</ci><cn id="S1.SS5.p3.6.m6.1.1.3.cmml" type="integer" xref="S1.SS5.p3.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS5.p3.6.m6.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS5.p3.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S1.SS5.p3.6.3">-GridContractionFixpoint</span> in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS3" title="4.3 Total Search Version ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Preliminaries</h2> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Definition 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2"> </span>(Contraction Map)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.9.9">Given a metric space <math alttext="(X,d_{X})" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.1.m1.2"><semantics id="S2.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem1.p1.1.1.m1.2.2.1" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml">X</mi><mo id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.3" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.2.cmml">,</mo><msub id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.cmml"><mi id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.2" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.2.cmml">d</mi><mi id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.3" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.3.cmml">X</mi></msub><mo id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.4" stretchy="false" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.1.m1.2b"><interval closure="open" id="S2.Thmtheorem1.p1.1.1.m1.2.2.2.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1"><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1">𝑋</ci><apply id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.2.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.2">𝑑</ci><ci id="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.3.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.2.2.1.1.3">𝑋</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.1.m1.2c">(X,d_{X})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.1.m1.2d">( italic_X , italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT )</annotation></semantics></math> and a contraction factor <math alttext="0\leq\lambda<1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem1.p1.2.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mn id="S2.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">0</mn><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">≤</mo><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1.4" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml">λ</mi><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.5" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.5.cmml"><</mo><mn id="S2.Thmtheorem1.p1.2.2.m2.1.1.6" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.2.m2.1b"><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><and id="S2.Thmtheorem1.p1.2.2.m2.1.1a.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></and><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1b.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><leq id="S2.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.3"></leq><cn id="S2.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.2">0</cn><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4">𝜆</ci></apply><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1c.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><lt id="S2.Thmtheorem1.p1.2.2.m2.1.1.5.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2503.16089v1#S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml" id="S2.Thmtheorem1.p1.2.2.m2.1.1d.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></share><cn id="S2.Thmtheorem1.p1.2.2.m2.1.1.6.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.2.m2.1c">0\leq\lambda<1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.2.m2.1d">0 ≤ italic_λ < 1</annotation></semantics></math>, a function <math alttext="f:X\rightarrow X" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem1.p1.3.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem1.p1.3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.2" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml">X</mi><mo id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.1" stretchy="false" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.3" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.3.m3.1b"><apply id="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1"><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.1">:</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.2">𝑓</ci><apply id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3"><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.1">→</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.2">𝑋</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.3.m3.1c">f:X\rightarrow X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.3.m3.1d">italic_f : italic_X → italic_X</annotation></semantics></math> is called a <em class="ltx_emph ltx_font_upright" id="S2.Thmtheorem1.p1.4.4.1"><math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.4.1.m1.1"><semantics id="S2.Thmtheorem1.p1.4.4.1.m1.1a"><mi id="S2.Thmtheorem1.p1.4.4.1.m1.1.1" xref="S2.Thmtheorem1.p1.4.4.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.4.1.m1.1b"><ci id="S2.Thmtheorem1.p1.4.4.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.4.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.4.1.m1.1d">italic_λ</annotation></semantics></math>-contraction map</em> (or <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.5.m4.1"><semantics id="S2.Thmtheorem1.p1.5.5.m4.1a"><mi id="S2.Thmtheorem1.p1.5.5.m4.1.1" xref="S2.Thmtheorem1.p1.5.5.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.5.m4.1b"><ci id="S2.Thmtheorem1.p1.5.5.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.5.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.5.m4.1d">italic_λ</annotation></semantics></math>-contracting) if <math alttext="d_{X}(f(x),f(y))\leq\lambda\cdot d_{X}(x,y)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.6.m5.6"><semantics id="S2.Thmtheorem1.p1.6.6.m5.6a"><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.cmml"><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.cmml"><msub id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.2.cmml">d</mi><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.3" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.3.cmml">X</mi></msub><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.3" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.3.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.3.cmml"><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.3.cmml">(</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.2" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.1" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.3.2" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.cmml">(</mo><mi id="S2.Thmtheorem1.p1.6.6.m5.1.1" xref="S2.Thmtheorem1.p1.6.6.m5.1.1.cmml">x</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.4" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.2.cmml">f</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.1" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.3.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.6.6.m5.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.2.2.cmml">y</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.5" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.3" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.3.cmml">≤</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.4" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.cmml"><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.2.cmml">λ</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.1.cmml">⋅</mo><msub id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.2.cmml">d</mi><mi id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.3" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.3.cmml">X</mi></msub></mrow><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.1" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.1.cmml"><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.1.cmml">(</mo><mi id="S2.Thmtheorem1.p1.6.6.m5.3.3" xref="S2.Thmtheorem1.p1.6.6.m5.3.3.cmml">x</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.2.2" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.1.cmml">,</mo><mi id="S2.Thmtheorem1.p1.6.6.m5.4.4" xref="S2.Thmtheorem1.p1.6.6.m5.4.4.cmml">y</mi><mo id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.6.m5.6b"><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6"><leq id="S2.Thmtheorem1.p1.6.6.m5.6.6.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.3"></leq><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2"><times id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.3"></times><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4">subscript</csymbol><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.2">𝑑</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.4.3">𝑋</ci></apply><interval closure="open" id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2"><apply id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1"><times id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.1"></times><ci id="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.5.5.1.1.1.1.2">𝑓</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.1.1">𝑥</ci></apply><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2"><times id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.1"></times><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.2.2.2.2.2">𝑓</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.2.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.2.2">𝑦</ci></apply></interval></apply><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4"><times id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.1"></times><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2"><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.1">⋅</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.2">𝜆</ci><apply id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.2">𝑑</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.2.3.3">𝑋</ci></apply></apply><interval closure="open" id="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.6.6.4.3.2"><ci id="S2.Thmtheorem1.p1.6.6.m5.3.3.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.3.3">𝑥</ci><ci id="S2.Thmtheorem1.p1.6.6.m5.4.4.cmml" xref="S2.Thmtheorem1.p1.6.6.m5.4.4">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.6.m5.6c">d_{X}(f(x),f(y))\leq\lambda\cdot d_{X}(x,y)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.6.m5.6d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_f ( italic_x ) , italic_f ( italic_y ) ) ≤ italic_λ ⋅ italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_x , italic_y )</annotation></semantics></math> holds for all <math alttext="x,y\in X" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.7.m6.2"><semantics id="S2.Thmtheorem1.p1.7.7.m6.2a"><mrow id="S2.Thmtheorem1.p1.7.7.m6.2.3" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.cmml"><mrow id="S2.Thmtheorem1.p1.7.7.m6.2.3.2.2" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.2.1.cmml"><mi id="S2.Thmtheorem1.p1.7.7.m6.1.1" xref="S2.Thmtheorem1.p1.7.7.m6.1.1.cmml">x</mi><mo id="S2.Thmtheorem1.p1.7.7.m6.2.3.2.2.1" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem1.p1.7.7.m6.2.2" xref="S2.Thmtheorem1.p1.7.7.m6.2.2.cmml">y</mi></mrow><mo id="S2.Thmtheorem1.p1.7.7.m6.2.3.1" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem1.p1.7.7.m6.2.3.3" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.7.7.m6.2b"><apply id="S2.Thmtheorem1.p1.7.7.m6.2.3.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.2.3"><in id="S2.Thmtheorem1.p1.7.7.m6.2.3.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.1"></in><list id="S2.Thmtheorem1.p1.7.7.m6.2.3.2.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.2.2"><ci id="S2.Thmtheorem1.p1.7.7.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.1.1">𝑥</ci><ci id="S2.Thmtheorem1.p1.7.7.m6.2.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.2.2">𝑦</ci></list><ci id="S2.Thmtheorem1.p1.7.7.m6.2.3.3.cmml" xref="S2.Thmtheorem1.p1.7.7.m6.2.3.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.7.m6.2c">x,y\in X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.7.m6.2d">italic_x , italic_y ∈ italic_X</annotation></semantics></math>. A function is called a <em class="ltx_emph ltx_font_upright" id="S2.Thmtheorem1.p1.9.9.2">contraction map</em> if it is <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.8.8.m7.1"><semantics id="S2.Thmtheorem1.p1.8.8.m7.1a"><mi id="S2.Thmtheorem1.p1.8.8.m7.1.1" xref="S2.Thmtheorem1.p1.8.8.m7.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.8.8.m7.1b"><ci id="S2.Thmtheorem1.p1.8.8.m7.1.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m7.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.8.8.m7.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.8.8.m7.1d">italic_λ</annotation></semantics></math>-contracting for some <math alttext="0\leq\lambda<1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.9.9.m8.1"><semantics id="S2.Thmtheorem1.p1.9.9.m8.1a"><mrow id="S2.Thmtheorem1.p1.9.9.m8.1.1" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.cmml"><mn id="S2.Thmtheorem1.p1.9.9.m8.1.1.2" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.2.cmml">0</mn><mo id="S2.Thmtheorem1.p1.9.9.m8.1.1.3" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.3.cmml">≤</mo><mi id="S2.Thmtheorem1.p1.9.9.m8.1.1.4" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.4.cmml">λ</mi><mo id="S2.Thmtheorem1.p1.9.9.m8.1.1.5" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.5.cmml"><</mo><mn id="S2.Thmtheorem1.p1.9.9.m8.1.1.6" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.9.9.m8.1b"><apply id="S2.Thmtheorem1.p1.9.9.m8.1.1.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1"><and id="S2.Thmtheorem1.p1.9.9.m8.1.1a.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1"></and><apply id="S2.Thmtheorem1.p1.9.9.m8.1.1b.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1"><leq id="S2.Thmtheorem1.p1.9.9.m8.1.1.3.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.3"></leq><cn id="S2.Thmtheorem1.p1.9.9.m8.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.2">0</cn><ci id="S2.Thmtheorem1.p1.9.9.m8.1.1.4.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.4">𝜆</ci></apply><apply id="S2.Thmtheorem1.p1.9.9.m8.1.1c.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1"><lt id="S2.Thmtheorem1.p1.9.9.m8.1.1.5.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.5"></lt><share href="https://arxiv.org/html/2503.16089v1#S2.Thmtheorem1.p1.9.9.m8.1.1.4.cmml" id="S2.Thmtheorem1.p1.9.9.m8.1.1d.cmml" xref="S2.Thmtheorem1.p1.9.9.m8.1.1"></share><cn id="S2.Thmtheorem1.p1.9.9.m8.1.1.6.cmml" type="integer" xref="S2.Thmtheorem1.p1.9.9.m8.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.9.9.m8.1c">0\leq\lambda<1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.9.9.m8.1d">0 ≤ italic_λ < 1</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Theorem 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2"> </span>(Banach Fixpoint Theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib1" title="">1</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem2.p1.4.4">Every contraction map <math alttext="f:X\rightarrow X" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.1.m1.1"><semantics id="S2.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem2.p1.1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">X</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.1.m1.1b"><apply id="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.1">:</ci><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3"><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.1">→</ci><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.2">𝑋</ci><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.1.m1.1c">f:X\rightarrow X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.1.m1.1d">italic_f : italic_X → italic_X</annotation></semantics></math> on a non-empty complete metric space <math alttext="(X,d_{X})" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.2.m2.2"><semantics id="S2.Thmtheorem2.p1.2.2.m2.2a"><mrow id="S2.Thmtheorem2.p1.2.2.m2.2.2.1" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.2.cmml"><mo id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.2.cmml">(</mo><mi id="S2.Thmtheorem2.p1.2.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml">X</mi><mo id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.3" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.2.cmml">,</mo><msub id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.cmml"><mi id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2.cmml">d</mi><mi id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3.cmml">X</mi></msub><mo id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.4" stretchy="false" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.2.m2.2b"><interval closure="open" id="S2.Thmtheorem2.p1.2.2.m2.2.2.2.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1"><ci id="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1">𝑋</ci><apply id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1">subscript</csymbol><ci id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.2">𝑑</ci><ci id="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.1.1.3">𝑋</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.2.m2.2c">(X,d_{X})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.2.m2.2d">( italic_X , italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT )</annotation></semantics></math> admits a unique fixpoint <math alttext="x^{\star}\in X" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem2.p1.3.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml"><msup id="S2.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.2" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.2.cmml">x</mi><mo id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.3" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.3.cmml">⋆</mo></msup><mo id="S2.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.3.m3.1b"><apply id="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1"><in id="S2.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.1"></in><apply id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.2">𝑥</ci><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.3">⋆</ci></apply><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.3.m3.1c">x^{\star}\in X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.3.m3.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∈ italic_X</annotation></semantics></math>, i.e., a unique point satisfying <math alttext="f(x^{\star})=x^{\star}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.4.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml">f</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2.cmml">x</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">=</mo><msup id="S2.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.2.cmml">x</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.4.4.m4.1b"><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1"><eq id="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2"></eq><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1"><times id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2"></times><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3">𝑓</ci><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2">𝑥</ci><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3">⋆</ci></apply></apply><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.2">𝑥</ci><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.4.m4.1c">f(x^{\star})=x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.4.m4.1d">italic_f ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) = italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.4">In this paper, we will consider metric spaces of the form <math alttext="([0,1]^{d},\ell_{p})" class="ltx_Math" display="inline" id="S2.p1.1.m1.4"><semantics id="S2.p1.1.m1.4a"><mrow id="S2.p1.1.m1.4.4.2" xref="S2.p1.1.m1.4.4.3.cmml"><mo id="S2.p1.1.m1.4.4.2.3" stretchy="false" xref="S2.p1.1.m1.4.4.3.cmml">(</mo><msup id="S2.p1.1.m1.3.3.1.1" xref="S2.p1.1.m1.3.3.1.1.cmml"><mrow id="S2.p1.1.m1.3.3.1.1.2.2" xref="S2.p1.1.m1.3.3.1.1.2.1.cmml"><mo id="S2.p1.1.m1.3.3.1.1.2.2.1" stretchy="false" xref="S2.p1.1.m1.3.3.1.1.2.1.cmml">[</mo><mn id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">0</mn><mo id="S2.p1.1.m1.3.3.1.1.2.2.2" xref="S2.p1.1.m1.3.3.1.1.2.1.cmml">,</mo><mn id="S2.p1.1.m1.2.2" xref="S2.p1.1.m1.2.2.cmml">1</mn><mo id="S2.p1.1.m1.3.3.1.1.2.2.3" stretchy="false" xref="S2.p1.1.m1.3.3.1.1.2.1.cmml">]</mo></mrow><mi id="S2.p1.1.m1.3.3.1.1.3" xref="S2.p1.1.m1.3.3.1.1.3.cmml">d</mi></msup><mo id="S2.p1.1.m1.4.4.2.4" xref="S2.p1.1.m1.4.4.3.cmml">,</mo><msub id="S2.p1.1.m1.4.4.2.2" xref="S2.p1.1.m1.4.4.2.2.cmml"><mi id="S2.p1.1.m1.4.4.2.2.2" mathvariant="normal" xref="S2.p1.1.m1.4.4.2.2.2.cmml">ℓ</mi><mi id="S2.p1.1.m1.4.4.2.2.3" xref="S2.p1.1.m1.4.4.2.2.3.cmml">p</mi></msub><mo id="S2.p1.1.m1.4.4.2.5" stretchy="false" xref="S2.p1.1.m1.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.4b"><interval closure="open" id="S2.p1.1.m1.4.4.3.cmml" xref="S2.p1.1.m1.4.4.2"><apply id="S2.p1.1.m1.3.3.1.1.cmml" xref="S2.p1.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S2.p1.1.m1.3.3.1.1.1.cmml" xref="S2.p1.1.m1.3.3.1.1">superscript</csymbol><interval closure="closed" id="S2.p1.1.m1.3.3.1.1.2.1.cmml" xref="S2.p1.1.m1.3.3.1.1.2.2"><cn id="S2.p1.1.m1.1.1.cmml" type="integer" xref="S2.p1.1.m1.1.1">0</cn><cn id="S2.p1.1.m1.2.2.cmml" type="integer" xref="S2.p1.1.m1.2.2">1</cn></interval><ci id="S2.p1.1.m1.3.3.1.1.3.cmml" xref="S2.p1.1.m1.3.3.1.1.3">𝑑</ci></apply><apply id="S2.p1.1.m1.4.4.2.2.cmml" xref="S2.p1.1.m1.4.4.2.2"><csymbol cd="ambiguous" id="S2.p1.1.m1.4.4.2.2.1.cmml" xref="S2.p1.1.m1.4.4.2.2">subscript</csymbol><ci id="S2.p1.1.m1.4.4.2.2.2.cmml" xref="S2.p1.1.m1.4.4.2.2.2">ℓ</ci><ci id="S2.p1.1.m1.4.4.2.2.3.cmml" xref="S2.p1.1.m1.4.4.2.2.3">𝑝</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.4c">([0,1]^{d},\ell_{p})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.4d">( [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math> for some <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.p1.2.m2.3"><semantics id="S2.p1.2.m2.3a"><mrow id="S2.p1.2.m2.3.4" xref="S2.p1.2.m2.3.4.cmml"><mi id="S2.p1.2.m2.3.4.2" xref="S2.p1.2.m2.3.4.2.cmml">p</mi><mo id="S2.p1.2.m2.3.4.1" xref="S2.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S2.p1.2.m2.3.4.3" xref="S2.p1.2.m2.3.4.3.cmml"><mrow id="S2.p1.2.m2.3.4.3.2.2" xref="S2.p1.2.m2.3.4.3.2.1.cmml"><mo id="S2.p1.2.m2.3.4.3.2.2.1" stretchy="false" xref="S2.p1.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">1</mn><mo id="S2.p1.2.m2.3.4.3.2.2.2" xref="S2.p1.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S2.p1.2.m2.2.2" mathvariant="normal" xref="S2.p1.2.m2.2.2.cmml">∞</mi><mo id="S2.p1.2.m2.3.4.3.2.2.3" stretchy="false" xref="S2.p1.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S2.p1.2.m2.3.4.3.1" xref="S2.p1.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S2.p1.2.m2.3.4.3.3.2" xref="S2.p1.2.m2.3.4.3.3.1.cmml"><mo id="S2.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="S2.p1.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S2.p1.2.m2.3.3" mathvariant="normal" xref="S2.p1.2.m2.3.3.cmml">∞</mi><mo id="S2.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="S2.p1.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.3b"><apply id="S2.p1.2.m2.3.4.cmml" xref="S2.p1.2.m2.3.4"><in id="S2.p1.2.m2.3.4.1.cmml" xref="S2.p1.2.m2.3.4.1"></in><ci id="S2.p1.2.m2.3.4.2.cmml" xref="S2.p1.2.m2.3.4.2">𝑝</ci><apply id="S2.p1.2.m2.3.4.3.cmml" xref="S2.p1.2.m2.3.4.3"><union id="S2.p1.2.m2.3.4.3.1.cmml" xref="S2.p1.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S2.p1.2.m2.3.4.3.2.1.cmml" xref="S2.p1.2.m2.3.4.3.2.2"><cn id="S2.p1.2.m2.1.1.cmml" type="integer" xref="S2.p1.2.m2.1.1">1</cn><infinity id="S2.p1.2.m2.2.2.cmml" xref="S2.p1.2.m2.2.2"></infinity></interval><set id="S2.p1.2.m2.3.4.3.3.1.cmml" xref="S2.p1.2.m2.3.4.3.3.2"><infinity id="S2.p1.2.m2.3.3.cmml" xref="S2.p1.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, where <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><msub id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml"><mi id="S2.p1.3.m3.1.1.2" mathvariant="normal" xref="S2.p1.3.m3.1.1.2.cmml">ℓ</mi><mi id="S2.p1.3.m3.1.1.3" xref="S2.p1.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><apply id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p1.3.m3.1.1.1.cmml" xref="S2.p1.3.m3.1.1">subscript</csymbol><ci id="S2.p1.3.m3.1.1.2.cmml" xref="S2.p1.3.m3.1.1.2">ℓ</ci><ci id="S2.p1.3.m3.1.1.3.cmml" xref="S2.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> denotes the metric induced by the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><msub id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" mathvariant="normal" xref="S2.p1.4.m4.1.1.2.cmml">ℓ</mi><mi id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1">subscript</csymbol><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">ℓ</ci><ci id="S2.p1.4.m4.1.1.3.cmml" xref="S2.p1.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.1.1">Definition 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.3.2"> </span>(<math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.1.m1.1"><semantics id="S2.Thmtheorem3.1.m1.1b"><msub id="S2.Thmtheorem3.1.m1.1.1" xref="S2.Thmtheorem3.1.m1.1.1.cmml"><mi id="S2.Thmtheorem3.1.m1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.Thmtheorem3.1.m1.1.1.3" xref="S2.Thmtheorem3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.1.m1.1c"><apply id="S2.Thmtheorem3.1.m1.1.1.cmml" xref="S2.Thmtheorem3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.1.m1.1.1.1.cmml" xref="S2.Thmtheorem3.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.1.m1.1.1.2.cmml" xref="S2.Thmtheorem3.1.m1.1.1.2">ℓ</ci><ci id="S2.Thmtheorem3.1.m1.1.1.3.cmml" xref="S2.Thmtheorem3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Norm)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.4.3">.</span> </h6> <div 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xref="S2.Thmtheorem3.p1.3.3.m3.2.2.1.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.3.m3.2c">||x||_{p}:=\left(\sum_{i=1}^{d}|x_{i}|^{p}\right)^{1/p}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.3.m3.2d">| | italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT := ( ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.4.m4.2"><semantics id="S2.Thmtheorem3.p1.4.4.m4.2a"><mrow id="S2.Thmtheorem3.p1.4.4.m4.2.3" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.cmml"><mi id="S2.Thmtheorem3.p1.4.4.m4.2.3.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.2.cmml">p</mi><mo id="S2.Thmtheorem3.p1.4.4.m4.2.3.1" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p1.4.4.m4.2.3.3.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml"><mo id="S2.Thmtheorem3.p1.4.4.m4.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">[</mo><mn id="S2.Thmtheorem3.p1.4.4.m4.1.1" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml">1</mn><mo id="S2.Thmtheorem3.p1.4.4.m4.2.3.3.2.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem3.p1.4.4.m4.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.cmml">∞</mi><mo id="S2.Thmtheorem3.p1.4.4.m4.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.4.m4.2b"><apply id="S2.Thmtheorem3.p1.4.4.m4.2.3.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.3"><in id="S2.Thmtheorem3.p1.4.4.m4.2.3.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.1"></in><ci id="S2.Thmtheorem3.p1.4.4.m4.2.3.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.2">𝑝</ci><interval closure="closed-open" id="S2.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.3.3.2"><cn id="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml" type="integer" xref="S2.Thmtheorem3.p1.4.4.m4.1.1">1</cn><infinity id="S2.Thmtheorem3.p1.4.4.m4.2.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.4.m4.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.4.m4.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math> and <math alttext="||x||_{p}:=\max_{i}|x_{i}|" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.5.m5.2"><semantics id="S2.Thmtheorem3.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem3.p1.5.5.m5.2.2" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.cmml"><msub id="S2.Thmtheorem3.p1.5.5.m5.2.2.3" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.cmml"><mrow id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.2" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.1.cmml"><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.1.1.cmml">‖</mo><mi id="S2.Thmtheorem3.p1.5.5.m5.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml">x</mi><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.1.1.cmml">‖</mo></mrow><mi id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.3" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.3.cmml">p</mi></msub><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.2" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.2.cmml">:=</mo><mrow id="S2.Thmtheorem3.p1.5.5.m5.2.2.1" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.cmml"><msub id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.cmml"><mi id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.2" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.2.cmml">max</mi><mi id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.3" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.3.cmml">i</mi></msub><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.1a" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.cmml">⁡</mo><mrow id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.cmml"><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.1.cmml">|</mo><msub id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.2" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.2.cmml">x</mi><mi id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.3" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.5.m5.2b"><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.5.5.m5.2.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.2">assign</csymbol><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3">subscript</csymbol><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.2.2.1">norm</csymbol><ci id="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.1">𝑥</ci></apply><ci id="S2.Thmtheorem3.p1.5.5.m5.2.2.3.3.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.3.3">𝑝</ci></apply><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1"><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2">subscript</csymbol><max id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.2"></max><ci id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.3.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.2.3">𝑖</ci></apply><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1"><abs id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.2"></abs><apply id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.2">𝑥</ci><ci id="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.5.m5.2c">||x||_{p}:=\max_{i}|x_{i}|</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.5.m5.2d">| | italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT := roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math> for <math alttext="p=\infty" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.6.6.m6.1"><semantics id="S2.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p1.6.6.m6.1.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi id="S2.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">p</mi><mo id="S2.Thmtheorem3.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.1.cmml">=</mo><mi id="S2.Thmtheorem3.p1.6.6.m6.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.6.6.m6.1b"><apply id="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1"><eq id="S2.Thmtheorem3.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.1"></eq><ci id="S2.Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.2">𝑝</ci><infinity id="S2.Thmtheorem3.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.6.6.m6.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.6.6.m6.1d">italic_p = ∞</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.6">Concretely, the distance between two points <math alttext="x,y\in([0,1]^{d},\ell_{p})" class="ltx_Math" display="inline" id="S2.p2.1.m1.6"><semantics id="S2.p2.1.m1.6a"><mrow id="S2.p2.1.m1.6.6" xref="S2.p2.1.m1.6.6.cmml"><mrow id="S2.p2.1.m1.6.6.4.2" xref="S2.p2.1.m1.6.6.4.1.cmml"><mi id="S2.p2.1.m1.3.3" xref="S2.p2.1.m1.3.3.cmml">x</mi><mo id="S2.p2.1.m1.6.6.4.2.1" xref="S2.p2.1.m1.6.6.4.1.cmml">,</mo><mi id="S2.p2.1.m1.4.4" xref="S2.p2.1.m1.4.4.cmml">y</mi></mrow><mo id="S2.p2.1.m1.6.6.3" xref="S2.p2.1.m1.6.6.3.cmml">∈</mo><mrow id="S2.p2.1.m1.6.6.2.2" xref="S2.p2.1.m1.6.6.2.3.cmml"><mo id="S2.p2.1.m1.6.6.2.2.3" stretchy="false" xref="S2.p2.1.m1.6.6.2.3.cmml">(</mo><msup id="S2.p2.1.m1.5.5.1.1.1" xref="S2.p2.1.m1.5.5.1.1.1.cmml"><mrow id="S2.p2.1.m1.5.5.1.1.1.2.2" xref="S2.p2.1.m1.5.5.1.1.1.2.1.cmml"><mo id="S2.p2.1.m1.5.5.1.1.1.2.2.1" stretchy="false" xref="S2.p2.1.m1.5.5.1.1.1.2.1.cmml">[</mo><mn id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">0</mn><mo id="S2.p2.1.m1.5.5.1.1.1.2.2.2" xref="S2.p2.1.m1.5.5.1.1.1.2.1.cmml">,</mo><mn id="S2.p2.1.m1.2.2" xref="S2.p2.1.m1.2.2.cmml">1</mn><mo id="S2.p2.1.m1.5.5.1.1.1.2.2.3" stretchy="false" xref="S2.p2.1.m1.5.5.1.1.1.2.1.cmml">]</mo></mrow><mi id="S2.p2.1.m1.5.5.1.1.1.3" xref="S2.p2.1.m1.5.5.1.1.1.3.cmml">d</mi></msup><mo id="S2.p2.1.m1.6.6.2.2.4" xref="S2.p2.1.m1.6.6.2.3.cmml">,</mo><msub id="S2.p2.1.m1.6.6.2.2.2" xref="S2.p2.1.m1.6.6.2.2.2.cmml"><mi id="S2.p2.1.m1.6.6.2.2.2.2" mathvariant="normal" xref="S2.p2.1.m1.6.6.2.2.2.2.cmml">ℓ</mi><mi id="S2.p2.1.m1.6.6.2.2.2.3" xref="S2.p2.1.m1.6.6.2.2.2.3.cmml">p</mi></msub><mo id="S2.p2.1.m1.6.6.2.2.5" stretchy="false" xref="S2.p2.1.m1.6.6.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.6b"><apply id="S2.p2.1.m1.6.6.cmml" xref="S2.p2.1.m1.6.6"><in id="S2.p2.1.m1.6.6.3.cmml" xref="S2.p2.1.m1.6.6.3"></in><list id="S2.p2.1.m1.6.6.4.1.cmml" xref="S2.p2.1.m1.6.6.4.2"><ci id="S2.p2.1.m1.3.3.cmml" xref="S2.p2.1.m1.3.3">𝑥</ci><ci id="S2.p2.1.m1.4.4.cmml" xref="S2.p2.1.m1.4.4">𝑦</ci></list><interval closure="open" id="S2.p2.1.m1.6.6.2.3.cmml" xref="S2.p2.1.m1.6.6.2.2"><apply id="S2.p2.1.m1.5.5.1.1.1.cmml" xref="S2.p2.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.p2.1.m1.5.5.1.1.1.1.cmml" xref="S2.p2.1.m1.5.5.1.1.1">superscript</csymbol><interval closure="closed" id="S2.p2.1.m1.5.5.1.1.1.2.1.cmml" xref="S2.p2.1.m1.5.5.1.1.1.2.2"><cn id="S2.p2.1.m1.1.1.cmml" type="integer" xref="S2.p2.1.m1.1.1">0</cn><cn id="S2.p2.1.m1.2.2.cmml" type="integer" xref="S2.p2.1.m1.2.2">1</cn></interval><ci id="S2.p2.1.m1.5.5.1.1.1.3.cmml" xref="S2.p2.1.m1.5.5.1.1.1.3">𝑑</ci></apply><apply id="S2.p2.1.m1.6.6.2.2.2.cmml" xref="S2.p2.1.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="S2.p2.1.m1.6.6.2.2.2.1.cmml" xref="S2.p2.1.m1.6.6.2.2.2">subscript</csymbol><ci id="S2.p2.1.m1.6.6.2.2.2.2.cmml" xref="S2.p2.1.m1.6.6.2.2.2.2">ℓ</ci><ci id="S2.p2.1.m1.6.6.2.2.2.3.cmml" xref="S2.p2.1.m1.6.6.2.2.2.3">𝑝</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.6c">x,y\in([0,1]^{d},\ell_{p})</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.6d">italic_x , italic_y ∈ ( [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math> is given by <math alttext="||y-x||_{p}" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><msub id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml"><mrow id="S2.p2.2.m2.1.1.1.1" xref="S2.p2.2.m2.1.1.1.2.cmml"><mo id="S2.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S2.p2.2.m2.1.1.1.2.1.cmml">‖</mo><mrow id="S2.p2.2.m2.1.1.1.1.1" xref="S2.p2.2.m2.1.1.1.1.1.cmml"><mi id="S2.p2.2.m2.1.1.1.1.1.2" xref="S2.p2.2.m2.1.1.1.1.1.2.cmml">y</mi><mo id="S2.p2.2.m2.1.1.1.1.1.1" xref="S2.p2.2.m2.1.1.1.1.1.1.cmml">−</mo><mi id="S2.p2.2.m2.1.1.1.1.1.3" xref="S2.p2.2.m2.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S2.p2.2.m2.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S2.p2.2.m2.1.1.3" xref="S2.p2.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><apply id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p2.2.m2.1.1.2.cmml" xref="S2.p2.2.m2.1.1">subscript</csymbol><apply id="S2.p2.2.m2.1.1.1.2.cmml" xref="S2.p2.2.m2.1.1.1.1"><csymbol cd="latexml" id="S2.p2.2.m2.1.1.1.2.1.cmml" xref="S2.p2.2.m2.1.1.1.1.2">norm</csymbol><apply id="S2.p2.2.m2.1.1.1.1.1.cmml" xref="S2.p2.2.m2.1.1.1.1.1"><minus id="S2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S2.p2.2.m2.1.1.1.1.1.1"></minus><ci id="S2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S2.p2.2.m2.1.1.1.1.1.2">𝑦</ci><ci id="S2.p2.2.m2.1.1.1.1.1.3.cmml" xref="S2.p2.2.m2.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S2.p2.2.m2.1.1.3.cmml" xref="S2.p2.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">||y-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">| | italic_y - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. We also use <math alttext="B^{p}(x,r)\coloneqq\{y\in\mathbb{R}^{d}\;|\;||y-x||_{p}\leq r\}" class="ltx_Math" display="inline" id="S2.p2.3.m3.4"><semantics id="S2.p2.3.m3.4a"><mrow id="S2.p2.3.m3.4.4" xref="S2.p2.3.m3.4.4.cmml"><mrow id="S2.p2.3.m3.4.4.4" xref="S2.p2.3.m3.4.4.4.cmml"><msup id="S2.p2.3.m3.4.4.4.2" xref="S2.p2.3.m3.4.4.4.2.cmml"><mi id="S2.p2.3.m3.4.4.4.2.2" xref="S2.p2.3.m3.4.4.4.2.2.cmml">B</mi><mi id="S2.p2.3.m3.4.4.4.2.3" xref="S2.p2.3.m3.4.4.4.2.3.cmml">p</mi></msup><mo id="S2.p2.3.m3.4.4.4.1" xref="S2.p2.3.m3.4.4.4.1.cmml">⁢</mo><mrow id="S2.p2.3.m3.4.4.4.3.2" xref="S2.p2.3.m3.4.4.4.3.1.cmml"><mo id="S2.p2.3.m3.4.4.4.3.2.1" stretchy="false" xref="S2.p2.3.m3.4.4.4.3.1.cmml">(</mo><mi id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml">x</mi><mo id="S2.p2.3.m3.4.4.4.3.2.2" xref="S2.p2.3.m3.4.4.4.3.1.cmml">,</mo><mi id="S2.p2.3.m3.2.2" xref="S2.p2.3.m3.2.2.cmml">r</mi><mo id="S2.p2.3.m3.4.4.4.3.2.3" stretchy="false" xref="S2.p2.3.m3.4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.3.m3.4.4.3" xref="S2.p2.3.m3.4.4.3.cmml">≔</mo><mrow id="S2.p2.3.m3.4.4.2.2" xref="S2.p2.3.m3.4.4.2.3.cmml"><mo id="S2.p2.3.m3.4.4.2.2.3" stretchy="false" xref="S2.p2.3.m3.4.4.2.3.1.cmml">{</mo><mrow id="S2.p2.3.m3.3.3.1.1.1" xref="S2.p2.3.m3.3.3.1.1.1.cmml"><mi id="S2.p2.3.m3.3.3.1.1.1.2" xref="S2.p2.3.m3.3.3.1.1.1.2.cmml">y</mi><mo id="S2.p2.3.m3.3.3.1.1.1.1" xref="S2.p2.3.m3.3.3.1.1.1.1.cmml">∈</mo><msup id="S2.p2.3.m3.3.3.1.1.1.3" xref="S2.p2.3.m3.3.3.1.1.1.3.cmml"><mi id="S2.p2.3.m3.3.3.1.1.1.3.2" xref="S2.p2.3.m3.3.3.1.1.1.3.2.cmml">ℝ</mi><mi id="S2.p2.3.m3.3.3.1.1.1.3.3" xref="S2.p2.3.m3.3.3.1.1.1.3.3.cmml">d</mi></msup></mrow><mo id="S2.p2.3.m3.4.4.2.2.4" lspace="0em" xref="S2.p2.3.m3.4.4.2.3.1.cmml">|</mo><mrow id="S2.p2.3.m3.4.4.2.2.2" xref="S2.p2.3.m3.4.4.2.2.2.cmml"><msub id="S2.p2.3.m3.4.4.2.2.2.1" xref="S2.p2.3.m3.4.4.2.2.2.1.cmml"><mrow id="S2.p2.3.m3.4.4.2.2.2.1.1.1" xref="S2.p2.3.m3.4.4.2.2.2.1.1.2.cmml"><mo id="S2.p2.3.m3.4.4.2.2.2.1.1.1.2" stretchy="false" xref="S2.p2.3.m3.4.4.2.2.2.1.1.2.1.cmml">‖</mo><mrow id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.cmml"><mi id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.2" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.2.cmml">y</mi><mo id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.1" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.3" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.p2.3.m3.4.4.2.2.2.1.1.1.3" stretchy="false" xref="S2.p2.3.m3.4.4.2.2.2.1.1.2.1.cmml">‖</mo></mrow><mi id="S2.p2.3.m3.4.4.2.2.2.1.3" xref="S2.p2.3.m3.4.4.2.2.2.1.3.cmml">p</mi></msub><mo id="S2.p2.3.m3.4.4.2.2.2.2" xref="S2.p2.3.m3.4.4.2.2.2.2.cmml">≤</mo><mi id="S2.p2.3.m3.4.4.2.2.2.3" xref="S2.p2.3.m3.4.4.2.2.2.3.cmml">r</mi></mrow><mo id="S2.p2.3.m3.4.4.2.2.5" stretchy="false" xref="S2.p2.3.m3.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.4b"><apply id="S2.p2.3.m3.4.4.cmml" xref="S2.p2.3.m3.4.4"><ci id="S2.p2.3.m3.4.4.3.cmml" xref="S2.p2.3.m3.4.4.3">≔</ci><apply id="S2.p2.3.m3.4.4.4.cmml" xref="S2.p2.3.m3.4.4.4"><times id="S2.p2.3.m3.4.4.4.1.cmml" xref="S2.p2.3.m3.4.4.4.1"></times><apply id="S2.p2.3.m3.4.4.4.2.cmml" xref="S2.p2.3.m3.4.4.4.2"><csymbol cd="ambiguous" id="S2.p2.3.m3.4.4.4.2.1.cmml" xref="S2.p2.3.m3.4.4.4.2">superscript</csymbol><ci id="S2.p2.3.m3.4.4.4.2.2.cmml" xref="S2.p2.3.m3.4.4.4.2.2">𝐵</ci><ci id="S2.p2.3.m3.4.4.4.2.3.cmml" xref="S2.p2.3.m3.4.4.4.2.3">𝑝</ci></apply><interval closure="open" id="S2.p2.3.m3.4.4.4.3.1.cmml" xref="S2.p2.3.m3.4.4.4.3.2"><ci id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1">𝑥</ci><ci id="S2.p2.3.m3.2.2.cmml" xref="S2.p2.3.m3.2.2">𝑟</ci></interval></apply><apply id="S2.p2.3.m3.4.4.2.3.cmml" xref="S2.p2.3.m3.4.4.2.2"><csymbol cd="latexml" id="S2.p2.3.m3.4.4.2.3.1.cmml" xref="S2.p2.3.m3.4.4.2.2.3">conditional-set</csymbol><apply id="S2.p2.3.m3.3.3.1.1.1.cmml" xref="S2.p2.3.m3.3.3.1.1.1"><in id="S2.p2.3.m3.3.3.1.1.1.1.cmml" xref="S2.p2.3.m3.3.3.1.1.1.1"></in><ci id="S2.p2.3.m3.3.3.1.1.1.2.cmml" xref="S2.p2.3.m3.3.3.1.1.1.2">𝑦</ci><apply id="S2.p2.3.m3.3.3.1.1.1.3.cmml" xref="S2.p2.3.m3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.p2.3.m3.3.3.1.1.1.3.1.cmml" xref="S2.p2.3.m3.3.3.1.1.1.3">superscript</csymbol><ci id="S2.p2.3.m3.3.3.1.1.1.3.2.cmml" xref="S2.p2.3.m3.3.3.1.1.1.3.2">ℝ</ci><ci id="S2.p2.3.m3.3.3.1.1.1.3.3.cmml" xref="S2.p2.3.m3.3.3.1.1.1.3.3">𝑑</ci></apply></apply><apply id="S2.p2.3.m3.4.4.2.2.2.cmml" xref="S2.p2.3.m3.4.4.2.2.2"><leq id="S2.p2.3.m3.4.4.2.2.2.2.cmml" xref="S2.p2.3.m3.4.4.2.2.2.2"></leq><apply id="S2.p2.3.m3.4.4.2.2.2.1.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1"><csymbol cd="ambiguous" id="S2.p2.3.m3.4.4.2.2.2.1.2.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1">subscript</csymbol><apply id="S2.p2.3.m3.4.4.2.2.2.1.1.2.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1"><csymbol cd="latexml" id="S2.p2.3.m3.4.4.2.2.2.1.1.2.1.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.2">norm</csymbol><apply id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1"><minus id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.1.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.1"></minus><ci id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.2.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.2">𝑦</ci><ci id="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.3.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S2.p2.3.m3.4.4.2.2.2.1.3.cmml" xref="S2.p2.3.m3.4.4.2.2.2.1.3">𝑝</ci></apply><ci id="S2.p2.3.m3.4.4.2.2.2.3.cmml" xref="S2.p2.3.m3.4.4.2.2.2.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.4c">B^{p}(x,r)\coloneqq\{y\in\mathbb{R}^{d}\;|\;||y-x||_{p}\leq r\}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.4d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x , italic_r ) ≔ { italic_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | | | italic_y - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_r }</annotation></semantics></math> to denote the <em class="ltx_emph ltx_font_italic" id="S2.p2.4.1"><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p2.4.1.m1.1"><semantics id="S2.p2.4.1.m1.1a"><msub id="S2.p2.4.1.m1.1.1" xref="S2.p2.4.1.m1.1.1.cmml"><mi id="S2.p2.4.1.m1.1.1.2" mathvariant="normal" xref="S2.p2.4.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.p2.4.1.m1.1.1.3" xref="S2.p2.4.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p2.4.1.m1.1b"><apply id="S2.p2.4.1.m1.1.1.cmml" xref="S2.p2.4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p2.4.1.m1.1.1.1.cmml" xref="S2.p2.4.1.m1.1.1">subscript</csymbol><ci id="S2.p2.4.1.m1.1.1.2.cmml" xref="S2.p2.4.1.m1.1.1.2">ℓ</ci><ci id="S2.p2.4.1.m1.1.1.3.cmml" xref="S2.p2.4.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-ball</em> of radius <math alttext="r" class="ltx_Math" display="inline" id="S2.p2.5.m4.1"><semantics id="S2.p2.5.m4.1a"><mi id="S2.p2.5.m4.1.1" xref="S2.p2.5.m4.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.p2.5.m4.1b"><ci id="S2.p2.5.m4.1.1.cmml" xref="S2.p2.5.m4.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m4.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m4.1d">italic_r</annotation></semantics></math> around <math alttext="x" class="ltx_Math" display="inline" id="S2.p2.6.m5.1"><semantics id="S2.p2.6.m5.1a"><mi id="S2.p2.6.m5.1.1" xref="S2.p2.6.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p2.6.m5.1b"><ci id="S2.p2.6.m5.1.1.cmml" xref="S2.p2.6.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m5.1d">italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1">Finding the exact fixpoint of a contraction map is often infeasible, which is why one usually considers the problem of finding <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mi id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoints instead.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.1.1">Definition 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.3.2"> </span>(<math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem4.1.m1.1"><semantics id="S2.Thmtheorem4.1.m1.1b"><mi id="S2.Thmtheorem4.1.m1.1.1" xref="S2.Thmtheorem4.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.1.m1.1c"><ci id="S2.Thmtheorem4.1.m1.1.1.cmml" xref="S2.Thmtheorem4.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.1.m1.1d">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.1.m1.1e">italic_ε</annotation></semantics></math>-Approximate Fixpoint)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.4.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.5.5">Given a contraction map <math alttext="f:X\rightarrow X" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.1.m1.1"><semantics id="S2.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem4.p1.1.1.m1.1.1" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.2" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">X</mi><mo id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.1.m1.1b"><apply id="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.1">:</ci><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3"><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.1">→</ci><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.2">𝑋</ci><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.1.m1.1c">f:X\rightarrow X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.1.m1.1d">italic_f : italic_X → italic_X</annotation></semantics></math> and an <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.2.2.m2.1"><semantics id="S2.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem4.p1.2.2.m2.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S2.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S2.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.2.2.m2.1b"><apply id="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1"><gt id="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1"></gt><ci id="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math>, a point <math alttext="x\in X" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.3.3.m3.1"><semantics id="S2.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem4.p1.3.3.m3.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">x</mi><mo id="S2.Thmtheorem4.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem4.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.3.m3.1b"><apply id="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1"><in id="S2.Thmtheorem4.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem4.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.2">𝑥</ci><ci id="S2.Thmtheorem4.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.3.m3.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.3.m3.1d">italic_x ∈ italic_X</annotation></semantics></math> is called an <em class="ltx_emph ltx_font_upright" id="S2.Thmtheorem4.p1.4.4.1"><math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.4.4.1.m1.1"><semantics id="S2.Thmtheorem4.p1.4.4.1.m1.1a"><mi id="S2.Thmtheorem4.p1.4.4.1.m1.1.1" xref="S2.Thmtheorem4.p1.4.4.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.4.4.1.m1.1b"><ci id="S2.Thmtheorem4.p1.4.4.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.4.4.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.4.4.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.4.4.1.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint</em> if <math alttext="d_{X}(x,f(x))\leq\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.5.5.m4.3"><semantics id="S2.Thmtheorem4.p1.5.5.m4.3a"><mrow id="S2.Thmtheorem4.p1.5.5.m4.3.3" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.cmml"><mrow id="S2.Thmtheorem4.p1.5.5.m4.3.3.1" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.cmml"><msub id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.cmml"><mi id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.2" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.2.cmml">d</mi><mi id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.3" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.3.cmml">X</mi></msub><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.2" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.2.cmml"><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.2.cmml">(</mo><mi id="S2.Thmtheorem4.p1.5.5.m4.2.2" xref="S2.Thmtheorem4.p1.5.5.m4.2.2.cmml">x</mi><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.3" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.2.cmml">,</mo><mrow id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.2" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.1" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.3.2" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.cmml"><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.cmml">(</mo><mi id="S2.Thmtheorem4.p1.5.5.m4.1.1" xref="S2.Thmtheorem4.p1.5.5.m4.1.1.cmml">x</mi><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.4" stretchy="false" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem4.p1.5.5.m4.3.3.2" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.2.cmml">≤</mo><mi id="S2.Thmtheorem4.p1.5.5.m4.3.3.3" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.5.5.m4.3b"><apply id="S2.Thmtheorem4.p1.5.5.m4.3.3.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3"><leq id="S2.Thmtheorem4.p1.5.5.m4.3.3.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.2"></leq><apply id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1"><times id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.2"></times><apply id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.1.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.2">𝑑</ci><ci id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.3.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.3.3">𝑋</ci></apply><interval closure="open" id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1"><ci id="S2.Thmtheorem4.p1.5.5.m4.2.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.2.2">𝑥</ci><apply id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1"><times id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.1"></times><ci id="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.1.1.1.1.2">𝑓</ci><ci id="S2.Thmtheorem4.p1.5.5.m4.1.1.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.1.1">𝑥</ci></apply></interval></apply><ci id="S2.Thmtheorem4.p1.5.5.m4.3.3.3.cmml" xref="S2.Thmtheorem4.p1.5.5.m4.3.3.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.5.5.m4.3c">d_{X}(x,f(x))\leq\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.5.5.m4.3d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_x , italic_f ( italic_x ) ) ≤ italic_ε</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.4">In the literature, this condition is often called the <em class="ltx_emph ltx_font_italic" id="S2.p4.4.1">residual error criterion</em>. Note that the contraction property ensures that any <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mi id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><ci id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint has a distance of at most <math alttext="\frac{\varepsilon}{1-\lambda}" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mfrac id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml"><mi id="S2.p4.2.m2.1.1.2" xref="S2.p4.2.m2.1.1.2.cmml">ε</mi><mrow id="S2.p4.2.m2.1.1.3" xref="S2.p4.2.m2.1.1.3.cmml"><mn id="S2.p4.2.m2.1.1.3.2" xref="S2.p4.2.m2.1.1.3.2.cmml">1</mn><mo id="S2.p4.2.m2.1.1.3.1" xref="S2.p4.2.m2.1.1.3.1.cmml">−</mo><mi id="S2.p4.2.m2.1.1.3.3" xref="S2.p4.2.m2.1.1.3.3.cmml">λ</mi></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><apply id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1"><divide id="S2.p4.2.m2.1.1.1.cmml" xref="S2.p4.2.m2.1.1"></divide><ci id="S2.p4.2.m2.1.1.2.cmml" xref="S2.p4.2.m2.1.1.2">𝜀</ci><apply id="S2.p4.2.m2.1.1.3.cmml" xref="S2.p4.2.m2.1.1.3"><minus id="S2.p4.2.m2.1.1.3.1.cmml" xref="S2.p4.2.m2.1.1.3.1"></minus><cn id="S2.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S2.p4.2.m2.1.1.3.2">1</cn><ci id="S2.p4.2.m2.1.1.3.3.cmml" xref="S2.p4.2.m2.1.1.3.3">𝜆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">\frac{\varepsilon}{1-\lambda}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">divide start_ARG italic_ε end_ARG start_ARG 1 - italic_λ end_ARG</annotation></semantics></math> to the unique fixpoint <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><msup id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml"><mi id="S2.p4.3.m3.1.1.2" xref="S2.p4.3.m3.1.1.2.cmml">x</mi><mo id="S2.p4.3.m3.1.1.3" xref="S2.p4.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><apply id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p4.3.m3.1.1.1.cmml" xref="S2.p4.3.m3.1.1">superscript</csymbol><ci id="S2.p4.3.m3.1.1.2.cmml" xref="S2.p4.3.m3.1.1.2">𝑥</ci><ci id="S2.p4.3.m3.1.1.3.cmml" xref="S2.p4.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, thus also bounding the so-called <em class="ltx_emph ltx_font_italic" id="S2.p4.4.2">absolute error</em> <math alttext="d_{X}(x,x^{\star})" class="ltx_Math" display="inline" id="S2.p4.4.m4.2"><semantics id="S2.p4.4.m4.2a"><mrow id="S2.p4.4.m4.2.2" xref="S2.p4.4.m4.2.2.cmml"><msub id="S2.p4.4.m4.2.2.3" xref="S2.p4.4.m4.2.2.3.cmml"><mi id="S2.p4.4.m4.2.2.3.2" xref="S2.p4.4.m4.2.2.3.2.cmml">d</mi><mi id="S2.p4.4.m4.2.2.3.3" xref="S2.p4.4.m4.2.2.3.3.cmml">X</mi></msub><mo id="S2.p4.4.m4.2.2.2" xref="S2.p4.4.m4.2.2.2.cmml">⁢</mo><mrow id="S2.p4.4.m4.2.2.1.1" xref="S2.p4.4.m4.2.2.1.2.cmml"><mo id="S2.p4.4.m4.2.2.1.1.2" stretchy="false" xref="S2.p4.4.m4.2.2.1.2.cmml">(</mo><mi id="S2.p4.4.m4.1.1" xref="S2.p4.4.m4.1.1.cmml">x</mi><mo id="S2.p4.4.m4.2.2.1.1.3" xref="S2.p4.4.m4.2.2.1.2.cmml">,</mo><msup id="S2.p4.4.m4.2.2.1.1.1" xref="S2.p4.4.m4.2.2.1.1.1.cmml"><mi id="S2.p4.4.m4.2.2.1.1.1.2" xref="S2.p4.4.m4.2.2.1.1.1.2.cmml">x</mi><mo id="S2.p4.4.m4.2.2.1.1.1.3" xref="S2.p4.4.m4.2.2.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.p4.4.m4.2.2.1.1.4" stretchy="false" xref="S2.p4.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" 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id="S2.p4.4.m4.2c">d_{X}(x,x^{\star})</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.2d">italic_d start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_x , italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.1.1.1">Definition 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.9"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem5.p1.9.9">The <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.1.1.m1.1"><semantics id="S2.Thmtheorem5.p1.1.1.m1.1a"><msub id="S2.Thmtheorem5.p1.1.1.m1.1.1" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem5.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.Thmtheorem5.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.1.1.m1.1b"><apply id="S2.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.2">ℓ</ci><ci id="S2.Thmtheorem5.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S2.Thmtheorem5.p1.9.9.1">-ContractionFixpoint</span> problem is to find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.2.2.m2.1"><semantics id="S2.Thmtheorem5.p1.2.2.m2.1a"><mi id="S2.Thmtheorem5.p1.2.2.m2.1.1" xref="S2.Thmtheorem5.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.2.2.m2.1b"><ci id="S2.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem5.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint of a <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.3.3.m3.1"><semantics id="S2.Thmtheorem5.p1.3.3.m3.1a"><mi id="S2.Thmtheorem5.p1.3.3.m3.1.1" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.3.3.m3.1b"><ci id="S2.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem5.p1.3.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.3.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.3.3.m3.1d">italic_λ</annotation></semantics></math>-contraction map <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.4.4.m4.1"><semantics id="S2.Thmtheorem5.p1.4.4.m4.1a"><mi id="S2.Thmtheorem5.p1.4.4.m4.1.1" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.4.4.m4.1b"><ci id="S2.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.4.4.m4.1d">italic_f</annotation></semantics></math> on <math 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id="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.1.1.3">0</cn></apply><apply id="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2"><in id="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.1.cmml" xref="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.1"></in><ci id="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.2.cmml" xref="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.2">𝜆</ci><interval closure="closed-open" id="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.3.1.cmml" xref="S2.Thmtheorem5.p1.6.6.m6.4.4.2.2.2.2.3.2"><cn id="S2.Thmtheorem5.p1.6.6.m6.1.1.cmml" type="integer" xref="S2.Thmtheorem5.p1.6.6.m6.1.1">0</cn><cn id="S2.Thmtheorem5.p1.6.6.m6.2.2.cmml" type="integer" xref="S2.Thmtheorem5.p1.6.6.m6.2.2">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.6.6.m6.4c">d\in\mathbb{N},\varepsilon>0,\lambda\in[0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.6.6.m6.4d">italic_d ∈ blackboard_N , italic_ε > 0 , italic_λ ∈ [ 0 , 1 )</annotation></semantics></math> and <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.7.7.m7.3"><semantics id="S2.Thmtheorem5.p1.7.7.m7.3a"><mrow id="S2.Thmtheorem5.p1.7.7.m7.3.4" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.cmml"><mi id="S2.Thmtheorem5.p1.7.7.m7.3.4.2" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.2.cmml">p</mi><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.1" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.1.cmml">∈</mo><mrow id="S2.Thmtheorem5.p1.7.7.m7.3.4.3" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.cmml"><mrow id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.2" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.1.cmml"><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem5.p1.7.7.m7.1.1" xref="S2.Thmtheorem5.p1.7.7.m7.1.1.cmml">1</mn><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.2.2" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem5.p1.7.7.m7.2.2" mathvariant="normal" xref="S2.Thmtheorem5.p1.7.7.m7.2.2.cmml">∞</mi><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.2.3" stretchy="false" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.1" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.1.cmml">∪</mo><mrow id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.2" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.1.cmml"><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.1.cmml">{</mo><mi id="S2.Thmtheorem5.p1.7.7.m7.3.3" mathvariant="normal" xref="S2.Thmtheorem5.p1.7.7.m7.3.3.cmml">∞</mi><mo id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.2.2" stretchy="false" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.7.7.m7.3b"><apply id="S2.Thmtheorem5.p1.7.7.m7.3.4.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4"><in id="S2.Thmtheorem5.p1.7.7.m7.3.4.1.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.1"></in><ci id="S2.Thmtheorem5.p1.7.7.m7.3.4.2.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.2">𝑝</ci><apply id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3"><union id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.1.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.1"></union><interval closure="closed-open" id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.1.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.2.2"><cn id="S2.Thmtheorem5.p1.7.7.m7.1.1.cmml" type="integer" xref="S2.Thmtheorem5.p1.7.7.m7.1.1">1</cn><infinity id="S2.Thmtheorem5.p1.7.7.m7.2.2.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.2.2"></infinity></interval><set id="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.1.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.4.3.3.2"><infinity id="S2.Thmtheorem5.p1.7.7.m7.3.3.cmml" xref="S2.Thmtheorem5.p1.7.7.m7.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.7.7.m7.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.7.7.m7.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> as well as black-box query access to <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.8.8.m8.1"><semantics id="S2.Thmtheorem5.p1.8.8.m8.1a"><mi id="S2.Thmtheorem5.p1.8.8.m8.1.1" xref="S2.Thmtheorem5.p1.8.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.8.8.m8.1b"><ci id="S2.Thmtheorem5.p1.8.8.m8.1.1.cmml" xref="S2.Thmtheorem5.p1.8.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.8.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.8.8.m8.1d">italic_f</annotation></semantics></math>. The goal is to minimize the number of queries made to <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.9.9.m9.1"><semantics id="S2.Thmtheorem5.p1.9.9.m9.1a"><mi id="S2.Thmtheorem5.p1.9.9.m9.1.1" xref="S2.Thmtheorem5.p1.9.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.9.9.m9.1b"><ci id="S2.Thmtheorem5.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem5.p1.9.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.9.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.9.9.m9.1d">italic_f</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.6">We consider an algorithm for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><msub id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml"><mi id="S2.p5.1.m1.1.1.2" mathvariant="normal" xref="S2.p5.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.p5.1.m1.1.1.3" xref="S2.p5.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.1.cmml" xref="S2.p5.1.m1.1.1">subscript</csymbol><ci id="S2.p5.1.m1.1.1.2.cmml" xref="S2.p5.1.m1.1.1.2">ℓ</ci><ci id="S2.p5.1.m1.1.1.3.cmml" xref="S2.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S2.p5.6.1">-ContractionFixpoint</span> to be <em class="ltx_emph ltx_font_italic" id="S2.p5.6.2">query-efficient</em> if the number of queries made to <math alttext="f" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mi id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><ci id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_f</annotation></semantics></math> is polynomial in <math alttext="d" class="ltx_Math" display="inline" id="S2.p5.3.m3.1"><semantics id="S2.p5.3.m3.1a"><mi id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.1b"><ci id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.1d">italic_d</annotation></semantics></math>, <math alttext="\log(\frac{1}{\varepsilon})" class="ltx_Math" display="inline" id="S2.p5.4.m4.2"><semantics id="S2.p5.4.m4.2a"><mrow id="S2.p5.4.m4.2.3.2" xref="S2.p5.4.m4.2.3.1.cmml"><mi id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml">log</mi><mo id="S2.p5.4.m4.2.3.2a" xref="S2.p5.4.m4.2.3.1.cmml">⁡</mo><mrow id="S2.p5.4.m4.2.3.2.1" xref="S2.p5.4.m4.2.3.1.cmml"><mo id="S2.p5.4.m4.2.3.2.1.1" stretchy="false" xref="S2.p5.4.m4.2.3.1.cmml">(</mo><mfrac id="S2.p5.4.m4.2.2" xref="S2.p5.4.m4.2.2.cmml"><mn id="S2.p5.4.m4.2.2.2" xref="S2.p5.4.m4.2.2.2.cmml">1</mn><mi id="S2.p5.4.m4.2.2.3" xref="S2.p5.4.m4.2.2.3.cmml">ε</mi></mfrac><mo id="S2.p5.4.m4.2.3.2.1.2" stretchy="false" xref="S2.p5.4.m4.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.4.m4.2b"><apply id="S2.p5.4.m4.2.3.1.cmml" xref="S2.p5.4.m4.2.3.2"><log id="S2.p5.4.m4.1.1.cmml" xref="S2.p5.4.m4.1.1"></log><apply id="S2.p5.4.m4.2.2.cmml" xref="S2.p5.4.m4.2.2"><divide id="S2.p5.4.m4.2.2.1.cmml" xref="S2.p5.4.m4.2.2"></divide><cn id="S2.p5.4.m4.2.2.2.cmml" type="integer" xref="S2.p5.4.m4.2.2.2">1</cn><ci id="S2.p5.4.m4.2.2.3.cmml" xref="S2.p5.4.m4.2.2.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m4.2c">\log(\frac{1}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m4.2d">roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math>, and <math alttext="\log(\frac{1}{1-\lambda})" class="ltx_Math" display="inline" id="S2.p5.5.m5.2"><semantics id="S2.p5.5.m5.2a"><mrow id="S2.p5.5.m5.2.3.2" xref="S2.p5.5.m5.2.3.1.cmml"><mi id="S2.p5.5.m5.1.1" xref="S2.p5.5.m5.1.1.cmml">log</mi><mo id="S2.p5.5.m5.2.3.2a" xref="S2.p5.5.m5.2.3.1.cmml">⁡</mo><mrow id="S2.p5.5.m5.2.3.2.1" xref="S2.p5.5.m5.2.3.1.cmml"><mo id="S2.p5.5.m5.2.3.2.1.1" stretchy="false" xref="S2.p5.5.m5.2.3.1.cmml">(</mo><mfrac id="S2.p5.5.m5.2.2" xref="S2.p5.5.m5.2.2.cmml"><mn id="S2.p5.5.m5.2.2.2" xref="S2.p5.5.m5.2.2.2.cmml">1</mn><mrow id="S2.p5.5.m5.2.2.3" xref="S2.p5.5.m5.2.2.3.cmml"><mn id="S2.p5.5.m5.2.2.3.2" xref="S2.p5.5.m5.2.2.3.2.cmml">1</mn><mo id="S2.p5.5.m5.2.2.3.1" xref="S2.p5.5.m5.2.2.3.1.cmml">−</mo><mi id="S2.p5.5.m5.2.2.3.3" xref="S2.p5.5.m5.2.2.3.3.cmml">λ</mi></mrow></mfrac><mo id="S2.p5.5.m5.2.3.2.1.2" stretchy="false" xref="S2.p5.5.m5.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.5.m5.2b"><apply id="S2.p5.5.m5.2.3.1.cmml" xref="S2.p5.5.m5.2.3.2"><log id="S2.p5.5.m5.1.1.cmml" xref="S2.p5.5.m5.1.1"></log><apply id="S2.p5.5.m5.2.2.cmml" xref="S2.p5.5.m5.2.2"><divide id="S2.p5.5.m5.2.2.1.cmml" xref="S2.p5.5.m5.2.2"></divide><cn id="S2.p5.5.m5.2.2.2.cmml" type="integer" xref="S2.p5.5.m5.2.2.2">1</cn><apply id="S2.p5.5.m5.2.2.3.cmml" xref="S2.p5.5.m5.2.2.3"><minus id="S2.p5.5.m5.2.2.3.1.cmml" xref="S2.p5.5.m5.2.2.3.1"></minus><cn id="S2.p5.5.m5.2.2.3.2.cmml" type="integer" xref="S2.p5.5.m5.2.2.3.2">1</cn><ci id="S2.p5.5.m5.2.2.3.3.cmml" xref="S2.p5.5.m5.2.2.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.5.m5.2c">\log(\frac{1}{1-\lambda})</annotation><annotation encoding="application/x-llamapun" id="S2.p5.5.m5.2d">roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG )</annotation></semantics></math>, and independent of <math alttext="p" class="ltx_Math" display="inline" id="S2.p5.6.m6.1"><semantics id="S2.p5.6.m6.1a"><mi id="S2.p5.6.m6.1.1" xref="S2.p5.6.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p5.6.m6.1b"><ci id="S2.p5.6.m6.1.1.cmml" xref="S2.p5.6.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.6.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p5.6.m6.1d">italic_p</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.5">In <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><msub id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml"><mi id="S2.p6.1.m1.1.1.2" mathvariant="normal" xref="S2.p6.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.p6.1.m1.1.1.3" xref="S2.p6.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.1.cmml" xref="S2.p6.1.m1.1.1">subscript</csymbol><ci id="S2.p6.1.m1.1.1.2.cmml" xref="S2.p6.1.m1.1.1.2">ℓ</ci><ci id="S2.p6.1.m1.1.1.3.cmml" xref="S2.p6.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S2.p6.5.1">-ContractionFixpoint</span>, we are allowed to query <math alttext="f" class="ltx_Math" display="inline" id="S2.p6.2.m2.1"><semantics id="S2.p6.2.m2.1a"><mi id="S2.p6.2.m2.1.1" xref="S2.p6.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.p6.2.m2.1b"><ci id="S2.p6.2.m2.1.1.cmml" xref="S2.p6.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m2.1d">italic_f</annotation></semantics></math> at any point <math alttext="x\in[0,1]^{d}" class="ltx_Math" display="inline" id="S2.p6.3.m3.2"><semantics id="S2.p6.3.m3.2a"><mrow id="S2.p6.3.m3.2.3" xref="S2.p6.3.m3.2.3.cmml"><mi id="S2.p6.3.m3.2.3.2" xref="S2.p6.3.m3.2.3.2.cmml">x</mi><mo id="S2.p6.3.m3.2.3.1" xref="S2.p6.3.m3.2.3.1.cmml">∈</mo><msup id="S2.p6.3.m3.2.3.3" xref="S2.p6.3.m3.2.3.3.cmml"><mrow id="S2.p6.3.m3.2.3.3.2.2" xref="S2.p6.3.m3.2.3.3.2.1.cmml"><mo id="S2.p6.3.m3.2.3.3.2.2.1" stretchy="false" xref="S2.p6.3.m3.2.3.3.2.1.cmml">[</mo><mn id="S2.p6.3.m3.1.1" xref="S2.p6.3.m3.1.1.cmml">0</mn><mo id="S2.p6.3.m3.2.3.3.2.2.2" xref="S2.p6.3.m3.2.3.3.2.1.cmml">,</mo><mn id="S2.p6.3.m3.2.2" xref="S2.p6.3.m3.2.2.cmml">1</mn><mo id="S2.p6.3.m3.2.3.3.2.2.3" stretchy="false" xref="S2.p6.3.m3.2.3.3.2.1.cmml">]</mo></mrow><mi id="S2.p6.3.m3.2.3.3.3" xref="S2.p6.3.m3.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.3.m3.2b"><apply id="S2.p6.3.m3.2.3.cmml" xref="S2.p6.3.m3.2.3"><in id="S2.p6.3.m3.2.3.1.cmml" xref="S2.p6.3.m3.2.3.1"></in><ci id="S2.p6.3.m3.2.3.2.cmml" xref="S2.p6.3.m3.2.3.2">𝑥</ci><apply id="S2.p6.3.m3.2.3.3.cmml" xref="S2.p6.3.m3.2.3.3"><csymbol cd="ambiguous" id="S2.p6.3.m3.2.3.3.1.cmml" xref="S2.p6.3.m3.2.3.3">superscript</csymbol><interval closure="closed" id="S2.p6.3.m3.2.3.3.2.1.cmml" xref="S2.p6.3.m3.2.3.3.2.2"><cn id="S2.p6.3.m3.1.1.cmml" type="integer" xref="S2.p6.3.m3.1.1">0</cn><cn id="S2.p6.3.m3.2.2.cmml" type="integer" xref="S2.p6.3.m3.2.2">1</cn></interval><ci id="S2.p6.3.m3.2.3.3.3.cmml" xref="S2.p6.3.m3.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.3.m3.2c">x\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.3.m3.2d">italic_x ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, even at points with irrational coordinates. This might be unsuitable in some applications, and it seems reasonable to also consider a discretized version of this problem, where we are only allowed to make queries to points on a discrete grid. Although we define this discretized version for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.p6.4.m4.3"><semantics id="S2.p6.4.m4.3a"><mrow id="S2.p6.4.m4.3.4" xref="S2.p6.4.m4.3.4.cmml"><mi id="S2.p6.4.m4.3.4.2" xref="S2.p6.4.m4.3.4.2.cmml">p</mi><mo id="S2.p6.4.m4.3.4.1" xref="S2.p6.4.m4.3.4.1.cmml">∈</mo><mrow id="S2.p6.4.m4.3.4.3" xref="S2.p6.4.m4.3.4.3.cmml"><mrow id="S2.p6.4.m4.3.4.3.2.2" xref="S2.p6.4.m4.3.4.3.2.1.cmml"><mo id="S2.p6.4.m4.3.4.3.2.2.1" stretchy="false" xref="S2.p6.4.m4.3.4.3.2.1.cmml">[</mo><mn id="S2.p6.4.m4.1.1" xref="S2.p6.4.m4.1.1.cmml">1</mn><mo id="S2.p6.4.m4.3.4.3.2.2.2" xref="S2.p6.4.m4.3.4.3.2.1.cmml">,</mo><mi id="S2.p6.4.m4.2.2" mathvariant="normal" xref="S2.p6.4.m4.2.2.cmml">∞</mi><mo id="S2.p6.4.m4.3.4.3.2.2.3" stretchy="false" xref="S2.p6.4.m4.3.4.3.2.1.cmml">)</mo></mrow><mo id="S2.p6.4.m4.3.4.3.1" xref="S2.p6.4.m4.3.4.3.1.cmml">∪</mo><mrow id="S2.p6.4.m4.3.4.3.3.2" xref="S2.p6.4.m4.3.4.3.3.1.cmml"><mo id="S2.p6.4.m4.3.4.3.3.2.1" stretchy="false" xref="S2.p6.4.m4.3.4.3.3.1.cmml">{</mo><mi id="S2.p6.4.m4.3.3" mathvariant="normal" xref="S2.p6.4.m4.3.3.cmml">∞</mi><mo id="S2.p6.4.m4.3.4.3.3.2.2" stretchy="false" xref="S2.p6.4.m4.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.4.m4.3b"><apply id="S2.p6.4.m4.3.4.cmml" xref="S2.p6.4.m4.3.4"><in id="S2.p6.4.m4.3.4.1.cmml" xref="S2.p6.4.m4.3.4.1"></in><ci id="S2.p6.4.m4.3.4.2.cmml" xref="S2.p6.4.m4.3.4.2">𝑝</ci><apply id="S2.p6.4.m4.3.4.3.cmml" xref="S2.p6.4.m4.3.4.3"><union id="S2.p6.4.m4.3.4.3.1.cmml" xref="S2.p6.4.m4.3.4.3.1"></union><interval closure="closed-open" id="S2.p6.4.m4.3.4.3.2.1.cmml" xref="S2.p6.4.m4.3.4.3.2.2"><cn id="S2.p6.4.m4.1.1.cmml" type="integer" xref="S2.p6.4.m4.1.1">1</cn><infinity id="S2.p6.4.m4.2.2.cmml" xref="S2.p6.4.m4.2.2"></infinity></interval><set id="S2.p6.4.m4.3.4.3.3.1.cmml" xref="S2.p6.4.m4.3.4.3.3.2"><infinity id="S2.p6.4.m4.3.3.cmml" xref="S2.p6.4.m4.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.4.m4.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.4.m4.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, we will only really study it for the cases <math alttext="p\in\{1,\infty\}" class="ltx_Math" display="inline" id="S2.p6.5.m5.2"><semantics id="S2.p6.5.m5.2a"><mrow id="S2.p6.5.m5.2.3" xref="S2.p6.5.m5.2.3.cmml"><mi id="S2.p6.5.m5.2.3.2" xref="S2.p6.5.m5.2.3.2.cmml">p</mi><mo id="S2.p6.5.m5.2.3.1" xref="S2.p6.5.m5.2.3.1.cmml">∈</mo><mrow id="S2.p6.5.m5.2.3.3.2" xref="S2.p6.5.m5.2.3.3.1.cmml"><mo id="S2.p6.5.m5.2.3.3.2.1" stretchy="false" xref="S2.p6.5.m5.2.3.3.1.cmml">{</mo><mn id="S2.p6.5.m5.1.1" xref="S2.p6.5.m5.1.1.cmml">1</mn><mo id="S2.p6.5.m5.2.3.3.2.2" xref="S2.p6.5.m5.2.3.3.1.cmml">,</mo><mi id="S2.p6.5.m5.2.2" mathvariant="normal" xref="S2.p6.5.m5.2.2.cmml">∞</mi><mo id="S2.p6.5.m5.2.3.3.2.3" stretchy="false" xref="S2.p6.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.5.m5.2b"><apply id="S2.p6.5.m5.2.3.cmml" xref="S2.p6.5.m5.2.3"><in id="S2.p6.5.m5.2.3.1.cmml" xref="S2.p6.5.m5.2.3.1"></in><ci id="S2.p6.5.m5.2.3.2.cmml" xref="S2.p6.5.m5.2.3.2">𝑝</ci><set id="S2.p6.5.m5.2.3.3.1.cmml" xref="S2.p6.5.m5.2.3.3.2"><cn id="S2.p6.5.m5.1.1.cmml" type="integer" xref="S2.p6.5.m5.1.1">1</cn><infinity id="S2.p6.5.m5.2.2.cmml" xref="S2.p6.5.m5.2.2"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.5.m5.2c">p\in\{1,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.5.m5.2d">italic_p ∈ { 1 , ∞ }</annotation></semantics></math> (motivated by applications).</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.1.1.1">Definition 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem6.p1"> <p class="ltx_p" id="S2.Thmtheorem6.p1.6"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem6.p1.6.6">Given an integer <math alttext="b\geq 1" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem6.p1.1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">b</mi><mo id="S2.Thmtheorem6.p1.1.1.m1.1.1.1" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">≥</mo><mn id="S2.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.1.1.m1.1b"><apply id="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1"><geq id="S2.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.1"></geq><ci id="S2.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.2">𝑏</ci><cn id="S2.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.1.m1.1c">b\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.1.m1.1d">italic_b ≥ 1</annotation></semantics></math>, the grid <math alttext="G^{d}_{b}\subseteq[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.2.2.m2.2"><semantics id="S2.Thmtheorem6.p1.2.2.m2.2a"><mrow id="S2.Thmtheorem6.p1.2.2.m2.2.3" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.cmml"><msubsup id="S2.Thmtheorem6.p1.2.2.m2.2.3.2" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.cmml"><mi id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.2" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.2.cmml">G</mi><mi id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.3" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.3.cmml">b</mi><mi id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.3" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.3.cmml">d</mi></msubsup><mo id="S2.Thmtheorem6.p1.2.2.m2.2.3.1" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.1.cmml">⊆</mo><msup id="S2.Thmtheorem6.p1.2.2.m2.2.3.3" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.cmml"><mrow id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.2" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.1.cmml"><mo id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem6.p1.2.2.m2.1.1" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml">0</mn><mo id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.2.2" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem6.p1.2.2.m2.2.2" xref="S2.Thmtheorem6.p1.2.2.m2.2.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.3" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.2.2.m2.2b"><apply id="S2.Thmtheorem6.p1.2.2.m2.2.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3"><subset id="S2.Thmtheorem6.p1.2.2.m2.2.3.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.1"></subset><apply id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2">subscript</csymbol><apply id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.2.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.2">𝐺</ci><ci id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem6.p1.2.2.m2.2.3.2.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.2.3">𝑏</ci></apply><apply id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.2.2"><cn id="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml" type="integer" xref="S2.Thmtheorem6.p1.2.2.m2.1.1">0</cn><cn id="S2.Thmtheorem6.p1.2.2.m2.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.2.2.m2.2.2">1</cn></interval><ci id="S2.Thmtheorem6.p1.2.2.m2.2.3.3.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.2.2.m2.2c">G^{d}_{b}\subseteq[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.2.2.m2.2d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is the set of points <math alttext="x\in[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.3.3.m3.2"><semantics id="S2.Thmtheorem6.p1.3.3.m3.2a"><mrow id="S2.Thmtheorem6.p1.3.3.m3.2.3" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.cmml"><mi id="S2.Thmtheorem6.p1.3.3.m3.2.3.2" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.2.cmml">x</mi><mo id="S2.Thmtheorem6.p1.3.3.m3.2.3.1" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.1.cmml">∈</mo><msup id="S2.Thmtheorem6.p1.3.3.m3.2.3.3" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.cmml"><mrow id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.2" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml"><mo id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem6.p1.3.3.m3.1.1" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml">0</mn><mo id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.2.2" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem6.p1.3.3.m3.2.2" xref="S2.Thmtheorem6.p1.3.3.m3.2.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.3" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.3.3.m3.2b"><apply id="S2.Thmtheorem6.p1.3.3.m3.2.3.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3"><in id="S2.Thmtheorem6.p1.3.3.m3.2.3.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.1"></in><ci id="S2.Thmtheorem6.p1.3.3.m3.2.3.2.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.2">𝑥</ci><apply id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.2.2"><cn id="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml" type="integer" xref="S2.Thmtheorem6.p1.3.3.m3.1.1">0</cn><cn id="S2.Thmtheorem6.p1.3.3.m3.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.3.3.m3.2.2">1</cn></interval><ci id="S2.Thmtheorem6.p1.3.3.m3.2.3.3.3.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.3.3.m3.2c">x\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.3.3.m3.2d">italic_x ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with rational coordinates <math alttext="x_{1},\dots,x_{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.4.4.m4.3"><semantics id="S2.Thmtheorem6.p1.4.4.m4.3a"><mrow id="S2.Thmtheorem6.p1.4.4.m4.3.3.2" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.3.cmml"><msub id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.cmml"><mi id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.2" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.2.cmml">x</mi><mn id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.3" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.3" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.3.cmml">,</mo><mi id="S2.Thmtheorem6.p1.4.4.m4.1.1" mathvariant="normal" xref="S2.Thmtheorem6.p1.4.4.m4.1.1.cmml">…</mi><mo id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.4" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.3.cmml">,</mo><msub id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.cmml"><mi id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.2" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.2.cmml">x</mi><mi id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.3" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.4.4.m4.3b"><list id="S2.Thmtheorem6.p1.4.4.m4.3.3.3.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2"><apply id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1">subscript</csymbol><ci id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.2.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.2">𝑥</ci><cn id="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.4.4.m4.2.2.1.1.3">1</cn></apply><ci id="S2.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.1.1">…</ci><apply id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.1.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.2.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.2">𝑥</ci><ci id="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.3.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.3.3.2.2.3">𝑑</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.4.4.m4.3c">x_{1},\dots,x_{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.4.4.m4.3d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> of the form <math alttext="x_{i}=\frac{k_{i}}{2^{b}}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.5.5.m5.1"><semantics id="S2.Thmtheorem6.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem6.p1.5.5.m5.1.1" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.cmml"><msub id="S2.Thmtheorem6.p1.5.5.m5.1.1.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.cmml"><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.2.cmml">x</mi><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S2.Thmtheorem6.p1.5.5.m5.1.1.1" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.1.cmml">=</mo><mfrac id="S2.Thmtheorem6.p1.5.5.m5.1.1.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.cmml"><msub id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.cmml"><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.2.cmml">k</mi><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.3.cmml">i</mi></msub><msup id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.cmml"><mn id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.2.cmml">2</mn><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.3.cmml">b</mi></msup></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.5.5.m5.1b"><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1"><eq id="S2.Thmtheorem6.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.1"></eq><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.2.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.2">𝑥</ci><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.3.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3"><divide id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3"></divide><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.2.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.2">𝑘</ci><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.3.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.2.3">𝑖</ci></apply><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3">superscript</csymbol><cn id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.2">2</cn><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.3.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.5.5.m5.1c">x_{i}=\frac{k_{i}}{2^{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.5.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> for integers <math alttext="k_{1},\dots,k_{d}\in\{0,1,\dots,2^{b}\}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.6.6.m6.7"><semantics id="S2.Thmtheorem6.p1.6.6.m6.7a"><mrow id="S2.Thmtheorem6.p1.6.6.m6.7.7" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.cmml"><mrow id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.3.cmml"><msub id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.2" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.2.cmml">k</mi><mn id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.3.cmml">1</mn></msub><mo id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.3" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.3.cmml">,</mo><mi id="S2.Thmtheorem6.p1.6.6.m6.4.4" mathvariant="normal" xref="S2.Thmtheorem6.p1.6.6.m6.4.4.cmml">…</mi><mo id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.4" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.3.cmml">,</mo><msub id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.cmml"><mi id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.2" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.2.cmml">k</mi><mi id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.3" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.3.cmml">d</mi></msub></mrow><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.4" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.4.cmml">∈</mo><mrow id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml"><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml">{</mo><mn id="S2.Thmtheorem6.p1.6.6.m6.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.cmml">0</mn><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml">,</mo><mn id="S2.Thmtheorem6.p1.6.6.m6.2.2" xref="S2.Thmtheorem6.p1.6.6.m6.2.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.4" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml">,</mo><mi id="S2.Thmtheorem6.p1.6.6.m6.3.3" mathvariant="normal" xref="S2.Thmtheorem6.p1.6.6.m6.3.3.cmml">…</mi><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.5" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml">,</mo><msup id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.cmml"><mn id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.2" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.2.cmml">2</mn><mi id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.3.cmml">b</mi></msup><mo id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.6" stretchy="false" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.6.6.m6.7b"><apply id="S2.Thmtheorem6.p1.6.6.m6.7.7.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7"><in id="S2.Thmtheorem6.p1.6.6.m6.7.7.4.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.4"></in><list id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2"><apply id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.2">𝑘</ci><cn id="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.6.6.m6.5.5.1.1.1.3">1</cn></apply><ci id="S2.Thmtheorem6.p1.6.6.m6.4.4.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.4.4">…</ci><apply id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.2">𝑘</ci><ci id="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.6.6.2.2.2.3">𝑑</ci></apply></list><set id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1"><cn id="S2.Thmtheorem6.p1.6.6.m6.1.1.cmml" type="integer" xref="S2.Thmtheorem6.p1.6.6.m6.1.1">0</cn><cn id="S2.Thmtheorem6.p1.6.6.m6.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.6.6.m6.2.2">1</cn><ci id="S2.Thmtheorem6.p1.6.6.m6.3.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.3.3">…</ci><apply id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1">superscript</csymbol><cn id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.2">2</cn><ci id="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.7.7.3.1.1.3">𝑏</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.6.6.m6.7c">k_{1},\dots,k_{d}\in\{0,1,\dots,2^{b}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.6.6.m6.7d">italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_k start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ∈ { 0 , 1 , … , 2 start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT }</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.1.1.1">Definition 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem7.p1"> <p class="ltx_p" id="S2.Thmtheorem7.p1.9"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.9.9">A function <math alttext="f:G^{d}_{b}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.1.1.m1.2"><semantics id="S2.Thmtheorem7.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem7.p1.1.1.m1.2.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.2.cmml">f</mi><mo id="S2.Thmtheorem7.p1.1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem7.p1.1.1.m1.2.3.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.cmml"><msubsup id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.2.cmml">G</mi><mi id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.3.cmml">b</mi><mi id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.1" stretchy="false" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">→</mo><msup id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.cmml"><mrow id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.1.cmml"><mo id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem7.p1.1.1.m1.1.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml">0</mn><mo id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem7.p1.1.1.m1.2.2" xref="S2.Thmtheorem7.p1.1.1.m1.2.2.cmml">1</mn><mo id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.3" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.1.1.m1.2b"><apply id="S2.Thmtheorem7.p1.1.1.m1.2.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3"><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.1">:</ci><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.2">𝑓</ci><apply id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3"><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.1">→</ci><apply id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2">subscript</csymbol><apply id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2">superscript</csymbol><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.2">𝐺</ci><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.2.3">𝑏</ci></apply><apply id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.2.2"><cn id="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S2.Thmtheorem7.p1.1.1.m1.1.1">0</cn><cn id="S2.Thmtheorem7.p1.1.1.m1.2.2.cmml" type="integer" xref="S2.Thmtheorem7.p1.1.1.m1.2.2">1</cn></interval><ci id="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.2.3.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.1.1.m1.2c">f:G^{d}_{b}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.1.1.m1.2d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is called a <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.2.2.m2.1"><semantics id="S2.Thmtheorem7.p1.2.2.m2.1a"><mi id="S2.Thmtheorem7.p1.2.2.m2.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.2.2.m2.1b"><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.2.2.m2.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.2.2.m2.1d">italic_λ</annotation></semantics></math>-contraction grid-map if there exists a <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.3.3.m3.1"><semantics id="S2.Thmtheorem7.p1.3.3.m3.1a"><mi id="S2.Thmtheorem7.p1.3.3.m3.1.1" xref="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.3.3.m3.1b"><ci id="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem7.p1.3.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.3.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.3.3.m3.1d">italic_λ</annotation></semantics></math>-contraction map <math alttext="f^{\prime}:[0,1]^{d}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.4.4.m4.4"><semantics id="S2.Thmtheorem7.p1.4.4.m4.4a"><mrow id="S2.Thmtheorem7.p1.4.4.m4.4.5" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.cmml"><msup id="S2.Thmtheorem7.p1.4.4.m4.4.5.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2.cmml"><mi id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2.2.cmml">f</mi><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.3" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.1.cmml">:</mo><mrow id="S2.Thmtheorem7.p1.4.4.m4.4.5.3" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.cmml"><msup id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.cmml"><mrow id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.1.cmml"><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.2.1" stretchy="false" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.1.cmml">[</mo><mn id="S2.Thmtheorem7.p1.4.4.m4.1.1" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml">0</mn><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.1.cmml">,</mo><mn id="S2.Thmtheorem7.p1.4.4.m4.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.2.2.cmml">1</mn><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.2.3" stretchy="false" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.3" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.3.cmml">d</mi></msup><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.1" stretchy="false" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.1.cmml">→</mo><msup id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.cmml"><mrow id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.1.cmml"><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem7.p1.4.4.m4.3.3" xref="S2.Thmtheorem7.p1.4.4.m4.3.3.cmml">0</mn><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.2.2" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem7.p1.4.4.m4.4.4" xref="S2.Thmtheorem7.p1.4.4.m4.4.4.cmml">1</mn><mo id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.3" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.4.4.m4.4b"><apply id="S2.Thmtheorem7.p1.4.4.m4.4.5.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5"><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.1">:</ci><apply id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2">superscript</csymbol><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.2.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2.2">𝑓</ci><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.2.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.2.3">′</ci></apply><apply id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3"><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.1">→</ci><apply id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.2.2"><cn id="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml" type="integer" xref="S2.Thmtheorem7.p1.4.4.m4.1.1">0</cn><cn id="S2.Thmtheorem7.p1.4.4.m4.2.2.cmml" type="integer" xref="S2.Thmtheorem7.p1.4.4.m4.2.2">1</cn></interval><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.2.3">𝑑</ci></apply><apply id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.2.2"><cn id="S2.Thmtheorem7.p1.4.4.m4.3.3.cmml" type="integer" xref="S2.Thmtheorem7.p1.4.4.m4.3.3">0</cn><cn id="S2.Thmtheorem7.p1.4.4.m4.4.4.cmml" type="integer" xref="S2.Thmtheorem7.p1.4.4.m4.4.4">1</cn></interval><ci id="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.4.5.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.4.4.m4.4c">f^{\prime}:[0,1]^{d}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.4.4.m4.4d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="f(x)=f^{\prime}(x)" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.5.5.m5.2"><semantics id="S2.Thmtheorem7.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem7.p1.5.5.m5.2.3" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.cmml"><mrow id="S2.Thmtheorem7.p1.5.5.m5.2.3.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.cmml"><mi id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.2.cmml">f</mi><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.1" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.3.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.cmml"><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.cmml">(</mo><mi id="S2.Thmtheorem7.p1.5.5.m5.1.1" xref="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml">x</mi><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.3.2.2" stretchy="false" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.1" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem7.p1.5.5.m5.2.3.3" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.cmml"><msup id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.cmml"><mi id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.2.cmml">f</mi><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.3" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.3.cmml">′</mo></msup><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.1" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.3.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.cmml"><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.cmml">(</mo><mi id="S2.Thmtheorem7.p1.5.5.m5.2.2" xref="S2.Thmtheorem7.p1.5.5.m5.2.2.cmml">x</mi><mo id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.3.2.2" stretchy="false" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.5.5.m5.2b"><apply id="S2.Thmtheorem7.p1.5.5.m5.2.3.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem7.p1.5.5.m5.2.3.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.1"></eq><apply id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2"><times id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.1"></times><ci id="S2.Thmtheorem7.p1.5.5.m5.2.3.2.2.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.2.2">𝑓</ci><ci id="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.1.1">𝑥</ci></apply><apply id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3"><times id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.1"></times><apply id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2">superscript</csymbol><ci id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.2.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.2">𝑓</ci><ci id="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.3.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.3.3.2.3">′</ci></apply><ci id="S2.Thmtheorem7.p1.5.5.m5.2.2.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.5.5.m5.2c">f(x)=f^{\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.5.5.m5.2d">italic_f ( italic_x ) = italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> for all <math alttext="x\in G^{d}_{b}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.6.6.m6.1"><semantics id="S2.Thmtheorem7.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem7.p1.6.6.m6.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml"><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.2.cmml">x</mi><mo id="S2.Thmtheorem7.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.1.cmml">∈</mo><msubsup id="S2.Thmtheorem7.p1.6.6.m6.1.1.3" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.cmml"><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.2" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.2.cmml">G</mi><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.3" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.3.cmml">b</mi><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.3" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.6.6.m6.1b"><apply id="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1"><in id="S2.Thmtheorem7.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.1"></in><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.2">𝑥</ci><apply id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.2">𝐺</ci><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.3.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.3.3.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.6.6.m6.1c">x\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.6.6.m6.1d">italic_x ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> (both <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.7.7.m7.1"><semantics id="S2.Thmtheorem7.p1.7.7.m7.1a"><msup id="S2.Thmtheorem7.p1.7.7.m7.1.1" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.cmml"><mi id="S2.Thmtheorem7.p1.7.7.m7.1.1.2" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.2.cmml">f</mi><mo id="S2.Thmtheorem7.p1.7.7.m7.1.1.3" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.7.7.m7.1b"><apply id="S2.Thmtheorem7.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem7.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem7.p1.7.7.m7.1.1">superscript</csymbol><ci id="S2.Thmtheorem7.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.2">𝑓</ci><ci id="S2.Thmtheorem7.p1.7.7.m7.1.1.3.cmml" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.7.7.m7.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.7.7.m7.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.8.8.m8.1"><semantics id="S2.Thmtheorem7.p1.8.8.m8.1a"><mi id="S2.Thmtheorem7.p1.8.8.m8.1.1" xref="S2.Thmtheorem7.p1.8.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.8.8.m8.1b"><ci id="S2.Thmtheorem7.p1.8.8.m8.1.1.cmml" xref="S2.Thmtheorem7.p1.8.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.8.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.8.8.m8.1d">italic_f</annotation></semantics></math> are contracting with respect to the same fixed <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.9.9.m9.1"><semantics id="S2.Thmtheorem7.p1.9.9.m9.1a"><msub id="S2.Thmtheorem7.p1.9.9.m9.1.1" xref="S2.Thmtheorem7.p1.9.9.m9.1.1.cmml"><mi id="S2.Thmtheorem7.p1.9.9.m9.1.1.2" mathvariant="normal" xref="S2.Thmtheorem7.p1.9.9.m9.1.1.2.cmml">ℓ</mi><mi id="S2.Thmtheorem7.p1.9.9.m9.1.1.3" xref="S2.Thmtheorem7.p1.9.9.m9.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.9.9.m9.1b"><apply id="S2.Thmtheorem7.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem7.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.9.9.m9.1.1.1.cmml" xref="S2.Thmtheorem7.p1.9.9.m9.1.1">subscript</csymbol><ci id="S2.Thmtheorem7.p1.9.9.m9.1.1.2.cmml" xref="S2.Thmtheorem7.p1.9.9.m9.1.1.2">ℓ</ci><ci id="S2.Thmtheorem7.p1.9.9.m9.1.1.3.cmml" xref="S2.Thmtheorem7.p1.9.9.m9.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.9.9.m9.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.9.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-metric).</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.1.1.1">Definition 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem8.p1"> <p class="ltx_p" id="S2.Thmtheorem8.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.5.5">The <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.1.1.m1.1"><semantics id="S2.Thmtheorem8.p1.1.1.m1.1a"><msub id="S2.Thmtheorem8.p1.1.1.m1.1.1" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.1.1.m1.1b"><apply id="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.2">ℓ</ci><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S2.Thmtheorem8.p1.5.5.1">-GridContractionFixpoint</span> problem is to find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.2.2.m2.1"><semantics id="S2.Thmtheorem8.p1.2.2.m2.1a"><mi id="S2.Thmtheorem8.p1.2.2.m2.1.1" xref="S2.Thmtheorem8.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.2.2.m2.1b"><ci id="S2.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem8.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint <math alttext="x\in G^{d}_{b}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.3.3.m3.1"><semantics id="S2.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem8.p1.3.3.m3.1.1" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">x</mi><mo id="S2.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">∈</mo><msubsup id="S2.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.2" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.2.cmml">G</mi><mi id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.3" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml">b</mi><mi id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.3" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.3.3.m3.1b"><apply id="S2.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1"><in id="S2.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.2">𝑥</ci><apply id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.2.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.2">𝐺</ci><ci id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.3.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.3.3.m3.1c">x\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.3.3.m3.1d">italic_x ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> of a <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.4.4.m4.1"><semantics id="S2.Thmtheorem8.p1.4.4.m4.1a"><mi id="S2.Thmtheorem8.p1.4.4.m4.1.1" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.4.4.m4.1b"><ci id="S2.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem8.p1.4.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.4.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.4.4.m4.1d">italic_λ</annotation></semantics></math>-contraction grid-map <math alttext="f:G^{d}_{b}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.5.5.m5.2"><semantics id="S2.Thmtheorem8.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem8.p1.5.5.m5.2.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.cmml"><mi id="S2.Thmtheorem8.p1.5.5.m5.2.3.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.2.cmml">f</mi><mo id="S2.Thmtheorem8.p1.5.5.m5.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem8.p1.5.5.m5.2.3.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.cmml"><msubsup id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.cmml"><mi id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.2.cmml">G</mi><mi id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.3.cmml">b</mi><mi id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.1" stretchy="false" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.1.cmml">→</mo><msup id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.cmml"><mrow id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.1.cmml"><mo id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem8.p1.5.5.m5.1.1" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.cmml">0</mn><mo id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.2.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem8.p1.5.5.m5.2.2" xref="S2.Thmtheorem8.p1.5.5.m5.2.2.cmml">1</mn><mo id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.3" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.5.5.m5.2b"><apply id="S2.Thmtheorem8.p1.5.5.m5.2.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3"><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.1">:</ci><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.2">𝑓</ci><apply id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3"><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.1">→</ci><apply id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2">subscript</csymbol><apply id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2">superscript</csymbol><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.2">𝐺</ci><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.2.3">𝑏</ci></apply><apply id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.2.2"><cn id="S2.Thmtheorem8.p1.5.5.m5.1.1.cmml" type="integer" xref="S2.Thmtheorem8.p1.5.5.m5.1.1">0</cn><cn id="S2.Thmtheorem8.p1.5.5.m5.2.2.cmml" type="integer" xref="S2.Thmtheorem8.p1.5.5.m5.2.2">1</cn></interval><ci id="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.2.3.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.5.5.m5.2c">f:G^{d}_{b}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.5.5.m5.2d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.2">A priori, <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S2.p7.1.m1.1"><semantics id="S2.p7.1.m1.1a"><msub id="S2.p7.1.m1.1.1" xref="S2.p7.1.m1.1.1.cmml"><mi id="S2.p7.1.m1.1.1.2" mathvariant="normal" xref="S2.p7.1.m1.1.1.2.cmml">ℓ</mi><mi id="S2.p7.1.m1.1.1.3" xref="S2.p7.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p7.1.m1.1b"><apply id="S2.p7.1.m1.1.1.cmml" xref="S2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p7.1.m1.1.1.1.cmml" xref="S2.p7.1.m1.1.1">subscript</csymbol><ci id="S2.p7.1.m1.1.1.2.cmml" xref="S2.p7.1.m1.1.1.2">ℓ</ci><ci id="S2.p7.1.m1.1.1.3.cmml" xref="S2.p7.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p7.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S2.p7.2.1">-GridContractionFixpoint</span> is not guaranteed to have a solution and might therefore not be well-defined. However, it is not hard to see that an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.p7.2.m2.1"><semantics id="S2.p7.2.m2.1a"><mi id="S2.p7.2.m2.1.1" xref="S2.p7.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.p7.2.m2.1b"><ci id="S2.p7.2.m2.1.1.cmml" xref="S2.p7.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.p7.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint on the grid must exist if the input grid is fine enough.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.1.1.1">Lemma 2.9</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem9.p1"> <p class="ltx_p" id="S2.Thmtheorem9.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem9.p1.4.4">For <math alttext="b\geq\log_{2}(\frac{d+d\lambda}{2\varepsilon})" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.1.1.m1.2"><semantics id="S2.Thmtheorem9.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem9.p1.1.1.m1.2.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.cmml"><mi id="S2.Thmtheorem9.p1.1.1.m1.2.2.3" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.3.cmml">b</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.2.2.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.2.cmml">≥</mo><mrow id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml"><msub id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.2.cmml">log</mi><mn id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.3" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1a" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml"><mo id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.2.1" stretchy="false" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml">(</mo><mfrac id="S2.Thmtheorem9.p1.1.1.m1.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml"><mrow id="S2.Thmtheorem9.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.cmml"><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.2" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.2.cmml">d</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.1.cmml">+</mo><mrow id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.cmml"><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.2" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.2.cmml">d</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.3" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.3.cmml">λ</mi></mrow></mrow><mrow id="S2.Thmtheorem9.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.cmml"><mn id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.2" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.2.cmml">2</mn><mo id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.3.cmml">ε</mi></mrow></mfrac><mo id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.2.2" stretchy="false" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.1.1.m1.2b"><apply id="S2.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2"><geq id="S2.Thmtheorem9.p1.1.1.m1.2.2.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.2"></geq><ci id="S2.Thmtheorem9.p1.1.1.m1.2.2.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.3">𝑏</ci><apply id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1"><apply id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1">subscript</csymbol><log id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.2"></log><cn id="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.1.1.1.3">2</cn></apply><apply id="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1"><divide id="S2.Thmtheorem9.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1"></divide><apply id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2"><plus id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.1"></plus><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.2">𝑑</ci><apply id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3"><times id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.1"></times><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.2">𝑑</ci><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.2.3.3">𝜆</ci></apply></apply><apply id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3"><times id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.1"></times><cn id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.2">2</cn><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.3.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.1.1.m1.2c">b\geq\log_{2}(\frac{d+d\lambda}{2\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.1.1.m1.2d">italic_b ≥ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_d + italic_d italic_λ end_ARG start_ARG 2 italic_ε end_ARG )</annotation></semantics></math>, any <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.2.2.m2.1"><semantics id="S2.Thmtheorem9.p1.2.2.m2.1a"><mi id="S2.Thmtheorem9.p1.2.2.m2.1.1" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.2.2.m2.1b"><ci id="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.2.2.m2.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.2.2.m2.1d">italic_λ</annotation></semantics></math>-contraction grid-map <math alttext="f:G^{d}_{b}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.3.3.m3.2"><semantics id="S2.Thmtheorem9.p1.3.3.m3.2a"><mrow id="S2.Thmtheorem9.p1.3.3.m3.2.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.cmml"><mi id="S2.Thmtheorem9.p1.3.3.m3.2.3.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.2.cmml">f</mi><mo id="S2.Thmtheorem9.p1.3.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem9.p1.3.3.m3.2.3.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.cmml"><msubsup id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.cmml"><mi id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.2.cmml">G</mi><mi id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.3.cmml">b</mi><mi id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.1" stretchy="false" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml">→</mo><msup id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.cmml"><mrow id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.1.cmml"><mo id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.2.1" stretchy="false" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.1.cmml">[</mo><mn id="S2.Thmtheorem9.p1.3.3.m3.1.1" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml">0</mn><mo id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.2.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.1.cmml">,</mo><mn id="S2.Thmtheorem9.p1.3.3.m3.2.2" xref="S2.Thmtheorem9.p1.3.3.m3.2.2.cmml">1</mn><mo id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.2.3" stretchy="false" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.1.cmml">]</mo></mrow><mi id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.3" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.3.3.m3.2b"><apply id="S2.Thmtheorem9.p1.3.3.m3.2.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3"><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.1">:</ci><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.2">𝑓</ci><apply id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3"><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.1">→</ci><apply id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2">subscript</csymbol><apply id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2">superscript</csymbol><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.2">𝐺</ci><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.2.3">𝑑</ci></apply><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.2.3">𝑏</ci></apply><apply id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3">superscript</csymbol><interval closure="closed" id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.2.2"><cn id="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml" type="integer" xref="S2.Thmtheorem9.p1.3.3.m3.1.1">0</cn><cn id="S2.Thmtheorem9.p1.3.3.m3.2.2.cmml" type="integer" xref="S2.Thmtheorem9.p1.3.3.m3.2.2">1</cn></interval><ci id="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.2.3.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.3.3.m3.2c">f:G^{d}_{b}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.3.3.m3.2d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> admits an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.4.4.m4.1"><semantics id="S2.Thmtheorem9.p1.4.4.m4.1a"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.4.4.m4.1b"><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.4.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.4.4.m4.1d">italic_ε</annotation></semantics></math>-approximate fixpoint.</span></p> </div> </div> <div class="ltx_proof" id="S2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.1.p1"> <p class="ltx_p" id="S2.1.p1.7">Let <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S2.1.p1.1.m1.1"><semantics id="S2.1.p1.1.m1.1a"><msup id="S2.1.p1.1.m1.1.1" xref="S2.1.p1.1.m1.1.1.cmml"><mi id="S2.1.p1.1.m1.1.1.2" xref="S2.1.p1.1.m1.1.1.2.cmml">x</mi><mo id="S2.1.p1.1.m1.1.1.3" xref="S2.1.p1.1.m1.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.1.p1.1.m1.1b"><apply id="S2.1.p1.1.m1.1.1.cmml" xref="S2.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.1.m1.1.1.1.cmml" xref="S2.1.p1.1.m1.1.1">superscript</csymbol><ci id="S2.1.p1.1.m1.1.1.2.cmml" xref="S2.1.p1.1.m1.1.1.2">𝑥</ci><ci id="S2.1.p1.1.m1.1.1.3.cmml" xref="S2.1.p1.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.1.m1.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.1.m1.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> be the unique fixpoint of some <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.1.p1.2.m2.1"><semantics id="S2.1.p1.2.m2.1a"><mi id="S2.1.p1.2.m2.1.1" xref="S2.1.p1.2.m2.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.2.m2.1b"><ci id="S2.1.p1.2.m2.1.1.cmml" xref="S2.1.p1.2.m2.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.2.m2.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.2.m2.1d">italic_λ</annotation></semantics></math>-contraction map <math alttext="f^{\prime}:[0,1]^{d}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S2.1.p1.3.m3.4"><semantics id="S2.1.p1.3.m3.4a"><mrow id="S2.1.p1.3.m3.4.5" xref="S2.1.p1.3.m3.4.5.cmml"><msup id="S2.1.p1.3.m3.4.5.2" xref="S2.1.p1.3.m3.4.5.2.cmml"><mi id="S2.1.p1.3.m3.4.5.2.2" xref="S2.1.p1.3.m3.4.5.2.2.cmml">f</mi><mo id="S2.1.p1.3.m3.4.5.2.3" xref="S2.1.p1.3.m3.4.5.2.3.cmml">′</mo></msup><mo id="S2.1.p1.3.m3.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.1.p1.3.m3.4.5.1.cmml">:</mo><mrow id="S2.1.p1.3.m3.4.5.3" xref="S2.1.p1.3.m3.4.5.3.cmml"><msup id="S2.1.p1.3.m3.4.5.3.2" xref="S2.1.p1.3.m3.4.5.3.2.cmml"><mrow id="S2.1.p1.3.m3.4.5.3.2.2.2" xref="S2.1.p1.3.m3.4.5.3.2.2.1.cmml"><mo id="S2.1.p1.3.m3.4.5.3.2.2.2.1" stretchy="false" xref="S2.1.p1.3.m3.4.5.3.2.2.1.cmml">[</mo><mn id="S2.1.p1.3.m3.1.1" xref="S2.1.p1.3.m3.1.1.cmml">0</mn><mo id="S2.1.p1.3.m3.4.5.3.2.2.2.2" xref="S2.1.p1.3.m3.4.5.3.2.2.1.cmml">,</mo><mn id="S2.1.p1.3.m3.2.2" xref="S2.1.p1.3.m3.2.2.cmml">1</mn><mo id="S2.1.p1.3.m3.4.5.3.2.2.2.3" stretchy="false" xref="S2.1.p1.3.m3.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="S2.1.p1.3.m3.4.5.3.2.3" xref="S2.1.p1.3.m3.4.5.3.2.3.cmml">d</mi></msup><mo id="S2.1.p1.3.m3.4.5.3.1" stretchy="false" xref="S2.1.p1.3.m3.4.5.3.1.cmml">→</mo><msup id="S2.1.p1.3.m3.4.5.3.3" xref="S2.1.p1.3.m3.4.5.3.3.cmml"><mrow id="S2.1.p1.3.m3.4.5.3.3.2.2" xref="S2.1.p1.3.m3.4.5.3.3.2.1.cmml"><mo id="S2.1.p1.3.m3.4.5.3.3.2.2.1" stretchy="false" xref="S2.1.p1.3.m3.4.5.3.3.2.1.cmml">[</mo><mn id="S2.1.p1.3.m3.3.3" xref="S2.1.p1.3.m3.3.3.cmml">0</mn><mo id="S2.1.p1.3.m3.4.5.3.3.2.2.2" xref="S2.1.p1.3.m3.4.5.3.3.2.1.cmml">,</mo><mn id="S2.1.p1.3.m3.4.4" xref="S2.1.p1.3.m3.4.4.cmml">1</mn><mo id="S2.1.p1.3.m3.4.5.3.3.2.2.3" stretchy="false" xref="S2.1.p1.3.m3.4.5.3.3.2.1.cmml">]</mo></mrow><mi id="S2.1.p1.3.m3.4.5.3.3.3" xref="S2.1.p1.3.m3.4.5.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.3.m3.4b"><apply id="S2.1.p1.3.m3.4.5.cmml" xref="S2.1.p1.3.m3.4.5"><ci id="S2.1.p1.3.m3.4.5.1.cmml" xref="S2.1.p1.3.m3.4.5.1">:</ci><apply id="S2.1.p1.3.m3.4.5.2.cmml" xref="S2.1.p1.3.m3.4.5.2"><csymbol cd="ambiguous" id="S2.1.p1.3.m3.4.5.2.1.cmml" xref="S2.1.p1.3.m3.4.5.2">superscript</csymbol><ci id="S2.1.p1.3.m3.4.5.2.2.cmml" xref="S2.1.p1.3.m3.4.5.2.2">𝑓</ci><ci id="S2.1.p1.3.m3.4.5.2.3.cmml" xref="S2.1.p1.3.m3.4.5.2.3">′</ci></apply><apply id="S2.1.p1.3.m3.4.5.3.cmml" xref="S2.1.p1.3.m3.4.5.3"><ci id="S2.1.p1.3.m3.4.5.3.1.cmml" xref="S2.1.p1.3.m3.4.5.3.1">→</ci><apply id="S2.1.p1.3.m3.4.5.3.2.cmml" xref="S2.1.p1.3.m3.4.5.3.2"><csymbol cd="ambiguous" id="S2.1.p1.3.m3.4.5.3.2.1.cmml" xref="S2.1.p1.3.m3.4.5.3.2">superscript</csymbol><interval closure="closed" id="S2.1.p1.3.m3.4.5.3.2.2.1.cmml" xref="S2.1.p1.3.m3.4.5.3.2.2.2"><cn id="S2.1.p1.3.m3.1.1.cmml" type="integer" xref="S2.1.p1.3.m3.1.1">0</cn><cn id="S2.1.p1.3.m3.2.2.cmml" type="integer" xref="S2.1.p1.3.m3.2.2">1</cn></interval><ci id="S2.1.p1.3.m3.4.5.3.2.3.cmml" xref="S2.1.p1.3.m3.4.5.3.2.3">𝑑</ci></apply><apply id="S2.1.p1.3.m3.4.5.3.3.cmml" xref="S2.1.p1.3.m3.4.5.3.3"><csymbol cd="ambiguous" id="S2.1.p1.3.m3.4.5.3.3.1.cmml" xref="S2.1.p1.3.m3.4.5.3.3">superscript</csymbol><interval closure="closed" id="S2.1.p1.3.m3.4.5.3.3.2.1.cmml" xref="S2.1.p1.3.m3.4.5.3.3.2.2"><cn id="S2.1.p1.3.m3.3.3.cmml" type="integer" xref="S2.1.p1.3.m3.3.3">0</cn><cn id="S2.1.p1.3.m3.4.4.cmml" type="integer" xref="S2.1.p1.3.m3.4.4">1</cn></interval><ci id="S2.1.p1.3.m3.4.5.3.3.3.cmml" xref="S2.1.p1.3.m3.4.5.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.3.m3.4c">f^{\prime}:[0,1]^{d}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.3.m3.4d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> extending <math alttext="f" class="ltx_Math" display="inline" id="S2.1.p1.4.m4.1"><semantics id="S2.1.p1.4.m4.1a"><mi id="S2.1.p1.4.m4.1.1" xref="S2.1.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.4.m4.1b"><ci id="S2.1.p1.4.m4.1.1.cmml" xref="S2.1.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.4.m4.1d">italic_f</annotation></semantics></math>. The following calculation shows that any point <math alttext="x\in[0,1]^{d}" class="ltx_Math" display="inline" id="S2.1.p1.5.m5.2"><semantics id="S2.1.p1.5.m5.2a"><mrow id="S2.1.p1.5.m5.2.3" xref="S2.1.p1.5.m5.2.3.cmml"><mi id="S2.1.p1.5.m5.2.3.2" xref="S2.1.p1.5.m5.2.3.2.cmml">x</mi><mo id="S2.1.p1.5.m5.2.3.1" xref="S2.1.p1.5.m5.2.3.1.cmml">∈</mo><msup id="S2.1.p1.5.m5.2.3.3" xref="S2.1.p1.5.m5.2.3.3.cmml"><mrow id="S2.1.p1.5.m5.2.3.3.2.2" xref="S2.1.p1.5.m5.2.3.3.2.1.cmml"><mo id="S2.1.p1.5.m5.2.3.3.2.2.1" stretchy="false" xref="S2.1.p1.5.m5.2.3.3.2.1.cmml">[</mo><mn id="S2.1.p1.5.m5.1.1" xref="S2.1.p1.5.m5.1.1.cmml">0</mn><mo id="S2.1.p1.5.m5.2.3.3.2.2.2" xref="S2.1.p1.5.m5.2.3.3.2.1.cmml">,</mo><mn id="S2.1.p1.5.m5.2.2" xref="S2.1.p1.5.m5.2.2.cmml">1</mn><mo id="S2.1.p1.5.m5.2.3.3.2.2.3" stretchy="false" xref="S2.1.p1.5.m5.2.3.3.2.1.cmml">]</mo></mrow><mi id="S2.1.p1.5.m5.2.3.3.3" xref="S2.1.p1.5.m5.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.5.m5.2b"><apply id="S2.1.p1.5.m5.2.3.cmml" xref="S2.1.p1.5.m5.2.3"><in id="S2.1.p1.5.m5.2.3.1.cmml" xref="S2.1.p1.5.m5.2.3.1"></in><ci id="S2.1.p1.5.m5.2.3.2.cmml" xref="S2.1.p1.5.m5.2.3.2">𝑥</ci><apply id="S2.1.p1.5.m5.2.3.3.cmml" xref="S2.1.p1.5.m5.2.3.3"><csymbol cd="ambiguous" id="S2.1.p1.5.m5.2.3.3.1.cmml" xref="S2.1.p1.5.m5.2.3.3">superscript</csymbol><interval closure="closed" id="S2.1.p1.5.m5.2.3.3.2.1.cmml" xref="S2.1.p1.5.m5.2.3.3.2.2"><cn id="S2.1.p1.5.m5.1.1.cmml" type="integer" xref="S2.1.p1.5.m5.1.1">0</cn><cn id="S2.1.p1.5.m5.2.2.cmml" type="integer" xref="S2.1.p1.5.m5.2.2">1</cn></interval><ci id="S2.1.p1.5.m5.2.3.3.3.cmml" xref="S2.1.p1.5.m5.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.5.m5.2c">x\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.5.m5.2d">italic_x ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||x-x^{\star}||_{p}\leq\frac{\varepsilon}{1+\lambda}" class="ltx_Math" display="inline" id="S2.1.p1.6.m6.1"><semantics id="S2.1.p1.6.m6.1a"><mrow id="S2.1.p1.6.m6.1.1" xref="S2.1.p1.6.m6.1.1.cmml"><msub id="S2.1.p1.6.m6.1.1.1" xref="S2.1.p1.6.m6.1.1.1.cmml"><mrow id="S2.1.p1.6.m6.1.1.1.1.1" xref="S2.1.p1.6.m6.1.1.1.1.2.cmml"><mo id="S2.1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S2.1.p1.6.m6.1.1.1.1.2.1.cmml">‖</mo><mrow id="S2.1.p1.6.m6.1.1.1.1.1.1" xref="S2.1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S2.1.p1.6.m6.1.1.1.1.1.1.2" xref="S2.1.p1.6.m6.1.1.1.1.1.1.2.cmml">x</mi><mo id="S2.1.p1.6.m6.1.1.1.1.1.1.1" xref="S2.1.p1.6.m6.1.1.1.1.1.1.1.cmml">−</mo><msup id="S2.1.p1.6.m6.1.1.1.1.1.1.3" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3.cmml"><mi id="S2.1.p1.6.m6.1.1.1.1.1.1.3.2" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3.2.cmml">x</mi><mo id="S2.1.p1.6.m6.1.1.1.1.1.1.3.3" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3.3.cmml">⋆</mo></msup></mrow><mo id="S2.1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S2.1.p1.6.m6.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S2.1.p1.6.m6.1.1.1.3" xref="S2.1.p1.6.m6.1.1.1.3.cmml">p</mi></msub><mo id="S2.1.p1.6.m6.1.1.2" xref="S2.1.p1.6.m6.1.1.2.cmml">≤</mo><mfrac id="S2.1.p1.6.m6.1.1.3" xref="S2.1.p1.6.m6.1.1.3.cmml"><mi id="S2.1.p1.6.m6.1.1.3.2" xref="S2.1.p1.6.m6.1.1.3.2.cmml">ε</mi><mrow id="S2.1.p1.6.m6.1.1.3.3" xref="S2.1.p1.6.m6.1.1.3.3.cmml"><mn id="S2.1.p1.6.m6.1.1.3.3.2" xref="S2.1.p1.6.m6.1.1.3.3.2.cmml">1</mn><mo id="S2.1.p1.6.m6.1.1.3.3.1" xref="S2.1.p1.6.m6.1.1.3.3.1.cmml">+</mo><mi id="S2.1.p1.6.m6.1.1.3.3.3" xref="S2.1.p1.6.m6.1.1.3.3.3.cmml">λ</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.6.m6.1b"><apply id="S2.1.p1.6.m6.1.1.cmml" xref="S2.1.p1.6.m6.1.1"><leq id="S2.1.p1.6.m6.1.1.2.cmml" xref="S2.1.p1.6.m6.1.1.2"></leq><apply id="S2.1.p1.6.m6.1.1.1.cmml" xref="S2.1.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.6.m6.1.1.1.2.cmml" xref="S2.1.p1.6.m6.1.1.1">subscript</csymbol><apply id="S2.1.p1.6.m6.1.1.1.1.2.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1"><csymbol cd="latexml" id="S2.1.p1.6.m6.1.1.1.1.2.1.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.2">norm</csymbol><apply id="S2.1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1"><minus id="S2.1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.1"></minus><ci id="S2.1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.2">𝑥</ci><apply id="S2.1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.1.p1.6.m6.1.1.1.1.1.1.3.1.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.1.p1.6.m6.1.1.1.1.1.1.3.2.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3.2">𝑥</ci><ci id="S2.1.p1.6.m6.1.1.1.1.1.1.3.3.cmml" xref="S2.1.p1.6.m6.1.1.1.1.1.1.3.3">⋆</ci></apply></apply></apply><ci id="S2.1.p1.6.m6.1.1.1.3.cmml" xref="S2.1.p1.6.m6.1.1.1.3">𝑝</ci></apply><apply id="S2.1.p1.6.m6.1.1.3.cmml" xref="S2.1.p1.6.m6.1.1.3"><divide id="S2.1.p1.6.m6.1.1.3.1.cmml" xref="S2.1.p1.6.m6.1.1.3"></divide><ci id="S2.1.p1.6.m6.1.1.3.2.cmml" xref="S2.1.p1.6.m6.1.1.3.2">𝜀</ci><apply id="S2.1.p1.6.m6.1.1.3.3.cmml" xref="S2.1.p1.6.m6.1.1.3.3"><plus id="S2.1.p1.6.m6.1.1.3.3.1.cmml" xref="S2.1.p1.6.m6.1.1.3.3.1"></plus><cn id="S2.1.p1.6.m6.1.1.3.3.2.cmml" type="integer" xref="S2.1.p1.6.m6.1.1.3.3.2">1</cn><ci id="S2.1.p1.6.m6.1.1.3.3.3.cmml" xref="S2.1.p1.6.m6.1.1.3.3.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.6.m6.1c">||x-x^{\star}||_{p}\leq\frac{\varepsilon}{1+\lambda}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.6.m6.1d">| | italic_x - italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ divide start_ARG italic_ε end_ARG start_ARG 1 + italic_λ end_ARG</annotation></semantics></math> must be an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.1.p1.7.m7.1"><semantics id="S2.1.p1.7.m7.1a"><mi id="S2.1.p1.7.m7.1.1" xref="S2.1.p1.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.7.m7.1b"><ci id="S2.1.p1.7.m7.1.1.cmml" xref="S2.1.p1.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.7.m7.1d">italic_ε</annotation></semantics></math>-approximate fixpoint:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx1"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||f(x)-x||_{p}" class="ltx_Math" display="inline" id="S2.Ex1.m1.2"><semantics id="S2.Ex1.m1.2a"><msub id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml"><mrow id="S2.Ex1.m1.2.2.1.1" xref="S2.Ex1.m1.2.2.1.2.cmml"><mo id="S2.Ex1.m1.2.2.1.1.2" stretchy="false" xref="S2.Ex1.m1.2.2.1.2.1.cmml">‖</mo><mrow id="S2.Ex1.m1.2.2.1.1.1" xref="S2.Ex1.m1.2.2.1.1.1.cmml"><mrow id="S2.Ex1.m1.2.2.1.1.1.2" xref="S2.Ex1.m1.2.2.1.1.1.2.cmml"><mi id="S2.Ex1.m1.2.2.1.1.1.2.2" xref="S2.Ex1.m1.2.2.1.1.1.2.2.cmml">f</mi><mo id="S2.Ex1.m1.2.2.1.1.1.2.1" xref="S2.Ex1.m1.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.2.2.1.1.1.2.3.2" xref="S2.Ex1.m1.2.2.1.1.1.2.cmml"><mo id="S2.Ex1.m1.2.2.1.1.1.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.1.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">x</mi><mo id="S2.Ex1.m1.2.2.1.1.1.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.2.2.1.1.1.1" xref="S2.Ex1.m1.2.2.1.1.1.1.cmml">−</mo><mi id="S2.Ex1.m1.2.2.1.1.1.3" xref="S2.Ex1.m1.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S2.Ex1.m1.2.2.1.1.3" stretchy="false" xref="S2.Ex1.m1.2.2.1.2.1.cmml">‖</mo></mrow><mi id="S2.Ex1.m1.2.2.3" xref="S2.Ex1.m1.2.2.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.2b"><apply id="S2.Ex1.m1.2.2.cmml" xref="S2.Ex1.m1.2.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.2.2.2.cmml" xref="S2.Ex1.m1.2.2">subscript</csymbol><apply id="S2.Ex1.m1.2.2.1.2.cmml" xref="S2.Ex1.m1.2.2.1.1"><csymbol cd="latexml" id="S2.Ex1.m1.2.2.1.2.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2">norm</csymbol><apply id="S2.Ex1.m1.2.2.1.1.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1"><minus id="S2.Ex1.m1.2.2.1.1.1.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.1"></minus><apply id="S2.Ex1.m1.2.2.1.1.1.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2"><times id="S2.Ex1.m1.2.2.1.1.1.2.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.1"></times><ci id="S2.Ex1.m1.2.2.1.1.1.2.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.2.2">𝑓</ci><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">𝑥</ci></apply><ci id="S2.Ex1.m1.2.2.1.1.1.3.cmml" xref="S2.Ex1.m1.2.2.1.1.1.3">𝑥</ci></apply></apply><ci id="S2.Ex1.m1.2.2.3.cmml" xref="S2.Ex1.m1.2.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.2c">\displaystyle||f(x)-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.2d">| | italic_f ( italic_x ) - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq||f(x)-f(x^{\star})||_{p}+||f(x^{\star})-x||_{p}" class="ltx_Math" display="inline" id="S2.Ex1.m2.3"><semantics id="S2.Ex1.m2.3a"><mrow id="S2.Ex1.m2.3.3" xref="S2.Ex1.m2.3.3.cmml"><mi id="S2.Ex1.m2.3.3.4" xref="S2.Ex1.m2.3.3.4.cmml"></mi><mo id="S2.Ex1.m2.3.3.3" xref="S2.Ex1.m2.3.3.3.cmml">≤</mo><mrow id="S2.Ex1.m2.3.3.2" xref="S2.Ex1.m2.3.3.2.cmml"><msub id="S2.Ex1.m2.2.2.1.1" xref="S2.Ex1.m2.2.2.1.1.cmml"><mrow id="S2.Ex1.m2.2.2.1.1.1.1" xref="S2.Ex1.m2.2.2.1.1.1.2.cmml"><mo id="S2.Ex1.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m2.2.2.1.1.1.2.1.cmml">‖</mo><mrow id="S2.Ex1.m2.2.2.1.1.1.1.1" xref="S2.Ex1.m2.2.2.1.1.1.1.1.cmml"><mrow id="S2.Ex1.m2.2.2.1.1.1.1.1.3" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.cmml"><mi id="S2.Ex1.m2.2.2.1.1.1.1.1.3.2" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.2.cmml">f</mi><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.3.1" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex1.m2.2.2.1.1.1.1.1.3.3.2" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.cmml"><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.cmml">(</mo><mi id="S2.Ex1.m2.1.1" xref="S2.Ex1.m2.1.1.cmml">x</mi><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.Ex1.m2.2.2.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.2" xref="S2.Ex1.m2.2.2.1.1.1.1.1.2.cmml">−</mo><mrow id="S2.Ex1.m2.2.2.1.1.1.1.1.1" xref="S2.Ex1.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m2.2.2.1.1.1.1.1.1.3" xref="S2.Ex1.m2.2.2.1.1.1.1.1.1.3.cmml">f</mi><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.1.2" xref="S2.Ex1.m2.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m2.2.2.1.1.1.1.1.1.1.1" xref="S2.Ex1.m2.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex1.m2.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" 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href="https://arxiv.org/html/2503.16089v1#S2.Ex3.m1.1.1.1.1.2.cmml" id="S2.Ex3.m1.1.1.1.1d.cmml" xref="S2.Ex3.m1.1.1.1"></share><ci id="S2.Ex3.m1.1.1.1.1.7.cmml" xref="S2.Ex3.m1.1.1.1.1.7">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.1c">\displaystyle=(1+\lambda)||x-x^{\star}||_{p}\leq\varepsilon.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.1d">= ( 1 + italic_λ ) | | italic_x - italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.2.p2"> <p class="ltx_p" id="S2.2.p2.11">We now determine a lower bound on <math alttext="b" class="ltx_Math" display="inline" id="S2.2.p2.1.m1.1"><semantics id="S2.2.p2.1.m1.1a"><mi id="S2.2.p2.1.m1.1.1" xref="S2.2.p2.1.m1.1.1.cmml">b</mi><annotation-xml 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id="S2.2.p2.2.m2.1b"><apply id="S2.2.p2.2.m2.1.1.cmml" xref="S2.2.p2.2.m2.1.1"><in id="S2.2.p2.2.m2.1.1.1.cmml" xref="S2.2.p2.2.m2.1.1.1"></in><ci id="S2.2.p2.2.m2.1.1.2.cmml" xref="S2.2.p2.2.m2.1.1.2">𝑥</ci><apply id="S2.2.p2.2.m2.1.1.3.cmml" xref="S2.2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.2.m2.1.1.3.1.cmml" xref="S2.2.p2.2.m2.1.1.3">subscript</csymbol><apply id="S2.2.p2.2.m2.1.1.3.2.cmml" xref="S2.2.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.2.m2.1.1.3.2.1.cmml" xref="S2.2.p2.2.m2.1.1.3">superscript</csymbol><ci id="S2.2.p2.2.m2.1.1.3.2.2.cmml" xref="S2.2.p2.2.m2.1.1.3.2.2">𝐺</ci><ci id="S2.2.p2.2.m2.1.1.3.2.3.cmml" xref="S2.2.p2.2.m2.1.1.3.2.3">𝑑</ci></apply><ci id="S2.2.p2.2.m2.1.1.3.3.cmml" xref="S2.2.p2.2.m2.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.2.m2.1c">x\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.2.m2.1d">italic_x ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Observe that since <math alttext="B^{\infty}(x^{\star},\frac{r}{d})\subseteq B^{p}(x^{\star},r)" class="ltx_Math" display="inline" id="S2.2.p2.3.m3.4"><semantics id="S2.2.p2.3.m3.4a"><mrow id="S2.2.p2.3.m3.4.4" xref="S2.2.p2.3.m3.4.4.cmml"><mrow id="S2.2.p2.3.m3.3.3.1" xref="S2.2.p2.3.m3.3.3.1.cmml"><msup id="S2.2.p2.3.m3.3.3.1.3" xref="S2.2.p2.3.m3.3.3.1.3.cmml"><mi id="S2.2.p2.3.m3.3.3.1.3.2" xref="S2.2.p2.3.m3.3.3.1.3.2.cmml">B</mi><mi id="S2.2.p2.3.m3.3.3.1.3.3" mathvariant="normal" xref="S2.2.p2.3.m3.3.3.1.3.3.cmml">∞</mi></msup><mo id="S2.2.p2.3.m3.3.3.1.2" xref="S2.2.p2.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S2.2.p2.3.m3.3.3.1.1.1" xref="S2.2.p2.3.m3.3.3.1.1.2.cmml"><mo id="S2.2.p2.3.m3.3.3.1.1.1.2" stretchy="false" xref="S2.2.p2.3.m3.3.3.1.1.2.cmml">(</mo><msup id="S2.2.p2.3.m3.3.3.1.1.1.1" xref="S2.2.p2.3.m3.3.3.1.1.1.1.cmml"><mi id="S2.2.p2.3.m3.3.3.1.1.1.1.2" xref="S2.2.p2.3.m3.3.3.1.1.1.1.2.cmml">x</mi><mo id="S2.2.p2.3.m3.3.3.1.1.1.1.3" xref="S2.2.p2.3.m3.3.3.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.2.p2.3.m3.3.3.1.1.1.3" xref="S2.2.p2.3.m3.3.3.1.1.2.cmml">,</mo><mfrac id="S2.2.p2.3.m3.1.1" xref="S2.2.p2.3.m3.1.1.cmml"><mi id="S2.2.p2.3.m3.1.1.2" xref="S2.2.p2.3.m3.1.1.2.cmml">r</mi><mi id="S2.2.p2.3.m3.1.1.3" xref="S2.2.p2.3.m3.1.1.3.cmml">d</mi></mfrac><mo id="S2.2.p2.3.m3.3.3.1.1.1.4" stretchy="false" xref="S2.2.p2.3.m3.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.2.p2.3.m3.4.4.3" xref="S2.2.p2.3.m3.4.4.3.cmml">⊆</mo><mrow id="S2.2.p2.3.m3.4.4.2" xref="S2.2.p2.3.m3.4.4.2.cmml"><msup id="S2.2.p2.3.m3.4.4.2.3" xref="S2.2.p2.3.m3.4.4.2.3.cmml"><mi id="S2.2.p2.3.m3.4.4.2.3.2" xref="S2.2.p2.3.m3.4.4.2.3.2.cmml">B</mi><mi id="S2.2.p2.3.m3.4.4.2.3.3" xref="S2.2.p2.3.m3.4.4.2.3.3.cmml">p</mi></msup><mo id="S2.2.p2.3.m3.4.4.2.2" xref="S2.2.p2.3.m3.4.4.2.2.cmml">⁢</mo><mrow id="S2.2.p2.3.m3.4.4.2.1.1" xref="S2.2.p2.3.m3.4.4.2.1.2.cmml"><mo id="S2.2.p2.3.m3.4.4.2.1.1.2" stretchy="false" xref="S2.2.p2.3.m3.4.4.2.1.2.cmml">(</mo><msup id="S2.2.p2.3.m3.4.4.2.1.1.1" xref="S2.2.p2.3.m3.4.4.2.1.1.1.cmml"><mi id="S2.2.p2.3.m3.4.4.2.1.1.1.2" xref="S2.2.p2.3.m3.4.4.2.1.1.1.2.cmml">x</mi><mo id="S2.2.p2.3.m3.4.4.2.1.1.1.3" xref="S2.2.p2.3.m3.4.4.2.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.2.p2.3.m3.4.4.2.1.1.3" xref="S2.2.p2.3.m3.4.4.2.1.2.cmml">,</mo><mi id="S2.2.p2.3.m3.2.2" xref="S2.2.p2.3.m3.2.2.cmml">r</mi><mo id="S2.2.p2.3.m3.4.4.2.1.1.4" stretchy="false" xref="S2.2.p2.3.m3.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.3.m3.4b"><apply id="S2.2.p2.3.m3.4.4.cmml" xref="S2.2.p2.3.m3.4.4"><subset id="S2.2.p2.3.m3.4.4.3.cmml" xref="S2.2.p2.3.m3.4.4.3"></subset><apply id="S2.2.p2.3.m3.3.3.1.cmml" xref="S2.2.p2.3.m3.3.3.1"><times id="S2.2.p2.3.m3.3.3.1.2.cmml" xref="S2.2.p2.3.m3.3.3.1.2"></times><apply id="S2.2.p2.3.m3.3.3.1.3.cmml" xref="S2.2.p2.3.m3.3.3.1.3"><csymbol cd="ambiguous" id="S2.2.p2.3.m3.3.3.1.3.1.cmml" xref="S2.2.p2.3.m3.3.3.1.3">superscript</csymbol><ci id="S2.2.p2.3.m3.3.3.1.3.2.cmml" xref="S2.2.p2.3.m3.3.3.1.3.2">𝐵</ci><infinity id="S2.2.p2.3.m3.3.3.1.3.3.cmml" xref="S2.2.p2.3.m3.3.3.1.3.3"></infinity></apply><interval closure="open" id="S2.2.p2.3.m3.3.3.1.1.2.cmml" xref="S2.2.p2.3.m3.3.3.1.1.1"><apply id="S2.2.p2.3.m3.3.3.1.1.1.1.cmml" xref="S2.2.p2.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.3.m3.3.3.1.1.1.1.1.cmml" xref="S2.2.p2.3.m3.3.3.1.1.1.1">superscript</csymbol><ci id="S2.2.p2.3.m3.3.3.1.1.1.1.2.cmml" xref="S2.2.p2.3.m3.3.3.1.1.1.1.2">𝑥</ci><ci id="S2.2.p2.3.m3.3.3.1.1.1.1.3.cmml" xref="S2.2.p2.3.m3.3.3.1.1.1.1.3">⋆</ci></apply><apply id="S2.2.p2.3.m3.1.1.cmml" xref="S2.2.p2.3.m3.1.1"><divide id="S2.2.p2.3.m3.1.1.1.cmml" xref="S2.2.p2.3.m3.1.1"></divide><ci id="S2.2.p2.3.m3.1.1.2.cmml" xref="S2.2.p2.3.m3.1.1.2">𝑟</ci><ci id="S2.2.p2.3.m3.1.1.3.cmml" xref="S2.2.p2.3.m3.1.1.3">𝑑</ci></apply></interval></apply><apply id="S2.2.p2.3.m3.4.4.2.cmml" xref="S2.2.p2.3.m3.4.4.2"><times id="S2.2.p2.3.m3.4.4.2.2.cmml" xref="S2.2.p2.3.m3.4.4.2.2"></times><apply id="S2.2.p2.3.m3.4.4.2.3.cmml" xref="S2.2.p2.3.m3.4.4.2.3"><csymbol cd="ambiguous" id="S2.2.p2.3.m3.4.4.2.3.1.cmml" xref="S2.2.p2.3.m3.4.4.2.3">superscript</csymbol><ci id="S2.2.p2.3.m3.4.4.2.3.2.cmml" xref="S2.2.p2.3.m3.4.4.2.3.2">𝐵</ci><ci id="S2.2.p2.3.m3.4.4.2.3.3.cmml" xref="S2.2.p2.3.m3.4.4.2.3.3">𝑝</ci></apply><interval closure="open" id="S2.2.p2.3.m3.4.4.2.1.2.cmml" xref="S2.2.p2.3.m3.4.4.2.1.1"><apply id="S2.2.p2.3.m3.4.4.2.1.1.1.cmml" xref="S2.2.p2.3.m3.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.3.m3.4.4.2.1.1.1.1.cmml" xref="S2.2.p2.3.m3.4.4.2.1.1.1">superscript</csymbol><ci id="S2.2.p2.3.m3.4.4.2.1.1.1.2.cmml" xref="S2.2.p2.3.m3.4.4.2.1.1.1.2">𝑥</ci><ci id="S2.2.p2.3.m3.4.4.2.1.1.1.3.cmml" xref="S2.2.p2.3.m3.4.4.2.1.1.1.3">⋆</ci></apply><ci id="S2.2.p2.3.m3.2.2.cmml" xref="S2.2.p2.3.m3.2.2">𝑟</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.3.m3.4c">B^{\infty}(x^{\star},\frac{r}{d})\subseteq B^{p}(x^{\star},r)</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.3.m3.4d">italic_B start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , divide start_ARG italic_r end_ARG start_ARG italic_d end_ARG ) ⊆ italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_r )</annotation></semantics></math> (follows from <math alttext="||x||_{p}\leq d||x||_{\infty}" class="ltx_Math" display="inline" id="S2.2.p2.4.m4.2"><semantics id="S2.2.p2.4.m4.2a"><mrow id="S2.2.p2.4.m4.2.3" xref="S2.2.p2.4.m4.2.3.cmml"><msub id="S2.2.p2.4.m4.2.3.2" xref="S2.2.p2.4.m4.2.3.2.cmml"><mrow id="S2.2.p2.4.m4.2.3.2.2.2" xref="S2.2.p2.4.m4.2.3.2.2.1.cmml"><mo id="S2.2.p2.4.m4.2.3.2.2.2.1" stretchy="false" xref="S2.2.p2.4.m4.2.3.2.2.1.1.cmml">‖</mo><mi id="S2.2.p2.4.m4.1.1" xref="S2.2.p2.4.m4.1.1.cmml">x</mi><mo id="S2.2.p2.4.m4.2.3.2.2.2.2" stretchy="false" xref="S2.2.p2.4.m4.2.3.2.2.1.1.cmml">‖</mo></mrow><mi id="S2.2.p2.4.m4.2.3.2.3" xref="S2.2.p2.4.m4.2.3.2.3.cmml">p</mi></msub><mo id="S2.2.p2.4.m4.2.3.1" xref="S2.2.p2.4.m4.2.3.1.cmml">≤</mo><mrow id="S2.2.p2.4.m4.2.3.3" xref="S2.2.p2.4.m4.2.3.3.cmml"><mi id="S2.2.p2.4.m4.2.3.3.2" xref="S2.2.p2.4.m4.2.3.3.2.cmml">d</mi><mo id="S2.2.p2.4.m4.2.3.3.1" xref="S2.2.p2.4.m4.2.3.3.1.cmml">⁢</mo><msub id="S2.2.p2.4.m4.2.3.3.3" xref="S2.2.p2.4.m4.2.3.3.3.cmml"><mrow id="S2.2.p2.4.m4.2.3.3.3.2.2" xref="S2.2.p2.4.m4.2.3.3.3.2.1.cmml"><mo id="S2.2.p2.4.m4.2.3.3.3.2.2.1" stretchy="false" xref="S2.2.p2.4.m4.2.3.3.3.2.1.1.cmml">‖</mo><mi id="S2.2.p2.4.m4.2.2" xref="S2.2.p2.4.m4.2.2.cmml">x</mi><mo id="S2.2.p2.4.m4.2.3.3.3.2.2.2" stretchy="false" xref="S2.2.p2.4.m4.2.3.3.3.2.1.1.cmml">‖</mo></mrow><mi id="S2.2.p2.4.m4.2.3.3.3.3" mathvariant="normal" xref="S2.2.p2.4.m4.2.3.3.3.3.cmml">∞</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.4.m4.2b"><apply id="S2.2.p2.4.m4.2.3.cmml" xref="S2.2.p2.4.m4.2.3"><leq id="S2.2.p2.4.m4.2.3.1.cmml" xref="S2.2.p2.4.m4.2.3.1"></leq><apply id="S2.2.p2.4.m4.2.3.2.cmml" xref="S2.2.p2.4.m4.2.3.2"><csymbol cd="ambiguous" id="S2.2.p2.4.m4.2.3.2.1.cmml" xref="S2.2.p2.4.m4.2.3.2">subscript</csymbol><apply id="S2.2.p2.4.m4.2.3.2.2.1.cmml" xref="S2.2.p2.4.m4.2.3.2.2.2"><csymbol cd="latexml" id="S2.2.p2.4.m4.2.3.2.2.1.1.cmml" xref="S2.2.p2.4.m4.2.3.2.2.2.1">norm</csymbol><ci id="S2.2.p2.4.m4.1.1.cmml" xref="S2.2.p2.4.m4.1.1">𝑥</ci></apply><ci id="S2.2.p2.4.m4.2.3.2.3.cmml" xref="S2.2.p2.4.m4.2.3.2.3">𝑝</ci></apply><apply id="S2.2.p2.4.m4.2.3.3.cmml" xref="S2.2.p2.4.m4.2.3.3"><times id="S2.2.p2.4.m4.2.3.3.1.cmml" xref="S2.2.p2.4.m4.2.3.3.1"></times><ci id="S2.2.p2.4.m4.2.3.3.2.cmml" xref="S2.2.p2.4.m4.2.3.3.2">𝑑</ci><apply id="S2.2.p2.4.m4.2.3.3.3.cmml" xref="S2.2.p2.4.m4.2.3.3.3"><csymbol cd="ambiguous" id="S2.2.p2.4.m4.2.3.3.3.1.cmml" xref="S2.2.p2.4.m4.2.3.3.3">subscript</csymbol><apply id="S2.2.p2.4.m4.2.3.3.3.2.1.cmml" xref="S2.2.p2.4.m4.2.3.3.3.2.2"><csymbol cd="latexml" id="S2.2.p2.4.m4.2.3.3.3.2.1.1.cmml" xref="S2.2.p2.4.m4.2.3.3.3.2.2.1">norm</csymbol><ci id="S2.2.p2.4.m4.2.2.cmml" xref="S2.2.p2.4.m4.2.2">𝑥</ci></apply><infinity id="S2.2.p2.4.m4.2.3.3.3.3.cmml" xref="S2.2.p2.4.m4.2.3.3.3.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.4.m4.2c">||x||_{p}\leq d||x||_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.4.m4.2d">| | italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_d | | italic_x | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>) for all <math alttext="r>0" class="ltx_Math" display="inline" id="S2.2.p2.5.m5.1"><semantics id="S2.2.p2.5.m5.1a"><mrow id="S2.2.p2.5.m5.1.1" xref="S2.2.p2.5.m5.1.1.cmml"><mi id="S2.2.p2.5.m5.1.1.2" xref="S2.2.p2.5.m5.1.1.2.cmml">r</mi><mo id="S2.2.p2.5.m5.1.1.1" xref="S2.2.p2.5.m5.1.1.1.cmml">></mo><mn id="S2.2.p2.5.m5.1.1.3" xref="S2.2.p2.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.5.m5.1b"><apply id="S2.2.p2.5.m5.1.1.cmml" xref="S2.2.p2.5.m5.1.1"><gt id="S2.2.p2.5.m5.1.1.1.cmml" xref="S2.2.p2.5.m5.1.1.1"></gt><ci id="S2.2.p2.5.m5.1.1.2.cmml" xref="S2.2.p2.5.m5.1.1.2">𝑟</ci><cn id="S2.2.p2.5.m5.1.1.3.cmml" type="integer" xref="S2.2.p2.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.5.m5.1c">r>0</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.5.m5.1d">italic_r > 0</annotation></semantics></math> and all <math alttext="p" class="ltx_Math" display="inline" id="S2.2.p2.6.m6.1"><semantics id="S2.2.p2.6.m6.1a"><mi id="S2.2.p2.6.m6.1.1" xref="S2.2.p2.6.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.2.p2.6.m6.1b"><ci id="S2.2.p2.6.m6.1.1.cmml" xref="S2.2.p2.6.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.6.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.6.m6.1d">italic_p</annotation></semantics></math>, it suffices to prove existence of a grid point in <math alttext="B^{\infty}(x^{\star},\frac{\varepsilon}{d+d\lambda})" class="ltx_Math" display="inline" id="S2.2.p2.7.m7.2"><semantics id="S2.2.p2.7.m7.2a"><mrow id="S2.2.p2.7.m7.2.2" xref="S2.2.p2.7.m7.2.2.cmml"><msup id="S2.2.p2.7.m7.2.2.3" xref="S2.2.p2.7.m7.2.2.3.cmml"><mi id="S2.2.p2.7.m7.2.2.3.2" xref="S2.2.p2.7.m7.2.2.3.2.cmml">B</mi><mi id="S2.2.p2.7.m7.2.2.3.3" mathvariant="normal" xref="S2.2.p2.7.m7.2.2.3.3.cmml">∞</mi></msup><mo id="S2.2.p2.7.m7.2.2.2" xref="S2.2.p2.7.m7.2.2.2.cmml">⁢</mo><mrow id="S2.2.p2.7.m7.2.2.1.1" xref="S2.2.p2.7.m7.2.2.1.2.cmml"><mo id="S2.2.p2.7.m7.2.2.1.1.2" stretchy="false" xref="S2.2.p2.7.m7.2.2.1.2.cmml">(</mo><msup id="S2.2.p2.7.m7.2.2.1.1.1" xref="S2.2.p2.7.m7.2.2.1.1.1.cmml"><mi id="S2.2.p2.7.m7.2.2.1.1.1.2" xref="S2.2.p2.7.m7.2.2.1.1.1.2.cmml">x</mi><mo id="S2.2.p2.7.m7.2.2.1.1.1.3" xref="S2.2.p2.7.m7.2.2.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.2.p2.7.m7.2.2.1.1.3" xref="S2.2.p2.7.m7.2.2.1.2.cmml">,</mo><mfrac id="S2.2.p2.7.m7.1.1" xref="S2.2.p2.7.m7.1.1.cmml"><mi id="S2.2.p2.7.m7.1.1.2" xref="S2.2.p2.7.m7.1.1.2.cmml">ε</mi><mrow id="S2.2.p2.7.m7.1.1.3" xref="S2.2.p2.7.m7.1.1.3.cmml"><mi id="S2.2.p2.7.m7.1.1.3.2" xref="S2.2.p2.7.m7.1.1.3.2.cmml">d</mi><mo id="S2.2.p2.7.m7.1.1.3.1" xref="S2.2.p2.7.m7.1.1.3.1.cmml">+</mo><mrow id="S2.2.p2.7.m7.1.1.3.3" xref="S2.2.p2.7.m7.1.1.3.3.cmml"><mi id="S2.2.p2.7.m7.1.1.3.3.2" xref="S2.2.p2.7.m7.1.1.3.3.2.cmml">d</mi><mo id="S2.2.p2.7.m7.1.1.3.3.1" xref="S2.2.p2.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S2.2.p2.7.m7.1.1.3.3.3" xref="S2.2.p2.7.m7.1.1.3.3.3.cmml">λ</mi></mrow></mrow></mfrac><mo id="S2.2.p2.7.m7.2.2.1.1.4" stretchy="false" xref="S2.2.p2.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.7.m7.2b"><apply id="S2.2.p2.7.m7.2.2.cmml" xref="S2.2.p2.7.m7.2.2"><times id="S2.2.p2.7.m7.2.2.2.cmml" xref="S2.2.p2.7.m7.2.2.2"></times><apply id="S2.2.p2.7.m7.2.2.3.cmml" xref="S2.2.p2.7.m7.2.2.3"><csymbol cd="ambiguous" id="S2.2.p2.7.m7.2.2.3.1.cmml" xref="S2.2.p2.7.m7.2.2.3">superscript</csymbol><ci id="S2.2.p2.7.m7.2.2.3.2.cmml" xref="S2.2.p2.7.m7.2.2.3.2">𝐵</ci><infinity id="S2.2.p2.7.m7.2.2.3.3.cmml" xref="S2.2.p2.7.m7.2.2.3.3"></infinity></apply><interval closure="open" id="S2.2.p2.7.m7.2.2.1.2.cmml" xref="S2.2.p2.7.m7.2.2.1.1"><apply id="S2.2.p2.7.m7.2.2.1.1.1.cmml" xref="S2.2.p2.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.7.m7.2.2.1.1.1.1.cmml" xref="S2.2.p2.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S2.2.p2.7.m7.2.2.1.1.1.2.cmml" xref="S2.2.p2.7.m7.2.2.1.1.1.2">𝑥</ci><ci id="S2.2.p2.7.m7.2.2.1.1.1.3.cmml" xref="S2.2.p2.7.m7.2.2.1.1.1.3">⋆</ci></apply><apply id="S2.2.p2.7.m7.1.1.cmml" xref="S2.2.p2.7.m7.1.1"><divide id="S2.2.p2.7.m7.1.1.1.cmml" xref="S2.2.p2.7.m7.1.1"></divide><ci id="S2.2.p2.7.m7.1.1.2.cmml" xref="S2.2.p2.7.m7.1.1.2">𝜀</ci><apply id="S2.2.p2.7.m7.1.1.3.cmml" xref="S2.2.p2.7.m7.1.1.3"><plus id="S2.2.p2.7.m7.1.1.3.1.cmml" xref="S2.2.p2.7.m7.1.1.3.1"></plus><ci id="S2.2.p2.7.m7.1.1.3.2.cmml" xref="S2.2.p2.7.m7.1.1.3.2">𝑑</ci><apply id="S2.2.p2.7.m7.1.1.3.3.cmml" xref="S2.2.p2.7.m7.1.1.3.3"><times id="S2.2.p2.7.m7.1.1.3.3.1.cmml" xref="S2.2.p2.7.m7.1.1.3.3.1"></times><ci id="S2.2.p2.7.m7.1.1.3.3.2.cmml" xref="S2.2.p2.7.m7.1.1.3.3.2">𝑑</ci><ci id="S2.2.p2.7.m7.1.1.3.3.3.cmml" xref="S2.2.p2.7.m7.1.1.3.3.3">𝜆</ci></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.7.m7.2c">B^{\infty}(x^{\star},\frac{\varepsilon}{d+d\lambda})</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.7.m7.2d">italic_B start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , divide start_ARG italic_ε end_ARG start_ARG italic_d + italic_d italic_λ end_ARG )</annotation></semantics></math>. Observe that this is the axis-aligned cube <math alttext="x^{\star}+[-\frac{\varepsilon}{d+d\lambda},\frac{\varepsilon}{d+d\lambda}]^{d}" class="ltx_Math" display="inline" id="S2.2.p2.8.m8.2"><semantics id="S2.2.p2.8.m8.2a"><mrow id="S2.2.p2.8.m8.2.2" xref="S2.2.p2.8.m8.2.2.cmml"><msup id="S2.2.p2.8.m8.2.2.3" xref="S2.2.p2.8.m8.2.2.3.cmml"><mi id="S2.2.p2.8.m8.2.2.3.2" xref="S2.2.p2.8.m8.2.2.3.2.cmml">x</mi><mo id="S2.2.p2.8.m8.2.2.3.3" xref="S2.2.p2.8.m8.2.2.3.3.cmml">⋆</mo></msup><mo id="S2.2.p2.8.m8.2.2.2" xref="S2.2.p2.8.m8.2.2.2.cmml">+</mo><msup id="S2.2.p2.8.m8.2.2.1" xref="S2.2.p2.8.m8.2.2.1.cmml"><mrow id="S2.2.p2.8.m8.2.2.1.1.1" xref="S2.2.p2.8.m8.2.2.1.1.2.cmml"><mo id="S2.2.p2.8.m8.2.2.1.1.1.2" stretchy="false" xref="S2.2.p2.8.m8.2.2.1.1.2.cmml">[</mo><mrow id="S2.2.p2.8.m8.2.2.1.1.1.1" xref="S2.2.p2.8.m8.2.2.1.1.1.1.cmml"><mo id="S2.2.p2.8.m8.2.2.1.1.1.1a" xref="S2.2.p2.8.m8.2.2.1.1.1.1.cmml">−</mo><mfrac id="S2.2.p2.8.m8.2.2.1.1.1.1.2" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.cmml"><mi id="S2.2.p2.8.m8.2.2.1.1.1.1.2.2" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.2.cmml">ε</mi><mrow id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.cmml"><mi id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.2" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.2.cmml">d</mi><mo id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.1" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.1.cmml">+</mo><mrow id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.cmml"><mi id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.2" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.2.cmml">d</mi><mo id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.1" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.1.cmml">⁢</mo><mi id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.3" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.3.cmml">λ</mi></mrow></mrow></mfrac></mrow><mo id="S2.2.p2.8.m8.2.2.1.1.1.3" xref="S2.2.p2.8.m8.2.2.1.1.2.cmml">,</mo><mfrac id="S2.2.p2.8.m8.1.1" xref="S2.2.p2.8.m8.1.1.cmml"><mi id="S2.2.p2.8.m8.1.1.2" xref="S2.2.p2.8.m8.1.1.2.cmml">ε</mi><mrow id="S2.2.p2.8.m8.1.1.3" xref="S2.2.p2.8.m8.1.1.3.cmml"><mi id="S2.2.p2.8.m8.1.1.3.2" xref="S2.2.p2.8.m8.1.1.3.2.cmml">d</mi><mo id="S2.2.p2.8.m8.1.1.3.1" xref="S2.2.p2.8.m8.1.1.3.1.cmml">+</mo><mrow id="S2.2.p2.8.m8.1.1.3.3" xref="S2.2.p2.8.m8.1.1.3.3.cmml"><mi id="S2.2.p2.8.m8.1.1.3.3.2" xref="S2.2.p2.8.m8.1.1.3.3.2.cmml">d</mi><mo id="S2.2.p2.8.m8.1.1.3.3.1" xref="S2.2.p2.8.m8.1.1.3.3.1.cmml">⁢</mo><mi id="S2.2.p2.8.m8.1.1.3.3.3" xref="S2.2.p2.8.m8.1.1.3.3.3.cmml">λ</mi></mrow></mrow></mfrac><mo id="S2.2.p2.8.m8.2.2.1.1.1.4" stretchy="false" xref="S2.2.p2.8.m8.2.2.1.1.2.cmml">]</mo></mrow><mi id="S2.2.p2.8.m8.2.2.1.3" xref="S2.2.p2.8.m8.2.2.1.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.8.m8.2b"><apply id="S2.2.p2.8.m8.2.2.cmml" xref="S2.2.p2.8.m8.2.2"><plus id="S2.2.p2.8.m8.2.2.2.cmml" xref="S2.2.p2.8.m8.2.2.2"></plus><apply id="S2.2.p2.8.m8.2.2.3.cmml" xref="S2.2.p2.8.m8.2.2.3"><csymbol cd="ambiguous" id="S2.2.p2.8.m8.2.2.3.1.cmml" xref="S2.2.p2.8.m8.2.2.3">superscript</csymbol><ci id="S2.2.p2.8.m8.2.2.3.2.cmml" xref="S2.2.p2.8.m8.2.2.3.2">𝑥</ci><ci id="S2.2.p2.8.m8.2.2.3.3.cmml" xref="S2.2.p2.8.m8.2.2.3.3">⋆</ci></apply><apply id="S2.2.p2.8.m8.2.2.1.cmml" xref="S2.2.p2.8.m8.2.2.1"><csymbol cd="ambiguous" id="S2.2.p2.8.m8.2.2.1.2.cmml" xref="S2.2.p2.8.m8.2.2.1">superscript</csymbol><interval closure="closed" id="S2.2.p2.8.m8.2.2.1.1.2.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1"><apply id="S2.2.p2.8.m8.2.2.1.1.1.1.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1"><minus id="S2.2.p2.8.m8.2.2.1.1.1.1.1.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1"></minus><apply id="S2.2.p2.8.m8.2.2.1.1.1.1.2.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2"><divide id="S2.2.p2.8.m8.2.2.1.1.1.1.2.1.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2"></divide><ci id="S2.2.p2.8.m8.2.2.1.1.1.1.2.2.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.2">𝜀</ci><apply id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3"><plus id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.1.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.1"></plus><ci id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.2.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.2">𝑑</ci><apply id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3"><times id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.1.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.1"></times><ci id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.2.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.2">𝑑</ci><ci id="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.3.cmml" xref="S2.2.p2.8.m8.2.2.1.1.1.1.2.3.3.3">𝜆</ci></apply></apply></apply></apply><apply id="S2.2.p2.8.m8.1.1.cmml" xref="S2.2.p2.8.m8.1.1"><divide id="S2.2.p2.8.m8.1.1.1.cmml" xref="S2.2.p2.8.m8.1.1"></divide><ci id="S2.2.p2.8.m8.1.1.2.cmml" xref="S2.2.p2.8.m8.1.1.2">𝜀</ci><apply id="S2.2.p2.8.m8.1.1.3.cmml" xref="S2.2.p2.8.m8.1.1.3"><plus id="S2.2.p2.8.m8.1.1.3.1.cmml" xref="S2.2.p2.8.m8.1.1.3.1"></plus><ci id="S2.2.p2.8.m8.1.1.3.2.cmml" xref="S2.2.p2.8.m8.1.1.3.2">𝑑</ci><apply id="S2.2.p2.8.m8.1.1.3.3.cmml" xref="S2.2.p2.8.m8.1.1.3.3"><times id="S2.2.p2.8.m8.1.1.3.3.1.cmml" xref="S2.2.p2.8.m8.1.1.3.3.1"></times><ci id="S2.2.p2.8.m8.1.1.3.3.2.cmml" xref="S2.2.p2.8.m8.1.1.3.3.2">𝑑</ci><ci id="S2.2.p2.8.m8.1.1.3.3.3.cmml" xref="S2.2.p2.8.m8.1.1.3.3.3">𝜆</ci></apply></apply></apply></interval><ci id="S2.2.p2.8.m8.2.2.1.3.cmml" xref="S2.2.p2.8.m8.2.2.1.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.8.m8.2c">x^{\star}+[-\frac{\varepsilon}{d+d\lambda},\frac{\varepsilon}{d+d\lambda}]^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.8.m8.2d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + [ - divide start_ARG italic_ε end_ARG start_ARG italic_d + italic_d italic_λ end_ARG , divide start_ARG italic_ε end_ARG start_ARG italic_d + italic_d italic_λ end_ARG ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. If <math alttext="2^{-b}\leq\frac{2\varepsilon}{d+d\lambda}" class="ltx_Math" display="inline" id="S2.2.p2.9.m9.1"><semantics id="S2.2.p2.9.m9.1a"><mrow id="S2.2.p2.9.m9.1.1" xref="S2.2.p2.9.m9.1.1.cmml"><msup id="S2.2.p2.9.m9.1.1.2" xref="S2.2.p2.9.m9.1.1.2.cmml"><mn id="S2.2.p2.9.m9.1.1.2.2" xref="S2.2.p2.9.m9.1.1.2.2.cmml">2</mn><mrow id="S2.2.p2.9.m9.1.1.2.3" xref="S2.2.p2.9.m9.1.1.2.3.cmml"><mo id="S2.2.p2.9.m9.1.1.2.3a" xref="S2.2.p2.9.m9.1.1.2.3.cmml">−</mo><mi id="S2.2.p2.9.m9.1.1.2.3.2" xref="S2.2.p2.9.m9.1.1.2.3.2.cmml">b</mi></mrow></msup><mo id="S2.2.p2.9.m9.1.1.1" xref="S2.2.p2.9.m9.1.1.1.cmml">≤</mo><mfrac id="S2.2.p2.9.m9.1.1.3" xref="S2.2.p2.9.m9.1.1.3.cmml"><mrow id="S2.2.p2.9.m9.1.1.3.2" xref="S2.2.p2.9.m9.1.1.3.2.cmml"><mn id="S2.2.p2.9.m9.1.1.3.2.2" xref="S2.2.p2.9.m9.1.1.3.2.2.cmml">2</mn><mo id="S2.2.p2.9.m9.1.1.3.2.1" xref="S2.2.p2.9.m9.1.1.3.2.1.cmml">⁢</mo><mi id="S2.2.p2.9.m9.1.1.3.2.3" xref="S2.2.p2.9.m9.1.1.3.2.3.cmml">ε</mi></mrow><mrow id="S2.2.p2.9.m9.1.1.3.3" xref="S2.2.p2.9.m9.1.1.3.3.cmml"><mi id="S2.2.p2.9.m9.1.1.3.3.2" xref="S2.2.p2.9.m9.1.1.3.3.2.cmml">d</mi><mo id="S2.2.p2.9.m9.1.1.3.3.1" xref="S2.2.p2.9.m9.1.1.3.3.1.cmml">+</mo><mrow id="S2.2.p2.9.m9.1.1.3.3.3" xref="S2.2.p2.9.m9.1.1.3.3.3.cmml"><mi id="S2.2.p2.9.m9.1.1.3.3.3.2" xref="S2.2.p2.9.m9.1.1.3.3.3.2.cmml">d</mi><mo id="S2.2.p2.9.m9.1.1.3.3.3.1" xref="S2.2.p2.9.m9.1.1.3.3.3.1.cmml">⁢</mo><mi id="S2.2.p2.9.m9.1.1.3.3.3.3" xref="S2.2.p2.9.m9.1.1.3.3.3.3.cmml">λ</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.9.m9.1b"><apply id="S2.2.p2.9.m9.1.1.cmml" xref="S2.2.p2.9.m9.1.1"><leq id="S2.2.p2.9.m9.1.1.1.cmml" xref="S2.2.p2.9.m9.1.1.1"></leq><apply id="S2.2.p2.9.m9.1.1.2.cmml" xref="S2.2.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.2.p2.9.m9.1.1.2.1.cmml" xref="S2.2.p2.9.m9.1.1.2">superscript</csymbol><cn id="S2.2.p2.9.m9.1.1.2.2.cmml" type="integer" xref="S2.2.p2.9.m9.1.1.2.2">2</cn><apply id="S2.2.p2.9.m9.1.1.2.3.cmml" xref="S2.2.p2.9.m9.1.1.2.3"><minus id="S2.2.p2.9.m9.1.1.2.3.1.cmml" xref="S2.2.p2.9.m9.1.1.2.3"></minus><ci id="S2.2.p2.9.m9.1.1.2.3.2.cmml" xref="S2.2.p2.9.m9.1.1.2.3.2">𝑏</ci></apply></apply><apply id="S2.2.p2.9.m9.1.1.3.cmml" xref="S2.2.p2.9.m9.1.1.3"><divide id="S2.2.p2.9.m9.1.1.3.1.cmml" xref="S2.2.p2.9.m9.1.1.3"></divide><apply id="S2.2.p2.9.m9.1.1.3.2.cmml" xref="S2.2.p2.9.m9.1.1.3.2"><times id="S2.2.p2.9.m9.1.1.3.2.1.cmml" xref="S2.2.p2.9.m9.1.1.3.2.1"></times><cn id="S2.2.p2.9.m9.1.1.3.2.2.cmml" type="integer" xref="S2.2.p2.9.m9.1.1.3.2.2">2</cn><ci id="S2.2.p2.9.m9.1.1.3.2.3.cmml" xref="S2.2.p2.9.m9.1.1.3.2.3">𝜀</ci></apply><apply id="S2.2.p2.9.m9.1.1.3.3.cmml" xref="S2.2.p2.9.m9.1.1.3.3"><plus id="S2.2.p2.9.m9.1.1.3.3.1.cmml" xref="S2.2.p2.9.m9.1.1.3.3.1"></plus><ci id="S2.2.p2.9.m9.1.1.3.3.2.cmml" xref="S2.2.p2.9.m9.1.1.3.3.2">𝑑</ci><apply id="S2.2.p2.9.m9.1.1.3.3.3.cmml" xref="S2.2.p2.9.m9.1.1.3.3.3"><times id="S2.2.p2.9.m9.1.1.3.3.3.1.cmml" xref="S2.2.p2.9.m9.1.1.3.3.3.1"></times><ci id="S2.2.p2.9.m9.1.1.3.3.3.2.cmml" xref="S2.2.p2.9.m9.1.1.3.3.3.2">𝑑</ci><ci id="S2.2.p2.9.m9.1.1.3.3.3.3.cmml" xref="S2.2.p2.9.m9.1.1.3.3.3.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.9.m9.1c">2^{-b}\leq\frac{2\varepsilon}{d+d\lambda}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.9.m9.1d">2 start_POSTSUPERSCRIPT - italic_b end_POSTSUPERSCRIPT ≤ divide start_ARG 2 italic_ε end_ARG start_ARG italic_d + italic_d italic_λ end_ARG</annotation></semantics></math>, this cube must always contain at least one point of <math alttext="G^{d}_{b}" class="ltx_Math" display="inline" id="S2.2.p2.10.m10.1"><semantics id="S2.2.p2.10.m10.1a"><msubsup id="S2.2.p2.10.m10.1.1" xref="S2.2.p2.10.m10.1.1.cmml"><mi id="S2.2.p2.10.m10.1.1.2.2" xref="S2.2.p2.10.m10.1.1.2.2.cmml">G</mi><mi id="S2.2.p2.10.m10.1.1.3" xref="S2.2.p2.10.m10.1.1.3.cmml">b</mi><mi id="S2.2.p2.10.m10.1.1.2.3" xref="S2.2.p2.10.m10.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.2.p2.10.m10.1b"><apply id="S2.2.p2.10.m10.1.1.cmml" xref="S2.2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.2.p2.10.m10.1.1.1.cmml" xref="S2.2.p2.10.m10.1.1">subscript</csymbol><apply id="S2.2.p2.10.m10.1.1.2.cmml" xref="S2.2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.2.p2.10.m10.1.1.2.1.cmml" xref="S2.2.p2.10.m10.1.1">superscript</csymbol><ci id="S2.2.p2.10.m10.1.1.2.2.cmml" xref="S2.2.p2.10.m10.1.1.2.2">𝐺</ci><ci id="S2.2.p2.10.m10.1.1.2.3.cmml" xref="S2.2.p2.10.m10.1.1.2.3">𝑑</ci></apply><ci id="S2.2.p2.10.m10.1.1.3.cmml" xref="S2.2.p2.10.m10.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.10.m10.1c">G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.10.m10.1d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, we get the lower bound <math alttext="b\geq\log_{2}(\frac{d+d\lambda}{2\varepsilon})" class="ltx_Math" display="inline" id="S2.2.p2.11.m11.2"><semantics id="S2.2.p2.11.m11.2a"><mrow id="S2.2.p2.11.m11.2.2" xref="S2.2.p2.11.m11.2.2.cmml"><mi id="S2.2.p2.11.m11.2.2.3" xref="S2.2.p2.11.m11.2.2.3.cmml">b</mi><mo id="S2.2.p2.11.m11.2.2.2" xref="S2.2.p2.11.m11.2.2.2.cmml">≥</mo><mrow id="S2.2.p2.11.m11.2.2.1.1" xref="S2.2.p2.11.m11.2.2.1.2.cmml"><msub id="S2.2.p2.11.m11.2.2.1.1.1" xref="S2.2.p2.11.m11.2.2.1.1.1.cmml"><mi id="S2.2.p2.11.m11.2.2.1.1.1.2" xref="S2.2.p2.11.m11.2.2.1.1.1.2.cmml">log</mi><mn id="S2.2.p2.11.m11.2.2.1.1.1.3" xref="S2.2.p2.11.m11.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S2.2.p2.11.m11.2.2.1.1a" xref="S2.2.p2.11.m11.2.2.1.2.cmml">⁡</mo><mrow id="S2.2.p2.11.m11.2.2.1.1.2" xref="S2.2.p2.11.m11.2.2.1.2.cmml"><mo id="S2.2.p2.11.m11.2.2.1.1.2.1" stretchy="false" xref="S2.2.p2.11.m11.2.2.1.2.cmml">(</mo><mfrac id="S2.2.p2.11.m11.1.1" xref="S2.2.p2.11.m11.1.1.cmml"><mrow id="S2.2.p2.11.m11.1.1.2" xref="S2.2.p2.11.m11.1.1.2.cmml"><mi id="S2.2.p2.11.m11.1.1.2.2" xref="S2.2.p2.11.m11.1.1.2.2.cmml">d</mi><mo id="S2.2.p2.11.m11.1.1.2.1" xref="S2.2.p2.11.m11.1.1.2.1.cmml">+</mo><mrow id="S2.2.p2.11.m11.1.1.2.3" xref="S2.2.p2.11.m11.1.1.2.3.cmml"><mi id="S2.2.p2.11.m11.1.1.2.3.2" xref="S2.2.p2.11.m11.1.1.2.3.2.cmml">d</mi><mo id="S2.2.p2.11.m11.1.1.2.3.1" xref="S2.2.p2.11.m11.1.1.2.3.1.cmml">⁢</mo><mi id="S2.2.p2.11.m11.1.1.2.3.3" xref="S2.2.p2.11.m11.1.1.2.3.3.cmml">λ</mi></mrow></mrow><mrow id="S2.2.p2.11.m11.1.1.3" xref="S2.2.p2.11.m11.1.1.3.cmml"><mn id="S2.2.p2.11.m11.1.1.3.2" xref="S2.2.p2.11.m11.1.1.3.2.cmml">2</mn><mo id="S2.2.p2.11.m11.1.1.3.1" xref="S2.2.p2.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S2.2.p2.11.m11.1.1.3.3" xref="S2.2.p2.11.m11.1.1.3.3.cmml">ε</mi></mrow></mfrac><mo id="S2.2.p2.11.m11.2.2.1.1.2.2" stretchy="false" xref="S2.2.p2.11.m11.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.11.m11.2b"><apply id="S2.2.p2.11.m11.2.2.cmml" xref="S2.2.p2.11.m11.2.2"><geq id="S2.2.p2.11.m11.2.2.2.cmml" xref="S2.2.p2.11.m11.2.2.2"></geq><ci id="S2.2.p2.11.m11.2.2.3.cmml" xref="S2.2.p2.11.m11.2.2.3">𝑏</ci><apply id="S2.2.p2.11.m11.2.2.1.2.cmml" xref="S2.2.p2.11.m11.2.2.1.1"><apply id="S2.2.p2.11.m11.2.2.1.1.1.cmml" xref="S2.2.p2.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.11.m11.2.2.1.1.1.1.cmml" xref="S2.2.p2.11.m11.2.2.1.1.1">subscript</csymbol><log id="S2.2.p2.11.m11.2.2.1.1.1.2.cmml" xref="S2.2.p2.11.m11.2.2.1.1.1.2"></log><cn id="S2.2.p2.11.m11.2.2.1.1.1.3.cmml" type="integer" xref="S2.2.p2.11.m11.2.2.1.1.1.3">2</cn></apply><apply id="S2.2.p2.11.m11.1.1.cmml" xref="S2.2.p2.11.m11.1.1"><divide id="S2.2.p2.11.m11.1.1.1.cmml" xref="S2.2.p2.11.m11.1.1"></divide><apply id="S2.2.p2.11.m11.1.1.2.cmml" xref="S2.2.p2.11.m11.1.1.2"><plus id="S2.2.p2.11.m11.1.1.2.1.cmml" xref="S2.2.p2.11.m11.1.1.2.1"></plus><ci id="S2.2.p2.11.m11.1.1.2.2.cmml" xref="S2.2.p2.11.m11.1.1.2.2">𝑑</ci><apply id="S2.2.p2.11.m11.1.1.2.3.cmml" xref="S2.2.p2.11.m11.1.1.2.3"><times id="S2.2.p2.11.m11.1.1.2.3.1.cmml" xref="S2.2.p2.11.m11.1.1.2.3.1"></times><ci id="S2.2.p2.11.m11.1.1.2.3.2.cmml" xref="S2.2.p2.11.m11.1.1.2.3.2">𝑑</ci><ci id="S2.2.p2.11.m11.1.1.2.3.3.cmml" xref="S2.2.p2.11.m11.1.1.2.3.3">𝜆</ci></apply></apply><apply id="S2.2.p2.11.m11.1.1.3.cmml" xref="S2.2.p2.11.m11.1.1.3"><times id="S2.2.p2.11.m11.1.1.3.1.cmml" xref="S2.2.p2.11.m11.1.1.3.1"></times><cn id="S2.2.p2.11.m11.1.1.3.2.cmml" type="integer" xref="S2.2.p2.11.m11.1.1.3.2">2</cn><ci id="S2.2.p2.11.m11.1.1.3.3.cmml" xref="S2.2.p2.11.m11.1.1.3.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.11.m11.2c">b\geq\log_{2}(\frac{d+d\lambda}{2\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.11.m11.2d">italic_b ≥ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG italic_d + italic_d italic_λ end_ARG start_ARG 2 italic_ε end_ARG )</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.1.m1.1"><semantics id="S3.1.m1.1b"><msub id="S3.1.m1.1.1" xref="S3.1.m1.1.1.cmml"><mi id="S3.1.m1.1.1.2" mathvariant="normal" xref="S3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.1.m1.1.1.3" xref="S3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.m1.1c"><apply id="S3.1.m1.1.1.cmml" xref="S3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.1.m1.1.1.1.cmml" xref="S3.1.m1.1.1">subscript</csymbol><ci id="S3.1.m1.1.1.2.cmml" xref="S3.1.m1.1.1.2">ℓ</ci><ci id="S3.1.m1.1.1.3.cmml" xref="S3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces and <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.2.m2.1"><semantics id="S3.2.m2.1b"><msub id="S3.2.m2.1.1" xref="S3.2.m2.1.1.cmml"><mi id="S3.2.m2.1.1.2" mathvariant="normal" xref="S3.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.2.m2.1.1.3" xref="S3.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.m2.1c"><apply id="S3.2.m2.1.1.cmml" xref="S3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.2.m2.1.1.1.cmml" xref="S3.2.m2.1.1">subscript</csymbol><ci id="S3.2.m2.1.1.2.cmml" xref="S3.2.m2.1.1.2">ℓ</ci><ci id="S3.2.m2.1.1.3.cmml" xref="S3.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.m2.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.2.m2.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoints</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.9">In this section, we generalize halfspaces and centerpoints from Euclidean geometry to <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><msub id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" mathvariant="normal" xref="S3.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">ℓ</ci><ci id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norms for any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.p1.2.m2.3"><semantics id="S3.p1.2.m2.3a"><mrow id="S3.p1.2.m2.3.4" xref="S3.p1.2.m2.3.4.cmml"><mi id="S3.p1.2.m2.3.4.2" xref="S3.p1.2.m2.3.4.2.cmml">p</mi><mo id="S3.p1.2.m2.3.4.1" xref="S3.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S3.p1.2.m2.3.4.3" xref="S3.p1.2.m2.3.4.3.cmml"><mrow id="S3.p1.2.m2.3.4.3.2.2" xref="S3.p1.2.m2.3.4.3.2.1.cmml"><mo id="S3.p1.2.m2.3.4.3.2.2.1" stretchy="false" xref="S3.p1.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">1</mn><mo id="S3.p1.2.m2.3.4.3.2.2.2" xref="S3.p1.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S3.p1.2.m2.2.2" mathvariant="normal" xref="S3.p1.2.m2.2.2.cmml">∞</mi><mo id="S3.p1.2.m2.3.4.3.2.2.3" stretchy="false" xref="S3.p1.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.p1.2.m2.3.4.3.1" xref="S3.p1.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S3.p1.2.m2.3.4.3.3.2" xref="S3.p1.2.m2.3.4.3.3.1.cmml"><mo id="S3.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="S3.p1.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S3.p1.2.m2.3.3" mathvariant="normal" xref="S3.p1.2.m2.3.3.cmml">∞</mi><mo id="S3.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="S3.p1.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.3b"><apply id="S3.p1.2.m2.3.4.cmml" xref="S3.p1.2.m2.3.4"><in id="S3.p1.2.m2.3.4.1.cmml" xref="S3.p1.2.m2.3.4.1"></in><ci id="S3.p1.2.m2.3.4.2.cmml" xref="S3.p1.2.m2.3.4.2">𝑝</ci><apply id="S3.p1.2.m2.3.4.3.cmml" xref="S3.p1.2.m2.3.4.3"><union id="S3.p1.2.m2.3.4.3.1.cmml" xref="S3.p1.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S3.p1.2.m2.3.4.3.2.1.cmml" xref="S3.p1.2.m2.3.4.3.2.2"><cn id="S3.p1.2.m2.1.1.cmml" type="integer" xref="S3.p1.2.m2.1.1">1</cn><infinity id="S3.p1.2.m2.2.2.cmml" xref="S3.p1.2.m2.2.2"></infinity></interval><set id="S3.p1.2.m2.3.4.3.3.1.cmml" xref="S3.p1.2.m2.3.4.3.3.2"><infinity id="S3.p1.2.m2.3.3.cmml" xref="S3.p1.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. Note that we sometimes use <math alttext="\measuredangle(v,w)" class="ltx_Math" display="inline" id="S3.p1.3.m3.2"><semantics id="S3.p1.3.m3.2a"><mrow id="S3.p1.3.m3.2.3" xref="S3.p1.3.m3.2.3.cmml"><mi id="S3.p1.3.m3.2.3.2" mathvariant="normal" xref="S3.p1.3.m3.2.3.2.cmml">∡</mi><mo id="S3.p1.3.m3.2.3.1" xref="S3.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S3.p1.3.m3.2.3.3.2" xref="S3.p1.3.m3.2.3.3.1.cmml"><mo id="S3.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S3.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">v</mi><mo id="S3.p1.3.m3.2.3.3.2.2" xref="S3.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.p1.3.m3.2.2" xref="S3.p1.3.m3.2.2.cmml">w</mi><mo id="S3.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S3.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.3.cmml" xref="S3.p1.3.m3.2.3"><times id="S3.p1.3.m3.2.3.1.cmml" xref="S3.p1.3.m3.2.3.1"></times><ci id="S3.p1.3.m3.2.3.2.cmml" xref="S3.p1.3.m3.2.3.2">∡</ci><interval closure="open" id="S3.p1.3.m3.2.3.3.1.cmml" xref="S3.p1.3.m3.2.3.3.2"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝑣</ci><ci id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2">𝑤</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.2c">\measuredangle(v,w)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.2d">∡ ( italic_v , italic_w )</annotation></semantics></math> to denote the angle between two vectors <math alttext="v,w\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.p1.4.m4.2"><semantics id="S3.p1.4.m4.2a"><mrow id="S3.p1.4.m4.2.3" xref="S3.p1.4.m4.2.3.cmml"><mrow id="S3.p1.4.m4.2.3.2.2" xref="S3.p1.4.m4.2.3.2.1.cmml"><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">v</mi><mo id="S3.p1.4.m4.2.3.2.2.1" xref="S3.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S3.p1.4.m4.2.2" xref="S3.p1.4.m4.2.2.cmml">w</mi></mrow><mo id="S3.p1.4.m4.2.3.1" xref="S3.p1.4.m4.2.3.1.cmml">∈</mo><msup id="S3.p1.4.m4.2.3.3" xref="S3.p1.4.m4.2.3.3.cmml"><mi id="S3.p1.4.m4.2.3.3.2" xref="S3.p1.4.m4.2.3.3.2.cmml">ℝ</mi><mi id="S3.p1.4.m4.2.3.3.3" xref="S3.p1.4.m4.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.2b"><apply id="S3.p1.4.m4.2.3.cmml" xref="S3.p1.4.m4.2.3"><in id="S3.p1.4.m4.2.3.1.cmml" xref="S3.p1.4.m4.2.3.1"></in><list id="S3.p1.4.m4.2.3.2.1.cmml" xref="S3.p1.4.m4.2.3.2.2"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">𝑣</ci><ci id="S3.p1.4.m4.2.2.cmml" xref="S3.p1.4.m4.2.2">𝑤</ci></list><apply id="S3.p1.4.m4.2.3.3.cmml" xref="S3.p1.4.m4.2.3.3"><csymbol cd="ambiguous" id="S3.p1.4.m4.2.3.3.1.cmml" xref="S3.p1.4.m4.2.3.3">superscript</csymbol><ci id="S3.p1.4.m4.2.3.3.2.cmml" xref="S3.p1.4.m4.2.3.3.2">ℝ</ci><ci id="S3.p1.4.m4.2.3.3.3.cmml" xref="S3.p1.4.m4.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.2c">v,w\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.2d">italic_v , italic_w ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. We also sometimes use <math alttext="\overrightarrow{xy}" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mover accent="true" id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mrow id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml"><mi id="S3.p1.5.m5.1.1.2.2" xref="S3.p1.5.m5.1.1.2.2.cmml">x</mi><mo id="S3.p1.5.m5.1.1.2.1" xref="S3.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S3.p1.5.m5.1.1.2.3" xref="S3.p1.5.m5.1.1.2.3.cmml">y</mi></mrow><mo id="S3.p1.5.m5.1.1.1" stretchy="false" xref="S3.p1.5.m5.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1"><ci id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1">→</ci><apply id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2"><times id="S3.p1.5.m5.1.1.2.1.cmml" xref="S3.p1.5.m5.1.1.2.1"></times><ci id="S3.p1.5.m5.1.1.2.2.cmml" xref="S3.p1.5.m5.1.1.2.2">𝑥</ci><ci id="S3.p1.5.m5.1.1.2.3.cmml" xref="S3.p1.5.m5.1.1.2.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\overrightarrow{xy}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">over→ start_ARG italic_x italic_y end_ARG</annotation></semantics></math> for the vector <math alttext="y-x" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mrow id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml"><mi id="S3.p1.6.m6.1.1.2" xref="S3.p1.6.m6.1.1.2.cmml">y</mi><mo id="S3.p1.6.m6.1.1.1" xref="S3.p1.6.m6.1.1.1.cmml">−</mo><mi id="S3.p1.6.m6.1.1.3" xref="S3.p1.6.m6.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><apply id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1"><minus id="S3.p1.6.m6.1.1.1.cmml" xref="S3.p1.6.m6.1.1.1"></minus><ci id="S3.p1.6.m6.1.1.2.cmml" xref="S3.p1.6.m6.1.1.2">𝑦</ci><ci id="S3.p1.6.m6.1.1.3.cmml" xref="S3.p1.6.m6.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">y-x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_y - italic_x</annotation></semantics></math>. For example, <math alttext="\measuredangle(\overrightarrow{xy},v)" class="ltx_Math" display="inline" id="S3.p1.7.m7.2"><semantics id="S3.p1.7.m7.2a"><mrow id="S3.p1.7.m7.2.3" xref="S3.p1.7.m7.2.3.cmml"><mi id="S3.p1.7.m7.2.3.2" mathvariant="normal" xref="S3.p1.7.m7.2.3.2.cmml">∡</mi><mo id="S3.p1.7.m7.2.3.1" xref="S3.p1.7.m7.2.3.1.cmml">⁢</mo><mrow id="S3.p1.7.m7.2.3.3.2" xref="S3.p1.7.m7.2.3.3.1.cmml"><mo id="S3.p1.7.m7.2.3.3.2.1" stretchy="false" xref="S3.p1.7.m7.2.3.3.1.cmml">(</mo><mover accent="true" id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml"><mrow id="S3.p1.7.m7.1.1.2" xref="S3.p1.7.m7.1.1.2.cmml"><mi id="S3.p1.7.m7.1.1.2.2" xref="S3.p1.7.m7.1.1.2.2.cmml">x</mi><mo id="S3.p1.7.m7.1.1.2.1" xref="S3.p1.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S3.p1.7.m7.1.1.2.3" xref="S3.p1.7.m7.1.1.2.3.cmml">y</mi></mrow><mo id="S3.p1.7.m7.1.1.1" stretchy="false" xref="S3.p1.7.m7.1.1.1.cmml">→</mo></mover><mo id="S3.p1.7.m7.2.3.3.2.2" xref="S3.p1.7.m7.2.3.3.1.cmml">,</mo><mi id="S3.p1.7.m7.2.2" xref="S3.p1.7.m7.2.2.cmml">v</mi><mo id="S3.p1.7.m7.2.3.3.2.3" stretchy="false" xref="S3.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.2b"><apply id="S3.p1.7.m7.2.3.cmml" xref="S3.p1.7.m7.2.3"><times id="S3.p1.7.m7.2.3.1.cmml" xref="S3.p1.7.m7.2.3.1"></times><ci id="S3.p1.7.m7.2.3.2.cmml" xref="S3.p1.7.m7.2.3.2">∡</ci><interval closure="open" id="S3.p1.7.m7.2.3.3.1.cmml" xref="S3.p1.7.m7.2.3.3.2"><apply id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1"><ci id="S3.p1.7.m7.1.1.1.cmml" xref="S3.p1.7.m7.1.1.1">→</ci><apply id="S3.p1.7.m7.1.1.2.cmml" xref="S3.p1.7.m7.1.1.2"><times id="S3.p1.7.m7.1.1.2.1.cmml" xref="S3.p1.7.m7.1.1.2.1"></times><ci id="S3.p1.7.m7.1.1.2.2.cmml" xref="S3.p1.7.m7.1.1.2.2">𝑥</ci><ci id="S3.p1.7.m7.1.1.2.3.cmml" xref="S3.p1.7.m7.1.1.2.3">𝑦</ci></apply></apply><ci id="S3.p1.7.m7.2.2.cmml" xref="S3.p1.7.m7.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.2c">\measuredangle(\overrightarrow{xy},v)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.2d">∡ ( over→ start_ARG italic_x italic_y end_ARG , italic_v )</annotation></semantics></math> denotes the angle between the vectors <math alttext="y-x" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><mrow id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><mi id="S3.p1.8.m8.1.1.2" xref="S3.p1.8.m8.1.1.2.cmml">y</mi><mo id="S3.p1.8.m8.1.1.1" xref="S3.p1.8.m8.1.1.1.cmml">−</mo><mi id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1"><minus id="S3.p1.8.m8.1.1.1.cmml" xref="S3.p1.8.m8.1.1.1"></minus><ci id="S3.p1.8.m8.1.1.2.cmml" xref="S3.p1.8.m8.1.1.2">𝑦</ci><ci id="S3.p1.8.m8.1.1.3.cmml" xref="S3.p1.8.m8.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">y-x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.1d">italic_y - italic_x</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S3.p1.9.m9.1"><semantics id="S3.p1.9.m9.1a"><mi id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.1b"><ci id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.1d">italic_v</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.1.m1.1"><semantics id="S3.SS1.1.m1.1b"><msub id="S3.SS1.1.m1.1.1" xref="S3.SS1.1.m1.1.1.cmml"><mi id="S3.SS1.1.m1.1.1.2" mathvariant="normal" xref="S3.SS1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.1.m1.1.1.3" xref="S3.SS1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.1.m1.1c"><apply id="S3.SS1.1.m1.1.1.cmml" xref="S3.SS1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.1.m1.1.1.1.cmml" xref="S3.SS1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.1.m1.1.1.2.cmml" xref="S3.SS1.1.m1.1.1.2">ℓ</ci><ci id="S3.SS1.1.m1.1.1.3.cmml" xref="S3.SS1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.5">There are many different ways of defining a halfspace in Euclidean geometry (<math alttext="p=2" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">p</mi><mo id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><eq id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></eq><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑝</ci><cn id="S3.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">p=2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_p = 2</annotation></semantics></math>). One natural way that will appear in our algorithms for finding the fixpoints of contraction maps (see <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4" title="4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a>) is to define a halfspace using two distinct points <math alttext="x,y\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.2"><semantics id="S3.SS1.p1.2.m2.2a"><mrow id="S3.SS1.p1.2.m2.2.3" xref="S3.SS1.p1.2.m2.2.3.cmml"><mrow id="S3.SS1.p1.2.m2.2.3.2.2" xref="S3.SS1.p1.2.m2.2.3.2.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">x</mi><mo id="S3.SS1.p1.2.m2.2.3.2.2.1" xref="S3.SS1.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S3.SS1.p1.2.m2.2.2" xref="S3.SS1.p1.2.m2.2.2.cmml">y</mi></mrow><mo id="S3.SS1.p1.2.m2.2.3.1" xref="S3.SS1.p1.2.m2.2.3.1.cmml">∈</mo><msup id="S3.SS1.p1.2.m2.2.3.3" xref="S3.SS1.p1.2.m2.2.3.3.cmml"><mi id="S3.SS1.p1.2.m2.2.3.3.2" xref="S3.SS1.p1.2.m2.2.3.3.2.cmml">ℝ</mi><mi id="S3.SS1.p1.2.m2.2.3.3.3" xref="S3.SS1.p1.2.m2.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.2b"><apply id="S3.SS1.p1.2.m2.2.3.cmml" xref="S3.SS1.p1.2.m2.2.3"><in id="S3.SS1.p1.2.m2.2.3.1.cmml" xref="S3.SS1.p1.2.m2.2.3.1"></in><list id="S3.SS1.p1.2.m2.2.3.2.1.cmml" xref="S3.SS1.p1.2.m2.2.3.2.2"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑥</ci><ci id="S3.SS1.p1.2.m2.2.2.cmml" xref="S3.SS1.p1.2.m2.2.2">𝑦</ci></list><apply id="S3.SS1.p1.2.m2.2.3.3.cmml" xref="S3.SS1.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.2.3.3.1.cmml" xref="S3.SS1.p1.2.m2.2.3.3">superscript</csymbol><ci id="S3.SS1.p1.2.m2.2.3.3.2.cmml" xref="S3.SS1.p1.2.m2.2.3.3.2">ℝ</ci><ci id="S3.SS1.p1.2.m2.2.3.3.3.cmml" xref="S3.SS1.p1.2.m2.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.2c">x,y\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.2d">italic_x , italic_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>: a point <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">z</mi><mo id="S3.SS1.p1.3.m3.1.1.1" xref="S3.SS1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml"><mi id="S3.SS1.p1.3.m3.1.1.3.2" xref="S3.SS1.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS1.p1.3.m3.1.1.3.3" xref="S3.SS1.p1.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><in id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1.1"></in><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝑧</ci><apply id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.p1.3.m3.1.1.3.2">ℝ</ci><ci id="S3.SS1.p1.3.m3.1.1.3.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is considered to be in the halfspace if and only if <math alttext="\lVert x-z\rVert_{2}\leq\lVert y-z\rVert_{2}" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.2"><semantics id="S3.SS1.p1.4.m4.2a"><mrow id="S3.SS1.p1.4.m4.2.2" xref="S3.SS1.p1.4.m4.2.2.cmml"><msub id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml"><mrow id="S3.SS1.p1.4.m4.1.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.2.cmml"><mo fence="true" id="S3.SS1.p1.4.m4.1.1.1.1.1.2" rspace="0em" xref="S3.SS1.p1.4.m4.1.1.1.1.2.1.cmml">∥</mo><mrow id="S3.SS1.p1.4.m4.1.1.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.4.m4.1.1.1.1.1.1.2" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml">x</mi><mo id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p1.4.m4.1.1.1.1.1.1.3" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="S3.SS1.p1.4.m4.1.1.1.1.1.3" lspace="0em" xref="S3.SS1.p1.4.m4.1.1.1.1.2.1.cmml">∥</mo></mrow><mn id="S3.SS1.p1.4.m4.1.1.1.3" xref="S3.SS1.p1.4.m4.1.1.1.3.cmml">2</mn></msub><mo id="S3.SS1.p1.4.m4.2.2.3" rspace="0.1389em" xref="S3.SS1.p1.4.m4.2.2.3.cmml">≤</mo><msub id="S3.SS1.p1.4.m4.2.2.2" xref="S3.SS1.p1.4.m4.2.2.2.cmml"><mrow id="S3.SS1.p1.4.m4.2.2.2.1.1" xref="S3.SS1.p1.4.m4.2.2.2.1.2.cmml"><mo fence="true" id="S3.SS1.p1.4.m4.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="S3.SS1.p1.4.m4.2.2.2.1.2.1.cmml">∥</mo><mrow id="S3.SS1.p1.4.m4.2.2.2.1.1.1" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.cmml"><mi id="S3.SS1.p1.4.m4.2.2.2.1.1.1.2" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.2.cmml">y</mi><mo id="S3.SS1.p1.4.m4.2.2.2.1.1.1.1" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p1.4.m4.2.2.2.1.1.1.3" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="S3.SS1.p1.4.m4.2.2.2.1.1.3" lspace="0em" xref="S3.SS1.p1.4.m4.2.2.2.1.2.1.cmml">∥</mo></mrow><mn id="S3.SS1.p1.4.m4.2.2.2.3" xref="S3.SS1.p1.4.m4.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.2b"><apply id="S3.SS1.p1.4.m4.2.2.cmml" xref="S3.SS1.p1.4.m4.2.2"><leq id="S3.SS1.p1.4.m4.2.2.3.cmml" xref="S3.SS1.p1.4.m4.2.2.3"></leq><apply id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1">subscript</csymbol><apply id="S3.SS1.p1.4.m4.1.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS1.p1.4.m4.1.1.1.1.2.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="S3.SS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1"><minus id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1"></minus><ci id="S3.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.2">𝑥</ci><ci id="S3.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S3.SS1.p1.4.m4.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.1.1.1.3">2</cn></apply><apply id="S3.SS1.p1.4.m4.2.2.2.cmml" xref="S3.SS1.p1.4.m4.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.2.2.2.2.cmml" xref="S3.SS1.p1.4.m4.2.2.2">subscript</csymbol><apply id="S3.SS1.p1.4.m4.2.2.2.1.2.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1"><csymbol cd="latexml" id="S3.SS1.p1.4.m4.2.2.2.1.2.1.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="S3.SS1.p1.4.m4.2.2.2.1.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1"><minus id="S3.SS1.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.1"></minus><ci id="S3.SS1.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.2">𝑦</ci><ci id="S3.SS1.p1.4.m4.2.2.2.1.1.1.3.cmml" xref="S3.SS1.p1.4.m4.2.2.2.1.1.1.3">𝑧</ci></apply></apply><cn id="S3.SS1.p1.4.m4.2.2.2.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.2c">\lVert x-z\rVert_{2}\leq\lVert y-z\rVert_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.2d">∥ italic_x - italic_z ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ∥ italic_y - italic_z ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. This directly generalizes to arbitrary <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.3"><semantics id="S3.SS1.p1.5.m5.3a"><mrow id="S3.SS1.p1.5.m5.3.4" xref="S3.SS1.p1.5.m5.3.4.cmml"><mi id="S3.SS1.p1.5.m5.3.4.2" xref="S3.SS1.p1.5.m5.3.4.2.cmml">p</mi><mo id="S3.SS1.p1.5.m5.3.4.1" xref="S3.SS1.p1.5.m5.3.4.1.cmml">∈</mo><mrow id="S3.SS1.p1.5.m5.3.4.3" xref="S3.SS1.p1.5.m5.3.4.3.cmml"><mrow id="S3.SS1.p1.5.m5.3.4.3.2.2" xref="S3.SS1.p1.5.m5.3.4.3.2.1.cmml"><mo id="S3.SS1.p1.5.m5.3.4.3.2.2.1" stretchy="false" xref="S3.SS1.p1.5.m5.3.4.3.2.1.cmml">[</mo><mn id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">1</mn><mo id="S3.SS1.p1.5.m5.3.4.3.2.2.2" xref="S3.SS1.p1.5.m5.3.4.3.2.1.cmml">,</mo><mi id="S3.SS1.p1.5.m5.2.2" mathvariant="normal" xref="S3.SS1.p1.5.m5.2.2.cmml">∞</mi><mo id="S3.SS1.p1.5.m5.3.4.3.2.2.3" stretchy="false" xref="S3.SS1.p1.5.m5.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.SS1.p1.5.m5.3.4.3.1" xref="S3.SS1.p1.5.m5.3.4.3.1.cmml">∪</mo><mrow id="S3.SS1.p1.5.m5.3.4.3.3.2" xref="S3.SS1.p1.5.m5.3.4.3.3.1.cmml"><mo id="S3.SS1.p1.5.m5.3.4.3.3.2.1" stretchy="false" xref="S3.SS1.p1.5.m5.3.4.3.3.1.cmml">{</mo><mi id="S3.SS1.p1.5.m5.3.3" mathvariant="normal" xref="S3.SS1.p1.5.m5.3.3.cmml">∞</mi><mo id="S3.SS1.p1.5.m5.3.4.3.3.2.2" stretchy="false" xref="S3.SS1.p1.5.m5.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.3b"><apply id="S3.SS1.p1.5.m5.3.4.cmml" xref="S3.SS1.p1.5.m5.3.4"><in id="S3.SS1.p1.5.m5.3.4.1.cmml" xref="S3.SS1.p1.5.m5.3.4.1"></in><ci id="S3.SS1.p1.5.m5.3.4.2.cmml" xref="S3.SS1.p1.5.m5.3.4.2">𝑝</ci><apply id="S3.SS1.p1.5.m5.3.4.3.cmml" xref="S3.SS1.p1.5.m5.3.4.3"><union id="S3.SS1.p1.5.m5.3.4.3.1.cmml" xref="S3.SS1.p1.5.m5.3.4.3.1"></union><interval closure="closed-open" id="S3.SS1.p1.5.m5.3.4.3.2.1.cmml" xref="S3.SS1.p1.5.m5.3.4.3.2.2"><cn id="S3.SS1.p1.5.m5.1.1.cmml" type="integer" xref="S3.SS1.p1.5.m5.1.1">1</cn><infinity id="S3.SS1.p1.5.m5.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2"></infinity></interval><set id="S3.SS1.p1.5.m5.3.4.3.3.1.cmml" xref="S3.SS1.p1.5.m5.3.4.3.3.2"><infinity id="S3.SS1.p1.5.m5.3.3.cmml" xref="S3.SS1.p1.5.m5.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> as follows.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.1.1">Definition 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.3.2"> </span>(Bisector <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.1.m1.1"><semantics id="S3.Thmtheorem1.1.m1.1b"><msub id="S3.Thmtheorem1.1.m1.1.1" xref="S3.Thmtheorem1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem1.1.m1.1.1.3" xref="S3.Thmtheorem1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.1.m1.1c"><apply id="S3.Thmtheorem1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.1.m1.1.1.2">ℓ</ci><ci id="S3.Thmtheorem1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspace)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.4.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.4.4">For fixed <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.3"><semantics id="S3.Thmtheorem1.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.3.4" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem1.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem1.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.3b"><apply id="S3.Thmtheorem1.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4"><in id="S3.Thmtheorem1.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem1.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem1.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and distinct points <math alttext="x,y\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.2"><semantics id="S3.Thmtheorem1.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem1.p1.2.2.m2.2.3" xref="S3.Thmtheorem1.p1.2.2.m2.2.3.cmml"><mrow id="S3.Thmtheorem1.p1.2.2.m2.2.3.2.2" xref="S3.Thmtheorem1.p1.2.2.m2.2.3.2.1.cmml"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml">x</mi><mo id="S3.Thmtheorem1.p1.2.2.m2.2.3.2.2.1" xref="S3.Thmtheorem1.p1.2.2.m2.2.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.2.2.m2.2.2" xref="S3.Thmtheorem1.p1.2.2.m2.2.2.cmml">y</mi></mrow><mo id="S3.Thmtheorem1.p1.2.2.m2.2.3.1" 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xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2">ℓ</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="H^{p}_{x,y}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.4.m4.2"><semantics id="S3.Thmtheorem1.p1.4.4.m4.2a"><mrow id="S3.Thmtheorem1.p1.4.4.m4.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.cmml"><msubsup id="S3.Thmtheorem1.p1.4.4.m4.2.3.2" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.2.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.2.3.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.2.2.2.cmml">H</mi><mrow id="S3.Thmtheorem1.p1.4.4.m4.2.2.2.4" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.2.2.2.4.1" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem1.p1.4.4.m4.2.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.2.2.cmml">y</mi></mrow><mi id="S3.Thmtheorem1.p1.4.4.m4.2.3.2.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem1.p1.4.4.m4.2.3.1" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.1.cmml">⊆</mo><msup id="S3.Thmtheorem1.p1.4.4.m4.2.3.3" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.3.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.2.3.3.2" 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xref="S3.Thmtheorem1.p1.4.4.m4.2.3.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.2.4"><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.2.2">𝑦</ci></list></apply><apply id="S3.Thmtheorem1.p1.4.4.m4.2.3.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.3">superscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.2.3.3.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.3.2">ℝ</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.2.3.3.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.2c">H^{p}_{x,y}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.2d">italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_y end_POSTSUBSCRIPT ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is defined as</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{p}_{x,y}\coloneqq\{z\in\mathbb{R}^{d}\mid||x-z||_{p}\leq||y-z||_{p}\}." class="ltx_Math" display="block" id="S3.Ex1.m1.3"><semantics id="S3.Ex1.m1.3a"><mrow id="S3.Ex1.m1.3.3.1" xref="S3.Ex1.m1.3.3.1.1.cmml"><mrow id="S3.Ex1.m1.3.3.1.1" xref="S3.Ex1.m1.3.3.1.1.cmml"><msubsup id="S3.Ex1.m1.3.3.1.1.4" xref="S3.Ex1.m1.3.3.1.1.4.cmml"><mi id="S3.Ex1.m1.3.3.1.1.4.2.2" xref="S3.Ex1.m1.3.3.1.1.4.2.2.cmml">H</mi><mrow id="S3.Ex1.m1.2.2.2.4" xref="S3.Ex1.m1.2.2.2.3.cmml"><mi id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml">x</mi><mo 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xref="S3.Ex1.m1.3.3.1.1.2.2.2.3.cmml">≤</mo><msub id="S3.Ex1.m1.3.3.1.1.2.2.2.2" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.cmml"><mrow id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.2.cmml"><mo id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.2" stretchy="false" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.2.1.cmml">‖</mo><mrow id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.cmml"><mi id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.2" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.2.cmml">y</mi><mo id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.1" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.1.cmml">−</mo><mi id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.3" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.1.3.cmml">z</mi></mrow><mo id="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.1.3" stretchy="false" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.1.2.1.cmml">‖</mo></mrow><mi id="S3.Ex1.m1.3.3.1.1.2.2.2.2.3" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.3.cmml">p</mi></msub></mrow><mo id="S3.Ex1.m1.3.3.1.1.2.2.5" stretchy="false" 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encoding="application/x-llamapun" id="S3.Ex1.m1.3d">italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_y end_POSTSUBSCRIPT ≔ { italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∣ | | italic_x - italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ | | italic_y - italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.13">In Euclidean geometry, rather than defining a halfspace using the principle of bisection, we can alternatively define it by a point <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_x</annotation></semantics></math> on the boundary and a normal vector <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">v</mi><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml">S</mi><mrow id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.3.2" xref="S3.SS1.p2.2.m2.1.1.3.3.2.cmml">d</mi><mo id="S3.SS1.p2.2.m2.1.1.3.3.1" xref="S3.SS1.p2.2.m2.1.1.3.3.1.cmml">−</mo><mn id="S3.SS1.p2.2.m2.1.1.3.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><in id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></in><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">𝑣</ci><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3">superscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">𝑆</ci><apply id="S3.SS1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3"><minus id="S3.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.1"></minus><ci id="S3.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.2">𝑑</ci><cn id="S3.SS1.p2.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, making use of the Euclidean inner product: a point <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mrow id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">z</mi><mo id="S3.SS1.p2.3.m3.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.cmml">∈</mo><msup id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml"><mi id="S3.SS1.p2.3.m3.1.1.3.2" xref="S3.SS1.p2.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS1.p2.3.m3.1.1.3.3" xref="S3.SS1.p2.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><in id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1.1"></in><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝑧</ci><apply id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.3.1.cmml" xref="S3.SS1.p2.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.3.2.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2">ℝ</ci><ci id="S3.SS1.p2.3.m3.1.1.3.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> belongs to the halfspace if and only if <math alttext="\langle v,z-x\rangle\geq 0" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.2"><semantics id="S3.SS1.p2.4.m4.2a"><mrow id="S3.SS1.p2.4.m4.2.2" xref="S3.SS1.p2.4.m4.2.2.cmml"><mrow id="S3.SS1.p2.4.m4.2.2.1.1" xref="S3.SS1.p2.4.m4.2.2.1.2.cmml"><mo id="S3.SS1.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.1.2.cmml">⟨</mo><mi id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml">v</mi><mo id="S3.SS1.p2.4.m4.2.2.1.1.3" xref="S3.SS1.p2.4.m4.2.2.1.2.cmml">,</mo><mrow id="S3.SS1.p2.4.m4.2.2.1.1.1" xref="S3.SS1.p2.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS1.p2.4.m4.2.2.1.1.1.2" xref="S3.SS1.p2.4.m4.2.2.1.1.1.2.cmml">z</mi><mo id="S3.SS1.p2.4.m4.2.2.1.1.1.1" xref="S3.SS1.p2.4.m4.2.2.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p2.4.m4.2.2.1.1.1.3" xref="S3.SS1.p2.4.m4.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S3.SS1.p2.4.m4.2.2.1.1.4" stretchy="false" xref="S3.SS1.p2.4.m4.2.2.1.2.cmml">⟩</mo></mrow><mo id="S3.SS1.p2.4.m4.2.2.2" xref="S3.SS1.p2.4.m4.2.2.2.cmml">≥</mo><mn id="S3.SS1.p2.4.m4.2.2.3" xref="S3.SS1.p2.4.m4.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.2b"><apply id="S3.SS1.p2.4.m4.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2"><geq id="S3.SS1.p2.4.m4.2.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2.2"></geq><list id="S3.SS1.p2.4.m4.2.2.1.2.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1"><ci id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">𝑣</ci><apply id="S3.SS1.p2.4.m4.2.2.1.1.1.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1"><minus id="S3.SS1.p2.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1.1"></minus><ci id="S3.SS1.p2.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1.2">𝑧</ci><ci id="S3.SS1.p2.4.m4.2.2.1.1.1.3.cmml" xref="S3.SS1.p2.4.m4.2.2.1.1.1.3">𝑥</ci></apply></list><cn id="S3.SS1.p2.4.m4.2.2.3.cmml" type="integer" xref="S3.SS1.p2.4.m4.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.2c">\langle v,z-x\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.2d">⟨ italic_v , italic_z - italic_x ⟩ ≥ 0</annotation></semantics></math>. In fact, the <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.1"><semantics id="S3.SS1.p2.5.m5.1a"><msub id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml"><mi id="S3.SS1.p2.5.m5.1.1.2" mathvariant="normal" xref="S3.SS1.p2.5.m5.1.1.2.cmml">ℓ</mi><mn id="S3.SS1.p2.5.m5.1.1.3" xref="S3.SS1.p2.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.1b"><apply id="S3.SS1.p2.5.m5.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.5.m5.1.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.p2.5.m5.1.1.2.cmml" xref="S3.SS1.p2.5.m5.1.1.2">ℓ</ci><cn id="S3.SS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.p2.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-norm is special because it is induced by an inner product. In general, <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.p2.6.m6.1"><semantics id="S3.SS1.p2.6.m6.1a"><msub id="S3.SS1.p2.6.m6.1.1" xref="S3.SS1.p2.6.m6.1.1.cmml"><mi id="S3.SS1.p2.6.m6.1.1.2" mathvariant="normal" xref="S3.SS1.p2.6.m6.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.p2.6.m6.1.1.3" xref="S3.SS1.p2.6.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m6.1b"><apply id="S3.SS1.p2.6.m6.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.6.m6.1.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.p2.6.m6.1.1.2.cmml" xref="S3.SS1.p2.6.m6.1.1.2">ℓ</ci><ci id="S3.SS1.p2.6.m6.1.1.3.cmml" xref="S3.SS1.p2.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norms are not induced by an inner product, and therefore this definition does not directly generalize. However, we can avoid the inner product with the following observation: a point <math alttext="z" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.1"><semantics id="S3.SS1.p2.7.m7.1a"><mi id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.1b"><ci id="S3.SS1.p2.7.m7.1.1.cmml" xref="S3.SS1.p2.7.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.1d">italic_z</annotation></semantics></math> satisfies <math alttext="\langle v,z-x\rangle\geq 0" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.2"><semantics id="S3.SS1.p2.8.m8.2a"><mrow id="S3.SS1.p2.8.m8.2.2" xref="S3.SS1.p2.8.m8.2.2.cmml"><mrow id="S3.SS1.p2.8.m8.2.2.1.1" xref="S3.SS1.p2.8.m8.2.2.1.2.cmml"><mo id="S3.SS1.p2.8.m8.2.2.1.1.2" stretchy="false" xref="S3.SS1.p2.8.m8.2.2.1.2.cmml">⟨</mo><mi id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml">v</mi><mo id="S3.SS1.p2.8.m8.2.2.1.1.3" xref="S3.SS1.p2.8.m8.2.2.1.2.cmml">,</mo><mrow id="S3.SS1.p2.8.m8.2.2.1.1.1" xref="S3.SS1.p2.8.m8.2.2.1.1.1.cmml"><mi id="S3.SS1.p2.8.m8.2.2.1.1.1.2" xref="S3.SS1.p2.8.m8.2.2.1.1.1.2.cmml">z</mi><mo id="S3.SS1.p2.8.m8.2.2.1.1.1.1" xref="S3.SS1.p2.8.m8.2.2.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p2.8.m8.2.2.1.1.1.3" xref="S3.SS1.p2.8.m8.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S3.SS1.p2.8.m8.2.2.1.1.4" stretchy="false" xref="S3.SS1.p2.8.m8.2.2.1.2.cmml">⟩</mo></mrow><mo id="S3.SS1.p2.8.m8.2.2.2" xref="S3.SS1.p2.8.m8.2.2.2.cmml">≥</mo><mn id="S3.SS1.p2.8.m8.2.2.3" xref="S3.SS1.p2.8.m8.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.2b"><apply id="S3.SS1.p2.8.m8.2.2.cmml" xref="S3.SS1.p2.8.m8.2.2"><geq id="S3.SS1.p2.8.m8.2.2.2.cmml" xref="S3.SS1.p2.8.m8.2.2.2"></geq><list id="S3.SS1.p2.8.m8.2.2.1.2.cmml" xref="S3.SS1.p2.8.m8.2.2.1.1"><ci id="S3.SS1.p2.8.m8.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1">𝑣</ci><apply id="S3.SS1.p2.8.m8.2.2.1.1.1.cmml" xref="S3.SS1.p2.8.m8.2.2.1.1.1"><minus id="S3.SS1.p2.8.m8.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.8.m8.2.2.1.1.1.1"></minus><ci id="S3.SS1.p2.8.m8.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.8.m8.2.2.1.1.1.2">𝑧</ci><ci id="S3.SS1.p2.8.m8.2.2.1.1.1.3.cmml" xref="S3.SS1.p2.8.m8.2.2.1.1.1.3">𝑥</ci></apply></list><cn id="S3.SS1.p2.8.m8.2.2.3.cmml" type="integer" xref="S3.SS1.p2.8.m8.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m8.2c">\langle v,z-x\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.2d">⟨ italic_v , italic_z - italic_x ⟩ ≥ 0</annotation></semantics></math> if and only if <math alttext="\lVert x-z\rVert_{2}\leq\lVert x-\varepsilon v-z\rVert_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m9.2"><semantics id="S3.SS1.p2.9.m9.2a"><mrow id="S3.SS1.p2.9.m9.2.2" xref="S3.SS1.p2.9.m9.2.2.cmml"><msub id="S3.SS1.p2.9.m9.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.cmml"><mrow id="S3.SS1.p2.9.m9.1.1.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.1.2.cmml"><mo fence="true" id="S3.SS1.p2.9.m9.1.1.1.1.1.2" rspace="0em" xref="S3.SS1.p2.9.m9.1.1.1.1.2.1.cmml">∥</mo><mrow id="S3.SS1.p2.9.m9.1.1.1.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p2.9.m9.1.1.1.1.1.1.2" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.2.cmml">x</mi><mo id="S3.SS1.p2.9.m9.1.1.1.1.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p2.9.m9.1.1.1.1.1.1.3" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="S3.SS1.p2.9.m9.1.1.1.1.1.3" lspace="0em" xref="S3.SS1.p2.9.m9.1.1.1.1.2.1.cmml">∥</mo></mrow><mn id="S3.SS1.p2.9.m9.1.1.1.3" xref="S3.SS1.p2.9.m9.1.1.1.3.cmml">2</mn></msub><mo id="S3.SS1.p2.9.m9.2.2.3" rspace="0.1389em" xref="S3.SS1.p2.9.m9.2.2.3.cmml">≤</mo><msub id="S3.SS1.p2.9.m9.2.2.2" xref="S3.SS1.p2.9.m9.2.2.2.cmml"><mrow id="S3.SS1.p2.9.m9.2.2.2.1.1" xref="S3.SS1.p2.9.m9.2.2.2.1.2.cmml"><mo fence="true" id="S3.SS1.p2.9.m9.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="S3.SS1.p2.9.m9.2.2.2.1.2.1.cmml">∥</mo><mrow id="S3.SS1.p2.9.m9.2.2.2.1.1.1" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.cmml"><mi id="S3.SS1.p2.9.m9.2.2.2.1.1.1.2" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.2.cmml">x</mi><mo id="S3.SS1.p2.9.m9.2.2.2.1.1.1.1" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.1.cmml">−</mo><mrow id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.cmml"><mi id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.2" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.2.cmml">ε</mi><mo id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.1" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.3" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.3.cmml">v</mi></mrow><mo id="S3.SS1.p2.9.m9.2.2.2.1.1.1.1a" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.1.cmml">−</mo><mi id="S3.SS1.p2.9.m9.2.2.2.1.1.1.4" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="S3.SS1.p2.9.m9.2.2.2.1.1.3" lspace="0em" xref="S3.SS1.p2.9.m9.2.2.2.1.2.1.cmml">∥</mo></mrow><mn id="S3.SS1.p2.9.m9.2.2.2.3" xref="S3.SS1.p2.9.m9.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m9.2b"><apply id="S3.SS1.p2.9.m9.2.2.cmml" xref="S3.SS1.p2.9.m9.2.2"><leq id="S3.SS1.p2.9.m9.2.2.3.cmml" xref="S3.SS1.p2.9.m9.2.2.3"></leq><apply id="S3.SS1.p2.9.m9.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.1">subscript</csymbol><apply id="S3.SS1.p2.9.m9.1.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS1.p2.9.m9.1.1.1.1.2.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="S3.SS1.p2.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1"><minus id="S3.SS1.p2.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.1"></minus><ci id="S3.SS1.p2.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.2">𝑥</ci><ci id="S3.SS1.p2.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p2.9.m9.1.1.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S3.SS1.p2.9.m9.1.1.1.3.cmml" type="integer" xref="S3.SS1.p2.9.m9.1.1.1.3">2</cn></apply><apply id="S3.SS1.p2.9.m9.2.2.2.cmml" xref="S3.SS1.p2.9.m9.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.9.m9.2.2.2.2.cmml" xref="S3.SS1.p2.9.m9.2.2.2">subscript</csymbol><apply id="S3.SS1.p2.9.m9.2.2.2.1.2.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1"><csymbol cd="latexml" id="S3.SS1.p2.9.m9.2.2.2.1.2.1.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="S3.SS1.p2.9.m9.2.2.2.1.1.1.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1"><minus id="S3.SS1.p2.9.m9.2.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.1"></minus><ci id="S3.SS1.p2.9.m9.2.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.2">𝑥</ci><apply id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3"><times id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.1.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.1"></times><ci id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.2.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.2">𝜀</ci><ci id="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.3.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.3.3">𝑣</ci></apply><ci id="S3.SS1.p2.9.m9.2.2.2.1.1.1.4.cmml" xref="S3.SS1.p2.9.m9.2.2.2.1.1.1.4">𝑧</ci></apply></apply><cn id="S3.SS1.p2.9.m9.2.2.2.3.cmml" type="integer" xref="S3.SS1.p2.9.m9.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m9.2c">\lVert x-z\rVert_{2}\leq\lVert x-\varepsilon v-z\rVert_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m9.2d">∥ italic_x - italic_z ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> holds for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m10.1"><semantics id="S3.SS1.p2.10.m10.1a"><mrow id="S3.SS1.p2.10.m10.1.1" xref="S3.SS1.p2.10.m10.1.1.cmml"><mi id="S3.SS1.p2.10.m10.1.1.2" xref="S3.SS1.p2.10.m10.1.1.2.cmml">ε</mi><mo id="S3.SS1.p2.10.m10.1.1.1" xref="S3.SS1.p2.10.m10.1.1.1.cmml">></mo><mn id="S3.SS1.p2.10.m10.1.1.3" xref="S3.SS1.p2.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m10.1b"><apply id="S3.SS1.p2.10.m10.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1"><gt id="S3.SS1.p2.10.m10.1.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1.1"></gt><ci id="S3.SS1.p2.10.m10.1.1.2.cmml" xref="S3.SS1.p2.10.m10.1.1.2">𝜀</ci><cn id="S3.SS1.p2.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m10.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m10.1d">italic_ε > 0</annotation></semantics></math>. In a way, this definition can be seen as the limit of the bisector definition with <math alttext="y=x-\varepsilon v" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m11.1"><semantics id="S3.SS1.p2.11.m11.1a"><mrow id="S3.SS1.p2.11.m11.1.1" xref="S3.SS1.p2.11.m11.1.1.cmml"><mi id="S3.SS1.p2.11.m11.1.1.2" xref="S3.SS1.p2.11.m11.1.1.2.cmml">y</mi><mo id="S3.SS1.p2.11.m11.1.1.1" xref="S3.SS1.p2.11.m11.1.1.1.cmml">=</mo><mrow id="S3.SS1.p2.11.m11.1.1.3" xref="S3.SS1.p2.11.m11.1.1.3.cmml"><mi id="S3.SS1.p2.11.m11.1.1.3.2" xref="S3.SS1.p2.11.m11.1.1.3.2.cmml">x</mi><mo id="S3.SS1.p2.11.m11.1.1.3.1" xref="S3.SS1.p2.11.m11.1.1.3.1.cmml">−</mo><mrow id="S3.SS1.p2.11.m11.1.1.3.3" xref="S3.SS1.p2.11.m11.1.1.3.3.cmml"><mi id="S3.SS1.p2.11.m11.1.1.3.3.2" xref="S3.SS1.p2.11.m11.1.1.3.3.2.cmml">ε</mi><mo id="S3.SS1.p2.11.m11.1.1.3.3.1" xref="S3.SS1.p2.11.m11.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.11.m11.1.1.3.3.3" xref="S3.SS1.p2.11.m11.1.1.3.3.3.cmml">v</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m11.1b"><apply id="S3.SS1.p2.11.m11.1.1.cmml" xref="S3.SS1.p2.11.m11.1.1"><eq id="S3.SS1.p2.11.m11.1.1.1.cmml" xref="S3.SS1.p2.11.m11.1.1.1"></eq><ci id="S3.SS1.p2.11.m11.1.1.2.cmml" xref="S3.SS1.p2.11.m11.1.1.2">𝑦</ci><apply id="S3.SS1.p2.11.m11.1.1.3.cmml" xref="S3.SS1.p2.11.m11.1.1.3"><minus id="S3.SS1.p2.11.m11.1.1.3.1.cmml" xref="S3.SS1.p2.11.m11.1.1.3.1"></minus><ci id="S3.SS1.p2.11.m11.1.1.3.2.cmml" xref="S3.SS1.p2.11.m11.1.1.3.2">𝑥</ci><apply id="S3.SS1.p2.11.m11.1.1.3.3.cmml" xref="S3.SS1.p2.11.m11.1.1.3.3"><times id="S3.SS1.p2.11.m11.1.1.3.3.1.cmml" xref="S3.SS1.p2.11.m11.1.1.3.3.1"></times><ci id="S3.SS1.p2.11.m11.1.1.3.3.2.cmml" xref="S3.SS1.p2.11.m11.1.1.3.3.2">𝜀</ci><ci id="S3.SS1.p2.11.m11.1.1.3.3.3.cmml" xref="S3.SS1.p2.11.m11.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m11.1c">y=x-\varepsilon v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m11.1d">italic_y = italic_x - italic_ε italic_v</annotation></semantics></math> and <math alttext="\varepsilon\rightarrow 0" class="ltx_Math" display="inline" id="S3.SS1.p2.12.m12.1"><semantics id="S3.SS1.p2.12.m12.1a"><mrow id="S3.SS1.p2.12.m12.1.1" xref="S3.SS1.p2.12.m12.1.1.cmml"><mi id="S3.SS1.p2.12.m12.1.1.2" xref="S3.SS1.p2.12.m12.1.1.2.cmml">ε</mi><mo id="S3.SS1.p2.12.m12.1.1.1" stretchy="false" xref="S3.SS1.p2.12.m12.1.1.1.cmml">→</mo><mn id="S3.SS1.p2.12.m12.1.1.3" xref="S3.SS1.p2.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.12.m12.1b"><apply id="S3.SS1.p2.12.m12.1.1.cmml" xref="S3.SS1.p2.12.m12.1.1"><ci id="S3.SS1.p2.12.m12.1.1.1.cmml" xref="S3.SS1.p2.12.m12.1.1.1">→</ci><ci id="S3.SS1.p2.12.m12.1.1.2.cmml" xref="S3.SS1.p2.12.m12.1.1.2">𝜀</ci><cn id="S3.SS1.p2.12.m12.1.1.3.cmml" type="integer" xref="S3.SS1.p2.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.12.m12.1c">\varepsilon\rightarrow 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.12.m12.1d">italic_ε → 0</annotation></semantics></math>. Stated like this, we can generalize the definition to arbitrary <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.SS1.p2.13.m13.3"><semantics id="S3.SS1.p2.13.m13.3a"><mrow id="S3.SS1.p2.13.m13.3.4" xref="S3.SS1.p2.13.m13.3.4.cmml"><mi id="S3.SS1.p2.13.m13.3.4.2" xref="S3.SS1.p2.13.m13.3.4.2.cmml">p</mi><mo id="S3.SS1.p2.13.m13.3.4.1" xref="S3.SS1.p2.13.m13.3.4.1.cmml">∈</mo><mrow id="S3.SS1.p2.13.m13.3.4.3" xref="S3.SS1.p2.13.m13.3.4.3.cmml"><mrow id="S3.SS1.p2.13.m13.3.4.3.2.2" xref="S3.SS1.p2.13.m13.3.4.3.2.1.cmml"><mo id="S3.SS1.p2.13.m13.3.4.3.2.2.1" stretchy="false" xref="S3.SS1.p2.13.m13.3.4.3.2.1.cmml">[</mo><mn id="S3.SS1.p2.13.m13.1.1" xref="S3.SS1.p2.13.m13.1.1.cmml">1</mn><mo id="S3.SS1.p2.13.m13.3.4.3.2.2.2" xref="S3.SS1.p2.13.m13.3.4.3.2.1.cmml">,</mo><mi id="S3.SS1.p2.13.m13.2.2" mathvariant="normal" xref="S3.SS1.p2.13.m13.2.2.cmml">∞</mi><mo id="S3.SS1.p2.13.m13.3.4.3.2.2.3" stretchy="false" xref="S3.SS1.p2.13.m13.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.SS1.p2.13.m13.3.4.3.1" xref="S3.SS1.p2.13.m13.3.4.3.1.cmml">∪</mo><mrow id="S3.SS1.p2.13.m13.3.4.3.3.2" xref="S3.SS1.p2.13.m13.3.4.3.3.1.cmml"><mo id="S3.SS1.p2.13.m13.3.4.3.3.2.1" stretchy="false" xref="S3.SS1.p2.13.m13.3.4.3.3.1.cmml">{</mo><mi id="S3.SS1.p2.13.m13.3.3" mathvariant="normal" xref="S3.SS1.p2.13.m13.3.3.cmml">∞</mi><mo id="S3.SS1.p2.13.m13.3.4.3.3.2.2" stretchy="false" xref="S3.SS1.p2.13.m13.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.13.m13.3b"><apply id="S3.SS1.p2.13.m13.3.4.cmml" xref="S3.SS1.p2.13.m13.3.4"><in id="S3.SS1.p2.13.m13.3.4.1.cmml" xref="S3.SS1.p2.13.m13.3.4.1"></in><ci id="S3.SS1.p2.13.m13.3.4.2.cmml" xref="S3.SS1.p2.13.m13.3.4.2">𝑝</ci><apply id="S3.SS1.p2.13.m13.3.4.3.cmml" xref="S3.SS1.p2.13.m13.3.4.3"><union id="S3.SS1.p2.13.m13.3.4.3.1.cmml" xref="S3.SS1.p2.13.m13.3.4.3.1"></union><interval closure="closed-open" id="S3.SS1.p2.13.m13.3.4.3.2.1.cmml" xref="S3.SS1.p2.13.m13.3.4.3.2.2"><cn id="S3.SS1.p2.13.m13.1.1.cmml" type="integer" xref="S3.SS1.p2.13.m13.1.1">1</cn><infinity id="S3.SS1.p2.13.m13.2.2.cmml" xref="S3.SS1.p2.13.m13.2.2"></infinity></interval><set id="S3.SS1.p2.13.m13.3.4.3.3.1.cmml" xref="S3.SS1.p2.13.m13.3.4.3.3.2"><infinity id="S3.SS1.p2.13.m13.3.3.cmml" xref="S3.SS1.p2.13.m13.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.13.m13.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.13.m13.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.1.1">Definition 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.3.2"> </span>(Limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.1.m1.1"><semantics id="S3.Thmtheorem2.1.m1.1b"><msub id="S3.Thmtheorem2.1.m1.1.1" xref="S3.Thmtheorem2.1.m1.1.1.cmml"><mi id="S3.Thmtheorem2.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem2.1.m1.1.1.3" xref="S3.Thmtheorem2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.1.m1.1c"><apply id="S3.Thmtheorem2.1.m1.1.1.cmml" xref="S3.Thmtheorem2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.1.m1.1.1.2">ℓ</ci><ci id="S3.Thmtheorem2.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspace)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.4.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.8.8">For fixed <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.3"><semantics id="S3.Thmtheorem2.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.4" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem2.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.3b"><apply id="S3.Thmtheorem2.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4"><in id="S3.Thmtheorem2.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem2.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem2.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, point <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem2.p1.2.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.2.m2.1b"><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1"><in id="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2">𝑥</ci><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, and direction <math alttext="v\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.1b"><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1"><in id="S3.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.2">𝑣</ci><apply id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.1c">v\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.1d">italic_v ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="v\neq 0" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem2.p1.4.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">≠</mo><mn id="S3.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.4.m4.1b"><apply id="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1"><neq id="S3.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.1"></neq><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.2">𝑣</ci><cn id="S3.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.4.m4.1c">v\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.1d">italic_v ≠ 0</annotation></semantics></math>, the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.5.m5.1a"><msub id="S3.Thmtheorem2.p1.5.5.m5.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.2" mathvariant="normal" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.5.m5.1b"><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.2">ℓ</ci><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.6.6.m6.2"><semantics id="S3.Thmtheorem2.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem2.p1.6.6.m6.2.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.cmml"><msubsup id="S3.Thmtheorem2.p1.6.6.m6.2.3.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.4" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.4.1" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem2.p1.6.6.m6.2.3.1" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.1.cmml">⊆</mo><msup id="S3.Thmtheorem2.p1.6.6.m6.2.3.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.6.6.m6.2b"><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3"><subset id="S3.Thmtheorem2.p1.6.6.m6.2.3.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.1"></subset><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2">subscript</csymbol><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.2">ℋ</ci><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.4"><ci id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2">𝑣</ci></list></apply><apply id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.2">ℝ</ci><ci id="S3.Thmtheorem2.p1.6.6.m6.2.3.3.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.6.6.m6.2c">\mathcal{H}^{p}_{x,v}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.6.6.m6.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> through <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.7.7.m7.1"><semantics id="S3.Thmtheorem2.p1.7.7.m7.1a"><mi id="S3.Thmtheorem2.p1.7.7.m7.1.1" xref="S3.Thmtheorem2.p1.7.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.7.7.m7.1b"><ci id="S3.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p1.7.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.7.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.7.7.m7.1d">italic_x</annotation></semantics></math> in the direction of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.8.8.m8.1"><semantics id="S3.Thmtheorem2.p1.8.8.m8.1a"><mi id="S3.Thmtheorem2.p1.8.8.m8.1.1" xref="S3.Thmtheorem2.p1.8.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.8.8.m8.1b"><ci id="S3.Thmtheorem2.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem2.p1.8.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.8.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.8.8.m8.1d">italic_v</annotation></semantics></math> is defined as</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{H}^{p}_{x,v}\coloneqq\{z\in\mathbb{R}^{d}\mid\forall\varepsilon>0\,:% \,||x-z||_{p}\leq||x-\varepsilon v-z||_{p}\}." class="ltx_Math" display="block" id="S3.Ex2.m1.3"><semantics id="S3.Ex2.m1.3a"><mrow id="S3.Ex2.m1.3.3.1" xref="S3.Ex2.m1.3.3.1.1.cmml"><mrow id="S3.Ex2.m1.3.3.1.1" xref="S3.Ex2.m1.3.3.1.1.cmml"><msubsup id="S3.Ex2.m1.3.3.1.1.4" xref="S3.Ex2.m1.3.3.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex2.m1.3.3.1.1.4.2.2" xref="S3.Ex2.m1.3.3.1.1.4.2.2.cmml">ℋ</mi><mrow id="S3.Ex2.m1.2.2.2.4" xref="S3.Ex2.m1.2.2.2.3.cmml"><mi id="S3.Ex2.m1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.cmml">x</mi><mo id="S3.Ex2.m1.2.2.2.4.1" xref="S3.Ex2.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex2.m1.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Ex2.m1.3.3.1.1.4.2.3" xref="S3.Ex2.m1.3.3.1.1.4.2.3.cmml">p</mi></msubsup><mo id="S3.Ex2.m1.3.3.1.1.3" xref="S3.Ex2.m1.3.3.1.1.3.cmml">≔</mo><mrow id="S3.Ex2.m1.3.3.1.1.2.2" xref="S3.Ex2.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex2.m1.3.3.1.1.2.2.3" stretchy="false" xref="S3.Ex2.m1.3.3.1.1.2.3.1.cmml">{</mo><mrow id="S3.Ex2.m1.3.3.1.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.Ex2.m1.3.3.1.1.1.1.1.2" xref="S3.Ex2.m1.3.3.1.1.1.1.1.2.cmml">z</mi><mo id="S3.Ex2.m1.3.3.1.1.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.cmml">∈</mo><msup id="S3.Ex2.m1.3.3.1.1.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.1.1.3.cmml"><mi id="S3.Ex2.m1.3.3.1.1.1.1.1.3.2" xref="S3.Ex2.m1.3.3.1.1.1.1.1.3.2.cmml">ℝ</mi><mi id="S3.Ex2.m1.3.3.1.1.1.1.1.3.3" xref="S3.Ex2.m1.3.3.1.1.1.1.1.3.3.cmml">d</mi></msup></mrow><mo 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italic_p end_POSTSUBSCRIPT ≤ | | italic_x - italic_ε italic_v - italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.4">Observe that scaling the direction <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mi id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><ci id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_v</annotation></semantics></math> with a positive scalar does not change the halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.2"><semantics id="S3.SS1.p3.2.m2.2a"><msubsup id="S3.SS1.p3.2.m2.2.3" xref="S3.SS1.p3.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.2.m2.2.3.2.2" xref="S3.SS1.p3.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS1.p3.2.m2.2.2.2.4" xref="S3.SS1.p3.2.m2.2.2.2.3.cmml"><mi id="S3.SS1.p3.2.m2.1.1.1.1" xref="S3.SS1.p3.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.SS1.p3.2.m2.2.2.2.4.1" xref="S3.SS1.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.2.m2.2.2.2.2" xref="S3.SS1.p3.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS1.p3.2.m2.2.3.2.3" xref="S3.SS1.p3.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.2b"><apply id="S3.SS1.p3.2.m2.2.3.cmml" xref="S3.SS1.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.2.m2.2.3.1.cmml" xref="S3.SS1.p3.2.m2.2.3">subscript</csymbol><apply id="S3.SS1.p3.2.m2.2.3.2.cmml" xref="S3.SS1.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.2.m2.2.3.2.1.cmml" xref="S3.SS1.p3.2.m2.2.3">superscript</csymbol><ci id="S3.SS1.p3.2.m2.2.3.2.2.cmml" xref="S3.SS1.p3.2.m2.2.3.2.2">ℋ</ci><ci id="S3.SS1.p3.2.m2.2.3.2.3.cmml" xref="S3.SS1.p3.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.SS1.p3.2.m2.2.2.2.3.cmml" xref="S3.SS1.p3.2.m2.2.2.2.4"><ci id="S3.SS1.p3.2.m2.1.1.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1.1.1">𝑥</ci><ci id="S3.SS1.p3.2.m2.2.2.2.2.cmml" xref="S3.SS1.p3.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Hence, we usually assume <math alttext="v\in S^{d-1}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mrow id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">v</mi><mo id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">∈</mo><msup id="S3.SS1.p3.3.m3.1.1.4" xref="S3.SS1.p3.3.m3.1.1.4.cmml"><mi id="S3.SS1.p3.3.m3.1.1.4.2" xref="S3.SS1.p3.3.m3.1.1.4.2.cmml">S</mi><mrow id="S3.SS1.p3.3.m3.1.1.4.3" xref="S3.SS1.p3.3.m3.1.1.4.3.cmml"><mi id="S3.SS1.p3.3.m3.1.1.4.3.2" xref="S3.SS1.p3.3.m3.1.1.4.3.2.cmml">d</mi><mo id="S3.SS1.p3.3.m3.1.1.4.3.1" xref="S3.SS1.p3.3.m3.1.1.4.3.1.cmml">−</mo><mn id="S3.SS1.p3.3.m3.1.1.4.3.3" xref="S3.SS1.p3.3.m3.1.1.4.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.p3.3.m3.1.1.5" xref="S3.SS1.p3.3.m3.1.1.5.cmml">⊆</mo><msup id="S3.SS1.p3.3.m3.1.1.6" xref="S3.SS1.p3.3.m3.1.1.6.cmml"><mi id="S3.SS1.p3.3.m3.1.1.6.2" xref="S3.SS1.p3.3.m3.1.1.6.2.cmml">ℝ</mi><mi id="S3.SS1.p3.3.m3.1.1.6.3" xref="S3.SS1.p3.3.m3.1.1.6.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><and id="S3.SS1.p3.3.m3.1.1a.cmml" xref="S3.SS1.p3.3.m3.1.1"></and><apply id="S3.SS1.p3.3.m3.1.1b.cmml" xref="S3.SS1.p3.3.m3.1.1"><in id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3"></in><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">𝑣</ci><apply id="S3.SS1.p3.3.m3.1.1.4.cmml" xref="S3.SS1.p3.3.m3.1.1.4"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.4.1.cmml" xref="S3.SS1.p3.3.m3.1.1.4">superscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.4.2.cmml" xref="S3.SS1.p3.3.m3.1.1.4.2">𝑆</ci><apply id="S3.SS1.p3.3.m3.1.1.4.3.cmml" xref="S3.SS1.p3.3.m3.1.1.4.3"><minus id="S3.SS1.p3.3.m3.1.1.4.3.1.cmml" xref="S3.SS1.p3.3.m3.1.1.4.3.1"></minus><ci id="S3.SS1.p3.3.m3.1.1.4.3.2.cmml" xref="S3.SS1.p3.3.m3.1.1.4.3.2">𝑑</ci><cn id="S3.SS1.p3.3.m3.1.1.4.3.3.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1.4.3.3">1</cn></apply></apply></apply><apply id="S3.SS1.p3.3.m3.1.1c.cmml" xref="S3.SS1.p3.3.m3.1.1"><subset id="S3.SS1.p3.3.m3.1.1.5.cmml" xref="S3.SS1.p3.3.m3.1.1.5"></subset><share href="https://arxiv.org/html/2503.16089v1#S3.SS1.p3.3.m3.1.1.4.cmml" id="S3.SS1.p3.3.m3.1.1d.cmml" xref="S3.SS1.p3.3.m3.1.1"></share><apply id="S3.SS1.p3.3.m3.1.1.6.cmml" xref="S3.SS1.p3.3.m3.1.1.6"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.6.1.cmml" xref="S3.SS1.p3.3.m3.1.1.6">superscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.6.2.cmml" xref="S3.SS1.p3.3.m3.1.1.6.2">ℝ</ci><ci id="S3.SS1.p3.3.m3.1.1.6.3.cmml" xref="S3.SS1.p3.3.m3.1.1.6.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">v\in S^{d-1}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. In fact, we will frequently use the following characterization for containment in a limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><msub id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml"><mi id="S3.SS1.p3.4.m4.1.1.2" mathvariant="normal" xref="S3.SS1.p3.4.m4.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.p3.4.m4.1.1.3" xref="S3.SS1.p3.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.4.m4.1.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.p3.4.m4.1.1.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2">ℓ</ci><ci id="S3.SS1.p3.4.m4.1.1.3.cmml" xref="S3.SS1.p3.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace.</p> </div> <div class="ltx_theorem ltx_theorem_observation" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Observation 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.10.10">For a given limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><msub id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.1.m1.1b"><apply id="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.2">ℓ</ci><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.2"><semantics id="S3.Thmtheorem3.p1.2.2.m2.2a"><msubsup id="S3.Thmtheorem3.p1.2.2.m2.2.3" xref="S3.Thmtheorem3.p1.2.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.2" xref="S3.Thmtheorem3.p1.2.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem3.p1.2.2.m2.2.2.2.4" xref="S3.Thmtheorem3.p1.2.2.m2.2.2.2.3.cmml"><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem3.p1.2.2.m2.2.2.2.4.1" xref="S3.Thmtheorem3.p1.2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem3.p1.2.2.m2.2.2.2.2" xref="S3.Thmtheorem3.p1.2.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.3" xref="S3.Thmtheorem3.p1.2.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.2.m2.2b"><apply id="S3.Thmtheorem3.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3">subscript</csymbol><apply id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem3.p1.2.2.m2.2.3.2.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem3.p1.2.2.m2.2.2.2.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.2.2.4"><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem3.p1.2.2.m2.2.2.2.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.2.m2.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and point <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.3.m3.1"><semantics id="S3.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem3.p1.3.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">z</mi><mo id="S3.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.3.m3.1b"><apply id="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1"><in id="S3.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.2">𝑧</ci><apply id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.3.m3.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.3.m3.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="R_{-}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.4.m4.1a"><msub id="S3.Thmtheorem3.p1.4.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">R</mi><mo id="S3.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.4.m4.1b"><apply id="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.2">𝑅</ci><minus id="S3.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.4.m4.1c">R_{-}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.4.m4.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> be the open ray from <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.5.5.m5.1"><semantics id="S3.Thmtheorem3.p1.5.5.m5.1a"><mi id="S3.Thmtheorem3.p1.5.5.m5.1.1" xref="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.5.5.m5.1b"><ci id="S3.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem3.p1.5.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.5.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.5.5.m5.1d">italic_x</annotation></semantics></math> in direction <math alttext="-v" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.6.6.m6.1"><semantics id="S3.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem3.p1.6.6.m6.1.1" xref="S3.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mo id="S3.Thmtheorem3.p1.6.6.m6.1.1a" xref="S3.Thmtheorem3.p1.6.6.m6.1.1.cmml">−</mo><mi id="S3.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.6.6.m6.1b"><apply id="S3.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.1.1"><minus id="S3.Thmtheorem3.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.1.1"></minus><ci id="S3.Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem3.p1.6.6.m6.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.6.6.m6.1c">-v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.6.6.m6.1d">- italic_v</annotation></semantics></math>, and let <math alttext="B_{z}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.7.7.m7.1"><semantics id="S3.Thmtheorem3.p1.7.7.m7.1a"><msub id="S3.Thmtheorem3.p1.7.7.m7.1.1" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem3.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.2.cmml">B</mi><mi id="S3.Thmtheorem3.p1.7.7.m7.1.1.3" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.7.7.m7.1b"><apply id="S3.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.2">𝐵</ci><ci id="S3.Thmtheorem3.p1.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem3.p1.7.7.m7.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.7.7.m7.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.7.7.m7.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> be the smallest closed <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.8.8.m8.1"><semantics id="S3.Thmtheorem3.p1.8.8.m8.1a"><msub id="S3.Thmtheorem3.p1.8.8.m8.1.1" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem3.p1.8.8.m8.1.1.2" mathvariant="normal" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem3.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.8.8.m8.1b"><apply id="S3.Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.2">ℓ</ci><ci id="S3.Thmtheorem3.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem3.p1.8.8.m8.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.8.8.m8.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.8.8.m8.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-ball with center <math alttext="z" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.9.9.m9.1"><semantics id="S3.Thmtheorem3.p1.9.9.m9.1a"><mi id="S3.Thmtheorem3.p1.9.9.m9.1.1" xref="S3.Thmtheorem3.p1.9.9.m9.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.9.9.m9.1b"><ci id="S3.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem3.p1.9.9.m9.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.9.9.m9.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.9.9.m9.1d">italic_z</annotation></semantics></math> that contains <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.10.10.m10.1"><semantics id="S3.Thmtheorem3.p1.10.10.m10.1a"><mi id="S3.Thmtheorem3.p1.10.10.m10.1.1" xref="S3.Thmtheorem3.p1.10.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.10.10.m10.1b"><ci id="S3.Thmtheorem3.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem3.p1.10.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.10.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.10.10.m10.1d">italic_x</annotation></semantics></math>. Then we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z\in\mathcal{H}^{p}_{x,v}\iff R_{-}\cap B_{z}^{\circ}=\varnothing," class="ltx_Math" display="block" id="S3.Ex3.m1.3"><semantics id="S3.Ex3.m1.3a"><mrow id="S3.Ex3.m1.3.3.1" xref="S3.Ex3.m1.3.3.1.1.cmml"><mrow id="S3.Ex3.m1.3.3.1.1" xref="S3.Ex3.m1.3.3.1.1.cmml"><mrow id="S3.Ex3.m1.3.3.1.1.2" xref="S3.Ex3.m1.3.3.1.1.2.cmml"><mi id="S3.Ex3.m1.3.3.1.1.2.2" xref="S3.Ex3.m1.3.3.1.1.2.2.cmml">z</mi><mo id="S3.Ex3.m1.3.3.1.1.2.1" xref="S3.Ex3.m1.3.3.1.1.2.1.cmml">∈</mo><msubsup id="S3.Ex3.m1.3.3.1.1.2.3" xref="S3.Ex3.m1.3.3.1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex3.m1.3.3.1.1.2.3.2.2" xref="S3.Ex3.m1.3.3.1.1.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Ex3.m1.2.2.2.4" xref="S3.Ex3.m1.2.2.2.3.cmml"><mi id="S3.Ex3.m1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml">x</mi><mo id="S3.Ex3.m1.2.2.2.4.1" xref="S3.Ex3.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex3.m1.2.2.2.2" xref="S3.Ex3.m1.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Ex3.m1.3.3.1.1.2.3.2.3" xref="S3.Ex3.m1.3.3.1.1.2.3.2.3.cmml">p</mi></msubsup></mrow><mo id="S3.Ex3.m1.3.3.1.1.1" stretchy="false" xref="S3.Ex3.m1.3.3.1.1.1.cmml">⇔</mo><mrow id="S3.Ex3.m1.3.3.1.1.3" xref="S3.Ex3.m1.3.3.1.1.3.cmml"><mrow id="S3.Ex3.m1.3.3.1.1.3.2" xref="S3.Ex3.m1.3.3.1.1.3.2.cmml"><msub id="S3.Ex3.m1.3.3.1.1.3.2.2" xref="S3.Ex3.m1.3.3.1.1.3.2.2.cmml"><mi id="S3.Ex3.m1.3.3.1.1.3.2.2.2" xref="S3.Ex3.m1.3.3.1.1.3.2.2.2.cmml">R</mi><mo id="S3.Ex3.m1.3.3.1.1.3.2.2.3" xref="S3.Ex3.m1.3.3.1.1.3.2.2.3.cmml">−</mo></msub><mo id="S3.Ex3.m1.3.3.1.1.3.2.1" xref="S3.Ex3.m1.3.3.1.1.3.2.1.cmml">∩</mo><msubsup id="S3.Ex3.m1.3.3.1.1.3.2.3" xref="S3.Ex3.m1.3.3.1.1.3.2.3.cmml"><mi id="S3.Ex3.m1.3.3.1.1.3.2.3.2.2" xref="S3.Ex3.m1.3.3.1.1.3.2.3.2.2.cmml">B</mi><mi id="S3.Ex3.m1.3.3.1.1.3.2.3.2.3" xref="S3.Ex3.m1.3.3.1.1.3.2.3.2.3.cmml">z</mi><mo id="S3.Ex3.m1.3.3.1.1.3.2.3.3" xref="S3.Ex3.m1.3.3.1.1.3.2.3.3.cmml">∘</mo></msubsup></mrow><mo id="S3.Ex3.m1.3.3.1.1.3.1" xref="S3.Ex3.m1.3.3.1.1.3.1.cmml">=</mo><mi id="S3.Ex3.m1.3.3.1.1.3.3" mathvariant="normal" xref="S3.Ex3.m1.3.3.1.1.3.3.cmml">∅</mi></mrow></mrow><mo id="S3.Ex3.m1.3.3.1.2" xref="S3.Ex3.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex3.m1.3b"><apply id="S3.Ex3.m1.3.3.1.1.cmml" xref="S3.Ex3.m1.3.3.1"><csymbol cd="latexml" id="S3.Ex3.m1.3.3.1.1.1.cmml" xref="S3.Ex3.m1.3.3.1.1.1">iff</csymbol><apply id="S3.Ex3.m1.3.3.1.1.2.cmml" xref="S3.Ex3.m1.3.3.1.1.2"><in id="S3.Ex3.m1.3.3.1.1.2.1.cmml" xref="S3.Ex3.m1.3.3.1.1.2.1"></in><ci id="S3.Ex3.m1.3.3.1.1.2.2.cmml" xref="S3.Ex3.m1.3.3.1.1.2.2">𝑧</ci><apply id="S3.Ex3.m1.3.3.1.1.2.3.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.Ex3.m1.3.3.1.1.2.3.1.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3">subscript</csymbol><apply id="S3.Ex3.m1.3.3.1.1.2.3.2.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.Ex3.m1.3.3.1.1.2.3.2.1.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S3.Ex3.m1.3.3.1.1.2.3.2.2.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3.2.2">ℋ</ci><ci id="S3.Ex3.m1.3.3.1.1.2.3.2.3.cmml" xref="S3.Ex3.m1.3.3.1.1.2.3.2.3">𝑝</ci></apply><list id="S3.Ex3.m1.2.2.2.3.cmml" xref="S3.Ex3.m1.2.2.2.4"><ci id="S3.Ex3.m1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1">𝑥</ci><ci id="S3.Ex3.m1.2.2.2.2.cmml" xref="S3.Ex3.m1.2.2.2.2">𝑣</ci></list></apply></apply><apply id="S3.Ex3.m1.3.3.1.1.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3"><eq id="S3.Ex3.m1.3.3.1.1.3.1.cmml" xref="S3.Ex3.m1.3.3.1.1.3.1"></eq><apply id="S3.Ex3.m1.3.3.1.1.3.2.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2"><intersect id="S3.Ex3.m1.3.3.1.1.3.2.1.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.1"></intersect><apply id="S3.Ex3.m1.3.3.1.1.3.2.2.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.Ex3.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.2">subscript</csymbol><ci id="S3.Ex3.m1.3.3.1.1.3.2.2.2.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.2.2">𝑅</ci><minus id="S3.Ex3.m1.3.3.1.1.3.2.2.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.2.3"></minus></apply><apply id="S3.Ex3.m1.3.3.1.1.3.2.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Ex3.m1.3.3.1.1.3.2.3.1.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3">superscript</csymbol><apply id="S3.Ex3.m1.3.3.1.1.3.2.3.2.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Ex3.m1.3.3.1.1.3.2.3.2.1.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3">subscript</csymbol><ci id="S3.Ex3.m1.3.3.1.1.3.2.3.2.2.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3.2.2">𝐵</ci><ci id="S3.Ex3.m1.3.3.1.1.3.2.3.2.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3.2.3">𝑧</ci></apply><compose id="S3.Ex3.m1.3.3.1.1.3.2.3.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3.2.3.3"></compose></apply></apply><emptyset id="S3.Ex3.m1.3.3.1.1.3.3.cmml" xref="S3.Ex3.m1.3.3.1.1.3.3"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.3c">z\in\mathcal{H}^{p}_{x,v}\iff R_{-}\cap B_{z}^{\circ}=\varnothing,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.3d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ⇔ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT = ∅ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem3.p1.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.12.2">where <math alttext="B_{z}^{\circ}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.11.1.m1.1"><semantics id="S3.Thmtheorem3.p1.11.1.m1.1a"><msubsup id="S3.Thmtheorem3.p1.11.1.m1.1.1" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.cmml"><mi id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.2" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.2.2.cmml">B</mi><mi id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.3" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.2.3.cmml">z</mi><mo id="S3.Thmtheorem3.p1.11.1.m1.1.1.3" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.11.1.m1.1b"><apply id="S3.Thmtheorem3.p1.11.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.11.1.m1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1">superscript</csymbol><apply id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.2.2">𝐵</ci><ci id="S3.Thmtheorem3.p1.11.1.m1.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.2.3">𝑧</ci></apply><compose id="S3.Thmtheorem3.p1.11.1.m1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.11.1.m1.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.11.1.m1.1c">B_{z}^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.11.1.m1.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> denotes the interior of <math alttext="B_{z}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.12.2.m2.1"><semantics id="S3.Thmtheorem3.p1.12.2.m2.1a"><msub id="S3.Thmtheorem3.p1.12.2.m2.1.1" xref="S3.Thmtheorem3.p1.12.2.m2.1.1.cmml"><mi id="S3.Thmtheorem3.p1.12.2.m2.1.1.2" xref="S3.Thmtheorem3.p1.12.2.m2.1.1.2.cmml">B</mi><mi id="S3.Thmtheorem3.p1.12.2.m2.1.1.3" xref="S3.Thmtheorem3.p1.12.2.m2.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.12.2.m2.1b"><apply id="S3.Thmtheorem3.p1.12.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.12.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.12.2.m2.1.1.1.cmml" xref="S3.Thmtheorem3.p1.12.2.m2.1.1">subscript</csymbol><ci id="S3.Thmtheorem3.p1.12.2.m2.1.1.2.cmml" xref="S3.Thmtheorem3.p1.12.2.m2.1.1.2">𝐵</ci><ci id="S3.Thmtheorem3.p1.12.2.m2.1.1.3.cmml" xref="S3.Thmtheorem3.p1.12.2.m2.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.12.2.m2.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.12.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.1">For our centerpoint theorem, we will exclusively work with limit halfspaces. However, as mentioned before, bisector halfspaces naturally appear in the analysis of our algorithms in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4" title="4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a>. Thus, we will need the following observation that allows us to translate between the two.</p> </div> <div class="ltx_theorem ltx_theorem_observation" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Observation 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.10.10">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.3"><semantics id="S3.Thmtheorem4.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem4.p1.1.1.m1.3.4" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem4.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem4.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem4.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem4.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.3b"><apply id="S3.Thmtheorem4.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4"><in id="S3.Thmtheorem4.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem4.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem4.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, <math alttext="x\in\mathbb{R}^{n}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><in id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2">𝑥</ci><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">x\in\mathbb{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.3.m3.1"><semantics id="S3.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem4.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.3.m3.1b"><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"><in id="S3.Thmtheorem4.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2">𝑣</ci><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3"><minus id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.1"></minus><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. The limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><msub id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.2" mathvariant="normal" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2">ℓ</ci><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.5.m5.2"><semantics id="S3.Thmtheorem4.p1.5.5.m5.2a"><msubsup id="S3.Thmtheorem4.p1.5.5.m5.2.3" xref="S3.Thmtheorem4.p1.5.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.2" xref="S3.Thmtheorem4.p1.5.5.m5.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.4" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.3.cmml"><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.4.1" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.2" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.3" xref="S3.Thmtheorem4.p1.5.5.m5.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.5.m5.2b"><apply id="S3.Thmtheorem4.p1.5.5.m5.2.3.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.5.5.m5.2.3.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3">subscript</csymbol><apply id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem4.p1.5.5.m5.2.3.2.3.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.4"><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.5.m5.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.5.m5.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is the intersection of all bisector <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.6.m6.1a"><msub id="S3.Thmtheorem4.p1.6.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.2" mathvariant="normal" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1.3" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.6.m6.1b"><apply id="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.2">ℓ</ci><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.6.m6.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.6.m6.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces <math alttext="H^{p}_{x,x-\varepsilon v}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.7.7.m7.2"><semantics id="S3.Thmtheorem4.p1.7.7.m7.2a"><msubsup id="S3.Thmtheorem4.p1.7.7.m7.2.3" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.cmml"><mi id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2.2.cmml">H</mi><mrow id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml"><mi id="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1" xref="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml">,</mo><mrow id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.cmml"><mi id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.2.cmml">x</mi><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.1" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.1.cmml">−</mo><mrow id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.cmml"><mi id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.2" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.2.cmml">ε</mi><mo id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.1" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.3" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.3.cmml">v</mi></mrow></mrow></mrow><mi id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.3" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.7.7.m7.2b"><apply id="S3.Thmtheorem4.p1.7.7.m7.2.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.7.7.m7.2.3.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3">subscript</csymbol><apply id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2.2">𝐻</ci><ci id="S3.Thmtheorem4.p1.7.7.m7.2.3.2.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2"><ci id="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.1.1.1.1">𝑥</ci><apply id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1"><minus id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.1"></minus><ci id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.2">𝑥</ci><apply id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3"><times id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.1"></times><ci id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.2.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.2">𝜀</ci><ci id="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.3.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.2.2.2.2.1.3.3">𝑣</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.7.7.m7.2c">H^{p}_{x,x-\varepsilon v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.7.7.m7.2d">italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_x - italic_ε italic_v end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.8.8.m8.1"><semantics id="S3.Thmtheorem4.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem4.p1.8.8.m8.1.1" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem4.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.2.cmml">ε</mi><mo id="S3.Thmtheorem4.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.1.cmml">></mo><mn id="S3.Thmtheorem4.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.8.8.m8.1b"><apply id="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1"><gt id="S3.Thmtheorem4.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.1"></gt><ci id="S3.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.2">𝜀</ci><cn id="S3.Thmtheorem4.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.8.8.m8.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.8.8.m8.1d">italic_ε > 0</annotation></semantics></math>. Thus, we have <math alttext="\mathcal{H}^{p}_{x,v}\subseteq H^{p}_{x,x-\varepsilon v}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.9.9.m9.4"><semantics id="S3.Thmtheorem4.p1.9.9.m9.4a"><mrow id="S3.Thmtheorem4.p1.9.9.m9.4.5" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.cmml"><msubsup id="S3.Thmtheorem4.p1.9.9.m9.4.5.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem4.p1.9.9.m9.2.2.2.4" xref="S3.Thmtheorem4.p1.9.9.m9.2.2.2.3.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.1.1.1.1" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem4.p1.9.9.m9.2.2.2.4.1" xref="S3.Thmtheorem4.p1.9.9.m9.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem4.p1.9.9.m9.2.2.2.2" xref="S3.Thmtheorem4.p1.9.9.m9.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.3" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem4.p1.9.9.m9.4.5.1" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.1.cmml">⊆</mo><msubsup id="S3.Thmtheorem4.p1.9.9.m9.4.5.3" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.2.cmml">H</mi><mrow id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.3.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.3.3.1.1" xref="S3.Thmtheorem4.p1.9.9.m9.3.3.1.1.cmml">x</mi><mo id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.3.cmml">,</mo><mrow id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.2.cmml">x</mi><mo id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.1" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.1.cmml">−</mo><mrow id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.cmml"><mi id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.2" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.2.cmml">ε</mi><mo id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.1" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.3" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.3.cmml">v</mi></mrow></mrow></mrow><mi id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.3" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.9.9.m9.4b"><apply id="S3.Thmtheorem4.p1.9.9.m9.4.5.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5"><subset id="S3.Thmtheorem4.p1.9.9.m9.4.5.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.1"></subset><apply id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2">subscript</csymbol><apply id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2">superscript</csymbol><ci id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.2">ℋ</ci><ci id="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem4.p1.9.9.m9.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.2.2.2.4"><ci id="S3.Thmtheorem4.p1.9.9.m9.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem4.p1.9.9.m9.2.2.2.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.2.2.2.2">𝑣</ci></list></apply><apply id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3">subscript</csymbol><apply id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.2">𝐻</ci><ci id="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.5.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2"><ci id="S3.Thmtheorem4.p1.9.9.m9.3.3.1.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.3.3.1.1">𝑥</ci><apply id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1"><minus id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.1"></minus><ci id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.2">𝑥</ci><apply id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3"><times id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.1.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.1"></times><ci id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.2.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.2">𝜀</ci><ci id="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.3.cmml" xref="S3.Thmtheorem4.p1.9.9.m9.4.4.2.2.1.3.3">𝑣</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.9.9.m9.4c">\mathcal{H}^{p}_{x,v}\subseteq H^{p}_{x,x-\varepsilon v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.9.9.m9.4d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ⊆ italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_x - italic_ε italic_v end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.10.10.m10.1"><semantics id="S3.Thmtheorem4.p1.10.10.m10.1a"><mrow id="S3.Thmtheorem4.p1.10.10.m10.1.1" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem4.p1.10.10.m10.1.1.2" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.2.cmml">ε</mi><mo id="S3.Thmtheorem4.p1.10.10.m10.1.1.1" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.1.cmml">></mo><mn id="S3.Thmtheorem4.p1.10.10.m10.1.1.3" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.10.10.m10.1b"><apply id="S3.Thmtheorem4.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem4.p1.10.10.m10.1.1"><gt id="S3.Thmtheorem4.p1.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.1"></gt><ci id="S3.Thmtheorem4.p1.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.2">𝜀</ci><cn id="S3.Thmtheorem4.p1.10.10.m10.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.10.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.10.10.m10.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.10.10.m10.1d">italic_ε > 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.3">For the rest of this section, we will exclusively work with limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><msub id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml"><mi id="S3.SS1.p5.1.m1.1.1.2" mathvariant="normal" xref="S3.SS1.p5.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.p5.1.m1.1.1.3" xref="S3.SS1.p5.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><apply id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.1.m1.1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.p5.1.m1.1.1.2.cmml" xref="S3.SS1.p5.1.m1.1.1.2">ℓ</ci><ci id="S3.SS1.p5.1.m1.1.1.3.cmml" xref="S3.SS1.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces and refer to them simply as <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><msub id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml"><mi id="S3.SS1.p5.2.m2.1.1.2" mathvariant="normal" xref="S3.SS1.p5.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.p5.2.m2.1.1.3" xref="S3.SS1.p5.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><apply id="S3.SS1.p5.2.m2.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.p5.2.m2.1.1.2.cmml" xref="S3.SS1.p5.2.m2.1.1.2">ℓ</ci><ci id="S3.SS1.p5.2.m2.1.1.3.cmml" xref="S3.SS1.p5.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces. In particular, we will first establish some properties of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><msub id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml"><mi id="S3.SS1.p5.3.m3.1.1.2" mathvariant="normal" xref="S3.SS1.p5.3.m3.1.1.2.cmml">ℓ</mi><mi id="S3.SS1.p5.3.m3.1.1.3" xref="S3.SS1.p5.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><apply id="S3.SS1.p5.3.m3.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.3.m3.1.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p5.3.m3.1.1.2.cmml" xref="S3.SS1.p5.3.m3.1.1.2">ℓ</ci><ci id="S3.SS1.p5.3.m3.1.1.3.cmml" xref="S3.SS1.p5.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces that we will then use to prove our centerpoint theorem.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Properties of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.1.m1.1"><semantics id="S3.SS2.1.m1.1b"><msub id="S3.SS2.1.m1.1.1" xref="S3.SS2.1.m1.1.1.cmml"><mi id="S3.SS2.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.1.m1.1.1.3" xref="S3.SS2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.1.m1.1c"><apply id="S3.SS2.1.m1.1.1.cmml" xref="S3.SS2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.m1.1.1.1.cmml" xref="S3.SS2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.1.m1.1.1.2.cmml" xref="S3.SS2.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.1.m1.1.1.3.cmml" xref="S3.SS2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.2">We now collect some useful properties of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><msub id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces. The proofs of all the following properties and additional insights into <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><msub id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><mi id="S3.SS2.p1.2.m2.1.1.2" mathvariant="normal" xref="S3.SS2.p1.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><apply id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p1.2.m2.1.1.2.cmml" xref="S3.SS2.p1.2.m2.1.1.2">ℓ</ci><ci id="S3.SS2.p1.2.m2.1.1.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces can be found in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1" title="Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">A</span></a>.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.4">We start with a simple observation for the case when the direction <math alttext="v" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_v</annotation></semantics></math> is a standard unit vector (i.e., <math alttext="v" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_v</annotation></semantics></math> is parallel to one of the coordinate axes). It turns out that for those directions, <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><msub id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml"><mi id="S3.SS2.p2.3.m3.1.1.2" mathvariant="normal" xref="S3.SS2.p2.3.m3.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p2.3.m3.1.1.3" xref="S3.SS2.p2.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><apply id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p2.3.m3.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.2">ℓ</ci><ci id="S3.SS2.p2.3.m3.1.1.3.cmml" xref="S3.SS2.p2.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces are no different from the classical <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><msub id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml"><mi id="S3.SS2.p2.4.m4.1.1.2" mathvariant="normal" xref="S3.SS2.p2.4.m4.1.1.2.cmml">ℓ</mi><mn id="S3.SS2.p2.4.m4.1.1.3" xref="S3.SS2.p2.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><apply id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.p2.4.m4.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.2">ℓ</ci><cn id="S3.SS2.p2.4.m4.1.1.3.cmml" type="integer" xref="S3.SS2.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Lemma 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.4.4">For any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.1.m1.3"><semantics id="S3.Thmtheorem5.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem5.p1.1.1.m1.3.4" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem5.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem5.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem5.p1.1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem5.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem5.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem5.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.1.m1.3b"><apply id="S3.Thmtheorem5.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4"><in id="S3.Thmtheorem5.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem5.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem5.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, any <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem5.p1.2.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1"><in id="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2">𝑥</ci><apply id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.2.m2.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, and any direction <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem5.p1.3.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.2" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.3" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.3.m3.1b"><apply id="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1"><in id="S3.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.2">𝑣</ci><apply id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3"><minus id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.1"></minus><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.3.m3.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.3.m3.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> parallel to one of the coordinate axes, we have <math alttext="\mathcal{H}^{p}_{x,v}=\mathcal{H}^{2}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.4.m4.4"><semantics id="S3.Thmtheorem5.p1.4.4.m4.4a"><mrow id="S3.Thmtheorem5.p1.4.4.m4.4.5" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.cmml"><msubsup id="S3.Thmtheorem5.p1.4.4.m4.4.5.2" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem5.p1.4.4.m4.2.2.2.4" xref="S3.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.2.2.2.4.1" xref="S3.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem5.p1.4.4.m4.2.2.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.3" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem5.p1.4.4.m4.4.5.1" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.1.cmml">=</mo><msubsup id="S3.Thmtheorem5.p1.4.4.m4.4.5.3" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem5.p1.4.4.m4.4.4.2.4" xref="S3.Thmtheorem5.p1.4.4.m4.4.4.2.3.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.3.3.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.3.3.1.1.cmml">x</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.4.4.2.4.1" xref="S3.Thmtheorem5.p1.4.4.m4.4.4.2.3.cmml">,</mo><mi id="S3.Thmtheorem5.p1.4.4.m4.4.4.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.4.4.2.2.cmml">v</mi></mrow><mn id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.3" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.3.cmml">2</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.4.m4.4b"><apply id="S3.Thmtheorem5.p1.4.4.m4.4.5.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5"><eq id="S3.Thmtheorem5.p1.4.4.m4.4.5.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.1"></eq><apply id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2">subscript</csymbol><apply id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2">superscript</csymbol><ci id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.2">ℋ</ci><ci id="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.2.2.2.4"><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.2.2.2.2">𝑣</ci></list></apply><apply id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3">subscript</csymbol><apply id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.2">ℋ</ci><cn id="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.4.m4.4.5.3.2.3">2</cn></apply><list id="S3.Thmtheorem5.p1.4.4.m4.4.4.2.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.4.2.4"><ci id="S3.Thmtheorem5.p1.4.4.m4.3.3.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.3.3.1.1">𝑥</ci><ci id="S3.Thmtheorem5.p1.4.4.m4.4.4.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.4.4.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.4.m4.4c">\mathcal{H}^{p}_{x,v}=\mathcal{H}^{2}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.4.m4.4d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT = caligraphic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.3"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem5" title="Lemma 3.5. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.5</span></a> really only describes a handful of special cases. For most directions <math alttext="v" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_v</annotation></semantics></math>, <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><msub id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.2" mathvariant="normal" xref="S3.SS2.p3.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2">ℓ</ci><ci id="S3.SS2.p3.2.m2.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces are very different to classical <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><msub id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><mi id="S3.SS2.p3.3.m3.1.1.2" mathvariant="normal" xref="S3.SS2.p3.3.m3.1.1.2.cmml">ℓ</mi><mn id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><apply id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.3.m3.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p3.3.m3.1.1.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2">ℓ</ci><cn id="S3.SS2.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces. In particular, they are in general not convex. Still, we can give a somewhat nice qualitative description of their shape.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Lemma 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem6.p1.6.6">For any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.1.m1.3"><semantics id="S3.Thmtheorem6.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem6.p1.1.1.m1.3.4" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem6.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem6.p1.1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem6.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem6.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.1.m1.3b"><apply id="S3.Thmtheorem6.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4"><in id="S3.Thmtheorem6.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem6.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem6.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem6.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem6.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, any limit halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.2.m2.2"><semantics id="S3.Thmtheorem6.p1.2.2.m2.2a"><msubsup id="S3.Thmtheorem6.p1.2.2.m2.2.3" xref="S3.Thmtheorem6.p1.2.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.2" xref="S3.Thmtheorem6.p1.2.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem6.p1.2.2.m2.2.2.2.4" xref="S3.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml"><mi id="S3.Thmtheorem6.p1.2.2.m2.1.1.1.1" xref="S3.Thmtheorem6.p1.2.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem6.p1.2.2.m2.2.2.2.4.1" xref="S3.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem6.p1.2.2.m2.2.2.2.2" xref="S3.Thmtheorem6.p1.2.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.3" xref="S3.Thmtheorem6.p1.2.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.2.m2.2b"><apply id="S3.Thmtheorem6.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3">subscript</csymbol><apply id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3">superscript</csymbol><ci id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.2.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem6.p1.2.2.m2.2.3.2.3.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.2.2.4"><ci id="S3.Thmtheorem6.p1.2.2.m2.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem6.p1.2.2.m2.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.2.m2.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is a union of rays originating in <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.3.m3.1"><semantics id="S3.Thmtheorem6.p1.3.3.m3.1a"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.1" xref="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.3.m3.1b"><ci id="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.3.m3.1d">italic_x</annotation></semantics></math>. For <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.4.m4.2"><semantics id="S3.Thmtheorem6.p1.4.4.m4.2a"><mrow id="S3.Thmtheorem6.p1.4.4.m4.2.3" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.cmml"><mi id="S3.Thmtheorem6.p1.4.4.m4.2.3.2" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem6.p1.4.4.m4.2.3.1" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.4.4.m4.2.3.3.2" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml"><mo id="S3.Thmtheorem6.p1.4.4.m4.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem6.p1.4.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml">1</mn><mo id="S3.Thmtheorem6.p1.4.4.m4.2.3.3.2.2" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.4.4.m4.2.2" mathvariant="normal" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.cmml">∞</mi><mo id="S3.Thmtheorem6.p1.4.4.m4.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.4.m4.2b"><apply id="S3.Thmtheorem6.p1.4.4.m4.2.3.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.3"><in id="S3.Thmtheorem6.p1.4.4.m4.2.3.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.1"></in><ci id="S3.Thmtheorem6.p1.4.4.m4.2.3.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem6.p1.4.4.m4.2.3.3.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.3.3.2"><cn id="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml" type="integer" xref="S3.Thmtheorem6.p1.4.4.m4.1.1">1</cn><infinity id="S3.Thmtheorem6.p1.4.4.m4.2.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.4.m4.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.4.m4.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, the boundary of any limit halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.5.m5.2"><semantics id="S3.Thmtheorem6.p1.5.5.m5.2a"><msubsup id="S3.Thmtheorem6.p1.5.5.m5.2.3" xref="S3.Thmtheorem6.p1.5.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.2" xref="S3.Thmtheorem6.p1.5.5.m5.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem6.p1.5.5.m5.2.2.2.4" xref="S3.Thmtheorem6.p1.5.5.m5.2.2.2.3.cmml"><mi id="S3.Thmtheorem6.p1.5.5.m5.1.1.1.1" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem6.p1.5.5.m5.2.2.2.4.1" xref="S3.Thmtheorem6.p1.5.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem6.p1.5.5.m5.2.2.2.2" xref="S3.Thmtheorem6.p1.5.5.m5.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.3" xref="S3.Thmtheorem6.p1.5.5.m5.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.5.m5.2b"><apply id="S3.Thmtheorem6.p1.5.5.m5.2.3.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.5.5.m5.2.3.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3">subscript</csymbol><apply id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3">superscript</csymbol><ci id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.2.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem6.p1.5.5.m5.2.3.2.3.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem6.p1.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.2.2.4"><ci id="S3.Thmtheorem6.p1.5.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem6.p1.5.5.m5.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.5.m5.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.5.m5.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is a union of lines through <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.6.m6.1"><semantics id="S3.Thmtheorem6.p1.6.6.m6.1a"><mi id="S3.Thmtheorem6.p1.6.6.m6.1.1" xref="S3.Thmtheorem6.p1.6.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.6.m6.1b"><ci id="S3.Thmtheorem6.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem6.p1.6.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.6.m6.1d">italic_x</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.3">Note that we will not really use the observation about the boundary of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><msub id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml"><mi id="S3.SS2.p4.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p4.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p4.1.m1.1.1.3" xref="S3.SS2.p4.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><apply id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p4.1.m1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.p4.1.m1.1.1.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces for <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.2"><semantics id="S3.SS2.p4.2.m2.2a"><mrow id="S3.SS2.p4.2.m2.2.3" xref="S3.SS2.p4.2.m2.2.3.cmml"><mi id="S3.SS2.p4.2.m2.2.3.2" xref="S3.SS2.p4.2.m2.2.3.2.cmml">p</mi><mo id="S3.SS2.p4.2.m2.2.3.1" xref="S3.SS2.p4.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.SS2.p4.2.m2.2.3.3.2" xref="S3.SS2.p4.2.m2.2.3.3.1.cmml"><mo id="S3.SS2.p4.2.m2.2.3.3.2.1" stretchy="false" xref="S3.SS2.p4.2.m2.2.3.3.1.cmml">(</mo><mn id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">1</mn><mo id="S3.SS2.p4.2.m2.2.3.3.2.2" xref="S3.SS2.p4.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.SS2.p4.2.m2.2.2" mathvariant="normal" xref="S3.SS2.p4.2.m2.2.2.cmml">∞</mi><mo id="S3.SS2.p4.2.m2.2.3.3.2.3" stretchy="false" xref="S3.SS2.p4.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.2b"><apply id="S3.SS2.p4.2.m2.2.3.cmml" xref="S3.SS2.p4.2.m2.2.3"><in id="S3.SS2.p4.2.m2.2.3.1.cmml" xref="S3.SS2.p4.2.m2.2.3.1"></in><ci id="S3.SS2.p4.2.m2.2.3.2.cmml" xref="S3.SS2.p4.2.m2.2.3.2">𝑝</ci><interval closure="open" id="S3.SS2.p4.2.m2.2.3.3.1.cmml" xref="S3.SS2.p4.2.m2.2.3.3.2"><cn id="S3.SS2.p4.2.m2.1.1.cmml" type="integer" xref="S3.SS2.p4.2.m2.1.1">1</cn><infinity id="S3.SS2.p4.2.m2.2.2.cmml" xref="S3.SS2.p4.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, but find it an interesting addition to the lemma. Unfortunately, it breaks down in some degenerate cases if <math alttext="p\in\{1,\infty\}" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.2"><semantics id="S3.SS2.p4.3.m3.2a"><mrow id="S3.SS2.p4.3.m3.2.3" xref="S3.SS2.p4.3.m3.2.3.cmml"><mi id="S3.SS2.p4.3.m3.2.3.2" xref="S3.SS2.p4.3.m3.2.3.2.cmml">p</mi><mo id="S3.SS2.p4.3.m3.2.3.1" xref="S3.SS2.p4.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.SS2.p4.3.m3.2.3.3.2" xref="S3.SS2.p4.3.m3.2.3.3.1.cmml"><mo id="S3.SS2.p4.3.m3.2.3.3.2.1" stretchy="false" xref="S3.SS2.p4.3.m3.2.3.3.1.cmml">{</mo><mn id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml">1</mn><mo id="S3.SS2.p4.3.m3.2.3.3.2.2" xref="S3.SS2.p4.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.SS2.p4.3.m3.2.2" mathvariant="normal" xref="S3.SS2.p4.3.m3.2.2.cmml">∞</mi><mo id="S3.SS2.p4.3.m3.2.3.3.2.3" stretchy="false" xref="S3.SS2.p4.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.2b"><apply id="S3.SS2.p4.3.m3.2.3.cmml" xref="S3.SS2.p4.3.m3.2.3"><in id="S3.SS2.p4.3.m3.2.3.1.cmml" xref="S3.SS2.p4.3.m3.2.3.1"></in><ci id="S3.SS2.p4.3.m3.2.3.2.cmml" xref="S3.SS2.p4.3.m3.2.3.2">𝑝</ci><set id="S3.SS2.p4.3.m3.2.3.3.1.cmml" xref="S3.SS2.p4.3.m3.2.3.3.2"><cn id="S3.SS2.p4.3.m3.1.1.cmml" type="integer" xref="S3.SS2.p4.3.m3.1.1">1</cn><infinity id="S3.SS2.p4.3.m3.2.2.cmml" xref="S3.SS2.p4.3.m3.2.2"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.2c">p\in\{1,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.2d">italic_p ∈ { 1 , ∞ }</annotation></semantics></math> (see, e.g., <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.F2" title="In 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="200" id="S3.F2.g1" src="x1.png" width="312"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.10.5.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.8.4" style="font-size:90%;">Examples of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a>. Two <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S3.F2.5.1.m1.1"><semantics id="S3.F2.5.1.m1.1b"><msub id="S3.F2.5.1.m1.1.1" xref="S3.F2.5.1.m1.1.1.cmml"><mi id="S3.F2.5.1.m1.1.1.2" mathvariant="normal" xref="S3.F2.5.1.m1.1.1.2.cmml">ℓ</mi><mn id="S3.F2.5.1.m1.1.1.3" xref="S3.F2.5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F2.5.1.m1.1c"><apply id="S3.F2.5.1.m1.1.1.cmml" xref="S3.F2.5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F2.5.1.m1.1.1.1.cmml" xref="S3.F2.5.1.m1.1.1">subscript</csymbol><ci id="S3.F2.5.1.m1.1.1.2.cmml" xref="S3.F2.5.1.m1.1.1.2">ℓ</ci><cn id="S3.F2.5.1.m1.1.1.3.cmml" type="integer" xref="S3.F2.5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.5.1.m1.1d">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.5.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, both are unions of rays starting at <math alttext="x" class="ltx_Math" display="inline" id="S3.F2.6.2.m2.1"><semantics id="S3.F2.6.2.m2.1b"><mi id="S3.F2.6.2.m2.1.1" xref="S3.F2.6.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.F2.6.2.m2.1c"><ci id="S3.F2.6.2.m2.1.1.cmml" xref="S3.F2.6.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.6.2.m2.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.F2.6.2.m2.1e">italic_x</annotation></semantics></math>, but only the left one (with non-degenerate <math alttext="v" class="ltx_Math" display="inline" id="S3.F2.7.3.m3.1"><semantics id="S3.F2.7.3.m3.1b"><mi id="S3.F2.7.3.m3.1.1" xref="S3.F2.7.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.F2.7.3.m3.1c"><ci id="S3.F2.7.3.m3.1.1.cmml" xref="S3.F2.7.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.7.3.m3.1d">v</annotation><annotation encoding="application/x-llamapun" id="S3.F2.7.3.m3.1e">italic_v</annotation></semantics></math>) has a boundary consisting of a union of lines through <math alttext="x" class="ltx_Math" display="inline" id="S3.F2.8.4.m4.1"><semantics id="S3.F2.8.4.m4.1b"><mi id="S3.F2.8.4.m4.1.1" xref="S3.F2.8.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.F2.8.4.m4.1c"><ci id="S3.F2.8.4.m4.1.1.cmml" xref="S3.F2.8.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.8.4.m4.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.F2.8.4.m4.1e">italic_x</annotation></semantics></math>. </span></figcaption> </figure> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.7"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a> provides a qualitative description of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p5.1.m1.1"><semantics id="S3.SS2.p5.1.m1.1a"><msub id="S3.SS2.p5.1.m1.1.1" xref="S3.SS2.p5.1.m1.1.1.cmml"><mi id="S3.SS2.p5.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p5.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p5.1.m1.1.1.3" xref="S3.SS2.p5.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.1.m1.1b"><apply id="S3.SS2.p5.1.m1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.1.m1.1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p5.1.m1.1.1.2.cmml" xref="S3.SS2.p5.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.p5.1.m1.1.1.3.cmml" xref="S3.SS2.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, but does not really give us any concrete points that must be contained in or outside of a given <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p5.2.m2.1"><semantics id="S3.SS2.p5.2.m2.1a"><msub id="S3.SS2.p5.2.m2.1.1" xref="S3.SS2.p5.2.m2.1.1.cmml"><mi id="S3.SS2.p5.2.m2.1.1.2" mathvariant="normal" xref="S3.SS2.p5.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p5.2.m2.1.1.3" xref="S3.SS2.p5.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.2.m2.1b"><apply id="S3.SS2.p5.2.m2.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.2.m2.1.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p5.2.m2.1.1.2.cmml" xref="S3.SS2.p5.2.m2.1.1.2">ℓ</ci><ci id="S3.SS2.p5.2.m2.1.1.3.cmml" xref="S3.SS2.p5.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace. For example, it seems intuitive that points lying roughly in the direction of <math alttext="v" class="ltx_Math" display="inline" id="S3.SS2.p5.3.m3.1"><semantics id="S3.SS2.p5.3.m3.1a"><mi id="S3.SS2.p5.3.m3.1.1" xref="S3.SS2.p5.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.3.m3.1b"><ci id="S3.SS2.p5.3.m3.1.1.cmml" xref="S3.SS2.p5.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.3.m3.1d">italic_v</annotation></semantics></math> from <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p5.4.m4.1"><semantics id="S3.SS2.p5.4.m4.1a"><mi id="S3.SS2.p5.4.m4.1.1" xref="S3.SS2.p5.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.4.m4.1b"><ci id="S3.SS2.p5.4.m4.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.4.m4.1d">italic_x</annotation></semantics></math> should be contained in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.SS2.p5.5.m5.2"><semantics id="S3.SS2.p5.5.m5.2a"><msubsup id="S3.SS2.p5.5.m5.2.3" xref="S3.SS2.p5.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.5.m5.2.3.2.2" xref="S3.SS2.p5.5.m5.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS2.p5.5.m5.2.2.2.4" xref="S3.SS2.p5.5.m5.2.2.2.3.cmml"><mi id="S3.SS2.p5.5.m5.1.1.1.1" xref="S3.SS2.p5.5.m5.1.1.1.1.cmml">x</mi><mo id="S3.SS2.p5.5.m5.2.2.2.4.1" xref="S3.SS2.p5.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p5.5.m5.2.2.2.2" xref="S3.SS2.p5.5.m5.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS2.p5.5.m5.2.3.2.3" xref="S3.SS2.p5.5.m5.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.5.m5.2b"><apply id="S3.SS2.p5.5.m5.2.3.cmml" xref="S3.SS2.p5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS2.p5.5.m5.2.3.1.cmml" xref="S3.SS2.p5.5.m5.2.3">subscript</csymbol><apply id="S3.SS2.p5.5.m5.2.3.2.cmml" xref="S3.SS2.p5.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS2.p5.5.m5.2.3.2.1.cmml" xref="S3.SS2.p5.5.m5.2.3">superscript</csymbol><ci id="S3.SS2.p5.5.m5.2.3.2.2.cmml" xref="S3.SS2.p5.5.m5.2.3.2.2">ℋ</ci><ci id="S3.SS2.p5.5.m5.2.3.2.3.cmml" xref="S3.SS2.p5.5.m5.2.3.2.3">𝑝</ci></apply><list id="S3.SS2.p5.5.m5.2.2.2.3.cmml" xref="S3.SS2.p5.5.m5.2.2.2.4"><ci id="S3.SS2.p5.5.m5.1.1.1.1.cmml" xref="S3.SS2.p5.5.m5.1.1.1.1">𝑥</ci><ci id="S3.SS2.p5.5.m5.2.2.2.2.cmml" xref="S3.SS2.p5.5.m5.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.5.m5.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.5.m5.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>, while points lying roughly in the direction of <math alttext="-v" class="ltx_Math" display="inline" id="S3.SS2.p5.6.m6.1"><semantics id="S3.SS2.p5.6.m6.1a"><mrow id="S3.SS2.p5.6.m6.1.1" xref="S3.SS2.p5.6.m6.1.1.cmml"><mo id="S3.SS2.p5.6.m6.1.1a" xref="S3.SS2.p5.6.m6.1.1.cmml">−</mo><mi id="S3.SS2.p5.6.m6.1.1.2" xref="S3.SS2.p5.6.m6.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.6.m6.1b"><apply id="S3.SS2.p5.6.m6.1.1.cmml" xref="S3.SS2.p5.6.m6.1.1"><minus id="S3.SS2.p5.6.m6.1.1.1.cmml" xref="S3.SS2.p5.6.m6.1.1"></minus><ci id="S3.SS2.p5.6.m6.1.1.2.cmml" xref="S3.SS2.p5.6.m6.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.6.m6.1c">-v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.6.m6.1d">- italic_v</annotation></semantics></math> from <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p5.7.m7.1"><semantics id="S3.SS2.p5.7.m7.1a"><mi id="S3.SS2.p5.7.m7.1.1" xref="S3.SS2.p5.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.7.m7.1b"><ci id="S3.SS2.p5.7.m7.1.1.cmml" xref="S3.SS2.p5.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.7.m7.1d">italic_x</annotation></semantics></math> should never be contained. The following lemma formalizes this (see also <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.F3" title="In 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.7.7">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.3"><semantics id="S3.Thmtheorem7.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem7.p1.1.1.m1.3.4" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem7.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem7.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem7.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem7.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem7.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.3b"><apply id="S3.Thmtheorem7.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4"><in id="S3.Thmtheorem7.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem7.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem7.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem7.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.2"><semantics id="S3.Thmtheorem7.p1.2.2.m2.2a"><msubsup id="S3.Thmtheorem7.p1.2.2.m2.2.3" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.2" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem7.p1.2.2.m2.2.2.2.4" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.1.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.2.2.2.4.1" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem7.p1.2.2.m2.2.2.2.2" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.3" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.2b"><apply id="S3.Thmtheorem7.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3">subscript</csymbol><apply id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.2.4"><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem7.p1.2.2.m2.2.2.2.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be arbitrary. All <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.3.m3.2"><semantics id="S3.Thmtheorem7.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem7.p1.3.3.m3.2.3" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.cmml"><mi id="S3.Thmtheorem7.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.2.cmml">z</mi><mo id="S3.Thmtheorem7.p1.3.3.m3.2.3.1" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.1.cmml">∈</mo><msubsup id="S3.Thmtheorem7.p1.3.3.m3.2.3.3" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.2" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem7.p1.3.3.m3.2.2.2.4" xref="S3.Thmtheorem7.p1.3.3.m3.2.2.2.3.cmml"><mi id="S3.Thmtheorem7.p1.3.3.m3.1.1.1.1" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem7.p1.3.3.m3.2.2.2.4.1" xref="S3.Thmtheorem7.p1.3.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem7.p1.3.3.m3.2.2.2.2" xref="S3.Thmtheorem7.p1.3.3.m3.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.3" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.3.3.m3.2b"><apply id="S3.Thmtheorem7.p1.3.3.m3.2.3.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3"><in id="S3.Thmtheorem7.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.1"></in><ci id="S3.Thmtheorem7.p1.3.3.m3.2.3.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.2">𝑧</ci><apply id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3">subscript</csymbol><apply id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.2">ℋ</ci><ci id="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.3.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.3.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem7.p1.3.3.m3.2.2.2.3.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.2.2.4"><ci id="S3.Thmtheorem7.p1.3.3.m3.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem7.p1.3.3.m3.2.2.2.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.3.m3.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.3.m3.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> satisfy <math alttext="\measuredangle(\overrightarrow{xz},v)\leq\pi-\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.4.4.m4.2"><semantics id="S3.Thmtheorem7.p1.4.4.m4.2a"><mrow id="S3.Thmtheorem7.p1.4.4.m4.2.3" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.cmml"><mrow id="S3.Thmtheorem7.p1.4.4.m4.2.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.cmml"><mi id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.2" mathvariant="normal" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.2.cmml">∡</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.1" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.1.cmml"><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem7.p1.4.4.m4.1.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml"><mrow id="S3.Thmtheorem7.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.cmml"><mi id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.2.cmml">x</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.3" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.3.cmml">z</mi></mrow><mo id="S3.Thmtheorem7.p1.4.4.m4.1.1.1" stretchy="false" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.2.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.4.4.m4.2.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.2.cmml">v</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.2.3" stretchy="false" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.1" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.1.cmml">≤</mo><mrow id="S3.Thmtheorem7.p1.4.4.m4.2.3.3" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.cmml"><mi id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.2.cmml">π</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.1" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.1.cmml">−</mo><msqrt id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.cmml"><mrow id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.cmml"><mpadded id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2" voffset="0.3em" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2.cmml"><mn id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2a" mathsize="70%" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2.cmml">1</mn></mpadded><mpadded id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1" lspace="-0.1em" width="-0.15em" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1.cmml"><mo id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1a" stretchy="true" symmetric="true" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1.cmml">/</mo></mpadded><mi id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.3" mathsize="70%" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.3.cmml">d</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.4.4.m4.2b"><apply id="S3.Thmtheorem7.p1.4.4.m4.2.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3"><leq id="S3.Thmtheorem7.p1.4.4.m4.2.3.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.1"></leq><apply id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2"><times id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.1"></times><ci id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.2">∡</ci><interval closure="open" id="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.2.3.2"><apply id="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1"><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.1">→</ci><apply id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2"><times id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.1"></times><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.2">𝑥</ci><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.3">𝑧</ci></apply></apply><ci id="S3.Thmtheorem7.p1.4.4.m4.2.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.2">𝑣</ci></interval></apply><apply id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3"><minus id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.1"></minus><ci id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.2">𝜋</ci><apply id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3"><root id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3a.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3"></root><apply id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2"><divide id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.1"></divide><cn id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.2">1</cn><ci id="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.2.3.3.3.2.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.4.m4.2c">\measuredangle(\overrightarrow{xz},v)\leq\pi-\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.4.m4.2d">∡ ( over→ start_ARG italic_x italic_z end_ARG , italic_v ) ≤ italic_π - square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. Similarly, all <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.5.5.m5.1"><semantics id="S3.Thmtheorem7.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.2.cmml">z</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem7.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.5.5.m5.1b"><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1"><in id="S3.Thmtheorem7.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.1"></in><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.2">𝑧</ci><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.5.5.m5.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.5.5.m5.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\measuredangle(\overrightarrow{xz},v)\leq\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.6.6.m6.2"><semantics id="S3.Thmtheorem7.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem7.p1.6.6.m6.2.3" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.cmml"><mrow id="S3.Thmtheorem7.p1.6.6.m6.2.3.2" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.cmml"><mi id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.2" mathvariant="normal" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.2.cmml">∡</mi><mo id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.1" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.2" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.1.cmml"><mo id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem7.p1.6.6.m6.1.1" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml"><mrow id="S3.Thmtheorem7.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.cmml"><mi id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.2.cmml">x</mi><mo id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.1" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.3" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.3.cmml">z</mi></mrow><mo id="S3.Thmtheorem7.p1.6.6.m6.1.1.1" stretchy="false" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.2.2" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.6.6.m6.2.2" xref="S3.Thmtheorem7.p1.6.6.m6.2.2.cmml">v</mi><mo id="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.2.3" stretchy="false" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem7.p1.6.6.m6.2.3.1" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.1.cmml">≤</mo><msqrt id="S3.Thmtheorem7.p1.6.6.m6.2.3.3" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3.cmml"><mrow id="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.cmml"><mpadded id="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.2" voffset="0.3em" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.2.cmml"><mn id="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.2a" mathsize="70%" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.2.cmml">1</mn></mpadded><mpadded 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xref="S3.Thmtheorem7.p1.6.6.m6.2.3.2.3.2"><apply id="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1"><ci id="S3.Thmtheorem7.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.1">→</ci><apply id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2"><times id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.1"></times><ci id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.2">𝑥</ci><ci id="S3.Thmtheorem7.p1.6.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.2.3">𝑧</ci></apply></apply><ci id="S3.Thmtheorem7.p1.6.6.m6.2.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.2.2">𝑣</ci></interval></apply><apply id="S3.Thmtheorem7.p1.6.6.m6.2.3.3.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3"><root id="S3.Thmtheorem7.p1.6.6.m6.2.3.3a.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3"></root><apply id="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.2.3.3.2"><divide 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xref="S3.Thmtheorem7.p1.7.7.m7.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.7.7.m7.2.3.2.1.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.2.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.7.7.m7.2.3.2.2.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.2.3.2.2">ℋ</ci><ci id="S3.Thmtheorem7.p1.7.7.m7.2.3.2.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.2.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem7.p1.7.7.m7.2.2.2.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.2.2.2.4"><ci id="S3.Thmtheorem7.p1.7.7.m7.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem7.p1.7.7.m7.2.2.2.2.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.7.7.m7.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.7.7.m7.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="202" id="S3.F3.g1" src="x2.png" width="312"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F3.6.3.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S3.F3.4.2" style="font-size:90%;">Sketch of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a>. No point in the red cone is contained in the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.F3.3.1.m1.1"><semantics id="S3.F3.3.1.m1.1b"><msub id="S3.F3.3.1.m1.1.1" xref="S3.F3.3.1.m1.1.1.cmml"><mi id="S3.F3.3.1.m1.1.1.2" mathvariant="normal" xref="S3.F3.3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.F3.3.1.m1.1.1.3" xref="S3.F3.3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F3.3.1.m1.1c"><apply id="S3.F3.3.1.m1.1.1.cmml" xref="S3.F3.3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F3.3.1.m1.1.1.1.cmml" xref="S3.F3.3.1.m1.1.1">subscript</csymbol><ci id="S3.F3.3.1.m1.1.1.2.cmml" xref="S3.F3.3.1.m1.1.1.2">ℓ</ci><ci id="S3.F3.3.1.m1.1.1.3.cmml" xref="S3.F3.3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.3.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.3.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.F3.4.2.m2.2"><semantics id="S3.F3.4.2.m2.2b"><msubsup id="S3.F3.4.2.m2.2.3" xref="S3.F3.4.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F3.4.2.m2.2.3.2.2" xref="S3.F3.4.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.F3.4.2.m2.2.2.2.4" xref="S3.F3.4.2.m2.2.2.2.3.cmml"><mi id="S3.F3.4.2.m2.1.1.1.1" xref="S3.F3.4.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.F3.4.2.m2.2.2.2.4.1" xref="S3.F3.4.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.F3.4.2.m2.2.2.2.2" xref="S3.F3.4.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.F3.4.2.m2.2.3.2.3" xref="S3.F3.4.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.F3.4.2.m2.2c"><apply id="S3.F3.4.2.m2.2.3.cmml" xref="S3.F3.4.2.m2.2.3"><csymbol cd="ambiguous" id="S3.F3.4.2.m2.2.3.1.cmml" xref="S3.F3.4.2.m2.2.3">subscript</csymbol><apply id="S3.F3.4.2.m2.2.3.2.cmml" xref="S3.F3.4.2.m2.2.3"><csymbol cd="ambiguous" id="S3.F3.4.2.m2.2.3.2.1.cmml" xref="S3.F3.4.2.m2.2.3">superscript</csymbol><ci id="S3.F3.4.2.m2.2.3.2.2.cmml" xref="S3.F3.4.2.m2.2.3.2.2">ℋ</ci><ci id="S3.F3.4.2.m2.2.3.2.3.cmml" xref="S3.F3.4.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.F3.4.2.m2.2.2.2.3.cmml" xref="S3.F3.4.2.m2.2.2.2.4"><ci id="S3.F3.4.2.m2.1.1.1.1.cmml" xref="S3.F3.4.2.m2.1.1.1.1">𝑥</ci><ci id="S3.F3.4.2.m2.2.2.2.2.cmml" xref="S3.F3.4.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.4.2.m2.2d">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.4.2.m2.2e">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>, but all points in the green cone are. </span></figcaption> </figure> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.5">With <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a>, it now also seems clear that any <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p6.1.m1.1"><semantics id="S3.SS2.p6.1.m1.1a"><msub id="S3.SS2.p6.1.m1.1.1" xref="S3.SS2.p6.1.m1.1.1.cmml"><mi id="S3.SS2.p6.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p6.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p6.1.m1.1.1.3" xref="S3.SS2.p6.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.1.m1.1b"><apply id="S3.SS2.p6.1.m1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.1.m1.1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p6.1.m1.1.1.2.cmml" xref="S3.SS2.p6.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.p6.1.m1.1.1.3.cmml" xref="S3.SS2.p6.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,-x}" class="ltx_Math" display="inline" id="S3.SS2.p6.2.m2.2"><semantics id="S3.SS2.p6.2.m2.2a"><msubsup id="S3.SS2.p6.2.m2.2.3" xref="S3.SS2.p6.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p6.2.m2.2.3.2.2" xref="S3.SS2.p6.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS2.p6.2.m2.2.2.2.2" xref="S3.SS2.p6.2.m2.2.2.2.3.cmml"><mi id="S3.SS2.p6.2.m2.1.1.1.1" xref="S3.SS2.p6.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.SS2.p6.2.m2.2.2.2.2.2" xref="S3.SS2.p6.2.m2.2.2.2.3.cmml">,</mo><mrow id="S3.SS2.p6.2.m2.2.2.2.2.1" xref="S3.SS2.p6.2.m2.2.2.2.2.1.cmml"><mo id="S3.SS2.p6.2.m2.2.2.2.2.1a" xref="S3.SS2.p6.2.m2.2.2.2.2.1.cmml">−</mo><mi id="S3.SS2.p6.2.m2.2.2.2.2.1.2" xref="S3.SS2.p6.2.m2.2.2.2.2.1.2.cmml">x</mi></mrow></mrow><mi id="S3.SS2.p6.2.m2.2.3.2.3" xref="S3.SS2.p6.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.2.m2.2b"><apply id="S3.SS2.p6.2.m2.2.3.cmml" xref="S3.SS2.p6.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p6.2.m2.2.3.1.cmml" xref="S3.SS2.p6.2.m2.2.3">subscript</csymbol><apply id="S3.SS2.p6.2.m2.2.3.2.cmml" xref="S3.SS2.p6.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p6.2.m2.2.3.2.1.cmml" xref="S3.SS2.p6.2.m2.2.3">superscript</csymbol><ci id="S3.SS2.p6.2.m2.2.3.2.2.cmml" xref="S3.SS2.p6.2.m2.2.3.2.2">ℋ</ci><ci id="S3.SS2.p6.2.m2.2.3.2.3.cmml" xref="S3.SS2.p6.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.SS2.p6.2.m2.2.2.2.3.cmml" xref="S3.SS2.p6.2.m2.2.2.2.2"><ci id="S3.SS2.p6.2.m2.1.1.1.1.cmml" xref="S3.SS2.p6.2.m2.1.1.1.1">𝑥</ci><apply id="S3.SS2.p6.2.m2.2.2.2.2.1.cmml" xref="S3.SS2.p6.2.m2.2.2.2.2.1"><minus id="S3.SS2.p6.2.m2.2.2.2.2.1.1.cmml" xref="S3.SS2.p6.2.m2.2.2.2.2.1"></minus><ci id="S3.SS2.p6.2.m2.2.2.2.2.1.2.cmml" xref="S3.SS2.p6.2.m2.2.2.2.2.1.2">𝑥</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.2.m2.2c">\mathcal{H}^{p}_{x,-x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_x end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p6.3.m3.1"><semantics id="S3.SS2.p6.3.m3.1a"><mi id="S3.SS2.p6.3.m3.1.1" xref="S3.SS2.p6.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.3.m3.1b"><ci id="S3.SS2.p6.3.m3.1.1.cmml" xref="S3.SS2.p6.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.3.m3.1d">italic_x</annotation></semantics></math> far away from the origin should contain every small compact set <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p6.4.m4.1"><semantics id="S3.SS2.p6.4.m4.1a"><mi id="S3.SS2.p6.4.m4.1.1" xref="S3.SS2.p6.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.4.m4.1b"><ci id="S3.SS2.p6.4.m4.1.1.cmml" xref="S3.SS2.p6.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.4.m4.1d">italic_C</annotation></semantics></math> around the origin. <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.8</span></a> makes this precise for <math alttext="C=[0,1]^{d}" class="ltx_Math" display="inline" id="S3.SS2.p6.5.m5.2"><semantics id="S3.SS2.p6.5.m5.2a"><mrow id="S3.SS2.p6.5.m5.2.3" xref="S3.SS2.p6.5.m5.2.3.cmml"><mi id="S3.SS2.p6.5.m5.2.3.2" xref="S3.SS2.p6.5.m5.2.3.2.cmml">C</mi><mo id="S3.SS2.p6.5.m5.2.3.1" xref="S3.SS2.p6.5.m5.2.3.1.cmml">=</mo><msup id="S3.SS2.p6.5.m5.2.3.3" xref="S3.SS2.p6.5.m5.2.3.3.cmml"><mrow id="S3.SS2.p6.5.m5.2.3.3.2.2" xref="S3.SS2.p6.5.m5.2.3.3.2.1.cmml"><mo id="S3.SS2.p6.5.m5.2.3.3.2.2.1" stretchy="false" xref="S3.SS2.p6.5.m5.2.3.3.2.1.cmml">[</mo><mn id="S3.SS2.p6.5.m5.1.1" xref="S3.SS2.p6.5.m5.1.1.cmml">0</mn><mo id="S3.SS2.p6.5.m5.2.3.3.2.2.2" xref="S3.SS2.p6.5.m5.2.3.3.2.1.cmml">,</mo><mn id="S3.SS2.p6.5.m5.2.2" xref="S3.SS2.p6.5.m5.2.2.cmml">1</mn><mo id="S3.SS2.p6.5.m5.2.3.3.2.2.3" stretchy="false" xref="S3.SS2.p6.5.m5.2.3.3.2.1.cmml">]</mo></mrow><mi id="S3.SS2.p6.5.m5.2.3.3.3" xref="S3.SS2.p6.5.m5.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.5.m5.2b"><apply id="S3.SS2.p6.5.m5.2.3.cmml" xref="S3.SS2.p6.5.m5.2.3"><eq id="S3.SS2.p6.5.m5.2.3.1.cmml" xref="S3.SS2.p6.5.m5.2.3.1"></eq><ci id="S3.SS2.p6.5.m5.2.3.2.cmml" xref="S3.SS2.p6.5.m5.2.3.2">𝐶</ci><apply id="S3.SS2.p6.5.m5.2.3.3.cmml" xref="S3.SS2.p6.5.m5.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p6.5.m5.2.3.3.1.cmml" xref="S3.SS2.p6.5.m5.2.3.3">superscript</csymbol><interval closure="closed" id="S3.SS2.p6.5.m5.2.3.3.2.1.cmml" xref="S3.SS2.p6.5.m5.2.3.3.2.2"><cn id="S3.SS2.p6.5.m5.1.1.cmml" type="integer" xref="S3.SS2.p6.5.m5.1.1">0</cn><cn id="S3.SS2.p6.5.m5.2.2.cmml" type="integer" xref="S3.SS2.p6.5.m5.2.2">1</cn></interval><ci id="S3.SS2.p6.5.m5.2.3.3.3.cmml" xref="S3.SS2.p6.5.m5.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.5.m5.2c">C=[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.5.m5.2d">italic_C = [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.1.1.1">Corollary 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem8.p1"> <p class="ltx_p" id="S3.Thmtheorem8.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.4.4">For arbitrary <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.1.m1.3"><semantics id="S3.Thmtheorem8.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem8.p1.1.1.m1.3.4" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem8.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem8.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem8.p1.1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem8.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem8.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem8.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.1.m1.3b"><apply id="S3.Thmtheorem8.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4"><in id="S3.Thmtheorem8.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem8.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem8.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem8.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.2.m2.1"><semantics id="S3.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem8.p1.2.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.2.m2.1b"><apply id="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1"><in id="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2">𝑥</ci><apply id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.2.m2.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\lVert x\rVert_{2}>2d" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.cmml"><msub id="S3.Thmtheorem8.p1.3.3.m3.1.2.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.cmml"><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.1.cmml"><mo fence="true" id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.2.1" rspace="0em" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.1.1.cmml">∥</mo><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml">x</mi><mo fence="true" id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.2.2" lspace="0em" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.1.1.cmml">∥</mo></mrow><mn id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.3.cmml">2</mn></msub><mo id="S3.Thmtheorem8.p1.3.3.m3.1.2.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.1.cmml">></mo><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.2.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.cmml"><mn id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.2.cmml">2</mn><mo id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mi id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.3.m3.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2"><gt id="S3.Thmtheorem8.p1.3.3.m3.1.2.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.1"></gt><apply id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2">subscript</csymbol><apply id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.2"><csymbol cd="latexml" id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.2.2.1">delimited-∥∥</csymbol><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1">𝑥</ci></apply><cn id="S3.Thmtheorem8.p1.3.3.m3.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.2.3">2</cn></apply><apply id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3"><times id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.1"></times><cn id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.2">2</cn><ci id="S3.Thmtheorem8.p1.3.3.m3.1.2.3.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.3.m3.1c">\lVert x\rVert_{2}>2d</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.3.m3.1d">∥ italic_x ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 2 italic_d</annotation></semantics></math>, we have <math alttext="[0,1]^{d}\subseteq\mathcal{H}^{p}_{x,-x}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.4.m4.4"><semantics id="S3.Thmtheorem8.p1.4.4.m4.4a"><mrow id="S3.Thmtheorem8.p1.4.4.m4.4.5" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.cmml"><msup id="S3.Thmtheorem8.p1.4.4.m4.4.5.2" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.cmml"><mrow id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.1.cmml"><mo id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.1.cmml">[</mo><mn id="S3.Thmtheorem8.p1.4.4.m4.3.3" xref="S3.Thmtheorem8.p1.4.4.m4.3.3.cmml">0</mn><mo id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.1.cmml">,</mo><mn id="S3.Thmtheorem8.p1.4.4.m4.4.4" xref="S3.Thmtheorem8.p1.4.4.m4.4.4.cmml">1</mn><mo id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.1.cmml">]</mo></mrow><mi id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.3" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.3.cmml">d</mi></msup><mo id="S3.Thmtheorem8.p1.4.4.m4.4.5.1" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.1.cmml">⊆</mo><msubsup id="S3.Thmtheorem8.p1.4.4.m4.4.5.3" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.3.cmml">,</mo><mrow id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.cmml"><mo id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1a" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.cmml">−</mo><mi id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.2.cmml">x</mi></mrow></mrow><mi id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.3" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.4.m4.4b"><apply id="S3.Thmtheorem8.p1.4.4.m4.4.5.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5"><subset id="S3.Thmtheorem8.p1.4.4.m4.4.5.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.1"></subset><apply id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2">superscript</csymbol><interval closure="closed" id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.2.2"><cn id="S3.Thmtheorem8.p1.4.4.m4.3.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.4.4.m4.3.3">0</cn><cn id="S3.Thmtheorem8.p1.4.4.m4.4.4.cmml" type="integer" xref="S3.Thmtheorem8.p1.4.4.m4.4.4">1</cn></interval><ci id="S3.Thmtheorem8.p1.4.4.m4.4.5.2.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.2.3">𝑑</ci></apply><apply id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3">subscript</csymbol><apply id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3">superscript</csymbol><ci id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.2">ℋ</ci><ci id="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.4.5.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2"><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1.1">𝑥</ci><apply id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1"><minus id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1"></minus><ci id="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.2.2.1.2">𝑥</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.4.m4.4c">[0,1]^{d}\subseteq\mathcal{H}^{p}_{x,-x}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.4.m4.4d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ⊆ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_x end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="202" id="S3.F4.g1" src="x3.png" width="312"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F4.8.4.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S3.F4.6.3" style="font-size:90%;">Sketch of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.8</span></a>. The cube <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S3.F4.4.1.m1.2"><semantics id="S3.F4.4.1.m1.2b"><msup id="S3.F4.4.1.m1.2.3" xref="S3.F4.4.1.m1.2.3.cmml"><mrow id="S3.F4.4.1.m1.2.3.2.2" xref="S3.F4.4.1.m1.2.3.2.1.cmml"><mo id="S3.F4.4.1.m1.2.3.2.2.1" stretchy="false" xref="S3.F4.4.1.m1.2.3.2.1.cmml">[</mo><mn id="S3.F4.4.1.m1.1.1" xref="S3.F4.4.1.m1.1.1.cmml">0</mn><mo id="S3.F4.4.1.m1.2.3.2.2.2" xref="S3.F4.4.1.m1.2.3.2.1.cmml">,</mo><mn id="S3.F4.4.1.m1.2.2" xref="S3.F4.4.1.m1.2.2.cmml">1</mn><mo id="S3.F4.4.1.m1.2.3.2.2.3" stretchy="false" xref="S3.F4.4.1.m1.2.3.2.1.cmml">]</mo></mrow><mi id="S3.F4.4.1.m1.2.3.3" xref="S3.F4.4.1.m1.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.F4.4.1.m1.2c"><apply id="S3.F4.4.1.m1.2.3.cmml" xref="S3.F4.4.1.m1.2.3"><csymbol cd="ambiguous" id="S3.F4.4.1.m1.2.3.1.cmml" xref="S3.F4.4.1.m1.2.3">superscript</csymbol><interval closure="closed" id="S3.F4.4.1.m1.2.3.2.1.cmml" xref="S3.F4.4.1.m1.2.3.2.2"><cn id="S3.F4.4.1.m1.1.1.cmml" type="integer" xref="S3.F4.4.1.m1.1.1">0</cn><cn id="S3.F4.4.1.m1.2.2.cmml" type="integer" xref="S3.F4.4.1.m1.2.2">1</cn></interval><ci id="S3.F4.4.1.m1.2.3.3.cmml" xref="S3.F4.4.1.m1.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.4.1.m1.2d">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.4.1.m1.2e">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is contained in all <math alttext="\mathcal{H}^{p}_{x,-x}" class="ltx_Math" display="inline" id="S3.F4.5.2.m2.2"><semantics id="S3.F4.5.2.m2.2b"><msubsup id="S3.F4.5.2.m2.2.3" xref="S3.F4.5.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.F4.5.2.m2.2.3.2.2" xref="S3.F4.5.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="S3.F4.5.2.m2.2.2.2.2" xref="S3.F4.5.2.m2.2.2.2.3.cmml"><mi id="S3.F4.5.2.m2.1.1.1.1" xref="S3.F4.5.2.m2.1.1.1.1.cmml">x</mi><mo id="S3.F4.5.2.m2.2.2.2.2.2" xref="S3.F4.5.2.m2.2.2.2.3.cmml">,</mo><mrow id="S3.F4.5.2.m2.2.2.2.2.1" xref="S3.F4.5.2.m2.2.2.2.2.1.cmml"><mo id="S3.F4.5.2.m2.2.2.2.2.1b" xref="S3.F4.5.2.m2.2.2.2.2.1.cmml">−</mo><mi id="S3.F4.5.2.m2.2.2.2.2.1.2" xref="S3.F4.5.2.m2.2.2.2.2.1.2.cmml">x</mi></mrow></mrow><mi id="S3.F4.5.2.m2.2.3.2.3" xref="S3.F4.5.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.F4.5.2.m2.2c"><apply id="S3.F4.5.2.m2.2.3.cmml" xref="S3.F4.5.2.m2.2.3"><csymbol cd="ambiguous" id="S3.F4.5.2.m2.2.3.1.cmml" xref="S3.F4.5.2.m2.2.3">subscript</csymbol><apply id="S3.F4.5.2.m2.2.3.2.cmml" xref="S3.F4.5.2.m2.2.3"><csymbol cd="ambiguous" id="S3.F4.5.2.m2.2.3.2.1.cmml" xref="S3.F4.5.2.m2.2.3">superscript</csymbol><ci id="S3.F4.5.2.m2.2.3.2.2.cmml" xref="S3.F4.5.2.m2.2.3.2.2">ℋ</ci><ci id="S3.F4.5.2.m2.2.3.2.3.cmml" xref="S3.F4.5.2.m2.2.3.2.3">𝑝</ci></apply><list id="S3.F4.5.2.m2.2.2.2.3.cmml" xref="S3.F4.5.2.m2.2.2.2.2"><ci id="S3.F4.5.2.m2.1.1.1.1.cmml" 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end_POSTSUBSCRIPT</annotation></semantics></math> large enough.</span></figcaption> </figure> <div class="ltx_para" id="S3.SS2.p7"> <p class="ltx_p" id="S3.SS2.p7.1">Finally, we will need some results on the interaction of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p7.1.m1.1"><semantics id="S3.SS2.p7.1.m1.1a"><msub id="S3.SS2.p7.1.m1.1.1" xref="S3.SS2.p7.1.m1.1.1.cmml"><mi id="S3.SS2.p7.1.m1.1.1.2" mathvariant="normal" xref="S3.SS2.p7.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p7.1.m1.1.1.3" xref="S3.SS2.p7.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.1.m1.1b"><apply id="S3.SS2.p7.1.m1.1.1.cmml" xref="S3.SS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p7.1.m1.1.1.1.cmml" xref="S3.SS2.p7.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p7.1.m1.1.1.2.cmml" xref="S3.SS2.p7.1.m1.1.1.2">ℓ</ci><ci id="S3.SS2.p7.1.m1.1.1.3.cmml" xref="S3.SS2.p7.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces with mass distributions, defined as follows.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.1.1.1">Definition 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2"> </span>(Mass Distribution)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.4.4">We call a measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.1.1.m1.1"><semantics id="S3.Thmtheorem9.p1.1.1.m1.1a"><mi id="S3.Thmtheorem9.p1.1.1.m1.1.1" xref="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.1.1.m1.1b"><ci id="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.2.m2.1"><semantics id="S3.Thmtheorem9.p1.2.2.m2.1a"><msup id="S3.Thmtheorem9.p1.2.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.2.m2.1b"><apply id="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.2">ℝ</ci><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> a mass distribution if it is absolutely continuous with respect to the Lebesgue measure on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.3.m3.1"><semantics id="S3.Thmtheorem9.p1.3.3.m3.1a"><msup id="S3.Thmtheorem9.p1.3.3.m3.1.1" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem9.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.3.3.m3.1b"><apply id="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.2">ℝ</ci><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.3.m3.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and satisfies <math alttext="\mu(\mathbb{R}^{d})<\infty" class="ltx_Math" display="inline" 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xref="S3.Thmtheorem9.p1.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply><infinity id="S3.Thmtheorem9.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.4.m4.1c">\mu(\mathbb{R}^{d})<\infty</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.4.m4.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) < ∞</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p8"> <p class="ltx_p" id="S3.SS2.p8.5">Note that absolute continuity of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p8.1.m1.1"><semantics id="S3.SS2.p8.1.m1.1a"><mi id="S3.SS2.p8.1.m1.1.1" xref="S3.SS2.p8.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.1.m1.1b"><ci id="S3.SS2.p8.1.m1.1.1.cmml" xref="S3.SS2.p8.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.1.m1.1d">italic_μ</annotation></semantics></math> says that all measurable sets <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.p8.2.m2.1"><semantics id="S3.SS2.p8.2.m2.1a"><mi id="S3.SS2.p8.2.m2.1.1" xref="S3.SS2.p8.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.2.m2.1b"><ci id="S3.SS2.p8.2.m2.1.1.cmml" xref="S3.SS2.p8.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.2.m2.1d">italic_A</annotation></semantics></math> with Lebesgue measure <math alttext="0" class="ltx_Math" display="inline" id="S3.SS2.p8.3.m3.1"><semantics id="S3.SS2.p8.3.m3.1a"><mn id="S3.SS2.p8.3.m3.1.1" xref="S3.SS2.p8.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.3.m3.1b"><cn id="S3.SS2.p8.3.m3.1.1.cmml" type="integer" xref="S3.SS2.p8.3.m3.1.1">0</cn></annotation-xml></semantics></math> must also satisfy <math alttext="\mu(A)=0" class="ltx_Math" display="inline" id="S3.SS2.p8.4.m4.1"><semantics id="S3.SS2.p8.4.m4.1a"><mrow id="S3.SS2.p8.4.m4.1.2" xref="S3.SS2.p8.4.m4.1.2.cmml"><mrow id="S3.SS2.p8.4.m4.1.2.2" xref="S3.SS2.p8.4.m4.1.2.2.cmml"><mi id="S3.SS2.p8.4.m4.1.2.2.2" xref="S3.SS2.p8.4.m4.1.2.2.2.cmml">μ</mi><mo id="S3.SS2.p8.4.m4.1.2.2.1" xref="S3.SS2.p8.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS2.p8.4.m4.1.2.2.3.2" xref="S3.SS2.p8.4.m4.1.2.2.cmml"><mo id="S3.SS2.p8.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS2.p8.4.m4.1.2.2.cmml">(</mo><mi id="S3.SS2.p8.4.m4.1.1" xref="S3.SS2.p8.4.m4.1.1.cmml">A</mi><mo id="S3.SS2.p8.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS2.p8.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p8.4.m4.1.2.1" xref="S3.SS2.p8.4.m4.1.2.1.cmml">=</mo><mn id="S3.SS2.p8.4.m4.1.2.3" xref="S3.SS2.p8.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.4.m4.1b"><apply id="S3.SS2.p8.4.m4.1.2.cmml" xref="S3.SS2.p8.4.m4.1.2"><eq id="S3.SS2.p8.4.m4.1.2.1.cmml" xref="S3.SS2.p8.4.m4.1.2.1"></eq><apply id="S3.SS2.p8.4.m4.1.2.2.cmml" xref="S3.SS2.p8.4.m4.1.2.2"><times id="S3.SS2.p8.4.m4.1.2.2.1.cmml" xref="S3.SS2.p8.4.m4.1.2.2.1"></times><ci id="S3.SS2.p8.4.m4.1.2.2.2.cmml" xref="S3.SS2.p8.4.m4.1.2.2.2">𝜇</ci><ci id="S3.SS2.p8.4.m4.1.1.cmml" xref="S3.SS2.p8.4.m4.1.1">𝐴</ci></apply><cn id="S3.SS2.p8.4.m4.1.2.3.cmml" type="integer" xref="S3.SS2.p8.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.4.m4.1c">\mu(A)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.4.m4.1d">italic_μ ( italic_A ) = 0</annotation></semantics></math>. By the Radon–Nikodym theorem, any such mass distribution must have a density, and so one can think of mass distributions as probability distributions with a density (by assuming <math alttext="\mu(\mathbb{R}^{d})=1" class="ltx_Math" display="inline" id="S3.SS2.p8.5.m5.1"><semantics id="S3.SS2.p8.5.m5.1a"><mrow id="S3.SS2.p8.5.m5.1.1" xref="S3.SS2.p8.5.m5.1.1.cmml"><mrow id="S3.SS2.p8.5.m5.1.1.1" xref="S3.SS2.p8.5.m5.1.1.1.cmml"><mi id="S3.SS2.p8.5.m5.1.1.1.3" xref="S3.SS2.p8.5.m5.1.1.1.3.cmml">μ</mi><mo id="S3.SS2.p8.5.m5.1.1.1.2" xref="S3.SS2.p8.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p8.5.m5.1.1.1.1.1" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p8.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS2.p8.5.m5.1.1.1.1.1.1" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p8.5.m5.1.1.1.1.1.1.2" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.p8.5.m5.1.1.1.1.1.1.3" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.p8.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p8.5.m5.1.1.2" xref="S3.SS2.p8.5.m5.1.1.2.cmml">=</mo><mn id="S3.SS2.p8.5.m5.1.1.3" xref="S3.SS2.p8.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.5.m5.1b"><apply id="S3.SS2.p8.5.m5.1.1.cmml" xref="S3.SS2.p8.5.m5.1.1"><eq id="S3.SS2.p8.5.m5.1.1.2.cmml" xref="S3.SS2.p8.5.m5.1.1.2"></eq><apply id="S3.SS2.p8.5.m5.1.1.1.cmml" xref="S3.SS2.p8.5.m5.1.1.1"><times id="S3.SS2.p8.5.m5.1.1.1.2.cmml" xref="S3.SS2.p8.5.m5.1.1.1.2"></times><ci id="S3.SS2.p8.5.m5.1.1.1.3.cmml" xref="S3.SS2.p8.5.m5.1.1.1.3">𝜇</ci><apply id="S3.SS2.p8.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p8.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p8.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S3.SS2.p8.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.SS2.p8.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p8.5.m5.1.1.1.1.1.1.3">𝑑</ci></apply></apply><cn id="S3.SS2.p8.5.m5.1.1.3.cmml" type="integer" xref="S3.SS2.p8.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.5.m5.1c">\mu(\mathbb{R}^{d})=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.5.m5.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>, without loss of generality). Concretely, we will adopt this point of view in the proofs of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.10</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem11" title="Lemma 3.11. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.11</span></a> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS2" title="A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.2</span></a>).</p> </div> <div class="ltx_para" id="S3.SS2.p9"> <p class="ltx_p" id="S3.SS2.p9.6">We can now study how <math alttext="\mu(\mathcal{H}^{p}_{x,v})" class="ltx_Math" display="inline" id="S3.SS2.p9.1.m1.3"><semantics id="S3.SS2.p9.1.m1.3a"><mrow id="S3.SS2.p9.1.m1.3.3" xref="S3.SS2.p9.1.m1.3.3.cmml"><mi id="S3.SS2.p9.1.m1.3.3.3" xref="S3.SS2.p9.1.m1.3.3.3.cmml">μ</mi><mo id="S3.SS2.p9.1.m1.3.3.2" xref="S3.SS2.p9.1.m1.3.3.2.cmml">⁢</mo><mrow id="S3.SS2.p9.1.m1.3.3.1.1" xref="S3.SS2.p9.1.m1.3.3.1.1.1.cmml"><mo id="S3.SS2.p9.1.m1.3.3.1.1.2" stretchy="false" xref="S3.SS2.p9.1.m1.3.3.1.1.1.cmml">(</mo><msubsup id="S3.SS2.p9.1.m1.3.3.1.1.1" xref="S3.SS2.p9.1.m1.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p9.1.m1.3.3.1.1.1.2.2" xref="S3.SS2.p9.1.m1.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.SS2.p9.1.m1.2.2.2.4" xref="S3.SS2.p9.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.p9.1.m1.1.1.1.1" xref="S3.SS2.p9.1.m1.1.1.1.1.cmml">x</mi><mo id="S3.SS2.p9.1.m1.2.2.2.4.1" xref="S3.SS2.p9.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p9.1.m1.2.2.2.2" xref="S3.SS2.p9.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS2.p9.1.m1.3.3.1.1.1.2.3" xref="S3.SS2.p9.1.m1.3.3.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.SS2.p9.1.m1.3.3.1.1.3" stretchy="false" xref="S3.SS2.p9.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.1.m1.3b"><apply id="S3.SS2.p9.1.m1.3.3.cmml" xref="S3.SS2.p9.1.m1.3.3"><times id="S3.SS2.p9.1.m1.3.3.2.cmml" xref="S3.SS2.p9.1.m1.3.3.2"></times><ci id="S3.SS2.p9.1.m1.3.3.3.cmml" xref="S3.SS2.p9.1.m1.3.3.3">𝜇</ci><apply id="S3.SS2.p9.1.m1.3.3.1.1.1.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.1.m1.3.3.1.1.1.1.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1">subscript</csymbol><apply id="S3.SS2.p9.1.m1.3.3.1.1.1.2.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.1.m1.3.3.1.1.1.2.1.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1">superscript</csymbol><ci id="S3.SS2.p9.1.m1.3.3.1.1.1.2.2.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1.1.2.2">ℋ</ci><ci id="S3.SS2.p9.1.m1.3.3.1.1.1.2.3.cmml" xref="S3.SS2.p9.1.m1.3.3.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS2.p9.1.m1.2.2.2.3.cmml" xref="S3.SS2.p9.1.m1.2.2.2.4"><ci id="S3.SS2.p9.1.m1.1.1.1.1.cmml" xref="S3.SS2.p9.1.m1.1.1.1.1">𝑥</ci><ci id="S3.SS2.p9.1.m1.2.2.2.2.cmml" xref="S3.SS2.p9.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.1.m1.3c">\mu(\mathcal{H}^{p}_{x,v})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.1.m1.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> behaves with respect to its arguments <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p9.2.m2.1"><semantics id="S3.SS2.p9.2.m2.1a"><mi id="S3.SS2.p9.2.m2.1.1" xref="S3.SS2.p9.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.2.m2.1b"><ci id="S3.SS2.p9.2.m2.1.1.cmml" xref="S3.SS2.p9.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.2.m2.1d">italic_x</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S3.SS2.p9.3.m3.1"><semantics id="S3.SS2.p9.3.m3.1a"><mi id="S3.SS2.p9.3.m3.1.1" xref="S3.SS2.p9.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.3.m3.1b"><ci id="S3.SS2.p9.3.m3.1.1.cmml" xref="S3.SS2.p9.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.3.m3.1d">italic_v</annotation></semantics></math>. For example, the next lemma says that the mass <math alttext="\mu(\mathcal{H}^{p}_{x,v})" class="ltx_Math" display="inline" id="S3.SS2.p9.4.m4.3"><semantics id="S3.SS2.p9.4.m4.3a"><mrow id="S3.SS2.p9.4.m4.3.3" xref="S3.SS2.p9.4.m4.3.3.cmml"><mi id="S3.SS2.p9.4.m4.3.3.3" xref="S3.SS2.p9.4.m4.3.3.3.cmml">μ</mi><mo id="S3.SS2.p9.4.m4.3.3.2" xref="S3.SS2.p9.4.m4.3.3.2.cmml">⁢</mo><mrow id="S3.SS2.p9.4.m4.3.3.1.1" xref="S3.SS2.p9.4.m4.3.3.1.1.1.cmml"><mo id="S3.SS2.p9.4.m4.3.3.1.1.2" stretchy="false" xref="S3.SS2.p9.4.m4.3.3.1.1.1.cmml">(</mo><msubsup id="S3.SS2.p9.4.m4.3.3.1.1.1" xref="S3.SS2.p9.4.m4.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p9.4.m4.3.3.1.1.1.2.2" xref="S3.SS2.p9.4.m4.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.SS2.p9.4.m4.2.2.2.4" xref="S3.SS2.p9.4.m4.2.2.2.3.cmml"><mi id="S3.SS2.p9.4.m4.1.1.1.1" xref="S3.SS2.p9.4.m4.1.1.1.1.cmml">x</mi><mo id="S3.SS2.p9.4.m4.2.2.2.4.1" xref="S3.SS2.p9.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p9.4.m4.2.2.2.2" xref="S3.SS2.p9.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS2.p9.4.m4.3.3.1.1.1.2.3" xref="S3.SS2.p9.4.m4.3.3.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.SS2.p9.4.m4.3.3.1.1.3" stretchy="false" xref="S3.SS2.p9.4.m4.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.4.m4.3b"><apply id="S3.SS2.p9.4.m4.3.3.cmml" xref="S3.SS2.p9.4.m4.3.3"><times id="S3.SS2.p9.4.m4.3.3.2.cmml" xref="S3.SS2.p9.4.m4.3.3.2"></times><ci id="S3.SS2.p9.4.m4.3.3.3.cmml" xref="S3.SS2.p9.4.m4.3.3.3">𝜇</ci><apply id="S3.SS2.p9.4.m4.3.3.1.1.1.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1">subscript</csymbol><apply id="S3.SS2.p9.4.m4.3.3.1.1.1.2.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.4.m4.3.3.1.1.1.2.1.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1">superscript</csymbol><ci id="S3.SS2.p9.4.m4.3.3.1.1.1.2.2.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1.1.2.2">ℋ</ci><ci id="S3.SS2.p9.4.m4.3.3.1.1.1.2.3.cmml" xref="S3.SS2.p9.4.m4.3.3.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS2.p9.4.m4.2.2.2.3.cmml" xref="S3.SS2.p9.4.m4.2.2.2.4"><ci id="S3.SS2.p9.4.m4.1.1.1.1.cmml" xref="S3.SS2.p9.4.m4.1.1.1.1">𝑥</ci><ci id="S3.SS2.p9.4.m4.2.2.2.2.cmml" xref="S3.SS2.p9.4.m4.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.4.m4.3c">\mu(\mathcal{H}^{p}_{x,v})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.4.m4.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> of an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS2.p9.5.m5.1"><semantics id="S3.SS2.p9.5.m5.1a"><msub id="S3.SS2.p9.5.m5.1.1" xref="S3.SS2.p9.5.m5.1.1.cmml"><mi id="S3.SS2.p9.5.m5.1.1.2" mathvariant="normal" xref="S3.SS2.p9.5.m5.1.1.2.cmml">ℓ</mi><mi id="S3.SS2.p9.5.m5.1.1.3" xref="S3.SS2.p9.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.5.m5.1b"><apply id="S3.SS2.p9.5.m5.1.1.cmml" xref="S3.SS2.p9.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p9.5.m5.1.1.1.cmml" xref="S3.SS2.p9.5.m5.1.1">subscript</csymbol><ci id="S3.SS2.p9.5.m5.1.1.2.cmml" xref="S3.SS2.p9.5.m5.1.1.2">ℓ</ci><ci id="S3.SS2.p9.5.m5.1.1.3.cmml" xref="S3.SS2.p9.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace is continuous under translation of the defining point <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p9.6.m6.1"><semantics id="S3.SS2.p9.6.m6.1a"><mi id="S3.SS2.p9.6.m6.1.1" xref="S3.SS2.p9.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p9.6.m6.1b"><ci id="S3.SS2.p9.6.m6.1.1.cmml" xref="S3.SS2.p9.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p9.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p9.6.m6.1d">italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.1.1.1">Lemma 3.10</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem10.p1"> <p class="ltx_p" id="S3.Thmtheorem10.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem10.p1.5.5">The function <math alttext="f(x)\coloneqq\mu(\mathcal{H}^{p}_{x,v})" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.1.1.m1.4"><semantics id="S3.Thmtheorem10.p1.1.1.m1.4a"><mrow id="S3.Thmtheorem10.p1.1.1.m1.4.4" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.cmml"><mrow id="S3.Thmtheorem10.p1.1.1.m1.4.4.3" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.cmml"><mi id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.2" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.2.cmml">f</mi><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.1" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.3.2" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.cmml"><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.cmml">(</mo><mi id="S3.Thmtheorem10.p1.1.1.m1.3.3" xref="S3.Thmtheorem10.p1.1.1.m1.3.3.cmml">x</mi><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.2" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.2.cmml">≔</mo><mrow id="S3.Thmtheorem10.p1.1.1.m1.4.4.1" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.cmml"><mi id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.3" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.3.cmml">μ</mi><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.2" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.cmml"><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.2" stretchy="false" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.cmml">(</mo><msubsup id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.2" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem10.p1.1.1.m1.2.2.2.4" xref="S3.Thmtheorem10.p1.1.1.m1.2.2.2.3.cmml"><mi id="S3.Thmtheorem10.p1.1.1.m1.1.1.1.1" xref="S3.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml">x</mi><mo id="S3.Thmtheorem10.p1.1.1.m1.2.2.2.4.1" xref="S3.Thmtheorem10.p1.1.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem10.p1.1.1.m1.2.2.2.2" xref="S3.Thmtheorem10.p1.1.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.3" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.3" stretchy="false" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.1.1.m1.4b"><apply id="S3.Thmtheorem10.p1.1.1.m1.4.4.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4"><ci id="S3.Thmtheorem10.p1.1.1.m1.4.4.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.2">≔</ci><apply id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3"><times id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.1"></times><ci id="S3.Thmtheorem10.p1.1.1.m1.4.4.3.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.3.2">𝑓</ci><ci id="S3.Thmtheorem10.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.3.3">𝑥</ci></apply><apply id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1"><times id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.2"></times><ci id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.3.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.3">𝜇</ci><apply id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.2">ℋ</ci><ci id="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.4.4.1.1.1.1.2.3">𝑝</ci></apply><list id="S3.Thmtheorem10.p1.1.1.m1.2.2.2.3.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.2.2.2.4"><ci id="S3.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem10.p1.1.1.m1.2.2.2.2.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.1.1.m1.4c">f(x)\coloneqq\mu(\mathcal{H}^{p}_{x,v})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.1.1.m1.4d">italic_f ( italic_x ) ≔ italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> is continuous in <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.2.2.m2.1"><semantics id="S3.Thmtheorem10.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem10.p1.2.2.m2.1.1" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem10.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem10.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem10.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.2.2.m2.1b"><apply id="S3.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1"><in id="S3.Thmtheorem10.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem10.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.2">𝑥</ci><apply id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.2.2.m2.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> for all mass distributions <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.3.3.m3.1"><semantics id="S3.Thmtheorem10.p1.3.3.m3.1a"><mi id="S3.Thmtheorem10.p1.3.3.m3.1.1" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.3.3.m3.1b"><ci id="S3.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem10.p1.3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.3.3.m3.1d">italic_μ</annotation></semantics></math>, all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.4.4.m4.1"><semantics id="S3.Thmtheorem10.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem10.p1.4.4.m4.1.1" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem10.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem10.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem10.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.cmml"><mi id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.2" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.1" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.3" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.4.4.m4.1b"><apply id="S3.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1"><in id="S3.Thmtheorem10.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem10.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.2">𝑣</ci><apply id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3"><minus id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.1.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.1"></minus><ci id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.2.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.4.4.m4.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.4.4.m4.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, and all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.5.5.m5.3"><semantics id="S3.Thmtheorem10.p1.5.5.m5.3a"><mrow id="S3.Thmtheorem10.p1.5.5.m5.3.4" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.cmml"><mi id="S3.Thmtheorem10.p1.5.5.m5.3.4.2" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.1" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem10.p1.5.5.m5.3.4.3" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.cmml"><mrow id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.2" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem10.p1.5.5.m5.1.1" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.cmml">1</mn><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.2.2" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem10.p1.5.5.m5.2.2" mathvariant="normal" xref="S3.Thmtheorem10.p1.5.5.m5.2.2.cmml">∞</mi><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.1" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.2" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem10.p1.5.5.m5.3.3" mathvariant="normal" xref="S3.Thmtheorem10.p1.5.5.m5.3.3.cmml">∞</mi><mo id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.5.5.m5.3b"><apply id="S3.Thmtheorem10.p1.5.5.m5.3.4.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4"><in id="S3.Thmtheorem10.p1.5.5.m5.3.4.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.1"></in><ci id="S3.Thmtheorem10.p1.5.5.m5.3.4.2.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.2">𝑝</ci><apply id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3"><union id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.2.2"><cn id="S3.Thmtheorem10.p1.5.5.m5.1.1.cmml" type="integer" xref="S3.Thmtheorem10.p1.5.5.m5.1.1">1</cn><infinity id="S3.Thmtheorem10.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.2.2"></infinity></interval><set id="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.4.3.3.2"><infinity id="S3.Thmtheorem10.p1.5.5.m5.3.3.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.5.5.m5.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.5.5.m5.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p10"> <p class="ltx_p" id="S3.SS2.p10.11">For <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.SS2.p10.1.m1.2"><semantics id="S3.SS2.p10.1.m1.2a"><mrow id="S3.SS2.p10.1.m1.2.3" xref="S3.SS2.p10.1.m1.2.3.cmml"><mi id="S3.SS2.p10.1.m1.2.3.2" xref="S3.SS2.p10.1.m1.2.3.2.cmml">p</mi><mo id="S3.SS2.p10.1.m1.2.3.1" xref="S3.SS2.p10.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.SS2.p10.1.m1.2.3.3.2" xref="S3.SS2.p10.1.m1.2.3.3.1.cmml"><mo id="S3.SS2.p10.1.m1.2.3.3.2.1" stretchy="false" xref="S3.SS2.p10.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.SS2.p10.1.m1.1.1" xref="S3.SS2.p10.1.m1.1.1.cmml">1</mn><mo id="S3.SS2.p10.1.m1.2.3.3.2.2" xref="S3.SS2.p10.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS2.p10.1.m1.2.2" mathvariant="normal" xref="S3.SS2.p10.1.m1.2.2.cmml">∞</mi><mo id="S3.SS2.p10.1.m1.2.3.3.2.3" stretchy="false" xref="S3.SS2.p10.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.1.m1.2b"><apply id="S3.SS2.p10.1.m1.2.3.cmml" xref="S3.SS2.p10.1.m1.2.3"><in id="S3.SS2.p10.1.m1.2.3.1.cmml" xref="S3.SS2.p10.1.m1.2.3.1"></in><ci id="S3.SS2.p10.1.m1.2.3.2.cmml" xref="S3.SS2.p10.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.SS2.p10.1.m1.2.3.3.1.cmml" xref="S3.SS2.p10.1.m1.2.3.3.2"><cn id="S3.SS2.p10.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p10.1.m1.1.1">1</cn><infinity id="S3.SS2.p10.1.m1.2.2.cmml" xref="S3.SS2.p10.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, one could also show continuity with respect to <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.SS2.p10.2.m2.1"><semantics id="S3.SS2.p10.2.m2.1a"><mrow id="S3.SS2.p10.2.m2.1.1" xref="S3.SS2.p10.2.m2.1.1.cmml"><mi id="S3.SS2.p10.2.m2.1.1.2" xref="S3.SS2.p10.2.m2.1.1.2.cmml">v</mi><mo id="S3.SS2.p10.2.m2.1.1.1" xref="S3.SS2.p10.2.m2.1.1.1.cmml">∈</mo><msup id="S3.SS2.p10.2.m2.1.1.3" xref="S3.SS2.p10.2.m2.1.1.3.cmml"><mi id="S3.SS2.p10.2.m2.1.1.3.2" xref="S3.SS2.p10.2.m2.1.1.3.2.cmml">S</mi><mrow id="S3.SS2.p10.2.m2.1.1.3.3" xref="S3.SS2.p10.2.m2.1.1.3.3.cmml"><mi id="S3.SS2.p10.2.m2.1.1.3.3.2" xref="S3.SS2.p10.2.m2.1.1.3.3.2.cmml">d</mi><mo id="S3.SS2.p10.2.m2.1.1.3.3.1" xref="S3.SS2.p10.2.m2.1.1.3.3.1.cmml">−</mo><mn id="S3.SS2.p10.2.m2.1.1.3.3.3" xref="S3.SS2.p10.2.m2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.2.m2.1b"><apply id="S3.SS2.p10.2.m2.1.1.cmml" xref="S3.SS2.p10.2.m2.1.1"><in id="S3.SS2.p10.2.m2.1.1.1.cmml" xref="S3.SS2.p10.2.m2.1.1.1"></in><ci id="S3.SS2.p10.2.m2.1.1.2.cmml" xref="S3.SS2.p10.2.m2.1.1.2">𝑣</ci><apply id="S3.SS2.p10.2.m2.1.1.3.cmml" xref="S3.SS2.p10.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p10.2.m2.1.1.3.1.cmml" xref="S3.SS2.p10.2.m2.1.1.3">superscript</csymbol><ci id="S3.SS2.p10.2.m2.1.1.3.2.cmml" xref="S3.SS2.p10.2.m2.1.1.3.2">𝑆</ci><apply id="S3.SS2.p10.2.m2.1.1.3.3.cmml" xref="S3.SS2.p10.2.m2.1.1.3.3"><minus id="S3.SS2.p10.2.m2.1.1.3.3.1.cmml" xref="S3.SS2.p10.2.m2.1.1.3.3.1"></minus><ci id="S3.SS2.p10.2.m2.1.1.3.3.2.cmml" xref="S3.SS2.p10.2.m2.1.1.3.3.2">𝑑</ci><cn id="S3.SS2.p10.2.m2.1.1.3.3.3.cmml" type="integer" xref="S3.SS2.p10.2.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.2.m2.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.2.m2.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> rather than <math alttext="x" class="ltx_Math" display="inline" id="S3.SS2.p10.3.m3.1"><semantics id="S3.SS2.p10.3.m3.1a"><mi id="S3.SS2.p10.3.m3.1.1" xref="S3.SS2.p10.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.3.m3.1b"><ci id="S3.SS2.p10.3.m3.1.1.cmml" xref="S3.SS2.p10.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.3.m3.1d">italic_x</annotation></semantics></math>. Unfortunately, this breaks down for <math alttext="p\in\{1,\infty\}" class="ltx_Math" display="inline" id="S3.SS2.p10.4.m4.2"><semantics id="S3.SS2.p10.4.m4.2a"><mrow id="S3.SS2.p10.4.m4.2.3" xref="S3.SS2.p10.4.m4.2.3.cmml"><mi id="S3.SS2.p10.4.m4.2.3.2" xref="S3.SS2.p10.4.m4.2.3.2.cmml">p</mi><mo id="S3.SS2.p10.4.m4.2.3.1" xref="S3.SS2.p10.4.m4.2.3.1.cmml">∈</mo><mrow id="S3.SS2.p10.4.m4.2.3.3.2" xref="S3.SS2.p10.4.m4.2.3.3.1.cmml"><mo id="S3.SS2.p10.4.m4.2.3.3.2.1" stretchy="false" xref="S3.SS2.p10.4.m4.2.3.3.1.cmml">{</mo><mn id="S3.SS2.p10.4.m4.1.1" xref="S3.SS2.p10.4.m4.1.1.cmml">1</mn><mo id="S3.SS2.p10.4.m4.2.3.3.2.2" xref="S3.SS2.p10.4.m4.2.3.3.1.cmml">,</mo><mi id="S3.SS2.p10.4.m4.2.2" mathvariant="normal" xref="S3.SS2.p10.4.m4.2.2.cmml">∞</mi><mo id="S3.SS2.p10.4.m4.2.3.3.2.3" stretchy="false" xref="S3.SS2.p10.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.4.m4.2b"><apply id="S3.SS2.p10.4.m4.2.3.cmml" xref="S3.SS2.p10.4.m4.2.3"><in id="S3.SS2.p10.4.m4.2.3.1.cmml" xref="S3.SS2.p10.4.m4.2.3.1"></in><ci id="S3.SS2.p10.4.m4.2.3.2.cmml" xref="S3.SS2.p10.4.m4.2.3.2">𝑝</ci><set id="S3.SS2.p10.4.m4.2.3.3.1.cmml" xref="S3.SS2.p10.4.m4.2.3.3.2"><cn id="S3.SS2.p10.4.m4.1.1.cmml" type="integer" xref="S3.SS2.p10.4.m4.1.1">1</cn><infinity id="S3.SS2.p10.4.m4.2.2.cmml" xref="S3.SS2.p10.4.m4.2.2"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.4.m4.2c">p\in\{1,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.4.m4.2d">italic_p ∈ { 1 , ∞ }</annotation></semantics></math> again in some degenerate cases. However, it turns out that a weaker property will suffice for our needs. Concretely, for fixed <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS2.p10.5.m5.1"><semantics id="S3.SS2.p10.5.m5.1a"><mrow id="S3.SS2.p10.5.m5.1.1" xref="S3.SS2.p10.5.m5.1.1.cmml"><mi id="S3.SS2.p10.5.m5.1.1.2" xref="S3.SS2.p10.5.m5.1.1.2.cmml">x</mi><mo id="S3.SS2.p10.5.m5.1.1.1" xref="S3.SS2.p10.5.m5.1.1.1.cmml">∈</mo><msup id="S3.SS2.p10.5.m5.1.1.3" xref="S3.SS2.p10.5.m5.1.1.3.cmml"><mi id="S3.SS2.p10.5.m5.1.1.3.2" xref="S3.SS2.p10.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS2.p10.5.m5.1.1.3.3" xref="S3.SS2.p10.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.5.m5.1b"><apply id="S3.SS2.p10.5.m5.1.1.cmml" xref="S3.SS2.p10.5.m5.1.1"><in id="S3.SS2.p10.5.m5.1.1.1.cmml" xref="S3.SS2.p10.5.m5.1.1.1"></in><ci id="S3.SS2.p10.5.m5.1.1.2.cmml" xref="S3.SS2.p10.5.m5.1.1.2">𝑥</ci><apply id="S3.SS2.p10.5.m5.1.1.3.cmml" xref="S3.SS2.p10.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p10.5.m5.1.1.3.1.cmml" xref="S3.SS2.p10.5.m5.1.1.3">superscript</csymbol><ci id="S3.SS2.p10.5.m5.1.1.3.2.cmml" xref="S3.SS2.p10.5.m5.1.1.3.2">ℝ</ci><ci id="S3.SS2.p10.5.m5.1.1.3.3.cmml" xref="S3.SS2.p10.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.5.m5.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.5.m5.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and some threshold <math alttext="t>0" class="ltx_Math" display="inline" id="S3.SS2.p10.6.m6.1"><semantics id="S3.SS2.p10.6.m6.1a"><mrow id="S3.SS2.p10.6.m6.1.1" xref="S3.SS2.p10.6.m6.1.1.cmml"><mi id="S3.SS2.p10.6.m6.1.1.2" xref="S3.SS2.p10.6.m6.1.1.2.cmml">t</mi><mo id="S3.SS2.p10.6.m6.1.1.1" xref="S3.SS2.p10.6.m6.1.1.1.cmml">></mo><mn id="S3.SS2.p10.6.m6.1.1.3" xref="S3.SS2.p10.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.6.m6.1b"><apply id="S3.SS2.p10.6.m6.1.1.cmml" xref="S3.SS2.p10.6.m6.1.1"><gt id="S3.SS2.p10.6.m6.1.1.1.cmml" xref="S3.SS2.p10.6.m6.1.1.1"></gt><ci id="S3.SS2.p10.6.m6.1.1.2.cmml" xref="S3.SS2.p10.6.m6.1.1.2">𝑡</ci><cn id="S3.SS2.p10.6.m6.1.1.3.cmml" type="integer" xref="S3.SS2.p10.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.6.m6.1c">t>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.6.m6.1d">italic_t > 0</annotation></semantics></math>, consider the set of directions <math alttext="V_{x}\coloneqq\{v\in S^{d-1}\mid\mu(\mathcal{H}^{p}_{x,v})<t\}" class="ltx_Math" display="inline" id="S3.SS2.p10.7.m7.4"><semantics id="S3.SS2.p10.7.m7.4a"><mrow id="S3.SS2.p10.7.m7.4.4" xref="S3.SS2.p10.7.m7.4.4.cmml"><msub id="S3.SS2.p10.7.m7.4.4.4" xref="S3.SS2.p10.7.m7.4.4.4.cmml"><mi id="S3.SS2.p10.7.m7.4.4.4.2" xref="S3.SS2.p10.7.m7.4.4.4.2.cmml">V</mi><mi id="S3.SS2.p10.7.m7.4.4.4.3" xref="S3.SS2.p10.7.m7.4.4.4.3.cmml">x</mi></msub><mo 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S^{d-1}\mid\mu(\mathcal{H}^{p}_{x,v})<t\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.7.m7.4d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ≔ { italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) < italic_t }</annotation></semantics></math> corresponding to halfspaces <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S3.SS2.p10.8.m8.2"><semantics id="S3.SS2.p10.8.m8.2a"><msubsup id="S3.SS2.p10.8.m8.2.3" xref="S3.SS2.p10.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p10.8.m8.2.3.2.2" xref="S3.SS2.p10.8.m8.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS2.p10.8.m8.2.2.2.4" xref="S3.SS2.p10.8.m8.2.2.2.3.cmml"><mi id="S3.SS2.p10.8.m8.1.1.1.1" xref="S3.SS2.p10.8.m8.1.1.1.1.cmml">x</mi><mo id="S3.SS2.p10.8.m8.2.2.2.4.1" xref="S3.SS2.p10.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p10.8.m8.2.2.2.2" xref="S3.SS2.p10.8.m8.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS2.p10.8.m8.2.3.2.3" xref="S3.SS2.p10.8.m8.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.8.m8.2b"><apply id="S3.SS2.p10.8.m8.2.3.cmml" xref="S3.SS2.p10.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS2.p10.8.m8.2.3.1.cmml" xref="S3.SS2.p10.8.m8.2.3">subscript</csymbol><apply id="S3.SS2.p10.8.m8.2.3.2.cmml" xref="S3.SS2.p10.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS2.p10.8.m8.2.3.2.1.cmml" xref="S3.SS2.p10.8.m8.2.3">superscript</csymbol><ci id="S3.SS2.p10.8.m8.2.3.2.2.cmml" xref="S3.SS2.p10.8.m8.2.3.2.2">ℋ</ci><ci id="S3.SS2.p10.8.m8.2.3.2.3.cmml" xref="S3.SS2.p10.8.m8.2.3.2.3">𝑝</ci></apply><list id="S3.SS2.p10.8.m8.2.2.2.3.cmml" xref="S3.SS2.p10.8.m8.2.2.2.4"><ci id="S3.SS2.p10.8.m8.1.1.1.1.cmml" xref="S3.SS2.p10.8.m8.1.1.1.1">𝑥</ci><ci id="S3.SS2.p10.8.m8.2.2.2.2.cmml" xref="S3.SS2.p10.8.m8.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.8.m8.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.8.m8.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> with strictly less than <math alttext="t" class="ltx_Math" display="inline" id="S3.SS2.p10.9.m9.1"><semantics id="S3.SS2.p10.9.m9.1a"><mi id="S3.SS2.p10.9.m9.1.1" xref="S3.SS2.p10.9.m9.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.9.m9.1b"><ci id="S3.SS2.p10.9.m9.1.1.cmml" xref="S3.SS2.p10.9.m9.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.9.m9.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.9.m9.1d">italic_t</annotation></semantics></math> mass. The following lemma says that <math alttext="V_{x}" class="ltx_Math" display="inline" id="S3.SS2.p10.10.m10.1"><semantics id="S3.SS2.p10.10.m10.1a"><msub id="S3.SS2.p10.10.m10.1.1" xref="S3.SS2.p10.10.m10.1.1.cmml"><mi id="S3.SS2.p10.10.m10.1.1.2" xref="S3.SS2.p10.10.m10.1.1.2.cmml">V</mi><mi id="S3.SS2.p10.10.m10.1.1.3" xref="S3.SS2.p10.10.m10.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.10.m10.1b"><apply id="S3.SS2.p10.10.m10.1.1.cmml" xref="S3.SS2.p10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.10.m10.1.1.1.cmml" xref="S3.SS2.p10.10.m10.1.1">subscript</csymbol><ci id="S3.SS2.p10.10.m10.1.1.2.cmml" xref="S3.SS2.p10.10.m10.1.1.2">𝑉</ci><ci id="S3.SS2.p10.10.m10.1.1.3.cmml" xref="S3.SS2.p10.10.m10.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.10.m10.1c">V_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.10.m10.1d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is an open subset of <math alttext="S^{d-1}" class="ltx_Math" display="inline" id="S3.SS2.p10.11.m11.1"><semantics id="S3.SS2.p10.11.m11.1a"><msup id="S3.SS2.p10.11.m11.1.1" xref="S3.SS2.p10.11.m11.1.1.cmml"><mi id="S3.SS2.p10.11.m11.1.1.2" xref="S3.SS2.p10.11.m11.1.1.2.cmml">S</mi><mrow id="S3.SS2.p10.11.m11.1.1.3" xref="S3.SS2.p10.11.m11.1.1.3.cmml"><mi id="S3.SS2.p10.11.m11.1.1.3.2" xref="S3.SS2.p10.11.m11.1.1.3.2.cmml">d</mi><mo id="S3.SS2.p10.11.m11.1.1.3.1" xref="S3.SS2.p10.11.m11.1.1.3.1.cmml">−</mo><mn id="S3.SS2.p10.11.m11.1.1.3.3" xref="S3.SS2.p10.11.m11.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p10.11.m11.1b"><apply id="S3.SS2.p10.11.m11.1.1.cmml" xref="S3.SS2.p10.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS2.p10.11.m11.1.1.1.cmml" xref="S3.SS2.p10.11.m11.1.1">superscript</csymbol><ci id="S3.SS2.p10.11.m11.1.1.2.cmml" xref="S3.SS2.p10.11.m11.1.1.2">𝑆</ci><apply id="S3.SS2.p10.11.m11.1.1.3.cmml" xref="S3.SS2.p10.11.m11.1.1.3"><minus id="S3.SS2.p10.11.m11.1.1.3.1.cmml" xref="S3.SS2.p10.11.m11.1.1.3.1"></minus><ci id="S3.SS2.p10.11.m11.1.1.3.2.cmml" xref="S3.SS2.p10.11.m11.1.1.3.2">𝑑</ci><cn id="S3.SS2.p10.11.m11.1.1.3.3.cmml" type="integer" xref="S3.SS2.p10.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p10.11.m11.1c">S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p10.11.m11.1d">italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem11.1.1.1">Lemma 3.11</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem11.p1"> <p 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id="S3.Thmtheorem11.p1.1.1.m1.3.3.1.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.1.1.m1.3.3.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2"><lt id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.2"></lt><apply id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1"><times id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.2"></times><ci id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.3.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.3">𝜇</ci><apply id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.2">ℋ</ci><ci id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.1.1.1.1.2.3">𝑝</ci></apply><list id="S3.Thmtheorem11.p1.1.1.m1.2.2.2.3.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.2.2.2.4"><ci id="S3.Thmtheorem11.p1.1.1.m1.1.1.1.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.1.1.1.1">𝑥</ci><ci id="S3.Thmtheorem11.p1.1.1.m1.2.2.2.2.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.2.2.2.2">𝑣</ci></list></apply></apply><ci id="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.3.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.4.4.2.2.2.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.1.1.m1.4c">V_{x}=\{v\in S^{d-1}\mid\mu(\mathcal{H}^{p}_{x,v})<t\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.1.1.m1.4d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = { italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) < italic_t }</annotation></semantics></math> is an open subset of <math alttext="S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.2.2.m2.1"><semantics id="S3.Thmtheorem11.p1.2.2.m2.1a"><msup id="S3.Thmtheorem11.p1.2.2.m2.1.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.2.cmml">S</mi><mrow id="S3.Thmtheorem11.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.2.cmml">d</mi><mo id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.1.cmml">−</mo><mn id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.2.2.m2.1b"><apply id="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem11.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.2">𝑆</ci><apply id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3"><minus id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.1"></minus><ci id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.2">𝑑</ci><cn id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.2.2.m2.1c">S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.2.2.m2.1d">italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> for all mass distributions <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.3.3.m3.1"><semantics id="S3.Thmtheorem11.p1.3.3.m3.1a"><mi id="S3.Thmtheorem11.p1.3.3.m3.1.1" xref="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.3.3.m3.1b"><ci id="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.3.3.m3.1d">italic_μ</annotation></semantics></math>, all <math alttext="t>0" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.4.4.m4.1"><semantics id="S3.Thmtheorem11.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem11.p1.4.4.m4.1.1" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem11.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.2.cmml">t</mi><mo id="S3.Thmtheorem11.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml">></mo><mn id="S3.Thmtheorem11.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.4.4.m4.1b"><apply id="S3.Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.2">𝑡</ci><cn id="S3.Thmtheorem11.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.4.4.m4.1c">t>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.4.4.m4.1d">italic_t > 0</annotation></semantics></math>, all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.5.5.m5.3"><semantics id="S3.Thmtheorem11.p1.5.5.m5.3a"><mrow id="S3.Thmtheorem11.p1.5.5.m5.3.4" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.cmml"><mi id="S3.Thmtheorem11.p1.5.5.m5.3.4.2" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.1" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem11.p1.5.5.m5.3.4.3" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.cmml"><mrow id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.2" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem11.p1.5.5.m5.1.1" xref="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml">1</mn><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.2.2" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem11.p1.5.5.m5.2.2" mathvariant="normal" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.cmml">∞</mi><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.1" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.2" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem11.p1.5.5.m5.3.3" mathvariant="normal" xref="S3.Thmtheorem11.p1.5.5.m5.3.3.cmml">∞</mi><mo id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.5.5.m5.3b"><apply id="S3.Thmtheorem11.p1.5.5.m5.3.4.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4"><in id="S3.Thmtheorem11.p1.5.5.m5.3.4.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.1"></in><ci id="S3.Thmtheorem11.p1.5.5.m5.3.4.2.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.2">𝑝</ci><apply id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3"><union id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.2.2"><cn id="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml" type="integer" xref="S3.Thmtheorem11.p1.5.5.m5.1.1">1</cn><infinity id="S3.Thmtheorem11.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.2.2"></infinity></interval><set id="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.4.3.3.2"><infinity id="S3.Thmtheorem11.p1.5.5.m5.3.3.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.5.5.m5.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.5.5.m5.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, and all <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.6.6.m6.1"><semantics id="S3.Thmtheorem11.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem11.p1.6.6.m6.1.1" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem11.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem11.p1.6.6.m6.1.1.1" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem11.p1.6.6.m6.1.1.3" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3.cmml"><mi id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.2" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.3" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.6.6.m6.1b"><apply id="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1"><in id="S3.Thmtheorem11.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.1"></in><ci id="S3.Thmtheorem11.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.2">𝑥</ci><apply id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.1.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.2.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem11.p1.6.6.m6.1.1.3.3.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.6.6.m6.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.6.6.m6.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.1.m1.1"><semantics id="S3.SS3.1.m1.1b"><msub id="S3.SS3.1.m1.1.1" xref="S3.SS3.1.m1.1.1.cmml"><mi id="S3.SS3.1.m1.1.1.2" mathvariant="normal" xref="S3.SS3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.1.m1.1.1.3" xref="S3.SS3.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.1.m1.1c"><apply id="S3.SS3.1.m1.1.1.cmml" xref="S3.SS3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.1.m1.1.1.1.cmml" xref="S3.SS3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.1.m1.1.1.2.cmml" xref="S3.SS3.1.m1.1.1.2">ℓ</ci><ci id="S3.SS3.1.m1.1.1.3.cmml" xref="S3.SS3.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoints of Mass Distributions</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">Using our notion of limit halfspaces, we can state the classical (Euclidean) centerpoint theorem for mass distributions as follows.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem12.1.1.1">Theorem 3.12</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem12.2.2"> </span>(Euclidean Centerpoint Theorem for Mass Distributions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib31" title="">31</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem12.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem12.p1"> <p class="ltx_p" id="S3.Thmtheorem12.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem12.p1.5.5">Let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem12.p1.1.1.m1.1"><semantics id="S3.Thmtheorem12.p1.1.1.m1.1a"><mi id="S3.Thmtheorem12.p1.1.1.m1.1.1" xref="S3.Thmtheorem12.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem12.p1.1.1.m1.1b"><ci id="S3.Thmtheorem12.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem12.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem12.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem12.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> be a mass distribution on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem12.p1.2.2.m2.1"><semantics id="S3.Thmtheorem12.p1.2.2.m2.1a"><msup id="S3.Thmtheorem12.p1.2.2.m2.1.1" xref="S3.Thmtheorem12.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem12.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem12.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem12.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem12.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem12.p1.2.2.m2.1b"><apply id="S3.Thmtheorem12.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem12.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem12.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem12.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem12.p1.2.2.m2.1.1.2">ℝ</ci><ci id="S3.Thmtheorem12.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem12.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem12.p1.2.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem12.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with bounded support. There exists <math alttext="c\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem12.p1.3.3.m3.1"><semantics id="S3.Thmtheorem12.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem12.p1.3.3.m3.1.1" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem12.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem12.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem12.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem12.p1.3.3.m3.1b"><apply id="S3.Thmtheorem12.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1"><in id="S3.Thmtheorem12.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem12.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.2">𝑐</ci><apply id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem12.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem12.p1.3.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem12.p1.3.3.m3.1c">c\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem12.p1.3.3.m3.1d">italic_c ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\mu(\mathcal{H}^{2}_{c,v})\geq\frac{1}{d+1}\mu(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.Thmtheorem12.p1.4.4.m4.4"><semantics id="S3.Thmtheorem12.p1.4.4.m4.4a"><mrow id="S3.Thmtheorem12.p1.4.4.m4.4.4" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.cmml"><mrow id="S3.Thmtheorem12.p1.4.4.m4.3.3.1" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.cmml"><mi id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.3" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.3.cmml">μ</mi><mo id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.2" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.cmml"><mo id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.2" stretchy="false" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.cmml">(</mo><msubsup id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.2" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem12.p1.4.4.m4.2.2.2.4" xref="S3.Thmtheorem12.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem12.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem12.p1.4.4.m4.1.1.1.1.cmml">c</mi><mo id="S3.Thmtheorem12.p1.4.4.m4.2.2.2.4.1" xref="S3.Thmtheorem12.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem12.p1.4.4.m4.2.2.2.2" xref="S3.Thmtheorem12.p1.4.4.m4.2.2.2.2.cmml">v</mi></mrow><mn id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.3" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.3" stretchy="false" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.3" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.3.cmml">≥</mo><mrow id="S3.Thmtheorem12.p1.4.4.m4.4.4.2" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.cmml"><mfrac id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.cmml"><mn id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.2" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.2.cmml">1</mn><mrow id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.cmml"><mi id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.2" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.1" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.1.cmml">+</mo><mn id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.3" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2.cmml">⁢</mo><mi id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.4" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.4.cmml">μ</mi><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2a" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.cmml"><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.2" stretchy="false" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.cmml"><mi id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.2" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.3" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.3" stretchy="false" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem12.p1.4.4.m4.4b"><apply id="S3.Thmtheorem12.p1.4.4.m4.4.4.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4"><geq id="S3.Thmtheorem12.p1.4.4.m4.4.4.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.3"></geq><apply id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1"><times id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.2"></times><ci id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.3">𝜇</ci><apply id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.2">ℋ</ci><cn id="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem12.p1.4.4.m4.3.3.1.1.1.1.2.3">2</cn></apply><list id="S3.Thmtheorem12.p1.4.4.m4.2.2.2.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.2.2.2.4"><ci id="S3.Thmtheorem12.p1.4.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.1.1.1.1">𝑐</ci><ci id="S3.Thmtheorem12.p1.4.4.m4.2.2.2.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.2.2.2.2">𝑣</ci></list></apply></apply><apply id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2"><times id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.2"></times><apply id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3"><divide id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3"></divide><cn id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.2.cmml" type="integer" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.2">1</cn><apply id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3"><plus id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.1"></plus><ci id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.2">𝑑</ci><cn id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.3.cmml" type="integer" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.3.3.3">1</cn></apply></apply><ci id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.4.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.4">𝜇</ci><apply id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.1.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1">superscript</csymbol><ci id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.2.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.3.cmml" xref="S3.Thmtheorem12.p1.4.4.m4.4.4.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem12.p1.4.4.m4.4c">\mu(\mathcal{H}^{2}_{c,v})\geq\frac{1}{d+1}\mu(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem12.p1.4.4.m4.4d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ) ≥ divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem12.p1.5.5.m5.1"><semantics id="S3.Thmtheorem12.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem12.p1.5.5.m5.1.1" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem12.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem12.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem12.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.cmml"><mi id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.1" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem12.p1.5.5.m5.1b"><apply id="S3.Thmtheorem12.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1"><in id="S3.Thmtheorem12.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.1"></in><ci id="S3.Thmtheorem12.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.2">𝑣</ci><apply id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3"><minus id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.1"></minus><ci id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem12.p1.5.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem12.p1.5.5.m5.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem12.p1.5.5.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.6">We call such a point <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mi id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><ci id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_c</annotation></semantics></math> an <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.2.1"><math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.1.m1.1"><semantics id="S3.SS3.p2.2.1.m1.1a"><msub id="S3.SS3.p2.2.1.m1.1.1" xref="S3.SS3.p2.2.1.m1.1.1.cmml"><mi id="S3.SS3.p2.2.1.m1.1.1.2" mathvariant="normal" xref="S3.SS3.p2.2.1.m1.1.1.2.cmml">ℓ</mi><mn id="S3.SS3.p2.2.1.m1.1.1.3" xref="S3.SS3.p2.2.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.1.m1.1b"><apply id="S3.SS3.p2.2.1.m1.1.1.cmml" xref="S3.SS3.p2.2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.2.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p2.2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.2.1.m1.1.1.2">ℓ</ci><cn id="S3.SS3.p2.2.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p2.2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.1.m1.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint</em> of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m2.1"><semantics id="S3.SS3.p2.3.m2.1a"><mi id="S3.SS3.p2.3.m2.1.1" xref="S3.SS3.p2.3.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m2.1b"><ci id="S3.SS3.p2.3.m2.1.1.cmml" xref="S3.SS3.p2.3.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m2.1d">italic_μ</annotation></semantics></math>. In this section, we will prove existence of an <em class="ltx_emph ltx_font_italic" id="S3.SS3.p2.4.2"><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p2.4.2.m1.1"><semantics id="S3.SS3.p2.4.2.m1.1a"><msub id="S3.SS3.p2.4.2.m1.1.1" xref="S3.SS3.p2.4.2.m1.1.1.cmml"><mi id="S3.SS3.p2.4.2.m1.1.1.2" mathvariant="normal" xref="S3.SS3.p2.4.2.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p2.4.2.m1.1.1.3" xref="S3.SS3.p2.4.2.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.4.2.m1.1b"><apply id="S3.SS3.p2.4.2.m1.1.1.cmml" xref="S3.SS3.p2.4.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.4.2.m1.1.1.1.cmml" xref="S3.SS3.p2.4.2.m1.1.1">subscript</csymbol><ci id="S3.SS3.p2.4.2.m1.1.1.2.cmml" xref="S3.SS3.p2.4.2.m1.1.1.2">ℓ</ci><ci id="S3.SS3.p2.4.2.m1.1.1.3.cmml" xref="S3.SS3.p2.4.2.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.2.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.2.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint</em> of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m3.1"><semantics id="S3.SS3.p2.5.m3.1a"><mi id="S3.SS3.p2.5.m3.1.1" xref="S3.SS3.p2.5.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m3.1b"><ci id="S3.SS3.p2.5.m3.1.1.cmml" xref="S3.SS3.p2.5.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m3.1d">italic_μ</annotation></semantics></math> for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m4.3"><semantics id="S3.SS3.p2.6.m4.3a"><mrow id="S3.SS3.p2.6.m4.3.4" xref="S3.SS3.p2.6.m4.3.4.cmml"><mi id="S3.SS3.p2.6.m4.3.4.2" xref="S3.SS3.p2.6.m4.3.4.2.cmml">p</mi><mo id="S3.SS3.p2.6.m4.3.4.1" xref="S3.SS3.p2.6.m4.3.4.1.cmml">∈</mo><mrow id="S3.SS3.p2.6.m4.3.4.3" xref="S3.SS3.p2.6.m4.3.4.3.cmml"><mrow id="S3.SS3.p2.6.m4.3.4.3.2.2" xref="S3.SS3.p2.6.m4.3.4.3.2.1.cmml"><mo id="S3.SS3.p2.6.m4.3.4.3.2.2.1" stretchy="false" xref="S3.SS3.p2.6.m4.3.4.3.2.1.cmml">[</mo><mn id="S3.SS3.p2.6.m4.1.1" xref="S3.SS3.p2.6.m4.1.1.cmml">1</mn><mo id="S3.SS3.p2.6.m4.3.4.3.2.2.2" xref="S3.SS3.p2.6.m4.3.4.3.2.1.cmml">,</mo><mi id="S3.SS3.p2.6.m4.2.2" mathvariant="normal" xref="S3.SS3.p2.6.m4.2.2.cmml">∞</mi><mo id="S3.SS3.p2.6.m4.3.4.3.2.2.3" stretchy="false" xref="S3.SS3.p2.6.m4.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.SS3.p2.6.m4.3.4.3.1" xref="S3.SS3.p2.6.m4.3.4.3.1.cmml">∪</mo><mrow id="S3.SS3.p2.6.m4.3.4.3.3.2" xref="S3.SS3.p2.6.m4.3.4.3.3.1.cmml"><mo id="S3.SS3.p2.6.m4.3.4.3.3.2.1" stretchy="false" xref="S3.SS3.p2.6.m4.3.4.3.3.1.cmml">{</mo><mi id="S3.SS3.p2.6.m4.3.3" mathvariant="normal" xref="S3.SS3.p2.6.m4.3.3.cmml">∞</mi><mo id="S3.SS3.p2.6.m4.3.4.3.3.2.2" stretchy="false" xref="S3.SS3.p2.6.m4.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m4.3b"><apply id="S3.SS3.p2.6.m4.3.4.cmml" xref="S3.SS3.p2.6.m4.3.4"><in id="S3.SS3.p2.6.m4.3.4.1.cmml" xref="S3.SS3.p2.6.m4.3.4.1"></in><ci id="S3.SS3.p2.6.m4.3.4.2.cmml" xref="S3.SS3.p2.6.m4.3.4.2">𝑝</ci><apply id="S3.SS3.p2.6.m4.3.4.3.cmml" xref="S3.SS3.p2.6.m4.3.4.3"><union id="S3.SS3.p2.6.m4.3.4.3.1.cmml" xref="S3.SS3.p2.6.m4.3.4.3.1"></union><interval closure="closed-open" id="S3.SS3.p2.6.m4.3.4.3.2.1.cmml" xref="S3.SS3.p2.6.m4.3.4.3.2.2"><cn id="S3.SS3.p2.6.m4.1.1.cmml" type="integer" xref="S3.SS3.p2.6.m4.1.1">1</cn><infinity id="S3.SS3.p2.6.m4.2.2.cmml" xref="S3.SS3.p2.6.m4.2.2"></infinity></interval><set id="S3.SS3.p2.6.m4.3.4.3.3.1.cmml" xref="S3.SS3.p2.6.m4.3.4.3.3.2"><infinity id="S3.SS3.p2.6.m4.3.3.cmml" xref="S3.SS3.p2.6.m4.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m4.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m4.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, as stated in the following theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem13.2.1.1">Theorem 3.13</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem13.3.2"> </span>(<math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem13.1.m1.1"><semantics id="S3.Thmtheorem13.1.m1.1b"><msub id="S3.Thmtheorem13.1.m1.1.1" xref="S3.Thmtheorem13.1.m1.1.1.cmml"><mi id="S3.Thmtheorem13.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem13.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem13.1.m1.1.1.3" xref="S3.Thmtheorem13.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.1.m1.1c"><apply id="S3.Thmtheorem13.1.m1.1.1.cmml" xref="S3.Thmtheorem13.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem13.1.m1.1.1.1.cmml" xref="S3.Thmtheorem13.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem13.1.m1.1.1.2.cmml" xref="S3.Thmtheorem13.1.m1.1.1.2">ℓ</ci><ci id="S3.Thmtheorem13.1.m1.1.1.3.cmml" xref="S3.Thmtheorem13.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoint Theorem for Mass Distributions)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem13.4.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem13.p1"> <p class="ltx_p" id="S3.Thmtheorem13.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem13.p1.6.6">Let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.1.1.m1.1"><semantics id="S3.Thmtheorem13.p1.1.1.m1.1a"><mi id="S3.Thmtheorem13.p1.1.1.m1.1.1" xref="S3.Thmtheorem13.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.1.1.m1.1b"><ci id="S3.Thmtheorem13.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem13.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> be a mass distribution on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.2.2.m2.1"><semantics id="S3.Thmtheorem13.p1.2.2.m2.1a"><msup id="S3.Thmtheorem13.p1.2.2.m2.1.1" xref="S3.Thmtheorem13.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem13.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem13.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem13.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem13.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.2.2.m2.1b"><apply id="S3.Thmtheorem13.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem13.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem13.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem13.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem13.p1.2.2.m2.1.1.2">ℝ</ci><ci id="S3.Thmtheorem13.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem13.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.2.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with bounded support, and let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.3.3.m3.3"><semantics id="S3.Thmtheorem13.p1.3.3.m3.3a"><mrow id="S3.Thmtheorem13.p1.3.3.m3.3.4" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.cmml"><mi id="S3.Thmtheorem13.p1.3.3.m3.3.4.2" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.1" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem13.p1.3.3.m3.3.4.3" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.cmml"><mrow id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.2" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem13.p1.3.3.m3.1.1" xref="S3.Thmtheorem13.p1.3.3.m3.1.1.cmml">1</mn><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.2.2" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem13.p1.3.3.m3.2.2" mathvariant="normal" xref="S3.Thmtheorem13.p1.3.3.m3.2.2.cmml">∞</mi><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.1" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.2" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem13.p1.3.3.m3.3.3" mathvariant="normal" xref="S3.Thmtheorem13.p1.3.3.m3.3.3.cmml">∞</mi><mo id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.3.3.m3.3b"><apply id="S3.Thmtheorem13.p1.3.3.m3.3.4.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4"><in id="S3.Thmtheorem13.p1.3.3.m3.3.4.1.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.1"></in><ci id="S3.Thmtheorem13.p1.3.3.m3.3.4.2.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.2">𝑝</ci><apply id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3"><union id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.1.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.1.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.2.2"><cn id="S3.Thmtheorem13.p1.3.3.m3.1.1.cmml" type="integer" xref="S3.Thmtheorem13.p1.3.3.m3.1.1">1</cn><infinity id="S3.Thmtheorem13.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.2.2"></infinity></interval><set id="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.1.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.4.3.3.2"><infinity id="S3.Thmtheorem13.p1.3.3.m3.3.3.cmml" xref="S3.Thmtheorem13.p1.3.3.m3.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.3.3.m3.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.3.3.m3.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary. There exists <math alttext="c\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.4.4.m4.1"><semantics id="S3.Thmtheorem13.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem13.p1.4.4.m4.1.1" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem13.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem13.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem13.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.4.4.m4.1b"><apply id="S3.Thmtheorem13.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1"><in id="S3.Thmtheorem13.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem13.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.2">𝑐</ci><apply id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem13.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem13.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.4.4.m4.1c">c\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.4.4.m4.1d">italic_c ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\mu(\mathcal{H}^{p}_{c,v})\geq\frac{1}{d+1}\mu(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.5.5.m5.4"><semantics id="S3.Thmtheorem13.p1.5.5.m5.4a"><mrow id="S3.Thmtheorem13.p1.5.5.m5.4.4" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.cmml"><mrow id="S3.Thmtheorem13.p1.5.5.m5.3.3.1" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.cmml"><mi id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.3" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.3.cmml">μ</mi><mo id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.2" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.cmml"><mo id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.2" stretchy="false" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.cmml">(</mo><msubsup id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.2" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem13.p1.5.5.m5.2.2.2.4" xref="S3.Thmtheorem13.p1.5.5.m5.2.2.2.3.cmml"><mi id="S3.Thmtheorem13.p1.5.5.m5.1.1.1.1" xref="S3.Thmtheorem13.p1.5.5.m5.1.1.1.1.cmml">c</mi><mo id="S3.Thmtheorem13.p1.5.5.m5.2.2.2.4.1" xref="S3.Thmtheorem13.p1.5.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem13.p1.5.5.m5.2.2.2.2" xref="S3.Thmtheorem13.p1.5.5.m5.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.3" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.3" stretchy="false" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.3" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.3.cmml">≥</mo><mrow id="S3.Thmtheorem13.p1.5.5.m5.4.4.2" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.cmml"><mfrac id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.cmml"><mn id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.2" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.2.cmml">1</mn><mrow id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.cmml"><mi id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.2" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.1" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.1.cmml">+</mo><mn id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.3" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2.cmml">⁢</mo><mi id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.4" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.4.cmml">μ</mi><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2a" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.cmml"><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.2" stretchy="false" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.cmml"><mi id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.2" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.3" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.3" stretchy="false" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.5.5.m5.4b"><apply id="S3.Thmtheorem13.p1.5.5.m5.4.4.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4"><geq id="S3.Thmtheorem13.p1.5.5.m5.4.4.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.3"></geq><apply id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1"><times id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.2"></times><ci id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.3">𝜇</ci><apply id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.2">ℋ</ci><ci id="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.3.3.1.1.1.1.2.3">𝑝</ci></apply><list id="S3.Thmtheorem13.p1.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.2.2.2.4"><ci id="S3.Thmtheorem13.p1.5.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.1.1.1.1">𝑐</ci><ci id="S3.Thmtheorem13.p1.5.5.m5.2.2.2.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.2.2.2.2">𝑣</ci></list></apply></apply><apply id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2"><times id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.2"></times><apply id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3"><divide id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3"></divide><cn id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.2.cmml" type="integer" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.2">1</cn><apply id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3"><plus id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.1"></plus><ci id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.2">𝑑</ci><cn id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.3.cmml" type="integer" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.3.3.3">1</cn></apply></apply><ci id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.4.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.4">𝜇</ci><apply id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.1.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1">superscript</csymbol><ci id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.2.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.3.cmml" xref="S3.Thmtheorem13.p1.5.5.m5.4.4.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.5.5.m5.4c">\mu(\mathcal{H}^{p}_{c,v})\geq\frac{1}{d+1}\mu(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.5.5.m5.4d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ) ≥ divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem13.p1.6.6.m6.1"><semantics id="S3.Thmtheorem13.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem13.p1.6.6.m6.1.1" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem13.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem13.p1.6.6.m6.1.1.1" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem13.p1.6.6.m6.1.1.3" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.cmml"><mi id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.2" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.cmml"><mi id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.2" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.1" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.3" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem13.p1.6.6.m6.1b"><apply id="S3.Thmtheorem13.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1"><in id="S3.Thmtheorem13.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.1"></in><ci id="S3.Thmtheorem13.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.2">𝑣</ci><apply id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.1.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.2.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3"><minus id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.1.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.1"></minus><ci id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.2.cmml" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem13.p1.6.6.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem13.p1.6.6.m6.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem13.p1.6.6.m6.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.5">A classical and simple proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem12" title="Theorem 3.12 (Euclidean Centerpoint Theorem for Mass Distributions [31]). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.12</span></a> uses Helly’s theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib18" title="">18</a>]</cite> to show that all <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><msub id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" mathvariant="normal" xref="S3.SS3.p3.1.m1.1.1.2.cmml">ℓ</mi><mn id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">ℓ</ci><cn id="S3.SS3.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces containing strictly more than a <math alttext="\frac{d}{d+1}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mfrac id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml"><mi id="S3.SS3.p3.2.m2.1.1.2" xref="S3.SS3.p3.2.m2.1.1.2.cmml">d</mi><mrow id="S3.SS3.p3.2.m2.1.1.3" xref="S3.SS3.p3.2.m2.1.1.3.cmml"><mi id="S3.SS3.p3.2.m2.1.1.3.2" xref="S3.SS3.p3.2.m2.1.1.3.2.cmml">d</mi><mo id="S3.SS3.p3.2.m2.1.1.3.1" xref="S3.SS3.p3.2.m2.1.1.3.1.cmml">+</mo><mn id="S3.SS3.p3.2.m2.1.1.3.3" xref="S3.SS3.p3.2.m2.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><apply id="S3.SS3.p3.2.m2.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1"><divide id="S3.SS3.p3.2.m2.1.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1"></divide><ci id="S3.SS3.p3.2.m2.1.1.2.cmml" xref="S3.SS3.p3.2.m2.1.1.2">𝑑</ci><apply id="S3.SS3.p3.2.m2.1.1.3.cmml" xref="S3.SS3.p3.2.m2.1.1.3"><plus id="S3.SS3.p3.2.m2.1.1.3.1.cmml" xref="S3.SS3.p3.2.m2.1.1.3.1"></plus><ci id="S3.SS3.p3.2.m2.1.1.3.2.cmml" xref="S3.SS3.p3.2.m2.1.1.3.2">𝑑</ci><cn id="S3.SS3.p3.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS3.p3.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">\frac{d}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of <math alttext="\mu(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mrow id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml"><mi id="S3.SS3.p3.3.m3.1.1.3" xref="S3.SS3.p3.3.m3.1.1.3.cmml">μ</mi><mo id="S3.SS3.p3.3.m3.1.1.2" xref="S3.SS3.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p3.3.m3.1.1.1.1" xref="S3.SS3.p3.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS3.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS3.p3.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S3.SS3.p3.3.m3.1.1.1.1.1" xref="S3.SS3.p3.3.m3.1.1.1.1.1.cmml"><mi id="S3.SS3.p3.3.m3.1.1.1.1.1.2" xref="S3.SS3.p3.3.m3.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS3.p3.3.m3.1.1.1.1.1.3" xref="S3.SS3.p3.3.m3.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS3.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S3.SS3.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><apply id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1"><times id="S3.SS3.p3.3.m3.1.1.2.cmml" xref="S3.SS3.p3.3.m3.1.1.2"></times><ci id="S3.SS3.p3.3.m3.1.1.3.cmml" xref="S3.SS3.p3.3.m3.1.1.3">𝜇</ci><apply id="S3.SS3.p3.3.m3.1.1.1.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1.1.1">superscript</csymbol><ci id="S3.SS3.p3.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS3.p3.3.m3.1.1.1.1.1.2">ℝ</ci><ci id="S3.SS3.p3.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS3.p3.3.m3.1.1.1.1.1.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">\mu(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> must have a non-empty intersection. Choosing <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_c</annotation></semantics></math> in this intersection guarantees the required property. However, this proof breaks down if we consider <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p3.5.m5.1"><semantics id="S3.SS3.p3.5.m5.1a"><msub id="S3.SS3.p3.5.m5.1.1" xref="S3.SS3.p3.5.m5.1.1.cmml"><mi id="S3.SS3.p3.5.m5.1.1.2" mathvariant="normal" xref="S3.SS3.p3.5.m5.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p3.5.m5.1.1.3" xref="S3.SS3.p3.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.5.m5.1b"><apply id="S3.SS3.p3.5.m5.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.1.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.p3.5.m5.1.1.2.cmml" xref="S3.SS3.p3.5.m5.1.1.2">ℓ</ci><ci id="S3.SS3.p3.5.m5.1.1.3.cmml" xref="S3.SS3.p3.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces: they are in general non-convex and thus Helly’s theorem is no longer applicable. Instead, we prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a> with Brouwer’s fixpoint theorem. The existence of a proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem12" title="Theorem 3.12 (Euclidean Centerpoint Theorem for Mass Distributions [31]). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.12</span></a> via Brouwer’s fixpoint theorem seems to be folklore<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Such a proof is vaguely outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib38" title="">38</a>]</cite>.</span></span></span>, but we were unable to find a reference containing the full proof. We briefly recall Brouwer’s fixpoint theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem14.1.1.1">Theorem 3.14</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem14.2.2"> </span>(Brouwer’s fixpoint theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib3" title="">3</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem14.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem14.p1"> <p class="ltx_p" id="S3.Thmtheorem14.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem14.p1.3.3">Let <math alttext="C\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem14.p1.1.1.m1.1"><semantics id="S3.Thmtheorem14.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem14.p1.1.1.m1.1.1" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem14.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.2.cmml">C</mi><mo id="S3.Thmtheorem14.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.1.cmml">⊆</mo><msup id="S3.Thmtheorem14.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.2" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.3" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem14.p1.1.1.m1.1b"><apply id="S3.Thmtheorem14.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1"><subset id="S3.Thmtheorem14.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.1"></subset><ci id="S3.Thmtheorem14.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.2">𝐶</ci><apply id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem14.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmtheorem14.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem14.p1.1.1.m1.1c">C\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem14.p1.1.1.m1.1d">italic_C ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-empty compact convex set, and let <math alttext="f:C\rightarrow C" class="ltx_Math" display="inline" id="S3.Thmtheorem14.p1.2.2.m2.1"><semantics id="S3.Thmtheorem14.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem14.p1.2.2.m2.1.1" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem14.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.2.cmml">f</mi><mo id="S3.Thmtheorem14.p1.2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem14.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.2.cmml">C</mi><mo id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.1" stretchy="false" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.1.cmml">→</mo><mi id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.3.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem14.p1.2.2.m2.1b"><apply id="S3.Thmtheorem14.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1"><ci id="S3.Thmtheorem14.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.1">:</ci><ci id="S3.Thmtheorem14.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.2">𝑓</ci><apply id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3"><ci id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.1">→</ci><ci id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.2">𝐶</ci><ci id="S3.Thmtheorem14.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem14.p1.2.2.m2.1.1.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem14.p1.2.2.m2.1c">f:C\rightarrow C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem14.p1.2.2.m2.1d">italic_f : italic_C → italic_C</annotation></semantics></math> be a continuous function. Then <math alttext="f" class="ltx_Math" display="inline" id="S3.Thmtheorem14.p1.3.3.m3.1"><semantics id="S3.Thmtheorem14.p1.3.3.m3.1a"><mi id="S3.Thmtheorem14.p1.3.3.m3.1.1" xref="S3.Thmtheorem14.p1.3.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem14.p1.3.3.m3.1b"><ci id="S3.Thmtheorem14.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem14.p1.3.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem14.p1.3.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem14.p1.3.3.m3.1d">italic_f</annotation></semantics></math> has a fixpoint.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.9">We will now use Brouwer’s fixpoint theorem to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a>. For this, we first define a function <math alttext="F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mrow id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">F</mi><mo id="S3.SS3.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS3.p4.1.m1.1.1.3" xref="S3.SS3.p4.1.m1.1.1.3.cmml"><msup id="S3.SS3.p4.1.m1.1.1.3.2" xref="S3.SS3.p4.1.m1.1.1.3.2.cmml"><mi id="S3.SS3.p4.1.m1.1.1.3.2.2" xref="S3.SS3.p4.1.m1.1.1.3.2.2.cmml">ℝ</mi><mi id="S3.SS3.p4.1.m1.1.1.3.2.3" xref="S3.SS3.p4.1.m1.1.1.3.2.3.cmml">d</mi></msup><mo id="S3.SS3.p4.1.m1.1.1.3.1" stretchy="false" xref="S3.SS3.p4.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.SS3.p4.1.m1.1.1.3.3" xref="S3.SS3.p4.1.m1.1.1.3.3.cmml"><mi id="S3.SS3.p4.1.m1.1.1.3.3.2" xref="S3.SS3.p4.1.m1.1.1.3.3.2.cmml">ℝ</mi><mi id="S3.SS3.p4.1.m1.1.1.3.3.3" xref="S3.SS3.p4.1.m1.1.1.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><ci id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1.1">:</ci><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">𝐹</ci><apply id="S3.SS3.p4.1.m1.1.1.3.cmml" xref="S3.SS3.p4.1.m1.1.1.3"><ci id="S3.SS3.p4.1.m1.1.1.3.1.cmml" xref="S3.SS3.p4.1.m1.1.1.3.1">→</ci><apply id="S3.SS3.p4.1.m1.1.1.3.2.cmml" xref="S3.SS3.p4.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.p4.1.m1.1.1.3.2.1.cmml" xref="S3.SS3.p4.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.SS3.p4.1.m1.1.1.3.2.2.cmml" xref="S3.SS3.p4.1.m1.1.1.3.2.2">ℝ</ci><ci id="S3.SS3.p4.1.m1.1.1.3.2.3.cmml" xref="S3.SS3.p4.1.m1.1.1.3.2.3">𝑑</ci></apply><apply id="S3.SS3.p4.1.m1.1.1.3.3.cmml" xref="S3.SS3.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.p4.1.m1.1.1.3.3.1.cmml" xref="S3.SS3.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS3.p4.1.m1.1.1.3.3.2.cmml" xref="S3.SS3.p4.1.m1.1.1.3.3.2">ℝ</ci><ci id="S3.SS3.p4.1.m1.1.1.3.3.3.cmml" xref="S3.SS3.p4.1.m1.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_F : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> such that any fixpoint of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mi id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><ci id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">italic_F</annotation></semantics></math> must be an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><msub id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml"><mi id="S3.SS3.p4.3.m3.1.1.2" mathvariant="normal" xref="S3.SS3.p4.3.m3.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p4.3.m3.1.1.3" xref="S3.SS3.p4.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><apply id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p4.3.m3.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2">ℓ</ci><ci id="S3.SS3.p4.3.m3.1.1.3.cmml" xref="S3.SS3.p4.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p4.4.m4.1"><semantics id="S3.SS3.p4.4.m4.1a"><mi id="S3.SS3.p4.4.m4.1.1" xref="S3.SS3.p4.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.4.m4.1b"><ci id="S3.SS3.p4.4.m4.1.1.cmml" xref="S3.SS3.p4.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.4.m4.1d">italic_μ</annotation></semantics></math>. Concretely, the intuition behind this function is that any point <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.p4.5.m5.1"><semantics id="S3.SS3.p4.5.m5.1a"><mi id="S3.SS3.p4.5.m5.1.1" xref="S3.SS3.p4.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.5.m5.1b"><ci id="S3.SS3.p4.5.m5.1.1.cmml" xref="S3.SS3.p4.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.5.m5.1d">italic_x</annotation></semantics></math> is “pushed” in all directions <math alttext="v" class="ltx_Math" display="inline" id="S3.SS3.p4.6.m6.1"><semantics id="S3.SS3.p4.6.m6.1a"><mi id="S3.SS3.p4.6.m6.1.1" xref="S3.SS3.p4.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.6.m6.1b"><ci id="S3.SS3.p4.6.m6.1.1.cmml" xref="S3.SS3.p4.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.6.m6.1d">italic_v</annotation></semantics></math> for which <math alttext="\mathcal{H}^{p}_{x,-v}" class="ltx_Math" display="inline" id="S3.SS3.p4.7.m7.2"><semantics id="S3.SS3.p4.7.m7.2a"><msubsup id="S3.SS3.p4.7.m7.2.3" xref="S3.SS3.p4.7.m7.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p4.7.m7.2.3.2.2" xref="S3.SS3.p4.7.m7.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS3.p4.7.m7.2.2.2.2" xref="S3.SS3.p4.7.m7.2.2.2.3.cmml"><mi id="S3.SS3.p4.7.m7.1.1.1.1" xref="S3.SS3.p4.7.m7.1.1.1.1.cmml">x</mi><mo id="S3.SS3.p4.7.m7.2.2.2.2.2" xref="S3.SS3.p4.7.m7.2.2.2.3.cmml">,</mo><mrow id="S3.SS3.p4.7.m7.2.2.2.2.1" xref="S3.SS3.p4.7.m7.2.2.2.2.1.cmml"><mo id="S3.SS3.p4.7.m7.2.2.2.2.1a" xref="S3.SS3.p4.7.m7.2.2.2.2.1.cmml">−</mo><mi id="S3.SS3.p4.7.m7.2.2.2.2.1.2" xref="S3.SS3.p4.7.m7.2.2.2.2.1.2.cmml">v</mi></mrow></mrow><mi id="S3.SS3.p4.7.m7.2.3.2.3" xref="S3.SS3.p4.7.m7.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.7.m7.2b"><apply id="S3.SS3.p4.7.m7.2.3.cmml" xref="S3.SS3.p4.7.m7.2.3"><csymbol cd="ambiguous" id="S3.SS3.p4.7.m7.2.3.1.cmml" xref="S3.SS3.p4.7.m7.2.3">subscript</csymbol><apply id="S3.SS3.p4.7.m7.2.3.2.cmml" xref="S3.SS3.p4.7.m7.2.3"><csymbol cd="ambiguous" id="S3.SS3.p4.7.m7.2.3.2.1.cmml" xref="S3.SS3.p4.7.m7.2.3">superscript</csymbol><ci id="S3.SS3.p4.7.m7.2.3.2.2.cmml" xref="S3.SS3.p4.7.m7.2.3.2.2">ℋ</ci><ci id="S3.SS3.p4.7.m7.2.3.2.3.cmml" xref="S3.SS3.p4.7.m7.2.3.2.3">𝑝</ci></apply><list id="S3.SS3.p4.7.m7.2.2.2.3.cmml" xref="S3.SS3.p4.7.m7.2.2.2.2"><ci id="S3.SS3.p4.7.m7.1.1.1.1.cmml" xref="S3.SS3.p4.7.m7.1.1.1.1">𝑥</ci><apply id="S3.SS3.p4.7.m7.2.2.2.2.1.cmml" xref="S3.SS3.p4.7.m7.2.2.2.2.1"><minus id="S3.SS3.p4.7.m7.2.2.2.2.1.1.cmml" xref="S3.SS3.p4.7.m7.2.2.2.2.1"></minus><ci id="S3.SS3.p4.7.m7.2.2.2.2.1.2.cmml" xref="S3.SS3.p4.7.m7.2.2.2.2.1.2">𝑣</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.7.m7.2c">\mathcal{H}^{p}_{x,-v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.7.m7.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT</annotation></semantics></math> does not contain a sufficient fraction of mass simultaneously. We achieve this by defining <math alttext="F_{i}(x)" class="ltx_Math" display="inline" id="S3.SS3.p4.8.m8.1"><semantics id="S3.SS3.p4.8.m8.1a"><mrow id="S3.SS3.p4.8.m8.1.2" xref="S3.SS3.p4.8.m8.1.2.cmml"><msub id="S3.SS3.p4.8.m8.1.2.2" xref="S3.SS3.p4.8.m8.1.2.2.cmml"><mi id="S3.SS3.p4.8.m8.1.2.2.2" xref="S3.SS3.p4.8.m8.1.2.2.2.cmml">F</mi><mi id="S3.SS3.p4.8.m8.1.2.2.3" xref="S3.SS3.p4.8.m8.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS3.p4.8.m8.1.2.1" xref="S3.SS3.p4.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.SS3.p4.8.m8.1.2.3.2" xref="S3.SS3.p4.8.m8.1.2.cmml"><mo id="S3.SS3.p4.8.m8.1.2.3.2.1" stretchy="false" xref="S3.SS3.p4.8.m8.1.2.cmml">(</mo><mi id="S3.SS3.p4.8.m8.1.1" xref="S3.SS3.p4.8.m8.1.1.cmml">x</mi><mo id="S3.SS3.p4.8.m8.1.2.3.2.2" stretchy="false" xref="S3.SS3.p4.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.8.m8.1b"><apply id="S3.SS3.p4.8.m8.1.2.cmml" xref="S3.SS3.p4.8.m8.1.2"><times id="S3.SS3.p4.8.m8.1.2.1.cmml" xref="S3.SS3.p4.8.m8.1.2.1"></times><apply id="S3.SS3.p4.8.m8.1.2.2.cmml" xref="S3.SS3.p4.8.m8.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.p4.8.m8.1.2.2.1.cmml" xref="S3.SS3.p4.8.m8.1.2.2">subscript</csymbol><ci id="S3.SS3.p4.8.m8.1.2.2.2.cmml" xref="S3.SS3.p4.8.m8.1.2.2.2">𝐹</ci><ci id="S3.SS3.p4.8.m8.1.2.2.3.cmml" xref="S3.SS3.p4.8.m8.1.2.2.3">𝑖</ci></apply><ci id="S3.SS3.p4.8.m8.1.1.cmml" xref="S3.SS3.p4.8.m8.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.8.m8.1c">F_{i}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.8.m8.1d">italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S3.SS3.p4.9.m9.1"><semantics id="S3.SS3.p4.9.m9.1a"><mrow id="S3.SS3.p4.9.m9.1.2" xref="S3.SS3.p4.9.m9.1.2.cmml"><mi id="S3.SS3.p4.9.m9.1.2.2" xref="S3.SS3.p4.9.m9.1.2.2.cmml">i</mi><mo id="S3.SS3.p4.9.m9.1.2.1" xref="S3.SS3.p4.9.m9.1.2.1.cmml">∈</mo><mrow id="S3.SS3.p4.9.m9.1.2.3.2" xref="S3.SS3.p4.9.m9.1.2.3.1.cmml"><mo id="S3.SS3.p4.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS3.p4.9.m9.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.p4.9.m9.1.1" xref="S3.SS3.p4.9.m9.1.1.cmml">d</mi><mo id="S3.SS3.p4.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS3.p4.9.m9.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.9.m9.1b"><apply id="S3.SS3.p4.9.m9.1.2.cmml" xref="S3.SS3.p4.9.m9.1.2"><in id="S3.SS3.p4.9.m9.1.2.1.cmml" xref="S3.SS3.p4.9.m9.1.2.1"></in><ci id="S3.SS3.p4.9.m9.1.2.2.cmml" xref="S3.SS3.p4.9.m9.1.2.2">𝑖</ci><apply id="S3.SS3.p4.9.m9.1.2.3.1.cmml" xref="S3.SS3.p4.9.m9.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.p4.9.m9.1.2.3.1.1.cmml" xref="S3.SS3.p4.9.m9.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.p4.9.m9.1.1.cmml" xref="S3.SS3.p4.9.m9.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.9.m9.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.9.m9.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math> using an integral over all directions, i.e.,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{i}(x)\coloneqq x_{i}+\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{% R}^{d})-\mu(\mathcal{H}^{p}_{x,-v}),0\right)dv." class="ltx_Math" display="block" id="S3.Ex4.m1.6"><semantics id="S3.Ex4.m1.6a"><mrow id="S3.Ex4.m1.6.6.1" xref="S3.Ex4.m1.6.6.1.1.cmml"><mrow id="S3.Ex4.m1.6.6.1.1" xref="S3.Ex4.m1.6.6.1.1.cmml"><mrow id="S3.Ex4.m1.6.6.1.1.3" xref="S3.Ex4.m1.6.6.1.1.3.cmml"><msub id="S3.Ex4.m1.6.6.1.1.3.2" xref="S3.Ex4.m1.6.6.1.1.3.2.cmml"><mi id="S3.Ex4.m1.6.6.1.1.3.2.2" xref="S3.Ex4.m1.6.6.1.1.3.2.2.cmml">F</mi><mi 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xref="S3.Ex4.m1.6.6.1.1.1.1.1.4.1">differential-d</csymbol><ci id="S3.Ex4.m1.6.6.1.1.1.1.1.4.2.cmml" xref="S3.Ex4.m1.6.6.1.1.1.1.1.4.2">𝑣</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m1.6c">F_{i}(x)\coloneqq x_{i}+\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{% R}^{d})-\mu(\mathcal{H}^{p}_{x,-v}),0\right)dv.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m1.6d">italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ≔ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∫ start_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT ) , 0 ) italic_d italic_v .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.2">Since we eventually want to apply Brouwer’s fixpoint theorem to <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><mi id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><ci id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.1.m1.1d">italic_F</annotation></semantics></math>, let us check that <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p5.2.m2.1"><semantics id="S3.SS3.p5.2.m2.1a"><mi id="S3.SS3.p5.2.m2.1.1" xref="S3.SS3.p5.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.1b"><ci id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.1d">italic_F</annotation></semantics></math> is continuous.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem15"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem15.1.1.1">Lemma 3.15</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem15.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem15.p1"> <p class="ltx_p" id="S3.Thmtheorem15.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem15.p1.4.4">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem15.p1.1.1.m1.3"><semantics 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xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem15.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem15.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem15.p1.1.1.m1.3b"><apply id="S3.Thmtheorem15.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4"><in id="S3.Thmtheorem15.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem15.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem15.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem15.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem15.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem15.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem15.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem15.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem15.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary. Let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem15.p1.2.2.m2.1"><semantics id="S3.Thmtheorem15.p1.2.2.m2.1a"><mi id="S3.Thmtheorem15.p1.2.2.m2.1.1" xref="S3.Thmtheorem15.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem15.p1.2.2.m2.1b"><ci id="S3.Thmtheorem15.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem15.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem15.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem15.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> be a mass distribution on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem15.p1.3.3.m3.1"><semantics id="S3.Thmtheorem15.p1.3.3.m3.1a"><msup id="S3.Thmtheorem15.p1.3.3.m3.1.1" xref="S3.Thmtheorem15.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem15.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem15.p1.3.3.m3.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem15.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem15.p1.3.3.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem15.p1.3.3.m3.1b"><apply id="S3.Thmtheorem15.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem15.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem15.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem15.p1.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem15.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem15.p1.3.3.m3.1.1.2">ℝ</ci><ci id="S3.Thmtheorem15.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem15.p1.3.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem15.p1.3.3.m3.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem15.p1.3.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with bounded support. Then the function <math alttext="F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem15.p1.4.4.m4.1"><semantics id="S3.Thmtheorem15.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem15.p1.4.4.m4.1.1" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem15.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.2.cmml">F</mi><mo id="S3.Thmtheorem15.p1.4.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem15.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.cmml"><msup id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.cmml"><mi id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.2" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.2.cmml">ℝ</mi><mi id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.3" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.3.cmml">d</mi></msup><mo id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.1" stretchy="false" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.1.cmml">→</mo><msup id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.cmml"><mi id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.2" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.3" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem15.p1.4.4.m4.1b"><apply id="S3.Thmtheorem15.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1"><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.1">:</ci><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.2">𝐹</ci><apply id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3"><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.1">→</ci><apply id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.1.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.2.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.2">ℝ</ci><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.3.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.2.3">𝑑</ci></apply><apply id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.1.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3">superscript</csymbol><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.2.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.2">ℝ</ci><ci id="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.3.cmml" xref="S3.Thmtheorem15.p1.4.4.m4.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem15.p1.4.4.m4.1c">F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem15.p1.4.4.m4.1d">italic_F : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> defined above is continuous.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.2.p1"> <p class="ltx_p" id="S3.SS3.2.p1.3">Fix <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.2.p1.1.m1.1"><semantics id="S3.SS3.2.p1.1.m1.1a"><mrow id="S3.SS3.2.p1.1.m1.1.1" xref="S3.SS3.2.p1.1.m1.1.1.cmml"><mi id="S3.SS3.2.p1.1.m1.1.1.2" xref="S3.SS3.2.p1.1.m1.1.1.2.cmml">x</mi><mo id="S3.SS3.2.p1.1.m1.1.1.1" xref="S3.SS3.2.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS3.2.p1.1.m1.1.1.3" xref="S3.SS3.2.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.2.p1.1.m1.1.1.3.2" xref="S3.SS3.2.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS3.2.p1.1.m1.1.1.3.3" xref="S3.SS3.2.p1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p1.1.m1.1b"><apply id="S3.SS3.2.p1.1.m1.1.1.cmml" xref="S3.SS3.2.p1.1.m1.1.1"><in id="S3.SS3.2.p1.1.m1.1.1.1.cmml" xref="S3.SS3.2.p1.1.m1.1.1.1"></in><ci id="S3.SS3.2.p1.1.m1.1.1.2.cmml" xref="S3.SS3.2.p1.1.m1.1.1.2">𝑥</ci><apply id="S3.SS3.2.p1.1.m1.1.1.3.cmml" xref="S3.SS3.2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.2.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS3.2.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.2.p1.1.m1.1.1.3.2">ℝ</ci><ci id="S3.SS3.2.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.2.p1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.1.m1.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.1.m1.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and let <math alttext="(x_{n})_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="S3.SS3.2.p1.2.m2.1"><semantics id="S3.SS3.2.p1.2.m2.1a"><msub id="S3.SS3.2.p1.2.m2.1.1" xref="S3.SS3.2.p1.2.m2.1.1.cmml"><mrow id="S3.SS3.2.p1.2.m2.1.1.1.1" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS3.2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.2.p1.2.m2.1.1.1.1.1" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.2.p1.2.m2.1.1.1.1.1.2" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.2.cmml">x</mi><mi id="S3.SS3.2.p1.2.m2.1.1.1.1.1.3" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.SS3.2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS3.2.p1.2.m2.1.1.3" xref="S3.SS3.2.p1.2.m2.1.1.3.cmml"><mi id="S3.SS3.2.p1.2.m2.1.1.3.2" xref="S3.SS3.2.p1.2.m2.1.1.3.2.cmml">n</mi><mo id="S3.SS3.2.p1.2.m2.1.1.3.1" xref="S3.SS3.2.p1.2.m2.1.1.3.1.cmml">∈</mo><mi id="S3.SS3.2.p1.2.m2.1.1.3.3" xref="S3.SS3.2.p1.2.m2.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p1.2.m2.1b"><apply id="S3.SS3.2.p1.2.m2.1.1.cmml" xref="S3.SS3.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.2.p1.2.m2.1.1.2.cmml" xref="S3.SS3.2.p1.2.m2.1.1">subscript</csymbol><apply id="S3.SS3.2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.SS3.2.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.2.p1.2.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS3.2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.2">𝑥</ci><ci id="S3.SS3.2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS3.2.p1.2.m2.1.1.1.1.1.3">𝑛</ci></apply><apply id="S3.SS3.2.p1.2.m2.1.1.3.cmml" xref="S3.SS3.2.p1.2.m2.1.1.3"><in id="S3.SS3.2.p1.2.m2.1.1.3.1.cmml" xref="S3.SS3.2.p1.2.m2.1.1.3.1"></in><ci id="S3.SS3.2.p1.2.m2.1.1.3.2.cmml" xref="S3.SS3.2.p1.2.m2.1.1.3.2">𝑛</ci><ci id="S3.SS3.2.p1.2.m2.1.1.3.3.cmml" xref="S3.SS3.2.p1.2.m2.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.2.m2.1c">(x_{n})_{n\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.2.m2.1d">( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> be an arbitrary sequence converging to <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.2.p1.3.m3.1"><semantics id="S3.SS3.2.p1.3.m3.1a"><mi id="S3.SS3.2.p1.3.m3.1.1" xref="S3.SS3.2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p1.3.m3.1b"><ci id="S3.SS3.2.p1.3.m3.1.1.cmml" xref="S3.SS3.2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.3.m3.1d">italic_x</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.10</span></a>, the functions</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{n}(v)\coloneqq v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal% {H}^{p}_{x_{n},-v}),0\right)" class="ltx_Math" display="block" id="S3.Ex5.m1.6"><semantics id="S3.Ex5.m1.6a"><mrow id="S3.Ex5.m1.6.6" xref="S3.Ex5.m1.6.6.cmml"><mrow id="S3.Ex5.m1.6.6.3" xref="S3.Ex5.m1.6.6.3.cmml"><msub id="S3.Ex5.m1.6.6.3.2" xref="S3.Ex5.m1.6.6.3.2.cmml"><mi id="S3.Ex5.m1.6.6.3.2.2" xref="S3.Ex5.m1.6.6.3.2.2.cmml">g</mi><mi id="S3.Ex5.m1.6.6.3.2.3" xref="S3.Ex5.m1.6.6.3.2.3.cmml">n</mi></msub><mo id="S3.Ex5.m1.6.6.3.1" xref="S3.Ex5.m1.6.6.3.1.cmml">⁢</mo><mrow id="S3.Ex5.m1.6.6.3.3.2" xref="S3.Ex5.m1.6.6.3.cmml"><mo id="S3.Ex5.m1.6.6.3.3.2.1" stretchy="false" xref="S3.Ex5.m1.6.6.3.cmml">(</mo><mi id="S3.Ex5.m1.3.3" xref="S3.Ex5.m1.3.3.cmml">v</mi><mo id="S3.Ex5.m1.6.6.3.3.2.2" stretchy="false" xref="S3.Ex5.m1.6.6.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex5.m1.6.6.2" xref="S3.Ex5.m1.6.6.2.cmml">≔</mo><mrow id="S3.Ex5.m1.6.6.1" xref="S3.Ex5.m1.6.6.1.cmml"><msub id="S3.Ex5.m1.6.6.1.3" xref="S3.Ex5.m1.6.6.1.3.cmml"><mi id="S3.Ex5.m1.6.6.1.3.2" xref="S3.Ex5.m1.6.6.1.3.2.cmml">v</mi><mi id="S3.Ex5.m1.6.6.1.3.3" xref="S3.Ex5.m1.6.6.1.3.3.cmml">i</mi></msub><mo id="S3.Ex5.m1.6.6.1.2" lspace="0.167em" xref="S3.Ex5.m1.6.6.1.2.cmml">⁢</mo><mrow id="S3.Ex5.m1.6.6.1.1.1" xref="S3.Ex5.m1.6.6.1.1.2.cmml"><mi id="S3.Ex5.m1.4.4" xref="S3.Ex5.m1.4.4.cmml">max</mi><mo id="S3.Ex5.m1.6.6.1.1.1a" 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v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal% {H}^{p}_{x_{n},-v}),0\right)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.6d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_v ) ≔ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , - italic_v end_POSTSUBSCRIPT ) , 0 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.2.p1.6">converge pointwise to <math alttext="g(v)=v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^{p}_{x,-v% }),0\right)" class="ltx_Math" display="inline" id="S3.SS3.2.p1.4.m1.6"><semantics 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xref="S3.SS3.2.p1.4.m1.6.6.1.1.1.1.1.2.1.1">superscript</csymbol><ci id="S3.SS3.2.p1.4.m1.6.6.1.1.1.1.1.2.1.1.1.2.2.cmml" xref="S3.SS3.2.p1.4.m1.6.6.1.1.1.1.1.2.1.1.1.2.2">ℋ</ci><ci id="S3.SS3.2.p1.4.m1.6.6.1.1.1.1.1.2.1.1.1.2.3.cmml" xref="S3.SS3.2.p1.4.m1.6.6.1.1.1.1.1.2.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS3.2.p1.4.m1.2.2.2.3.cmml" xref="S3.SS3.2.p1.4.m1.2.2.2.2"><ci id="S3.SS3.2.p1.4.m1.1.1.1.1.cmml" xref="S3.SS3.2.p1.4.m1.1.1.1.1">𝑥</ci><apply id="S3.SS3.2.p1.4.m1.2.2.2.2.1.cmml" xref="S3.SS3.2.p1.4.m1.2.2.2.2.1"><minus id="S3.SS3.2.p1.4.m1.2.2.2.2.1.1.cmml" xref="S3.SS3.2.p1.4.m1.2.2.2.2.1"></minus><ci id="S3.SS3.2.p1.4.m1.2.2.2.2.1.2.cmml" xref="S3.SS3.2.p1.4.m1.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></apply><cn id="S3.SS3.2.p1.4.m1.5.5.cmml" type="integer" xref="S3.SS3.2.p1.4.m1.5.5">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.4.m1.6c">g(v)=v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^{p}_{x,-v% }),0\right)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.4.m1.6d">italic_g ( italic_v ) = italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT ) , 0 )</annotation></semantics></math> for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S3.SS3.2.p1.5.m2.1"><semantics id="S3.SS3.2.p1.5.m2.1a"><mrow id="S3.SS3.2.p1.5.m2.1.2" xref="S3.SS3.2.p1.5.m2.1.2.cmml"><mi id="S3.SS3.2.p1.5.m2.1.2.2" xref="S3.SS3.2.p1.5.m2.1.2.2.cmml">i</mi><mo id="S3.SS3.2.p1.5.m2.1.2.1" xref="S3.SS3.2.p1.5.m2.1.2.1.cmml">∈</mo><mrow id="S3.SS3.2.p1.5.m2.1.2.3.2" xref="S3.SS3.2.p1.5.m2.1.2.3.1.cmml"><mo id="S3.SS3.2.p1.5.m2.1.2.3.2.1" stretchy="false" xref="S3.SS3.2.p1.5.m2.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.2.p1.5.m2.1.1" xref="S3.SS3.2.p1.5.m2.1.1.cmml">d</mi><mo id="S3.SS3.2.p1.5.m2.1.2.3.2.2" stretchy="false" xref="S3.SS3.2.p1.5.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p1.5.m2.1b"><apply id="S3.SS3.2.p1.5.m2.1.2.cmml" xref="S3.SS3.2.p1.5.m2.1.2"><in id="S3.SS3.2.p1.5.m2.1.2.1.cmml" xref="S3.SS3.2.p1.5.m2.1.2.1"></in><ci id="S3.SS3.2.p1.5.m2.1.2.2.cmml" xref="S3.SS3.2.p1.5.m2.1.2.2">𝑖</ci><apply id="S3.SS3.2.p1.5.m2.1.2.3.1.cmml" xref="S3.SS3.2.p1.5.m2.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.2.p1.5.m2.1.2.3.1.1.cmml" xref="S3.SS3.2.p1.5.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.2.p1.5.m2.1.1.cmml" xref="S3.SS3.2.p1.5.m2.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.5.m2.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.5.m2.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. Since they are also globally bounded, we can apply the dominated convergence theorem to exchange limit and integral, yielding continuity of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.2.p1.6.m3.1"><semantics id="S3.SS3.2.p1.6.m3.1a"><mi id="S3.SS3.2.p1.6.m3.1.1" xref="S3.SS3.2.p1.6.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p1.6.m3.1b"><ci id="S3.SS3.2.p1.6.m3.1.1.cmml" xref="S3.SS3.2.p1.6.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p1.6.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p1.6.m3.1d">italic_F</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p6"> <p class="ltx_p" id="S3.SS3.p6.5">We next turn our attention to the fixpoints of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p6.1.m1.1"><semantics id="S3.SS3.p6.1.m1.1a"><mi id="S3.SS3.p6.1.m1.1.1" xref="S3.SS3.p6.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.1.m1.1b"><ci id="S3.SS3.p6.1.m1.1.1.cmml" xref="S3.SS3.p6.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.1.m1.1d">italic_F</annotation></semantics></math>. Concretely, we wish to show that any fixpoint <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.p6.2.m2.1"><semantics id="S3.SS3.p6.2.m2.1a"><mi id="S3.SS3.p6.2.m2.1.1" xref="S3.SS3.p6.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.2.m2.1b"><ci id="S3.SS3.p6.2.m2.1.1.cmml" xref="S3.SS3.p6.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.2.m2.1d">italic_x</annotation></semantics></math> of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p6.3.m3.1"><semantics id="S3.SS3.p6.3.m3.1a"><mi id="S3.SS3.p6.3.m3.1.1" xref="S3.SS3.p6.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.3.m3.1b"><ci id="S3.SS3.p6.3.m3.1.1.cmml" xref="S3.SS3.p6.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.3.m3.1d">italic_F</annotation></semantics></math> must be an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p6.4.m4.1"><semantics id="S3.SS3.p6.4.m4.1a"><msub id="S3.SS3.p6.4.m4.1.1" xref="S3.SS3.p6.4.m4.1.1.cmml"><mi id="S3.SS3.p6.4.m4.1.1.2" mathvariant="normal" xref="S3.SS3.p6.4.m4.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p6.4.m4.1.1.3" xref="S3.SS3.p6.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.4.m4.1b"><apply id="S3.SS3.p6.4.m4.1.1.cmml" xref="S3.SS3.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.p6.4.m4.1.1.1.cmml" xref="S3.SS3.p6.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.p6.4.m4.1.1.2.cmml" xref="S3.SS3.p6.4.m4.1.1.2">ℓ</ci><ci id="S3.SS3.p6.4.m4.1.1.3.cmml" xref="S3.SS3.p6.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p6.5.m5.1"><semantics id="S3.SS3.p6.5.m5.1a"><mi id="S3.SS3.p6.5.m5.1.1" xref="S3.SS3.p6.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.5.m5.1b"><ci id="S3.SS3.p6.5.m5.1.1.cmml" xref="S3.SS3.p6.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.5.m5.1d">italic_μ</annotation></semantics></math>. To argue this, we analyze the set</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="V_{x}\coloneqq\{v\in S^{d-1}\mid\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu% (\mathcal{H}^{p}_{x,-v}),0\right)>0\}" class="ltx_Math" display="block" id="S3.Ex6.m1.6"><semantics id="S3.Ex6.m1.6a"><mrow id="S3.Ex6.m1.6.6" xref="S3.Ex6.m1.6.6.cmml"><msub id="S3.Ex6.m1.6.6.4" xref="S3.Ex6.m1.6.6.4.cmml"><mi id="S3.Ex6.m1.6.6.4.2" xref="S3.Ex6.m1.6.6.4.2.cmml">V</mi><mi id="S3.Ex6.m1.6.6.4.3" xref="S3.Ex6.m1.6.6.4.3.cmml">x</mi></msub><mo id="S3.Ex6.m1.6.6.3" xref="S3.Ex6.m1.6.6.3.cmml">≔</mo><mrow id="S3.Ex6.m1.6.6.2.2" xref="S3.Ex6.m1.6.6.2.3.cmml"><mo id="S3.Ex6.m1.6.6.2.2.3" stretchy="false" xref="S3.Ex6.m1.6.6.2.3.1.cmml">{</mo><mrow id="S3.Ex6.m1.5.5.1.1.1" xref="S3.Ex6.m1.5.5.1.1.1.cmml"><mi id="S3.Ex6.m1.5.5.1.1.1.2" 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italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT ) , 0 ) > 0 }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p6.14">of directions which contribute towards the integral. We will show that it is impossible for <math alttext="\text{conv}(V_{x})" class="ltx_Math" display="inline" id="S3.SS3.p6.6.m1.1"><semantics id="S3.SS3.p6.6.m1.1a"><mrow id="S3.SS3.p6.6.m1.1.1" xref="S3.SS3.p6.6.m1.1.1.cmml"><mtext id="S3.SS3.p6.6.m1.1.1.3" xref="S3.SS3.p6.6.m1.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.p6.6.m1.1.1.2" xref="S3.SS3.p6.6.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p6.6.m1.1.1.1.1" xref="S3.SS3.p6.6.m1.1.1.1.1.1.cmml"><mo id="S3.SS3.p6.6.m1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p6.6.m1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p6.6.m1.1.1.1.1.1" xref="S3.SS3.p6.6.m1.1.1.1.1.1.cmml"><mi id="S3.SS3.p6.6.m1.1.1.1.1.1.2" xref="S3.SS3.p6.6.m1.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.p6.6.m1.1.1.1.1.1.3" xref="S3.SS3.p6.6.m1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S3.SS3.p6.6.m1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p6.6.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.6.m1.1b"><apply id="S3.SS3.p6.6.m1.1.1.cmml" xref="S3.SS3.p6.6.m1.1.1"><times id="S3.SS3.p6.6.m1.1.1.2.cmml" xref="S3.SS3.p6.6.m1.1.1.2"></times><ci id="S3.SS3.p6.6.m1.1.1.3a.cmml" xref="S3.SS3.p6.6.m1.1.1.3"><mtext id="S3.SS3.p6.6.m1.1.1.3.cmml" xref="S3.SS3.p6.6.m1.1.1.3">conv</mtext></ci><apply id="S3.SS3.p6.6.m1.1.1.1.1.1.cmml" xref="S3.SS3.p6.6.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p6.6.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.p6.6.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p6.6.m1.1.1.1.1.1.2.cmml" xref="S3.SS3.p6.6.m1.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.p6.6.m1.1.1.1.1.1.3.cmml" xref="S3.SS3.p6.6.m1.1.1.1.1.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.6.m1.1c">\text{conv}(V_{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.6.m1.1d">conv ( italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> to contain <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.p6.7.m2.1"><semantics id="S3.SS3.p6.7.m2.1a"><mn id="S3.SS3.p6.7.m2.1.1" xref="S3.SS3.p6.7.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.7.m2.1b"><cn id="S3.SS3.p6.7.m2.1.1.cmml" type="integer" xref="S3.SS3.p6.7.m2.1.1">0</cn></annotation-xml></semantics></math> (for any <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.p6.8.m3.1"><semantics id="S3.SS3.p6.8.m3.1a"><mrow id="S3.SS3.p6.8.m3.1.1" xref="S3.SS3.p6.8.m3.1.1.cmml"><mi id="S3.SS3.p6.8.m3.1.1.2" xref="S3.SS3.p6.8.m3.1.1.2.cmml">x</mi><mo id="S3.SS3.p6.8.m3.1.1.1" xref="S3.SS3.p6.8.m3.1.1.1.cmml">∈</mo><msup id="S3.SS3.p6.8.m3.1.1.3" xref="S3.SS3.p6.8.m3.1.1.3.cmml"><mi id="S3.SS3.p6.8.m3.1.1.3.2" xref="S3.SS3.p6.8.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS3.p6.8.m3.1.1.3.3" xref="S3.SS3.p6.8.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.8.m3.1b"><apply id="S3.SS3.p6.8.m3.1.1.cmml" xref="S3.SS3.p6.8.m3.1.1"><in id="S3.SS3.p6.8.m3.1.1.1.cmml" xref="S3.SS3.p6.8.m3.1.1.1"></in><ci id="S3.SS3.p6.8.m3.1.1.2.cmml" xref="S3.SS3.p6.8.m3.1.1.2">𝑥</ci><apply id="S3.SS3.p6.8.m3.1.1.3.cmml" xref="S3.SS3.p6.8.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p6.8.m3.1.1.3.1.cmml" xref="S3.SS3.p6.8.m3.1.1.3">superscript</csymbol><ci id="S3.SS3.p6.8.m3.1.1.3.2.cmml" xref="S3.SS3.p6.8.m3.1.1.3.2">ℝ</ci><ci id="S3.SS3.p6.8.m3.1.1.3.3.cmml" xref="S3.SS3.p6.8.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.8.m3.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.8.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>). Later, we will then use this to argue that the integral in the definition of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p6.9.m4.1"><semantics id="S3.SS3.p6.9.m4.1a"><mi id="S3.SS3.p6.9.m4.1.1" xref="S3.SS3.p6.9.m4.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.9.m4.1b"><ci id="S3.SS3.p6.9.m4.1.1.cmml" xref="S3.SS3.p6.9.m4.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.9.m4.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.9.m4.1d">italic_F</annotation></semantics></math> cannot be <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.p6.10.m5.1"><semantics id="S3.SS3.p6.10.m5.1a"><mn id="S3.SS3.p6.10.m5.1.1" xref="S3.SS3.p6.10.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.10.m5.1b"><cn id="S3.SS3.p6.10.m5.1.1.cmml" type="integer" xref="S3.SS3.p6.10.m5.1.1">0</cn></annotation-xml></semantics></math> without <math alttext="V_{x}" class="ltx_Math" display="inline" id="S3.SS3.p6.11.m6.1"><semantics id="S3.SS3.p6.11.m6.1a"><msub id="S3.SS3.p6.11.m6.1.1" xref="S3.SS3.p6.11.m6.1.1.cmml"><mi id="S3.SS3.p6.11.m6.1.1.2" xref="S3.SS3.p6.11.m6.1.1.2.cmml">V</mi><mi id="S3.SS3.p6.11.m6.1.1.3" xref="S3.SS3.p6.11.m6.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.11.m6.1b"><apply id="S3.SS3.p6.11.m6.1.1.cmml" xref="S3.SS3.p6.11.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p6.11.m6.1.1.1.cmml" xref="S3.SS3.p6.11.m6.1.1">subscript</csymbol><ci id="S3.SS3.p6.11.m6.1.1.2.cmml" xref="S3.SS3.p6.11.m6.1.1.2">𝑉</ci><ci id="S3.SS3.p6.11.m6.1.1.3.cmml" xref="S3.SS3.p6.11.m6.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.11.m6.1c">V_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.11.m6.1d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> being empty. This in turn allows us to conclude that fixpoints of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p6.12.m7.1"><semantics id="S3.SS3.p6.12.m7.1a"><mi id="S3.SS3.p6.12.m7.1.1" xref="S3.SS3.p6.12.m7.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.12.m7.1b"><ci id="S3.SS3.p6.12.m7.1.1.cmml" xref="S3.SS3.p6.12.m7.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.12.m7.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.12.m7.1d">italic_F</annotation></semantics></math> are <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p6.13.m8.1"><semantics id="S3.SS3.p6.13.m8.1a"><msub id="S3.SS3.p6.13.m8.1.1" xref="S3.SS3.p6.13.m8.1.1.cmml"><mi id="S3.SS3.p6.13.m8.1.1.2" mathvariant="normal" xref="S3.SS3.p6.13.m8.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p6.13.m8.1.1.3" xref="S3.SS3.p6.13.m8.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.13.m8.1b"><apply id="S3.SS3.p6.13.m8.1.1.cmml" xref="S3.SS3.p6.13.m8.1.1"><csymbol cd="ambiguous" id="S3.SS3.p6.13.m8.1.1.1.cmml" xref="S3.SS3.p6.13.m8.1.1">subscript</csymbol><ci id="S3.SS3.p6.13.m8.1.1.2.cmml" xref="S3.SS3.p6.13.m8.1.1.2">ℓ</ci><ci id="S3.SS3.p6.13.m8.1.1.3.cmml" xref="S3.SS3.p6.13.m8.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.13.m8.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.13.m8.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoints of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p6.14.m9.1"><semantics id="S3.SS3.p6.14.m9.1a"><mi id="S3.SS3.p6.14.m9.1.1" xref="S3.SS3.p6.14.m9.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.14.m9.1b"><ci id="S3.SS3.p6.14.m9.1.1.cmml" xref="S3.SS3.p6.14.m9.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.14.m9.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.14.m9.1d">italic_μ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.p7"> <p class="ltx_p" id="S3.SS3.p7.1">To prove that <math alttext="0\not\in\text{conv}(V_{x})" class="ltx_Math" display="inline" id="S3.SS3.p7.1.m1.1"><semantics id="S3.SS3.p7.1.m1.1a"><mrow id="S3.SS3.p7.1.m1.1.1" xref="S3.SS3.p7.1.m1.1.1.cmml"><mn id="S3.SS3.p7.1.m1.1.1.3" xref="S3.SS3.p7.1.m1.1.1.3.cmml">0</mn><mo id="S3.SS3.p7.1.m1.1.1.2" xref="S3.SS3.p7.1.m1.1.1.2.cmml">∉</mo><mrow id="S3.SS3.p7.1.m1.1.1.1" xref="S3.SS3.p7.1.m1.1.1.1.cmml"><mtext id="S3.SS3.p7.1.m1.1.1.1.3" xref="S3.SS3.p7.1.m1.1.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.p7.1.m1.1.1.1.2" xref="S3.SS3.p7.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p7.1.m1.1.1.1.1.1" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p7.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p7.1.m1.1.1.1.1.1.1" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p7.1.m1.1.1.1.1.1.1.2" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.p7.1.m1.1.1.1.1.1.1.3" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S3.SS3.p7.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.1.m1.1b"><apply id="S3.SS3.p7.1.m1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1"><notin id="S3.SS3.p7.1.m1.1.1.2.cmml" xref="S3.SS3.p7.1.m1.1.1.2"></notin><cn id="S3.SS3.p7.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p7.1.m1.1.1.3">0</cn><apply id="S3.SS3.p7.1.m1.1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1.1"><times id="S3.SS3.p7.1.m1.1.1.1.2.cmml" xref="S3.SS3.p7.1.m1.1.1.1.2"></times><ci id="S3.SS3.p7.1.m1.1.1.1.3a.cmml" xref="S3.SS3.p7.1.m1.1.1.1.3"><mtext id="S3.SS3.p7.1.m1.1.1.1.3.cmml" xref="S3.SS3.p7.1.m1.1.1.1.3">conv</mtext></ci><apply id="S3.SS3.p7.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p7.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p7.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.p7.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p7.1.m1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.1.m1.1c">0\not\in\text{conv}(V_{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.1.m1.1d">0 ∉ conv ( italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>, we need the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem16"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem16.1.1.1">Lemma 3.16</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem16.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem16.p1"> <p class="ltx_p" id="S3.Thmtheorem16.p1.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem16.p1.7.7">Assume that <math alttext="0\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.1.1.m1.1"><semantics id="S3.Thmtheorem16.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem16.p1.1.1.m1.1.1" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.cmml"><mn id="S3.Thmtheorem16.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.2.cmml">0</mn><mo id="S3.Thmtheorem16.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem16.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.2" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.3" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.1.1.m1.1b"><apply id="S3.Thmtheorem16.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1"><in id="S3.Thmtheorem16.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.1"></in><cn id="S3.Thmtheorem16.p1.1.1.m1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.2">0</cn><apply id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem16.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmtheorem16.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.1.1.m1.1c">0\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.1.1.m1.1d">0 ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is contained in the convex hull of <math alttext="k\leq d+1" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.2.2.m2.1"><semantics id="S3.Thmtheorem16.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem16.p1.2.2.m2.1.1" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem16.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.2.cmml">k</mi><mo id="S3.Thmtheorem16.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.1.cmml">≤</mo><mrow id="S3.Thmtheorem16.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.2.cmml">d</mi><mo id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.1" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.1.cmml">+</mo><mn id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.2.2.m2.1b"><apply id="S3.Thmtheorem16.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1"><leq id="S3.Thmtheorem16.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.1"></leq><ci id="S3.Thmtheorem16.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.2">𝑘</ci><apply id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3"><plus id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.1"></plus><ci id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.2">𝑑</ci><cn id="S3.Thmtheorem16.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem16.p1.2.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.2.2.m2.1c">k\leq d+1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.2.2.m2.1d">italic_k ≤ italic_d + 1</annotation></semantics></math> points <math alttext="v_{1},\dots,v_{k}\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.3.3.m3.3"><semantics id="S3.Thmtheorem16.p1.3.3.m3.3a"><mrow id="S3.Thmtheorem16.p1.3.3.m3.3.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.cmml"><mrow id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.3.cmml"><msub id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.2" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.2.cmml">v</mi><mn id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.3" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.3.cmml">,</mo><mi id="S3.Thmtheorem16.p1.3.3.m3.1.1" mathvariant="normal" xref="S3.Thmtheorem16.p1.3.3.m3.1.1.cmml">…</mi><mo id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.4" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.3.cmml">,</mo><msub id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.cmml"><mi id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.2" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.2.cmml">v</mi><mi id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.Thmtheorem16.p1.3.3.m3.3.3.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.3.cmml">∈</mo><msup id="S3.Thmtheorem16.p1.3.3.m3.3.3.4" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.cmml"><mi id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.2" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.2.cmml">S</mi><mrow id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.cmml"><mi id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.2" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.2.cmml">d</mi><mo id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.1" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.1.cmml">−</mo><mn id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.3" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.3.3.m3.3b"><apply id="S3.Thmtheorem16.p1.3.3.m3.3.3.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3"><in id="S3.Thmtheorem16.p1.3.3.m3.3.3.3.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.3"></in><list id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.3.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2"><apply id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.2">𝑣</ci><cn id="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem16.p1.3.3.m3.2.2.1.1.1.3">1</cn></apply><ci id="S3.Thmtheorem16.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.1.1">…</ci><apply id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.2.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.2">𝑣</ci><ci id="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.3.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.2.2.2.3">𝑘</ci></apply></list><apply id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4">superscript</csymbol><ci id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.2.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.2">𝑆</ci><apply id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3"><minus id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.1.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.1"></minus><ci id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.2.cmml" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.2">𝑑</ci><cn id="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.3.cmml" type="integer" xref="S3.Thmtheorem16.p1.3.3.m3.3.3.4.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.3.3.m3.3c">v_{1},\dots,v_{k}\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.3.3.m3.3d">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.4.4.m4.1"><semantics id="S3.Thmtheorem16.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem16.p1.4.4.m4.1.1" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem16.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.2.cmml">z</mi><mo id="S3.Thmtheorem16.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem16.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.4.4.m4.1b"><apply id="S3.Thmtheorem16.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1"><in id="S3.Thmtheorem16.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem16.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.2">𝑧</ci><apply id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem16.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem16.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.4.4.m4.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.4.4.m4.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.5.5.m5.3"><semantics id="S3.Thmtheorem16.p1.5.5.m5.3a"><mrow id="S3.Thmtheorem16.p1.5.5.m5.3.4" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.cmml"><mi id="S3.Thmtheorem16.p1.5.5.m5.3.4.2" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.1" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem16.p1.5.5.m5.3.4.3" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.cmml"><mrow id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.2" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem16.p1.5.5.m5.1.1" xref="S3.Thmtheorem16.p1.5.5.m5.1.1.cmml">1</mn><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.2.2" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem16.p1.5.5.m5.2.2" mathvariant="normal" xref="S3.Thmtheorem16.p1.5.5.m5.2.2.cmml">∞</mi><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.1" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.2" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem16.p1.5.5.m5.3.3" mathvariant="normal" xref="S3.Thmtheorem16.p1.5.5.m5.3.3.cmml">∞</mi><mo id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.5.5.m5.3b"><apply id="S3.Thmtheorem16.p1.5.5.m5.3.4.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4"><in id="S3.Thmtheorem16.p1.5.5.m5.3.4.1.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.1"></in><ci id="S3.Thmtheorem16.p1.5.5.m5.3.4.2.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.2">𝑝</ci><apply id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3"><union id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.1.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.1.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.2.2"><cn id="S3.Thmtheorem16.p1.5.5.m5.1.1.cmml" type="integer" xref="S3.Thmtheorem16.p1.5.5.m5.1.1">1</cn><infinity id="S3.Thmtheorem16.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.2.2"></infinity></interval><set id="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.1.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.4.3.3.2"><infinity id="S3.Thmtheorem16.p1.5.5.m5.3.3.cmml" xref="S3.Thmtheorem16.p1.5.5.m5.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.5.5.m5.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.5.5.m5.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary. Then there exists <math alttext="j\in[k]" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.6.6.m6.1"><semantics id="S3.Thmtheorem16.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem16.p1.6.6.m6.1.2" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.cmml"><mi id="S3.Thmtheorem16.p1.6.6.m6.1.2.2" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.2.cmml">j</mi><mo id="S3.Thmtheorem16.p1.6.6.m6.1.2.1" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem16.p1.6.6.m6.1.2.3.2" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.3.1.cmml"><mo id="S3.Thmtheorem16.p1.6.6.m6.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S3.Thmtheorem16.p1.6.6.m6.1.1" xref="S3.Thmtheorem16.p1.6.6.m6.1.1.cmml">k</mi><mo id="S3.Thmtheorem16.p1.6.6.m6.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.6.6.m6.1b"><apply id="S3.Thmtheorem16.p1.6.6.m6.1.2.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.2"><in id="S3.Thmtheorem16.p1.6.6.m6.1.2.1.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.1"></in><ci id="S3.Thmtheorem16.p1.6.6.m6.1.2.2.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.2">𝑗</ci><apply id="S3.Thmtheorem16.p1.6.6.m6.1.2.3.1.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.3.2"><csymbol cd="latexml" id="S3.Thmtheorem16.p1.6.6.m6.1.2.3.1.1.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem16.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem16.p1.6.6.m6.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.6.6.m6.1c">j\in[k]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.6.6.m6.1d">italic_j ∈ [ italic_k ]</annotation></semantics></math> such that <math alttext="z\in\mathcal{H}^{p}_{0,-v_{j}}" class="ltx_Math" display="inline" id="S3.Thmtheorem16.p1.7.7.m7.2"><semantics id="S3.Thmtheorem16.p1.7.7.m7.2a"><mrow id="S3.Thmtheorem16.p1.7.7.m7.2.3" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.cmml"><mi id="S3.Thmtheorem16.p1.7.7.m7.2.3.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.2.cmml">z</mi><mo id="S3.Thmtheorem16.p1.7.7.m7.2.3.1" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.1.cmml">∈</mo><msubsup id="S3.Thmtheorem16.p1.7.7.m7.2.3.3" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.3.cmml"><mn id="S3.Thmtheorem16.p1.7.7.m7.1.1.1.1" xref="S3.Thmtheorem16.p1.7.7.m7.1.1.1.1.cmml">0</mn><mo id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.3.cmml">,</mo><mrow id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.cmml"><mo id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1a" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.cmml">−</mo><msub id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.cmml"><mi id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.2" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.2.cmml">v</mi><mi id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.3" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.3.cmml">j</mi></msub></mrow></mrow><mi id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.3" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem16.p1.7.7.m7.2b"><apply id="S3.Thmtheorem16.p1.7.7.m7.2.3.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3"><in id="S3.Thmtheorem16.p1.7.7.m7.2.3.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.1"></in><ci id="S3.Thmtheorem16.p1.7.7.m7.2.3.2.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.2">𝑧</ci><apply id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3">subscript</csymbol><apply id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3">superscript</csymbol><ci id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.2.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.2">ℋ</ci><ci id="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.3.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.3.3.2.3">𝑝</ci></apply><list id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.3.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2"><cn id="S3.Thmtheorem16.p1.7.7.m7.1.1.1.1.cmml" type="integer" xref="S3.Thmtheorem16.p1.7.7.m7.1.1.1.1">0</cn><apply id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1"><minus id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1"></minus><apply id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.1.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2">subscript</csymbol><ci id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.2.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.2">𝑣</ci><ci id="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.3.cmml" xref="S3.Thmtheorem16.p1.7.7.m7.2.2.2.2.1.2.3">𝑗</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem16.p1.7.7.m7.2c">z\in\mathcal{H}^{p}_{0,-v_{j}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem16.p1.7.7.m7.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , - italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.3.p1"> <p class="ltx_p" id="S3.SS3.3.p1.17">Let <math alttext="R_{1},\ldots,R_{k}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.1.m1.3"><semantics id="S3.SS3.3.p1.1.m1.3a"><mrow id="S3.SS3.3.p1.1.m1.3.3.2" xref="S3.SS3.3.p1.1.m1.3.3.3.cmml"><msub id="S3.SS3.3.p1.1.m1.2.2.1.1" xref="S3.SS3.3.p1.1.m1.2.2.1.1.cmml"><mi id="S3.SS3.3.p1.1.m1.2.2.1.1.2" xref="S3.SS3.3.p1.1.m1.2.2.1.1.2.cmml">R</mi><mn id="S3.SS3.3.p1.1.m1.2.2.1.1.3" xref="S3.SS3.3.p1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.3.p1.1.m1.3.3.2.3" xref="S3.SS3.3.p1.1.m1.3.3.3.cmml">,</mo><mi id="S3.SS3.3.p1.1.m1.1.1" mathvariant="normal" xref="S3.SS3.3.p1.1.m1.1.1.cmml">…</mi><mo id="S3.SS3.3.p1.1.m1.3.3.2.4" xref="S3.SS3.3.p1.1.m1.3.3.3.cmml">,</mo><msub id="S3.SS3.3.p1.1.m1.3.3.2.2" xref="S3.SS3.3.p1.1.m1.3.3.2.2.cmml"><mi id="S3.SS3.3.p1.1.m1.3.3.2.2.2" xref="S3.SS3.3.p1.1.m1.3.3.2.2.2.cmml">R</mi><mi id="S3.SS3.3.p1.1.m1.3.3.2.2.3" xref="S3.SS3.3.p1.1.m1.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.1.m1.3b"><list id="S3.SS3.3.p1.1.m1.3.3.3.cmml" xref="S3.SS3.3.p1.1.m1.3.3.2"><apply id="S3.SS3.3.p1.1.m1.2.2.1.1.cmml" xref="S3.SS3.3.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.1.m1.2.2.1.1.1.cmml" xref="S3.SS3.3.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.1.m1.2.2.1.1.2.cmml" xref="S3.SS3.3.p1.1.m1.2.2.1.1.2">𝑅</ci><cn id="S3.SS3.3.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.3.p1.1.m1.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.3.p1.1.m1.1.1.cmml" xref="S3.SS3.3.p1.1.m1.1.1">…</ci><apply id="S3.SS3.3.p1.1.m1.3.3.2.2.cmml" xref="S3.SS3.3.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.1.m1.3.3.2.2.1.cmml" xref="S3.SS3.3.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="S3.SS3.3.p1.1.m1.3.3.2.2.2.cmml" xref="S3.SS3.3.p1.1.m1.3.3.2.2.2">𝑅</ci><ci id="S3.SS3.3.p1.1.m1.3.3.2.2.3.cmml" xref="S3.SS3.3.p1.1.m1.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.1.m1.3c">R_{1},\ldots,R_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.1.m1.3d">italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> be the open rays from <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.2.m2.1"><semantics id="S3.SS3.3.p1.2.m2.1a"><mn id="S3.SS3.3.p1.2.m2.1.1" xref="S3.SS3.3.p1.2.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.2.m2.1b"><cn id="S3.SS3.3.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS3.3.p1.2.m2.1.1">0</cn></annotation-xml></semantics></math> in the directions <math alttext="v_{1},\ldots,v_{k}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.3.m3.3"><semantics id="S3.SS3.3.p1.3.m3.3a"><mrow id="S3.SS3.3.p1.3.m3.3.3.2" xref="S3.SS3.3.p1.3.m3.3.3.3.cmml"><msub id="S3.SS3.3.p1.3.m3.2.2.1.1" xref="S3.SS3.3.p1.3.m3.2.2.1.1.cmml"><mi id="S3.SS3.3.p1.3.m3.2.2.1.1.2" xref="S3.SS3.3.p1.3.m3.2.2.1.1.2.cmml">v</mi><mn id="S3.SS3.3.p1.3.m3.2.2.1.1.3" xref="S3.SS3.3.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.3.p1.3.m3.3.3.2.3" xref="S3.SS3.3.p1.3.m3.3.3.3.cmml">,</mo><mi id="S3.SS3.3.p1.3.m3.1.1" mathvariant="normal" xref="S3.SS3.3.p1.3.m3.1.1.cmml">…</mi><mo id="S3.SS3.3.p1.3.m3.3.3.2.4" xref="S3.SS3.3.p1.3.m3.3.3.3.cmml">,</mo><msub id="S3.SS3.3.p1.3.m3.3.3.2.2" xref="S3.SS3.3.p1.3.m3.3.3.2.2.cmml"><mi id="S3.SS3.3.p1.3.m3.3.3.2.2.2" xref="S3.SS3.3.p1.3.m3.3.3.2.2.2.cmml">v</mi><mi id="S3.SS3.3.p1.3.m3.3.3.2.2.3" xref="S3.SS3.3.p1.3.m3.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.3.m3.3b"><list id="S3.SS3.3.p1.3.m3.3.3.3.cmml" xref="S3.SS3.3.p1.3.m3.3.3.2"><apply id="S3.SS3.3.p1.3.m3.2.2.1.1.cmml" xref="S3.SS3.3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.3.m3.2.2.1.1.1.cmml" xref="S3.SS3.3.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.3.m3.2.2.1.1.2.cmml" xref="S3.SS3.3.p1.3.m3.2.2.1.1.2">𝑣</ci><cn id="S3.SS3.3.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.3.p1.3.m3.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.3.p1.3.m3.1.1.cmml" xref="S3.SS3.3.p1.3.m3.1.1">…</ci><apply id="S3.SS3.3.p1.3.m3.3.3.2.2.cmml" xref="S3.SS3.3.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.3.m3.3.3.2.2.1.cmml" xref="S3.SS3.3.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="S3.SS3.3.p1.3.m3.3.3.2.2.2.cmml" xref="S3.SS3.3.p1.3.m3.3.3.2.2.2">𝑣</ci><ci id="S3.SS3.3.p1.3.m3.3.3.2.2.3.cmml" xref="S3.SS3.3.p1.3.m3.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.3.m3.3c">v_{1},\ldots,v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.3.m3.3d">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, respectively. Let us now grow an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.4.m4.1"><semantics id="S3.SS3.3.p1.4.m4.1a"><msub id="S3.SS3.3.p1.4.m4.1.1" xref="S3.SS3.3.p1.4.m4.1.1.cmml"><mi id="S3.SS3.3.p1.4.m4.1.1.2" mathvariant="normal" xref="S3.SS3.3.p1.4.m4.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.3.p1.4.m4.1.1.3" xref="S3.SS3.3.p1.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.4.m4.1b"><apply id="S3.SS3.3.p1.4.m4.1.1.cmml" xref="S3.SS3.3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.4.m4.1.1.1.cmml" xref="S3.SS3.3.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.4.m4.1.1.2.cmml" xref="S3.SS3.3.p1.4.m4.1.1.2">ℓ</ci><ci id="S3.SS3.3.p1.4.m4.1.1.3.cmml" xref="S3.SS3.3.p1.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-ball <math alttext="B_{z}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.5.m5.1"><semantics id="S3.SS3.3.p1.5.m5.1a"><msub id="S3.SS3.3.p1.5.m5.1.1" xref="S3.SS3.3.p1.5.m5.1.1.cmml"><mi id="S3.SS3.3.p1.5.m5.1.1.2" xref="S3.SS3.3.p1.5.m5.1.1.2.cmml">B</mi><mi id="S3.SS3.3.p1.5.m5.1.1.3" xref="S3.SS3.3.p1.5.m5.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.5.m5.1b"><apply id="S3.SS3.3.p1.5.m5.1.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.5.m5.1.1.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.5.m5.1.1.2.cmml" xref="S3.SS3.3.p1.5.m5.1.1.2">𝐵</ci><ci id="S3.SS3.3.p1.5.m5.1.1.3.cmml" xref="S3.SS3.3.p1.5.m5.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.5.m5.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.5.m5.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> centered at <math alttext="z" class="ltx_Math" display="inline" id="S3.SS3.3.p1.6.m6.1"><semantics id="S3.SS3.3.p1.6.m6.1a"><mi id="S3.SS3.3.p1.6.m6.1.1" xref="S3.SS3.3.p1.6.m6.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.6.m6.1b"><ci id="S3.SS3.3.p1.6.m6.1.1.cmml" xref="S3.SS3.3.p1.6.m6.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.6.m6.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.6.m6.1d">italic_z</annotation></semantics></math>, exactly until the ball contains <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.7.m7.1"><semantics id="S3.SS3.3.p1.7.m7.1a"><mn id="S3.SS3.3.p1.7.m7.1.1" xref="S3.SS3.3.p1.7.m7.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.7.m7.1b"><cn id="S3.SS3.3.p1.7.m7.1.1.cmml" type="integer" xref="S3.SS3.3.p1.7.m7.1.1">0</cn></annotation-xml></semantics></math>. Recall that by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>, we have <math alttext="z\in\mathcal{H}^{p}_{0,-v_{j}}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.8.m8.2"><semantics id="S3.SS3.3.p1.8.m8.2a"><mrow id="S3.SS3.3.p1.8.m8.2.3" xref="S3.SS3.3.p1.8.m8.2.3.cmml"><mi id="S3.SS3.3.p1.8.m8.2.3.2" xref="S3.SS3.3.p1.8.m8.2.3.2.cmml">z</mi><mo id="S3.SS3.3.p1.8.m8.2.3.1" xref="S3.SS3.3.p1.8.m8.2.3.1.cmml">∈</mo><msubsup id="S3.SS3.3.p1.8.m8.2.3.3" xref="S3.SS3.3.p1.8.m8.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.3.p1.8.m8.2.3.3.2.2" xref="S3.SS3.3.p1.8.m8.2.3.3.2.2.cmml">ℋ</mi><mrow id="S3.SS3.3.p1.8.m8.2.2.2.2" xref="S3.SS3.3.p1.8.m8.2.2.2.3.cmml"><mn id="S3.SS3.3.p1.8.m8.1.1.1.1" xref="S3.SS3.3.p1.8.m8.1.1.1.1.cmml">0</mn><mo id="S3.SS3.3.p1.8.m8.2.2.2.2.2" xref="S3.SS3.3.p1.8.m8.2.2.2.3.cmml">,</mo><mrow id="S3.SS3.3.p1.8.m8.2.2.2.2.1" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.cmml"><mo id="S3.SS3.3.p1.8.m8.2.2.2.2.1a" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.cmml">−</mo><msub id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.cmml"><mi id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.2" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.2.cmml">v</mi><mi id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.3" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.3.cmml">j</mi></msub></mrow></mrow><mi id="S3.SS3.3.p1.8.m8.2.3.3.2.3" xref="S3.SS3.3.p1.8.m8.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.8.m8.2b"><apply id="S3.SS3.3.p1.8.m8.2.3.cmml" xref="S3.SS3.3.p1.8.m8.2.3"><in id="S3.SS3.3.p1.8.m8.2.3.1.cmml" xref="S3.SS3.3.p1.8.m8.2.3.1"></in><ci id="S3.SS3.3.p1.8.m8.2.3.2.cmml" xref="S3.SS3.3.p1.8.m8.2.3.2">𝑧</ci><apply id="S3.SS3.3.p1.8.m8.2.3.3.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.8.m8.2.3.3.1.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3">subscript</csymbol><apply id="S3.SS3.3.p1.8.m8.2.3.3.2.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.8.m8.2.3.3.2.1.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3">superscript</csymbol><ci id="S3.SS3.3.p1.8.m8.2.3.3.2.2.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3.2.2">ℋ</ci><ci id="S3.SS3.3.p1.8.m8.2.3.3.2.3.cmml" xref="S3.SS3.3.p1.8.m8.2.3.3.2.3">𝑝</ci></apply><list id="S3.SS3.3.p1.8.m8.2.2.2.3.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2"><cn id="S3.SS3.3.p1.8.m8.1.1.1.1.cmml" type="integer" xref="S3.SS3.3.p1.8.m8.1.1.1.1">0</cn><apply id="S3.SS3.3.p1.8.m8.2.2.2.2.1.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1"><minus id="S3.SS3.3.p1.8.m8.2.2.2.2.1.1.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1"></minus><apply id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.1.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2">subscript</csymbol><ci id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.2.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.2">𝑣</ci><ci id="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.3.cmml" xref="S3.SS3.3.p1.8.m8.2.2.2.2.1.2.3">𝑗</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.8.m8.2c">z\in\mathcal{H}^{p}_{0,-v_{j}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.8.m8.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , - italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.9.m9.1"><semantics id="S3.SS3.3.p1.9.m9.1a"><msubsup id="S3.SS3.3.p1.9.m9.1.1" xref="S3.SS3.3.p1.9.m9.1.1.cmml"><mi id="S3.SS3.3.p1.9.m9.1.1.2.2" xref="S3.SS3.3.p1.9.m9.1.1.2.2.cmml">B</mi><mi id="S3.SS3.3.p1.9.m9.1.1.3" xref="S3.SS3.3.p1.9.m9.1.1.3.cmml">z</mi><mo id="S3.SS3.3.p1.9.m9.1.1.2.3" xref="S3.SS3.3.p1.9.m9.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.9.m9.1b"><apply id="S3.SS3.3.p1.9.m9.1.1.cmml" xref="S3.SS3.3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.9.m9.1.1.1.cmml" xref="S3.SS3.3.p1.9.m9.1.1">subscript</csymbol><apply id="S3.SS3.3.p1.9.m9.1.1.2.cmml" xref="S3.SS3.3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.9.m9.1.1.2.1.cmml" xref="S3.SS3.3.p1.9.m9.1.1">superscript</csymbol><ci id="S3.SS3.3.p1.9.m9.1.1.2.2.cmml" xref="S3.SS3.3.p1.9.m9.1.1.2.2">𝐵</ci><compose id="S3.SS3.3.p1.9.m9.1.1.2.3.cmml" xref="S3.SS3.3.p1.9.m9.1.1.2.3"></compose></apply><ci id="S3.SS3.3.p1.9.m9.1.1.3.cmml" xref="S3.SS3.3.p1.9.m9.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.9.m9.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.9.m9.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> does not intersect <math alttext="R_{j}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.10.m10.1"><semantics id="S3.SS3.3.p1.10.m10.1a"><msub id="S3.SS3.3.p1.10.m10.1.1" xref="S3.SS3.3.p1.10.m10.1.1.cmml"><mi id="S3.SS3.3.p1.10.m10.1.1.2" xref="S3.SS3.3.p1.10.m10.1.1.2.cmml">R</mi><mi id="S3.SS3.3.p1.10.m10.1.1.3" xref="S3.SS3.3.p1.10.m10.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.10.m10.1b"><apply id="S3.SS3.3.p1.10.m10.1.1.cmml" xref="S3.SS3.3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.10.m10.1.1.1.cmml" xref="S3.SS3.3.p1.10.m10.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.10.m10.1.1.2.cmml" xref="S3.SS3.3.p1.10.m10.1.1.2">𝑅</ci><ci id="S3.SS3.3.p1.10.m10.1.1.3.cmml" xref="S3.SS3.3.p1.10.m10.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.10.m10.1c">R_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.10.m10.1d">italic_R start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Now observe that if we had <math alttext="B^{\circ}_{z}\cap R_{j}\neq\varnothing" class="ltx_Math" display="inline" id="S3.SS3.3.p1.11.m11.1"><semantics id="S3.SS3.3.p1.11.m11.1a"><mrow id="S3.SS3.3.p1.11.m11.1.1" xref="S3.SS3.3.p1.11.m11.1.1.cmml"><mrow id="S3.SS3.3.p1.11.m11.1.1.2" xref="S3.SS3.3.p1.11.m11.1.1.2.cmml"><msubsup id="S3.SS3.3.p1.11.m11.1.1.2.2" xref="S3.SS3.3.p1.11.m11.1.1.2.2.cmml"><mi id="S3.SS3.3.p1.11.m11.1.1.2.2.2.2" xref="S3.SS3.3.p1.11.m11.1.1.2.2.2.2.cmml">B</mi><mi id="S3.SS3.3.p1.11.m11.1.1.2.2.3" xref="S3.SS3.3.p1.11.m11.1.1.2.2.3.cmml">z</mi><mo id="S3.SS3.3.p1.11.m11.1.1.2.2.2.3" xref="S3.SS3.3.p1.11.m11.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="S3.SS3.3.p1.11.m11.1.1.2.1" xref="S3.SS3.3.p1.11.m11.1.1.2.1.cmml">∩</mo><msub id="S3.SS3.3.p1.11.m11.1.1.2.3" xref="S3.SS3.3.p1.11.m11.1.1.2.3.cmml"><mi id="S3.SS3.3.p1.11.m11.1.1.2.3.2" xref="S3.SS3.3.p1.11.m11.1.1.2.3.2.cmml">R</mi><mi id="S3.SS3.3.p1.11.m11.1.1.2.3.3" xref="S3.SS3.3.p1.11.m11.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S3.SS3.3.p1.11.m11.1.1.1" xref="S3.SS3.3.p1.11.m11.1.1.1.cmml">≠</mo><mi id="S3.SS3.3.p1.11.m11.1.1.3" mathvariant="normal" xref="S3.SS3.3.p1.11.m11.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.11.m11.1b"><apply id="S3.SS3.3.p1.11.m11.1.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1"><neq id="S3.SS3.3.p1.11.m11.1.1.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1.1"></neq><apply id="S3.SS3.3.p1.11.m11.1.1.2.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2"><intersect id="S3.SS3.3.p1.11.m11.1.1.2.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.1"></intersect><apply id="S3.SS3.3.p1.11.m11.1.1.2.2.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.11.m11.1.1.2.2.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2">subscript</csymbol><apply id="S3.SS3.3.p1.11.m11.1.1.2.2.2.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.11.m11.1.1.2.2.2.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2">superscript</csymbol><ci id="S3.SS3.3.p1.11.m11.1.1.2.2.2.2.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2.2.2">𝐵</ci><compose id="S3.SS3.3.p1.11.m11.1.1.2.2.2.3.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2.2.3"></compose></apply><ci id="S3.SS3.3.p1.11.m11.1.1.2.2.3.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.2.3">𝑧</ci></apply><apply id="S3.SS3.3.p1.11.m11.1.1.2.3.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.11.m11.1.1.2.3.1.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.3">subscript</csymbol><ci id="S3.SS3.3.p1.11.m11.1.1.2.3.2.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.3.2">𝑅</ci><ci id="S3.SS3.3.p1.11.m11.1.1.2.3.3.cmml" xref="S3.SS3.3.p1.11.m11.1.1.2.3.3">𝑗</ci></apply></apply><emptyset id="S3.SS3.3.p1.11.m11.1.1.3.cmml" xref="S3.SS3.3.p1.11.m11.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.11.m11.1c">B^{\circ}_{z}\cap R_{j}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.11.m11.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∩ italic_R start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math> for all <math alttext="j\in[k]" class="ltx_Math" display="inline" id="S3.SS3.3.p1.12.m12.1"><semantics id="S3.SS3.3.p1.12.m12.1a"><mrow id="S3.SS3.3.p1.12.m12.1.2" xref="S3.SS3.3.p1.12.m12.1.2.cmml"><mi id="S3.SS3.3.p1.12.m12.1.2.2" xref="S3.SS3.3.p1.12.m12.1.2.2.cmml">j</mi><mo id="S3.SS3.3.p1.12.m12.1.2.1" xref="S3.SS3.3.p1.12.m12.1.2.1.cmml">∈</mo><mrow id="S3.SS3.3.p1.12.m12.1.2.3.2" xref="S3.SS3.3.p1.12.m12.1.2.3.1.cmml"><mo id="S3.SS3.3.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S3.SS3.3.p1.12.m12.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.3.p1.12.m12.1.1" xref="S3.SS3.3.p1.12.m12.1.1.cmml">k</mi><mo id="S3.SS3.3.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S3.SS3.3.p1.12.m12.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.12.m12.1b"><apply id="S3.SS3.3.p1.12.m12.1.2.cmml" xref="S3.SS3.3.p1.12.m12.1.2"><in id="S3.SS3.3.p1.12.m12.1.2.1.cmml" xref="S3.SS3.3.p1.12.m12.1.2.1"></in><ci id="S3.SS3.3.p1.12.m12.1.2.2.cmml" xref="S3.SS3.3.p1.12.m12.1.2.2">𝑗</ci><apply id="S3.SS3.3.p1.12.m12.1.2.3.1.cmml" xref="S3.SS3.3.p1.12.m12.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.3.p1.12.m12.1.2.3.1.1.cmml" xref="S3.SS3.3.p1.12.m12.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.3.p1.12.m12.1.1.cmml" xref="S3.SS3.3.p1.12.m12.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.12.m12.1c">j\in[k]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.12.m12.1d">italic_j ∈ [ italic_k ]</annotation></semantics></math>, then this would imply <math alttext="0\in B^{\circ}_{z}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.13.m13.1"><semantics id="S3.SS3.3.p1.13.m13.1a"><mrow id="S3.SS3.3.p1.13.m13.1.1" xref="S3.SS3.3.p1.13.m13.1.1.cmml"><mn id="S3.SS3.3.p1.13.m13.1.1.2" xref="S3.SS3.3.p1.13.m13.1.1.2.cmml">0</mn><mo id="S3.SS3.3.p1.13.m13.1.1.1" xref="S3.SS3.3.p1.13.m13.1.1.1.cmml">∈</mo><msubsup id="S3.SS3.3.p1.13.m13.1.1.3" xref="S3.SS3.3.p1.13.m13.1.1.3.cmml"><mi id="S3.SS3.3.p1.13.m13.1.1.3.2.2" xref="S3.SS3.3.p1.13.m13.1.1.3.2.2.cmml">B</mi><mi id="S3.SS3.3.p1.13.m13.1.1.3.3" xref="S3.SS3.3.p1.13.m13.1.1.3.3.cmml">z</mi><mo id="S3.SS3.3.p1.13.m13.1.1.3.2.3" xref="S3.SS3.3.p1.13.m13.1.1.3.2.3.cmml">∘</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.13.m13.1b"><apply id="S3.SS3.3.p1.13.m13.1.1.cmml" xref="S3.SS3.3.p1.13.m13.1.1"><in id="S3.SS3.3.p1.13.m13.1.1.1.cmml" xref="S3.SS3.3.p1.13.m13.1.1.1"></in><cn id="S3.SS3.3.p1.13.m13.1.1.2.cmml" type="integer" xref="S3.SS3.3.p1.13.m13.1.1.2">0</cn><apply id="S3.SS3.3.p1.13.m13.1.1.3.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.13.m13.1.1.3.1.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3">subscript</csymbol><apply id="S3.SS3.3.p1.13.m13.1.1.3.2.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.13.m13.1.1.3.2.1.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3">superscript</csymbol><ci id="S3.SS3.3.p1.13.m13.1.1.3.2.2.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3.2.2">𝐵</ci><compose id="S3.SS3.3.p1.13.m13.1.1.3.2.3.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3.2.3"></compose></apply><ci id="S3.SS3.3.p1.13.m13.1.1.3.3.cmml" xref="S3.SS3.3.p1.13.m13.1.1.3.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.13.m13.1c">0\in B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.13.m13.1d">0 ∈ italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> by convexity of <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.14.m14.1"><semantics id="S3.SS3.3.p1.14.m14.1a"><msubsup id="S3.SS3.3.p1.14.m14.1.1" xref="S3.SS3.3.p1.14.m14.1.1.cmml"><mi id="S3.SS3.3.p1.14.m14.1.1.2.2" xref="S3.SS3.3.p1.14.m14.1.1.2.2.cmml">B</mi><mi id="S3.SS3.3.p1.14.m14.1.1.3" xref="S3.SS3.3.p1.14.m14.1.1.3.cmml">z</mi><mo id="S3.SS3.3.p1.14.m14.1.1.2.3" xref="S3.SS3.3.p1.14.m14.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.14.m14.1b"><apply id="S3.SS3.3.p1.14.m14.1.1.cmml" xref="S3.SS3.3.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.14.m14.1.1.1.cmml" xref="S3.SS3.3.p1.14.m14.1.1">subscript</csymbol><apply id="S3.SS3.3.p1.14.m14.1.1.2.cmml" xref="S3.SS3.3.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.14.m14.1.1.2.1.cmml" xref="S3.SS3.3.p1.14.m14.1.1">superscript</csymbol><ci id="S3.SS3.3.p1.14.m14.1.1.2.2.cmml" xref="S3.SS3.3.p1.14.m14.1.1.2.2">𝐵</ci><compose id="S3.SS3.3.p1.14.m14.1.1.2.3.cmml" xref="S3.SS3.3.p1.14.m14.1.1.2.3"></compose></apply><ci id="S3.SS3.3.p1.14.m14.1.1.3.cmml" xref="S3.SS3.3.p1.14.m14.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.14.m14.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.14.m14.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>. But then, <math alttext="B_{z}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.15.m15.1"><semantics id="S3.SS3.3.p1.15.m15.1a"><msub id="S3.SS3.3.p1.15.m15.1.1" xref="S3.SS3.3.p1.15.m15.1.1.cmml"><mi id="S3.SS3.3.p1.15.m15.1.1.2" xref="S3.SS3.3.p1.15.m15.1.1.2.cmml">B</mi><mi id="S3.SS3.3.p1.15.m15.1.1.3" xref="S3.SS3.3.p1.15.m15.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.15.m15.1b"><apply id="S3.SS3.3.p1.15.m15.1.1.cmml" xref="S3.SS3.3.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS3.3.p1.15.m15.1.1.1.cmml" xref="S3.SS3.3.p1.15.m15.1.1">subscript</csymbol><ci id="S3.SS3.3.p1.15.m15.1.1.2.cmml" xref="S3.SS3.3.p1.15.m15.1.1.2">𝐵</ci><ci id="S3.SS3.3.p1.15.m15.1.1.3.cmml" xref="S3.SS3.3.p1.15.m15.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.15.m15.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.15.m15.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> cannot have been the minimum-radius ball centered at <math alttext="z" class="ltx_Math" display="inline" id="S3.SS3.3.p1.16.m16.1"><semantics id="S3.SS3.3.p1.16.m16.1a"><mi id="S3.SS3.3.p1.16.m16.1.1" xref="S3.SS3.3.p1.16.m16.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.16.m16.1b"><ci id="S3.SS3.3.p1.16.m16.1.1.cmml" xref="S3.SS3.3.p1.16.m16.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.16.m16.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.16.m16.1d">italic_z</annotation></semantics></math> containing <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.17.m17.1"><semantics id="S3.SS3.3.p1.17.m17.1a"><mn id="S3.SS3.3.p1.17.m17.1.1" xref="S3.SS3.3.p1.17.m17.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.17.m17.1b"><cn id="S3.SS3.3.p1.17.m17.1.1.cmml" type="integer" xref="S3.SS3.3.p1.17.m17.1.1">0</cn></annotation-xml></semantics></math>, a contradiction. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p8"> <p class="ltx_p" id="S3.SS3.p8.2">Let us now see how this lemma implies <math alttext="0\not\in\text{conv}(V_{x})" class="ltx_Math" display="inline" id="S3.SS3.p8.1.m1.1"><semantics id="S3.SS3.p8.1.m1.1a"><mrow id="S3.SS3.p8.1.m1.1.1" xref="S3.SS3.p8.1.m1.1.1.cmml"><mn id="S3.SS3.p8.1.m1.1.1.3" xref="S3.SS3.p8.1.m1.1.1.3.cmml">0</mn><mo id="S3.SS3.p8.1.m1.1.1.2" xref="S3.SS3.p8.1.m1.1.1.2.cmml">∉</mo><mrow id="S3.SS3.p8.1.m1.1.1.1" xref="S3.SS3.p8.1.m1.1.1.1.cmml"><mtext id="S3.SS3.p8.1.m1.1.1.1.3" xref="S3.SS3.p8.1.m1.1.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.p8.1.m1.1.1.1.2" xref="S3.SS3.p8.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p8.1.m1.1.1.1.1.1" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p8.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p8.1.m1.1.1.1.1.1.1" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p8.1.m1.1.1.1.1.1.1.2" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.p8.1.m1.1.1.1.1.1.1.3" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S3.SS3.p8.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.1.m1.1b"><apply id="S3.SS3.p8.1.m1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1"><notin id="S3.SS3.p8.1.m1.1.1.2.cmml" xref="S3.SS3.p8.1.m1.1.1.2"></notin><cn id="S3.SS3.p8.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p8.1.m1.1.1.3">0</cn><apply id="S3.SS3.p8.1.m1.1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1.1"><times id="S3.SS3.p8.1.m1.1.1.1.2.cmml" xref="S3.SS3.p8.1.m1.1.1.1.2"></times><ci id="S3.SS3.p8.1.m1.1.1.1.3a.cmml" xref="S3.SS3.p8.1.m1.1.1.1.3"><mtext id="S3.SS3.p8.1.m1.1.1.1.3.cmml" xref="S3.SS3.p8.1.m1.1.1.1.3">conv</mtext></ci><apply id="S3.SS3.p8.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p8.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p8.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.p8.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p8.1.m1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.1.m1.1c">0\not\in\text{conv}(V_{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.1.m1.1d">0 ∉ conv ( italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> for all <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.p8.2.m2.1"><semantics id="S3.SS3.p8.2.m2.1a"><mrow id="S3.SS3.p8.2.m2.1.1" xref="S3.SS3.p8.2.m2.1.1.cmml"><mi id="S3.SS3.p8.2.m2.1.1.2" xref="S3.SS3.p8.2.m2.1.1.2.cmml">x</mi><mo id="S3.SS3.p8.2.m2.1.1.1" xref="S3.SS3.p8.2.m2.1.1.1.cmml">∈</mo><msup id="S3.SS3.p8.2.m2.1.1.3" xref="S3.SS3.p8.2.m2.1.1.3.cmml"><mi id="S3.SS3.p8.2.m2.1.1.3.2" xref="S3.SS3.p8.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS3.p8.2.m2.1.1.3.3" xref="S3.SS3.p8.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.2.m2.1b"><apply id="S3.SS3.p8.2.m2.1.1.cmml" xref="S3.SS3.p8.2.m2.1.1"><in id="S3.SS3.p8.2.m2.1.1.1.cmml" xref="S3.SS3.p8.2.m2.1.1.1"></in><ci id="S3.SS3.p8.2.m2.1.1.2.cmml" xref="S3.SS3.p8.2.m2.1.1.2">𝑥</ci><apply id="S3.SS3.p8.2.m2.1.1.3.cmml" xref="S3.SS3.p8.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p8.2.m2.1.1.3.1.cmml" xref="S3.SS3.p8.2.m2.1.1.3">superscript</csymbol><ci id="S3.SS3.p8.2.m2.1.1.3.2.cmml" xref="S3.SS3.p8.2.m2.1.1.3.2">ℝ</ci><ci id="S3.SS3.p8.2.m2.1.1.3.3.cmml" xref="S3.SS3.p8.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.2.m2.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem17"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem17.1.1.1">Corollary 3.17</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem17.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem17.p1"> <p class="ltx_p" id="S3.Thmtheorem17.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem17.p1.6.6">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.1.1.m1.3"><semantics id="S3.Thmtheorem17.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem17.p1.1.1.m1.3.4" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem17.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem17.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem17.p1.1.1.m1.1.1" xref="S3.Thmtheorem17.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem17.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem17.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem17.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem17.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem17.p1.1.1.m1.3b"><apply id="S3.Thmtheorem17.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4"><in id="S3.Thmtheorem17.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem17.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem17.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem17.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem17.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem17.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem17.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem17.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem17.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary and let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.2.2.m2.1"><semantics id="S3.Thmtheorem17.p1.2.2.m2.1a"><mi id="S3.Thmtheorem17.p1.2.2.m2.1.1" xref="S3.Thmtheorem17.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem17.p1.2.2.m2.1b"><ci id="S3.Thmtheorem17.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem17.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem17.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem17.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> be a mass distribution on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.3.3.m3.1"><semantics id="S3.Thmtheorem17.p1.3.3.m3.1a"><msup id="S3.Thmtheorem17.p1.3.3.m3.1.1" xref="S3.Thmtheorem17.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem17.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem17.p1.3.3.m3.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem17.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem17.p1.3.3.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem17.p1.3.3.m3.1b"><apply id="S3.Thmtheorem17.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem17.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem17.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem17.p1.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem17.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem17.p1.3.3.m3.1.1.2">ℝ</ci><ci id="S3.Thmtheorem17.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem17.p1.3.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem17.p1.3.3.m3.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem17.p1.3.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. For any point <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.4.4.m4.1"><semantics id="S3.Thmtheorem17.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem17.p1.4.4.m4.1.1" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem17.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem17.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem17.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem17.p1.4.4.m4.1b"><apply id="S3.Thmtheorem17.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1"><in id="S3.Thmtheorem17.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem17.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.2">𝑥</ci><apply id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem17.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem17.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem17.p1.4.4.m4.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem17.p1.4.4.m4.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, the set <math alttext="V_{x}=\{v\in S^{d-1}\mid\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(% \mathcal{H}^{p}_{x,-v}),0\right)>0\}" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.5.5.m5.6"><semantics id="S3.Thmtheorem17.p1.5.5.m5.6a"><mrow id="S3.Thmtheorem17.p1.5.5.m5.6.6" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.cmml"><msub id="S3.Thmtheorem17.p1.5.5.m5.6.6.4" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.4.cmml"><mi id="S3.Thmtheorem17.p1.5.5.m5.6.6.4.2" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.4.2.cmml">V</mi><mi id="S3.Thmtheorem17.p1.5.5.m5.6.6.4.3" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.4.3.cmml">x</mi></msub><mo id="S3.Thmtheorem17.p1.5.5.m5.6.6.3" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.3.cmml">=</mo><mrow id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.3.cmml"><mo id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.3" stretchy="false" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.3.1.cmml">{</mo><mrow id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.cmml"><mi id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.2" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.1" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.cmml"><mi id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.2" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.cmml"><mi id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.2" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.1" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.3" xref="S3.Thmtheorem17.p1.5.5.m5.5.5.1.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><mo fence="true" id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.4" lspace="0em" rspace="0.167em" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.3.1.cmml">∣</mo><mrow id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.cmml"><mrow 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id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.1"></plus><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.3.3.3">1</cn></apply></apply><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.4.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.4">𝜇</ci><apply id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2"><times id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.2"></times><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.3">𝜇</ci><apply id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1">subscript</csymbol><apply id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1">superscript</csymbol><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.2">ℋ</ci><ci id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.3.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.1.1.1.1.2.1.1.1.2.3">𝑝</ci></apply><list id="S3.Thmtheorem17.p1.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2"><ci id="S3.Thmtheorem17.p1.5.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.1.1.1.1">𝑥</ci><apply id="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1"><minus id="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1"></minus><ci id="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1.2.cmml" xref="S3.Thmtheorem17.p1.5.5.m5.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></apply><cn id="S3.Thmtheorem17.p1.5.5.m5.4.4.cmml" type="integer" xref="S3.Thmtheorem17.p1.5.5.m5.4.4">0</cn></apply><cn id="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.3.cmml" type="integer" xref="S3.Thmtheorem17.p1.5.5.m5.6.6.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem17.p1.5.5.m5.6c">V_{x}=\{v\in S^{d-1}\mid\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(% \mathcal{H}^{p}_{x,-v}),0\right)>0\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem17.p1.5.5.m5.6d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = { italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT ) , 0 ) > 0 }</annotation></semantics></math> does not contain <math alttext="0" class="ltx_Math" display="inline" id="S3.Thmtheorem17.p1.6.6.m6.1"><semantics id="S3.Thmtheorem17.p1.6.6.m6.1a"><mn id="S3.Thmtheorem17.p1.6.6.m6.1.1" xref="S3.Thmtheorem17.p1.6.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem17.p1.6.6.m6.1b"><cn id="S3.Thmtheorem17.p1.6.6.m6.1.1.cmml" type="integer" xref="S3.Thmtheorem17.p1.6.6.m6.1.1">0</cn></annotation-xml></semantics></math> in its convex hull.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.4.p1"> <p class="ltx_p" id="S3.SS3.4.p1.7">If <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.4.p1.1.m1.1"><semantics id="S3.SS3.4.p1.1.m1.1a"><mn id="S3.SS3.4.p1.1.m1.1.1" xref="S3.SS3.4.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.1.m1.1b"><cn id="S3.SS3.4.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS3.4.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math> would lie in <math alttext="\text{conv}(V_{x})" class="ltx_Math" display="inline" id="S3.SS3.4.p1.2.m2.1"><semantics id="S3.SS3.4.p1.2.m2.1a"><mrow id="S3.SS3.4.p1.2.m2.1.1" xref="S3.SS3.4.p1.2.m2.1.1.cmml"><mtext id="S3.SS3.4.p1.2.m2.1.1.3" xref="S3.SS3.4.p1.2.m2.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.4.p1.2.m2.1.1.2" xref="S3.SS3.4.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.4.p1.2.m2.1.1.1.1" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS3.4.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.4.p1.2.m2.1.1.1.1.1" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.4.p1.2.m2.1.1.1.1.1.2" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.4.p1.2.m2.1.1.1.1.1.3" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S3.SS3.4.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.2.m2.1b"><apply id="S3.SS3.4.p1.2.m2.1.1.cmml" xref="S3.SS3.4.p1.2.m2.1.1"><times id="S3.SS3.4.p1.2.m2.1.1.2.cmml" xref="S3.SS3.4.p1.2.m2.1.1.2"></times><ci id="S3.SS3.4.p1.2.m2.1.1.3a.cmml" xref="S3.SS3.4.p1.2.m2.1.1.3"><mtext id="S3.SS3.4.p1.2.m2.1.1.3.cmml" xref="S3.SS3.4.p1.2.m2.1.1.3">conv</mtext></ci><apply id="S3.SS3.4.p1.2.m2.1.1.1.1.1.cmml" xref="S3.SS3.4.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.4.p1.2.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS3.4.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.4.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS3.4.p1.2.m2.1.1.1.1.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.2.m2.1c">\text{conv}(V_{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.2.m2.1d">conv ( italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>, this would mean that <math alttext="V_{x}" class="ltx_Math" display="inline" id="S3.SS3.4.p1.3.m3.1"><semantics id="S3.SS3.4.p1.3.m3.1a"><msub id="S3.SS3.4.p1.3.m3.1.1" xref="S3.SS3.4.p1.3.m3.1.1.cmml"><mi id="S3.SS3.4.p1.3.m3.1.1.2" xref="S3.SS3.4.p1.3.m3.1.1.2.cmml">V</mi><mi id="S3.SS3.4.p1.3.m3.1.1.3" xref="S3.SS3.4.p1.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.3.m3.1b"><apply id="S3.SS3.4.p1.3.m3.1.1.cmml" xref="S3.SS3.4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.3.m3.1.1.1.cmml" xref="S3.SS3.4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.4.p1.3.m3.1.1.2.cmml" xref="S3.SS3.4.p1.3.m3.1.1.2">𝑉</ci><ci id="S3.SS3.4.p1.3.m3.1.1.3.cmml" xref="S3.SS3.4.p1.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.3.m3.1c">V_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.3.m3.1d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is non-empty. In particular, using Carathéodory’s theorem, we would get a set of <math alttext="k\leq d+1" class="ltx_Math" display="inline" id="S3.SS3.4.p1.4.m4.1"><semantics id="S3.SS3.4.p1.4.m4.1a"><mrow id="S3.SS3.4.p1.4.m4.1.1" xref="S3.SS3.4.p1.4.m4.1.1.cmml"><mi id="S3.SS3.4.p1.4.m4.1.1.2" xref="S3.SS3.4.p1.4.m4.1.1.2.cmml">k</mi><mo id="S3.SS3.4.p1.4.m4.1.1.1" xref="S3.SS3.4.p1.4.m4.1.1.1.cmml">≤</mo><mrow id="S3.SS3.4.p1.4.m4.1.1.3" xref="S3.SS3.4.p1.4.m4.1.1.3.cmml"><mi id="S3.SS3.4.p1.4.m4.1.1.3.2" xref="S3.SS3.4.p1.4.m4.1.1.3.2.cmml">d</mi><mo id="S3.SS3.4.p1.4.m4.1.1.3.1" xref="S3.SS3.4.p1.4.m4.1.1.3.1.cmml">+</mo><mn id="S3.SS3.4.p1.4.m4.1.1.3.3" xref="S3.SS3.4.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.4.m4.1b"><apply id="S3.SS3.4.p1.4.m4.1.1.cmml" xref="S3.SS3.4.p1.4.m4.1.1"><leq id="S3.SS3.4.p1.4.m4.1.1.1.cmml" xref="S3.SS3.4.p1.4.m4.1.1.1"></leq><ci id="S3.SS3.4.p1.4.m4.1.1.2.cmml" xref="S3.SS3.4.p1.4.m4.1.1.2">𝑘</ci><apply id="S3.SS3.4.p1.4.m4.1.1.3.cmml" xref="S3.SS3.4.p1.4.m4.1.1.3"><plus id="S3.SS3.4.p1.4.m4.1.1.3.1.cmml" xref="S3.SS3.4.p1.4.m4.1.1.3.1"></plus><ci id="S3.SS3.4.p1.4.m4.1.1.3.2.cmml" xref="S3.SS3.4.p1.4.m4.1.1.3.2">𝑑</ci><cn id="S3.SS3.4.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S3.SS3.4.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.4.m4.1c">k\leq d+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.4.m4.1d">italic_k ≤ italic_d + 1</annotation></semantics></math> vectors <math alttext="v_{1},\dots,v_{k}\in V_{x}" class="ltx_Math" display="inline" id="S3.SS3.4.p1.5.m5.3"><semantics id="S3.SS3.4.p1.5.m5.3a"><mrow id="S3.SS3.4.p1.5.m5.3.3" xref="S3.SS3.4.p1.5.m5.3.3.cmml"><mrow id="S3.SS3.4.p1.5.m5.3.3.2.2" xref="S3.SS3.4.p1.5.m5.3.3.2.3.cmml"><msub id="S3.SS3.4.p1.5.m5.2.2.1.1.1" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1.cmml"><mi id="S3.SS3.4.p1.5.m5.2.2.1.1.1.2" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1.2.cmml">v</mi><mn id="S3.SS3.4.p1.5.m5.2.2.1.1.1.3" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.4.p1.5.m5.3.3.2.2.3" xref="S3.SS3.4.p1.5.m5.3.3.2.3.cmml">,</mo><mi id="S3.SS3.4.p1.5.m5.1.1" mathvariant="normal" xref="S3.SS3.4.p1.5.m5.1.1.cmml">…</mi><mo id="S3.SS3.4.p1.5.m5.3.3.2.2.4" xref="S3.SS3.4.p1.5.m5.3.3.2.3.cmml">,</mo><msub id="S3.SS3.4.p1.5.m5.3.3.2.2.2" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2.cmml"><mi id="S3.SS3.4.p1.5.m5.3.3.2.2.2.2" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2.2.cmml">v</mi><mi id="S3.SS3.4.p1.5.m5.3.3.2.2.2.3" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.SS3.4.p1.5.m5.3.3.3" xref="S3.SS3.4.p1.5.m5.3.3.3.cmml">∈</mo><msub id="S3.SS3.4.p1.5.m5.3.3.4" xref="S3.SS3.4.p1.5.m5.3.3.4.cmml"><mi id="S3.SS3.4.p1.5.m5.3.3.4.2" xref="S3.SS3.4.p1.5.m5.3.3.4.2.cmml">V</mi><mi id="S3.SS3.4.p1.5.m5.3.3.4.3" xref="S3.SS3.4.p1.5.m5.3.3.4.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.5.m5.3b"><apply id="S3.SS3.4.p1.5.m5.3.3.cmml" xref="S3.SS3.4.p1.5.m5.3.3"><in id="S3.SS3.4.p1.5.m5.3.3.3.cmml" xref="S3.SS3.4.p1.5.m5.3.3.3"></in><list id="S3.SS3.4.p1.5.m5.3.3.2.3.cmml" xref="S3.SS3.4.p1.5.m5.3.3.2.2"><apply id="S3.SS3.4.p1.5.m5.2.2.1.1.1.cmml" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.5.m5.2.2.1.1.1.1.cmml" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S3.SS3.4.p1.5.m5.2.2.1.1.1.2.cmml" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1.2">𝑣</ci><cn id="S3.SS3.4.p1.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS3.4.p1.5.m5.2.2.1.1.1.3">1</cn></apply><ci id="S3.SS3.4.p1.5.m5.1.1.cmml" xref="S3.SS3.4.p1.5.m5.1.1">…</ci><apply id="S3.SS3.4.p1.5.m5.3.3.2.2.2.cmml" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.5.m5.3.3.2.2.2.1.cmml" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S3.SS3.4.p1.5.m5.3.3.2.2.2.2.cmml" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2.2">𝑣</ci><ci id="S3.SS3.4.p1.5.m5.3.3.2.2.2.3.cmml" xref="S3.SS3.4.p1.5.m5.3.3.2.2.2.3">𝑘</ci></apply></list><apply id="S3.SS3.4.p1.5.m5.3.3.4.cmml" xref="S3.SS3.4.p1.5.m5.3.3.4"><csymbol cd="ambiguous" id="S3.SS3.4.p1.5.m5.3.3.4.1.cmml" xref="S3.SS3.4.p1.5.m5.3.3.4">subscript</csymbol><ci id="S3.SS3.4.p1.5.m5.3.3.4.2.cmml" xref="S3.SS3.4.p1.5.m5.3.3.4.2">𝑉</ci><ci id="S3.SS3.4.p1.5.m5.3.3.4.3.cmml" xref="S3.SS3.4.p1.5.m5.3.3.4.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.5.m5.3c">v_{1},\dots,v_{k}\in V_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.5.m5.3d">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="0\in\text{conv}(v_{1},\dots,v_{k})" class="ltx_Math" display="inline" id="S3.SS3.4.p1.6.m6.3"><semantics id="S3.SS3.4.p1.6.m6.3a"><mrow id="S3.SS3.4.p1.6.m6.3.3" xref="S3.SS3.4.p1.6.m6.3.3.cmml"><mn id="S3.SS3.4.p1.6.m6.3.3.4" xref="S3.SS3.4.p1.6.m6.3.3.4.cmml">0</mn><mo id="S3.SS3.4.p1.6.m6.3.3.3" xref="S3.SS3.4.p1.6.m6.3.3.3.cmml">∈</mo><mrow id="S3.SS3.4.p1.6.m6.3.3.2" xref="S3.SS3.4.p1.6.m6.3.3.2.cmml"><mtext id="S3.SS3.4.p1.6.m6.3.3.2.4" xref="S3.SS3.4.p1.6.m6.3.3.2.4a.cmml">conv</mtext><mo id="S3.SS3.4.p1.6.m6.3.3.2.3" xref="S3.SS3.4.p1.6.m6.3.3.2.3.cmml">⁢</mo><mrow id="S3.SS3.4.p1.6.m6.3.3.2.2.2" xref="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml"><mo id="S3.SS3.4.p1.6.m6.3.3.2.2.2.3" stretchy="false" xref="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml">(</mo><msub id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.2" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.2.cmml">v</mi><mn id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.3" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.4.p1.6.m6.3.3.2.2.2.4" xref="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml">,</mo><mi id="S3.SS3.4.p1.6.m6.1.1" mathvariant="normal" xref="S3.SS3.4.p1.6.m6.1.1.cmml">…</mi><mo id="S3.SS3.4.p1.6.m6.3.3.2.2.2.5" xref="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml">,</mo><msub id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.cmml"><mi id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.2" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.2.cmml">v</mi><mi id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.3" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS3.4.p1.6.m6.3.3.2.2.2.6" stretchy="false" xref="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.6.m6.3b"><apply id="S3.SS3.4.p1.6.m6.3.3.cmml" xref="S3.SS3.4.p1.6.m6.3.3"><in id="S3.SS3.4.p1.6.m6.3.3.3.cmml" xref="S3.SS3.4.p1.6.m6.3.3.3"></in><cn id="S3.SS3.4.p1.6.m6.3.3.4.cmml" type="integer" xref="S3.SS3.4.p1.6.m6.3.3.4">0</cn><apply id="S3.SS3.4.p1.6.m6.3.3.2.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2"><times id="S3.SS3.4.p1.6.m6.3.3.2.3.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.3"></times><ci id="S3.SS3.4.p1.6.m6.3.3.2.4a.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.4"><mtext id="S3.SS3.4.p1.6.m6.3.3.2.4.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.4">conv</mtext></ci><vector id="S3.SS3.4.p1.6.m6.3.3.2.2.3.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2"><apply id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.cmml" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.2">𝑣</ci><cn id="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.SS3.4.p1.6.m6.2.2.1.1.1.1.3">1</cn></apply><ci id="S3.SS3.4.p1.6.m6.1.1.cmml" xref="S3.SS3.4.p1.6.m6.1.1">…</ci><apply id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.1.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2">subscript</csymbol><ci id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.2.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.2">𝑣</ci><ci id="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.3.cmml" xref="S3.SS3.4.p1.6.m6.3.3.2.2.2.2.3">𝑘</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.6.m6.3c">0\in\text{conv}(v_{1},\dots,v_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.6.m6.3d">0 ∈ conv ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. Since these vectors are in <math alttext="V_{x}" class="ltx_Math" display="inline" id="S3.SS3.4.p1.7.m7.1"><semantics id="S3.SS3.4.p1.7.m7.1a"><msub id="S3.SS3.4.p1.7.m7.1.1" xref="S3.SS3.4.p1.7.m7.1.1.cmml"><mi id="S3.SS3.4.p1.7.m7.1.1.2" xref="S3.SS3.4.p1.7.m7.1.1.2.cmml">V</mi><mi id="S3.SS3.4.p1.7.m7.1.1.3" xref="S3.SS3.4.p1.7.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.7.m7.1b"><apply id="S3.SS3.4.p1.7.m7.1.1.cmml" xref="S3.SS3.4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.7.m7.1.1.1.cmml" xref="S3.SS3.4.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.4.p1.7.m7.1.1.2.cmml" xref="S3.SS3.4.p1.7.m7.1.1.2">𝑉</ci><ci id="S3.SS3.4.p1.7.m7.1.1.3.cmml" xref="S3.SS3.4.p1.7.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.7.m7.1c">V_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.7.m7.1d">italic_V start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, we get on the one hand that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sum_{i=1}^{k}\mu(\mathcal{H}^{p}_{x,-v_{i}})<\frac{k}{d+1}\mu(\mathbb{R}^{d})% \leq\mu(\mathbb{R}^{d})." class="ltx_Math" display="block" id="S3.Ex7.m1.3"><semantics id="S3.Ex7.m1.3a"><mrow id="S3.Ex7.m1.3.3.1" xref="S3.Ex7.m1.3.3.1.1.cmml"><mrow id="S3.Ex7.m1.3.3.1.1" xref="S3.Ex7.m1.3.3.1.1.cmml"><mrow id="S3.Ex7.m1.3.3.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.cmml"><munderover id="S3.Ex7.m1.3.3.1.1.1.2" xref="S3.Ex7.m1.3.3.1.1.1.2.cmml"><mo id="S3.Ex7.m1.3.3.1.1.1.2.2.2" movablelimits="false" xref="S3.Ex7.m1.3.3.1.1.1.2.2.2.cmml">∑</mo><mrow id="S3.Ex7.m1.3.3.1.1.1.2.2.3" xref="S3.Ex7.m1.3.3.1.1.1.2.2.3.cmml"><mi id="S3.Ex7.m1.3.3.1.1.1.2.2.3.2" xref="S3.Ex7.m1.3.3.1.1.1.2.2.3.2.cmml">i</mi><mo id="S3.Ex7.m1.3.3.1.1.1.2.2.3.1" xref="S3.Ex7.m1.3.3.1.1.1.2.2.3.1.cmml">=</mo><mn id="S3.Ex7.m1.3.3.1.1.1.2.2.3.3" xref="S3.Ex7.m1.3.3.1.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="S3.Ex7.m1.3.3.1.1.1.2.3" xref="S3.Ex7.m1.3.3.1.1.1.2.3.cmml">k</mi></munderover><mrow id="S3.Ex7.m1.3.3.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.cmml"><mi id="S3.Ex7.m1.3.3.1.1.1.1.3" xref="S3.Ex7.m1.3.3.1.1.1.1.3.cmml">μ</mi><mo id="S3.Ex7.m1.3.3.1.1.1.1.2" xref="S3.Ex7.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex7.m1.3.3.1.1.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex7.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.Ex7.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.3.3.1.1.1.1.1.1.1.2.2" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.Ex7.m1.2.2.2.2" xref="S3.Ex7.m1.2.2.2.3.cmml"><mi id="S3.Ex7.m1.1.1.1.1" 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href="https://arxiv.org/html/2503.16089v1#S3.Ex7.m1.3.3.1.1.2.cmml" id="S3.Ex7.m1.3.3.1.1d.cmml" xref="S3.Ex7.m1.3.3.1"></share><apply id="S3.Ex7.m1.3.3.1.1.3.cmml" xref="S3.Ex7.m1.3.3.1.1.3"><times id="S3.Ex7.m1.3.3.1.1.3.2.cmml" xref="S3.Ex7.m1.3.3.1.1.3.2"></times><ci id="S3.Ex7.m1.3.3.1.1.3.3.cmml" xref="S3.Ex7.m1.3.3.1.1.3.3">𝜇</ci><apply id="S3.Ex7.m1.3.3.1.1.3.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.3.1.1"><csymbol cd="ambiguous" id="S3.Ex7.m1.3.3.1.1.3.1.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.3.1.1">superscript</csymbol><ci id="S3.Ex7.m1.3.3.1.1.3.1.1.1.2.cmml" xref="S3.Ex7.m1.3.3.1.1.3.1.1.1.2">ℝ</ci><ci id="S3.Ex7.m1.3.3.1.1.3.1.1.1.3.cmml" xref="S3.Ex7.m1.3.3.1.1.3.1.1.1.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.3c">\sum_{i=1}^{k}\mu(\mathcal{H}^{p}_{x,-v_{i}})<\frac{k}{d+1}\mu(\mathbb{R}^{d})% \leq\mu(\mathbb{R}^{d}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.3d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) < divide start_ARG italic_k end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) ≤ italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.4.p1.9">On the other hand, <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem16" title="Lemma 3.16. ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text 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id="S3.SS3.4.p1.8.m1.1.1.3.1.cmml" xref="S3.SS3.4.p1.8.m1.1.1.3">superscript</csymbol><ci id="S3.SS3.4.p1.8.m1.1.1.3.2.cmml" xref="S3.SS3.4.p1.8.m1.1.1.3.2">ℝ</ci><ci id="S3.SS3.4.p1.8.m1.1.1.3.3.cmml" xref="S3.SS3.4.p1.8.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.8.m1.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.8.m1.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is contained in at least one of the halfspaces <math alttext="\mathcal{H}^{p}_{x,-v_{1}},\dots,\mathcal{H}^{p}_{x,-v_{k}}" class="ltx_Math" display="inline" id="S3.SS3.4.p1.9.m2.7"><semantics id="S3.SS3.4.p1.9.m2.7a"><mrow id="S3.SS3.4.p1.9.m2.7.7.2" xref="S3.SS3.4.p1.9.m2.7.7.3.cmml"><msubsup id="S3.SS3.4.p1.9.m2.6.6.1.1" xref="S3.SS3.4.p1.9.m2.6.6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.4.p1.9.m2.6.6.1.1.2.2" xref="S3.SS3.4.p1.9.m2.6.6.1.1.2.2.cmml">ℋ</mi><mrow id="S3.SS3.4.p1.9.m2.2.2.2.2" xref="S3.SS3.4.p1.9.m2.2.2.2.3.cmml"><mi id="S3.SS3.4.p1.9.m2.1.1.1.1" xref="S3.SS3.4.p1.9.m2.1.1.1.1.cmml">x</mi><mo id="S3.SS3.4.p1.9.m2.2.2.2.2.2" xref="S3.SS3.4.p1.9.m2.2.2.2.3.cmml">,</mo><mrow id="S3.SS3.4.p1.9.m2.2.2.2.2.1" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.cmml"><mo id="S3.SS3.4.p1.9.m2.2.2.2.2.1a" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.cmml">−</mo><msub id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.cmml"><mi id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.2" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.2.cmml">v</mi><mn id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.3" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.3.cmml">1</mn></msub></mrow></mrow><mi id="S3.SS3.4.p1.9.m2.6.6.1.1.2.3" xref="S3.SS3.4.p1.9.m2.6.6.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.SS3.4.p1.9.m2.7.7.2.3" xref="S3.SS3.4.p1.9.m2.7.7.3.cmml">,</mo><mi id="S3.SS3.4.p1.9.m2.5.5" mathvariant="normal" xref="S3.SS3.4.p1.9.m2.5.5.cmml">…</mi><mo id="S3.SS3.4.p1.9.m2.7.7.2.4" xref="S3.SS3.4.p1.9.m2.7.7.3.cmml">,</mo><msubsup id="S3.SS3.4.p1.9.m2.7.7.2.2" xref="S3.SS3.4.p1.9.m2.7.7.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.4.p1.9.m2.7.7.2.2.2.2" xref="S3.SS3.4.p1.9.m2.7.7.2.2.2.2.cmml">ℋ</mi><mrow id="S3.SS3.4.p1.9.m2.4.4.2.2" xref="S3.SS3.4.p1.9.m2.4.4.2.3.cmml"><mi id="S3.SS3.4.p1.9.m2.3.3.1.1" xref="S3.SS3.4.p1.9.m2.3.3.1.1.cmml">x</mi><mo id="S3.SS3.4.p1.9.m2.4.4.2.2.2" xref="S3.SS3.4.p1.9.m2.4.4.2.3.cmml">,</mo><mrow id="S3.SS3.4.p1.9.m2.4.4.2.2.1" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.cmml"><mo id="S3.SS3.4.p1.9.m2.4.4.2.2.1a" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.cmml">−</mo><msub id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.cmml"><mi id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.2" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.2.cmml">v</mi><mi id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.3" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.3.cmml">k</mi></msub></mrow></mrow><mi id="S3.SS3.4.p1.9.m2.7.7.2.2.2.3" xref="S3.SS3.4.p1.9.m2.7.7.2.2.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.4.p1.9.m2.7b"><list id="S3.SS3.4.p1.9.m2.7.7.3.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2"><apply id="S3.SS3.4.p1.9.m2.6.6.1.1.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.6.6.1.1.1.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1">subscript</csymbol><apply id="S3.SS3.4.p1.9.m2.6.6.1.1.2.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.6.6.1.1.2.1.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1">superscript</csymbol><ci id="S3.SS3.4.p1.9.m2.6.6.1.1.2.2.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1.2.2">ℋ</ci><ci id="S3.SS3.4.p1.9.m2.6.6.1.1.2.3.cmml" xref="S3.SS3.4.p1.9.m2.6.6.1.1.2.3">𝑝</ci></apply><list id="S3.SS3.4.p1.9.m2.2.2.2.3.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2"><ci id="S3.SS3.4.p1.9.m2.1.1.1.1.cmml" xref="S3.SS3.4.p1.9.m2.1.1.1.1">𝑥</ci><apply id="S3.SS3.4.p1.9.m2.2.2.2.2.1.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1"><minus id="S3.SS3.4.p1.9.m2.2.2.2.2.1.1.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1"></minus><apply id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.1.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2">subscript</csymbol><ci id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.2.cmml" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.2">𝑣</ci><cn id="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.3.cmml" type="integer" xref="S3.SS3.4.p1.9.m2.2.2.2.2.1.2.3">1</cn></apply></apply></list></apply><ci id="S3.SS3.4.p1.9.m2.5.5.cmml" xref="S3.SS3.4.p1.9.m2.5.5">…</ci><apply id="S3.SS3.4.p1.9.m2.7.7.2.2.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.7.7.2.2.1.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2">subscript</csymbol><apply id="S3.SS3.4.p1.9.m2.7.7.2.2.2.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.7.7.2.2.2.1.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2">superscript</csymbol><ci id="S3.SS3.4.p1.9.m2.7.7.2.2.2.2.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2.2.2">ℋ</ci><ci id="S3.SS3.4.p1.9.m2.7.7.2.2.2.3.cmml" xref="S3.SS3.4.p1.9.m2.7.7.2.2.2.3">𝑝</ci></apply><list id="S3.SS3.4.p1.9.m2.4.4.2.3.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2"><ci id="S3.SS3.4.p1.9.m2.3.3.1.1.cmml" xref="S3.SS3.4.p1.9.m2.3.3.1.1">𝑥</ci><apply id="S3.SS3.4.p1.9.m2.4.4.2.2.1.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1"><minus id="S3.SS3.4.p1.9.m2.4.4.2.2.1.1.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1"></minus><apply id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.1.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2">subscript</csymbol><ci id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.2.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.2">𝑣</ci><ci id="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.3.cmml" xref="S3.SS3.4.p1.9.m2.4.4.2.2.1.2.3">𝑘</ci></apply></apply></list></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.4.p1.9.m2.7c">\mathcal{H}^{p}_{x,-v_{1}},\dots,\mathcal{H}^{p}_{x,-v_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.4.p1.9.m2.7d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, we can derive</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sum_{i=1}^{k}\mu(\mathcal{H}^{p}_{x,-v_{i}})\geq\mu\left(\bigcup_{i=1}^{k}% \mathcal{H}^{p}_{x,-v_{i}}\right)=\mu(\mathbb{R}^{d})," class="ltx_Math" display="block" id="S3.Ex8.m1.5"><semantics id="S3.Ex8.m1.5a"><mrow id="S3.Ex8.m1.5.5.1" xref="S3.Ex8.m1.5.5.1.1.cmml"><mrow id="S3.Ex8.m1.5.5.1.1" xref="S3.Ex8.m1.5.5.1.1.cmml"><mrow id="S3.Ex8.m1.5.5.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.cmml"><munderover id="S3.Ex8.m1.5.5.1.1.1.2" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml"><mo id="S3.Ex8.m1.5.5.1.1.1.2.2.2" movablelimits="false" xref="S3.Ex8.m1.5.5.1.1.1.2.2.2.cmml">∑</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.2.2.3" xref="S3.Ex8.m1.5.5.1.1.1.2.2.3.cmml"><mi id="S3.Ex8.m1.5.5.1.1.1.2.2.3.2" xref="S3.Ex8.m1.5.5.1.1.1.2.2.3.2.cmml">i</mi><mo id="S3.Ex8.m1.5.5.1.1.1.2.2.3.1" xref="S3.Ex8.m1.5.5.1.1.1.2.2.3.1.cmml">=</mo><mn id="S3.Ex8.m1.5.5.1.1.1.2.2.3.3" xref="S3.Ex8.m1.5.5.1.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="S3.Ex8.m1.5.5.1.1.1.2.3" xref="S3.Ex8.m1.5.5.1.1.1.2.3.cmml">k</mi></munderover><mrow id="S3.Ex8.m1.5.5.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.1.cmml"><mi id="S3.Ex8.m1.5.5.1.1.1.1.3" xref="S3.Ex8.m1.5.5.1.1.1.1.3.cmml">μ</mi><mo id="S3.Ex8.m1.5.5.1.1.1.1.2" xref="S3.Ex8.m1.5.5.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.Ex8.m1.5.5.1.1.1.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.2.2" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.Ex8.m1.2.2.2.2" xref="S3.Ex8.m1.2.2.2.3.cmml"><mi id="S3.Ex8.m1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.cmml">x</mi><mo id="S3.Ex8.m1.2.2.2.2.2" xref="S3.Ex8.m1.2.2.2.3.cmml">,</mo><mrow id="S3.Ex8.m1.2.2.2.2.1" xref="S3.Ex8.m1.2.2.2.2.1.cmml"><mo id="S3.Ex8.m1.2.2.2.2.1a" xref="S3.Ex8.m1.2.2.2.2.1.cmml">−</mo><msub id="S3.Ex8.m1.2.2.2.2.1.2" xref="S3.Ex8.m1.2.2.2.2.1.2.cmml"><mi id="S3.Ex8.m1.2.2.2.2.1.2.2" xref="S3.Ex8.m1.2.2.2.2.1.2.2.cmml">v</mi><mi id="S3.Ex8.m1.2.2.2.2.1.2.3" xref="S3.Ex8.m1.2.2.2.2.1.2.3.cmml">i</mi></msub></mrow></mrow><mi id="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.2.3" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.5.5.1.1.5" xref="S3.Ex8.m1.5.5.1.1.5.cmml">≥</mo><mrow id="S3.Ex8.m1.5.5.1.1.2" xref="S3.Ex8.m1.5.5.1.1.2.cmml"><mi id="S3.Ex8.m1.5.5.1.1.2.3" xref="S3.Ex8.m1.5.5.1.1.2.3.cmml">μ</mi><mo id="S3.Ex8.m1.5.5.1.1.2.2" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.5.5.1.1.2.1.1" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.cmml"><mo 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id="S3.Ex8.m1.5c">\sum_{i=1}^{k}\mu(\mathcal{H}^{p}_{x,-v_{i}})\geq\mu\left(\bigcup_{i=1}^{k}% \mathcal{H}^{p}_{x,-v_{i}}\right)=\mu(\mathbb{R}^{d}),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.5d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≥ italic_μ ( ⋃ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.4.p1.10">a contradiction. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p9"> <p class="ltx_p" id="S3.SS3.p9.1">We are now ready to put all of this together to get a proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a>. The main technical thing that we still have to do, is to restrict <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.p9.1.m1.1"><semantics id="S3.SS3.p9.1.m1.1a"><mi id="S3.SS3.p9.1.m1.1.1" xref="S3.SS3.p9.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p9.1.m1.1b"><ci id="S3.SS3.p9.1.m1.1.1.cmml" xref="S3.SS3.p9.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p9.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p9.1.m1.1d">italic_F</annotation></semantics></math> to a compact convex set (so that we can apply Brouwer’s fixpoint theorem).</p> </div> <div class="ltx_proof" id="S3.SS3.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.13</span></a>.</h6> <div class="ltx_para" id="S3.SS3.5.p1"> <p class="ltx_p" id="S3.SS3.5.p1.19">Without loss of generality, assume that the bounded support of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.5.p1.1.m1.1"><semantics id="S3.SS3.5.p1.1.m1.1a"><mi id="S3.SS3.5.p1.1.m1.1.1" xref="S3.SS3.5.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.1.m1.1b"><ci id="S3.SS3.5.p1.1.m1.1.1.cmml" xref="S3.SS3.5.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.1.m1.1d">italic_μ</annotation></semantics></math> is contained in the box <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S3.SS3.5.p1.2.m2.2"><semantics id="S3.SS3.5.p1.2.m2.2a"><msup id="S3.SS3.5.p1.2.m2.2.3" xref="S3.SS3.5.p1.2.m2.2.3.cmml"><mrow id="S3.SS3.5.p1.2.m2.2.3.2.2" xref="S3.SS3.5.p1.2.m2.2.3.2.1.cmml"><mo id="S3.SS3.5.p1.2.m2.2.3.2.2.1" stretchy="false" xref="S3.SS3.5.p1.2.m2.2.3.2.1.cmml">[</mo><mn id="S3.SS3.5.p1.2.m2.1.1" xref="S3.SS3.5.p1.2.m2.1.1.cmml">0</mn><mo id="S3.SS3.5.p1.2.m2.2.3.2.2.2" xref="S3.SS3.5.p1.2.m2.2.3.2.1.cmml">,</mo><mn id="S3.SS3.5.p1.2.m2.2.2" xref="S3.SS3.5.p1.2.m2.2.2.cmml">1</mn><mo id="S3.SS3.5.p1.2.m2.2.3.2.2.3" stretchy="false" xref="S3.SS3.5.p1.2.m2.2.3.2.1.cmml">]</mo></mrow><mi id="S3.SS3.5.p1.2.m2.2.3.3" xref="S3.SS3.5.p1.2.m2.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.2.m2.2b"><apply id="S3.SS3.5.p1.2.m2.2.3.cmml" xref="S3.SS3.5.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS3.5.p1.2.m2.2.3.1.cmml" xref="S3.SS3.5.p1.2.m2.2.3">superscript</csymbol><interval closure="closed" id="S3.SS3.5.p1.2.m2.2.3.2.1.cmml" xref="S3.SS3.5.p1.2.m2.2.3.2.2"><cn id="S3.SS3.5.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS3.5.p1.2.m2.1.1">0</cn><cn id="S3.SS3.5.p1.2.m2.2.2.cmml" type="integer" xref="S3.SS3.5.p1.2.m2.2.2">1</cn></interval><ci id="S3.SS3.5.p1.2.m2.2.3.3.cmml" xref="S3.SS3.5.p1.2.m2.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.2.m2.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.2.m2.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. To apply Brouwer’s fixpoint theorem, we need a function going from a compact convex set <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.3.m3.1"><semantics id="S3.SS3.5.p1.3.m3.1a"><mi id="S3.SS3.5.p1.3.m3.1.1" xref="S3.SS3.5.p1.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.3.m3.1b"><ci id="S3.SS3.5.p1.3.m3.1.1.cmml" xref="S3.SS3.5.p1.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.3.m3.1d">italic_C</annotation></semantics></math> to itself, rather than from <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.5.p1.4.m4.1"><semantics id="S3.SS3.5.p1.4.m4.1a"><msup id="S3.SS3.5.p1.4.m4.1.1" xref="S3.SS3.5.p1.4.m4.1.1.cmml"><mi id="S3.SS3.5.p1.4.m4.1.1.2" xref="S3.SS3.5.p1.4.m4.1.1.2.cmml">ℝ</mi><mi id="S3.SS3.5.p1.4.m4.1.1.3" xref="S3.SS3.5.p1.4.m4.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.4.m4.1b"><apply id="S3.SS3.5.p1.4.m4.1.1.cmml" xref="S3.SS3.5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.5.p1.4.m4.1.1.1.cmml" xref="S3.SS3.5.p1.4.m4.1.1">superscript</csymbol><ci id="S3.SS3.5.p1.4.m4.1.1.2.cmml" xref="S3.SS3.5.p1.4.m4.1.1.2">ℝ</ci><ci id="S3.SS3.5.p1.4.m4.1.1.3.cmml" xref="S3.SS3.5.p1.4.m4.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.4.m4.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS3.5.p1.5.m5.1"><semantics id="S3.SS3.5.p1.5.m5.1a"><msup id="S3.SS3.5.p1.5.m5.1.1" xref="S3.SS3.5.p1.5.m5.1.1.cmml"><mi id="S3.SS3.5.p1.5.m5.1.1.2" xref="S3.SS3.5.p1.5.m5.1.1.2.cmml">ℝ</mi><mi id="S3.SS3.5.p1.5.m5.1.1.3" xref="S3.SS3.5.p1.5.m5.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.5.m5.1b"><apply id="S3.SS3.5.p1.5.m5.1.1.cmml" xref="S3.SS3.5.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.5.p1.5.m5.1.1.1.cmml" xref="S3.SS3.5.p1.5.m5.1.1">superscript</csymbol><ci id="S3.SS3.5.p1.5.m5.1.1.2.cmml" xref="S3.SS3.5.p1.5.m5.1.1.2">ℝ</ci><ci id="S3.SS3.5.p1.5.m5.1.1.3.cmml" xref="S3.SS3.5.p1.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.5.m5.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> like our function <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.5.p1.6.m6.1"><semantics id="S3.SS3.5.p1.6.m6.1a"><mi id="S3.SS3.5.p1.6.m6.1.1" xref="S3.SS3.5.p1.6.m6.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.6.m6.1b"><ci id="S3.SS3.5.p1.6.m6.1.1.cmml" xref="S3.SS3.5.p1.6.m6.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.6.m6.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.6.m6.1d">italic_F</annotation></semantics></math>. We create such a function <math alttext="F_{C}" class="ltx_Math" display="inline" id="S3.SS3.5.p1.7.m7.1"><semantics id="S3.SS3.5.p1.7.m7.1a"><msub id="S3.SS3.5.p1.7.m7.1.1" xref="S3.SS3.5.p1.7.m7.1.1.cmml"><mi id="S3.SS3.5.p1.7.m7.1.1.2" xref="S3.SS3.5.p1.7.m7.1.1.2.cmml">F</mi><mi id="S3.SS3.5.p1.7.m7.1.1.3" xref="S3.SS3.5.p1.7.m7.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.7.m7.1b"><apply id="S3.SS3.5.p1.7.m7.1.1.cmml" xref="S3.SS3.5.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.5.p1.7.m7.1.1.1.cmml" xref="S3.SS3.5.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.5.p1.7.m7.1.1.2.cmml" xref="S3.SS3.5.p1.7.m7.1.1.2">𝐹</ci><ci id="S3.SS3.5.p1.7.m7.1.1.3.cmml" xref="S3.SS3.5.p1.7.m7.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.7.m7.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.7.m7.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> by defining a large compact convex set <math alttext="C\supseteq[0,1]^{d}" class="ltx_Math" display="inline" id="S3.SS3.5.p1.8.m8.2"><semantics id="S3.SS3.5.p1.8.m8.2a"><mrow id="S3.SS3.5.p1.8.m8.2.3" xref="S3.SS3.5.p1.8.m8.2.3.cmml"><mi id="S3.SS3.5.p1.8.m8.2.3.2" xref="S3.SS3.5.p1.8.m8.2.3.2.cmml">C</mi><mo id="S3.SS3.5.p1.8.m8.2.3.1" xref="S3.SS3.5.p1.8.m8.2.3.cmml">⊇</mo><msup id="S3.SS3.5.p1.8.m8.2.3.3" xref="S3.SS3.5.p1.8.m8.2.3.3.cmml"><mrow id="S3.SS3.5.p1.8.m8.2.3.3.2.2" xref="S3.SS3.5.p1.8.m8.2.3.3.2.1.cmml"><mo id="S3.SS3.5.p1.8.m8.2.3.3.2.2.1" stretchy="false" xref="S3.SS3.5.p1.8.m8.2.3.3.2.1.cmml">[</mo><mn id="S3.SS3.5.p1.8.m8.1.1" xref="S3.SS3.5.p1.8.m8.1.1.cmml">0</mn><mo id="S3.SS3.5.p1.8.m8.2.3.3.2.2.2" xref="S3.SS3.5.p1.8.m8.2.3.3.2.1.cmml">,</mo><mn id="S3.SS3.5.p1.8.m8.2.2" xref="S3.SS3.5.p1.8.m8.2.2.cmml">1</mn><mo id="S3.SS3.5.p1.8.m8.2.3.3.2.2.3" stretchy="false" xref="S3.SS3.5.p1.8.m8.2.3.3.2.1.cmml">]</mo></mrow><mi id="S3.SS3.5.p1.8.m8.2.3.3.3" xref="S3.SS3.5.p1.8.m8.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.8.m8.2b"><apply id="S3.SS3.5.p1.8.m8.2.3.cmml" xref="S3.SS3.5.p1.8.m8.2.3"><subset id="S3.SS3.5.p1.8.m8.2.3a.cmml" xref="S3.SS3.5.p1.8.m8.2.3"></subset><apply id="S3.SS3.5.p1.8.m8.2.3.3.cmml" xref="S3.SS3.5.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="S3.SS3.5.p1.8.m8.2.3.3.1.cmml" xref="S3.SS3.5.p1.8.m8.2.3.3">superscript</csymbol><interval closure="closed" id="S3.SS3.5.p1.8.m8.2.3.3.2.1.cmml" xref="S3.SS3.5.p1.8.m8.2.3.3.2.2"><cn id="S3.SS3.5.p1.8.m8.1.1.cmml" type="integer" xref="S3.SS3.5.p1.8.m8.1.1">0</cn><cn id="S3.SS3.5.p1.8.m8.2.2.cmml" type="integer" xref="S3.SS3.5.p1.8.m8.2.2">1</cn></interval><ci id="S3.SS3.5.p1.8.m8.2.3.3.3.cmml" xref="S3.SS3.5.p1.8.m8.2.3.3.3">𝑑</ci></apply><ci id="S3.SS3.5.p1.8.m8.2.3.2.cmml" xref="S3.SS3.5.p1.8.m8.2.3.2">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.8.m8.2c">C\supseteq[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.8.m8.2d">italic_C ⊇ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, and then restricting the function <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.5.p1.9.m9.1"><semantics id="S3.SS3.5.p1.9.m9.1a"><mi id="S3.SS3.5.p1.9.m9.1.1" xref="S3.SS3.5.p1.9.m9.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.9.m9.1b"><ci id="S3.SS3.5.p1.9.m9.1.1.cmml" xref="S3.SS3.5.p1.9.m9.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.9.m9.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.9.m9.1d">italic_F</annotation></semantics></math> to <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.10.m10.1"><semantics id="S3.SS3.5.p1.10.m10.1a"><mi id="S3.SS3.5.p1.10.m10.1.1" xref="S3.SS3.5.p1.10.m10.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.10.m10.1b"><ci id="S3.SS3.5.p1.10.m10.1.1.cmml" xref="S3.SS3.5.p1.10.m10.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.10.m10.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.10.m10.1d">italic_C</annotation></semantics></math> using projection. More formally, we use the restricted function <math alttext="F_{C}:C\rightarrow C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.11.m11.1"><semantics id="S3.SS3.5.p1.11.m11.1a"><mrow id="S3.SS3.5.p1.11.m11.1.1" xref="S3.SS3.5.p1.11.m11.1.1.cmml"><msub id="S3.SS3.5.p1.11.m11.1.1.2" xref="S3.SS3.5.p1.11.m11.1.1.2.cmml"><mi id="S3.SS3.5.p1.11.m11.1.1.2.2" xref="S3.SS3.5.p1.11.m11.1.1.2.2.cmml">F</mi><mi id="S3.SS3.5.p1.11.m11.1.1.2.3" xref="S3.SS3.5.p1.11.m11.1.1.2.3.cmml">C</mi></msub><mo id="S3.SS3.5.p1.11.m11.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.5.p1.11.m11.1.1.1.cmml">:</mo><mrow id="S3.SS3.5.p1.11.m11.1.1.3" xref="S3.SS3.5.p1.11.m11.1.1.3.cmml"><mi id="S3.SS3.5.p1.11.m11.1.1.3.2" xref="S3.SS3.5.p1.11.m11.1.1.3.2.cmml">C</mi><mo id="S3.SS3.5.p1.11.m11.1.1.3.1" stretchy="false" xref="S3.SS3.5.p1.11.m11.1.1.3.1.cmml">→</mo><mi id="S3.SS3.5.p1.11.m11.1.1.3.3" xref="S3.SS3.5.p1.11.m11.1.1.3.3.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.11.m11.1b"><apply id="S3.SS3.5.p1.11.m11.1.1.cmml" xref="S3.SS3.5.p1.11.m11.1.1"><ci id="S3.SS3.5.p1.11.m11.1.1.1.cmml" xref="S3.SS3.5.p1.11.m11.1.1.1">:</ci><apply id="S3.SS3.5.p1.11.m11.1.1.2.cmml" xref="S3.SS3.5.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.5.p1.11.m11.1.1.2.1.cmml" xref="S3.SS3.5.p1.11.m11.1.1.2">subscript</csymbol><ci id="S3.SS3.5.p1.11.m11.1.1.2.2.cmml" xref="S3.SS3.5.p1.11.m11.1.1.2.2">𝐹</ci><ci id="S3.SS3.5.p1.11.m11.1.1.2.3.cmml" xref="S3.SS3.5.p1.11.m11.1.1.2.3">𝐶</ci></apply><apply id="S3.SS3.5.p1.11.m11.1.1.3.cmml" xref="S3.SS3.5.p1.11.m11.1.1.3"><ci id="S3.SS3.5.p1.11.m11.1.1.3.1.cmml" xref="S3.SS3.5.p1.11.m11.1.1.3.1">→</ci><ci id="S3.SS3.5.p1.11.m11.1.1.3.2.cmml" xref="S3.SS3.5.p1.11.m11.1.1.3.2">𝐶</ci><ci id="S3.SS3.5.p1.11.m11.1.1.3.3.cmml" xref="S3.SS3.5.p1.11.m11.1.1.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.11.m11.1c">F_{C}:C\rightarrow C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.11.m11.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT : italic_C → italic_C</annotation></semantics></math> with <math alttext="F_{C}(x)" class="ltx_Math" display="inline" id="S3.SS3.5.p1.12.m12.1"><semantics id="S3.SS3.5.p1.12.m12.1a"><mrow id="S3.SS3.5.p1.12.m12.1.2" xref="S3.SS3.5.p1.12.m12.1.2.cmml"><msub id="S3.SS3.5.p1.12.m12.1.2.2" xref="S3.SS3.5.p1.12.m12.1.2.2.cmml"><mi id="S3.SS3.5.p1.12.m12.1.2.2.2" xref="S3.SS3.5.p1.12.m12.1.2.2.2.cmml">F</mi><mi id="S3.SS3.5.p1.12.m12.1.2.2.3" xref="S3.SS3.5.p1.12.m12.1.2.2.3.cmml">C</mi></msub><mo id="S3.SS3.5.p1.12.m12.1.2.1" xref="S3.SS3.5.p1.12.m12.1.2.1.cmml">⁢</mo><mrow id="S3.SS3.5.p1.12.m12.1.2.3.2" xref="S3.SS3.5.p1.12.m12.1.2.cmml"><mo id="S3.SS3.5.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S3.SS3.5.p1.12.m12.1.2.cmml">(</mo><mi id="S3.SS3.5.p1.12.m12.1.1" xref="S3.SS3.5.p1.12.m12.1.1.cmml">x</mi><mo id="S3.SS3.5.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S3.SS3.5.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.12.m12.1b"><apply id="S3.SS3.5.p1.12.m12.1.2.cmml" xref="S3.SS3.5.p1.12.m12.1.2"><times id="S3.SS3.5.p1.12.m12.1.2.1.cmml" xref="S3.SS3.5.p1.12.m12.1.2.1"></times><apply id="S3.SS3.5.p1.12.m12.1.2.2.cmml" xref="S3.SS3.5.p1.12.m12.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.5.p1.12.m12.1.2.2.1.cmml" xref="S3.SS3.5.p1.12.m12.1.2.2">subscript</csymbol><ci id="S3.SS3.5.p1.12.m12.1.2.2.2.cmml" xref="S3.SS3.5.p1.12.m12.1.2.2.2">𝐹</ci><ci id="S3.SS3.5.p1.12.m12.1.2.2.3.cmml" xref="S3.SS3.5.p1.12.m12.1.2.2.3">𝐶</ci></apply><ci id="S3.SS3.5.p1.12.m12.1.1.cmml" xref="S3.SS3.5.p1.12.m12.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.12.m12.1c">F_{C}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.12.m12.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> defined as the projection of <math alttext="F(x)" class="ltx_Math" display="inline" id="S3.SS3.5.p1.13.m13.1"><semantics id="S3.SS3.5.p1.13.m13.1a"><mrow id="S3.SS3.5.p1.13.m13.1.2" xref="S3.SS3.5.p1.13.m13.1.2.cmml"><mi id="S3.SS3.5.p1.13.m13.1.2.2" xref="S3.SS3.5.p1.13.m13.1.2.2.cmml">F</mi><mo id="S3.SS3.5.p1.13.m13.1.2.1" xref="S3.SS3.5.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S3.SS3.5.p1.13.m13.1.2.3.2" xref="S3.SS3.5.p1.13.m13.1.2.cmml"><mo id="S3.SS3.5.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S3.SS3.5.p1.13.m13.1.2.cmml">(</mo><mi id="S3.SS3.5.p1.13.m13.1.1" xref="S3.SS3.5.p1.13.m13.1.1.cmml">x</mi><mo id="S3.SS3.5.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S3.SS3.5.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.13.m13.1b"><apply id="S3.SS3.5.p1.13.m13.1.2.cmml" xref="S3.SS3.5.p1.13.m13.1.2"><times id="S3.SS3.5.p1.13.m13.1.2.1.cmml" xref="S3.SS3.5.p1.13.m13.1.2.1"></times><ci id="S3.SS3.5.p1.13.m13.1.2.2.cmml" xref="S3.SS3.5.p1.13.m13.1.2.2">𝐹</ci><ci id="S3.SS3.5.p1.13.m13.1.1.cmml" xref="S3.SS3.5.p1.13.m13.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.13.m13.1c">F(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.13.m13.1d">italic_F ( italic_x )</annotation></semantics></math> onto <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.14.m14.1"><semantics id="S3.SS3.5.p1.14.m14.1a"><mi id="S3.SS3.5.p1.14.m14.1.1" xref="S3.SS3.5.p1.14.m14.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.14.m14.1b"><ci id="S3.SS3.5.p1.14.m14.1.1.cmml" xref="S3.SS3.5.p1.14.m14.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.14.m14.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.14.m14.1d">italic_C</annotation></semantics></math> in the direction of the origin (for any <math alttext="x\in C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.15.m15.1"><semantics id="S3.SS3.5.p1.15.m15.1a"><mrow id="S3.SS3.5.p1.15.m15.1.1" xref="S3.SS3.5.p1.15.m15.1.1.cmml"><mi id="S3.SS3.5.p1.15.m15.1.1.2" xref="S3.SS3.5.p1.15.m15.1.1.2.cmml">x</mi><mo id="S3.SS3.5.p1.15.m15.1.1.1" xref="S3.SS3.5.p1.15.m15.1.1.1.cmml">∈</mo><mi id="S3.SS3.5.p1.15.m15.1.1.3" xref="S3.SS3.5.p1.15.m15.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.15.m15.1b"><apply id="S3.SS3.5.p1.15.m15.1.1.cmml" xref="S3.SS3.5.p1.15.m15.1.1"><in id="S3.SS3.5.p1.15.m15.1.1.1.cmml" xref="S3.SS3.5.p1.15.m15.1.1.1"></in><ci id="S3.SS3.5.p1.15.m15.1.1.2.cmml" xref="S3.SS3.5.p1.15.m15.1.1.2">𝑥</ci><ci id="S3.SS3.5.p1.15.m15.1.1.3.cmml" xref="S3.SS3.5.p1.15.m15.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.15.m15.1c">x\in C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.15.m15.1d">italic_x ∈ italic_C</annotation></semantics></math>). As our set <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.16.m16.1"><semantics id="S3.SS3.5.p1.16.m16.1a"><mi id="S3.SS3.5.p1.16.m16.1.1" xref="S3.SS3.5.p1.16.m16.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.16.m16.1b"><ci id="S3.SS3.5.p1.16.m16.1.1.cmml" xref="S3.SS3.5.p1.16.m16.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.16.m16.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.16.m16.1d">italic_C</annotation></semantics></math>, we choose the Euclidean ball of radius <math alttext="1+2d" class="ltx_Math" display="inline" id="S3.SS3.5.p1.17.m17.1"><semantics id="S3.SS3.5.p1.17.m17.1a"><mrow id="S3.SS3.5.p1.17.m17.1.1" xref="S3.SS3.5.p1.17.m17.1.1.cmml"><mn id="S3.SS3.5.p1.17.m17.1.1.2" xref="S3.SS3.5.p1.17.m17.1.1.2.cmml">1</mn><mo id="S3.SS3.5.p1.17.m17.1.1.1" xref="S3.SS3.5.p1.17.m17.1.1.1.cmml">+</mo><mrow id="S3.SS3.5.p1.17.m17.1.1.3" xref="S3.SS3.5.p1.17.m17.1.1.3.cmml"><mn id="S3.SS3.5.p1.17.m17.1.1.3.2" xref="S3.SS3.5.p1.17.m17.1.1.3.2.cmml">2</mn><mo id="S3.SS3.5.p1.17.m17.1.1.3.1" xref="S3.SS3.5.p1.17.m17.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.5.p1.17.m17.1.1.3.3" xref="S3.SS3.5.p1.17.m17.1.1.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.17.m17.1b"><apply id="S3.SS3.5.p1.17.m17.1.1.cmml" xref="S3.SS3.5.p1.17.m17.1.1"><plus id="S3.SS3.5.p1.17.m17.1.1.1.cmml" xref="S3.SS3.5.p1.17.m17.1.1.1"></plus><cn id="S3.SS3.5.p1.17.m17.1.1.2.cmml" type="integer" xref="S3.SS3.5.p1.17.m17.1.1.2">1</cn><apply id="S3.SS3.5.p1.17.m17.1.1.3.cmml" xref="S3.SS3.5.p1.17.m17.1.1.3"><times id="S3.SS3.5.p1.17.m17.1.1.3.1.cmml" xref="S3.SS3.5.p1.17.m17.1.1.3.1"></times><cn id="S3.SS3.5.p1.17.m17.1.1.3.2.cmml" type="integer" xref="S3.SS3.5.p1.17.m17.1.1.3.2">2</cn><ci id="S3.SS3.5.p1.17.m17.1.1.3.3.cmml" xref="S3.SS3.5.p1.17.m17.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.17.m17.1c">1+2d</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.17.m17.1d">1 + 2 italic_d</annotation></semantics></math> around the origin, since this allows us to apply <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.8</span></a> for all <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.5.p1.18.m18.1"><semantics id="S3.SS3.5.p1.18.m18.1a"><mi id="S3.SS3.5.p1.18.m18.1.1" xref="S3.SS3.5.p1.18.m18.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.18.m18.1b"><ci id="S3.SS3.5.p1.18.m18.1.1.cmml" xref="S3.SS3.5.p1.18.m18.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.18.m18.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.18.m18.1d">italic_x</annotation></semantics></math> on the boundary of <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.5.p1.19.m19.1"><semantics id="S3.SS3.5.p1.19.m19.1a"><mi id="S3.SS3.5.p1.19.m19.1.1" xref="S3.SS3.5.p1.19.m19.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.5.p1.19.m19.1b"><ci id="S3.SS3.5.p1.19.m19.1.1.cmml" xref="S3.SS3.5.p1.19.m19.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.5.p1.19.m19.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.5.p1.19.m19.1d">italic_C</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.6.p2"> <p class="ltx_p" id="S3.SS3.6.p2.7">By continuity of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.6.p2.1.m1.1"><semantics id="S3.SS3.6.p2.1.m1.1a"><mi id="S3.SS3.6.p2.1.m1.1.1" xref="S3.SS3.6.p2.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.1.m1.1b"><ci id="S3.SS3.6.p2.1.m1.1.1.cmml" xref="S3.SS3.6.p2.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.1.m1.1d">italic_F</annotation></semantics></math>, it follows that <math alttext="F_{C}" class="ltx_Math" display="inline" id="S3.SS3.6.p2.2.m2.1"><semantics id="S3.SS3.6.p2.2.m2.1a"><msub id="S3.SS3.6.p2.2.m2.1.1" xref="S3.SS3.6.p2.2.m2.1.1.cmml"><mi id="S3.SS3.6.p2.2.m2.1.1.2" xref="S3.SS3.6.p2.2.m2.1.1.2.cmml">F</mi><mi id="S3.SS3.6.p2.2.m2.1.1.3" xref="S3.SS3.6.p2.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.2.m2.1b"><apply id="S3.SS3.6.p2.2.m2.1.1.cmml" xref="S3.SS3.6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.6.p2.2.m2.1.1.1.cmml" xref="S3.SS3.6.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.6.p2.2.m2.1.1.2.cmml" xref="S3.SS3.6.p2.2.m2.1.1.2">𝐹</ci><ci id="S3.SS3.6.p2.2.m2.1.1.3.cmml" xref="S3.SS3.6.p2.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.2.m2.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> is also continuous. Thus, we can use Brouwer’s fixpoint theorem, which tells us that there exists <math alttext="c\in C" class="ltx_Math" display="inline" id="S3.SS3.6.p2.3.m3.1"><semantics id="S3.SS3.6.p2.3.m3.1a"><mrow id="S3.SS3.6.p2.3.m3.1.1" xref="S3.SS3.6.p2.3.m3.1.1.cmml"><mi id="S3.SS3.6.p2.3.m3.1.1.2" xref="S3.SS3.6.p2.3.m3.1.1.2.cmml">c</mi><mo id="S3.SS3.6.p2.3.m3.1.1.1" xref="S3.SS3.6.p2.3.m3.1.1.1.cmml">∈</mo><mi id="S3.SS3.6.p2.3.m3.1.1.3" xref="S3.SS3.6.p2.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.3.m3.1b"><apply id="S3.SS3.6.p2.3.m3.1.1.cmml" xref="S3.SS3.6.p2.3.m3.1.1"><in id="S3.SS3.6.p2.3.m3.1.1.1.cmml" xref="S3.SS3.6.p2.3.m3.1.1.1"></in><ci id="S3.SS3.6.p2.3.m3.1.1.2.cmml" xref="S3.SS3.6.p2.3.m3.1.1.2">𝑐</ci><ci id="S3.SS3.6.p2.3.m3.1.1.3.cmml" xref="S3.SS3.6.p2.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.3.m3.1c">c\in C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.3.m3.1d">italic_c ∈ italic_C</annotation></semantics></math> with <math alttext="F_{C}(c)=c" class="ltx_Math" display="inline" id="S3.SS3.6.p2.4.m4.1"><semantics id="S3.SS3.6.p2.4.m4.1a"><mrow id="S3.SS3.6.p2.4.m4.1.2" xref="S3.SS3.6.p2.4.m4.1.2.cmml"><mrow id="S3.SS3.6.p2.4.m4.1.2.2" xref="S3.SS3.6.p2.4.m4.1.2.2.cmml"><msub id="S3.SS3.6.p2.4.m4.1.2.2.2" xref="S3.SS3.6.p2.4.m4.1.2.2.2.cmml"><mi id="S3.SS3.6.p2.4.m4.1.2.2.2.2" xref="S3.SS3.6.p2.4.m4.1.2.2.2.2.cmml">F</mi><mi id="S3.SS3.6.p2.4.m4.1.2.2.2.3" xref="S3.SS3.6.p2.4.m4.1.2.2.2.3.cmml">C</mi></msub><mo id="S3.SS3.6.p2.4.m4.1.2.2.1" xref="S3.SS3.6.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS3.6.p2.4.m4.1.2.2.3.2" xref="S3.SS3.6.p2.4.m4.1.2.2.cmml"><mo id="S3.SS3.6.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS3.6.p2.4.m4.1.2.2.cmml">(</mo><mi id="S3.SS3.6.p2.4.m4.1.1" xref="S3.SS3.6.p2.4.m4.1.1.cmml">c</mi><mo id="S3.SS3.6.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS3.6.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.6.p2.4.m4.1.2.1" xref="S3.SS3.6.p2.4.m4.1.2.1.cmml">=</mo><mi id="S3.SS3.6.p2.4.m4.1.2.3" xref="S3.SS3.6.p2.4.m4.1.2.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.4.m4.1b"><apply id="S3.SS3.6.p2.4.m4.1.2.cmml" xref="S3.SS3.6.p2.4.m4.1.2"><eq id="S3.SS3.6.p2.4.m4.1.2.1.cmml" xref="S3.SS3.6.p2.4.m4.1.2.1"></eq><apply id="S3.SS3.6.p2.4.m4.1.2.2.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2"><times id="S3.SS3.6.p2.4.m4.1.2.2.1.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2.1"></times><apply id="S3.SS3.6.p2.4.m4.1.2.2.2.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.6.p2.4.m4.1.2.2.2.1.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2.2">subscript</csymbol><ci id="S3.SS3.6.p2.4.m4.1.2.2.2.2.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2.2.2">𝐹</ci><ci id="S3.SS3.6.p2.4.m4.1.2.2.2.3.cmml" xref="S3.SS3.6.p2.4.m4.1.2.2.2.3">𝐶</ci></apply><ci id="S3.SS3.6.p2.4.m4.1.1.cmml" xref="S3.SS3.6.p2.4.m4.1.1">𝑐</ci></apply><ci id="S3.SS3.6.p2.4.m4.1.2.3.cmml" xref="S3.SS3.6.p2.4.m4.1.2.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.4.m4.1c">F_{C}(c)=c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.4.m4.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c ) = italic_c</annotation></semantics></math>. We now prove that <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.6.p2.5.m5.1"><semantics id="S3.SS3.6.p2.5.m5.1a"><mi id="S3.SS3.6.p2.5.m5.1.1" xref="S3.SS3.6.p2.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.5.m5.1b"><ci id="S3.SS3.6.p2.5.m5.1.1.cmml" xref="S3.SS3.6.p2.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.5.m5.1d">italic_c</annotation></semantics></math> is a centerpoint by distinguishing the two cases <math alttext="F(c)=F_{C}(c)" class="ltx_Math" display="inline" id="S3.SS3.6.p2.6.m6.2"><semantics id="S3.SS3.6.p2.6.m6.2a"><mrow id="S3.SS3.6.p2.6.m6.2.3" xref="S3.SS3.6.p2.6.m6.2.3.cmml"><mrow id="S3.SS3.6.p2.6.m6.2.3.2" xref="S3.SS3.6.p2.6.m6.2.3.2.cmml"><mi id="S3.SS3.6.p2.6.m6.2.3.2.2" xref="S3.SS3.6.p2.6.m6.2.3.2.2.cmml">F</mi><mo id="S3.SS3.6.p2.6.m6.2.3.2.1" xref="S3.SS3.6.p2.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.6.p2.6.m6.2.3.2.3.2" xref="S3.SS3.6.p2.6.m6.2.3.2.cmml"><mo id="S3.SS3.6.p2.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.6.p2.6.m6.2.3.2.cmml">(</mo><mi id="S3.SS3.6.p2.6.m6.1.1" xref="S3.SS3.6.p2.6.m6.1.1.cmml">c</mi><mo id="S3.SS3.6.p2.6.m6.2.3.2.3.2.2" stretchy="false" xref="S3.SS3.6.p2.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.6.p2.6.m6.2.3.1" xref="S3.SS3.6.p2.6.m6.2.3.1.cmml">=</mo><mrow id="S3.SS3.6.p2.6.m6.2.3.3" xref="S3.SS3.6.p2.6.m6.2.3.3.cmml"><msub id="S3.SS3.6.p2.6.m6.2.3.3.2" xref="S3.SS3.6.p2.6.m6.2.3.3.2.cmml"><mi id="S3.SS3.6.p2.6.m6.2.3.3.2.2" xref="S3.SS3.6.p2.6.m6.2.3.3.2.2.cmml">F</mi><mi id="S3.SS3.6.p2.6.m6.2.3.3.2.3" xref="S3.SS3.6.p2.6.m6.2.3.3.2.3.cmml">C</mi></msub><mo id="S3.SS3.6.p2.6.m6.2.3.3.1" xref="S3.SS3.6.p2.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS3.6.p2.6.m6.2.3.3.3.2" xref="S3.SS3.6.p2.6.m6.2.3.3.cmml"><mo id="S3.SS3.6.p2.6.m6.2.3.3.3.2.1" stretchy="false" xref="S3.SS3.6.p2.6.m6.2.3.3.cmml">(</mo><mi id="S3.SS3.6.p2.6.m6.2.2" xref="S3.SS3.6.p2.6.m6.2.2.cmml">c</mi><mo id="S3.SS3.6.p2.6.m6.2.3.3.3.2.2" stretchy="false" xref="S3.SS3.6.p2.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.6.m6.2b"><apply id="S3.SS3.6.p2.6.m6.2.3.cmml" xref="S3.SS3.6.p2.6.m6.2.3"><eq id="S3.SS3.6.p2.6.m6.2.3.1.cmml" xref="S3.SS3.6.p2.6.m6.2.3.1"></eq><apply id="S3.SS3.6.p2.6.m6.2.3.2.cmml" xref="S3.SS3.6.p2.6.m6.2.3.2"><times id="S3.SS3.6.p2.6.m6.2.3.2.1.cmml" xref="S3.SS3.6.p2.6.m6.2.3.2.1"></times><ci id="S3.SS3.6.p2.6.m6.2.3.2.2.cmml" xref="S3.SS3.6.p2.6.m6.2.3.2.2">𝐹</ci><ci id="S3.SS3.6.p2.6.m6.1.1.cmml" xref="S3.SS3.6.p2.6.m6.1.1">𝑐</ci></apply><apply id="S3.SS3.6.p2.6.m6.2.3.3.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3"><times id="S3.SS3.6.p2.6.m6.2.3.3.1.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3.1"></times><apply id="S3.SS3.6.p2.6.m6.2.3.3.2.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS3.6.p2.6.m6.2.3.3.2.1.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3.2">subscript</csymbol><ci id="S3.SS3.6.p2.6.m6.2.3.3.2.2.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3.2.2">𝐹</ci><ci id="S3.SS3.6.p2.6.m6.2.3.3.2.3.cmml" xref="S3.SS3.6.p2.6.m6.2.3.3.2.3">𝐶</ci></apply><ci id="S3.SS3.6.p2.6.m6.2.2.cmml" xref="S3.SS3.6.p2.6.m6.2.2">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.6.m6.2c">F(c)=F_{C}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.6.m6.2d">italic_F ( italic_c ) = italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> and <math alttext="F(c)\neq F_{C}(c)" class="ltx_Math" display="inline" id="S3.SS3.6.p2.7.m7.2"><semantics id="S3.SS3.6.p2.7.m7.2a"><mrow id="S3.SS3.6.p2.7.m7.2.3" xref="S3.SS3.6.p2.7.m7.2.3.cmml"><mrow id="S3.SS3.6.p2.7.m7.2.3.2" xref="S3.SS3.6.p2.7.m7.2.3.2.cmml"><mi id="S3.SS3.6.p2.7.m7.2.3.2.2" xref="S3.SS3.6.p2.7.m7.2.3.2.2.cmml">F</mi><mo id="S3.SS3.6.p2.7.m7.2.3.2.1" xref="S3.SS3.6.p2.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.6.p2.7.m7.2.3.2.3.2" xref="S3.SS3.6.p2.7.m7.2.3.2.cmml"><mo id="S3.SS3.6.p2.7.m7.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.6.p2.7.m7.2.3.2.cmml">(</mo><mi id="S3.SS3.6.p2.7.m7.1.1" xref="S3.SS3.6.p2.7.m7.1.1.cmml">c</mi><mo id="S3.SS3.6.p2.7.m7.2.3.2.3.2.2" stretchy="false" xref="S3.SS3.6.p2.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.6.p2.7.m7.2.3.1" xref="S3.SS3.6.p2.7.m7.2.3.1.cmml">≠</mo><mrow id="S3.SS3.6.p2.7.m7.2.3.3" xref="S3.SS3.6.p2.7.m7.2.3.3.cmml"><msub id="S3.SS3.6.p2.7.m7.2.3.3.2" xref="S3.SS3.6.p2.7.m7.2.3.3.2.cmml"><mi id="S3.SS3.6.p2.7.m7.2.3.3.2.2" xref="S3.SS3.6.p2.7.m7.2.3.3.2.2.cmml">F</mi><mi id="S3.SS3.6.p2.7.m7.2.3.3.2.3" xref="S3.SS3.6.p2.7.m7.2.3.3.2.3.cmml">C</mi></msub><mo id="S3.SS3.6.p2.7.m7.2.3.3.1" xref="S3.SS3.6.p2.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS3.6.p2.7.m7.2.3.3.3.2" xref="S3.SS3.6.p2.7.m7.2.3.3.cmml"><mo id="S3.SS3.6.p2.7.m7.2.3.3.3.2.1" stretchy="false" xref="S3.SS3.6.p2.7.m7.2.3.3.cmml">(</mo><mi id="S3.SS3.6.p2.7.m7.2.2" xref="S3.SS3.6.p2.7.m7.2.2.cmml">c</mi><mo id="S3.SS3.6.p2.7.m7.2.3.3.3.2.2" stretchy="false" xref="S3.SS3.6.p2.7.m7.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.6.p2.7.m7.2b"><apply id="S3.SS3.6.p2.7.m7.2.3.cmml" xref="S3.SS3.6.p2.7.m7.2.3"><neq id="S3.SS3.6.p2.7.m7.2.3.1.cmml" xref="S3.SS3.6.p2.7.m7.2.3.1"></neq><apply id="S3.SS3.6.p2.7.m7.2.3.2.cmml" xref="S3.SS3.6.p2.7.m7.2.3.2"><times id="S3.SS3.6.p2.7.m7.2.3.2.1.cmml" xref="S3.SS3.6.p2.7.m7.2.3.2.1"></times><ci id="S3.SS3.6.p2.7.m7.2.3.2.2.cmml" xref="S3.SS3.6.p2.7.m7.2.3.2.2">𝐹</ci><ci id="S3.SS3.6.p2.7.m7.1.1.cmml" xref="S3.SS3.6.p2.7.m7.1.1">𝑐</ci></apply><apply id="S3.SS3.6.p2.7.m7.2.3.3.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3"><times id="S3.SS3.6.p2.7.m7.2.3.3.1.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3.1"></times><apply id="S3.SS3.6.p2.7.m7.2.3.3.2.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS3.6.p2.7.m7.2.3.3.2.1.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3.2">subscript</csymbol><ci id="S3.SS3.6.p2.7.m7.2.3.3.2.2.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3.2.2">𝐹</ci><ci id="S3.SS3.6.p2.7.m7.2.3.3.2.3.cmml" xref="S3.SS3.6.p2.7.m7.2.3.3.2.3">𝐶</ci></apply><ci id="S3.SS3.6.p2.7.m7.2.2.cmml" xref="S3.SS3.6.p2.7.m7.2.2">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.6.p2.7.m7.2c">F(c)\neq F_{C}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.6.p2.7.m7.2d">italic_F ( italic_c ) ≠ italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.7.p3"> <p class="ltx_p" id="S3.SS3.7.p3.16">In the first case, we must have that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^% {p}_{c,-v}),0\right)dv=0" class="ltx_Math" display="block" id="S3.Ex9.m1.5"><semantics id="S3.Ex9.m1.5a"><mrow id="S3.Ex9.m1.5.5" xref="S3.Ex9.m1.5.5.cmml"><mrow id="S3.Ex9.m1.5.5.1" xref="S3.Ex9.m1.5.5.1.cmml"><msub id="S3.Ex9.m1.5.5.1.2" xref="S3.Ex9.m1.5.5.1.2.cmml"><mo id="S3.Ex9.m1.5.5.1.2.2" xref="S3.Ex9.m1.5.5.1.2.2.cmml">∫</mo><msup id="S3.Ex9.m1.5.5.1.2.3" xref="S3.Ex9.m1.5.5.1.2.3.cmml"><mi id="S3.Ex9.m1.5.5.1.2.3.2" xref="S3.Ex9.m1.5.5.1.2.3.2.cmml">S</mi><mrow id="S3.Ex9.m1.5.5.1.2.3.3" xref="S3.Ex9.m1.5.5.1.2.3.3.cmml"><mi id="S3.Ex9.m1.5.5.1.2.3.3.2" xref="S3.Ex9.m1.5.5.1.2.3.3.2.cmml">d</mi><mo id="S3.Ex9.m1.5.5.1.2.3.3.1" xref="S3.Ex9.m1.5.5.1.2.3.3.1.cmml">−</mo><mn id="S3.Ex9.m1.5.5.1.2.3.3.3" xref="S3.Ex9.m1.5.5.1.2.3.3.3.cmml">1</mn></mrow></msup></msub><mrow id="S3.Ex9.m1.5.5.1.1" xref="S3.Ex9.m1.5.5.1.1.cmml"><msub id="S3.Ex9.m1.5.5.1.1.3" xref="S3.Ex9.m1.5.5.1.1.3.cmml"><mi id="S3.Ex9.m1.5.5.1.1.3.2" xref="S3.Ex9.m1.5.5.1.1.3.2.cmml">v</mi><mi id="S3.Ex9.m1.5.5.1.1.3.3" xref="S3.Ex9.m1.5.5.1.1.3.3.cmml">i</mi></msub><mo id="S3.Ex9.m1.5.5.1.1.2" lspace="0.167em" xref="S3.Ex9.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.2.cmml"><mi id="S3.Ex9.m1.3.3" xref="S3.Ex9.m1.3.3.cmml">max</mi><mo id="S3.Ex9.m1.5.5.1.1.1.1a" xref="S3.Ex9.m1.5.5.1.1.1.2.cmml">⁡</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.2.cmml"><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.2" xref="S3.Ex9.m1.5.5.1.1.1.2.cmml">(</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.cmml"><mfrac id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.cmml"><mn id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.2.cmml">1</mn><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.cmml"><mi id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.2.cmml">d</mi><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.4" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.4.cmml">μ</mi><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2a" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.3.cmml">−</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.3" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.3.cmml">μ</mi><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.cmml"><mo id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.2" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.2.cmml">ℋ</mi><mrow 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xref="S3.Ex9.m1.5.5.1.1.4.cmml"><mo id="S3.Ex9.m1.5.5.1.1.4.1" rspace="0em" xref="S3.Ex9.m1.5.5.1.1.4.1.cmml">𝑑</mo><mi id="S3.Ex9.m1.5.5.1.1.4.2" xref="S3.Ex9.m1.5.5.1.1.4.2.cmml">v</mi></mrow></mrow></mrow><mo id="S3.Ex9.m1.5.5.2" xref="S3.Ex9.m1.5.5.2.cmml">=</mo><mn id="S3.Ex9.m1.5.5.3" xref="S3.Ex9.m1.5.5.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex9.m1.5b"><apply id="S3.Ex9.m1.5.5.cmml" xref="S3.Ex9.m1.5.5"><eq id="S3.Ex9.m1.5.5.2.cmml" xref="S3.Ex9.m1.5.5.2"></eq><apply id="S3.Ex9.m1.5.5.1.cmml" xref="S3.Ex9.m1.5.5.1"><apply id="S3.Ex9.m1.5.5.1.2.cmml" xref="S3.Ex9.m1.5.5.1.2"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.1.2.1.cmml" xref="S3.Ex9.m1.5.5.1.2">subscript</csymbol><int id="S3.Ex9.m1.5.5.1.2.2.cmml" xref="S3.Ex9.m1.5.5.1.2.2"></int><apply id="S3.Ex9.m1.5.5.1.2.3.cmml" xref="S3.Ex9.m1.5.5.1.2.3"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.1.2.3.1.cmml" xref="S3.Ex9.m1.5.5.1.2.3">superscript</csymbol><ci id="S3.Ex9.m1.5.5.1.2.3.2.cmml" 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xref="S3.Ex9.m1.5.5.1.1.1.1.1.1"><minus id="S3.Ex9.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.3"></minus><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1"><times id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.2"></times><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3"><divide id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3"></divide><cn id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.2">1</cn><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3"><plus id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.1"></plus><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.2">𝑑</ci><cn id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.4.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.4">𝜇</ci><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2"><times id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.2"></times><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.3">𝜇</ci><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1">subscript</csymbol><apply id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.1.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1">superscript</csymbol><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.2.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.2">ℋ</ci><ci id="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.3.cmml" xref="S3.Ex9.m1.5.5.1.1.1.1.1.1.2.1.1.1.2.3">𝑝</ci></apply><list id="S3.Ex9.m1.2.2.2.3.cmml" xref="S3.Ex9.m1.2.2.2.2"><ci id="S3.Ex9.m1.1.1.1.1.cmml" xref="S3.Ex9.m1.1.1.1.1">𝑐</ci><apply id="S3.Ex9.m1.2.2.2.2.1.cmml" xref="S3.Ex9.m1.2.2.2.2.1"><minus id="S3.Ex9.m1.2.2.2.2.1.1.cmml" xref="S3.Ex9.m1.2.2.2.2.1"></minus><ci id="S3.Ex9.m1.2.2.2.2.1.2.cmml" xref="S3.Ex9.m1.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></apply><cn id="S3.Ex9.m1.4.4.cmml" type="integer" xref="S3.Ex9.m1.4.4">0</cn></apply><apply id="S3.Ex9.m1.5.5.1.1.4.cmml" xref="S3.Ex9.m1.5.5.1.1.4"><csymbol cd="latexml" id="S3.Ex9.m1.5.5.1.1.4.1.cmml" xref="S3.Ex9.m1.5.5.1.1.4.1">differential-d</csymbol><ci id="S3.Ex9.m1.5.5.1.1.4.2.cmml" xref="S3.Ex9.m1.5.5.1.1.4.2">𝑣</ci></apply></apply></apply><cn id="S3.Ex9.m1.5.5.3.cmml" type="integer" xref="S3.Ex9.m1.5.5.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex9.m1.5c">\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^% {p}_{c,-v}),0\right)dv=0</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.5d">∫ start_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , - italic_v end_POSTSUBSCRIPT ) , 0 ) italic_d italic_v = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.7.p3.15">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S3.SS3.7.p3.1.m1.1"><semantics id="S3.SS3.7.p3.1.m1.1a"><mrow id="S3.SS3.7.p3.1.m1.1.2" xref="S3.SS3.7.p3.1.m1.1.2.cmml"><mi id="S3.SS3.7.p3.1.m1.1.2.2" xref="S3.SS3.7.p3.1.m1.1.2.2.cmml">i</mi><mo id="S3.SS3.7.p3.1.m1.1.2.1" xref="S3.SS3.7.p3.1.m1.1.2.1.cmml">∈</mo><mrow id="S3.SS3.7.p3.1.m1.1.2.3.2" xref="S3.SS3.7.p3.1.m1.1.2.3.1.cmml"><mo id="S3.SS3.7.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS3.7.p3.1.m1.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.7.p3.1.m1.1.1" xref="S3.SS3.7.p3.1.m1.1.1.cmml">d</mi><mo id="S3.SS3.7.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS3.7.p3.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.1.m1.1b"><apply id="S3.SS3.7.p3.1.m1.1.2.cmml" xref="S3.SS3.7.p3.1.m1.1.2"><in id="S3.SS3.7.p3.1.m1.1.2.1.cmml" xref="S3.SS3.7.p3.1.m1.1.2.1"></in><ci id="S3.SS3.7.p3.1.m1.1.2.2.cmml" xref="S3.SS3.7.p3.1.m1.1.2.2">𝑖</ci><apply id="S3.SS3.7.p3.1.m1.1.2.3.1.cmml" xref="S3.SS3.7.p3.1.m1.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.7.p3.1.m1.1.2.3.1.1.cmml" xref="S3.SS3.7.p3.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.7.p3.1.m1.1.1.cmml" xref="S3.SS3.7.p3.1.m1.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.1.m1.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.1.m1.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. Now consider the set of directions <math alttext="V_{c}=\{v\in S^{d-1}\mid\max\big{(}\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(% \mathcal{H}^{p}_{c,-v}),0\big{)}>0\}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.2.m2.6"><semantics id="S3.SS3.7.p3.2.m2.6a"><mrow id="S3.SS3.7.p3.2.m2.6.6" xref="S3.SS3.7.p3.2.m2.6.6.cmml"><msub id="S3.SS3.7.p3.2.m2.6.6.4" xref="S3.SS3.7.p3.2.m2.6.6.4.cmml"><mi id="S3.SS3.7.p3.2.m2.6.6.4.2" xref="S3.SS3.7.p3.2.m2.6.6.4.2.cmml">V</mi><mi id="S3.SS3.7.p3.2.m2.6.6.4.3" xref="S3.SS3.7.p3.2.m2.6.6.4.3.cmml">c</mi></msub><mo id="S3.SS3.7.p3.2.m2.6.6.3" xref="S3.SS3.7.p3.2.m2.6.6.3.cmml">=</mo><mrow id="S3.SS3.7.p3.2.m2.6.6.2.2" xref="S3.SS3.7.p3.2.m2.6.6.2.3.cmml"><mo id="S3.SS3.7.p3.2.m2.6.6.2.2.3" stretchy="false" xref="S3.SS3.7.p3.2.m2.6.6.2.3.1.cmml">{</mo><mrow id="S3.SS3.7.p3.2.m2.5.5.1.1.1" xref="S3.SS3.7.p3.2.m2.5.5.1.1.1.cmml"><mi id="S3.SS3.7.p3.2.m2.5.5.1.1.1.2" xref="S3.SS3.7.p3.2.m2.5.5.1.1.1.2.cmml">v</mi><mo id="S3.SS3.7.p3.2.m2.5.5.1.1.1.1" 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xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1">subscript</csymbol><apply id="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.cmml" xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.1.cmml" xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1">superscript</csymbol><ci id="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.2.cmml" xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.2">ℋ</ci><ci id="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.3.cmml" xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.1.1.1.1.2.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS3.7.p3.2.m2.2.2.2.3.cmml" xref="S3.SS3.7.p3.2.m2.2.2.2.2"><ci id="S3.SS3.7.p3.2.m2.1.1.1.1.cmml" xref="S3.SS3.7.p3.2.m2.1.1.1.1">𝑐</ci><apply id="S3.SS3.7.p3.2.m2.2.2.2.2.1.cmml" xref="S3.SS3.7.p3.2.m2.2.2.2.2.1"><minus id="S3.SS3.7.p3.2.m2.2.2.2.2.1.1.cmml" xref="S3.SS3.7.p3.2.m2.2.2.2.2.1"></minus><ci id="S3.SS3.7.p3.2.m2.2.2.2.2.1.2.cmml" xref="S3.SS3.7.p3.2.m2.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></apply><cn id="S3.SS3.7.p3.2.m2.4.4.cmml" type="integer" xref="S3.SS3.7.p3.2.m2.4.4">0</cn></apply><cn id="S3.SS3.7.p3.2.m2.6.6.2.2.2.3.cmml" type="integer" xref="S3.SS3.7.p3.2.m2.6.6.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.2.m2.6c">V_{c}=\{v\in S^{d-1}\mid\max\big{(}\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(% \mathcal{H}^{p}_{c,-v}),0\big{)}>0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.2.m2.6d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = { italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , - italic_v end_POSTSUBSCRIPT ) , 0 ) > 0 }</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem11" title="Lemma 3.11. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.11</span></a>, this is an open subset of <math alttext="S^{d-1}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.3.m3.1"><semantics id="S3.SS3.7.p3.3.m3.1a"><msup id="S3.SS3.7.p3.3.m3.1.1" xref="S3.SS3.7.p3.3.m3.1.1.cmml"><mi id="S3.SS3.7.p3.3.m3.1.1.2" xref="S3.SS3.7.p3.3.m3.1.1.2.cmml">S</mi><mrow id="S3.SS3.7.p3.3.m3.1.1.3" xref="S3.SS3.7.p3.3.m3.1.1.3.cmml"><mi id="S3.SS3.7.p3.3.m3.1.1.3.2" xref="S3.SS3.7.p3.3.m3.1.1.3.2.cmml">d</mi><mo id="S3.SS3.7.p3.3.m3.1.1.3.1" xref="S3.SS3.7.p3.3.m3.1.1.3.1.cmml">−</mo><mn id="S3.SS3.7.p3.3.m3.1.1.3.3" xref="S3.SS3.7.p3.3.m3.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.3.m3.1b"><apply id="S3.SS3.7.p3.3.m3.1.1.cmml" xref="S3.SS3.7.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.3.m3.1.1.1.cmml" xref="S3.SS3.7.p3.3.m3.1.1">superscript</csymbol><ci id="S3.SS3.7.p3.3.m3.1.1.2.cmml" xref="S3.SS3.7.p3.3.m3.1.1.2">𝑆</ci><apply id="S3.SS3.7.p3.3.m3.1.1.3.cmml" xref="S3.SS3.7.p3.3.m3.1.1.3"><minus id="S3.SS3.7.p3.3.m3.1.1.3.1.cmml" xref="S3.SS3.7.p3.3.m3.1.1.3.1"></minus><ci id="S3.SS3.7.p3.3.m3.1.1.3.2.cmml" xref="S3.SS3.7.p3.3.m3.1.1.3.2">𝑑</ci><cn id="S3.SS3.7.p3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.SS3.7.p3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.3.m3.1c">S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.3.m3.1d">italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Moreover, <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.7.p3.4.m4.1"><semantics id="S3.SS3.7.p3.4.m4.1a"><mn id="S3.SS3.7.p3.4.m4.1.1" xref="S3.SS3.7.p3.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.4.m4.1b"><cn id="S3.SS3.7.p3.4.m4.1.1.cmml" type="integer" xref="S3.SS3.7.p3.4.m4.1.1">0</cn></annotation-xml></semantics></math> is not contained in <math alttext="\text{conv}(V_{c})" class="ltx_Math" display="inline" id="S3.SS3.7.p3.5.m5.1"><semantics id="S3.SS3.7.p3.5.m5.1a"><mrow id="S3.SS3.7.p3.5.m5.1.1" xref="S3.SS3.7.p3.5.m5.1.1.cmml"><mtext id="S3.SS3.7.p3.5.m5.1.1.3" xref="S3.SS3.7.p3.5.m5.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.7.p3.5.m5.1.1.2" xref="S3.SS3.7.p3.5.m5.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.7.p3.5.m5.1.1.1.1" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.cmml"><mo id="S3.SS3.7.p3.5.m5.1.1.1.1.2" stretchy="false" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.7.p3.5.m5.1.1.1.1.1" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.cmml"><mi id="S3.SS3.7.p3.5.m5.1.1.1.1.1.2" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.5.m5.1.1.1.1.1.3" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S3.SS3.7.p3.5.m5.1.1.1.1.3" stretchy="false" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.5.m5.1b"><apply id="S3.SS3.7.p3.5.m5.1.1.cmml" xref="S3.SS3.7.p3.5.m5.1.1"><times id="S3.SS3.7.p3.5.m5.1.1.2.cmml" xref="S3.SS3.7.p3.5.m5.1.1.2"></times><ci id="S3.SS3.7.p3.5.m5.1.1.3a.cmml" xref="S3.SS3.7.p3.5.m5.1.1.3"><mtext id="S3.SS3.7.p3.5.m5.1.1.3.cmml" xref="S3.SS3.7.p3.5.m5.1.1.3">conv</mtext></ci><apply id="S3.SS3.7.p3.5.m5.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.5.m5.1.1.1.1.1.2.cmml" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.5.m5.1.1.1.1.1.3.cmml" xref="S3.SS3.7.p3.5.m5.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.5.m5.1c">\text{conv}(V_{c})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.5.m5.1d">conv ( italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem17" title="Corollary 3.17. ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.17</span></a>. However, this implies that <math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.6.m6.1"><semantics id="S3.SS3.7.p3.6.m6.1a"><msub id="S3.SS3.7.p3.6.m6.1.1" xref="S3.SS3.7.p3.6.m6.1.1.cmml"><mi id="S3.SS3.7.p3.6.m6.1.1.2" xref="S3.SS3.7.p3.6.m6.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.6.m6.1.1.3" xref="S3.SS3.7.p3.6.m6.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.6.m6.1b"><apply id="S3.SS3.7.p3.6.m6.1.1.cmml" xref="S3.SS3.7.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.6.m6.1.1.1.cmml" xref="S3.SS3.7.p3.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.6.m6.1.1.2.cmml" xref="S3.SS3.7.p3.6.m6.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.6.m6.1.1.3.cmml" xref="S3.SS3.7.p3.6.m6.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.6.m6.1c">V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.6.m6.1d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> must be empty: if <math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.7.m7.1"><semantics id="S3.SS3.7.p3.7.m7.1a"><msub id="S3.SS3.7.p3.7.m7.1.1" xref="S3.SS3.7.p3.7.m7.1.1.cmml"><mi id="S3.SS3.7.p3.7.m7.1.1.2" xref="S3.SS3.7.p3.7.m7.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.7.m7.1.1.3" xref="S3.SS3.7.p3.7.m7.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.7.m7.1b"><apply id="S3.SS3.7.p3.7.m7.1.1.cmml" xref="S3.SS3.7.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.7.m7.1.1.1.cmml" xref="S3.SS3.7.p3.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.7.m7.1.1.2.cmml" xref="S3.SS3.7.p3.7.m7.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.7.m7.1.1.3.cmml" xref="S3.SS3.7.p3.7.m7.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.7.m7.1c">V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.7.m7.1d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> were non-empty, we could find a separating hyperplane between the two convex sets <math alttext="\{0\}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.8.m8.1"><semantics id="S3.SS3.7.p3.8.m8.1a"><mrow id="S3.SS3.7.p3.8.m8.1.2.2" xref="S3.SS3.7.p3.8.m8.1.2.1.cmml"><mo id="S3.SS3.7.p3.8.m8.1.2.2.1" stretchy="false" xref="S3.SS3.7.p3.8.m8.1.2.1.cmml">{</mo><mn id="S3.SS3.7.p3.8.m8.1.1" xref="S3.SS3.7.p3.8.m8.1.1.cmml">0</mn><mo id="S3.SS3.7.p3.8.m8.1.2.2.2" stretchy="false" xref="S3.SS3.7.p3.8.m8.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.8.m8.1b"><set id="S3.SS3.7.p3.8.m8.1.2.1.cmml" xref="S3.SS3.7.p3.8.m8.1.2.2"><cn id="S3.SS3.7.p3.8.m8.1.1.cmml" type="integer" xref="S3.SS3.7.p3.8.m8.1.1">0</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.8.m8.1c">\{0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.8.m8.1d">{ 0 }</annotation></semantics></math> and <math alttext="\text{conv}(V_{c})" class="ltx_Math" display="inline" id="S3.SS3.7.p3.9.m9.1"><semantics id="S3.SS3.7.p3.9.m9.1a"><mrow id="S3.SS3.7.p3.9.m9.1.1" xref="S3.SS3.7.p3.9.m9.1.1.cmml"><mtext id="S3.SS3.7.p3.9.m9.1.1.3" xref="S3.SS3.7.p3.9.m9.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.7.p3.9.m9.1.1.2" xref="S3.SS3.7.p3.9.m9.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.7.p3.9.m9.1.1.1.1" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.cmml"><mo id="S3.SS3.7.p3.9.m9.1.1.1.1.2" stretchy="false" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.7.p3.9.m9.1.1.1.1.1" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.cmml"><mi id="S3.SS3.7.p3.9.m9.1.1.1.1.1.2" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.9.m9.1.1.1.1.1.3" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S3.SS3.7.p3.9.m9.1.1.1.1.3" stretchy="false" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.9.m9.1b"><apply id="S3.SS3.7.p3.9.m9.1.1.cmml" xref="S3.SS3.7.p3.9.m9.1.1"><times id="S3.SS3.7.p3.9.m9.1.1.2.cmml" xref="S3.SS3.7.p3.9.m9.1.1.2"></times><ci id="S3.SS3.7.p3.9.m9.1.1.3a.cmml" xref="S3.SS3.7.p3.9.m9.1.1.3"><mtext id="S3.SS3.7.p3.9.m9.1.1.3.cmml" xref="S3.SS3.7.p3.9.m9.1.1.3">conv</mtext></ci><apply id="S3.SS3.7.p3.9.m9.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.9.m9.1.1.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.9.m9.1.1.1.1.1.2.cmml" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.9.m9.1.1.1.1.1.3.cmml" xref="S3.SS3.7.p3.9.m9.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.9.m9.1c">\text{conv}(V_{c})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.9.m9.1d">conv ( italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math>. But then, the definition of <math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.10.m10.1"><semantics id="S3.SS3.7.p3.10.m10.1a"><msub id="S3.SS3.7.p3.10.m10.1.1" xref="S3.SS3.7.p3.10.m10.1.1.cmml"><mi id="S3.SS3.7.p3.10.m10.1.1.2" xref="S3.SS3.7.p3.10.m10.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.10.m10.1.1.3" xref="S3.SS3.7.p3.10.m10.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.10.m10.1b"><apply id="S3.SS3.7.p3.10.m10.1.1.cmml" xref="S3.SS3.7.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.10.m10.1.1.1.cmml" xref="S3.SS3.7.p3.10.m10.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.10.m10.1.1.2.cmml" xref="S3.SS3.7.p3.10.m10.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.10.m10.1.1.3.cmml" xref="S3.SS3.7.p3.10.m10.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.10.m10.1c">V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.10.m10.1d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> and its openness contradicts that the above integral is zero (there would be a non-zero net push in the direction of the normal vector defining the separating hyperplane). Therefore, since <math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.11.m11.1"><semantics id="S3.SS3.7.p3.11.m11.1a"><msub id="S3.SS3.7.p3.11.m11.1.1" xref="S3.SS3.7.p3.11.m11.1.1.cmml"><mi id="S3.SS3.7.p3.11.m11.1.1.2" xref="S3.SS3.7.p3.11.m11.1.1.2.cmml">V</mi><mi id="S3.SS3.7.p3.11.m11.1.1.3" xref="S3.SS3.7.p3.11.m11.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.11.m11.1b"><apply id="S3.SS3.7.p3.11.m11.1.1.cmml" xref="S3.SS3.7.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.11.m11.1.1.1.cmml" xref="S3.SS3.7.p3.11.m11.1.1">subscript</csymbol><ci id="S3.SS3.7.p3.11.m11.1.1.2.cmml" xref="S3.SS3.7.p3.11.m11.1.1.2">𝑉</ci><ci id="S3.SS3.7.p3.11.m11.1.1.3.cmml" xref="S3.SS3.7.p3.11.m11.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.11.m11.1c">V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.11.m11.1d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> must be empty, the expression <math alttext="\max(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^{p}_{c,-v}),0)" class="ltx_Math" display="inline" id="S3.SS3.7.p3.12.m12.5"><semantics id="S3.SS3.7.p3.12.m12.5a"><mrow id="S3.SS3.7.p3.12.m12.5.5.1" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml"><mi id="S3.SS3.7.p3.12.m12.3.3" xref="S3.SS3.7.p3.12.m12.3.3.cmml">max</mi><mo id="S3.SS3.7.p3.12.m12.5.5.1a" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml">⁡</mo><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml"><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.2" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml">(</mo><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.cmml"><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.cmml"><mfrac id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.cmml"><mn id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.2.cmml">1</mn><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.cmml"><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.2.cmml">d</mi><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.1.cmml">+</mo><mn id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2.cmml">⁢</mo><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.4" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.4.cmml">μ</mi><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2a" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.3.cmml">−</mo><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.cmml"><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.3.cmml">μ</mi><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.2.cmml">⁢</mo><mrow id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.cmml"><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.2" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.2" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.SS3.7.p3.12.m12.2.2.2.2" xref="S3.SS3.7.p3.12.m12.2.2.2.3.cmml"><mi id="S3.SS3.7.p3.12.m12.1.1.1.1" xref="S3.SS3.7.p3.12.m12.1.1.1.1.cmml">c</mi><mo id="S3.SS3.7.p3.12.m12.2.2.2.2.2" xref="S3.SS3.7.p3.12.m12.2.2.2.3.cmml">,</mo><mrow id="S3.SS3.7.p3.12.m12.2.2.2.2.1" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1.cmml"><mo id="S3.SS3.7.p3.12.m12.2.2.2.2.1a" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1.cmml">−</mo><mi id="S3.SS3.7.p3.12.m12.2.2.2.2.1.2" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1.2.cmml">v</mi></mrow></mrow><mi id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.3" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.3" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.3" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml">,</mo><mn id="S3.SS3.7.p3.12.m12.4.4" xref="S3.SS3.7.p3.12.m12.4.4.cmml">0</mn><mo id="S3.SS3.7.p3.12.m12.5.5.1.1.4" stretchy="false" xref="S3.SS3.7.p3.12.m12.5.5.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.12.m12.5b"><apply id="S3.SS3.7.p3.12.m12.5.5.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1"><max id="S3.SS3.7.p3.12.m12.3.3.cmml" xref="S3.SS3.7.p3.12.m12.3.3"></max><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1"><minus id="S3.SS3.7.p3.12.m12.5.5.1.1.1.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.3"></minus><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1"><times id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.2"></times><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3"><divide id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3"></divide><cn id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.2.cmml" type="integer" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.2">1</cn><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3"><plus id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.1"></plus><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.2">𝑑</ci><cn id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.3.cmml" type="integer" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.3.3.3">1</cn></apply></apply><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.4.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.4">𝜇</ci><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2"><times id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.2"></times><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.3">𝜇</ci><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1">subscript</csymbol><apply id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.1.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1">superscript</csymbol><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.2.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.2">ℋ</ci><ci id="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.3.cmml" xref="S3.SS3.7.p3.12.m12.5.5.1.1.1.2.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS3.7.p3.12.m12.2.2.2.3.cmml" xref="S3.SS3.7.p3.12.m12.2.2.2.2"><ci id="S3.SS3.7.p3.12.m12.1.1.1.1.cmml" xref="S3.SS3.7.p3.12.m12.1.1.1.1">𝑐</ci><apply id="S3.SS3.7.p3.12.m12.2.2.2.2.1.cmml" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1"><minus id="S3.SS3.7.p3.12.m12.2.2.2.2.1.1.cmml" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1"></minus><ci id="S3.SS3.7.p3.12.m12.2.2.2.2.1.2.cmml" xref="S3.SS3.7.p3.12.m12.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></apply><cn id="S3.SS3.7.p3.12.m12.4.4.cmml" type="integer" xref="S3.SS3.7.p3.12.m12.4.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.12.m12.5c">\max(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^{p}_{c,-v}),0)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.12.m12.5d">roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , - italic_v end_POSTSUBSCRIPT ) , 0 )</annotation></semantics></math> must be <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.7.p3.13.m13.1"><semantics id="S3.SS3.7.p3.13.m13.1a"><mn id="S3.SS3.7.p3.13.m13.1.1" xref="S3.SS3.7.p3.13.m13.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.13.m13.1b"><cn id="S3.SS3.7.p3.13.m13.1.1.cmml" type="integer" xref="S3.SS3.7.p3.13.m13.1.1">0</cn></annotation-xml></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.SS3.7.p3.14.m14.1"><semantics id="S3.SS3.7.p3.14.m14.1a"><mrow id="S3.SS3.7.p3.14.m14.1.1" xref="S3.SS3.7.p3.14.m14.1.1.cmml"><mi id="S3.SS3.7.p3.14.m14.1.1.2" xref="S3.SS3.7.p3.14.m14.1.1.2.cmml">v</mi><mo id="S3.SS3.7.p3.14.m14.1.1.1" xref="S3.SS3.7.p3.14.m14.1.1.1.cmml">∈</mo><msup id="S3.SS3.7.p3.14.m14.1.1.3" xref="S3.SS3.7.p3.14.m14.1.1.3.cmml"><mi id="S3.SS3.7.p3.14.m14.1.1.3.2" xref="S3.SS3.7.p3.14.m14.1.1.3.2.cmml">S</mi><mrow id="S3.SS3.7.p3.14.m14.1.1.3.3" xref="S3.SS3.7.p3.14.m14.1.1.3.3.cmml"><mi id="S3.SS3.7.p3.14.m14.1.1.3.3.2" xref="S3.SS3.7.p3.14.m14.1.1.3.3.2.cmml">d</mi><mo id="S3.SS3.7.p3.14.m14.1.1.3.3.1" xref="S3.SS3.7.p3.14.m14.1.1.3.3.1.cmml">−</mo><mn id="S3.SS3.7.p3.14.m14.1.1.3.3.3" xref="S3.SS3.7.p3.14.m14.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.14.m14.1b"><apply id="S3.SS3.7.p3.14.m14.1.1.cmml" xref="S3.SS3.7.p3.14.m14.1.1"><in id="S3.SS3.7.p3.14.m14.1.1.1.cmml" xref="S3.SS3.7.p3.14.m14.1.1.1"></in><ci id="S3.SS3.7.p3.14.m14.1.1.2.cmml" xref="S3.SS3.7.p3.14.m14.1.1.2">𝑣</ci><apply id="S3.SS3.7.p3.14.m14.1.1.3.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.7.p3.14.m14.1.1.3.1.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3">superscript</csymbol><ci id="S3.SS3.7.p3.14.m14.1.1.3.2.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3.2">𝑆</ci><apply id="S3.SS3.7.p3.14.m14.1.1.3.3.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3.3"><minus id="S3.SS3.7.p3.14.m14.1.1.3.3.1.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3.3.1"></minus><ci id="S3.SS3.7.p3.14.m14.1.1.3.3.2.cmml" xref="S3.SS3.7.p3.14.m14.1.1.3.3.2">𝑑</ci><cn id="S3.SS3.7.p3.14.m14.1.1.3.3.3.cmml" type="integer" xref="S3.SS3.7.p3.14.m14.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.14.m14.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.14.m14.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.7.p3.15.m15.1"><semantics id="S3.SS3.7.p3.15.m15.1a"><mi id="S3.SS3.7.p3.15.m15.1.1" xref="S3.SS3.7.p3.15.m15.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.7.p3.15.m15.1b"><ci id="S3.SS3.7.p3.15.m15.1.1.cmml" xref="S3.SS3.7.p3.15.m15.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.7.p3.15.m15.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.7.p3.15.m15.1d">italic_c</annotation></semantics></math> is a centerpoint.</p> </div> <div class="ltx_para" id="S3.SS3.8.p4"> <p class="ltx_p" id="S3.SS3.8.p4.8">In the second case, <math alttext="F(c)\neq F_{C}(c)" class="ltx_Math" display="inline" id="S3.SS3.8.p4.1.m1.2"><semantics id="S3.SS3.8.p4.1.m1.2a"><mrow id="S3.SS3.8.p4.1.m1.2.3" xref="S3.SS3.8.p4.1.m1.2.3.cmml"><mrow id="S3.SS3.8.p4.1.m1.2.3.2" xref="S3.SS3.8.p4.1.m1.2.3.2.cmml"><mi id="S3.SS3.8.p4.1.m1.2.3.2.2" xref="S3.SS3.8.p4.1.m1.2.3.2.2.cmml">F</mi><mo id="S3.SS3.8.p4.1.m1.2.3.2.1" xref="S3.SS3.8.p4.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.8.p4.1.m1.2.3.2.3.2" xref="S3.SS3.8.p4.1.m1.2.3.2.cmml"><mo id="S3.SS3.8.p4.1.m1.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.8.p4.1.m1.2.3.2.cmml">(</mo><mi id="S3.SS3.8.p4.1.m1.1.1" xref="S3.SS3.8.p4.1.m1.1.1.cmml">c</mi><mo id="S3.SS3.8.p4.1.m1.2.3.2.3.2.2" stretchy="false" xref="S3.SS3.8.p4.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.8.p4.1.m1.2.3.1" xref="S3.SS3.8.p4.1.m1.2.3.1.cmml">≠</mo><mrow id="S3.SS3.8.p4.1.m1.2.3.3" xref="S3.SS3.8.p4.1.m1.2.3.3.cmml"><msub id="S3.SS3.8.p4.1.m1.2.3.3.2" xref="S3.SS3.8.p4.1.m1.2.3.3.2.cmml"><mi id="S3.SS3.8.p4.1.m1.2.3.3.2.2" xref="S3.SS3.8.p4.1.m1.2.3.3.2.2.cmml">F</mi><mi id="S3.SS3.8.p4.1.m1.2.3.3.2.3" xref="S3.SS3.8.p4.1.m1.2.3.3.2.3.cmml">C</mi></msub><mo id="S3.SS3.8.p4.1.m1.2.3.3.1" xref="S3.SS3.8.p4.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS3.8.p4.1.m1.2.3.3.3.2" xref="S3.SS3.8.p4.1.m1.2.3.3.cmml"><mo id="S3.SS3.8.p4.1.m1.2.3.3.3.2.1" stretchy="false" xref="S3.SS3.8.p4.1.m1.2.3.3.cmml">(</mo><mi id="S3.SS3.8.p4.1.m1.2.2" xref="S3.SS3.8.p4.1.m1.2.2.cmml">c</mi><mo id="S3.SS3.8.p4.1.m1.2.3.3.3.2.2" stretchy="false" xref="S3.SS3.8.p4.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.1.m1.2b"><apply id="S3.SS3.8.p4.1.m1.2.3.cmml" xref="S3.SS3.8.p4.1.m1.2.3"><neq id="S3.SS3.8.p4.1.m1.2.3.1.cmml" xref="S3.SS3.8.p4.1.m1.2.3.1"></neq><apply id="S3.SS3.8.p4.1.m1.2.3.2.cmml" xref="S3.SS3.8.p4.1.m1.2.3.2"><times id="S3.SS3.8.p4.1.m1.2.3.2.1.cmml" xref="S3.SS3.8.p4.1.m1.2.3.2.1"></times><ci id="S3.SS3.8.p4.1.m1.2.3.2.2.cmml" xref="S3.SS3.8.p4.1.m1.2.3.2.2">𝐹</ci><ci id="S3.SS3.8.p4.1.m1.1.1.cmml" xref="S3.SS3.8.p4.1.m1.1.1">𝑐</ci></apply><apply id="S3.SS3.8.p4.1.m1.2.3.3.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3"><times id="S3.SS3.8.p4.1.m1.2.3.3.1.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3.1"></times><apply id="S3.SS3.8.p4.1.m1.2.3.3.2.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS3.8.p4.1.m1.2.3.3.2.1.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3.2">subscript</csymbol><ci id="S3.SS3.8.p4.1.m1.2.3.3.2.2.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3.2.2">𝐹</ci><ci id="S3.SS3.8.p4.1.m1.2.3.3.2.3.cmml" xref="S3.SS3.8.p4.1.m1.2.3.3.2.3">𝐶</ci></apply><ci id="S3.SS3.8.p4.1.m1.2.2.cmml" xref="S3.SS3.8.p4.1.m1.2.2">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.1.m1.2c">F(c)\neq F_{C}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.1.m1.2d">italic_F ( italic_c ) ≠ italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math>, we know that <math alttext="F_{C}" class="ltx_Math" display="inline" id="S3.SS3.8.p4.2.m2.1"><semantics id="S3.SS3.8.p4.2.m2.1a"><msub id="S3.SS3.8.p4.2.m2.1.1" xref="S3.SS3.8.p4.2.m2.1.1.cmml"><mi id="S3.SS3.8.p4.2.m2.1.1.2" xref="S3.SS3.8.p4.2.m2.1.1.2.cmml">F</mi><mi id="S3.SS3.8.p4.2.m2.1.1.3" xref="S3.SS3.8.p4.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.2.m2.1b"><apply id="S3.SS3.8.p4.2.m2.1.1.cmml" xref="S3.SS3.8.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.2.m2.1.1.1.cmml" xref="S3.SS3.8.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.2.m2.1.1.2.cmml" xref="S3.SS3.8.p4.2.m2.1.1.2">𝐹</ci><ci id="S3.SS3.8.p4.2.m2.1.1.3.cmml" xref="S3.SS3.8.p4.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.2.m2.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> “used” the projection to map <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.8.p4.3.m3.1"><semantics id="S3.SS3.8.p4.3.m3.1a"><mi id="S3.SS3.8.p4.3.m3.1.1" xref="S3.SS3.8.p4.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.3.m3.1b"><ci id="S3.SS3.8.p4.3.m3.1.1.cmml" xref="S3.SS3.8.p4.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.3.m3.1d">italic_c</annotation></semantics></math>. We thus know that <math alttext="c=F_{C}(c)" class="ltx_Math" display="inline" id="S3.SS3.8.p4.4.m4.1"><semantics id="S3.SS3.8.p4.4.m4.1a"><mrow id="S3.SS3.8.p4.4.m4.1.2" xref="S3.SS3.8.p4.4.m4.1.2.cmml"><mi id="S3.SS3.8.p4.4.m4.1.2.2" xref="S3.SS3.8.p4.4.m4.1.2.2.cmml">c</mi><mo id="S3.SS3.8.p4.4.m4.1.2.1" xref="S3.SS3.8.p4.4.m4.1.2.1.cmml">=</mo><mrow id="S3.SS3.8.p4.4.m4.1.2.3" xref="S3.SS3.8.p4.4.m4.1.2.3.cmml"><msub id="S3.SS3.8.p4.4.m4.1.2.3.2" xref="S3.SS3.8.p4.4.m4.1.2.3.2.cmml"><mi id="S3.SS3.8.p4.4.m4.1.2.3.2.2" xref="S3.SS3.8.p4.4.m4.1.2.3.2.2.cmml">F</mi><mi id="S3.SS3.8.p4.4.m4.1.2.3.2.3" xref="S3.SS3.8.p4.4.m4.1.2.3.2.3.cmml">C</mi></msub><mo id="S3.SS3.8.p4.4.m4.1.2.3.1" xref="S3.SS3.8.p4.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS3.8.p4.4.m4.1.2.3.3.2" xref="S3.SS3.8.p4.4.m4.1.2.3.cmml"><mo id="S3.SS3.8.p4.4.m4.1.2.3.3.2.1" stretchy="false" xref="S3.SS3.8.p4.4.m4.1.2.3.cmml">(</mo><mi id="S3.SS3.8.p4.4.m4.1.1" xref="S3.SS3.8.p4.4.m4.1.1.cmml">c</mi><mo id="S3.SS3.8.p4.4.m4.1.2.3.3.2.2" stretchy="false" xref="S3.SS3.8.p4.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.4.m4.1b"><apply id="S3.SS3.8.p4.4.m4.1.2.cmml" xref="S3.SS3.8.p4.4.m4.1.2"><eq id="S3.SS3.8.p4.4.m4.1.2.1.cmml" xref="S3.SS3.8.p4.4.m4.1.2.1"></eq><ci id="S3.SS3.8.p4.4.m4.1.2.2.cmml" xref="S3.SS3.8.p4.4.m4.1.2.2">𝑐</ci><apply id="S3.SS3.8.p4.4.m4.1.2.3.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3"><times id="S3.SS3.8.p4.4.m4.1.2.3.1.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3.1"></times><apply id="S3.SS3.8.p4.4.m4.1.2.3.2.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S3.SS3.8.p4.4.m4.1.2.3.2.1.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3.2">subscript</csymbol><ci id="S3.SS3.8.p4.4.m4.1.2.3.2.2.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3.2.2">𝐹</ci><ci id="S3.SS3.8.p4.4.m4.1.2.3.2.3.cmml" xref="S3.SS3.8.p4.4.m4.1.2.3.2.3">𝐶</ci></apply><ci id="S3.SS3.8.p4.4.m4.1.1.cmml" xref="S3.SS3.8.p4.4.m4.1.1">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.4.m4.1c">c=F_{C}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.4.m4.1d">italic_c = italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> lies on the boundary of <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.8.p4.5.m5.1"><semantics id="S3.SS3.8.p4.5.m5.1a"><mi id="S3.SS3.8.p4.5.m5.1.1" xref="S3.SS3.8.p4.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.5.m5.1b"><ci id="S3.SS3.8.p4.5.m5.1.1.cmml" xref="S3.SS3.8.p4.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.5.m5.1d">italic_C</annotation></semantics></math>. Furthermore, since the projection onto <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.8.p4.6.m6.1"><semantics id="S3.SS3.8.p4.6.m6.1a"><mi id="S3.SS3.8.p4.6.m6.1.1" xref="S3.SS3.8.p4.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.6.m6.1b"><ci id="S3.SS3.8.p4.6.m6.1.1.cmml" xref="S3.SS3.8.p4.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.6.m6.1d">italic_C</annotation></semantics></math> is towards the origin, we get <math alttext="F(c)=(1+\varepsilon)c" class="ltx_Math" display="inline" id="S3.SS3.8.p4.7.m7.2"><semantics id="S3.SS3.8.p4.7.m7.2a"><mrow id="S3.SS3.8.p4.7.m7.2.2" xref="S3.SS3.8.p4.7.m7.2.2.cmml"><mrow id="S3.SS3.8.p4.7.m7.2.2.3" xref="S3.SS3.8.p4.7.m7.2.2.3.cmml"><mi id="S3.SS3.8.p4.7.m7.2.2.3.2" xref="S3.SS3.8.p4.7.m7.2.2.3.2.cmml">F</mi><mo id="S3.SS3.8.p4.7.m7.2.2.3.1" xref="S3.SS3.8.p4.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS3.8.p4.7.m7.2.2.3.3.2" xref="S3.SS3.8.p4.7.m7.2.2.3.cmml"><mo id="S3.SS3.8.p4.7.m7.2.2.3.3.2.1" stretchy="false" xref="S3.SS3.8.p4.7.m7.2.2.3.cmml">(</mo><mi id="S3.SS3.8.p4.7.m7.1.1" xref="S3.SS3.8.p4.7.m7.1.1.cmml">c</mi><mo id="S3.SS3.8.p4.7.m7.2.2.3.3.2.2" stretchy="false" xref="S3.SS3.8.p4.7.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS3.8.p4.7.m7.2.2.2" xref="S3.SS3.8.p4.7.m7.2.2.2.cmml">=</mo><mrow id="S3.SS3.8.p4.7.m7.2.2.1" xref="S3.SS3.8.p4.7.m7.2.2.1.cmml"><mrow id="S3.SS3.8.p4.7.m7.2.2.1.1.1" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.cmml"><mo id="S3.SS3.8.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.cmml"><mn id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.2" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.2.cmml">1</mn><mo id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.1" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.1.cmml">+</mo><mi id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.3" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S3.SS3.8.p4.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS3.8.p4.7.m7.2.2.1.2" xref="S3.SS3.8.p4.7.m7.2.2.1.2.cmml">⁢</mo><mi id="S3.SS3.8.p4.7.m7.2.2.1.3" xref="S3.SS3.8.p4.7.m7.2.2.1.3.cmml">c</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.7.m7.2b"><apply id="S3.SS3.8.p4.7.m7.2.2.cmml" xref="S3.SS3.8.p4.7.m7.2.2"><eq id="S3.SS3.8.p4.7.m7.2.2.2.cmml" xref="S3.SS3.8.p4.7.m7.2.2.2"></eq><apply id="S3.SS3.8.p4.7.m7.2.2.3.cmml" xref="S3.SS3.8.p4.7.m7.2.2.3"><times id="S3.SS3.8.p4.7.m7.2.2.3.1.cmml" xref="S3.SS3.8.p4.7.m7.2.2.3.1"></times><ci id="S3.SS3.8.p4.7.m7.2.2.3.2.cmml" xref="S3.SS3.8.p4.7.m7.2.2.3.2">𝐹</ci><ci id="S3.SS3.8.p4.7.m7.1.1.cmml" xref="S3.SS3.8.p4.7.m7.1.1">𝑐</ci></apply><apply id="S3.SS3.8.p4.7.m7.2.2.1.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1"><times id="S3.SS3.8.p4.7.m7.2.2.1.2.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1.2"></times><apply id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1"><plus id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.1"></plus><cn id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.2">1</cn><ci id="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1.1.1.1.3">𝜀</ci></apply><ci id="S3.SS3.8.p4.7.m7.2.2.1.3.cmml" xref="S3.SS3.8.p4.7.m7.2.2.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.7.m7.2c">F(c)=(1+\varepsilon)c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.7.m7.2d">italic_F ( italic_c ) = ( 1 + italic_ε ) italic_c</annotation></semantics></math> for some <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS3.8.p4.8.m8.1"><semantics id="S3.SS3.8.p4.8.m8.1a"><mrow id="S3.SS3.8.p4.8.m8.1.1" xref="S3.SS3.8.p4.8.m8.1.1.cmml"><mi id="S3.SS3.8.p4.8.m8.1.1.2" xref="S3.SS3.8.p4.8.m8.1.1.2.cmml">ε</mi><mo id="S3.SS3.8.p4.8.m8.1.1.1" xref="S3.SS3.8.p4.8.m8.1.1.1.cmml">></mo><mn id="S3.SS3.8.p4.8.m8.1.1.3" xref="S3.SS3.8.p4.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.8.m8.1b"><apply id="S3.SS3.8.p4.8.m8.1.1.cmml" xref="S3.SS3.8.p4.8.m8.1.1"><gt id="S3.SS3.8.p4.8.m8.1.1.1.cmml" xref="S3.SS3.8.p4.8.m8.1.1.1"></gt><ci id="S3.SS3.8.p4.8.m8.1.1.2.cmml" xref="S3.SS3.8.p4.8.m8.1.1.2">𝜀</ci><cn id="S3.SS3.8.p4.8.m8.1.1.3.cmml" type="integer" xref="S3.SS3.8.p4.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.8.m8.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.8.m8.1d">italic_ε > 0</annotation></semantics></math>. Concretely, we must have</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^% {p}_{c,-v}),0\right)dv=\varepsilon c_{i}" class="ltx_Math" display="block" id="S3.Ex10.m1.5"><semantics id="S3.Ex10.m1.5a"><mrow id="S3.Ex10.m1.5.5" xref="S3.Ex10.m1.5.5.cmml"><mrow id="S3.Ex10.m1.5.5.1" xref="S3.Ex10.m1.5.5.1.cmml"><msub id="S3.Ex10.m1.5.5.1.2" xref="S3.Ex10.m1.5.5.1.2.cmml"><mo id="S3.Ex10.m1.5.5.1.2.2" xref="S3.Ex10.m1.5.5.1.2.2.cmml">∫</mo><msup id="S3.Ex10.m1.5.5.1.2.3" xref="S3.Ex10.m1.5.5.1.2.3.cmml"><mi id="S3.Ex10.m1.5.5.1.2.3.2" xref="S3.Ex10.m1.5.5.1.2.3.2.cmml">S</mi><mrow id="S3.Ex10.m1.5.5.1.2.3.3" xref="S3.Ex10.m1.5.5.1.2.3.3.cmml"><mi id="S3.Ex10.m1.5.5.1.2.3.3.2" xref="S3.Ex10.m1.5.5.1.2.3.3.2.cmml">d</mi><mo id="S3.Ex10.m1.5.5.1.2.3.3.1" xref="S3.Ex10.m1.5.5.1.2.3.3.1.cmml">−</mo><mn id="S3.Ex10.m1.5.5.1.2.3.3.3" xref="S3.Ex10.m1.5.5.1.2.3.3.3.cmml">1</mn></mrow></msup></msub><mrow id="S3.Ex10.m1.5.5.1.1" xref="S3.Ex10.m1.5.5.1.1.cmml"><msub id="S3.Ex10.m1.5.5.1.1.3" xref="S3.Ex10.m1.5.5.1.1.3.cmml"><mi id="S3.Ex10.m1.5.5.1.1.3.2" xref="S3.Ex10.m1.5.5.1.1.3.2.cmml">v</mi><mi id="S3.Ex10.m1.5.5.1.1.3.3" xref="S3.Ex10.m1.5.5.1.1.3.3.cmml">i</mi></msub><mo id="S3.Ex10.m1.5.5.1.1.2" lspace="0.167em" xref="S3.Ex10.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.2.cmml"><mi id="S3.Ex10.m1.3.3" xref="S3.Ex10.m1.3.3.cmml">max</mi><mo id="S3.Ex10.m1.5.5.1.1.1.1a" xref="S3.Ex10.m1.5.5.1.1.1.2.cmml">⁡</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.2.cmml"><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.2" xref="S3.Ex10.m1.5.5.1.1.1.2.cmml">(</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.cmml"><mfrac id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.cmml"><mn id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.2.cmml">1</mn><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.cmml"><mi id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.2.cmml">d</mi><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.1" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.1.cmml">+</mo><mn id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.4" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.4.cmml">μ</mi><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.2a" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.3.cmml">−</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.3" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.3.cmml">μ</mi><mo id="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.2" xref="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex10.m1.5.5.1.1.1.1.1.1.2.1.1" 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xref="S3.Ex10.m1.5.5.1.1.4.2">𝑣</ci></apply></apply></apply><apply id="S3.Ex10.m1.5.5.3.cmml" xref="S3.Ex10.m1.5.5.3"><times id="S3.Ex10.m1.5.5.3.1.cmml" xref="S3.Ex10.m1.5.5.3.1"></times><ci id="S3.Ex10.m1.5.5.3.2.cmml" xref="S3.Ex10.m1.5.5.3.2">𝜀</ci><apply id="S3.Ex10.m1.5.5.3.3.cmml" xref="S3.Ex10.m1.5.5.3.3"><csymbol cd="ambiguous" id="S3.Ex10.m1.5.5.3.3.1.cmml" xref="S3.Ex10.m1.5.5.3.3">subscript</csymbol><ci id="S3.Ex10.m1.5.5.3.3.2.cmml" xref="S3.Ex10.m1.5.5.3.3.2">𝑐</ci><ci id="S3.Ex10.m1.5.5.3.3.3.cmml" xref="S3.Ex10.m1.5.5.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.5c">\int_{S^{d-1}}v_{i}\max\left(\frac{1}{d+1}\mu(\mathbb{R}^{d})-\mu(\mathcal{H}^% {p}_{c,-v}),0\right)dv=\varepsilon c_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.5d">∫ start_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) - italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , - italic_v end_POSTSUBSCRIPT ) , 0 ) italic_d italic_v = italic_ε italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.8.p4.20">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S3.SS3.8.p4.9.m1.1"><semantics id="S3.SS3.8.p4.9.m1.1a"><mrow id="S3.SS3.8.p4.9.m1.1.2" xref="S3.SS3.8.p4.9.m1.1.2.cmml"><mi id="S3.SS3.8.p4.9.m1.1.2.2" xref="S3.SS3.8.p4.9.m1.1.2.2.cmml">i</mi><mo id="S3.SS3.8.p4.9.m1.1.2.1" xref="S3.SS3.8.p4.9.m1.1.2.1.cmml">∈</mo><mrow id="S3.SS3.8.p4.9.m1.1.2.3.2" xref="S3.SS3.8.p4.9.m1.1.2.3.1.cmml"><mo id="S3.SS3.8.p4.9.m1.1.2.3.2.1" stretchy="false" xref="S3.SS3.8.p4.9.m1.1.2.3.1.1.cmml">[</mo><mi id="S3.SS3.8.p4.9.m1.1.1" xref="S3.SS3.8.p4.9.m1.1.1.cmml">d</mi><mo id="S3.SS3.8.p4.9.m1.1.2.3.2.2" stretchy="false" xref="S3.SS3.8.p4.9.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.9.m1.1b"><apply id="S3.SS3.8.p4.9.m1.1.2.cmml" xref="S3.SS3.8.p4.9.m1.1.2"><in id="S3.SS3.8.p4.9.m1.1.2.1.cmml" xref="S3.SS3.8.p4.9.m1.1.2.1"></in><ci id="S3.SS3.8.p4.9.m1.1.2.2.cmml" xref="S3.SS3.8.p4.9.m1.1.2.2">𝑖</ci><apply id="S3.SS3.8.p4.9.m1.1.2.3.1.cmml" xref="S3.SS3.8.p4.9.m1.1.2.3.2"><csymbol cd="latexml" id="S3.SS3.8.p4.9.m1.1.2.3.1.1.cmml" xref="S3.SS3.8.p4.9.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS3.8.p4.9.m1.1.1.cmml" xref="S3.SS3.8.p4.9.m1.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.9.m1.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.9.m1.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. Considering the set <math alttext="V_{c}" class="ltx_Math" display="inline" id="S3.SS3.8.p4.10.m2.1"><semantics id="S3.SS3.8.p4.10.m2.1a"><msub id="S3.SS3.8.p4.10.m2.1.1" xref="S3.SS3.8.p4.10.m2.1.1.cmml"><mi id="S3.SS3.8.p4.10.m2.1.1.2" xref="S3.SS3.8.p4.10.m2.1.1.2.cmml">V</mi><mi id="S3.SS3.8.p4.10.m2.1.1.3" xref="S3.SS3.8.p4.10.m2.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.10.m2.1b"><apply id="S3.SS3.8.p4.10.m2.1.1.cmml" xref="S3.SS3.8.p4.10.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.10.m2.1.1.1.cmml" xref="S3.SS3.8.p4.10.m2.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.10.m2.1.1.2.cmml" xref="S3.SS3.8.p4.10.m2.1.1.2">𝑉</ci><ci id="S3.SS3.8.p4.10.m2.1.1.3.cmml" xref="S3.SS3.8.p4.10.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.10.m2.1c">V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.10.m2.1d">italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> again, we can see that there hence must exist some <math alttext="\delta>0" class="ltx_Math" display="inline" id="S3.SS3.8.p4.11.m3.1"><semantics id="S3.SS3.8.p4.11.m3.1a"><mrow id="S3.SS3.8.p4.11.m3.1.1" xref="S3.SS3.8.p4.11.m3.1.1.cmml"><mi id="S3.SS3.8.p4.11.m3.1.1.2" xref="S3.SS3.8.p4.11.m3.1.1.2.cmml">δ</mi><mo id="S3.SS3.8.p4.11.m3.1.1.1" xref="S3.SS3.8.p4.11.m3.1.1.1.cmml">></mo><mn id="S3.SS3.8.p4.11.m3.1.1.3" xref="S3.SS3.8.p4.11.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.11.m3.1b"><apply id="S3.SS3.8.p4.11.m3.1.1.cmml" xref="S3.SS3.8.p4.11.m3.1.1"><gt id="S3.SS3.8.p4.11.m3.1.1.1.cmml" xref="S3.SS3.8.p4.11.m3.1.1.1"></gt><ci id="S3.SS3.8.p4.11.m3.1.1.2.cmml" xref="S3.SS3.8.p4.11.m3.1.1.2">𝛿</ci><cn id="S3.SS3.8.p4.11.m3.1.1.3.cmml" type="integer" xref="S3.SS3.8.p4.11.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.11.m3.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.11.m3.1d">italic_δ > 0</annotation></semantics></math> with <math alttext="\delta c\in\text{conv}(V_{c})" class="ltx_Math" display="inline" id="S3.SS3.8.p4.12.m4.1"><semantics id="S3.SS3.8.p4.12.m4.1a"><mrow id="S3.SS3.8.p4.12.m4.1.1" xref="S3.SS3.8.p4.12.m4.1.1.cmml"><mrow id="S3.SS3.8.p4.12.m4.1.1.3" xref="S3.SS3.8.p4.12.m4.1.1.3.cmml"><mi id="S3.SS3.8.p4.12.m4.1.1.3.2" xref="S3.SS3.8.p4.12.m4.1.1.3.2.cmml">δ</mi><mo id="S3.SS3.8.p4.12.m4.1.1.3.1" xref="S3.SS3.8.p4.12.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.8.p4.12.m4.1.1.3.3" xref="S3.SS3.8.p4.12.m4.1.1.3.3.cmml">c</mi></mrow><mo id="S3.SS3.8.p4.12.m4.1.1.2" xref="S3.SS3.8.p4.12.m4.1.1.2.cmml">∈</mo><mrow id="S3.SS3.8.p4.12.m4.1.1.1" xref="S3.SS3.8.p4.12.m4.1.1.1.cmml"><mtext id="S3.SS3.8.p4.12.m4.1.1.1.3" xref="S3.SS3.8.p4.12.m4.1.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.8.p4.12.m4.1.1.1.2" xref="S3.SS3.8.p4.12.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.8.p4.12.m4.1.1.1.1.1" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS3.8.p4.12.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.2" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.3" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S3.SS3.8.p4.12.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.12.m4.1b"><apply id="S3.SS3.8.p4.12.m4.1.1.cmml" xref="S3.SS3.8.p4.12.m4.1.1"><in id="S3.SS3.8.p4.12.m4.1.1.2.cmml" xref="S3.SS3.8.p4.12.m4.1.1.2"></in><apply id="S3.SS3.8.p4.12.m4.1.1.3.cmml" xref="S3.SS3.8.p4.12.m4.1.1.3"><times id="S3.SS3.8.p4.12.m4.1.1.3.1.cmml" xref="S3.SS3.8.p4.12.m4.1.1.3.1"></times><ci id="S3.SS3.8.p4.12.m4.1.1.3.2.cmml" xref="S3.SS3.8.p4.12.m4.1.1.3.2">𝛿</ci><ci id="S3.SS3.8.p4.12.m4.1.1.3.3.cmml" xref="S3.SS3.8.p4.12.m4.1.1.3.3">𝑐</ci></apply><apply id="S3.SS3.8.p4.12.m4.1.1.1.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1"><times id="S3.SS3.8.p4.12.m4.1.1.1.2.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.2"></times><ci id="S3.SS3.8.p4.12.m4.1.1.1.3a.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.3"><mtext id="S3.SS3.8.p4.12.m4.1.1.1.3.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.3">conv</mtext></ci><apply id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS3.8.p4.12.m4.1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.12.m4.1c">\delta c\in\text{conv}(V_{c})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.12.m4.1d">italic_δ italic_c ∈ conv ( italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math>: otherwise, we could find a hyperplane separating <math alttext="\text{conv}(V_{c})" class="ltx_Math" display="inline" id="S3.SS3.8.p4.13.m5.1"><semantics id="S3.SS3.8.p4.13.m5.1a"><mrow id="S3.SS3.8.p4.13.m5.1.1" xref="S3.SS3.8.p4.13.m5.1.1.cmml"><mtext id="S3.SS3.8.p4.13.m5.1.1.3" xref="S3.SS3.8.p4.13.m5.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.8.p4.13.m5.1.1.2" xref="S3.SS3.8.p4.13.m5.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.8.p4.13.m5.1.1.1.1" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.cmml"><mo id="S3.SS3.8.p4.13.m5.1.1.1.1.2" stretchy="false" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.8.p4.13.m5.1.1.1.1.1" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.cmml"><mi id="S3.SS3.8.p4.13.m5.1.1.1.1.1.2" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.8.p4.13.m5.1.1.1.1.1.3" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S3.SS3.8.p4.13.m5.1.1.1.1.3" stretchy="false" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.13.m5.1b"><apply id="S3.SS3.8.p4.13.m5.1.1.cmml" xref="S3.SS3.8.p4.13.m5.1.1"><times id="S3.SS3.8.p4.13.m5.1.1.2.cmml" xref="S3.SS3.8.p4.13.m5.1.1.2"></times><ci id="S3.SS3.8.p4.13.m5.1.1.3a.cmml" xref="S3.SS3.8.p4.13.m5.1.1.3"><mtext id="S3.SS3.8.p4.13.m5.1.1.3.cmml" xref="S3.SS3.8.p4.13.m5.1.1.3">conv</mtext></ci><apply id="S3.SS3.8.p4.13.m5.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.13.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.13.m5.1.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.13.m5.1.1.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.13.m5.1.1.1.1.1.2.cmml" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.8.p4.13.m5.1.1.1.1.1.3.cmml" xref="S3.SS3.8.p4.13.m5.1.1.1.1.1.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.13.m5.1c">\text{conv}(V_{c})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.13.m5.1d">conv ( italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> from the line segment <math alttext="[0,\frac{c}{\lVert c\rVert_{2}}]" class="ltx_Math" display="inline" id="S3.SS3.8.p4.14.m6.2"><semantics id="S3.SS3.8.p4.14.m6.2a"><mrow id="S3.SS3.8.p4.14.m6.2.3.2" xref="S3.SS3.8.p4.14.m6.2.3.1.cmml"><mo id="S3.SS3.8.p4.14.m6.2.3.2.1" stretchy="false" xref="S3.SS3.8.p4.14.m6.2.3.1.cmml">[</mo><mn id="S3.SS3.8.p4.14.m6.2.2" xref="S3.SS3.8.p4.14.m6.2.2.cmml">0</mn><mo id="S3.SS3.8.p4.14.m6.2.3.2.2" xref="S3.SS3.8.p4.14.m6.2.3.1.cmml">,</mo><mfrac id="S3.SS3.8.p4.14.m6.1.1" xref="S3.SS3.8.p4.14.m6.1.1.cmml"><mi id="S3.SS3.8.p4.14.m6.1.1.3" xref="S3.SS3.8.p4.14.m6.1.1.3.cmml">c</mi><msub id="S3.SS3.8.p4.14.m6.1.1.1" xref="S3.SS3.8.p4.14.m6.1.1.1.cmml"><mrow id="S3.SS3.8.p4.14.m6.1.1.1.3.2" xref="S3.SS3.8.p4.14.m6.1.1.1.3.1.cmml"><mo fence="true" id="S3.SS3.8.p4.14.m6.1.1.1.3.2.1" rspace="0em" xref="S3.SS3.8.p4.14.m6.1.1.1.3.1.1.cmml">∥</mo><mi id="S3.SS3.8.p4.14.m6.1.1.1.1" xref="S3.SS3.8.p4.14.m6.1.1.1.1.cmml">c</mi><mo fence="true" id="S3.SS3.8.p4.14.m6.1.1.1.3.2.2" lspace="0em" xref="S3.SS3.8.p4.14.m6.1.1.1.3.1.1.cmml">∥</mo></mrow><mn id="S3.SS3.8.p4.14.m6.1.1.1.4" xref="S3.SS3.8.p4.14.m6.1.1.1.4.cmml">2</mn></msub></mfrac><mo id="S3.SS3.8.p4.14.m6.2.3.2.3" stretchy="false" xref="S3.SS3.8.p4.14.m6.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.14.m6.2b"><interval closure="closed" id="S3.SS3.8.p4.14.m6.2.3.1.cmml" xref="S3.SS3.8.p4.14.m6.2.3.2"><cn id="S3.SS3.8.p4.14.m6.2.2.cmml" type="integer" xref="S3.SS3.8.p4.14.m6.2.2">0</cn><apply id="S3.SS3.8.p4.14.m6.1.1.cmml" xref="S3.SS3.8.p4.14.m6.1.1"><divide id="S3.SS3.8.p4.14.m6.1.1.2.cmml" xref="S3.SS3.8.p4.14.m6.1.1"></divide><ci id="S3.SS3.8.p4.14.m6.1.1.3.cmml" xref="S3.SS3.8.p4.14.m6.1.1.3">𝑐</ci><apply id="S3.SS3.8.p4.14.m6.1.1.1.cmml" xref="S3.SS3.8.p4.14.m6.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.14.m6.1.1.1.2.cmml" xref="S3.SS3.8.p4.14.m6.1.1.1">subscript</csymbol><apply id="S3.SS3.8.p4.14.m6.1.1.1.3.1.cmml" xref="S3.SS3.8.p4.14.m6.1.1.1.3.2"><csymbol cd="latexml" id="S3.SS3.8.p4.14.m6.1.1.1.3.1.1.cmml" xref="S3.SS3.8.p4.14.m6.1.1.1.3.2.1">delimited-∥∥</csymbol><ci id="S3.SS3.8.p4.14.m6.1.1.1.1.cmml" xref="S3.SS3.8.p4.14.m6.1.1.1.1">𝑐</ci></apply><cn id="S3.SS3.8.p4.14.m6.1.1.1.4.cmml" type="integer" xref="S3.SS3.8.p4.14.m6.1.1.1.4">2</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.14.m6.2c">[0,\frac{c}{\lVert c\rVert_{2}}]</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.14.m6.2d">[ 0 , divide start_ARG italic_c end_ARG start_ARG ∥ italic_c ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ]</annotation></semantics></math>, yielding a contradiction. But since <math alttext="C" class="ltx_Math" display="inline" id="S3.SS3.8.p4.15.m7.1"><semantics id="S3.SS3.8.p4.15.m7.1a"><mi id="S3.SS3.8.p4.15.m7.1.1" xref="S3.SS3.8.p4.15.m7.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.15.m7.1b"><ci id="S3.SS3.8.p4.15.m7.1.1.cmml" xref="S3.SS3.8.p4.15.m7.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.15.m7.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.15.m7.1d">italic_C</annotation></semantics></math> was chosen large enough, we also get <math alttext="-\frac{c}{||c||_{2}}\in V_{c}" class="ltx_Math" display="inline" id="S3.SS3.8.p4.16.m8.1"><semantics id="S3.SS3.8.p4.16.m8.1a"><mrow id="S3.SS3.8.p4.16.m8.1.2" xref="S3.SS3.8.p4.16.m8.1.2.cmml"><mrow id="S3.SS3.8.p4.16.m8.1.2.2" xref="S3.SS3.8.p4.16.m8.1.2.2.cmml"><mo id="S3.SS3.8.p4.16.m8.1.2.2a" xref="S3.SS3.8.p4.16.m8.1.2.2.cmml">−</mo><mfrac id="S3.SS3.8.p4.16.m8.1.1" xref="S3.SS3.8.p4.16.m8.1.1.cmml"><mi id="S3.SS3.8.p4.16.m8.1.1.3" xref="S3.SS3.8.p4.16.m8.1.1.3.cmml">c</mi><msub id="S3.SS3.8.p4.16.m8.1.1.1" xref="S3.SS3.8.p4.16.m8.1.1.1.cmml"><mrow id="S3.SS3.8.p4.16.m8.1.1.1.3.2" xref="S3.SS3.8.p4.16.m8.1.1.1.3.1.cmml"><mo id="S3.SS3.8.p4.16.m8.1.1.1.3.2.1" maxsize="142%" minsize="142%" xref="S3.SS3.8.p4.16.m8.1.1.1.3.1.1.cmml">‖</mo><mi id="S3.SS3.8.p4.16.m8.1.1.1.1" xref="S3.SS3.8.p4.16.m8.1.1.1.1.cmml">c</mi><mo id="S3.SS3.8.p4.16.m8.1.1.1.3.2.2" maxsize="142%" minsize="142%" xref="S3.SS3.8.p4.16.m8.1.1.1.3.1.1.cmml">‖</mo></mrow><mn id="S3.SS3.8.p4.16.m8.1.1.1.4" xref="S3.SS3.8.p4.16.m8.1.1.1.4.cmml">2</mn></msub></mfrac></mrow><mo id="S3.SS3.8.p4.16.m8.1.2.1" xref="S3.SS3.8.p4.16.m8.1.2.1.cmml">∈</mo><msub id="S3.SS3.8.p4.16.m8.1.2.3" xref="S3.SS3.8.p4.16.m8.1.2.3.cmml"><mi id="S3.SS3.8.p4.16.m8.1.2.3.2" xref="S3.SS3.8.p4.16.m8.1.2.3.2.cmml">V</mi><mi id="S3.SS3.8.p4.16.m8.1.2.3.3" xref="S3.SS3.8.p4.16.m8.1.2.3.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.16.m8.1b"><apply id="S3.SS3.8.p4.16.m8.1.2.cmml" xref="S3.SS3.8.p4.16.m8.1.2"><in id="S3.SS3.8.p4.16.m8.1.2.1.cmml" xref="S3.SS3.8.p4.16.m8.1.2.1"></in><apply id="S3.SS3.8.p4.16.m8.1.2.2.cmml" xref="S3.SS3.8.p4.16.m8.1.2.2"><minus id="S3.SS3.8.p4.16.m8.1.2.2.1.cmml" xref="S3.SS3.8.p4.16.m8.1.2.2"></minus><apply id="S3.SS3.8.p4.16.m8.1.1.cmml" xref="S3.SS3.8.p4.16.m8.1.1"><divide id="S3.SS3.8.p4.16.m8.1.1.2.cmml" xref="S3.SS3.8.p4.16.m8.1.1"></divide><ci id="S3.SS3.8.p4.16.m8.1.1.3.cmml" xref="S3.SS3.8.p4.16.m8.1.1.3">𝑐</ci><apply id="S3.SS3.8.p4.16.m8.1.1.1.cmml" xref="S3.SS3.8.p4.16.m8.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.16.m8.1.1.1.2.cmml" xref="S3.SS3.8.p4.16.m8.1.1.1">subscript</csymbol><apply id="S3.SS3.8.p4.16.m8.1.1.1.3.1.cmml" xref="S3.SS3.8.p4.16.m8.1.1.1.3.2"><csymbol cd="latexml" id="S3.SS3.8.p4.16.m8.1.1.1.3.1.1.cmml" xref="S3.SS3.8.p4.16.m8.1.1.1.3.2.1">norm</csymbol><ci id="S3.SS3.8.p4.16.m8.1.1.1.1.cmml" xref="S3.SS3.8.p4.16.m8.1.1.1.1">𝑐</ci></apply><cn id="S3.SS3.8.p4.16.m8.1.1.1.4.cmml" type="integer" xref="S3.SS3.8.p4.16.m8.1.1.1.4">2</cn></apply></apply></apply><apply id="S3.SS3.8.p4.16.m8.1.2.3.cmml" xref="S3.SS3.8.p4.16.m8.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.8.p4.16.m8.1.2.3.1.cmml" xref="S3.SS3.8.p4.16.m8.1.2.3">subscript</csymbol><ci id="S3.SS3.8.p4.16.m8.1.2.3.2.cmml" xref="S3.SS3.8.p4.16.m8.1.2.3.2">𝑉</ci><ci id="S3.SS3.8.p4.16.m8.1.2.3.3.cmml" xref="S3.SS3.8.p4.16.m8.1.2.3.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.16.m8.1c">-\frac{c}{||c||_{2}}\in V_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.16.m8.1d">- divide start_ARG italic_c end_ARG start_ARG | | italic_c | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ∈ italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.8</span></a>. Thus, we see that <math alttext="0\in\text{conv}(V_{c})" class="ltx_Math" display="inline" id="S3.SS3.8.p4.17.m9.1"><semantics id="S3.SS3.8.p4.17.m9.1a"><mrow id="S3.SS3.8.p4.17.m9.1.1" xref="S3.SS3.8.p4.17.m9.1.1.cmml"><mn id="S3.SS3.8.p4.17.m9.1.1.3" xref="S3.SS3.8.p4.17.m9.1.1.3.cmml">0</mn><mo id="S3.SS3.8.p4.17.m9.1.1.2" xref="S3.SS3.8.p4.17.m9.1.1.2.cmml">∈</mo><mrow id="S3.SS3.8.p4.17.m9.1.1.1" xref="S3.SS3.8.p4.17.m9.1.1.1.cmml"><mtext id="S3.SS3.8.p4.17.m9.1.1.1.3" xref="S3.SS3.8.p4.17.m9.1.1.1.3a.cmml">conv</mtext><mo id="S3.SS3.8.p4.17.m9.1.1.1.2" xref="S3.SS3.8.p4.17.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.8.p4.17.m9.1.1.1.1.1" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.cmml"><mo id="S3.SS3.8.p4.17.m9.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.cmml"><mi id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.2" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.2.cmml">V</mi><mi id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.3" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.3.cmml">c</mi></msub><mo id="S3.SS3.8.p4.17.m9.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.17.m9.1b"><apply id="S3.SS3.8.p4.17.m9.1.1.cmml" xref="S3.SS3.8.p4.17.m9.1.1"><in id="S3.SS3.8.p4.17.m9.1.1.2.cmml" xref="S3.SS3.8.p4.17.m9.1.1.2"></in><cn id="S3.SS3.8.p4.17.m9.1.1.3.cmml" type="integer" xref="S3.SS3.8.p4.17.m9.1.1.3">0</cn><apply id="S3.SS3.8.p4.17.m9.1.1.1.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1"><times id="S3.SS3.8.p4.17.m9.1.1.1.2.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.2"></times><ci id="S3.SS3.8.p4.17.m9.1.1.1.3a.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.3"><mtext id="S3.SS3.8.p4.17.m9.1.1.1.3.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.3">conv</mtext></ci><apply id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.1.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.2.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.2">𝑉</ci><ci id="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.3.cmml" xref="S3.SS3.8.p4.17.m9.1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.17.m9.1c">0\in\text{conv}(V_{c})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.17.m9.1d">0 ∈ conv ( italic_V start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math>. This contradicts <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem17" title="Corollary 3.17. ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.17</span></a>, and hence this case cannot occur. We conclude that every fixpoint of <math alttext="F_{C}" class="ltx_Math" display="inline" id="S3.SS3.8.p4.18.m10.1"><semantics id="S3.SS3.8.p4.18.m10.1a"><msub id="S3.SS3.8.p4.18.m10.1.1" xref="S3.SS3.8.p4.18.m10.1.1.cmml"><mi id="S3.SS3.8.p4.18.m10.1.1.2" xref="S3.SS3.8.p4.18.m10.1.1.2.cmml">F</mi><mi id="S3.SS3.8.p4.18.m10.1.1.3" xref="S3.SS3.8.p4.18.m10.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.18.m10.1b"><apply id="S3.SS3.8.p4.18.m10.1.1.cmml" xref="S3.SS3.8.p4.18.m10.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.18.m10.1.1.1.cmml" xref="S3.SS3.8.p4.18.m10.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.18.m10.1.1.2.cmml" xref="S3.SS3.8.p4.18.m10.1.1.2">𝐹</ci><ci id="S3.SS3.8.p4.18.m10.1.1.3.cmml" xref="S3.SS3.8.p4.18.m10.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.18.m10.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.18.m10.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> is a fixpoint of <math alttext="F" class="ltx_Math" display="inline" id="S3.SS3.8.p4.19.m11.1"><semantics id="S3.SS3.8.p4.19.m11.1a"><mi id="S3.SS3.8.p4.19.m11.1.1" xref="S3.SS3.8.p4.19.m11.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.19.m11.1b"><ci id="S3.SS3.8.p4.19.m11.1.1.cmml" xref="S3.SS3.8.p4.19.m11.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.19.m11.1c">F</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.19.m11.1d">italic_F</annotation></semantics></math>, and thus every fixpoint of <math alttext="F_{C}" class="ltx_Math" display="inline" id="S3.SS3.8.p4.20.m12.1"><semantics id="S3.SS3.8.p4.20.m12.1a"><msub id="S3.SS3.8.p4.20.m12.1.1" xref="S3.SS3.8.p4.20.m12.1.1.cmml"><mi id="S3.SS3.8.p4.20.m12.1.1.2" xref="S3.SS3.8.p4.20.m12.1.1.2.cmml">F</mi><mi id="S3.SS3.8.p4.20.m12.1.1.3" xref="S3.SS3.8.p4.20.m12.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.8.p4.20.m12.1b"><apply id="S3.SS3.8.p4.20.m12.1.1.cmml" xref="S3.SS3.8.p4.20.m12.1.1"><csymbol cd="ambiguous" id="S3.SS3.8.p4.20.m12.1.1.1.cmml" xref="S3.SS3.8.p4.20.m12.1.1">subscript</csymbol><ci id="S3.SS3.8.p4.20.m12.1.1.2.cmml" xref="S3.SS3.8.p4.20.m12.1.1.2">𝐹</ci><ci id="S3.SS3.8.p4.20.m12.1.1.3.cmml" xref="S3.SS3.8.p4.20.m12.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.8.p4.20.m12.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.8.p4.20.m12.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> must be a centerpoint. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p10"> <p class="ltx_p" id="S3.SS3.p10.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem12" title="Theorem 3.12 (Euclidean Centerpoint Theorem for Mass Distributions [31]). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.12</span></a> can be adapted to point sets instead of mass distributions (see <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a> below). The proof is not difficult but a bit technical, which is why we defer it to <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A2" title="Appendix B Proof of Theorem 3.18 ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">B</span></a>. The main idea is to put a ball of small radius around each point and to apply <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a>. Letting the radius go to zero, we obtain a sequence of centerpoints. A subsequence of this sequence must converge to a discrete <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.p10.1.m1.1"><semantics id="S3.SS3.p10.1.m1.1a"><msub id="S3.SS3.p10.1.m1.1.1" xref="S3.SS3.p10.1.m1.1.1.cmml"><mi id="S3.SS3.p10.1.m1.1.1.2" mathvariant="normal" xref="S3.SS3.p10.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.p10.1.m1.1.1.3" xref="S3.SS3.p10.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p10.1.m1.1b"><apply id="S3.SS3.p10.1.m1.1.1.cmml" xref="S3.SS3.p10.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p10.1.m1.1.1.1.cmml" xref="S3.SS3.p10.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p10.1.m1.1.1.2.cmml" xref="S3.SS3.p10.1.m1.1.1.2">ℓ</ci><ci id="S3.SS3.p10.1.m1.1.1.3.cmml" xref="S3.SS3.p10.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p10.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p10.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of the point set.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem18"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem18.2.1.1">Theorem 3.18</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem18.3.2"> </span>(<math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem18.1.m1.1"><semantics id="S3.Thmtheorem18.1.m1.1b"><msub id="S3.Thmtheorem18.1.m1.1.1" xref="S3.Thmtheorem18.1.m1.1.1.cmml"><mi id="S3.Thmtheorem18.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmtheorem18.1.m1.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem18.1.m1.1.1.3" xref="S3.Thmtheorem18.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.1.m1.1c"><apply id="S3.Thmtheorem18.1.m1.1.1.cmml" xref="S3.Thmtheorem18.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem18.1.m1.1.1.1.cmml" xref="S3.Thmtheorem18.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem18.1.m1.1.1.2.cmml" xref="S3.Thmtheorem18.1.m1.1.1.2">ℓ</ci><ci id="S3.Thmtheorem18.1.m1.1.1.3.cmml" xref="S3.Thmtheorem18.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Centerpoint Theorem for Finite Point Sets)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem18.4.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem18.p1"> <p class="ltx_p" id="S3.Thmtheorem18.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem18.p1.5.5">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem18.p1.1.1.m1.3"><semantics id="S3.Thmtheorem18.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem18.p1.1.1.m1.3.4" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.cmml"><mi id="S3.Thmtheorem18.p1.1.1.m1.3.4.2" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.1" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem18.p1.1.1.m1.3.4.3" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.cmml"><mrow id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.2" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem18.p1.1.1.m1.1.1" xref="S3.Thmtheorem18.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.2.2" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem18.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem18.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.1" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.2" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem18.p1.1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem18.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.p1.1.1.m1.3b"><apply id="S3.Thmtheorem18.p1.1.1.m1.3.4.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4"><in id="S3.Thmtheorem18.p1.1.1.m1.3.4.1.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.1"></in><ci id="S3.Thmtheorem18.p1.1.1.m1.3.4.2.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3"><union id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.1.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.1.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.2.2"><cn id="S3.Thmtheorem18.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem18.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem18.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.2.2"></infinity></interval><set id="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.1.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.4.3.3.2"><infinity id="S3.Thmtheorem18.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem18.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary, and let <math alttext="P\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem18.p1.2.2.m2.1"><semantics id="S3.Thmtheorem18.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem18.p1.2.2.m2.1.1" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem18.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem18.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S3.Thmtheorem18.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.p1.2.2.m2.1b"><apply id="S3.Thmtheorem18.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1"><subset id="S3.Thmtheorem18.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem18.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.2">𝑃</ci><apply id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem18.p1.2.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem18.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.p1.2.2.m2.1c">P\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.p1.2.2.m2.1d">italic_P ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be a finite set of points. There exists a point <math alttext="c" class="ltx_Math" display="inline" id="S3.Thmtheorem18.p1.3.3.m3.1"><semantics id="S3.Thmtheorem18.p1.3.3.m3.1a"><mi id="S3.Thmtheorem18.p1.3.3.m3.1.1" xref="S3.Thmtheorem18.p1.3.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.p1.3.3.m3.1b"><ci id="S3.Thmtheorem18.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem18.p1.3.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.p1.3.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.p1.3.3.m3.1d">italic_c</annotation></semantics></math> such that <math alttext="|\mathcal{H}^{p}_{c,v}\cap P|\geq\frac{|P|}{d+1}" class="ltx_Math" display="inline" id="S3.Thmtheorem18.p1.4.4.m4.4"><semantics id="S3.Thmtheorem18.p1.4.4.m4.4a"><mrow id="S3.Thmtheorem18.p1.4.4.m4.4.4" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.cmml"><mrow id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.2.cmml"><mo id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.2" stretchy="false" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.2.1.cmml">|</mo><mrow id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.cmml"><msubsup id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.2" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem18.p1.4.4.m4.2.2.2.4" xref="S3.Thmtheorem18.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem18.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem18.p1.4.4.m4.1.1.1.1.cmml">c</mi><mo id="S3.Thmtheorem18.p1.4.4.m4.2.2.2.4.1" xref="S3.Thmtheorem18.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem18.p1.4.4.m4.2.2.2.2" xref="S3.Thmtheorem18.p1.4.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.3" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.1" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.1.cmml">∩</mo><mi id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.3" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.3.cmml">P</mi></mrow><mo id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.3" stretchy="false" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.2.1.cmml">|</mo></mrow><mo id="S3.Thmtheorem18.p1.4.4.m4.4.4.2" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.2.cmml">≥</mo><mfrac id="S3.Thmtheorem18.p1.4.4.m4.3.3" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.cmml"><mrow id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.3" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.2.cmml"><mo id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.3.1" stretchy="false" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.2.1.cmml">|</mo><mi id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.1" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.1.cmml">P</mi><mo id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.3.2" stretchy="false" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.2.1.cmml">|</mo></mrow><mrow id="S3.Thmtheorem18.p1.4.4.m4.3.3.3" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.cmml"><mi id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.2" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.1" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.1.cmml">+</mo><mn id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.3" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.p1.4.4.m4.4b"><apply id="S3.Thmtheorem18.p1.4.4.m4.4.4.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4"><geq id="S3.Thmtheorem18.p1.4.4.m4.4.4.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.2"></geq><apply id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1"><abs id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.2.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.2"></abs><apply id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1"><intersect id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.1"></intersect><apply id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.2">ℋ</ci><ci id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.3.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem18.p1.4.4.m4.2.2.2.3.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.2.2.2.4"><ci id="S3.Thmtheorem18.p1.4.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.1.1.1.1">𝑐</ci><ci id="S3.Thmtheorem18.p1.4.4.m4.2.2.2.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.2.2.2.2">𝑣</ci></list></apply><ci id="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.3.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.4.4.1.1.1.3">𝑃</ci></apply></apply><apply id="S3.Thmtheorem18.p1.4.4.m4.3.3.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3"><divide id="S3.Thmtheorem18.p1.4.4.m4.3.3.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3"></divide><apply id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.3"><abs id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.2.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.3.1"></abs><ci id="S3.Thmtheorem18.p1.4.4.m4.3.3.1.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.1.1">𝑃</ci></apply><apply id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3"><plus id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.1.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.1"></plus><ci id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.2.cmml" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.2">𝑑</ci><cn id="S3.Thmtheorem18.p1.4.4.m4.3.3.3.3.cmml" type="integer" xref="S3.Thmtheorem18.p1.4.4.m4.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.p1.4.4.m4.4c">|\mathcal{H}^{p}_{c,v}\cap P|\geq\frac{|P|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.p1.4.4.m4.4d">| caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ∩ italic_P | ≥ divide start_ARG | italic_P | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem18.p1.5.5.m5.1"><semantics id="S3.Thmtheorem18.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem18.p1.5.5.m5.1.1" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem18.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem18.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem18.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.cmml"><mi id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.1" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem18.p1.5.5.m5.1b"><apply id="S3.Thmtheorem18.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1"><in id="S3.Thmtheorem18.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.1"></in><ci id="S3.Thmtheorem18.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.2">𝑣</ci><apply id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3"><minus id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.1"></minus><ci id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem18.p1.5.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem18.p1.5.5.m5.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem18.p1.5.5.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p11"> <p class="ltx_p" id="S3.SS3.p11.2">Finally, we observe that the centerpoint in both the discrete and continuous setting must lie inside any axis-aligned bounding box of the point set <math alttext="P" class="ltx_Math" display="inline" id="S3.SS3.p11.1.m1.1"><semantics id="S3.SS3.p11.1.m1.1a"><mi id="S3.SS3.p11.1.m1.1.1" xref="S3.SS3.p11.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p11.1.m1.1b"><ci id="S3.SS3.p11.1.m1.1.1.cmml" xref="S3.SS3.p11.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p11.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p11.1.m1.1d">italic_P</annotation></semantics></math> or the support of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p11.2.m2.1"><semantics id="S3.SS3.p11.2.m2.1a"><mi id="S3.SS3.p11.2.m2.1.1" xref="S3.SS3.p11.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p11.2.m2.1b"><ci id="S3.SS3.p11.2.m2.1.1.cmml" xref="S3.SS3.p11.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p11.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p11.2.m2.1d">italic_μ</annotation></semantics></math>, respectively.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem19"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem19.1.1.1">Lemma 3.19</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem19.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem19.p1"> <p class="ltx_p" id="S3.Thmtheorem19.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem19.p1.5.5">Let <math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem19.p1.1.1.m1.1"><semantics id="S3.Thmtheorem19.p1.1.1.m1.1a"><mi id="S3.Thmtheorem19.p1.1.1.m1.1.1" xref="S3.Thmtheorem19.p1.1.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem19.p1.1.1.m1.1b"><ci id="S3.Thmtheorem19.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem19.p1.1.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem19.p1.1.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem19.p1.1.1.m1.1d">italic_B</annotation></semantics></math> be an axis-aligned bounding box that contains the support of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem19.p1.2.2.m2.1"><semantics id="S3.Thmtheorem19.p1.2.2.m2.1a"><mi id="S3.Thmtheorem19.p1.2.2.m2.1.1" xref="S3.Thmtheorem19.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem19.p1.2.2.m2.1b"><ci id="S3.Thmtheorem19.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem19.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem19.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem19.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> (in the case of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a>) or all of <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem19.p1.3.3.m3.1"><semantics id="S3.Thmtheorem19.p1.3.3.m3.1a"><mi id="S3.Thmtheorem19.p1.3.3.m3.1.1" xref="S3.Thmtheorem19.p1.3.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem19.p1.3.3.m3.1b"><ci id="S3.Thmtheorem19.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem19.p1.3.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem19.p1.3.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem19.p1.3.3.m3.1d">italic_P</annotation></semantics></math> (in the case of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a>), respectively. Any <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem19.p1.4.4.m4.1"><semantics id="S3.Thmtheorem19.p1.4.4.m4.1a"><msub id="S3.Thmtheorem19.p1.4.4.m4.1.1" xref="S3.Thmtheorem19.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem19.p1.4.4.m4.1.1.2" mathvariant="normal" xref="S3.Thmtheorem19.p1.4.4.m4.1.1.2.cmml">ℓ</mi><mi id="S3.Thmtheorem19.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem19.p1.4.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem19.p1.4.4.m4.1b"><apply id="S3.Thmtheorem19.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem19.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem19.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem19.p1.4.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem19.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem19.p1.4.4.m4.1.1.2">ℓ</ci><ci id="S3.Thmtheorem19.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem19.p1.4.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem19.p1.4.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem19.p1.4.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint guaranteed by either theorem must lie inside <math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem19.p1.5.5.m5.1"><semantics id="S3.Thmtheorem19.p1.5.5.m5.1a"><mi id="S3.Thmtheorem19.p1.5.5.m5.1.1" xref="S3.Thmtheorem19.p1.5.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem19.p1.5.5.m5.1b"><ci id="S3.Thmtheorem19.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem19.p1.5.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem19.p1.5.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem19.p1.5.5.m5.1d">italic_B</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS3.9"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.9.p1"> <p class="ltx_p" id="S3.SS3.9.p1.13">Towards a contradiction, let <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.9.p1.1.m1.1"><semantics id="S3.SS3.9.p1.1.m1.1a"><mi id="S3.SS3.9.p1.1.m1.1.1" xref="S3.SS3.9.p1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.1.m1.1b"><ci id="S3.SS3.9.p1.1.m1.1.1.cmml" xref="S3.SS3.9.p1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.1.m1.1d">italic_c</annotation></semantics></math> be an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S3.SS3.9.p1.2.m2.1"><semantics id="S3.SS3.9.p1.2.m2.1a"><msub id="S3.SS3.9.p1.2.m2.1.1" xref="S3.SS3.9.p1.2.m2.1.1.cmml"><mi id="S3.SS3.9.p1.2.m2.1.1.2" mathvariant="normal" xref="S3.SS3.9.p1.2.m2.1.1.2.cmml">ℓ</mi><mi id="S3.SS3.9.p1.2.m2.1.1.3" xref="S3.SS3.9.p1.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.2.m2.1b"><apply id="S3.SS3.9.p1.2.m2.1.1.cmml" xref="S3.SS3.9.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.2.m2.1.1.1.cmml" xref="S3.SS3.9.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.9.p1.2.m2.1.1.2.cmml" xref="S3.SS3.9.p1.2.m2.1.1.2">ℓ</ci><ci id="S3.SS3.9.p1.2.m2.1.1.3.cmml" xref="S3.SS3.9.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.9.p1.3.m3.1"><semantics id="S3.SS3.9.p1.3.m3.1a"><mi id="S3.SS3.9.p1.3.m3.1.1" xref="S3.SS3.9.p1.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.3.m3.1b"><ci id="S3.SS3.9.p1.3.m3.1.1.cmml" xref="S3.SS3.9.p1.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.3.m3.1d">italic_μ</annotation></semantics></math> (or <math alttext="P" class="ltx_Math" display="inline" id="S3.SS3.9.p1.4.m4.1"><semantics id="S3.SS3.9.p1.4.m4.1a"><mi id="S3.SS3.9.p1.4.m4.1.1" xref="S3.SS3.9.p1.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.4.m4.1b"><ci id="S3.SS3.9.p1.4.m4.1.1.cmml" xref="S3.SS3.9.p1.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.4.m4.1d">italic_P</annotation></semantics></math>, respectively) not contained in <math alttext="B" class="ltx_Math" display="inline" id="S3.SS3.9.p1.5.m5.1"><semantics id="S3.SS3.9.p1.5.m5.1a"><mi id="S3.SS3.9.p1.5.m5.1.1" xref="S3.SS3.9.p1.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.5.m5.1b"><ci id="S3.SS3.9.p1.5.m5.1.1.cmml" xref="S3.SS3.9.p1.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.5.m5.1d">italic_B</annotation></semantics></math>. Assume without loss of generality that every point <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.9.p1.6.m6.1"><semantics id="S3.SS3.9.p1.6.m6.1a"><mi id="S3.SS3.9.p1.6.m6.1.1" xref="S3.SS3.9.p1.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.6.m6.1b"><ci id="S3.SS3.9.p1.6.m6.1.1.cmml" xref="S3.SS3.9.p1.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.6.m6.1d">italic_x</annotation></semantics></math> in the support of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.9.p1.7.m7.1"><semantics id="S3.SS3.9.p1.7.m7.1a"><mi id="S3.SS3.9.p1.7.m7.1.1" xref="S3.SS3.9.p1.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.7.m7.1b"><ci id="S3.SS3.9.p1.7.m7.1.1.cmml" xref="S3.SS3.9.p1.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.7.m7.1d">italic_μ</annotation></semantics></math> (or in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS3.9.p1.8.m8.1"><semantics id="S3.SS3.9.p1.8.m8.1a"><mi id="S3.SS3.9.p1.8.m8.1.1" xref="S3.SS3.9.p1.8.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.8.m8.1b"><ci id="S3.SS3.9.p1.8.m8.1.1.cmml" xref="S3.SS3.9.p1.8.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.8.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.8.m8.1d">italic_P</annotation></semantics></math>) satisfies <math alttext="x_{1}<c_{1}" class="ltx_Math" display="inline" id="S3.SS3.9.p1.9.m9.1"><semantics id="S3.SS3.9.p1.9.m9.1a"><mrow id="S3.SS3.9.p1.9.m9.1.1" xref="S3.SS3.9.p1.9.m9.1.1.cmml"><msub id="S3.SS3.9.p1.9.m9.1.1.2" xref="S3.SS3.9.p1.9.m9.1.1.2.cmml"><mi id="S3.SS3.9.p1.9.m9.1.1.2.2" xref="S3.SS3.9.p1.9.m9.1.1.2.2.cmml">x</mi><mn id="S3.SS3.9.p1.9.m9.1.1.2.3" xref="S3.SS3.9.p1.9.m9.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS3.9.p1.9.m9.1.1.1" xref="S3.SS3.9.p1.9.m9.1.1.1.cmml"><</mo><msub id="S3.SS3.9.p1.9.m9.1.1.3" xref="S3.SS3.9.p1.9.m9.1.1.3.cmml"><mi id="S3.SS3.9.p1.9.m9.1.1.3.2" xref="S3.SS3.9.p1.9.m9.1.1.3.2.cmml">c</mi><mn id="S3.SS3.9.p1.9.m9.1.1.3.3" xref="S3.SS3.9.p1.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.9.m9.1b"><apply id="S3.SS3.9.p1.9.m9.1.1.cmml" xref="S3.SS3.9.p1.9.m9.1.1"><lt id="S3.SS3.9.p1.9.m9.1.1.1.cmml" xref="S3.SS3.9.p1.9.m9.1.1.1"></lt><apply id="S3.SS3.9.p1.9.m9.1.1.2.cmml" xref="S3.SS3.9.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.9.p1.9.m9.1.1.2.1.cmml" xref="S3.SS3.9.p1.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS3.9.p1.9.m9.1.1.2.2.cmml" xref="S3.SS3.9.p1.9.m9.1.1.2.2">𝑥</ci><cn id="S3.SS3.9.p1.9.m9.1.1.2.3.cmml" type="integer" xref="S3.SS3.9.p1.9.m9.1.1.2.3">1</cn></apply><apply id="S3.SS3.9.p1.9.m9.1.1.3.cmml" xref="S3.SS3.9.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.9.p1.9.m9.1.1.3.1.cmml" xref="S3.SS3.9.p1.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS3.9.p1.9.m9.1.1.3.2.cmml" xref="S3.SS3.9.p1.9.m9.1.1.3.2">𝑐</ci><cn id="S3.SS3.9.p1.9.m9.1.1.3.3.cmml" type="integer" xref="S3.SS3.9.p1.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.9.m9.1c">x_{1}<c_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.9.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem5" title="Lemma 3.5. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.5</span></a>, we have that <math alttext="\mu(\mathcal{H}^{p}_{c,e_{1}})=0" class="ltx_Math" display="inline" id="S3.SS3.9.p1.10.m10.3"><semantics id="S3.SS3.9.p1.10.m10.3a"><mrow id="S3.SS3.9.p1.10.m10.3.3" xref="S3.SS3.9.p1.10.m10.3.3.cmml"><mrow id="S3.SS3.9.p1.10.m10.3.3.1" xref="S3.SS3.9.p1.10.m10.3.3.1.cmml"><mi id="S3.SS3.9.p1.10.m10.3.3.1.3" xref="S3.SS3.9.p1.10.m10.3.3.1.3.cmml">μ</mi><mo id="S3.SS3.9.p1.10.m10.3.3.1.2" xref="S3.SS3.9.p1.10.m10.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS3.9.p1.10.m10.3.3.1.1.1" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.cmml"><mo id="S3.SS3.9.p1.10.m10.3.3.1.1.1.2" stretchy="false" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.cmml">(</mo><msubsup id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.2" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="S3.SS3.9.p1.10.m10.2.2.2.2" xref="S3.SS3.9.p1.10.m10.2.2.2.3.cmml"><mi id="S3.SS3.9.p1.10.m10.1.1.1.1" xref="S3.SS3.9.p1.10.m10.1.1.1.1.cmml">c</mi><mo id="S3.SS3.9.p1.10.m10.2.2.2.2.2" xref="S3.SS3.9.p1.10.m10.2.2.2.3.cmml">,</mo><msub id="S3.SS3.9.p1.10.m10.2.2.2.2.1" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1.cmml"><mi id="S3.SS3.9.p1.10.m10.2.2.2.2.1.2" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1.2.cmml">e</mi><mn id="S3.SS3.9.p1.10.m10.2.2.2.2.1.3" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1.3.cmml">1</mn></msub></mrow><mi id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.3" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S3.SS3.9.p1.10.m10.3.3.1.1.1.3" stretchy="false" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.9.p1.10.m10.3.3.2" xref="S3.SS3.9.p1.10.m10.3.3.2.cmml">=</mo><mn id="S3.SS3.9.p1.10.m10.3.3.3" xref="S3.SS3.9.p1.10.m10.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.10.m10.3b"><apply id="S3.SS3.9.p1.10.m10.3.3.cmml" xref="S3.SS3.9.p1.10.m10.3.3"><eq id="S3.SS3.9.p1.10.m10.3.3.2.cmml" xref="S3.SS3.9.p1.10.m10.3.3.2"></eq><apply id="S3.SS3.9.p1.10.m10.3.3.1.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1"><times id="S3.SS3.9.p1.10.m10.3.3.1.2.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.2"></times><ci id="S3.SS3.9.p1.10.m10.3.3.1.3.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.3">𝜇</ci><apply id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.1.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1">subscript</csymbol><apply id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.1.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1">superscript</csymbol><ci id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.2.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.2">ℋ</ci><ci id="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.3.cmml" xref="S3.SS3.9.p1.10.m10.3.3.1.1.1.1.2.3">𝑝</ci></apply><list id="S3.SS3.9.p1.10.m10.2.2.2.3.cmml" xref="S3.SS3.9.p1.10.m10.2.2.2.2"><ci id="S3.SS3.9.p1.10.m10.1.1.1.1.cmml" xref="S3.SS3.9.p1.10.m10.1.1.1.1">𝑐</ci><apply id="S3.SS3.9.p1.10.m10.2.2.2.2.1.cmml" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.10.m10.2.2.2.2.1.1.cmml" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1">subscript</csymbol><ci id="S3.SS3.9.p1.10.m10.2.2.2.2.1.2.cmml" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1.2">𝑒</ci><cn id="S3.SS3.9.p1.10.m10.2.2.2.2.1.3.cmml" type="integer" xref="S3.SS3.9.p1.10.m10.2.2.2.2.1.3">1</cn></apply></list></apply></apply><cn id="S3.SS3.9.p1.10.m10.3.3.3.cmml" type="integer" xref="S3.SS3.9.p1.10.m10.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.10.m10.3c">\mu(\mathcal{H}^{p}_{c,e_{1}})=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.10.m10.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> (or <math alttext="\mathcal{H}^{p}_{c,e_{1}}\cap P=0" class="ltx_Math" display="inline" id="S3.SS3.9.p1.11.m11.2"><semantics id="S3.SS3.9.p1.11.m11.2a"><mrow id="S3.SS3.9.p1.11.m11.2.3" xref="S3.SS3.9.p1.11.m11.2.3.cmml"><mrow id="S3.SS3.9.p1.11.m11.2.3.2" xref="S3.SS3.9.p1.11.m11.2.3.2.cmml"><msubsup id="S3.SS3.9.p1.11.m11.2.3.2.2" xref="S3.SS3.9.p1.11.m11.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.9.p1.11.m11.2.3.2.2.2.2" xref="S3.SS3.9.p1.11.m11.2.3.2.2.2.2.cmml">ℋ</mi><mrow id="S3.SS3.9.p1.11.m11.2.2.2.2" xref="S3.SS3.9.p1.11.m11.2.2.2.3.cmml"><mi id="S3.SS3.9.p1.11.m11.1.1.1.1" xref="S3.SS3.9.p1.11.m11.1.1.1.1.cmml">c</mi><mo id="S3.SS3.9.p1.11.m11.2.2.2.2.2" xref="S3.SS3.9.p1.11.m11.2.2.2.3.cmml">,</mo><msub id="S3.SS3.9.p1.11.m11.2.2.2.2.1" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1.cmml"><mi id="S3.SS3.9.p1.11.m11.2.2.2.2.1.2" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1.2.cmml">e</mi><mn id="S3.SS3.9.p1.11.m11.2.2.2.2.1.3" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1.3.cmml">1</mn></msub></mrow><mi id="S3.SS3.9.p1.11.m11.2.3.2.2.2.3" xref="S3.SS3.9.p1.11.m11.2.3.2.2.2.3.cmml">p</mi></msubsup><mo id="S3.SS3.9.p1.11.m11.2.3.2.1" xref="S3.SS3.9.p1.11.m11.2.3.2.1.cmml">∩</mo><mi id="S3.SS3.9.p1.11.m11.2.3.2.3" xref="S3.SS3.9.p1.11.m11.2.3.2.3.cmml">P</mi></mrow><mo id="S3.SS3.9.p1.11.m11.2.3.1" xref="S3.SS3.9.p1.11.m11.2.3.1.cmml">=</mo><mn id="S3.SS3.9.p1.11.m11.2.3.3" xref="S3.SS3.9.p1.11.m11.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.11.m11.2b"><apply id="S3.SS3.9.p1.11.m11.2.3.cmml" xref="S3.SS3.9.p1.11.m11.2.3"><eq id="S3.SS3.9.p1.11.m11.2.3.1.cmml" xref="S3.SS3.9.p1.11.m11.2.3.1"></eq><apply id="S3.SS3.9.p1.11.m11.2.3.2.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2"><intersect id="S3.SS3.9.p1.11.m11.2.3.2.1.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.1"></intersect><apply id="S3.SS3.9.p1.11.m11.2.3.2.2.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.9.p1.11.m11.2.3.2.2.1.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2">subscript</csymbol><apply id="S3.SS3.9.p1.11.m11.2.3.2.2.2.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.9.p1.11.m11.2.3.2.2.2.1.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2">superscript</csymbol><ci id="S3.SS3.9.p1.11.m11.2.3.2.2.2.2.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2.2.2">ℋ</ci><ci id="S3.SS3.9.p1.11.m11.2.3.2.2.2.3.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.2.2.3">𝑝</ci></apply><list id="S3.SS3.9.p1.11.m11.2.2.2.3.cmml" xref="S3.SS3.9.p1.11.m11.2.2.2.2"><ci id="S3.SS3.9.p1.11.m11.1.1.1.1.cmml" xref="S3.SS3.9.p1.11.m11.1.1.1.1">𝑐</ci><apply id="S3.SS3.9.p1.11.m11.2.2.2.2.1.cmml" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.11.m11.2.2.2.2.1.1.cmml" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1">subscript</csymbol><ci id="S3.SS3.9.p1.11.m11.2.2.2.2.1.2.cmml" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1.2">𝑒</ci><cn id="S3.SS3.9.p1.11.m11.2.2.2.2.1.3.cmml" type="integer" xref="S3.SS3.9.p1.11.m11.2.2.2.2.1.3">1</cn></apply></list></apply><ci id="S3.SS3.9.p1.11.m11.2.3.2.3.cmml" xref="S3.SS3.9.p1.11.m11.2.3.2.3">𝑃</ci></apply><cn id="S3.SS3.9.p1.11.m11.2.3.3.cmml" type="integer" xref="S3.SS3.9.p1.11.m11.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.11.m11.2c">\mathcal{H}^{p}_{c,e_{1}}\cap P=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.11.m11.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ italic_P = 0</annotation></semantics></math>), where <math alttext="e_{1}" class="ltx_Math" display="inline" id="S3.SS3.9.p1.12.m12.1"><semantics id="S3.SS3.9.p1.12.m12.1a"><msub id="S3.SS3.9.p1.12.m12.1.1" xref="S3.SS3.9.p1.12.m12.1.1.cmml"><mi id="S3.SS3.9.p1.12.m12.1.1.2" xref="S3.SS3.9.p1.12.m12.1.1.2.cmml">e</mi><mn id="S3.SS3.9.p1.12.m12.1.1.3" xref="S3.SS3.9.p1.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.12.m12.1b"><apply id="S3.SS3.9.p1.12.m12.1.1.cmml" xref="S3.SS3.9.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS3.9.p1.12.m12.1.1.1.cmml" xref="S3.SS3.9.p1.12.m12.1.1">subscript</csymbol><ci id="S3.SS3.9.p1.12.m12.1.1.2.cmml" xref="S3.SS3.9.p1.12.m12.1.1.2">𝑒</ci><cn id="S3.SS3.9.p1.12.m12.1.1.3.cmml" type="integer" xref="S3.SS3.9.p1.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.12.m12.1c">e_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.12.m12.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is the first standard unit vector. This contradicts the centerpoint property of <math alttext="c" class="ltx_Math" display="inline" id="S3.SS3.9.p1.13.m13.1"><semantics id="S3.SS3.9.p1.13.m13.1a"><mi id="S3.SS3.9.p1.13.m13.1.1" xref="S3.SS3.9.p1.13.m13.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.9.p1.13.m13.1b"><ci id="S3.SS3.9.p1.13.m13.1.1.cmml" xref="S3.SS3.9.p1.13.m13.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.9.p1.13.m13.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.9.p1.13.m13.1d">italic_c</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4 </span>Tightness of Centerpoint Theorems</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.2">We want to remark that the fraction <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.1"><semantics id="S3.SS4.p1.1.m1.1a"><mfrac id="S3.SS4.p1.1.m1.1.1" xref="S3.SS4.p1.1.m1.1.1.cmml"><mn id="S3.SS4.p1.1.m1.1.1.2" xref="S3.SS4.p1.1.m1.1.1.2.cmml">1</mn><mrow id="S3.SS4.p1.1.m1.1.1.3" xref="S3.SS4.p1.1.m1.1.1.3.cmml"><mi id="S3.SS4.p1.1.m1.1.1.3.2" xref="S3.SS4.p1.1.m1.1.1.3.2.cmml">d</mi><mo id="S3.SS4.p1.1.m1.1.1.3.1" xref="S3.SS4.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S3.SS4.p1.1.m1.1.1.3.3" xref="S3.SS4.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.1.m1.1b"><apply id="S3.SS4.p1.1.m1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1"><divide id="S3.SS4.p1.1.m1.1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1"></divide><cn id="S3.SS4.p1.1.m1.1.1.2.cmml" type="integer" xref="S3.SS4.p1.1.m1.1.1.2">1</cn><apply id="S3.SS4.p1.1.m1.1.1.3.cmml" xref="S3.SS4.p1.1.m1.1.1.3"><plus id="S3.SS4.p1.1.m1.1.1.3.1.cmml" xref="S3.SS4.p1.1.m1.1.1.3.1"></plus><ci id="S3.SS4.p1.1.m1.1.1.3.2.cmml" xref="S3.SS4.p1.1.m1.1.1.3.2">𝑑</ci><cn id="S3.SS4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.1.m1.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.1.m1.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math> in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a> is tight for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.3"><semantics id="S3.SS4.p1.2.m2.3a"><mrow id="S3.SS4.p1.2.m2.3.4" xref="S3.SS4.p1.2.m2.3.4.cmml"><mi id="S3.SS4.p1.2.m2.3.4.2" xref="S3.SS4.p1.2.m2.3.4.2.cmml">p</mi><mo id="S3.SS4.p1.2.m2.3.4.1" xref="S3.SS4.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S3.SS4.p1.2.m2.3.4.3" xref="S3.SS4.p1.2.m2.3.4.3.cmml"><mrow id="S3.SS4.p1.2.m2.3.4.3.2.2" xref="S3.SS4.p1.2.m2.3.4.3.2.1.cmml"><mo id="S3.SS4.p1.2.m2.3.4.3.2.2.1" stretchy="false" xref="S3.SS4.p1.2.m2.3.4.3.2.1.cmml">[</mo><mn id="S3.SS4.p1.2.m2.1.1" xref="S3.SS4.p1.2.m2.1.1.cmml">1</mn><mo id="S3.SS4.p1.2.m2.3.4.3.2.2.2" xref="S3.SS4.p1.2.m2.3.4.3.2.1.cmml">,</mo><mi id="S3.SS4.p1.2.m2.2.2" mathvariant="normal" xref="S3.SS4.p1.2.m2.2.2.cmml">∞</mi><mo id="S3.SS4.p1.2.m2.3.4.3.2.2.3" stretchy="false" xref="S3.SS4.p1.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.SS4.p1.2.m2.3.4.3.1" xref="S3.SS4.p1.2.m2.3.4.3.1.cmml">∪</mo><mrow id="S3.SS4.p1.2.m2.3.4.3.3.2" xref="S3.SS4.p1.2.m2.3.4.3.3.1.cmml"><mo id="S3.SS4.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="S3.SS4.p1.2.m2.3.4.3.3.1.cmml">{</mo><mi id="S3.SS4.p1.2.m2.3.3" mathvariant="normal" xref="S3.SS4.p1.2.m2.3.3.cmml">∞</mi><mo id="S3.SS4.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="S3.SS4.p1.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.3b"><apply id="S3.SS4.p1.2.m2.3.4.cmml" xref="S3.SS4.p1.2.m2.3.4"><in id="S3.SS4.p1.2.m2.3.4.1.cmml" xref="S3.SS4.p1.2.m2.3.4.1"></in><ci id="S3.SS4.p1.2.m2.3.4.2.cmml" xref="S3.SS4.p1.2.m2.3.4.2">𝑝</ci><apply id="S3.SS4.p1.2.m2.3.4.3.cmml" xref="S3.SS4.p1.2.m2.3.4.3"><union id="S3.SS4.p1.2.m2.3.4.3.1.cmml" xref="S3.SS4.p1.2.m2.3.4.3.1"></union><interval closure="closed-open" id="S3.SS4.p1.2.m2.3.4.3.2.1.cmml" xref="S3.SS4.p1.2.m2.3.4.3.2.2"><cn id="S3.SS4.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS4.p1.2.m2.1.1">1</cn><infinity id="S3.SS4.p1.2.m2.2.2.cmml" xref="S3.SS4.p1.2.m2.2.2"></infinity></interval><set id="S3.SS4.p1.2.m2.3.4.3.3.1.cmml" xref="S3.SS4.p1.2.m2.3.4.3.3.2"><infinity id="S3.SS4.p1.2.m2.3.3.cmml" xref="S3.SS4.p1.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. To prove this, we use a construction that has also been used to prove tightness of the classical Euclidean centerpoint theorem. We will restrict ourselves to the discrete setting of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a>, but tightness of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a> follows as well because any better bound for mass distributions could be used to get a better bound for point sets by following the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A2" title="Appendix B Proof of Theorem 3.18 ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">B</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem20"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem20.1.1.1">Lemma 3.20</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem20.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem20.p1"> <p class="ltx_p" id="S3.Thmtheorem20.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem20.p1.6.6">For every <math alttext="d\in\mathbb{N}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.1.1.m1.1"><semantics id="S3.Thmtheorem20.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem20.p1.1.1.m1.1.1" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem20.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.2.cmml">d</mi><mo id="S3.Thmtheorem20.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem20.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.1.1.m1.1b"><apply id="S3.Thmtheorem20.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem20.p1.1.1.m1.1.1"><in id="S3.Thmtheorem20.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.1"></in><ci id="S3.Thmtheorem20.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.2">𝑑</ci><ci id="S3.Thmtheorem20.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem20.p1.1.1.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.1.1.m1.1c">d\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.1.1.m1.1d">italic_d ∈ blackboard_N</annotation></semantics></math>, there exists a point set <math alttext="P_{d}\subseteq[0,1]^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.2.2.m2.2"><semantics id="S3.Thmtheorem20.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem20.p1.2.2.m2.2.3" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.cmml"><msub id="S3.Thmtheorem20.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2.cmml"><mi id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.2" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2.2.cmml">P</mi><mi id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.3" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2.3.cmml">d</mi></msub><mo id="S3.Thmtheorem20.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.1.cmml">⊆</mo><msup id="S3.Thmtheorem20.p1.2.2.m2.2.3.3" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.cmml"><mrow id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.1.cmml"><mo id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.2.1" stretchy="false" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem20.p1.2.2.m2.1.1" xref="S3.Thmtheorem20.p1.2.2.m2.1.1.cmml">0</mn><mo id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.2.2" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.1.cmml">,</mo><mn id="S3.Thmtheorem20.p1.2.2.m2.2.2" xref="S3.Thmtheorem20.p1.2.2.m2.2.2.cmml">1</mn><mo id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.2.3" stretchy="false" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.1.cmml">]</mo></mrow><mi id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.3" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.2.2.m2.2b"><apply id="S3.Thmtheorem20.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3"><subset id="S3.Thmtheorem20.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.1"></subset><apply id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.1.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.2.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2.2">𝑃</ci><ci id="S3.Thmtheorem20.p1.2.2.m2.2.3.2.3.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.2.3">𝑑</ci></apply><apply id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3">superscript</csymbol><interval closure="closed" id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.1.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.2.2"><cn id="S3.Thmtheorem20.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem20.p1.2.2.m2.1.1">0</cn><cn id="S3.Thmtheorem20.p1.2.2.m2.2.2.cmml" type="integer" xref="S3.Thmtheorem20.p1.2.2.m2.2.2">1</cn></interval><ci id="S3.Thmtheorem20.p1.2.2.m2.2.3.3.3.cmml" xref="S3.Thmtheorem20.p1.2.2.m2.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.2.2.m2.2c">P_{d}\subseteq[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.2.2.m2.2d">italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, such that for every <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.3.3.m3.3"><semantics id="S3.Thmtheorem20.p1.3.3.m3.3a"><mrow id="S3.Thmtheorem20.p1.3.3.m3.3.4" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.cmml"><mi id="S3.Thmtheorem20.p1.3.3.m3.3.4.2" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.2.cmml">p</mi><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.1" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.1.cmml">∈</mo><mrow id="S3.Thmtheorem20.p1.3.3.m3.3.4.3" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.cmml"><mrow id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.2" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.1.cmml"><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.2.1" stretchy="false" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.1.cmml">[</mo><mn id="S3.Thmtheorem20.p1.3.3.m3.1.1" xref="S3.Thmtheorem20.p1.3.3.m3.1.1.cmml">1</mn><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.2.2" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.1.cmml">,</mo><mi id="S3.Thmtheorem20.p1.3.3.m3.2.2" mathvariant="normal" xref="S3.Thmtheorem20.p1.3.3.m3.2.2.cmml">∞</mi><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.2.3" stretchy="false" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.1" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.1.cmml">∪</mo><mrow id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.2" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.1.cmml"><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem20.p1.3.3.m3.3.3" mathvariant="normal" xref="S3.Thmtheorem20.p1.3.3.m3.3.3.cmml">∞</mi><mo id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.3.3.m3.3b"><apply id="S3.Thmtheorem20.p1.3.3.m3.3.4.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4"><in id="S3.Thmtheorem20.p1.3.3.m3.3.4.1.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.1"></in><ci id="S3.Thmtheorem20.p1.3.3.m3.3.4.2.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.2">𝑝</ci><apply id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3"><union id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.1.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.1"></union><interval closure="closed-open" id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.1.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.2.2"><cn id="S3.Thmtheorem20.p1.3.3.m3.1.1.cmml" type="integer" xref="S3.Thmtheorem20.p1.3.3.m3.1.1">1</cn><infinity id="S3.Thmtheorem20.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.2.2"></infinity></interval><set id="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.1.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.4.3.3.2"><infinity id="S3.Thmtheorem20.p1.3.3.m3.3.3.cmml" xref="S3.Thmtheorem20.p1.3.3.m3.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.3.3.m3.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.3.3.m3.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, every point <math alttext="c\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.4.4.m4.1"><semantics id="S3.Thmtheorem20.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem20.p1.4.4.m4.1.1" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem20.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.2.cmml">c</mi><mo id="S3.Thmtheorem20.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem20.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3.cmml"><mi id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.4.4.m4.1b"><apply id="S3.Thmtheorem20.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1"><in id="S3.Thmtheorem20.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem20.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.2">𝑐</ci><apply id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem20.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem20.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.4.4.m4.1c">c\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.4.4.m4.1d">italic_c ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> has a direction <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.5.5.m5.1"><semantics id="S3.Thmtheorem20.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem20.p1.5.5.m5.1.1" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem20.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem20.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem20.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.2.cmml">S</mi><mrow id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.cmml"><mi id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.1" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.1.cmml">−</mo><mn id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.5.5.m5.1b"><apply id="S3.Thmtheorem20.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1"><in id="S3.Thmtheorem20.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.1"></in><ci id="S3.Thmtheorem20.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.2">𝑣</ci><apply id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.2">𝑆</ci><apply id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3"><minus id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.1"></minus><ci id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.2">𝑑</ci><cn id="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.Thmtheorem20.p1.5.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.5.5.m5.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.5.5.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|\mathcal{H}^{p}_{c,v}\cap P_{d}|\leq\frac{|P_{d}|}{d+1}" class="ltx_Math" display="inline" id="S3.Thmtheorem20.p1.6.6.m6.4"><semantics id="S3.Thmtheorem20.p1.6.6.m6.4a"><mrow id="S3.Thmtheorem20.p1.6.6.m6.4.4" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.cmml"><mrow id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.2.cmml"><mo id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.2" stretchy="false" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.2.1.cmml">|</mo><mrow id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.cmml"><msubsup id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.2" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.2.cmml">ℋ</mi><mrow id="S3.Thmtheorem20.p1.6.6.m6.2.2.2.4" xref="S3.Thmtheorem20.p1.6.6.m6.2.2.2.3.cmml"><mi id="S3.Thmtheorem20.p1.6.6.m6.1.1.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.1.1.1.1.cmml">c</mi><mo id="S3.Thmtheorem20.p1.6.6.m6.2.2.2.4.1" xref="S3.Thmtheorem20.p1.6.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.Thmtheorem20.p1.6.6.m6.2.2.2.2" xref="S3.Thmtheorem20.p1.6.6.m6.2.2.2.2.cmml">v</mi></mrow><mi id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.3" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.3.cmml">p</mi></msubsup><mo id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.1.cmml">∩</mo><msub id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.cmml"><mi id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.2" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.2.cmml">P</mi><mi id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.3" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.3" stretchy="false" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.2.1.cmml">|</mo></mrow><mo id="S3.Thmtheorem20.p1.6.6.m6.4.4.2" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.2.cmml">≤</mo><mfrac id="S3.Thmtheorem20.p1.6.6.m6.3.3" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.cmml"><mrow id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.2.cmml"><mo id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.2" stretchy="false" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.2.1.cmml">|</mo><msub id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.cmml"><mi id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.2" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.2.cmml">P</mi><mi id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.3" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.3.cmml">d</mi></msub><mo id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.3" stretchy="false" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.2.1.cmml">|</mo></mrow><mrow id="S3.Thmtheorem20.p1.6.6.m6.3.3.3" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.cmml"><mi id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.2" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.2.cmml">d</mi><mo id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.1" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.1.cmml">+</mo><mn id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.3" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem20.p1.6.6.m6.4b"><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4"><leq id="S3.Thmtheorem20.p1.6.6.m6.4.4.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.2"></leq><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1"><abs id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.2.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.2"></abs><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1"><intersect id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.1"></intersect><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2">subscript</csymbol><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.2">ℋ</ci><ci id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.2.2.3">𝑝</ci></apply><list id="S3.Thmtheorem20.p1.6.6.m6.2.2.2.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.2.2.2.4"><ci id="S3.Thmtheorem20.p1.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.1.1.1.1">𝑐</ci><ci id="S3.Thmtheorem20.p1.6.6.m6.2.2.2.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.2.2.2.2">𝑣</ci></list></apply><apply id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.2">𝑃</ci><ci id="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.4.4.1.1.1.3.3">𝑑</ci></apply></apply></apply><apply id="S3.Thmtheorem20.p1.6.6.m6.3.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3"><divide id="S3.Thmtheorem20.p1.6.6.m6.3.3.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3"></divide><apply id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1"><abs id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.2.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.2"></abs><apply id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.2">𝑃</ci><ci id="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3"><plus id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.1.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.1"></plus><ci id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.2.cmml" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.2">𝑑</ci><cn id="S3.Thmtheorem20.p1.6.6.m6.3.3.3.3.cmml" type="integer" xref="S3.Thmtheorem20.p1.6.6.m6.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem20.p1.6.6.m6.4c">|\mathcal{H}^{p}_{c,v}\cap P_{d}|\leq\frac{|P_{d}|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem20.p1.6.6.m6.4d">| caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ∩ italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT | ≤ divide start_ARG | italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS4.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS4.1.p1"> <p class="ltx_p" id="S3.SS4.1.p1.10">Let <math alttext="P_{d}=\{0,e_{1},\ldots,e_{d}\}\subseteq[0,1]^{d}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.1.m1.6"><semantics id="S3.SS4.1.p1.1.m1.6a"><mrow id="S3.SS4.1.p1.1.m1.6.6" xref="S3.SS4.1.p1.1.m1.6.6.cmml"><msub id="S3.SS4.1.p1.1.m1.6.6.4" xref="S3.SS4.1.p1.1.m1.6.6.4.cmml"><mi id="S3.SS4.1.p1.1.m1.6.6.4.2" xref="S3.SS4.1.p1.1.m1.6.6.4.2.cmml">P</mi><mi id="S3.SS4.1.p1.1.m1.6.6.4.3" xref="S3.SS4.1.p1.1.m1.6.6.4.3.cmml">d</mi></msub><mo id="S3.SS4.1.p1.1.m1.6.6.5" xref="S3.SS4.1.p1.1.m1.6.6.5.cmml">=</mo><mrow id="S3.SS4.1.p1.1.m1.6.6.2.2" xref="S3.SS4.1.p1.1.m1.6.6.2.3.cmml"><mo 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id="S3.SS4.1.p1.1.m1.6.6.2.2.7" stretchy="false" xref="S3.SS4.1.p1.1.m1.6.6.2.3.cmml">}</mo></mrow><mo id="S3.SS4.1.p1.1.m1.6.6.6" xref="S3.SS4.1.p1.1.m1.6.6.6.cmml">⊆</mo><msup id="S3.SS4.1.p1.1.m1.6.6.7" xref="S3.SS4.1.p1.1.m1.6.6.7.cmml"><mrow id="S3.SS4.1.p1.1.m1.6.6.7.2.2" xref="S3.SS4.1.p1.1.m1.6.6.7.2.1.cmml"><mo id="S3.SS4.1.p1.1.m1.6.6.7.2.2.1" stretchy="false" xref="S3.SS4.1.p1.1.m1.6.6.7.2.1.cmml">[</mo><mn id="S3.SS4.1.p1.1.m1.3.3" xref="S3.SS4.1.p1.1.m1.3.3.cmml">0</mn><mo id="S3.SS4.1.p1.1.m1.6.6.7.2.2.2" xref="S3.SS4.1.p1.1.m1.6.6.7.2.1.cmml">,</mo><mn id="S3.SS4.1.p1.1.m1.4.4" xref="S3.SS4.1.p1.1.m1.4.4.cmml">1</mn><mo id="S3.SS4.1.p1.1.m1.6.6.7.2.2.3" stretchy="false" xref="S3.SS4.1.p1.1.m1.6.6.7.2.1.cmml">]</mo></mrow><mi id="S3.SS4.1.p1.1.m1.6.6.7.3" xref="S3.SS4.1.p1.1.m1.6.6.7.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.1.m1.6b"><apply id="S3.SS4.1.p1.1.m1.6.6.cmml" xref="S3.SS4.1.p1.1.m1.6.6"><and id="S3.SS4.1.p1.1.m1.6.6a.cmml" xref="S3.SS4.1.p1.1.m1.6.6"></and><apply id="S3.SS4.1.p1.1.m1.6.6b.cmml" xref="S3.SS4.1.p1.1.m1.6.6"><eq id="S3.SS4.1.p1.1.m1.6.6.5.cmml" xref="S3.SS4.1.p1.1.m1.6.6.5"></eq><apply id="S3.SS4.1.p1.1.m1.6.6.4.cmml" xref="S3.SS4.1.p1.1.m1.6.6.4"><csymbol cd="ambiguous" id="S3.SS4.1.p1.1.m1.6.6.4.1.cmml" xref="S3.SS4.1.p1.1.m1.6.6.4">subscript</csymbol><ci id="S3.SS4.1.p1.1.m1.6.6.4.2.cmml" xref="S3.SS4.1.p1.1.m1.6.6.4.2">𝑃</ci><ci id="S3.SS4.1.p1.1.m1.6.6.4.3.cmml" xref="S3.SS4.1.p1.1.m1.6.6.4.3">𝑑</ci></apply><set id="S3.SS4.1.p1.1.m1.6.6.2.3.cmml" xref="S3.SS4.1.p1.1.m1.6.6.2.2"><cn id="S3.SS4.1.p1.1.m1.1.1.cmml" type="integer" xref="S3.SS4.1.p1.1.m1.1.1">0</cn><apply id="S3.SS4.1.p1.1.m1.5.5.1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.1.m1.5.5.1.1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.5.5.1.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.1.m1.5.5.1.1.1.2.cmml" xref="S3.SS4.1.p1.1.m1.5.5.1.1.1.2">𝑒</ci><cn id="S3.SS4.1.p1.1.m1.5.5.1.1.1.3.cmml" type="integer" xref="S3.SS4.1.p1.1.m1.5.5.1.1.1.3">1</cn></apply><ci id="S3.SS4.1.p1.1.m1.2.2.cmml" xref="S3.SS4.1.p1.1.m1.2.2">…</ci><apply id="S3.SS4.1.p1.1.m1.6.6.2.2.2.cmml" xref="S3.SS4.1.p1.1.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.1.m1.6.6.2.2.2.1.cmml" xref="S3.SS4.1.p1.1.m1.6.6.2.2.2">subscript</csymbol><ci id="S3.SS4.1.p1.1.m1.6.6.2.2.2.2.cmml" xref="S3.SS4.1.p1.1.m1.6.6.2.2.2.2">𝑒</ci><ci id="S3.SS4.1.p1.1.m1.6.6.2.2.2.3.cmml" xref="S3.SS4.1.p1.1.m1.6.6.2.2.2.3">𝑑</ci></apply></set></apply><apply id="S3.SS4.1.p1.1.m1.6.6c.cmml" xref="S3.SS4.1.p1.1.m1.6.6"><subset id="S3.SS4.1.p1.1.m1.6.6.6.cmml" xref="S3.SS4.1.p1.1.m1.6.6.6"></subset><share href="https://arxiv.org/html/2503.16089v1#S3.SS4.1.p1.1.m1.6.6.2.cmml" id="S3.SS4.1.p1.1.m1.6.6d.cmml" xref="S3.SS4.1.p1.1.m1.6.6"></share><apply id="S3.SS4.1.p1.1.m1.6.6.7.cmml" xref="S3.SS4.1.p1.1.m1.6.6.7"><csymbol cd="ambiguous" id="S3.SS4.1.p1.1.m1.6.6.7.1.cmml" xref="S3.SS4.1.p1.1.m1.6.6.7">superscript</csymbol><interval closure="closed" id="S3.SS4.1.p1.1.m1.6.6.7.2.1.cmml" xref="S3.SS4.1.p1.1.m1.6.6.7.2.2"><cn id="S3.SS4.1.p1.1.m1.3.3.cmml" type="integer" xref="S3.SS4.1.p1.1.m1.3.3">0</cn><cn id="S3.SS4.1.p1.1.m1.4.4.cmml" type="integer" xref="S3.SS4.1.p1.1.m1.4.4">1</cn></interval><ci id="S3.SS4.1.p1.1.m1.6.6.7.3.cmml" xref="S3.SS4.1.p1.1.m1.6.6.7.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.1.m1.6c">P_{d}=\{0,e_{1},\ldots,e_{d}\}\subseteq[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.1.m1.6d">italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = { 0 , italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT } ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="e_{i}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.2.m2.1"><semantics id="S3.SS4.1.p1.2.m2.1a"><msub id="S3.SS4.1.p1.2.m2.1.1" xref="S3.SS4.1.p1.2.m2.1.1.cmml"><mi id="S3.SS4.1.p1.2.m2.1.1.2" xref="S3.SS4.1.p1.2.m2.1.1.2.cmml">e</mi><mi id="S3.SS4.1.p1.2.m2.1.1.3" xref="S3.SS4.1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.2.m2.1b"><apply id="S3.SS4.1.p1.2.m2.1.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.2.m2.1.1.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.2.m2.1.1.2.cmml" xref="S3.SS4.1.p1.2.m2.1.1.2">𝑒</ci><ci id="S3.SS4.1.p1.2.m2.1.1.3.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.2.m2.1c">e_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.2.m2.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="i" class="ltx_Math" display="inline" id="S3.SS4.1.p1.3.m3.1"><semantics id="S3.SS4.1.p1.3.m3.1a"><mi id="S3.SS4.1.p1.3.m3.1.1" xref="S3.SS4.1.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.3.m3.1b"><ci id="S3.SS4.1.p1.3.m3.1.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.3.m3.1d">italic_i</annotation></semantics></math>-th standard unit vector. Let <math alttext="c\neq 0" class="ltx_Math" display="inline" id="S3.SS4.1.p1.4.m4.1"><semantics id="S3.SS4.1.p1.4.m4.1a"><mrow id="S3.SS4.1.p1.4.m4.1.1" xref="S3.SS4.1.p1.4.m4.1.1.cmml"><mi id="S3.SS4.1.p1.4.m4.1.1.2" xref="S3.SS4.1.p1.4.m4.1.1.2.cmml">c</mi><mo id="S3.SS4.1.p1.4.m4.1.1.1" xref="S3.SS4.1.p1.4.m4.1.1.1.cmml">≠</mo><mn id="S3.SS4.1.p1.4.m4.1.1.3" xref="S3.SS4.1.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.4.m4.1b"><apply id="S3.SS4.1.p1.4.m4.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1"><neq id="S3.SS4.1.p1.4.m4.1.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1.1"></neq><ci id="S3.SS4.1.p1.4.m4.1.1.2.cmml" xref="S3.SS4.1.p1.4.m4.1.1.2">𝑐</ci><cn id="S3.SS4.1.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.SS4.1.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.4.m4.1c">c\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.4.m4.1d">italic_c ≠ 0</annotation></semantics></math> be arbitrary. Then <math alttext="c" class="ltx_Math" display="inline" id="S3.SS4.1.p1.5.m5.1"><semantics id="S3.SS4.1.p1.5.m5.1a"><mi id="S3.SS4.1.p1.5.m5.1.1" xref="S3.SS4.1.p1.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.5.m5.1b"><ci id="S3.SS4.1.p1.5.m5.1.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.5.m5.1d">italic_c</annotation></semantics></math> cannot lie on all axis-aligned facets of the convex hull of <math alttext="P_{d}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.6.m6.1"><semantics id="S3.SS4.1.p1.6.m6.1a"><msub id="S3.SS4.1.p1.6.m6.1.1" xref="S3.SS4.1.p1.6.m6.1.1.cmml"><mi id="S3.SS4.1.p1.6.m6.1.1.2" xref="S3.SS4.1.p1.6.m6.1.1.2.cmml">P</mi><mi id="S3.SS4.1.p1.6.m6.1.1.3" xref="S3.SS4.1.p1.6.m6.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.6.m6.1b"><apply id="S3.SS4.1.p1.6.m6.1.1.cmml" xref="S3.SS4.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.6.m6.1.1.1.cmml" xref="S3.SS4.1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.6.m6.1.1.2.cmml" xref="S3.SS4.1.p1.6.m6.1.1.2">𝑃</ci><ci id="S3.SS4.1.p1.6.m6.1.1.3.cmml" xref="S3.SS4.1.p1.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.6.m6.1c">P_{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.6.m6.1d">italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> simultaneously. Thus, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem5" title="Lemma 3.5. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.5</span></a> there must exist a direction <math alttext="v\in\{-e_{1},\ldots,-e_{d},e_{1},\ldots,e_{d}\}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.7.m7.6"><semantics id="S3.SS4.1.p1.7.m7.6a"><mrow id="S3.SS4.1.p1.7.m7.6.6" xref="S3.SS4.1.p1.7.m7.6.6.cmml"><mi id="S3.SS4.1.p1.7.m7.6.6.6" xref="S3.SS4.1.p1.7.m7.6.6.6.cmml">v</mi><mo id="S3.SS4.1.p1.7.m7.6.6.5" xref="S3.SS4.1.p1.7.m7.6.6.5.cmml">∈</mo><mrow id="S3.SS4.1.p1.7.m7.6.6.4.4" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml"><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.5" stretchy="false" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">{</mo><mrow id="S3.SS4.1.p1.7.m7.3.3.1.1.1" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.cmml"><mo id="S3.SS4.1.p1.7.m7.3.3.1.1.1a" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.cmml">−</mo><msub id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.cmml"><mi id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.2" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.2.cmml">e</mi><mn id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.3" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.6" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">,</mo><mi id="S3.SS4.1.p1.7.m7.1.1" mathvariant="normal" xref="S3.SS4.1.p1.7.m7.1.1.cmml">…</mi><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.7" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">,</mo><mrow id="S3.SS4.1.p1.7.m7.4.4.2.2.2" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.cmml"><mo id="S3.SS4.1.p1.7.m7.4.4.2.2.2a" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.cmml">−</mo><msub id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.cmml"><mi id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.2" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.2.cmml">e</mi><mi id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.3" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.3.cmml">d</mi></msub></mrow><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.8" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">,</mo><msub id="S3.SS4.1.p1.7.m7.5.5.3.3.3" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3.cmml"><mi id="S3.SS4.1.p1.7.m7.5.5.3.3.3.2" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3.2.cmml">e</mi><mn id="S3.SS4.1.p1.7.m7.5.5.3.3.3.3" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3.3.cmml">1</mn></msub><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.9" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">,</mo><mi id="S3.SS4.1.p1.7.m7.2.2" mathvariant="normal" xref="S3.SS4.1.p1.7.m7.2.2.cmml">…</mi><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.10" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">,</mo><msub id="S3.SS4.1.p1.7.m7.6.6.4.4.4" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4.cmml"><mi id="S3.SS4.1.p1.7.m7.6.6.4.4.4.2" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4.2.cmml">e</mi><mi id="S3.SS4.1.p1.7.m7.6.6.4.4.4.3" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4.3.cmml">d</mi></msub><mo id="S3.SS4.1.p1.7.m7.6.6.4.4.11" stretchy="false" xref="S3.SS4.1.p1.7.m7.6.6.4.5.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.7.m7.6b"><apply id="S3.SS4.1.p1.7.m7.6.6.cmml" xref="S3.SS4.1.p1.7.m7.6.6"><in id="S3.SS4.1.p1.7.m7.6.6.5.cmml" xref="S3.SS4.1.p1.7.m7.6.6.5"></in><ci id="S3.SS4.1.p1.7.m7.6.6.6.cmml" xref="S3.SS4.1.p1.7.m7.6.6.6">𝑣</ci><set id="S3.SS4.1.p1.7.m7.6.6.4.5.cmml" xref="S3.SS4.1.p1.7.m7.6.6.4.4"><apply id="S3.SS4.1.p1.7.m7.3.3.1.1.1.cmml" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1"><minus id="S3.SS4.1.p1.7.m7.3.3.1.1.1.1.cmml" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1"></minus><apply id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.cmml" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.1.cmml" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2">subscript</csymbol><ci id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.2.cmml" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.2">𝑒</ci><cn id="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.3.cmml" type="integer" xref="S3.SS4.1.p1.7.m7.3.3.1.1.1.2.3">1</cn></apply></apply><ci id="S3.SS4.1.p1.7.m7.1.1.cmml" xref="S3.SS4.1.p1.7.m7.1.1">…</ci><apply id="S3.SS4.1.p1.7.m7.4.4.2.2.2.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2"><minus id="S3.SS4.1.p1.7.m7.4.4.2.2.2.1.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2"></minus><apply id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.1.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2">subscript</csymbol><ci id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.2.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.2">𝑒</ci><ci id="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.3.cmml" xref="S3.SS4.1.p1.7.m7.4.4.2.2.2.2.3">𝑑</ci></apply></apply><apply id="S3.SS4.1.p1.7.m7.5.5.3.3.3.cmml" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.5.5.3.3.3.1.cmml" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3">subscript</csymbol><ci id="S3.SS4.1.p1.7.m7.5.5.3.3.3.2.cmml" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3.2">𝑒</ci><cn id="S3.SS4.1.p1.7.m7.5.5.3.3.3.3.cmml" type="integer" xref="S3.SS4.1.p1.7.m7.5.5.3.3.3.3">1</cn></apply><ci id="S3.SS4.1.p1.7.m7.2.2.cmml" xref="S3.SS4.1.p1.7.m7.2.2">…</ci><apply id="S3.SS4.1.p1.7.m7.6.6.4.4.4.cmml" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.6.6.4.4.4.1.cmml" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4">subscript</csymbol><ci id="S3.SS4.1.p1.7.m7.6.6.4.4.4.2.cmml" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4.2">𝑒</ci><ci id="S3.SS4.1.p1.7.m7.6.6.4.4.4.3.cmml" xref="S3.SS4.1.p1.7.m7.6.6.4.4.4.3">𝑑</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.7.m7.6c">v\in\{-e_{1},\ldots,-e_{d},e_{1},\ldots,e_{d}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.7.m7.6d">italic_v ∈ { - italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , - italic_e start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT }</annotation></semantics></math> such that <math alttext="\mathcal{H}^{p}_{c,v}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.8.m8.2"><semantics id="S3.SS4.1.p1.8.m8.2a"><msubsup id="S3.SS4.1.p1.8.m8.2.3" xref="S3.SS4.1.p1.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.8.m8.2.3.2.2" xref="S3.SS4.1.p1.8.m8.2.3.2.2.cmml">ℋ</mi><mrow id="S3.SS4.1.p1.8.m8.2.2.2.4" xref="S3.SS4.1.p1.8.m8.2.2.2.3.cmml"><mi id="S3.SS4.1.p1.8.m8.1.1.1.1" xref="S3.SS4.1.p1.8.m8.1.1.1.1.cmml">c</mi><mo id="S3.SS4.1.p1.8.m8.2.2.2.4.1" xref="S3.SS4.1.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS4.1.p1.8.m8.2.2.2.2" xref="S3.SS4.1.p1.8.m8.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS4.1.p1.8.m8.2.3.2.3" xref="S3.SS4.1.p1.8.m8.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.8.m8.2b"><apply id="S3.SS4.1.p1.8.m8.2.3.cmml" xref="S3.SS4.1.p1.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.8.m8.2.3.1.cmml" xref="S3.SS4.1.p1.8.m8.2.3">subscript</csymbol><apply id="S3.SS4.1.p1.8.m8.2.3.2.cmml" xref="S3.SS4.1.p1.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.8.m8.2.3.2.1.cmml" xref="S3.SS4.1.p1.8.m8.2.3">superscript</csymbol><ci id="S3.SS4.1.p1.8.m8.2.3.2.2.cmml" xref="S3.SS4.1.p1.8.m8.2.3.2.2">ℋ</ci><ci id="S3.SS4.1.p1.8.m8.2.3.2.3.cmml" xref="S3.SS4.1.p1.8.m8.2.3.2.3">𝑝</ci></apply><list id="S3.SS4.1.p1.8.m8.2.2.2.3.cmml" xref="S3.SS4.1.p1.8.m8.2.2.2.4"><ci id="S3.SS4.1.p1.8.m8.1.1.1.1.cmml" xref="S3.SS4.1.p1.8.m8.1.1.1.1">𝑐</ci><ci id="S3.SS4.1.p1.8.m8.2.2.2.2.cmml" xref="S3.SS4.1.p1.8.m8.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.8.m8.2c">\mathcal{H}^{p}_{c,v}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.8.m8.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> does not contain any vertices of a facet that <math alttext="c" class="ltx_Math" display="inline" id="S3.SS4.1.p1.9.m9.1"><semantics id="S3.SS4.1.p1.9.m9.1a"><mi id="S3.SS4.1.p1.9.m9.1.1" xref="S3.SS4.1.p1.9.m9.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.9.m9.1b"><ci id="S3.SS4.1.p1.9.m9.1.1.cmml" xref="S3.SS4.1.p1.9.m9.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.9.m9.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.9.m9.1d">italic_c</annotation></semantics></math> does not lie on, and therefore <math alttext="|\mathcal{H}^{p}_{c,v}\cap P_{d}|\leq 1=\frac{|P_{d}|}{d+1}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.10.m10.4"><semantics id="S3.SS4.1.p1.10.m10.4a"><mrow id="S3.SS4.1.p1.10.m10.4.4" xref="S3.SS4.1.p1.10.m10.4.4.cmml"><mrow id="S3.SS4.1.p1.10.m10.4.4.1.1" xref="S3.SS4.1.p1.10.m10.4.4.1.2.cmml"><mo id="S3.SS4.1.p1.10.m10.4.4.1.1.2" stretchy="false" xref="S3.SS4.1.p1.10.m10.4.4.1.2.1.cmml">|</mo><mrow id="S3.SS4.1.p1.10.m10.4.4.1.1.1" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.cmml"><msubsup id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.2" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.2.cmml">ℋ</mi><mrow id="S3.SS4.1.p1.10.m10.2.2.2.4" xref="S3.SS4.1.p1.10.m10.2.2.2.3.cmml"><mi id="S3.SS4.1.p1.10.m10.1.1.1.1" xref="S3.SS4.1.p1.10.m10.1.1.1.1.cmml">c</mi><mo id="S3.SS4.1.p1.10.m10.2.2.2.4.1" xref="S3.SS4.1.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S3.SS4.1.p1.10.m10.2.2.2.2" xref="S3.SS4.1.p1.10.m10.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.3" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.3.cmml">p</mi></msubsup><mo id="S3.SS4.1.p1.10.m10.4.4.1.1.1.1" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.1.cmml">∩</mo><msub id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.cmml"><mi id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.2" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.2.cmml">P</mi><mi id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.3" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S3.SS4.1.p1.10.m10.4.4.1.1.3" stretchy="false" xref="S3.SS4.1.p1.10.m10.4.4.1.2.1.cmml">|</mo></mrow><mo id="S3.SS4.1.p1.10.m10.4.4.3" xref="S3.SS4.1.p1.10.m10.4.4.3.cmml">≤</mo><mn id="S3.SS4.1.p1.10.m10.4.4.4" xref="S3.SS4.1.p1.10.m10.4.4.4.cmml">1</mn><mo id="S3.SS4.1.p1.10.m10.4.4.5" xref="S3.SS4.1.p1.10.m10.4.4.5.cmml">=</mo><mfrac id="S3.SS4.1.p1.10.m10.3.3" xref="S3.SS4.1.p1.10.m10.3.3.cmml"><mrow id="S3.SS4.1.p1.10.m10.3.3.1.1" xref="S3.SS4.1.p1.10.m10.3.3.1.2.cmml"><mo id="S3.SS4.1.p1.10.m10.3.3.1.1.2" stretchy="false" xref="S3.SS4.1.p1.10.m10.3.3.1.2.1.cmml">|</mo><msub id="S3.SS4.1.p1.10.m10.3.3.1.1.1" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1.cmml"><mi id="S3.SS4.1.p1.10.m10.3.3.1.1.1.2" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1.2.cmml">P</mi><mi id="S3.SS4.1.p1.10.m10.3.3.1.1.1.3" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1.3.cmml">d</mi></msub><mo id="S3.SS4.1.p1.10.m10.3.3.1.1.3" stretchy="false" xref="S3.SS4.1.p1.10.m10.3.3.1.2.1.cmml">|</mo></mrow><mrow id="S3.SS4.1.p1.10.m10.3.3.3" xref="S3.SS4.1.p1.10.m10.3.3.3.cmml"><mi id="S3.SS4.1.p1.10.m10.3.3.3.2" xref="S3.SS4.1.p1.10.m10.3.3.3.2.cmml">d</mi><mo id="S3.SS4.1.p1.10.m10.3.3.3.1" xref="S3.SS4.1.p1.10.m10.3.3.3.1.cmml">+</mo><mn id="S3.SS4.1.p1.10.m10.3.3.3.3" xref="S3.SS4.1.p1.10.m10.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.10.m10.4b"><apply id="S3.SS4.1.p1.10.m10.4.4.cmml" xref="S3.SS4.1.p1.10.m10.4.4"><and id="S3.SS4.1.p1.10.m10.4.4a.cmml" xref="S3.SS4.1.p1.10.m10.4.4"></and><apply id="S3.SS4.1.p1.10.m10.4.4b.cmml" xref="S3.SS4.1.p1.10.m10.4.4"><leq id="S3.SS4.1.p1.10.m10.4.4.3.cmml" xref="S3.SS4.1.p1.10.m10.4.4.3"></leq><apply id="S3.SS4.1.p1.10.m10.4.4.1.2.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1"><abs id="S3.SS4.1.p1.10.m10.4.4.1.2.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.2"></abs><apply id="S3.SS4.1.p1.10.m10.4.4.1.1.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1"><intersect id="S3.SS4.1.p1.10.m10.4.4.1.1.1.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.1"></intersect><apply id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2">subscript</csymbol><apply id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2">superscript</csymbol><ci id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.2.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.2">ℋ</ci><ci id="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.3.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.2.2.3">𝑝</ci></apply><list id="S3.SS4.1.p1.10.m10.2.2.2.3.cmml" xref="S3.SS4.1.p1.10.m10.2.2.2.4"><ci id="S3.SS4.1.p1.10.m10.1.1.1.1.cmml" xref="S3.SS4.1.p1.10.m10.1.1.1.1">𝑐</ci><ci id="S3.SS4.1.p1.10.m10.2.2.2.2.cmml" xref="S3.SS4.1.p1.10.m10.2.2.2.2">𝑣</ci></list></apply><apply id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.1.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3">subscript</csymbol><ci id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.2.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.2">𝑃</ci><ci id="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.3.cmml" xref="S3.SS4.1.p1.10.m10.4.4.1.1.1.3.3">𝑑</ci></apply></apply></apply><cn id="S3.SS4.1.p1.10.m10.4.4.4.cmml" type="integer" xref="S3.SS4.1.p1.10.m10.4.4.4">1</cn></apply><apply id="S3.SS4.1.p1.10.m10.4.4c.cmml" xref="S3.SS4.1.p1.10.m10.4.4"><eq id="S3.SS4.1.p1.10.m10.4.4.5.cmml" xref="S3.SS4.1.p1.10.m10.4.4.5"></eq><share href="https://arxiv.org/html/2503.16089v1#S3.SS4.1.p1.10.m10.4.4.4.cmml" id="S3.SS4.1.p1.10.m10.4.4d.cmml" xref="S3.SS4.1.p1.10.m10.4.4"></share><apply id="S3.SS4.1.p1.10.m10.3.3.cmml" xref="S3.SS4.1.p1.10.m10.3.3"><divide id="S3.SS4.1.p1.10.m10.3.3.2.cmml" xref="S3.SS4.1.p1.10.m10.3.3"></divide><apply id="S3.SS4.1.p1.10.m10.3.3.1.2.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1"><abs id="S3.SS4.1.p1.10.m10.3.3.1.2.1.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1.2"></abs><apply id="S3.SS4.1.p1.10.m10.3.3.1.1.1.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.10.m10.3.3.1.1.1.1.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.10.m10.3.3.1.1.1.2.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1.2">𝑃</ci><ci id="S3.SS4.1.p1.10.m10.3.3.1.1.1.3.cmml" xref="S3.SS4.1.p1.10.m10.3.3.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.SS4.1.p1.10.m10.3.3.3.cmml" xref="S3.SS4.1.p1.10.m10.3.3.3"><plus id="S3.SS4.1.p1.10.m10.3.3.3.1.cmml" xref="S3.SS4.1.p1.10.m10.3.3.3.1"></plus><ci id="S3.SS4.1.p1.10.m10.3.3.3.2.cmml" xref="S3.SS4.1.p1.10.m10.3.3.3.2">𝑑</ci><cn id="S3.SS4.1.p1.10.m10.3.3.3.3.cmml" type="integer" xref="S3.SS4.1.p1.10.m10.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.10.m10.4c">|\mathcal{H}^{p}_{c,v}\cap P_{d}|\leq 1=\frac{|P_{d}|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.10.m10.4d">| caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ∩ italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT | ≤ 1 = divide start_ARG | italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.2.p2"> <p class="ltx_p" id="S3.SS4.2.p2.3">If we instead have <math alttext="c=0" class="ltx_Math" display="inline" id="S3.SS4.2.p2.1.m1.1"><semantics id="S3.SS4.2.p2.1.m1.1a"><mrow id="S3.SS4.2.p2.1.m1.1.1" xref="S3.SS4.2.p2.1.m1.1.1.cmml"><mi id="S3.SS4.2.p2.1.m1.1.1.2" xref="S3.SS4.2.p2.1.m1.1.1.2.cmml">c</mi><mo id="S3.SS4.2.p2.1.m1.1.1.1" xref="S3.SS4.2.p2.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS4.2.p2.1.m1.1.1.3" xref="S3.SS4.2.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.1.m1.1b"><apply id="S3.SS4.2.p2.1.m1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1"><eq id="S3.SS4.2.p2.1.m1.1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.1"></eq><ci id="S3.SS4.2.p2.1.m1.1.1.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.2">𝑐</ci><cn id="S3.SS4.2.p2.1.m1.1.1.3.cmml" type="integer" xref="S3.SS4.2.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.1.m1.1c">c=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.1.m1.1d">italic_c = 0</annotation></semantics></math>, then <math alttext="\mathcal{H}^{p}_{c,v}\cap P_{d}=\{0\}" class="ltx_Math" display="inline" id="S3.SS4.2.p2.2.m2.3"><semantics id="S3.SS4.2.p2.2.m2.3a"><mrow id="S3.SS4.2.p2.2.m2.3.4" xref="S3.SS4.2.p2.2.m2.3.4.cmml"><mrow id="S3.SS4.2.p2.2.m2.3.4.2" xref="S3.SS4.2.p2.2.m2.3.4.2.cmml"><msubsup id="S3.SS4.2.p2.2.m2.3.4.2.2" xref="S3.SS4.2.p2.2.m2.3.4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.2.m2.3.4.2.2.2.2" xref="S3.SS4.2.p2.2.m2.3.4.2.2.2.2.cmml">ℋ</mi><mrow id="S3.SS4.2.p2.2.m2.2.2.2.4" xref="S3.SS4.2.p2.2.m2.2.2.2.3.cmml"><mi id="S3.SS4.2.p2.2.m2.1.1.1.1" xref="S3.SS4.2.p2.2.m2.1.1.1.1.cmml">c</mi><mo id="S3.SS4.2.p2.2.m2.2.2.2.4.1" xref="S3.SS4.2.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS4.2.p2.2.m2.2.2.2.2" xref="S3.SS4.2.p2.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="S3.SS4.2.p2.2.m2.3.4.2.2.2.3" xref="S3.SS4.2.p2.2.m2.3.4.2.2.2.3.cmml">p</mi></msubsup><mo id="S3.SS4.2.p2.2.m2.3.4.2.1" xref="S3.SS4.2.p2.2.m2.3.4.2.1.cmml">∩</mo><msub id="S3.SS4.2.p2.2.m2.3.4.2.3" xref="S3.SS4.2.p2.2.m2.3.4.2.3.cmml"><mi id="S3.SS4.2.p2.2.m2.3.4.2.3.2" xref="S3.SS4.2.p2.2.m2.3.4.2.3.2.cmml">P</mi><mi id="S3.SS4.2.p2.2.m2.3.4.2.3.3" xref="S3.SS4.2.p2.2.m2.3.4.2.3.3.cmml">d</mi></msub></mrow><mo id="S3.SS4.2.p2.2.m2.3.4.1" xref="S3.SS4.2.p2.2.m2.3.4.1.cmml">=</mo><mrow id="S3.SS4.2.p2.2.m2.3.4.3.2" xref="S3.SS4.2.p2.2.m2.3.4.3.1.cmml"><mo id="S3.SS4.2.p2.2.m2.3.4.3.2.1" stretchy="false" xref="S3.SS4.2.p2.2.m2.3.4.3.1.cmml">{</mo><mn id="S3.SS4.2.p2.2.m2.3.3" xref="S3.SS4.2.p2.2.m2.3.3.cmml">0</mn><mo id="S3.SS4.2.p2.2.m2.3.4.3.2.2" stretchy="false" xref="S3.SS4.2.p2.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.2.m2.3b"><apply id="S3.SS4.2.p2.2.m2.3.4.cmml" xref="S3.SS4.2.p2.2.m2.3.4"><eq id="S3.SS4.2.p2.2.m2.3.4.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.1"></eq><apply id="S3.SS4.2.p2.2.m2.3.4.2.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2"><intersect id="S3.SS4.2.p2.2.m2.3.4.2.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.1"></intersect><apply id="S3.SS4.2.p2.2.m2.3.4.2.2.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.3.4.2.2.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2">subscript</csymbol><apply id="S3.SS4.2.p2.2.m2.3.4.2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.3.4.2.2.2.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2">superscript</csymbol><ci id="S3.SS4.2.p2.2.m2.3.4.2.2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2.2.2">ℋ</ci><ci id="S3.SS4.2.p2.2.m2.3.4.2.2.2.3.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.2.2.3">𝑝</ci></apply><list id="S3.SS4.2.p2.2.m2.2.2.2.3.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.4"><ci id="S3.SS4.2.p2.2.m2.1.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1">𝑐</ci><ci id="S3.SS4.2.p2.2.m2.2.2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.2">𝑣</ci></list></apply><apply id="S3.SS4.2.p2.2.m2.3.4.2.3.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.3"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.3.4.2.3.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.3">subscript</csymbol><ci id="S3.SS4.2.p2.2.m2.3.4.2.3.2.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.3.2">𝑃</ci><ci id="S3.SS4.2.p2.2.m2.3.4.2.3.3.cmml" xref="S3.SS4.2.p2.2.m2.3.4.2.3.3">𝑑</ci></apply></apply><set id="S3.SS4.2.p2.2.m2.3.4.3.1.cmml" xref="S3.SS4.2.p2.2.m2.3.4.3.2"><cn id="S3.SS4.2.p2.2.m2.3.3.cmml" type="integer" xref="S3.SS4.2.p2.2.m2.3.3">0</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.2.m2.3c">\mathcal{H}^{p}_{c,v}\cap P_{d}=\{0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.2.m2.3d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ∩ italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = { 0 }</annotation></semantics></math> for <math alttext="v=(-\sqrt{\nicefrac{{1}}{{d}}},\ldots,-\sqrt{\nicefrac{{1}}{{d}}})\in S^{d-1}" class="ltx_Math" display="inline" id="S3.SS4.2.p2.3.m3.3"><semantics id="S3.SS4.2.p2.3.m3.3a"><mrow id="S3.SS4.2.p2.3.m3.3.3" xref="S3.SS4.2.p2.3.m3.3.3.cmml"><mi id="S3.SS4.2.p2.3.m3.3.3.4" xref="S3.SS4.2.p2.3.m3.3.3.4.cmml">v</mi><mo id="S3.SS4.2.p2.3.m3.3.3.5" xref="S3.SS4.2.p2.3.m3.3.3.5.cmml">=</mo><mrow id="S3.SS4.2.p2.3.m3.3.3.2.2" xref="S3.SS4.2.p2.3.m3.3.3.2.3.cmml"><mo id="S3.SS4.2.p2.3.m3.3.3.2.2.3" 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xref="S4.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.1.m1.1.1.3" xref="S4.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.1.m1.1c"><apply id="S4.1.m1.1.1.cmml" xref="S4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.1.m1.1.1.1.cmml" xref="S4.1.m1.1.1">subscript</csymbol><ci id="S4.1.m1.1.1.2.cmml" xref="S4.1.m1.1.1.2">ℓ</ci><ci id="S4.1.m1.1.1.3.cmml" xref="S4.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Contraction Maps</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.5">In this section, we describe our algorithms for the continuous problem <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><msub id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml"><mi id="S4.p1.1.m1.1.1.2" mathvariant="normal" xref="S4.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.1.cmml" xref="S4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.2">ℓ</ci><ci id="S4.p1.1.m1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.p1.5.1">-ContractionFixpoint</span> and explain how, in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><msub id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml"><mi id="S4.p1.2.m2.1.1.2" mathvariant="normal" xref="S4.p1.2.m2.1.1.2.cmml">ℓ</mi><mn id="S4.p1.2.m2.1.1.3" xref="S4.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.1.1.1.cmml" xref="S4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.2">ℓ</ci><cn id="S4.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case, the algorithm can be adapted to the discretized setting <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.p1.3.m3.1"><semantics id="S4.p1.3.m3.1a"><msub id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml"><mi id="S4.p1.3.m3.1.1.2" mathvariant="normal" xref="S4.p1.3.m3.1.1.2.cmml">ℓ</mi><mn id="S4.p1.3.m3.1.1.3" xref="S4.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.1b"><apply id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p1.3.m3.1.1.1.cmml" xref="S4.p1.3.m3.1.1">subscript</csymbol><ci id="S4.p1.3.m3.1.1.2.cmml" xref="S4.p1.3.m3.1.1.2">ℓ</ci><cn id="S4.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.p1.5.2">-GridContractionFixpoint</span>. Note that in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><msub id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml"><mi id="S4.p1.4.m4.1.1.2" mathvariant="normal" xref="S4.p1.4.m4.1.1.2.cmml">ℓ</mi><mi id="S4.p1.4.m4.1.1.3" mathvariant="normal" xref="S4.p1.4.m4.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><apply id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.p1.4.m4.1.1.1.cmml" xref="S4.p1.4.m4.1.1">subscript</csymbol><ci id="S4.p1.4.m4.1.1.2.cmml" xref="S4.p1.4.m4.1.1.2">ℓ</ci><infinity id="S4.p1.4.m4.1.1.3.cmml" xref="S4.p1.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case, a suitable rounding strategy has already been provided by Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite>. We will not repeat that, but note that their rounding can also be used in combination with our methods (solving <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><msub id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" mathvariant="normal" xref="S4.p1.5.m5.1.1.2.cmml">ℓ</mi><mi id="S4.p1.5.m5.1.1.3" mathvariant="normal" xref="S4.p1.5.m5.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1">subscript</csymbol><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">ℓ</ci><infinity id="S4.p1.5.m5.1.1.3.cmml" xref="S4.p1.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.p1.5.3">-GridContractionFixpoint</span>).</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Solving <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS1.1.m1.1"><semantics id="S4.SS1.1.m1.1b"><msub id="S4.SS1.1.m1.1.1" xref="S4.SS1.1.m1.1.1.cmml"><mi id="S4.SS1.1.m1.1.1.2" mathvariant="normal" xref="S4.SS1.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.SS1.1.m1.1.1.3" xref="S4.SS1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.1.m1.1c"><apply id="S4.SS1.1.m1.1.1.cmml" xref="S4.SS1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.1.m1.1.1.1.cmml" xref="S4.SS1.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.1.m1.1.1.2.cmml" xref="S4.SS1.1.m1.1.1.2">ℓ</ci><ci id="S4.SS1.1.m1.1.1.3.cmml" xref="S4.SS1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.SS1.5.1">-ContractionFixpoint</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.22">Our algorithm works as follows: we maintain a search space <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_M</annotation></semantics></math> that is guaranteed to always contain the fixpoint <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><msup id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">x</mi><mo id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝑥</ci><ci id="S4.SS1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. At the beginning of the algorithm, we simply set <math alttext="M=[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.2"><semantics id="S4.SS1.p1.3.m3.2a"><mrow id="S4.SS1.p1.3.m3.2.3" xref="S4.SS1.p1.3.m3.2.3.cmml"><mi id="S4.SS1.p1.3.m3.2.3.2" xref="S4.SS1.p1.3.m3.2.3.2.cmml">M</mi><mo id="S4.SS1.p1.3.m3.2.3.1" xref="S4.SS1.p1.3.m3.2.3.1.cmml">=</mo><msup id="S4.SS1.p1.3.m3.2.3.3" xref="S4.SS1.p1.3.m3.2.3.3.cmml"><mrow id="S4.SS1.p1.3.m3.2.3.3.2.2" xref="S4.SS1.p1.3.m3.2.3.3.2.1.cmml"><mo id="S4.SS1.p1.3.m3.2.3.3.2.2.1" stretchy="false" xref="S4.SS1.p1.3.m3.2.3.3.2.1.cmml">[</mo><mn id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.p1.3.m3.2.3.3.2.2.2" xref="S4.SS1.p1.3.m3.2.3.3.2.1.cmml">,</mo><mn id="S4.SS1.p1.3.m3.2.2" xref="S4.SS1.p1.3.m3.2.2.cmml">1</mn><mo id="S4.SS1.p1.3.m3.2.3.3.2.2.3" stretchy="false" xref="S4.SS1.p1.3.m3.2.3.3.2.1.cmml">]</mo></mrow><mi id="S4.SS1.p1.3.m3.2.3.3.3" xref="S4.SS1.p1.3.m3.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.2b"><apply id="S4.SS1.p1.3.m3.2.3.cmml" xref="S4.SS1.p1.3.m3.2.3"><eq id="S4.SS1.p1.3.m3.2.3.1.cmml" xref="S4.SS1.p1.3.m3.2.3.1"></eq><ci id="S4.SS1.p1.3.m3.2.3.2.cmml" xref="S4.SS1.p1.3.m3.2.3.2">𝑀</ci><apply id="S4.SS1.p1.3.m3.2.3.3.cmml" xref="S4.SS1.p1.3.m3.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.2.3.3.1.cmml" xref="S4.SS1.p1.3.m3.2.3.3">superscript</csymbol><interval closure="closed" id="S4.SS1.p1.3.m3.2.3.3.2.1.cmml" xref="S4.SS1.p1.3.m3.2.3.3.2.2"><cn id="S4.SS1.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p1.3.m3.1.1">0</cn><cn id="S4.SS1.p1.3.m3.2.2.cmml" type="integer" xref="S4.SS1.p1.3.m3.2.2">1</cn></interval><ci id="S4.SS1.p1.3.m3.2.3.3.3.cmml" xref="S4.SS1.p1.3.m3.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.2c">M=[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.2d">italic_M = [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. We then iteratively query the centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><mi id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><ci id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">italic_c</annotation></semantics></math> of our remaining search space <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mi id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><ci id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">italic_M</annotation></semantics></math>. Concretely, we use the measure <math alttext="\operatorname{vol}" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.1"><semantics id="S4.SS1.p1.6.m6.1a"><mi id="S4.SS1.p1.6.m6.1.1" xref="S4.SS1.p1.6.m6.1.1.cmml">vol</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.1b"><ci id="S4.SS1.p1.6.m6.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1">vol</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.1c">\operatorname{vol}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.1d">roman_vol</annotation></semantics></math> (which we simply call <em class="ltx_emph ltx_font_italic" id="S4.SS1.p1.22.1">volume</em>) defined by the Lebesgue measure (such that <math alttext="\operatorname{vol}([0,1]^{d})=1" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.4"><semantics id="S4.SS1.p1.7.m7.4a"><mrow id="S4.SS1.p1.7.m7.4.4" xref="S4.SS1.p1.7.m7.4.4.cmml"><mrow id="S4.SS1.p1.7.m7.4.4.1.1" xref="S4.SS1.p1.7.m7.4.4.1.2.cmml"><mi id="S4.SS1.p1.7.m7.3.3" xref="S4.SS1.p1.7.m7.3.3.cmml">vol</mi><mo id="S4.SS1.p1.7.m7.4.4.1.1a" xref="S4.SS1.p1.7.m7.4.4.1.2.cmml">⁡</mo><mrow id="S4.SS1.p1.7.m7.4.4.1.1.1" xref="S4.SS1.p1.7.m7.4.4.1.2.cmml"><mo id="S4.SS1.p1.7.m7.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.p1.7.m7.4.4.1.2.cmml">(</mo><msup id="S4.SS1.p1.7.m7.4.4.1.1.1.1" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.cmml"><mrow id="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.2" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.1.cmml"><mo id="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.2.1" stretchy="false" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.1.cmml">[</mo><mn id="S4.SS1.p1.7.m7.1.1" xref="S4.SS1.p1.7.m7.1.1.cmml">0</mn><mo id="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.2.2" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.1.cmml">,</mo><mn id="S4.SS1.p1.7.m7.2.2" xref="S4.SS1.p1.7.m7.2.2.cmml">1</mn><mo id="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.2.3" stretchy="false" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.1.cmml">]</mo></mrow><mi id="S4.SS1.p1.7.m7.4.4.1.1.1.1.3" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS1.p1.7.m7.4.4.1.1.1.3" stretchy="false" xref="S4.SS1.p1.7.m7.4.4.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p1.7.m7.4.4.2" xref="S4.SS1.p1.7.m7.4.4.2.cmml">=</mo><mn id="S4.SS1.p1.7.m7.4.4.3" xref="S4.SS1.p1.7.m7.4.4.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.4b"><apply id="S4.SS1.p1.7.m7.4.4.cmml" xref="S4.SS1.p1.7.m7.4.4"><eq id="S4.SS1.p1.7.m7.4.4.2.cmml" xref="S4.SS1.p1.7.m7.4.4.2"></eq><apply id="S4.SS1.p1.7.m7.4.4.1.2.cmml" xref="S4.SS1.p1.7.m7.4.4.1.1"><ci id="S4.SS1.p1.7.m7.3.3.cmml" xref="S4.SS1.p1.7.m7.3.3">vol</ci><apply id="S4.SS1.p1.7.m7.4.4.1.1.1.1.cmml" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1">superscript</csymbol><interval closure="closed" id="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.1.cmml" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.2.2"><cn id="S4.SS1.p1.7.m7.1.1.cmml" type="integer" xref="S4.SS1.p1.7.m7.1.1">0</cn><cn id="S4.SS1.p1.7.m7.2.2.cmml" type="integer" xref="S4.SS1.p1.7.m7.2.2">1</cn></interval><ci id="S4.SS1.p1.7.m7.4.4.1.1.1.1.3.cmml" xref="S4.SS1.p1.7.m7.4.4.1.1.1.1.3">𝑑</ci></apply></apply><cn id="S4.SS1.p1.7.m7.4.4.3.cmml" type="integer" xref="S4.SS1.p1.7.m7.4.4.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.4c">\operatorname{vol}([0,1]^{d})=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.4d">roman_vol ( [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>), with its support restricted to <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m8.1"><semantics id="S4.SS1.p1.8.m8.1a"><mi id="S4.SS1.p1.8.m8.1.1" xref="S4.SS1.p1.8.m8.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m8.1b"><ci id="S4.SS1.p1.8.m8.1.1.cmml" xref="S4.SS1.p1.8.m8.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m8.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m8.1d">italic_M</annotation></semantics></math>. Whenever we query the centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m9.1"><semantics id="S4.SS1.p1.9.m9.1a"><mi id="S4.SS1.p1.9.m9.1.1" xref="S4.SS1.p1.9.m9.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m9.1b"><ci id="S4.SS1.p1.9.m9.1.1.cmml" xref="S4.SS1.p1.9.m9.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m9.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m9.1d">italic_c</annotation></semantics></math> of <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m10.1"><semantics id="S4.SS1.p1.10.m10.1a"><mi id="S4.SS1.p1.10.m10.1.1" xref="S4.SS1.p1.10.m10.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m10.1b"><ci id="S4.SS1.p1.10.m10.1.1.cmml" xref="S4.SS1.p1.10.m10.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m10.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m10.1d">italic_M</annotation></semantics></math>, we get to discard at least a <math alttext="\frac{1}{d+1}" class="ltx_Math" display="inline" id="S4.SS1.p1.11.m11.1"><semantics id="S4.SS1.p1.11.m11.1a"><mfrac id="S4.SS1.p1.11.m11.1.1" xref="S4.SS1.p1.11.m11.1.1.cmml"><mn id="S4.SS1.p1.11.m11.1.1.2" xref="S4.SS1.p1.11.m11.1.1.2.cmml">1</mn><mrow id="S4.SS1.p1.11.m11.1.1.3" xref="S4.SS1.p1.11.m11.1.1.3.cmml"><mi id="S4.SS1.p1.11.m11.1.1.3.2" xref="S4.SS1.p1.11.m11.1.1.3.2.cmml">d</mi><mo id="S4.SS1.p1.11.m11.1.1.3.1" xref="S4.SS1.p1.11.m11.1.1.3.1.cmml">+</mo><mn id="S4.SS1.p1.11.m11.1.1.3.3" xref="S4.SS1.p1.11.m11.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.11.m11.1b"><apply id="S4.SS1.p1.11.m11.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1"><divide id="S4.SS1.p1.11.m11.1.1.1.cmml" xref="S4.SS1.p1.11.m11.1.1"></divide><cn id="S4.SS1.p1.11.m11.1.1.2.cmml" type="integer" xref="S4.SS1.p1.11.m11.1.1.2">1</cn><apply id="S4.SS1.p1.11.m11.1.1.3.cmml" xref="S4.SS1.p1.11.m11.1.1.3"><plus id="S4.SS1.p1.11.m11.1.1.3.1.cmml" xref="S4.SS1.p1.11.m11.1.1.3.1"></plus><ci id="S4.SS1.p1.11.m11.1.1.3.2.cmml" xref="S4.SS1.p1.11.m11.1.1.3.2">𝑑</ci><cn id="S4.SS1.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S4.SS1.p1.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.11.m11.1c">\frac{1}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.11.m11.1d">divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of the search space, because <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S4.SS1.p1.12.m12.1"><semantics id="S4.SS1.p1.12.m12.1a"><msup id="S4.SS1.p1.12.m12.1.1" xref="S4.SS1.p1.12.m12.1.1.cmml"><mi id="S4.SS1.p1.12.m12.1.1.2" xref="S4.SS1.p1.12.m12.1.1.2.cmml">x</mi><mo id="S4.SS1.p1.12.m12.1.1.3" xref="S4.SS1.p1.12.m12.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.12.m12.1b"><apply id="S4.SS1.p1.12.m12.1.1.cmml" xref="S4.SS1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.12.m12.1.1.1.cmml" xref="S4.SS1.p1.12.m12.1.1">superscript</csymbol><ci id="S4.SS1.p1.12.m12.1.1.2.cmml" xref="S4.SS1.p1.12.m12.1.1.2">𝑥</ci><ci id="S4.SS1.p1.12.m12.1.1.3.cmml" xref="S4.SS1.p1.12.m12.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.12.m12.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.12.m12.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> must lie closer to <math alttext="f(c)" class="ltx_Math" display="inline" id="S4.SS1.p1.13.m13.1"><semantics id="S4.SS1.p1.13.m13.1a"><mrow id="S4.SS1.p1.13.m13.1.2" xref="S4.SS1.p1.13.m13.1.2.cmml"><mi id="S4.SS1.p1.13.m13.1.2.2" xref="S4.SS1.p1.13.m13.1.2.2.cmml">f</mi><mo id="S4.SS1.p1.13.m13.1.2.1" xref="S4.SS1.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p1.13.m13.1.2.3.2" xref="S4.SS1.p1.13.m13.1.2.cmml"><mo id="S4.SS1.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S4.SS1.p1.13.m13.1.2.cmml">(</mo><mi id="S4.SS1.p1.13.m13.1.1" xref="S4.SS1.p1.13.m13.1.1.cmml">c</mi><mo id="S4.SS1.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S4.SS1.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.13.m13.1b"><apply id="S4.SS1.p1.13.m13.1.2.cmml" xref="S4.SS1.p1.13.m13.1.2"><times id="S4.SS1.p1.13.m13.1.2.1.cmml" xref="S4.SS1.p1.13.m13.1.2.1"></times><ci id="S4.SS1.p1.13.m13.1.2.2.cmml" xref="S4.SS1.p1.13.m13.1.2.2">𝑓</ci><ci id="S4.SS1.p1.13.m13.1.1.cmml" xref="S4.SS1.p1.13.m13.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.13.m13.1c">f(c)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.13.m13.1d">italic_f ( italic_c )</annotation></semantics></math> than to <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.p1.14.m14.1"><semantics id="S4.SS1.p1.14.m14.1a"><mi id="S4.SS1.p1.14.m14.1.1" xref="S4.SS1.p1.14.m14.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.14.m14.1b"><ci id="S4.SS1.p1.14.m14.1.1.cmml" xref="S4.SS1.p1.14.m14.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.14.m14.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.14.m14.1d">italic_c</annotation></semantics></math> itself. With each query, the volume of <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.15.m15.1"><semantics id="S4.SS1.p1.15.m15.1a"><mi id="S4.SS1.p1.15.m15.1.1" xref="S4.SS1.p1.15.m15.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.15.m15.1b"><ci id="S4.SS1.p1.15.m15.1.1.cmml" xref="S4.SS1.p1.15.m15.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.15.m15.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.15.m15.1d">italic_M</annotation></semantics></math> is thus multiplied with a factor of at most <math alttext="\frac{d}{d+1}" class="ltx_Math" display="inline" id="S4.SS1.p1.16.m16.1"><semantics id="S4.SS1.p1.16.m16.1a"><mfrac id="S4.SS1.p1.16.m16.1.1" xref="S4.SS1.p1.16.m16.1.1.cmml"><mi id="S4.SS1.p1.16.m16.1.1.2" xref="S4.SS1.p1.16.m16.1.1.2.cmml">d</mi><mrow id="S4.SS1.p1.16.m16.1.1.3" xref="S4.SS1.p1.16.m16.1.1.3.cmml"><mi id="S4.SS1.p1.16.m16.1.1.3.2" xref="S4.SS1.p1.16.m16.1.1.3.2.cmml">d</mi><mo id="S4.SS1.p1.16.m16.1.1.3.1" xref="S4.SS1.p1.16.m16.1.1.3.1.cmml">+</mo><mn id="S4.SS1.p1.16.m16.1.1.3.3" xref="S4.SS1.p1.16.m16.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.16.m16.1b"><apply id="S4.SS1.p1.16.m16.1.1.cmml" xref="S4.SS1.p1.16.m16.1.1"><divide id="S4.SS1.p1.16.m16.1.1.1.cmml" xref="S4.SS1.p1.16.m16.1.1"></divide><ci id="S4.SS1.p1.16.m16.1.1.2.cmml" xref="S4.SS1.p1.16.m16.1.1.2">𝑑</ci><apply id="S4.SS1.p1.16.m16.1.1.3.cmml" xref="S4.SS1.p1.16.m16.1.1.3"><plus id="S4.SS1.p1.16.m16.1.1.3.1.cmml" xref="S4.SS1.p1.16.m16.1.1.3.1"></plus><ci id="S4.SS1.p1.16.m16.1.1.3.2.cmml" xref="S4.SS1.p1.16.m16.1.1.3.2">𝑑</ci><cn id="S4.SS1.p1.16.m16.1.1.3.3.cmml" type="integer" xref="S4.SS1.p1.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.16.m16.1c">\frac{d}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.16.m16.1d">divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>. We terminate once we happen to query an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.p1.17.m17.1"><semantics id="S4.SS1.p1.17.m17.1a"><mi id="S4.SS1.p1.17.m17.1.1" xref="S4.SS1.p1.17.m17.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.17.m17.1b"><ci id="S4.SS1.p1.17.m17.1.1.cmml" xref="S4.SS1.p1.17.m17.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.17.m17.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.17.m17.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. It remains to prove that this must happen before the search space <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.18.m18.1"><semantics id="S4.SS1.p1.18.m18.1a"><mi id="S4.SS1.p1.18.m18.1.1" xref="S4.SS1.p1.18.m18.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.18.m18.1b"><ci id="S4.SS1.p1.18.m18.1.1.cmml" xref="S4.SS1.p1.18.m18.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.18.m18.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.18.m18.1d">italic_M</annotation></semantics></math> gets too small. To that end, we show in the next lemma that, whenever we query a point that is not an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.p1.19.m19.1"><semantics id="S4.SS1.p1.19.m19.1a"><mi id="S4.SS1.p1.19.m19.1.1" xref="S4.SS1.p1.19.m19.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.19.m19.1b"><ci id="S4.SS1.p1.19.m19.1.1.cmml" xref="S4.SS1.p1.19.m19.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.19.m19.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.19.m19.1d">italic_ε</annotation></semantics></math>-approximate fixpoint, a ball of some radius <math alttext="r_{\varepsilon,\lambda}" class="ltx_Math" display="inline" id="S4.SS1.p1.20.m20.2"><semantics id="S4.SS1.p1.20.m20.2a"><msub id="S4.SS1.p1.20.m20.2.3" xref="S4.SS1.p1.20.m20.2.3.cmml"><mi id="S4.SS1.p1.20.m20.2.3.2" xref="S4.SS1.p1.20.m20.2.3.2.cmml">r</mi><mrow id="S4.SS1.p1.20.m20.2.2.2.4" xref="S4.SS1.p1.20.m20.2.2.2.3.cmml"><mi id="S4.SS1.p1.20.m20.1.1.1.1" xref="S4.SS1.p1.20.m20.1.1.1.1.cmml">ε</mi><mo id="S4.SS1.p1.20.m20.2.2.2.4.1" xref="S4.SS1.p1.20.m20.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p1.20.m20.2.2.2.2" xref="S4.SS1.p1.20.m20.2.2.2.2.cmml">λ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.20.m20.2b"><apply id="S4.SS1.p1.20.m20.2.3.cmml" xref="S4.SS1.p1.20.m20.2.3"><csymbol cd="ambiguous" id="S4.SS1.p1.20.m20.2.3.1.cmml" xref="S4.SS1.p1.20.m20.2.3">subscript</csymbol><ci id="S4.SS1.p1.20.m20.2.3.2.cmml" xref="S4.SS1.p1.20.m20.2.3.2">𝑟</ci><list id="S4.SS1.p1.20.m20.2.2.2.3.cmml" xref="S4.SS1.p1.20.m20.2.2.2.4"><ci id="S4.SS1.p1.20.m20.1.1.1.1.cmml" xref="S4.SS1.p1.20.m20.1.1.1.1">𝜀</ci><ci id="S4.SS1.p1.20.m20.2.2.2.2.cmml" xref="S4.SS1.p1.20.m20.2.2.2.2">𝜆</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.20.m20.2c">r_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.20.m20.2d">italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math> around <math alttext="x^{\star}" class="ltx_Math" display="inline" id="S4.SS1.p1.21.m21.1"><semantics id="S4.SS1.p1.21.m21.1a"><msup id="S4.SS1.p1.21.m21.1.1" xref="S4.SS1.p1.21.m21.1.1.cmml"><mi id="S4.SS1.p1.21.m21.1.1.2" xref="S4.SS1.p1.21.m21.1.1.2.cmml">x</mi><mo id="S4.SS1.p1.21.m21.1.1.3" xref="S4.SS1.p1.21.m21.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.21.m21.1b"><apply id="S4.SS1.p1.21.m21.1.1.cmml" xref="S4.SS1.p1.21.m21.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.21.m21.1.1.1.cmml" xref="S4.SS1.p1.21.m21.1.1">superscript</csymbol><ci id="S4.SS1.p1.21.m21.1.1.2.cmml" xref="S4.SS1.p1.21.m21.1.1.2">𝑥</ci><ci id="S4.SS1.p1.21.m21.1.1.3.cmml" xref="S4.SS1.p1.21.m21.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.21.m21.1c">x^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.21.m21.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> cannot be discarded, and thus has to remain in the search space <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.p1.22.m22.1"><semantics id="S4.SS1.p1.22.m22.1a"><mi id="S4.SS1.p1.22.m22.1.1" xref="S4.SS1.p1.22.m22.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.22.m22.1b"><ci id="S4.SS1.p1.22.m22.1.1.cmml" xref="S4.SS1.p1.22.m22.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.22.m22.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.22.m22.1d">italic_M</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">Note that while the centerpoint theorems use limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msub id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" mathvariant="normal" xref="S4.SS1.p2.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">ℓ</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, we will now use bisector halfspaces in the analysis of our algorithm. <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem4" title="Observation 3.4. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.4</span></a> allows us to translate between the two.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Lemma 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.8.8">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.3"><semantics id="S4.Thmtheorem1.p1.1.1.m1.3a"><mrow id="S4.Thmtheorem1.p1.1.1.m1.3.4" 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xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.2" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S4.Thmtheorem1.p1.1.1.m1.3.3" mathvariant="normal" xref="S4.Thmtheorem1.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.3b"><apply id="S4.Thmtheorem1.p1.1.1.m1.3.4.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4"><in id="S4.Thmtheorem1.p1.1.1.m1.3.4.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.1"></in><ci id="S4.Thmtheorem1.p1.1.1.m1.3.4.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3"><union id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.2.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.2.2"><cn id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval><set id="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.4.3.3.2"><infinity id="S4.Thmtheorem1.p1.1.1.m1.3.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> be arbitrary. 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encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> be the unique fixpoint of the <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.3.m3.1a"><mi id="S4.Thmtheorem1.p1.3.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.1b"><ci id="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.1d">italic_λ</annotation></semantics></math>-contracting map <math alttext="f:[0,1]^{d}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.4"><semantics id="S4.Thmtheorem1.p1.4.4.m4.4a"><mrow 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stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.3.cmml">d</mi></msup><mo id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.1" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.1.cmml">→</mo><msup id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.cmml"><mrow id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.1.cmml"><mo id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem1.p1.4.4.m4.3.3" xref="S4.Thmtheorem1.p1.4.4.m4.3.3.cmml">0</mn><mo id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.1.cmml">,</mo><mn id="S4.Thmtheorem1.p1.4.4.m4.4.4" xref="S4.Thmtheorem1.p1.4.4.m4.4.4.cmml">1</mn><mo id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.3" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.4b"><apply id="S4.Thmtheorem1.p1.4.4.m4.4.5.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5"><ci id="S4.Thmtheorem1.p1.4.4.m4.4.5.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.1">:</ci><ci id="S4.Thmtheorem1.p1.4.4.m4.4.5.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.2">𝑓</ci><apply id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3"><ci id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.1">→</ci><apply id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.2.2"><cn id="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.1.1">0</cn><cn id="S4.Thmtheorem1.p1.4.4.m4.2.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.2.2">1</cn></interval><ci id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.2.3">𝑑</ci></apply><apply id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.2.2"><cn id="S4.Thmtheorem1.p1.4.4.m4.3.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.3.3">0</cn><cn id="S4.Thmtheorem1.p1.4.4.m4.4.4.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.4.4">1</cn></interval><ci id="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.4.5.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.4c">f:[0,1]^{d}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.4d">italic_f : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="x\in[0,1]^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.2"><semantics id="S4.Thmtheorem1.p1.5.5.m5.2a"><mrow id="S4.Thmtheorem1.p1.5.5.m5.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.cmml"><mi id="S4.Thmtheorem1.p1.5.5.m5.2.3.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2.cmml">x</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.1" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.1.cmml">∈</mo><msup id="S4.Thmtheorem1.p1.5.5.m5.2.3.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.cmml"><mrow id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.1.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.1.cmml">,</mo><mn id="S4.Thmtheorem1.p1.5.5.m5.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml">1</mn><mo id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.2b"><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3"><in id="S4.Thmtheorem1.p1.5.5.m5.2.3.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.1"></in><ci id="S4.Thmtheorem1.p1.5.5.m5.2.3.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.2">𝑥</ci><apply id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.2.2"><cn id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">0</cn><cn id="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.5.m5.2.2">1</cn></interval><ci id="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.2c">x\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.2d">italic_x ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be arbitrary. If <math alttext="x" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.6.m6.1"><semantics id="S4.Thmtheorem1.p1.6.6.m6.1a"><mi id="S4.Thmtheorem1.p1.6.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.6.m6.1b"><ci id="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.6.m6.1d">italic_x</annotation></semantics></math> is not an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.7.7.m7.1"><semantics id="S4.Thmtheorem1.p1.7.7.m7.1a"><mi id="S4.Thmtheorem1.p1.7.7.m7.1.1" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.7.7.m7.1b"><ci id="S4.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem1.p1.7.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.7.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.7.7.m7.1d">italic_ε</annotation></semantics></math>-approximate fixpoint of <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.8.8.m8.1"><semantics id="S4.Thmtheorem1.p1.8.8.m8.1a"><mi id="S4.Thmtheorem1.p1.8.8.m8.1.1" xref="S4.Thmtheorem1.p1.8.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.8.8.m8.1b"><ci id="S4.Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.8.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.8.8.m8.1d">italic_f</annotation></semantics></math>, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="B^{p}(x^{\star},r_{\varepsilon,\lambda})\cap H^{p}_{x,f(x)}=\varnothing" class="ltx_Math" display="block" id="S4.Ex1.m1.7"><semantics id="S4.Ex1.m1.7a"><mrow id="S4.Ex1.m1.7.7" xref="S4.Ex1.m1.7.7.cmml"><mrow id="S4.Ex1.m1.7.7.2" xref="S4.Ex1.m1.7.7.2.cmml"><mrow id="S4.Ex1.m1.7.7.2.2" xref="S4.Ex1.m1.7.7.2.2.cmml"><msup id="S4.Ex1.m1.7.7.2.2.4" xref="S4.Ex1.m1.7.7.2.2.4.cmml"><mi id="S4.Ex1.m1.7.7.2.2.4.2" xref="S4.Ex1.m1.7.7.2.2.4.2.cmml">B</mi><mi id="S4.Ex1.m1.7.7.2.2.4.3" xref="S4.Ex1.m1.7.7.2.2.4.3.cmml">p</mi></msup><mo id="S4.Ex1.m1.7.7.2.2.3" xref="S4.Ex1.m1.7.7.2.2.3.cmml">⁢</mo><mrow id="S4.Ex1.m1.7.7.2.2.2.2" xref="S4.Ex1.m1.7.7.2.2.2.3.cmml"><mo id="S4.Ex1.m1.7.7.2.2.2.2.3" stretchy="false" xref="S4.Ex1.m1.7.7.2.2.2.3.cmml">(</mo><msup id="S4.Ex1.m1.6.6.1.1.1.1.1" xref="S4.Ex1.m1.6.6.1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.6.6.1.1.1.1.1.2" xref="S4.Ex1.m1.6.6.1.1.1.1.1.2.cmml">x</mi><mo id="S4.Ex1.m1.6.6.1.1.1.1.1.3" xref="S4.Ex1.m1.6.6.1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S4.Ex1.m1.7.7.2.2.2.2.4" xref="S4.Ex1.m1.7.7.2.2.2.3.cmml">,</mo><msub id="S4.Ex1.m1.7.7.2.2.2.2.2" xref="S4.Ex1.m1.7.7.2.2.2.2.2.cmml"><mi id="S4.Ex1.m1.7.7.2.2.2.2.2.2" xref="S4.Ex1.m1.7.7.2.2.2.2.2.2.cmml">r</mi><mrow id="S4.Ex1.m1.2.2.2.4" xref="S4.Ex1.m1.2.2.2.3.cmml"><mi id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml">ε</mi><mo id="S4.Ex1.m1.2.2.2.4.1" xref="S4.Ex1.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex1.m1.2.2.2.2" xref="S4.Ex1.m1.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.Ex1.m1.7.7.2.2.2.2.5" stretchy="false" xref="S4.Ex1.m1.7.7.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex1.m1.7.7.2.3" xref="S4.Ex1.m1.7.7.2.3.cmml">∩</mo><msubsup id="S4.Ex1.m1.7.7.2.4" xref="S4.Ex1.m1.7.7.2.4.cmml"><mi id="S4.Ex1.m1.7.7.2.4.2.2" xref="S4.Ex1.m1.7.7.2.4.2.2.cmml">H</mi><mrow id="S4.Ex1.m1.5.5.3.3" xref="S4.Ex1.m1.5.5.3.4.cmml"><mi id="S4.Ex1.m1.4.4.2.2" xref="S4.Ex1.m1.4.4.2.2.cmml">x</mi><mo id="S4.Ex1.m1.5.5.3.3.2" xref="S4.Ex1.m1.5.5.3.4.cmml">,</mo><mrow id="S4.Ex1.m1.5.5.3.3.1" xref="S4.Ex1.m1.5.5.3.3.1.cmml"><mi id="S4.Ex1.m1.5.5.3.3.1.2" xref="S4.Ex1.m1.5.5.3.3.1.2.cmml">f</mi><mo id="S4.Ex1.m1.5.5.3.3.1.1" xref="S4.Ex1.m1.5.5.3.3.1.1.cmml">⁢</mo><mrow id="S4.Ex1.m1.5.5.3.3.1.3.2" xref="S4.Ex1.m1.5.5.3.3.1.cmml"><mo id="S4.Ex1.m1.5.5.3.3.1.3.2.1" stretchy="false" xref="S4.Ex1.m1.5.5.3.3.1.cmml">(</mo><mi id="S4.Ex1.m1.3.3.1.1" xref="S4.Ex1.m1.3.3.1.1.cmml">x</mi><mo id="S4.Ex1.m1.5.5.3.3.1.3.2.2" stretchy="false" xref="S4.Ex1.m1.5.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><mi id="S4.Ex1.m1.7.7.2.4.2.3" xref="S4.Ex1.m1.7.7.2.4.2.3.cmml">p</mi></msubsup></mrow><mo id="S4.Ex1.m1.7.7.3" xref="S4.Ex1.m1.7.7.3.cmml">=</mo><mi id="S4.Ex1.m1.7.7.4" mathvariant="normal" xref="S4.Ex1.m1.7.7.4.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex1.m1.7b"><apply id="S4.Ex1.m1.7.7.cmml" xref="S4.Ex1.m1.7.7"><eq id="S4.Ex1.m1.7.7.3.cmml" xref="S4.Ex1.m1.7.7.3"></eq><apply id="S4.Ex1.m1.7.7.2.cmml" xref="S4.Ex1.m1.7.7.2"><intersect id="S4.Ex1.m1.7.7.2.3.cmml" xref="S4.Ex1.m1.7.7.2.3"></intersect><apply id="S4.Ex1.m1.7.7.2.2.cmml" xref="S4.Ex1.m1.7.7.2.2"><times id="S4.Ex1.m1.7.7.2.2.3.cmml" xref="S4.Ex1.m1.7.7.2.2.3"></times><apply id="S4.Ex1.m1.7.7.2.2.4.cmml" xref="S4.Ex1.m1.7.7.2.2.4"><csymbol cd="ambiguous" id="S4.Ex1.m1.7.7.2.2.4.1.cmml" xref="S4.Ex1.m1.7.7.2.2.4">superscript</csymbol><ci id="S4.Ex1.m1.7.7.2.2.4.2.cmml" xref="S4.Ex1.m1.7.7.2.2.4.2">𝐵</ci><ci id="S4.Ex1.m1.7.7.2.2.4.3.cmml" xref="S4.Ex1.m1.7.7.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S4.Ex1.m1.7.7.2.2.2.3.cmml" xref="S4.Ex1.m1.7.7.2.2.2.2"><apply id="S4.Ex1.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex1.m1.6.6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex1.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.Ex1.m1.6.6.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex1.m1.6.6.1.1.1.1.1.2.cmml" xref="S4.Ex1.m1.6.6.1.1.1.1.1.2">𝑥</ci><ci id="S4.Ex1.m1.6.6.1.1.1.1.1.3.cmml" xref="S4.Ex1.m1.6.6.1.1.1.1.1.3">⋆</ci></apply><apply id="S4.Ex1.m1.7.7.2.2.2.2.2.cmml" xref="S4.Ex1.m1.7.7.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex1.m1.7.7.2.2.2.2.2.1.cmml" xref="S4.Ex1.m1.7.7.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex1.m1.7.7.2.2.2.2.2.2.cmml" xref="S4.Ex1.m1.7.7.2.2.2.2.2.2">𝑟</ci><list id="S4.Ex1.m1.2.2.2.3.cmml" xref="S4.Ex1.m1.2.2.2.4"><ci id="S4.Ex1.m1.1.1.1.1.cmml" xref="S4.Ex1.m1.1.1.1.1">𝜀</ci><ci id="S4.Ex1.m1.2.2.2.2.cmml" xref="S4.Ex1.m1.2.2.2.2">𝜆</ci></list></apply></interval></apply><apply id="S4.Ex1.m1.7.7.2.4.cmml" xref="S4.Ex1.m1.7.7.2.4"><csymbol cd="ambiguous" id="S4.Ex1.m1.7.7.2.4.1.cmml" xref="S4.Ex1.m1.7.7.2.4">subscript</csymbol><apply id="S4.Ex1.m1.7.7.2.4.2.cmml" xref="S4.Ex1.m1.7.7.2.4"><csymbol cd="ambiguous" id="S4.Ex1.m1.7.7.2.4.2.1.cmml" xref="S4.Ex1.m1.7.7.2.4">superscript</csymbol><ci id="S4.Ex1.m1.7.7.2.4.2.2.cmml" xref="S4.Ex1.m1.7.7.2.4.2.2">𝐻</ci><ci id="S4.Ex1.m1.7.7.2.4.2.3.cmml" xref="S4.Ex1.m1.7.7.2.4.2.3">𝑝</ci></apply><list id="S4.Ex1.m1.5.5.3.4.cmml" xref="S4.Ex1.m1.5.5.3.3"><ci id="S4.Ex1.m1.4.4.2.2.cmml" xref="S4.Ex1.m1.4.4.2.2">𝑥</ci><apply id="S4.Ex1.m1.5.5.3.3.1.cmml" xref="S4.Ex1.m1.5.5.3.3.1"><times id="S4.Ex1.m1.5.5.3.3.1.1.cmml" xref="S4.Ex1.m1.5.5.3.3.1.1"></times><ci id="S4.Ex1.m1.5.5.3.3.1.2.cmml" xref="S4.Ex1.m1.5.5.3.3.1.2">𝑓</ci><ci id="S4.Ex1.m1.3.3.1.1.cmml" xref="S4.Ex1.m1.3.3.1.1">𝑥</ci></apply></list></apply></apply><emptyset id="S4.Ex1.m1.7.7.4.cmml" xref="S4.Ex1.m1.7.7.4"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex1.m1.7c">B^{p}(x^{\star},r_{\varepsilon,\lambda})\cap H^{p}_{x,f(x)}=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S4.Ex1.m1.7d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT ) ∩ italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_f ( italic_x ) end_POSTSUBSCRIPT = ∅</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.9.1">for <math alttext="r_{\varepsilon,\lambda}=\frac{\varepsilon-\varepsilon\lambda}{2+2\lambda}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.9.1.m1.2"><semantics id="S4.Thmtheorem1.p1.9.1.m1.2a"><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.cmml"><msub id="S4.Thmtheorem1.p1.9.1.m1.2.3.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.2.cmml"><mi id="S4.Thmtheorem1.p1.9.1.m1.2.3.2.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.2.2.cmml">r</mi><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.2.2.4" xref="S4.Thmtheorem1.p1.9.1.m1.2.2.2.3.cmml"><mi id="S4.Thmtheorem1.p1.9.1.m1.1.1.1.1" xref="S4.Thmtheorem1.p1.9.1.m1.1.1.1.1.cmml">ε</mi><mo id="S4.Thmtheorem1.p1.9.1.m1.2.2.2.4.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem1.p1.9.1.m1.2.2.2.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.Thmtheorem1.p1.9.1.m1.2.3.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.1.cmml">=</mo><mfrac id="S4.Thmtheorem1.p1.9.1.m1.2.3.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.cmml"><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.cmml"><mi id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.2.cmml">ε</mi><mo id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.1.cmml">−</mo><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.cmml"><mi id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.2.cmml">ε</mi><mo id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.3.cmml">λ</mi></mrow></mrow><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.cmml"><mn id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.2.cmml">2</mn><mo id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.1.cmml">+</mo><mrow id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.cmml"><mn id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.2" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.2.cmml">2</mn><mo id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.1" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.3" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.3.cmml">λ</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.9.1.m1.2b"><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3"><eq id="S4.Thmtheorem1.p1.9.1.m1.2.3.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.1"></eq><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.9.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.9.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.2.2">𝑟</ci><list id="S4.Thmtheorem1.p1.9.1.m1.2.2.2.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.2.2.4"><ci id="S4.Thmtheorem1.p1.9.1.m1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.1.1.1.1">𝜀</ci><ci id="S4.Thmtheorem1.p1.9.1.m1.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.2.2.2">𝜆</ci></list></apply><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3"><divide id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3"></divide><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2"><minus id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.1"></minus><ci id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.2">𝜀</ci><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3"><times id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.1"></times><ci id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.2.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.2">𝜀</ci><ci id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.2.3.3">𝜆</ci></apply></apply><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3"><plus id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.1"></plus><cn id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.2">2</cn><apply id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3"><times id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.1.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.1"></times><cn id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.2">2</cn><ci id="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.3.cmml" xref="S4.Thmtheorem1.p1.9.1.m1.2.3.3.3.3.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.9.1.m1.2c">r_{\varepsilon,\lambda}=\frac{\varepsilon-\varepsilon\lambda}{2+2\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.9.1.m1.2d">italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT = divide start_ARG italic_ε - italic_ε italic_λ end_ARG start_ARG 2 + 2 italic_λ end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.2.p1"> <p class="ltx_p" id="S4.SS1.2.p1.7">Let <math alttext="z\in B^{p}(x^{\star},r_{\varepsilon,\lambda})" class="ltx_Math" display="inline" id="S4.SS1.2.p1.1.m1.4"><semantics id="S4.SS1.2.p1.1.m1.4a"><mrow id="S4.SS1.2.p1.1.m1.4.4" xref="S4.SS1.2.p1.1.m1.4.4.cmml"><mi id="S4.SS1.2.p1.1.m1.4.4.4" xref="S4.SS1.2.p1.1.m1.4.4.4.cmml">z</mi><mo id="S4.SS1.2.p1.1.m1.4.4.3" xref="S4.SS1.2.p1.1.m1.4.4.3.cmml">∈</mo><mrow id="S4.SS1.2.p1.1.m1.4.4.2" xref="S4.SS1.2.p1.1.m1.4.4.2.cmml"><msup id="S4.SS1.2.p1.1.m1.4.4.2.4" xref="S4.SS1.2.p1.1.m1.4.4.2.4.cmml"><mi id="S4.SS1.2.p1.1.m1.4.4.2.4.2" xref="S4.SS1.2.p1.1.m1.4.4.2.4.2.cmml">B</mi><mi id="S4.SS1.2.p1.1.m1.4.4.2.4.3" xref="S4.SS1.2.p1.1.m1.4.4.2.4.3.cmml">p</mi></msup><mo id="S4.SS1.2.p1.1.m1.4.4.2.3" xref="S4.SS1.2.p1.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="S4.SS1.2.p1.1.m1.4.4.2.2.2" xref="S4.SS1.2.p1.1.m1.4.4.2.2.3.cmml"><mo id="S4.SS1.2.p1.1.m1.4.4.2.2.2.3" stretchy="false" xref="S4.SS1.2.p1.1.m1.4.4.2.2.3.cmml">(</mo><msup id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.2" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.2.cmml">x</mi><mo id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.3" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S4.SS1.2.p1.1.m1.4.4.2.2.2.4" xref="S4.SS1.2.p1.1.m1.4.4.2.2.3.cmml">,</mo><msub id="S4.SS1.2.p1.1.m1.4.4.2.2.2.2" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.cmml"><mi id="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.2" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.2.cmml">r</mi><mrow id="S4.SS1.2.p1.1.m1.2.2.2.4" xref="S4.SS1.2.p1.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.2.p1.1.m1.1.1.1.1" xref="S4.SS1.2.p1.1.m1.1.1.1.1.cmml">ε</mi><mo id="S4.SS1.2.p1.1.m1.2.2.2.4.1" xref="S4.SS1.2.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.2.p1.1.m1.2.2.2.2" xref="S4.SS1.2.p1.1.m1.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS1.2.p1.1.m1.4.4.2.2.2.5" stretchy="false" xref="S4.SS1.2.p1.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.1.m1.4b"><apply id="S4.SS1.2.p1.1.m1.4.4.cmml" xref="S4.SS1.2.p1.1.m1.4.4"><in id="S4.SS1.2.p1.1.m1.4.4.3.cmml" xref="S4.SS1.2.p1.1.m1.4.4.3"></in><ci id="S4.SS1.2.p1.1.m1.4.4.4.cmml" xref="S4.SS1.2.p1.1.m1.4.4.4">𝑧</ci><apply id="S4.SS1.2.p1.1.m1.4.4.2.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2"><times id="S4.SS1.2.p1.1.m1.4.4.2.3.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.3"></times><apply id="S4.SS1.2.p1.1.m1.4.4.2.4.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.4"><csymbol cd="ambiguous" id="S4.SS1.2.p1.1.m1.4.4.2.4.1.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.4">superscript</csymbol><ci id="S4.SS1.2.p1.1.m1.4.4.2.4.2.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.4.2">𝐵</ci><ci id="S4.SS1.2.p1.1.m1.4.4.2.4.3.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.4.3">𝑝</ci></apply><interval closure="open" id="S4.SS1.2.p1.1.m1.4.4.2.2.3.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2"><apply id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.2">𝑥</ci><ci id="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.1.m1.3.3.1.1.1.1.3">⋆</ci></apply><apply id="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.1.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2.2">subscript</csymbol><ci id="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.2.cmml" xref="S4.SS1.2.p1.1.m1.4.4.2.2.2.2.2">𝑟</ci><list id="S4.SS1.2.p1.1.m1.2.2.2.3.cmml" xref="S4.SS1.2.p1.1.m1.2.2.2.4"><ci id="S4.SS1.2.p1.1.m1.1.1.1.1.cmml" xref="S4.SS1.2.p1.1.m1.1.1.1.1">𝜀</ci><ci id="S4.SS1.2.p1.1.m1.2.2.2.2.cmml" xref="S4.SS1.2.p1.1.m1.2.2.2.2">𝜆</ci></list></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.1.m1.4c">z\in B^{p}(x^{\star},r_{\varepsilon,\lambda})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.1.m1.4d">italic_z ∈ italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT )</annotation></semantics></math> be arbitrary. We need to show <math alttext="||z-x||_{p}>||z-f(x)||_{p}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.2.m2.3"><semantics id="S4.SS1.2.p1.2.m2.3a"><mrow id="S4.SS1.2.p1.2.m2.3.3" xref="S4.SS1.2.p1.2.m2.3.3.cmml"><msub id="S4.SS1.2.p1.2.m2.2.2.1" xref="S4.SS1.2.p1.2.m2.2.2.1.cmml"><mrow id="S4.SS1.2.p1.2.m2.2.2.1.1.1" xref="S4.SS1.2.p1.2.m2.2.2.1.1.2.cmml"><mo id="S4.SS1.2.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.2.m2.2.2.1.1.2.1.cmml">‖</mo><mrow id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.2" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.2.cmml">z</mi><mo id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.1" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.3" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.SS1.2.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.2.p1.2.m2.2.2.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.SS1.2.p1.2.m2.2.2.1.3" xref="S4.SS1.2.p1.2.m2.2.2.1.3.cmml">p</mi></msub><mo id="S4.SS1.2.p1.2.m2.3.3.3" xref="S4.SS1.2.p1.2.m2.3.3.3.cmml">></mo><msub id="S4.SS1.2.p1.2.m2.3.3.2" xref="S4.SS1.2.p1.2.m2.3.3.2.cmml"><mrow id="S4.SS1.2.p1.2.m2.3.3.2.1.1" xref="S4.SS1.2.p1.2.m2.3.3.2.1.2.cmml"><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.2" stretchy="false" xref="S4.SS1.2.p1.2.m2.3.3.2.1.2.1.cmml">‖</mo><mrow id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.cmml"><mi id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.2" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.2.cmml">z</mi><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.1" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.1.cmml">−</mo><mrow id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.cmml"><mi id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.2" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.2.cmml">f</mi><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.1" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.3.2" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.cmml"><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.3.2.1" stretchy="false" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.cmml">(</mo><mi id="S4.SS1.2.p1.2.m2.1.1" xref="S4.SS1.2.p1.2.m2.1.1.cmml">x</mi><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.3.2.2" stretchy="false" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.2.p1.2.m2.3.3.2.1.1.3" stretchy="false" xref="S4.SS1.2.p1.2.m2.3.3.2.1.2.1.cmml">‖</mo></mrow><mi id="S4.SS1.2.p1.2.m2.3.3.2.3" xref="S4.SS1.2.p1.2.m2.3.3.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.2.m2.3b"><apply id="S4.SS1.2.p1.2.m2.3.3.cmml" xref="S4.SS1.2.p1.2.m2.3.3"><gt id="S4.SS1.2.p1.2.m2.3.3.3.cmml" xref="S4.SS1.2.p1.2.m2.3.3.3"></gt><apply id="S4.SS1.2.p1.2.m2.2.2.1.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.2.m2.2.2.1.2.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1">subscript</csymbol><apply id="S4.SS1.2.p1.2.m2.2.2.1.1.2.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS1.2.p1.2.m2.2.2.1.1.2.1.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.2">norm</csymbol><apply id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1"><minus id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.1"></minus><ci id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.2">𝑧</ci><ci id="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.SS1.2.p1.2.m2.2.2.1.3.cmml" xref="S4.SS1.2.p1.2.m2.2.2.1.3">𝑝</ci></apply><apply id="S4.SS1.2.p1.2.m2.3.3.2.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.2.p1.2.m2.3.3.2.2.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2">subscript</csymbol><apply id="S4.SS1.2.p1.2.m2.3.3.2.1.2.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1"><csymbol cd="latexml" id="S4.SS1.2.p1.2.m2.3.3.2.1.2.1.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.2">norm</csymbol><apply id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1"><minus id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.1.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.1"></minus><ci id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.2.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.2">𝑧</ci><apply id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3"><times id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.1.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.1"></times><ci id="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.2.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.1.1.1.3.2">𝑓</ci><ci id="S4.SS1.2.p1.2.m2.1.1.cmml" xref="S4.SS1.2.p1.2.m2.1.1">𝑥</ci></apply></apply></apply><ci id="S4.SS1.2.p1.2.m2.3.3.2.3.cmml" xref="S4.SS1.2.p1.2.m2.3.3.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.2.m2.3c">||z-x||_{p}>||z-f(x)||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.2.m2.3d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > | | italic_z - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. Let us first collect some facts. Concretely, we know that <math alttext="x" class="ltx_Math" display="inline" id="S4.SS1.2.p1.3.m3.1"><semantics id="S4.SS1.2.p1.3.m3.1a"><mi id="S4.SS1.2.p1.3.m3.1.1" xref="S4.SS1.2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.3.m3.1b"><ci id="S4.SS1.2.p1.3.m3.1.1.cmml" xref="S4.SS1.2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.3.m3.1d">italic_x</annotation></semantics></math> is not an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.2.p1.4.m4.1"><semantics id="S4.SS1.2.p1.4.m4.1a"><mi id="S4.SS1.2.p1.4.m4.1.1" xref="S4.SS1.2.p1.4.m4.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.4.m4.1b"><ci id="S4.SS1.2.p1.4.m4.1.1.cmml" xref="S4.SS1.2.p1.4.m4.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.4.m4.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.4.m4.1d">italic_ε</annotation></semantics></math>-approximate fixpoint, <math alttext="z\in B^{p}(x^{\star},r_{\varepsilon,\lambda})" class="ltx_Math" display="inline" id="S4.SS1.2.p1.5.m5.4"><semantics id="S4.SS1.2.p1.5.m5.4a"><mrow id="S4.SS1.2.p1.5.m5.4.4" xref="S4.SS1.2.p1.5.m5.4.4.cmml"><mi id="S4.SS1.2.p1.5.m5.4.4.4" xref="S4.SS1.2.p1.5.m5.4.4.4.cmml">z</mi><mo id="S4.SS1.2.p1.5.m5.4.4.3" xref="S4.SS1.2.p1.5.m5.4.4.3.cmml">∈</mo><mrow id="S4.SS1.2.p1.5.m5.4.4.2" xref="S4.SS1.2.p1.5.m5.4.4.2.cmml"><msup id="S4.SS1.2.p1.5.m5.4.4.2.4" xref="S4.SS1.2.p1.5.m5.4.4.2.4.cmml"><mi id="S4.SS1.2.p1.5.m5.4.4.2.4.2" xref="S4.SS1.2.p1.5.m5.4.4.2.4.2.cmml">B</mi><mi id="S4.SS1.2.p1.5.m5.4.4.2.4.3" xref="S4.SS1.2.p1.5.m5.4.4.2.4.3.cmml">p</mi></msup><mo id="S4.SS1.2.p1.5.m5.4.4.2.3" xref="S4.SS1.2.p1.5.m5.4.4.2.3.cmml">⁢</mo><mrow id="S4.SS1.2.p1.5.m5.4.4.2.2.2" xref="S4.SS1.2.p1.5.m5.4.4.2.2.3.cmml"><mo id="S4.SS1.2.p1.5.m5.4.4.2.2.2.3" stretchy="false" xref="S4.SS1.2.p1.5.m5.4.4.2.2.3.cmml">(</mo><msup id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.cmml"><mi id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.2" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.2.cmml">x</mi><mo id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.3" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S4.SS1.2.p1.5.m5.4.4.2.2.2.4" xref="S4.SS1.2.p1.5.m5.4.4.2.2.3.cmml">,</mo><msub id="S4.SS1.2.p1.5.m5.4.4.2.2.2.2" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.cmml"><mi id="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.2" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.2.cmml">r</mi><mrow id="S4.SS1.2.p1.5.m5.2.2.2.4" xref="S4.SS1.2.p1.5.m5.2.2.2.3.cmml"><mi id="S4.SS1.2.p1.5.m5.1.1.1.1" xref="S4.SS1.2.p1.5.m5.1.1.1.1.cmml">ε</mi><mo id="S4.SS1.2.p1.5.m5.2.2.2.4.1" xref="S4.SS1.2.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS1.2.p1.5.m5.2.2.2.2" xref="S4.SS1.2.p1.5.m5.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS1.2.p1.5.m5.4.4.2.2.2.5" stretchy="false" xref="S4.SS1.2.p1.5.m5.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.5.m5.4b"><apply id="S4.SS1.2.p1.5.m5.4.4.cmml" xref="S4.SS1.2.p1.5.m5.4.4"><in id="S4.SS1.2.p1.5.m5.4.4.3.cmml" xref="S4.SS1.2.p1.5.m5.4.4.3"></in><ci id="S4.SS1.2.p1.5.m5.4.4.4.cmml" xref="S4.SS1.2.p1.5.m5.4.4.4">𝑧</ci><apply id="S4.SS1.2.p1.5.m5.4.4.2.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2"><times id="S4.SS1.2.p1.5.m5.4.4.2.3.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.3"></times><apply id="S4.SS1.2.p1.5.m5.4.4.2.4.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.4"><csymbol cd="ambiguous" id="S4.SS1.2.p1.5.m5.4.4.2.4.1.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.4">superscript</csymbol><ci id="S4.SS1.2.p1.5.m5.4.4.2.4.2.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.4.2">𝐵</ci><ci id="S4.SS1.2.p1.5.m5.4.4.2.4.3.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.4.3">𝑝</ci></apply><interval closure="open" id="S4.SS1.2.p1.5.m5.4.4.2.2.3.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2"><apply id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.cmml" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1">superscript</csymbol><ci id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.2">𝑥</ci><ci id="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.5.m5.3.3.1.1.1.1.3">⋆</ci></apply><apply id="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.1.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2.2">subscript</csymbol><ci id="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.2.cmml" xref="S4.SS1.2.p1.5.m5.4.4.2.2.2.2.2">𝑟</ci><list id="S4.SS1.2.p1.5.m5.2.2.2.3.cmml" xref="S4.SS1.2.p1.5.m5.2.2.2.4"><ci id="S4.SS1.2.p1.5.m5.1.1.1.1.cmml" xref="S4.SS1.2.p1.5.m5.1.1.1.1">𝜀</ci><ci id="S4.SS1.2.p1.5.m5.2.2.2.2.cmml" xref="S4.SS1.2.p1.5.m5.2.2.2.2">𝜆</ci></list></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.5.m5.4c">z\in B^{p}(x^{\star},r_{\varepsilon,\lambda})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.5.m5.4d">italic_z ∈ italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT )</annotation></semantics></math>, and that <math alttext="f" class="ltx_Math" display="inline" id="S4.SS1.2.p1.6.m6.1"><semantics id="S4.SS1.2.p1.6.m6.1a"><mi id="S4.SS1.2.p1.6.m6.1.1" xref="S4.SS1.2.p1.6.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.6.m6.1b"><ci id="S4.SS1.2.p1.6.m6.1.1.cmml" xref="S4.SS1.2.p1.6.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.6.m6.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.6.m6.1d">italic_f</annotation></semantics></math> is <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS1.2.p1.7.m7.1"><semantics id="S4.SS1.2.p1.7.m7.1a"><mi id="S4.SS1.2.p1.7.m7.1.1" xref="S4.SS1.2.p1.7.m7.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.7.m7.1b"><ci id="S4.SS1.2.p1.7.m7.1.1.cmml" xref="S4.SS1.2.p1.7.m7.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.7.m7.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.7.m7.1d">italic_λ</annotation></semantics></math>-contracting. This gives us the following inequalities:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx2"> <tbody id="S4.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||f(x)-x||_{p}" class="ltx_Math" display="inline" id="S4.E1.m1.2"><semantics id="S4.E1.m1.2a"><msub id="S4.E1.m1.2.2" xref="S4.E1.m1.2.2.cmml"><mrow id="S4.E1.m1.2.2.1.1" xref="S4.E1.m1.2.2.1.2.cmml"><mo id="S4.E1.m1.2.2.1.1.2" stretchy="false" xref="S4.E1.m1.2.2.1.2.1.cmml">‖</mo><mrow id="S4.E1.m1.2.2.1.1.1" xref="S4.E1.m1.2.2.1.1.1.cmml"><mrow id="S4.E1.m1.2.2.1.1.1.2" xref="S4.E1.m1.2.2.1.1.1.2.cmml"><mi id="S4.E1.m1.2.2.1.1.1.2.2" xref="S4.E1.m1.2.2.1.1.1.2.2.cmml">f</mi><mo id="S4.E1.m1.2.2.1.1.1.2.1" xref="S4.E1.m1.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.E1.m1.2.2.1.1.1.2.3.2" xref="S4.E1.m1.2.2.1.1.1.2.cmml"><mo id="S4.E1.m1.2.2.1.1.1.2.3.2.1" stretchy="false" xref="S4.E1.m1.2.2.1.1.1.2.cmml">(</mo><mi id="S4.E1.m1.1.1" xref="S4.E1.m1.1.1.cmml">x</mi><mo id="S4.E1.m1.2.2.1.1.1.2.3.2.2" stretchy="false" xref="S4.E1.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.2.2.1.1.1.1" xref="S4.E1.m1.2.2.1.1.1.1.cmml">−</mo><mi id="S4.E1.m1.2.2.1.1.1.3" xref="S4.E1.m1.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S4.E1.m1.2.2.1.1.3" stretchy="false" xref="S4.E1.m1.2.2.1.2.1.cmml">‖</mo></mrow><mi id="S4.E1.m1.2.2.3" xref="S4.E1.m1.2.2.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E1.m1.2b"><apply id="S4.E1.m1.2.2.cmml" xref="S4.E1.m1.2.2"><csymbol cd="ambiguous" id="S4.E1.m1.2.2.2.cmml" xref="S4.E1.m1.2.2">subscript</csymbol><apply id="S4.E1.m1.2.2.1.2.cmml" xref="S4.E1.m1.2.2.1.1"><csymbol cd="latexml" id="S4.E1.m1.2.2.1.2.1.cmml" xref="S4.E1.m1.2.2.1.1.2">norm</csymbol><apply id="S4.E1.m1.2.2.1.1.1.cmml" xref="S4.E1.m1.2.2.1.1.1"><minus id="S4.E1.m1.2.2.1.1.1.1.cmml" xref="S4.E1.m1.2.2.1.1.1.1"></minus><apply id="S4.E1.m1.2.2.1.1.1.2.cmml" xref="S4.E1.m1.2.2.1.1.1.2"><times id="S4.E1.m1.2.2.1.1.1.2.1.cmml" xref="S4.E1.m1.2.2.1.1.1.2.1"></times><ci id="S4.E1.m1.2.2.1.1.1.2.2.cmml" xref="S4.E1.m1.2.2.1.1.1.2.2">𝑓</ci><ci id="S4.E1.m1.1.1.cmml" xref="S4.E1.m1.1.1">𝑥</ci></apply><ci id="S4.E1.m1.2.2.1.1.1.3.cmml" xref="S4.E1.m1.2.2.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.E1.m1.2.2.3.cmml" xref="S4.E1.m1.2.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.2c">\displaystyle||f(x)-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.2d">| | italic_f ( italic_x ) - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle>\varepsilon" class="ltx_Math" display="inline" id="S4.E1.m2.1"><semantics id="S4.E1.m2.1a"><mrow id="S4.E1.m2.1.1" xref="S4.E1.m2.1.1.cmml"><mi id="S4.E1.m2.1.1.2" xref="S4.E1.m2.1.1.2.cmml"></mi><mo id="S4.E1.m2.1.1.1" xref="S4.E1.m2.1.1.1.cmml">></mo><mi id="S4.E1.m2.1.1.3" xref="S4.E1.m2.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.E1.m2.1b"><apply id="S4.E1.m2.1.1.cmml" xref="S4.E1.m2.1.1"><gt id="S4.E1.m2.1.1.1.cmml" xref="S4.E1.m2.1.1.1"></gt><csymbol cd="latexml" id="S4.E1.m2.1.1.2.cmml" xref="S4.E1.m2.1.1.2">absent</csymbol><ci id="S4.E1.m2.1.1.3.cmml" xref="S4.E1.m2.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m2.1c">\displaystyle>\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m2.1d">> italic_ε</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.1)</span></td> </tr></tbody> <tbody id="S4.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||z-x^{\star}||_{p}" class="ltx_Math" display="inline" id="S4.E2.m1.1"><semantics id="S4.E2.m1.1a"><msub id="S4.E2.m1.1.1" xref="S4.E2.m1.1.1.cmml"><mrow id="S4.E2.m1.1.1.1.1" xref="S4.E2.m1.1.1.1.2.cmml"><mo id="S4.E2.m1.1.1.1.1.2" stretchy="false" xref="S4.E2.m1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E2.m1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.1.1.1.1.1.2" xref="S4.E2.m1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.E2.m1.1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.E2.m1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.3.cmml"><mi id="S4.E2.m1.1.1.1.1.1.3.2" xref="S4.E2.m1.1.1.1.1.1.3.2.cmml">x</mi><mo id="S4.E2.m1.1.1.1.1.1.3.3" xref="S4.E2.m1.1.1.1.1.1.3.3.cmml">⋆</mo></msup></mrow><mo id="S4.E2.m1.1.1.1.1.3" stretchy="false" xref="S4.E2.m1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.E2.m1.1.1.3" xref="S4.E2.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E2.m1.1b"><apply id="S4.E2.m1.1.1.cmml" xref="S4.E2.m1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.1.1.2.cmml" xref="S4.E2.m1.1.1">subscript</csymbol><apply id="S4.E2.m1.1.1.1.2.cmml" xref="S4.E2.m1.1.1.1.1"><csymbol cd="latexml" id="S4.E2.m1.1.1.1.2.1.cmml" xref="S4.E2.m1.1.1.1.1.2">norm</csymbol><apply id="S4.E2.m1.1.1.1.1.1.cmml" xref="S4.E2.m1.1.1.1.1.1"><minus id="S4.E2.m1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.1.1.1.1.1.1"></minus><ci id="S4.E2.m1.1.1.1.1.1.2.cmml" xref="S4.E2.m1.1.1.1.1.1.2">𝑧</ci><apply id="S4.E2.m1.1.1.1.1.1.3.cmml" xref="S4.E2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E2.m1.1.1.1.1.1.3.1.cmml" xref="S4.E2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.E2.m1.1.1.1.1.1.3.2.cmml" xref="S4.E2.m1.1.1.1.1.1.3.2">𝑥</ci><ci id="S4.E2.m1.1.1.1.1.1.3.3.cmml" xref="S4.E2.m1.1.1.1.1.1.3.3">⋆</ci></apply></apply></apply><ci id="S4.E2.m1.1.1.3.cmml" xref="S4.E2.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.1c">\displaystyle||z-x^{\star}||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.1d">| | italic_z - italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq r_{\varepsilon,\lambda}" class="ltx_Math" display="inline" id="S4.E2.m2.2"><semantics id="S4.E2.m2.2a"><mrow id="S4.E2.m2.2.3" xref="S4.E2.m2.2.3.cmml"><mi id="S4.E2.m2.2.3.2" xref="S4.E2.m2.2.3.2.cmml"></mi><mo id="S4.E2.m2.2.3.1" xref="S4.E2.m2.2.3.1.cmml">≤</mo><msub id="S4.E2.m2.2.3.3" xref="S4.E2.m2.2.3.3.cmml"><mi id="S4.E2.m2.2.3.3.2" xref="S4.E2.m2.2.3.3.2.cmml">r</mi><mrow id="S4.E2.m2.2.2.2.4" xref="S4.E2.m2.2.2.2.3.cmml"><mi id="S4.E2.m2.1.1.1.1" xref="S4.E2.m2.1.1.1.1.cmml">ε</mi><mo id="S4.E2.m2.2.2.2.4.1" xref="S4.E2.m2.2.2.2.3.cmml">,</mo><mi id="S4.E2.m2.2.2.2.2" xref="S4.E2.m2.2.2.2.2.cmml">λ</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.E2.m2.2b"><apply id="S4.E2.m2.2.3.cmml" xref="S4.E2.m2.2.3"><leq id="S4.E2.m2.2.3.1.cmml" xref="S4.E2.m2.2.3.1"></leq><csymbol cd="latexml" id="S4.E2.m2.2.3.2.cmml" xref="S4.E2.m2.2.3.2">absent</csymbol><apply id="S4.E2.m2.2.3.3.cmml" xref="S4.E2.m2.2.3.3"><csymbol cd="ambiguous" id="S4.E2.m2.2.3.3.1.cmml" xref="S4.E2.m2.2.3.3">subscript</csymbol><ci id="S4.E2.m2.2.3.3.2.cmml" xref="S4.E2.m2.2.3.3.2">𝑟</ci><list id="S4.E2.m2.2.2.2.3.cmml" xref="S4.E2.m2.2.2.2.4"><ci id="S4.E2.m2.1.1.1.1.cmml" xref="S4.E2.m2.1.1.1.1">𝜀</ci><ci id="S4.E2.m2.2.2.2.2.cmml" xref="S4.E2.m2.2.2.2.2">𝜆</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m2.2c">\displaystyle\leq r_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m2.2d">≤ italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.2)</span></td> </tr></tbody> <tbody id="S4.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||x^{\star}-f(x)||_{p}" class="ltx_Math" display="inline" id="S4.E3.m1.2"><semantics id="S4.E3.m1.2a"><msub id="S4.E3.m1.2.2" xref="S4.E3.m1.2.2.cmml"><mrow id="S4.E3.m1.2.2.1.1" xref="S4.E3.m1.2.2.1.2.cmml"><mo id="S4.E3.m1.2.2.1.1.2" stretchy="false" xref="S4.E3.m1.2.2.1.2.1.cmml">‖</mo><mrow id="S4.E3.m1.2.2.1.1.1" xref="S4.E3.m1.2.2.1.1.1.cmml"><msup id="S4.E3.m1.2.2.1.1.1.2" xref="S4.E3.m1.2.2.1.1.1.2.cmml"><mi id="S4.E3.m1.2.2.1.1.1.2.2" xref="S4.E3.m1.2.2.1.1.1.2.2.cmml">x</mi><mo id="S4.E3.m1.2.2.1.1.1.2.3" xref="S4.E3.m1.2.2.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S4.E3.m1.2.2.1.1.1.1" xref="S4.E3.m1.2.2.1.1.1.1.cmml">−</mo><mrow id="S4.E3.m1.2.2.1.1.1.3" xref="S4.E3.m1.2.2.1.1.1.3.cmml"><mi id="S4.E3.m1.2.2.1.1.1.3.2" xref="S4.E3.m1.2.2.1.1.1.3.2.cmml">f</mi><mo id="S4.E3.m1.2.2.1.1.1.3.1" xref="S4.E3.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E3.m1.2.2.1.1.1.3.3.2" xref="S4.E3.m1.2.2.1.1.1.3.cmml"><mo id="S4.E3.m1.2.2.1.1.1.3.3.2.1" stretchy="false" xref="S4.E3.m1.2.2.1.1.1.3.cmml">(</mo><mi id="S4.E3.m1.1.1" xref="S4.E3.m1.1.1.cmml">x</mi><mo id="S4.E3.m1.2.2.1.1.1.3.3.2.2" stretchy="false" xref="S4.E3.m1.2.2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E3.m1.2.2.1.1.3" stretchy="false" xref="S4.E3.m1.2.2.1.2.1.cmml">‖</mo></mrow><mi id="S4.E3.m1.2.2.3" xref="S4.E3.m1.2.2.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E3.m1.2b"><apply id="S4.E3.m1.2.2.cmml" xref="S4.E3.m1.2.2"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.2.cmml" xref="S4.E3.m1.2.2">subscript</csymbol><apply id="S4.E3.m1.2.2.1.2.cmml" xref="S4.E3.m1.2.2.1.1"><csymbol cd="latexml" id="S4.E3.m1.2.2.1.2.1.cmml" xref="S4.E3.m1.2.2.1.1.2">norm</csymbol><apply id="S4.E3.m1.2.2.1.1.1.cmml" xref="S4.E3.m1.2.2.1.1.1"><minus id="S4.E3.m1.2.2.1.1.1.1.cmml" xref="S4.E3.m1.2.2.1.1.1.1"></minus><apply id="S4.E3.m1.2.2.1.1.1.2.cmml" xref="S4.E3.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.1.1.1.2.1.cmml" xref="S4.E3.m1.2.2.1.1.1.2">superscript</csymbol><ci id="S4.E3.m1.2.2.1.1.1.2.2.cmml" xref="S4.E3.m1.2.2.1.1.1.2.2">𝑥</ci><ci id="S4.E3.m1.2.2.1.1.1.2.3.cmml" xref="S4.E3.m1.2.2.1.1.1.2.3">⋆</ci></apply><apply id="S4.E3.m1.2.2.1.1.1.3.cmml" xref="S4.E3.m1.2.2.1.1.1.3"><times id="S4.E3.m1.2.2.1.1.1.3.1.cmml" xref="S4.E3.m1.2.2.1.1.1.3.1"></times><ci id="S4.E3.m1.2.2.1.1.1.3.2.cmml" xref="S4.E3.m1.2.2.1.1.1.3.2">𝑓</ci><ci id="S4.E3.m1.1.1.cmml" xref="S4.E3.m1.1.1">𝑥</ci></apply></apply></apply><ci id="S4.E3.m1.2.2.3.cmml" xref="S4.E3.m1.2.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.2c">\displaystyle||x^{\star}-f(x)||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.2d">| | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\lambda||x^{\star}-x||_{p}." class="ltx_Math" display="inline" id="S4.E3.m2.1"><semantics id="S4.E3.m2.1a"><mrow id="S4.E3.m2.1.1.1" xref="S4.E3.m2.1.1.1.1.cmml"><mrow id="S4.E3.m2.1.1.1.1" xref="S4.E3.m2.1.1.1.1.cmml"><mi id="S4.E3.m2.1.1.1.1.3" xref="S4.E3.m2.1.1.1.1.3.cmml"></mi><mo id="S4.E3.m2.1.1.1.1.2" xref="S4.E3.m2.1.1.1.1.2.cmml">≤</mo><mrow id="S4.E3.m2.1.1.1.1.1" xref="S4.E3.m2.1.1.1.1.1.cmml"><mi id="S4.E3.m2.1.1.1.1.1.3" xref="S4.E3.m2.1.1.1.1.1.3.cmml">λ</mi><mo id="S4.E3.m2.1.1.1.1.1.2" xref="S4.E3.m2.1.1.1.1.1.2.cmml">⁢</mo><msub id="S4.E3.m2.1.1.1.1.1.1" xref="S4.E3.m2.1.1.1.1.1.1.cmml"><mrow id="S4.E3.m2.1.1.1.1.1.1.1.1" xref="S4.E3.m2.1.1.1.1.1.1.1.2.cmml"><mo id="S4.E3.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E3.m2.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E3.m2.1.1.1.1.1.1.1.1.1" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.cmml"><msup id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mo id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.3" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S4.E3.m2.1.1.1.1.1.1.1.1.1.1" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E3.m2.1.1.1.1.1.1.1.1.1.3" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.E3.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E3.m2.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.E3.m2.1.1.1.1.1.1.3" xref="S4.E3.m2.1.1.1.1.1.1.3.cmml">p</mi></msub></mrow></mrow><mo id="S4.E3.m2.1.1.1.2" lspace="0em" xref="S4.E3.m2.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E3.m2.1b"><apply id="S4.E3.m2.1.1.1.1.cmml" xref="S4.E3.m2.1.1.1"><leq id="S4.E3.m2.1.1.1.1.2.cmml" xref="S4.E3.m2.1.1.1.1.2"></leq><csymbol cd="latexml" id="S4.E3.m2.1.1.1.1.3.cmml" xref="S4.E3.m2.1.1.1.1.3">absent</csymbol><apply id="S4.E3.m2.1.1.1.1.1.cmml" xref="S4.E3.m2.1.1.1.1.1"><times id="S4.E3.m2.1.1.1.1.1.2.cmml" xref="S4.E3.m2.1.1.1.1.1.2"></times><ci id="S4.E3.m2.1.1.1.1.1.3.cmml" xref="S4.E3.m2.1.1.1.1.1.3">𝜆</ci><apply id="S4.E3.m2.1.1.1.1.1.1.cmml" xref="S4.E3.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E3.m2.1.1.1.1.1.1.2.cmml" xref="S4.E3.m2.1.1.1.1.1.1">subscript</csymbol><apply id="S4.E3.m2.1.1.1.1.1.1.1.2.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.E3.m2.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="S4.E3.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1"><minus id="S4.E3.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.1"></minus><apply id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.2">𝑥</ci><ci id="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.2.3">⋆</ci></apply><ci id="S4.E3.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E3.m2.1.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.E3.m2.1.1.1.1.1.1.3.cmml" xref="S4.E3.m2.1.1.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m2.1c">\displaystyle\leq\lambda||x^{\star}-x||_{p}.</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m2.1d">≤ italic_λ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.2.p1.12">Combining these using the triangle inequality, we also get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx3"> <tbody id="S4.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||z-x||_{p}" class="ltx_Math" display="inline" id="S4.E4.m1.1"><semantics id="S4.E4.m1.1a"><msub id="S4.E4.m1.1.1" xref="S4.E4.m1.1.1.cmml"><mrow id="S4.E4.m1.1.1.1.1" xref="S4.E4.m1.1.1.1.2.cmml"><mo id="S4.E4.m1.1.1.1.1.2" stretchy="false" xref="S4.E4.m1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E4.m1.1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.1.cmml"><mi id="S4.E4.m1.1.1.1.1.1.2" xref="S4.E4.m1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.E4.m1.1.1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E4.m1.1.1.1.1.1.3" xref="S4.E4.m1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.E4.m1.1.1.1.1.3" stretchy="false" xref="S4.E4.m1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.E4.m1.1.1.3" xref="S4.E4.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E4.m1.1b"><apply id="S4.E4.m1.1.1.cmml" xref="S4.E4.m1.1.1"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.2.cmml" xref="S4.E4.m1.1.1">subscript</csymbol><apply id="S4.E4.m1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.1"><csymbol cd="latexml" id="S4.E4.m1.1.1.1.2.1.cmml" xref="S4.E4.m1.1.1.1.1.2">norm</csymbol><apply id="S4.E4.m1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1"><minus id="S4.E4.m1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1.1"></minus><ci id="S4.E4.m1.1.1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.1.1.2">𝑧</ci><ci id="S4.E4.m1.1.1.1.1.1.3.cmml" xref="S4.E4.m1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.E4.m1.1.1.3.cmml" xref="S4.E4.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.1c">\displaystyle||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.1d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq||x^{\star}-x||_{p}-r_{\varepsilon,\lambda}" class="ltx_Math" display="inline" id="S4.E4.m2.3"><semantics id="S4.E4.m2.3a"><mrow id="S4.E4.m2.3.3" xref="S4.E4.m2.3.3.cmml"><mi id="S4.E4.m2.3.3.3" xref="S4.E4.m2.3.3.3.cmml"></mi><mo id="S4.E4.m2.3.3.2" xref="S4.E4.m2.3.3.2.cmml">≥</mo><mrow id="S4.E4.m2.3.3.1" xref="S4.E4.m2.3.3.1.cmml"><msub id="S4.E4.m2.3.3.1.1" xref="S4.E4.m2.3.3.1.1.cmml"><mrow id="S4.E4.m2.3.3.1.1.1.1" xref="S4.E4.m2.3.3.1.1.1.2.cmml"><mo id="S4.E4.m2.3.3.1.1.1.1.2" stretchy="false" xref="S4.E4.m2.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E4.m2.3.3.1.1.1.1.1" xref="S4.E4.m2.3.3.1.1.1.1.1.cmml"><msup id="S4.E4.m2.3.3.1.1.1.1.1.2" xref="S4.E4.m2.3.3.1.1.1.1.1.2.cmml"><mi id="S4.E4.m2.3.3.1.1.1.1.1.2.2" xref="S4.E4.m2.3.3.1.1.1.1.1.2.2.cmml">x</mi><mo id="S4.E4.m2.3.3.1.1.1.1.1.2.3" xref="S4.E4.m2.3.3.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S4.E4.m2.3.3.1.1.1.1.1.1" xref="S4.E4.m2.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E4.m2.3.3.1.1.1.1.1.3" xref="S4.E4.m2.3.3.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.E4.m2.3.3.1.1.1.1.3" stretchy="false" xref="S4.E4.m2.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.E4.m2.3.3.1.1.3" xref="S4.E4.m2.3.3.1.1.3.cmml">p</mi></msub><mo id="S4.E4.m2.3.3.1.2" xref="S4.E4.m2.3.3.1.2.cmml">−</mo><msub id="S4.E4.m2.3.3.1.3" xref="S4.E4.m2.3.3.1.3.cmml"><mi id="S4.E4.m2.3.3.1.3.2" xref="S4.E4.m2.3.3.1.3.2.cmml">r</mi><mrow id="S4.E4.m2.2.2.2.4" xref="S4.E4.m2.2.2.2.3.cmml"><mi id="S4.E4.m2.1.1.1.1" xref="S4.E4.m2.1.1.1.1.cmml">ε</mi><mo 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encoding="application/x-tex" id="S4.E4.m2.3c">\displaystyle\geq||x^{\star}-x||_{p}-r_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m2.3d">≥ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.4)</span></td> </tr></tbody> <tbody id="S4.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||z-f(x)||_{p}" class="ltx_Math" display="inline" id="S4.E5.m1.2"><semantics id="S4.E5.m1.2a"><msub id="S4.E5.m1.2.2" xref="S4.E5.m1.2.2.cmml"><mrow 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xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.E5.m2.4.4.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.2.2">𝑥</ci><ci id="S4.E5.m2.4.4.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.2.3">⋆</ci></apply><apply id="S4.E5.m2.4.4.1.1.1.1.1.1.1.3.cmml" xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.3"><times id="S4.E5.m2.4.4.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.3.1"></times><ci id="S4.E5.m2.4.4.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E5.m2.4.4.1.1.1.1.1.1.1.3.2">𝑓</ci><ci id="S4.E5.m2.3.3.cmml" xref="S4.E5.m2.3.3">𝑥</ci></apply></apply></apply><ci id="S4.E5.m2.4.4.1.1.1.1.3.cmml" xref="S4.E5.m2.4.4.1.1.1.1.3">𝑝</ci></apply><apply id="S4.E5.m2.4.4.1.1.1.3.cmml" xref="S4.E5.m2.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S4.E5.m2.4.4.1.1.1.3.1.cmml" xref="S4.E5.m2.4.4.1.1.1.3">subscript</csymbol><ci id="S4.E5.m2.4.4.1.1.1.3.2.cmml" xref="S4.E5.m2.4.4.1.1.1.3.2">𝑟</ci><list id="S4.E5.m2.2.2.2.3.cmml" xref="S4.E5.m2.2.2.2.4"><ci id="S4.E5.m2.1.1.1.1.cmml" xref="S4.E5.m2.1.1.1.1">𝜀</ci><ci id="S4.E5.m2.2.2.2.2.cmml" xref="S4.E5.m2.2.2.2.2">𝜆</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m2.4c">\displaystyle\leq||x^{\star}-f(x)||_{p}+r_{\varepsilon,\lambda}.</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m2.4d">≤ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.2.p1.9">Before putting everything together, we also want to lower bound <math alttext="||z-x||_{p}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.8.m1.1"><semantics id="S4.SS1.2.p1.8.m1.1a"><msub id="S4.SS1.2.p1.8.m1.1.1" xref="S4.SS1.2.p1.8.m1.1.1.cmml"><mrow id="S4.SS1.2.p1.8.m1.1.1.1.1" xref="S4.SS1.2.p1.8.m1.1.1.1.2.cmml"><mo id="S4.SS1.2.p1.8.m1.1.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.8.m1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS1.2.p1.8.m1.1.1.1.1.1" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.cmml"><mi id="S4.SS1.2.p1.8.m1.1.1.1.1.1.2" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.SS1.2.p1.8.m1.1.1.1.1.1.1" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS1.2.p1.8.m1.1.1.1.1.1.3" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.SS1.2.p1.8.m1.1.1.1.1.3" stretchy="false" xref="S4.SS1.2.p1.8.m1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.SS1.2.p1.8.m1.1.1.3" xref="S4.SS1.2.p1.8.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.8.m1.1b"><apply id="S4.SS1.2.p1.8.m1.1.1.cmml" xref="S4.SS1.2.p1.8.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.8.m1.1.1.2.cmml" xref="S4.SS1.2.p1.8.m1.1.1">subscript</csymbol><apply id="S4.SS1.2.p1.8.m1.1.1.1.2.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.2.p1.8.m1.1.1.1.2.1.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1.2">norm</csymbol><apply id="S4.SS1.2.p1.8.m1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1"><minus id="S4.SS1.2.p1.8.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.1"></minus><ci id="S4.SS1.2.p1.8.m1.1.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.2">𝑧</ci><ci id="S4.SS1.2.p1.8.m1.1.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.8.m1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.SS1.2.p1.8.m1.1.1.3.cmml" xref="S4.SS1.2.p1.8.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.8.m1.1c">||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.8.m1.1d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. We first lower bound <math alttext="||x^{\star}-x||_{p}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.9.m2.1"><semantics id="S4.SS1.2.p1.9.m2.1a"><msub id="S4.SS1.2.p1.9.m2.1.1" xref="S4.SS1.2.p1.9.m2.1.1.cmml"><mrow id="S4.SS1.2.p1.9.m2.1.1.1.1" xref="S4.SS1.2.p1.9.m2.1.1.1.2.cmml"><mo id="S4.SS1.2.p1.9.m2.1.1.1.1.2" stretchy="false" xref="S4.SS1.2.p1.9.m2.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS1.2.p1.9.m2.1.1.1.1.1" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.cmml"><msup id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.cmml"><mi id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.2" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.2.cmml">x</mi><mo id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.3" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S4.SS1.2.p1.9.m2.1.1.1.1.1.1" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS1.2.p1.9.m2.1.1.1.1.1.3" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.SS1.2.p1.9.m2.1.1.1.1.3" stretchy="false" xref="S4.SS1.2.p1.9.m2.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.SS1.2.p1.9.m2.1.1.3" xref="S4.SS1.2.p1.9.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.9.m2.1b"><apply id="S4.SS1.2.p1.9.m2.1.1.cmml" xref="S4.SS1.2.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p1.9.m2.1.1.2.cmml" xref="S4.SS1.2.p1.9.m2.1.1">subscript</csymbol><apply id="S4.SS1.2.p1.9.m2.1.1.1.2.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.2.p1.9.m2.1.1.1.2.1.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.2">norm</csymbol><apply id="S4.SS1.2.p1.9.m2.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1"><minus id="S4.SS1.2.p1.9.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.1"></minus><apply id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.1.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2">superscript</csymbol><ci id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.2.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.2">𝑥</ci><ci id="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.3.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.2.3">⋆</ci></apply><ci id="S4.SS1.2.p1.9.m2.1.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.9.m2.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.SS1.2.p1.9.m2.1.1.3.cmml" xref="S4.SS1.2.p1.9.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.9.m2.1c">||x^{\star}-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.9.m2.1d">| | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> using the calculation</p> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varepsilon\overset{(\ref{ineq:noepsapproximate})}{<}||x-f(x)||_{p}\leq||x^{% \star}-x||_{p}+||x^{\star}-f(x)||_{p}\overset{(\ref{ineq:contractivefunction})% }{\leq}(1+\lambda)||x^{\star}-x||_{p}." class="ltx_Math" display="block" id="S4.E6.m1.5"><semantics id="S4.E6.m1.5a"><mrow id="S4.E6.m1.5.5.1" xref="S4.E6.m1.5.5.1.1.cmml"><mrow id="S4.E6.m1.5.5.1.1" xref="S4.E6.m1.5.5.1.1.cmml"><mrow id="S4.E6.m1.5.5.1.1.1" xref="S4.E6.m1.5.5.1.1.1.cmml"><mi id="S4.E6.m1.5.5.1.1.1.3" xref="S4.E6.m1.5.5.1.1.1.3.cmml">ε</mi><mo id="S4.E6.m1.5.5.1.1.1.2" xref="S4.E6.m1.5.5.1.1.1.2.cmml">⁢</mo><mover accent="true" id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml"><mo id="S4.E6.m1.1.1.2" xref="S4.E6.m1.1.1.2.cmml"><</mo><mrow id="S4.E6.m1.1.1.1.3" xref="S4.E6.m1.1.1.1.1c.cmml"><mo id="S4.E6.m1.1.1.1.3.1" stretchy="false" xref="S4.E6.m1.1.1.1.1c.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S4.E6.m1.1.1.1.1" xref="S4.E6.m1.1.1.1.1c.cmml"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2503.16089v1#S4.E1" title="Equation 4.1 ‣ Proof. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.1</span></a></mtext><mo id="S4.E6.m1.1.1.1.3.2" stretchy="false" xref="S4.E6.m1.1.1.1.1c.cmml">)</mo></mrow></mover><mo id="S4.E6.m1.5.5.1.1.1.2a" xref="S4.E6.m1.5.5.1.1.1.2.cmml">⁢</mo><msub id="S4.E6.m1.5.5.1.1.1.1" xref="S4.E6.m1.5.5.1.1.1.1.cmml"><mrow id="S4.E6.m1.5.5.1.1.1.1.1.1" xref="S4.E6.m1.5.5.1.1.1.1.1.2.cmml"><mo id="S4.E6.m1.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S4.E6.m1.5.5.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E6.m1.5.5.1.1.1.1.1.1.1" xref="S4.E6.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S4.E6.m1.5.5.1.1.1.1.1.1.1.2" xref="S4.E6.m1.5.5.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S4.E6.m1.5.5.1.1.1.1.1.1.1.1" xref="S4.E6.m1.5.5.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E6.m1.5.5.1.1.1.1.1.1.1.3" xref="S4.E6.m1.5.5.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E6.m1.5.5.1.1.1.1.1.1.1.3.2" xref="S4.E6.m1.5.5.1.1.1.1.1.1.1.3.2.cmml">f</mi><mo 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xref="S4.E6.m1.5.5.1.1.5.4.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.5c">\varepsilon\overset{(\ref{ineq:noepsapproximate})}{<}||x-f(x)||_{p}\leq||x^{% \star}-x||_{p}+||x^{\star}-f(x)||_{p}\overset{(\ref{ineq:contractivefunction})% }{\leq}(1+\lambda)||x^{\star}-x||_{p}.</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.5d">italic_ε start_OVERACCENT ( ) end_OVERACCENT start_ARG < end_ARG | | italic_x - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_OVERACCENT ( ) end_OVERACCENT start_ARG ≤ end_ARG ( 1 + italic_λ ) | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.2.p1.13">Combining <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.E4" title="In Proof. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Equations</span> <span class="ltx_text ltx_ref_tag">4.4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.E6" title="Equation 4.6 ‣ Proof. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.6</span></a>, we get</p> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="||z-x||_{p}>\frac{\varepsilon}{1+\lambda}-r_{\varepsilon,\lambda}." class="ltx_Math" display="block" id="S4.E7.m1.3"><semantics id="S4.E7.m1.3a"><mrow id="S4.E7.m1.3.3.1" xref="S4.E7.m1.3.3.1.1.cmml"><mrow id="S4.E7.m1.3.3.1.1" xref="S4.E7.m1.3.3.1.1.cmml"><msub id="S4.E7.m1.3.3.1.1.1" xref="S4.E7.m1.3.3.1.1.1.cmml"><mrow id="S4.E7.m1.3.3.1.1.1.1.1" xref="S4.E7.m1.3.3.1.1.1.1.2.cmml"><mo id="S4.E7.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.E7.m1.3.3.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E7.m1.3.3.1.1.1.1.1.1" xref="S4.E7.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.E7.m1.3.3.1.1.1.1.1.1.2" xref="S4.E7.m1.3.3.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.E7.m1.3.3.1.1.1.1.1.1.1" xref="S4.E7.m1.3.3.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E7.m1.3.3.1.1.1.1.1.1.3" xref="S4.E7.m1.3.3.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo 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xref="S4.E7.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S4.E7.m1.3.3.1.1.3.3.1.cmml" xref="S4.E7.m1.3.3.1.1.3.3">subscript</csymbol><ci id="S4.E7.m1.3.3.1.1.3.3.2.cmml" xref="S4.E7.m1.3.3.1.1.3.3.2">𝑟</ci><list id="S4.E7.m1.2.2.2.3.cmml" xref="S4.E7.m1.2.2.2.4"><ci id="S4.E7.m1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1">𝜀</ci><ci id="S4.E7.m1.2.2.2.2.cmml" xref="S4.E7.m1.2.2.2.2">𝜆</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.3c">||z-x||_{p}>\frac{\varepsilon}{1+\lambda}-r_{\varepsilon,\lambda}.</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.3d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > divide start_ARG italic_ε end_ARG start_ARG 1 + italic_λ end_ARG - italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.2.p1.14">We can now put everything together to obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx4"> <tbody id="S4.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle||z-f(x)||_{p}" class="ltx_Math" display="inline" id="S4.Ex2.m1.2"><semantics id="S4.Ex2.m1.2a"><msub id="S4.Ex2.m1.2.2" xref="S4.Ex2.m1.2.2.cmml"><mrow id="S4.Ex2.m1.2.2.1.1" xref="S4.Ex2.m1.2.2.1.2.cmml"><mo id="S4.Ex2.m1.2.2.1.1.2" stretchy="false" xref="S4.Ex2.m1.2.2.1.2.1.cmml">‖</mo><mrow id="S4.Ex2.m1.2.2.1.1.1" xref="S4.Ex2.m1.2.2.1.1.1.cmml"><mi id="S4.Ex2.m1.2.2.1.1.1.2" xref="S4.Ex2.m1.2.2.1.1.1.2.cmml">z</mi><mo id="S4.Ex2.m1.2.2.1.1.1.1" xref="S4.Ex2.m1.2.2.1.1.1.1.cmml">−</mo><mrow 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ltx_eqn_cell"><math alttext="\displaystyle{\leq}||x^{\star}-f(x)||_{p}+r_{\varepsilon,\lambda}" class="ltx_Math" display="inline" id="S4.Ex2.m2.4"><semantics id="S4.Ex2.m2.4a"><mrow id="S4.Ex2.m2.4.4" xref="S4.Ex2.m2.4.4.cmml"><mi id="S4.Ex2.m2.4.4.3" xref="S4.Ex2.m2.4.4.3.cmml"></mi><mo id="S4.Ex2.m2.4.4.2" xref="S4.Ex2.m2.4.4.2.cmml">≤</mo><mrow id="S4.Ex2.m2.4.4.1" xref="S4.Ex2.m2.4.4.1.cmml"><msub id="S4.Ex2.m2.4.4.1.1" xref="S4.Ex2.m2.4.4.1.1.cmml"><mrow id="S4.Ex2.m2.4.4.1.1.1.1" xref="S4.Ex2.m2.4.4.1.1.1.2.cmml"><mo id="S4.Ex2.m2.4.4.1.1.1.1.2" stretchy="false" xref="S4.Ex2.m2.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex2.m2.4.4.1.1.1.1.1" xref="S4.Ex2.m2.4.4.1.1.1.1.1.cmml"><msup id="S4.Ex2.m2.4.4.1.1.1.1.1.2" xref="S4.Ex2.m2.4.4.1.1.1.1.1.2.cmml"><mi id="S4.Ex2.m2.4.4.1.1.1.1.1.2.2" xref="S4.Ex2.m2.4.4.1.1.1.1.1.2.2.cmml">x</mi><mo id="S4.Ex2.m2.4.4.1.1.1.1.1.2.3" xref="S4.Ex2.m2.4.4.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S4.Ex2.m2.4.4.1.1.1.1.1.1" 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id="S4.Ex2.m2.4.4.1.3.cmml" xref="S4.Ex2.m2.4.4.1.3"><csymbol cd="ambiguous" id="S4.Ex2.m2.4.4.1.3.1.cmml" xref="S4.Ex2.m2.4.4.1.3">subscript</csymbol><ci id="S4.Ex2.m2.4.4.1.3.2.cmml" xref="S4.Ex2.m2.4.4.1.3.2">𝑟</ci><list id="S4.Ex2.m2.2.2.2.3.cmml" xref="S4.Ex2.m2.2.2.2.4"><ci id="S4.Ex2.m2.1.1.1.1.cmml" xref="S4.Ex2.m2.1.1.1.1">𝜀</ci><ci id="S4.Ex2.m2.2.2.2.2.cmml" xref="S4.Ex2.m2.2.2.2.2">𝜆</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m2.4c">\displaystyle{\leq}||x^{\star}-f(x)||_{p}+r_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m2.4d">≤ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex3"><tr 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id="S4.Ex3.m1.3.3.1.3.1.cmml" xref="S4.Ex3.m1.3.3.1.3">subscript</csymbol><ci id="S4.Ex3.m1.3.3.1.3.2.cmml" xref="S4.Ex3.m1.3.3.1.3.2">𝑟</ci><list id="S4.Ex3.m1.2.2.2.3.cmml" xref="S4.Ex3.m1.2.2.2.4"><ci id="S4.Ex3.m1.1.1.1.1.cmml" xref="S4.Ex3.m1.1.1.1.1">𝜀</ci><ci id="S4.Ex3.m1.2.2.2.2.cmml" xref="S4.Ex3.m1.2.2.2.2">𝜆</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex3.m1.3c">\displaystyle{\leq}\lambda||x^{\star}-x||_{p}+r_{\varepsilon,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex3.m1.3d">≤ italic_λ | | italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S4.SS1.2.p1.10.m1.2.2.1.1.1.1.1"></minus><ci id="S4.SS1.2.p1.10.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS1.2.p1.10.m1.2.2.1.1.1.1.2">𝑧</ci><ci id="S4.SS1.2.p1.10.m1.2.2.1.1.1.1.3.cmml" xref="S4.SS1.2.p1.10.m1.2.2.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.SS1.2.p1.10.m1.2.2.1.3.cmml" xref="S4.SS1.2.p1.10.m1.2.2.1.3">𝑝</ci></apply><apply id="S4.SS1.2.p1.10.m1.3.3.2.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.2.p1.10.m1.3.3.2.2.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2">subscript</csymbol><apply id="S4.SS1.2.p1.10.m1.3.3.2.1.2.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1"><csymbol cd="latexml" id="S4.SS1.2.p1.10.m1.3.3.2.1.2.1.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.2">norm</csymbol><apply id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1"><minus id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.1.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.1"></minus><ci id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.2.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.2">𝑧</ci><apply id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3"><times id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3.1.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3.1"></times><ci id="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3.2.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.1.1.1.3.2">𝑓</ci><ci id="S4.SS1.2.p1.10.m1.1.1.cmml" xref="S4.SS1.2.p1.10.m1.1.1">𝑥</ci></apply></apply></apply><ci id="S4.SS1.2.p1.10.m1.3.3.2.3.cmml" xref="S4.SS1.2.p1.10.m1.3.3.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.10.m1.3c">||z-x||_{p}>||z-f(x)||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.10.m1.3d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > | | italic_z - italic_f ( italic_x ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and thus <math alttext="z\notin H^{p}_{x,f(x)}" class="ltx_Math" display="inline" id="S4.SS1.2.p1.11.m2.3"><semantics id="S4.SS1.2.p1.11.m2.3a"><mrow id="S4.SS1.2.p1.11.m2.3.4" xref="S4.SS1.2.p1.11.m2.3.4.cmml"><mi id="S4.SS1.2.p1.11.m2.3.4.2" xref="S4.SS1.2.p1.11.m2.3.4.2.cmml">z</mi><mo id="S4.SS1.2.p1.11.m2.3.4.1" xref="S4.SS1.2.p1.11.m2.3.4.1.cmml">∉</mo><msubsup id="S4.SS1.2.p1.11.m2.3.4.3" xref="S4.SS1.2.p1.11.m2.3.4.3.cmml"><mi id="S4.SS1.2.p1.11.m2.3.4.3.2.2" xref="S4.SS1.2.p1.11.m2.3.4.3.2.2.cmml">H</mi><mrow id="S4.SS1.2.p1.11.m2.3.3.3.3" xref="S4.SS1.2.p1.11.m2.3.3.3.4.cmml"><mi id="S4.SS1.2.p1.11.m2.2.2.2.2" xref="S4.SS1.2.p1.11.m2.2.2.2.2.cmml">x</mi><mo id="S4.SS1.2.p1.11.m2.3.3.3.3.2" xref="S4.SS1.2.p1.11.m2.3.3.3.4.cmml">,</mo><mrow id="S4.SS1.2.p1.11.m2.3.3.3.3.1" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.cmml"><mi id="S4.SS1.2.p1.11.m2.3.3.3.3.1.2" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.2.cmml">f</mi><mo id="S4.SS1.2.p1.11.m2.3.3.3.3.1.1" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.1.cmml">⁢</mo><mrow id="S4.SS1.2.p1.11.m2.3.3.3.3.1.3.2" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.cmml"><mo id="S4.SS1.2.p1.11.m2.3.3.3.3.1.3.2.1" stretchy="false" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.cmml">(</mo><mi id="S4.SS1.2.p1.11.m2.1.1.1.1" xref="S4.SS1.2.p1.11.m2.1.1.1.1.cmml">x</mi><mo id="S4.SS1.2.p1.11.m2.3.3.3.3.1.3.2.2" stretchy="false" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><mi id="S4.SS1.2.p1.11.m2.3.4.3.2.3" xref="S4.SS1.2.p1.11.m2.3.4.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p1.11.m2.3b"><apply id="S4.SS1.2.p1.11.m2.3.4.cmml" xref="S4.SS1.2.p1.11.m2.3.4"><notin id="S4.SS1.2.p1.11.m2.3.4.1.cmml" xref="S4.SS1.2.p1.11.m2.3.4.1"></notin><ci id="S4.SS1.2.p1.11.m2.3.4.2.cmml" xref="S4.SS1.2.p1.11.m2.3.4.2">𝑧</ci><apply id="S4.SS1.2.p1.11.m2.3.4.3.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3"><csymbol cd="ambiguous" id="S4.SS1.2.p1.11.m2.3.4.3.1.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3">subscript</csymbol><apply id="S4.SS1.2.p1.11.m2.3.4.3.2.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3"><csymbol cd="ambiguous" id="S4.SS1.2.p1.11.m2.3.4.3.2.1.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3">superscript</csymbol><ci id="S4.SS1.2.p1.11.m2.3.4.3.2.2.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3.2.2">𝐻</ci><ci id="S4.SS1.2.p1.11.m2.3.4.3.2.3.cmml" xref="S4.SS1.2.p1.11.m2.3.4.3.2.3">𝑝</ci></apply><list id="S4.SS1.2.p1.11.m2.3.3.3.4.cmml" xref="S4.SS1.2.p1.11.m2.3.3.3.3"><ci id="S4.SS1.2.p1.11.m2.2.2.2.2.cmml" xref="S4.SS1.2.p1.11.m2.2.2.2.2">𝑥</ci><apply id="S4.SS1.2.p1.11.m2.3.3.3.3.1.cmml" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1"><times id="S4.SS1.2.p1.11.m2.3.3.3.3.1.1.cmml" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.1"></times><ci id="S4.SS1.2.p1.11.m2.3.3.3.3.1.2.cmml" xref="S4.SS1.2.p1.11.m2.3.3.3.3.1.2">𝑓</ci><ci id="S4.SS1.2.p1.11.m2.1.1.1.1.cmml" xref="S4.SS1.2.p1.11.m2.1.1.1.1">𝑥</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p1.11.m2.3c">z\notin H^{p}_{x,f(x)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p1.11.m2.3d">italic_z ∉ italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_f ( italic_x ) end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.1">We now have all the ingredients to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem2" title="Theorem 4.2. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Theorem 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p 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xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.1" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.2" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S4.Thmtheorem2.p1.1.1.m1.3.3" mathvariant="normal" xref="S4.Thmtheorem2.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.3b"><apply id="S4.Thmtheorem2.p1.1.1.m1.3.4.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4"><in id="S4.Thmtheorem2.p1.1.1.m1.3.4.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.1"></in><ci id="S4.Thmtheorem2.p1.1.1.m1.3.4.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3"><union id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.2.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.2.2"><cn id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval><set id="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.4.3.3.2"><infinity id="S4.Thmtheorem2.p1.1.1.m1.3.3.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint of a <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.3.m3.1a"><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.3.m3.1b"><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.1d">italic_λ</annotation></semantics></math>-contracting (in <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.4.m4.1a"><msub id="S4.Thmtheorem2.p1.4.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem2.p1.4.4.m4.1.1.2" mathvariant="normal" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">ℓ</mi><mi id="S4.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.4.m4.1b"><apply id="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.2">ℓ</ci><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm) function <math alttext="f:[0,1]^{d}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.5.m5.4"><semantics id="S4.Thmtheorem2.p1.5.5.m5.4a"><mrow id="S4.Thmtheorem2.p1.5.5.m5.4.5" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.cmml"><mi id="S4.Thmtheorem2.p1.5.5.m5.4.5.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.2.cmml">f</mi><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.1.cmml">:</mo><mrow id="S4.Thmtheorem2.p1.5.5.m5.4.5.3" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.cmml"><msup id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.cmml"><mrow id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.1.cmml"><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.1.cmml">[</mo><mn id="S4.Thmtheorem2.p1.5.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.1.cmml">,</mo><mn id="S4.Thmtheorem2.p1.5.5.m5.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.2.2.cmml">1</mn><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.3" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.3.cmml">d</mi></msup><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.1" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.1.cmml">→</mo><msup id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.cmml"><mrow id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.1.cmml"><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem2.p1.5.5.m5.3.3" xref="S4.Thmtheorem2.p1.5.5.m5.3.3.cmml">0</mn><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.2.2" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.1.cmml">,</mo><mn id="S4.Thmtheorem2.p1.5.5.m5.4.4" xref="S4.Thmtheorem2.p1.5.5.m5.4.4.cmml">1</mn><mo id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.3" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.5.m5.4b"><apply id="S4.Thmtheorem2.p1.5.5.m5.4.5.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5"><ci id="S4.Thmtheorem2.p1.5.5.m5.4.5.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.1">:</ci><ci id="S4.Thmtheorem2.p1.5.5.m5.4.5.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.2">𝑓</ci><apply id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3"><ci id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.1">→</ci><apply id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.2.2"><cn id="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.5.m5.1.1">0</cn><cn id="S4.Thmtheorem2.p1.5.5.m5.2.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.5.m5.2.2">1</cn></interval><ci id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.2.3">𝑑</ci></apply><apply id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.2.2"><cn id="S4.Thmtheorem2.p1.5.5.m5.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.5.m5.3.3">0</cn><cn id="S4.Thmtheorem2.p1.5.5.m5.4.4.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.5.m5.4.4">1</cn></interval><ci id="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.3.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.4.5.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.5.m5.4c">f:[0,1]^{d}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.5.m5.4d">italic_f : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> can be found using <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.6.m6.1"><semantics id="S4.Thmtheorem2.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3.cmml">𝒪</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml"><msup id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.2.cmml">d</mi><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1.cmml">log</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mfrac id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.cmml"><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2.cmml">1</mn><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml">ε</mi></mfrac></mrow><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mfrac id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml"><mn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml">1</mn><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml">−</mo><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mrow></mrow><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.6.m6.1b"><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1"><times id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2"></times><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.3">𝒪</ci><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1"><times id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.2"></times><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.2">𝑑</ci><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.3.3">2</cn></apply><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1"><plus id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.1"></plus><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2"><log id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1"></log><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2"><divide id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2"></divide><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2">1</cn><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3"><log id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1"></log><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2"><divide id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2"></divide><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3"><minus id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.6.m6.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.6.m6.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> queries.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.3.p1"> <p class="ltx_p" id="S4.SS1.3.p1.11">We begin with the search space <math alttext="M=[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS1.3.p1.1.m1.2"><semantics id="S4.SS1.3.p1.1.m1.2a"><mrow id="S4.SS1.3.p1.1.m1.2.3" xref="S4.SS1.3.p1.1.m1.2.3.cmml"><mi id="S4.SS1.3.p1.1.m1.2.3.2" xref="S4.SS1.3.p1.1.m1.2.3.2.cmml">M</mi><mo id="S4.SS1.3.p1.1.m1.2.3.1" xref="S4.SS1.3.p1.1.m1.2.3.1.cmml">=</mo><msup id="S4.SS1.3.p1.1.m1.2.3.3" xref="S4.SS1.3.p1.1.m1.2.3.3.cmml"><mrow id="S4.SS1.3.p1.1.m1.2.3.3.2.2" xref="S4.SS1.3.p1.1.m1.2.3.3.2.1.cmml"><mo id="S4.SS1.3.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S4.SS1.3.p1.1.m1.2.3.3.2.1.cmml">[</mo><mn id="S4.SS1.3.p1.1.m1.1.1" xref="S4.SS1.3.p1.1.m1.1.1.cmml">0</mn><mo id="S4.SS1.3.p1.1.m1.2.3.3.2.2.2" xref="S4.SS1.3.p1.1.m1.2.3.3.2.1.cmml">,</mo><mn id="S4.SS1.3.p1.1.m1.2.2" xref="S4.SS1.3.p1.1.m1.2.2.cmml">1</mn><mo id="S4.SS1.3.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="S4.SS1.3.p1.1.m1.2.3.3.2.1.cmml">]</mo></mrow><mi id="S4.SS1.3.p1.1.m1.2.3.3.3" xref="S4.SS1.3.p1.1.m1.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.1.m1.2b"><apply id="S4.SS1.3.p1.1.m1.2.3.cmml" xref="S4.SS1.3.p1.1.m1.2.3"><eq id="S4.SS1.3.p1.1.m1.2.3.1.cmml" xref="S4.SS1.3.p1.1.m1.2.3.1"></eq><ci id="S4.SS1.3.p1.1.m1.2.3.2.cmml" xref="S4.SS1.3.p1.1.m1.2.3.2">𝑀</ci><apply id="S4.SS1.3.p1.1.m1.2.3.3.cmml" xref="S4.SS1.3.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.3.p1.1.m1.2.3.3.1.cmml" xref="S4.SS1.3.p1.1.m1.2.3.3">superscript</csymbol><interval closure="closed" id="S4.SS1.3.p1.1.m1.2.3.3.2.1.cmml" xref="S4.SS1.3.p1.1.m1.2.3.3.2.2"><cn id="S4.SS1.3.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS1.3.p1.1.m1.1.1">0</cn><cn id="S4.SS1.3.p1.1.m1.2.2.cmml" type="integer" xref="S4.SS1.3.p1.1.m1.2.2">1</cn></interval><ci id="S4.SS1.3.p1.1.m1.2.3.3.3.cmml" xref="S4.SS1.3.p1.1.m1.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.1.m1.2c">M=[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.1.m1.2d">italic_M = [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\operatorname{vol}(M)=1" class="ltx_Math" display="inline" id="S4.SS1.3.p1.2.m2.2"><semantics id="S4.SS1.3.p1.2.m2.2a"><mrow id="S4.SS1.3.p1.2.m2.2.3" xref="S4.SS1.3.p1.2.m2.2.3.cmml"><mrow id="S4.SS1.3.p1.2.m2.2.3.2.2" xref="S4.SS1.3.p1.2.m2.2.3.2.1.cmml"><mi id="S4.SS1.3.p1.2.m2.1.1" xref="S4.SS1.3.p1.2.m2.1.1.cmml">vol</mi><mo id="S4.SS1.3.p1.2.m2.2.3.2.2a" xref="S4.SS1.3.p1.2.m2.2.3.2.1.cmml">⁡</mo><mrow id="S4.SS1.3.p1.2.m2.2.3.2.2.1" xref="S4.SS1.3.p1.2.m2.2.3.2.1.cmml"><mo id="S4.SS1.3.p1.2.m2.2.3.2.2.1.1" stretchy="false" xref="S4.SS1.3.p1.2.m2.2.3.2.1.cmml">(</mo><mi id="S4.SS1.3.p1.2.m2.2.2" xref="S4.SS1.3.p1.2.m2.2.2.cmml">M</mi><mo id="S4.SS1.3.p1.2.m2.2.3.2.2.1.2" stretchy="false" xref="S4.SS1.3.p1.2.m2.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p1.2.m2.2.3.1" xref="S4.SS1.3.p1.2.m2.2.3.1.cmml">=</mo><mn id="S4.SS1.3.p1.2.m2.2.3.3" xref="S4.SS1.3.p1.2.m2.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.2.m2.2b"><apply id="S4.SS1.3.p1.2.m2.2.3.cmml" xref="S4.SS1.3.p1.2.m2.2.3"><eq id="S4.SS1.3.p1.2.m2.2.3.1.cmml" xref="S4.SS1.3.p1.2.m2.2.3.1"></eq><apply id="S4.SS1.3.p1.2.m2.2.3.2.1.cmml" xref="S4.SS1.3.p1.2.m2.2.3.2.2"><ci id="S4.SS1.3.p1.2.m2.1.1.cmml" xref="S4.SS1.3.p1.2.m2.1.1">vol</ci><ci id="S4.SS1.3.p1.2.m2.2.2.cmml" xref="S4.SS1.3.p1.2.m2.2.2">𝑀</ci></apply><cn id="S4.SS1.3.p1.2.m2.2.3.3.cmml" type="integer" xref="S4.SS1.3.p1.2.m2.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.2.m2.2c">\operatorname{vol}(M)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.2.m2.2d">roman_vol ( italic_M ) = 1</annotation></semantics></math>. We then repeatedly query the centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.3.p1.3.m3.1"><semantics id="S4.SS1.3.p1.3.m3.1a"><mi id="S4.SS1.3.p1.3.m3.1.1" xref="S4.SS1.3.p1.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.3.m3.1b"><ci id="S4.SS1.3.p1.3.m3.1.1.cmml" xref="S4.SS1.3.p1.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.3.m3.1d">italic_c</annotation></semantics></math> of <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.3.p1.4.m4.1"><semantics id="S4.SS1.3.p1.4.m4.1a"><mi id="S4.SS1.3.p1.4.m4.1.1" xref="S4.SS1.3.p1.4.m4.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.4.m4.1b"><ci id="S4.SS1.3.p1.4.m4.1.1.cmml" xref="S4.SS1.3.p1.4.m4.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.4.m4.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.4.m4.1d">italic_M</annotation></semantics></math> (or rather the measure induced by <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.3.p1.5.m5.1"><semantics id="S4.SS1.3.p1.5.m5.1a"><mi id="S4.SS1.3.p1.5.m5.1.1" xref="S4.SS1.3.p1.5.m5.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.5.m5.1b"><ci id="S4.SS1.3.p1.5.m5.1.1.cmml" xref="S4.SS1.3.p1.5.m5.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.5.m5.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.5.m5.1d">italic_M</annotation></semantics></math>), guaranteed to exist by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a>. We terminate if <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.3.p1.6.m6.1"><semantics id="S4.SS1.3.p1.6.m6.1a"><mi id="S4.SS1.3.p1.6.m6.1.1" xref="S4.SS1.3.p1.6.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.6.m6.1b"><ci id="S4.SS1.3.p1.6.m6.1.1.cmml" xref="S4.SS1.3.p1.6.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.6.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.6.m6.1d">italic_c</annotation></semantics></math> is an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.3.p1.7.m7.1"><semantics id="S4.SS1.3.p1.7.m7.1a"><mi id="S4.SS1.3.p1.7.m7.1.1" xref="S4.SS1.3.p1.7.m7.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.7.m7.1b"><ci id="S4.SS1.3.p1.7.m7.1.1.cmml" xref="S4.SS1.3.p1.7.m7.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.7.m7.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.7.m7.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. Otherwise, we remove the bisector halfspace <math alttext="H_{c,f(c)}" class="ltx_Math" display="inline" id="S4.SS1.3.p1.8.m8.3"><semantics id="S4.SS1.3.p1.8.m8.3a"><msub id="S4.SS1.3.p1.8.m8.3.4" xref="S4.SS1.3.p1.8.m8.3.4.cmml"><mi id="S4.SS1.3.p1.8.m8.3.4.2" xref="S4.SS1.3.p1.8.m8.3.4.2.cmml">H</mi><mrow id="S4.SS1.3.p1.8.m8.3.3.3.3" xref="S4.SS1.3.p1.8.m8.3.3.3.4.cmml"><mi id="S4.SS1.3.p1.8.m8.2.2.2.2" xref="S4.SS1.3.p1.8.m8.2.2.2.2.cmml">c</mi><mo id="S4.SS1.3.p1.8.m8.3.3.3.3.2" xref="S4.SS1.3.p1.8.m8.3.3.3.4.cmml">,</mo><mrow id="S4.SS1.3.p1.8.m8.3.3.3.3.1" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.cmml"><mi id="S4.SS1.3.p1.8.m8.3.3.3.3.1.2" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.2.cmml">f</mi><mo id="S4.SS1.3.p1.8.m8.3.3.3.3.1.1" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.1.cmml">⁢</mo><mrow id="S4.SS1.3.p1.8.m8.3.3.3.3.1.3.2" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.cmml"><mo id="S4.SS1.3.p1.8.m8.3.3.3.3.1.3.2.1" stretchy="false" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.cmml">(</mo><mi id="S4.SS1.3.p1.8.m8.1.1.1.1" xref="S4.SS1.3.p1.8.m8.1.1.1.1.cmml">c</mi><mo id="S4.SS1.3.p1.8.m8.3.3.3.3.1.3.2.2" stretchy="false" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.8.m8.3b"><apply id="S4.SS1.3.p1.8.m8.3.4.cmml" xref="S4.SS1.3.p1.8.m8.3.4"><csymbol cd="ambiguous" id="S4.SS1.3.p1.8.m8.3.4.1.cmml" xref="S4.SS1.3.p1.8.m8.3.4">subscript</csymbol><ci id="S4.SS1.3.p1.8.m8.3.4.2.cmml" xref="S4.SS1.3.p1.8.m8.3.4.2">𝐻</ci><list id="S4.SS1.3.p1.8.m8.3.3.3.4.cmml" xref="S4.SS1.3.p1.8.m8.3.3.3.3"><ci id="S4.SS1.3.p1.8.m8.2.2.2.2.cmml" xref="S4.SS1.3.p1.8.m8.2.2.2.2">𝑐</ci><apply id="S4.SS1.3.p1.8.m8.3.3.3.3.1.cmml" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1"><times id="S4.SS1.3.p1.8.m8.3.3.3.3.1.1.cmml" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.1"></times><ci id="S4.SS1.3.p1.8.m8.3.3.3.3.1.2.cmml" xref="S4.SS1.3.p1.8.m8.3.3.3.3.1.2">𝑓</ci><ci id="S4.SS1.3.p1.8.m8.1.1.1.1.cmml" xref="S4.SS1.3.p1.8.m8.1.1.1.1">𝑐</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.8.m8.3c">H_{c,f(c)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.8.m8.3d">italic_H start_POSTSUBSCRIPT italic_c , italic_f ( italic_c ) end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.3.p1.9.m9.1"><semantics id="S4.SS1.3.p1.9.m9.1a"><mi id="S4.SS1.3.p1.9.m9.1.1" xref="S4.SS1.3.p1.9.m9.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.9.m9.1b"><ci id="S4.SS1.3.p1.9.m9.1.1.cmml" xref="S4.SS1.3.p1.9.m9.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.9.m9.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.9.m9.1d">italic_M</annotation></semantics></math>. For each non-terminating query, we get the guarantee that the volume of <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.3.p1.10.m10.1"><semantics id="S4.SS1.3.p1.10.m10.1a"><mi id="S4.SS1.3.p1.10.m10.1.1" xref="S4.SS1.3.p1.10.m10.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.10.m10.1b"><ci id="S4.SS1.3.p1.10.m10.1.1.cmml" xref="S4.SS1.3.p1.10.m10.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.10.m10.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.10.m10.1d">italic_M</annotation></semantics></math> decreases to at most a <math alttext="\frac{d}{d+1}" class="ltx_Math" display="inline" id="S4.SS1.3.p1.11.m11.1"><semantics id="S4.SS1.3.p1.11.m11.1a"><mfrac id="S4.SS1.3.p1.11.m11.1.1" xref="S4.SS1.3.p1.11.m11.1.1.cmml"><mi id="S4.SS1.3.p1.11.m11.1.1.2" xref="S4.SS1.3.p1.11.m11.1.1.2.cmml">d</mi><mrow id="S4.SS1.3.p1.11.m11.1.1.3" xref="S4.SS1.3.p1.11.m11.1.1.3.cmml"><mi id="S4.SS1.3.p1.11.m11.1.1.3.2" xref="S4.SS1.3.p1.11.m11.1.1.3.2.cmml">d</mi><mo id="S4.SS1.3.p1.11.m11.1.1.3.1" xref="S4.SS1.3.p1.11.m11.1.1.3.1.cmml">+</mo><mn id="S4.SS1.3.p1.11.m11.1.1.3.3" xref="S4.SS1.3.p1.11.m11.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p1.11.m11.1b"><apply id="S4.SS1.3.p1.11.m11.1.1.cmml" xref="S4.SS1.3.p1.11.m11.1.1"><divide id="S4.SS1.3.p1.11.m11.1.1.1.cmml" xref="S4.SS1.3.p1.11.m11.1.1"></divide><ci id="S4.SS1.3.p1.11.m11.1.1.2.cmml" xref="S4.SS1.3.p1.11.m11.1.1.2">𝑑</ci><apply id="S4.SS1.3.p1.11.m11.1.1.3.cmml" xref="S4.SS1.3.p1.11.m11.1.1.3"><plus id="S4.SS1.3.p1.11.m11.1.1.3.1.cmml" xref="S4.SS1.3.p1.11.m11.1.1.3.1"></plus><ci id="S4.SS1.3.p1.11.m11.1.1.3.2.cmml" xref="S4.SS1.3.p1.11.m11.1.1.3.2">𝑑</ci><cn id="S4.SS1.3.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S4.SS1.3.p1.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p1.11.m11.1c">\frac{d}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p1.11.m11.1d">divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>-fraction of its previous volume, where we use <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem4" title="Observation 3.4. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.4</span></a> to translate the centerpoint guarantee of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a> from limit halfspaces to bisector halfspaces.</p> </div> <div class="ltx_para" id="S4.SS1.4.p2"> <p class="ltx_p" id="S4.SS1.4.p2.6">Now recall that <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> guarantees that <math alttext="B^{p}(x^{*},r_{\varepsilon,\lambda})" class="ltx_Math" display="inline" id="S4.SS1.4.p2.1.m1.4"><semantics id="S4.SS1.4.p2.1.m1.4a"><mrow id="S4.SS1.4.p2.1.m1.4.4" xref="S4.SS1.4.p2.1.m1.4.4.cmml"><msup id="S4.SS1.4.p2.1.m1.4.4.4" xref="S4.SS1.4.p2.1.m1.4.4.4.cmml"><mi id="S4.SS1.4.p2.1.m1.4.4.4.2" xref="S4.SS1.4.p2.1.m1.4.4.4.2.cmml">B</mi><mi id="S4.SS1.4.p2.1.m1.4.4.4.3" xref="S4.SS1.4.p2.1.m1.4.4.4.3.cmml">p</mi></msup><mo id="S4.SS1.4.p2.1.m1.4.4.3" xref="S4.SS1.4.p2.1.m1.4.4.3.cmml">⁢</mo><mrow id="S4.SS1.4.p2.1.m1.4.4.2.2" xref="S4.SS1.4.p2.1.m1.4.4.2.3.cmml"><mo id="S4.SS1.4.p2.1.m1.4.4.2.2.3" stretchy="false" xref="S4.SS1.4.p2.1.m1.4.4.2.3.cmml">(</mo><msup id="S4.SS1.4.p2.1.m1.3.3.1.1.1" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS1.4.p2.1.m1.3.3.1.1.1.2" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1.2.cmml">x</mi><mo id="S4.SS1.4.p2.1.m1.3.3.1.1.1.3" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.4.p2.1.m1.4.4.2.2.4" xref="S4.SS1.4.p2.1.m1.4.4.2.3.cmml">,</mo><msub id="S4.SS1.4.p2.1.m1.4.4.2.2.2" xref="S4.SS1.4.p2.1.m1.4.4.2.2.2.cmml"><mi id="S4.SS1.4.p2.1.m1.4.4.2.2.2.2" xref="S4.SS1.4.p2.1.m1.4.4.2.2.2.2.cmml">r</mi><mrow id="S4.SS1.4.p2.1.m1.2.2.2.4" xref="S4.SS1.4.p2.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.4.p2.1.m1.1.1.1.1" xref="S4.SS1.4.p2.1.m1.1.1.1.1.cmml">ε</mi><mo id="S4.SS1.4.p2.1.m1.2.2.2.4.1" xref="S4.SS1.4.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.4.p2.1.m1.2.2.2.2" xref="S4.SS1.4.p2.1.m1.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS1.4.p2.1.m1.4.4.2.2.5" stretchy="false" xref="S4.SS1.4.p2.1.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.1.m1.4b"><apply id="S4.SS1.4.p2.1.m1.4.4.cmml" xref="S4.SS1.4.p2.1.m1.4.4"><times id="S4.SS1.4.p2.1.m1.4.4.3.cmml" xref="S4.SS1.4.p2.1.m1.4.4.3"></times><apply id="S4.SS1.4.p2.1.m1.4.4.4.cmml" xref="S4.SS1.4.p2.1.m1.4.4.4"><csymbol cd="ambiguous" id="S4.SS1.4.p2.1.m1.4.4.4.1.cmml" xref="S4.SS1.4.p2.1.m1.4.4.4">superscript</csymbol><ci id="S4.SS1.4.p2.1.m1.4.4.4.2.cmml" xref="S4.SS1.4.p2.1.m1.4.4.4.2">𝐵</ci><ci id="S4.SS1.4.p2.1.m1.4.4.4.3.cmml" xref="S4.SS1.4.p2.1.m1.4.4.4.3">𝑝</ci></apply><interval closure="open" id="S4.SS1.4.p2.1.m1.4.4.2.3.cmml" xref="S4.SS1.4.p2.1.m1.4.4.2.2"><apply id="S4.SS1.4.p2.1.m1.3.3.1.1.1.cmml" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1.2">𝑥</ci><times id="S4.SS1.4.p2.1.m1.3.3.1.1.1.3.cmml" xref="S4.SS1.4.p2.1.m1.3.3.1.1.1.3"></times></apply><apply id="S4.SS1.4.p2.1.m1.4.4.2.2.2.cmml" xref="S4.SS1.4.p2.1.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p2.1.m1.4.4.2.2.2.1.cmml" xref="S4.SS1.4.p2.1.m1.4.4.2.2.2">subscript</csymbol><ci id="S4.SS1.4.p2.1.m1.4.4.2.2.2.2.cmml" xref="S4.SS1.4.p2.1.m1.4.4.2.2.2.2">𝑟</ci><list id="S4.SS1.4.p2.1.m1.2.2.2.3.cmml" xref="S4.SS1.4.p2.1.m1.2.2.2.4"><ci id="S4.SS1.4.p2.1.m1.1.1.1.1.cmml" xref="S4.SS1.4.p2.1.m1.1.1.1.1">𝜀</ci><ci id="S4.SS1.4.p2.1.m1.2.2.2.2.cmml" xref="S4.SS1.4.p2.1.m1.2.2.2.2">𝜆</ci></list></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.1.m1.4c">B^{p}(x^{*},r_{\varepsilon,\lambda})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.1.m1.4d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT )</annotation></semantics></math> stays in <math alttext="M" class="ltx_Math" display="inline" id="S4.SS1.4.p2.2.m2.1"><semantics id="S4.SS1.4.p2.2.m2.1a"><mi id="S4.SS1.4.p2.2.m2.1.1" xref="S4.SS1.4.p2.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.2.m2.1b"><ci id="S4.SS1.4.p2.2.m2.1.1.cmml" xref="S4.SS1.4.p2.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.2.m2.1d">italic_M</annotation></semantics></math> as long as we do not query an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p2.3.m3.1"><semantics id="S4.SS1.4.p2.3.m3.1a"><mi id="S4.SS1.4.p2.3.m3.1.1" xref="S4.SS1.4.p2.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.3.m3.1b"><ci id="S4.SS1.4.p2.3.m3.1.1.cmml" xref="S4.SS1.4.p2.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.3.m3.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. Thus, we know that we must terminate before having reached <math alttext="\operatorname{vol}(M)<\operatorname{vol}(B^{p}(0,r_{\varepsilon,\lambda}))" class="ltx_Math" display="inline" id="S4.SS1.4.p2.4.m4.7"><semantics id="S4.SS1.4.p2.4.m4.7a"><mrow id="S4.SS1.4.p2.4.m4.7.7" xref="S4.SS1.4.p2.4.m4.7.7.cmml"><mrow id="S4.SS1.4.p2.4.m4.7.7.3.2" xref="S4.SS1.4.p2.4.m4.7.7.3.1.cmml"><mi id="S4.SS1.4.p2.4.m4.3.3" xref="S4.SS1.4.p2.4.m4.3.3.cmml">vol</mi><mo id="S4.SS1.4.p2.4.m4.7.7.3.2a" xref="S4.SS1.4.p2.4.m4.7.7.3.1.cmml">⁡</mo><mrow id="S4.SS1.4.p2.4.m4.7.7.3.2.1" xref="S4.SS1.4.p2.4.m4.7.7.3.1.cmml"><mo id="S4.SS1.4.p2.4.m4.7.7.3.2.1.1" stretchy="false" xref="S4.SS1.4.p2.4.m4.7.7.3.1.cmml">(</mo><mi id="S4.SS1.4.p2.4.m4.4.4" xref="S4.SS1.4.p2.4.m4.4.4.cmml">M</mi><mo id="S4.SS1.4.p2.4.m4.7.7.3.2.1.2" stretchy="false" xref="S4.SS1.4.p2.4.m4.7.7.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p2.4.m4.7.7.2" xref="S4.SS1.4.p2.4.m4.7.7.2.cmml"><</mo><mrow id="S4.SS1.4.p2.4.m4.7.7.1.1" 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xref="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.2.cmml">(</mo><mn id="S4.SS1.4.p2.4.m4.5.5" xref="S4.SS1.4.p2.4.m4.5.5.cmml">0</mn><mo id="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.3" xref="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.2.cmml">,</mo><msub id="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.1.2.cmml">r</mi><mrow id="S4.SS1.4.p2.4.m4.2.2.2.4" xref="S4.SS1.4.p2.4.m4.2.2.2.3.cmml"><mi id="S4.SS1.4.p2.4.m4.1.1.1.1" xref="S4.SS1.4.p2.4.m4.1.1.1.1.cmml">ε</mi><mo id="S4.SS1.4.p2.4.m4.2.2.2.4.1" xref="S4.SS1.4.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.4.p2.4.m4.2.2.2.2" xref="S4.SS1.4.p2.4.m4.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.1.4" stretchy="false" xref="S4.SS1.4.p2.4.m4.7.7.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p2.4.m4.7.7.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.4.m4.7.7.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml 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id="S4.SS1.4.p2.4.m4.7c">\operatorname{vol}(M)<\operatorname{vol}(B^{p}(0,r_{\varepsilon,\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.4.m4.7d">roman_vol ( italic_M ) < roman_vol ( italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( 0 , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT ) )</annotation></semantics></math>. Therefore, to compute our final query bound, we first note that it is well-known that the volume of the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS1.4.p2.5.m5.1"><semantics id="S4.SS1.4.p2.5.m5.1a"><msub id="S4.SS1.4.p2.5.m5.1.1" xref="S4.SS1.4.p2.5.m5.1.1.cmml"><mi id="S4.SS1.4.p2.5.m5.1.1.2" mathvariant="normal" xref="S4.SS1.4.p2.5.m5.1.1.2.cmml">ℓ</mi><mi id="S4.SS1.4.p2.5.m5.1.1.3" xref="S4.SS1.4.p2.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.5.m5.1b"><apply id="S4.SS1.4.p2.5.m5.1.1.cmml" xref="S4.SS1.4.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.5.m5.1.1.1.cmml" xref="S4.SS1.4.p2.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.4.p2.5.m5.1.1.2.cmml" xref="S4.SS1.4.p2.5.m5.1.1.2">ℓ</ci><ci id="S4.SS1.4.p2.5.m5.1.1.3.cmml" xref="S4.SS1.4.p2.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm ball <math alttext="B^{p}(0,r_{\varepsilon,\lambda})" class="ltx_Math" display="inline" id="S4.SS1.4.p2.6.m6.4"><semantics id="S4.SS1.4.p2.6.m6.4a"><mrow id="S4.SS1.4.p2.6.m6.4.4" xref="S4.SS1.4.p2.6.m6.4.4.cmml"><msup id="S4.SS1.4.p2.6.m6.4.4.3" xref="S4.SS1.4.p2.6.m6.4.4.3.cmml"><mi id="S4.SS1.4.p2.6.m6.4.4.3.2" xref="S4.SS1.4.p2.6.m6.4.4.3.2.cmml">B</mi><mi id="S4.SS1.4.p2.6.m6.4.4.3.3" xref="S4.SS1.4.p2.6.m6.4.4.3.3.cmml">p</mi></msup><mo id="S4.SS1.4.p2.6.m6.4.4.2" xref="S4.SS1.4.p2.6.m6.4.4.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.6.m6.4.4.1.1" xref="S4.SS1.4.p2.6.m6.4.4.1.2.cmml"><mo id="S4.SS1.4.p2.6.m6.4.4.1.1.2" stretchy="false" xref="S4.SS1.4.p2.6.m6.4.4.1.2.cmml">(</mo><mn id="S4.SS1.4.p2.6.m6.3.3" xref="S4.SS1.4.p2.6.m6.3.3.cmml">0</mn><mo id="S4.SS1.4.p2.6.m6.4.4.1.1.3" xref="S4.SS1.4.p2.6.m6.4.4.1.2.cmml">,</mo><msub 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id="S4.SS1.4.p2.6.m6.4c">B^{p}(0,r_{\varepsilon,\lambda})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.6.m6.4d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( 0 , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT )</annotation></semantics></math> can be bounded from below by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{vol}(B^{p}_{0,r_{\varepsilon,\lambda}})\geq\frac{2^{d}}{d!}r_{% \varepsilon,\lambda}^{d}=\frac{2^{d}}{d!}(\frac{\varepsilon-\varepsilon\lambda% }{2+2\lambda})^{d}." class="ltx_Math" display="block" id="S4.Ex9.m1.9"><semantics id="S4.Ex9.m1.9a"><mrow id="S4.Ex9.m1.9.9.1" xref="S4.Ex9.m1.9.9.1.1.cmml"><mrow id="S4.Ex9.m1.9.9.1.1" xref="S4.Ex9.m1.9.9.1.1.cmml"><mrow id="S4.Ex9.m1.9.9.1.1.1.1" xref="S4.Ex9.m1.9.9.1.1.1.2.cmml"><mi id="S4.Ex9.m1.7.7" xref="S4.Ex9.m1.7.7.cmml">vol</mi><mo id="S4.Ex9.m1.9.9.1.1.1.1a" xref="S4.Ex9.m1.9.9.1.1.1.2.cmml">⁡</mo><mrow id="S4.Ex9.m1.9.9.1.1.1.1.1" xref="S4.Ex9.m1.9.9.1.1.1.2.cmml"><mo id="S4.Ex9.m1.9.9.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.9.9.1.1.1.2.cmml">(</mo><msubsup id="S4.Ex9.m1.9.9.1.1.1.1.1.1" xref="S4.Ex9.m1.9.9.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.9.9.1.1.1.1.1.1.2.2" xref="S4.Ex9.m1.9.9.1.1.1.1.1.1.2.2.cmml">B</mi><mrow id="S4.Ex9.m1.4.4.4.4" xref="S4.Ex9.m1.4.4.4.5.cmml"><mn id="S4.Ex9.m1.3.3.3.3" xref="S4.Ex9.m1.3.3.3.3.cmml">0</mn><mo id="S4.Ex9.m1.4.4.4.4.2" xref="S4.Ex9.m1.4.4.4.5.cmml">,</mo><msub id="S4.Ex9.m1.4.4.4.4.1" xref="S4.Ex9.m1.4.4.4.4.1.cmml"><mi id="S4.Ex9.m1.4.4.4.4.1.2" xref="S4.Ex9.m1.4.4.4.4.1.2.cmml">r</mi><mrow id="S4.Ex9.m1.2.2.2.2.2.4" xref="S4.Ex9.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.cmml">ε</mi><mo id="S4.Ex9.m1.2.2.2.2.2.4.1" xref="S4.Ex9.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex9.m1.2.2.2.2.2.2" xref="S4.Ex9.m1.2.2.2.2.2.2.cmml">λ</mi></mrow></msub></mrow><mi id="S4.Ex9.m1.9.9.1.1.1.1.1.1.2.3" xref="S4.Ex9.m1.9.9.1.1.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="S4.Ex9.m1.9.9.1.1.1.1.1.3" stretchy="false" xref="S4.Ex9.m1.9.9.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.9.9.1.1.3" xref="S4.Ex9.m1.9.9.1.1.3.cmml">≥</mo><mrow id="S4.Ex9.m1.9.9.1.1.4" xref="S4.Ex9.m1.9.9.1.1.4.cmml"><mfrac id="S4.Ex9.m1.9.9.1.1.4.2" xref="S4.Ex9.m1.9.9.1.1.4.2.cmml"><msup id="S4.Ex9.m1.9.9.1.1.4.2.2" xref="S4.Ex9.m1.9.9.1.1.4.2.2.cmml"><mn id="S4.Ex9.m1.9.9.1.1.4.2.2.2" xref="S4.Ex9.m1.9.9.1.1.4.2.2.2.cmml">2</mn><mi id="S4.Ex9.m1.9.9.1.1.4.2.2.3" xref="S4.Ex9.m1.9.9.1.1.4.2.2.3.cmml">d</mi></msup><mrow id="S4.Ex9.m1.9.9.1.1.4.2.3" xref="S4.Ex9.m1.9.9.1.1.4.2.3.cmml"><mi id="S4.Ex9.m1.9.9.1.1.4.2.3.2" xref="S4.Ex9.m1.9.9.1.1.4.2.3.2.cmml">d</mi><mo id="S4.Ex9.m1.9.9.1.1.4.2.3.1" xref="S4.Ex9.m1.9.9.1.1.4.2.3.1.cmml">!</mo></mrow></mfrac><mo id="S4.Ex9.m1.9.9.1.1.4.1" xref="S4.Ex9.m1.9.9.1.1.4.1.cmml">⁢</mo><msubsup id="S4.Ex9.m1.9.9.1.1.4.3" xref="S4.Ex9.m1.9.9.1.1.4.3.cmml"><mi id="S4.Ex9.m1.9.9.1.1.4.3.2.2" xref="S4.Ex9.m1.9.9.1.1.4.3.2.2.cmml">r</mi><mrow id="S4.Ex9.m1.6.6.2.4" xref="S4.Ex9.m1.6.6.2.3.cmml"><mi id="S4.Ex9.m1.5.5.1.1" xref="S4.Ex9.m1.5.5.1.1.cmml">ε</mi><mo id="S4.Ex9.m1.6.6.2.4.1" xref="S4.Ex9.m1.6.6.2.3.cmml">,</mo><mi id="S4.Ex9.m1.6.6.2.2" xref="S4.Ex9.m1.6.6.2.2.cmml">λ</mi></mrow><mi id="S4.Ex9.m1.9.9.1.1.4.3.3" xref="S4.Ex9.m1.9.9.1.1.4.3.3.cmml">d</mi></msubsup></mrow><mo id="S4.Ex9.m1.9.9.1.1.5" xref="S4.Ex9.m1.9.9.1.1.5.cmml">=</mo><mrow id="S4.Ex9.m1.9.9.1.1.6" xref="S4.Ex9.m1.9.9.1.1.6.cmml"><mfrac id="S4.Ex9.m1.9.9.1.1.6.2" xref="S4.Ex9.m1.9.9.1.1.6.2.cmml"><msup id="S4.Ex9.m1.9.9.1.1.6.2.2" xref="S4.Ex9.m1.9.9.1.1.6.2.2.cmml"><mn id="S4.Ex9.m1.9.9.1.1.6.2.2.2" xref="S4.Ex9.m1.9.9.1.1.6.2.2.2.cmml">2</mn><mi 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xref="S4.Ex9.m1.8.8.2.3.1"></times><ci id="S4.Ex9.m1.8.8.2.3.2.cmml" xref="S4.Ex9.m1.8.8.2.3.2">𝜀</ci><ci id="S4.Ex9.m1.8.8.2.3.3.cmml" xref="S4.Ex9.m1.8.8.2.3.3">𝜆</ci></apply></apply><apply id="S4.Ex9.m1.8.8.3.cmml" xref="S4.Ex9.m1.8.8.3"><plus id="S4.Ex9.m1.8.8.3.1.cmml" xref="S4.Ex9.m1.8.8.3.1"></plus><cn id="S4.Ex9.m1.8.8.3.2.cmml" type="integer" xref="S4.Ex9.m1.8.8.3.2">2</cn><apply id="S4.Ex9.m1.8.8.3.3.cmml" xref="S4.Ex9.m1.8.8.3.3"><times id="S4.Ex9.m1.8.8.3.3.1.cmml" xref="S4.Ex9.m1.8.8.3.3.1"></times><cn id="S4.Ex9.m1.8.8.3.3.2.cmml" type="integer" xref="S4.Ex9.m1.8.8.3.3.2">2</cn><ci id="S4.Ex9.m1.8.8.3.3.3.cmml" xref="S4.Ex9.m1.8.8.3.3.3">𝜆</ci></apply></apply></apply><ci id="S4.Ex9.m1.9.9.1.1.6.3.3.cmml" xref="S4.Ex9.m1.9.9.1.1.6.3.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.9c">\operatorname{vol}(B^{p}_{0,r_{\varepsilon,\lambda}})\geq\frac{2^{d}}{d!}r_{% \varepsilon,\lambda}^{d}=\frac{2^{d}}{d!}(\frac{\varepsilon-\varepsilon\lambda% }{2+2\lambda})^{d}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.9d">roman_vol ( italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≥ divide start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG italic_d ! end_ARG italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT = divide start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG italic_d ! end_ARG ( divide start_ARG italic_ε - italic_ε italic_λ end_ARG start_ARG 2 + 2 italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.4.p2.7">Finally, we can upper bound the number <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.4.p2.7.m1.1"><semantics id="S4.SS1.4.p2.7.m1.1a"><mi id="S4.SS1.4.p2.7.m1.1.1" xref="S4.SS1.4.p2.7.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.7.m1.1b"><ci id="S4.SS1.4.p2.7.m1.1.1.cmml" xref="S4.SS1.4.p2.7.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.7.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.7.m1.1d">italic_k</annotation></semantics></math> of queries by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="k\leq\log_{\frac{d}{d+1}}\left(\frac{2^{d}}{d!}\left(\frac{\varepsilon-% \varepsilon\lambda}{2+2\lambda}\right)^{d}\right)=\frac{\log\left(\frac{2^{d}}% {d!}(\frac{\varepsilon-\varepsilon\lambda}{2+2\lambda})^{d}\right)}{\log\left(% \frac{d}{d+1}\right)}=\frac{\log\left(\frac{d!}{2^{d}}(\frac{2}{\varepsilon}% \frac{1+\lambda}{1-\lambda})^{d}\right)}{\log\left(\frac{d+1}{d}\right)}" class="ltx_Math" display="block" id="S4.Ex10.m1.12"><semantics id="S4.Ex10.m1.12a"><mrow id="S4.Ex10.m1.12.12" xref="S4.Ex10.m1.12.12.cmml"><mi id="S4.Ex10.m1.12.12.4" xref="S4.Ex10.m1.12.12.4.cmml">k</mi><mo id="S4.Ex10.m1.12.12.5" xref="S4.Ex10.m1.12.12.5.cmml">≤</mo><mrow id="S4.Ex10.m1.12.12.2.2" xref="S4.Ex10.m1.12.12.2.3.cmml"><msub id="S4.Ex10.m1.11.11.1.1.1" xref="S4.Ex10.m1.11.11.1.1.1.cmml"><mi id="S4.Ex10.m1.11.11.1.1.1.2" xref="S4.Ex10.m1.11.11.1.1.1.2.cmml">log</mi><mfrac id="S4.Ex10.m1.11.11.1.1.1.3" xref="S4.Ex10.m1.11.11.1.1.1.3.cmml"><mi id="S4.Ex10.m1.11.11.1.1.1.3.2" xref="S4.Ex10.m1.11.11.1.1.1.3.2.cmml">d</mi><mrow id="S4.Ex10.m1.11.11.1.1.1.3.3" xref="S4.Ex10.m1.11.11.1.1.1.3.3.cmml"><mi id="S4.Ex10.m1.11.11.1.1.1.3.3.2" xref="S4.Ex10.m1.11.11.1.1.1.3.3.2.cmml">d</mi><mo id="S4.Ex10.m1.11.11.1.1.1.3.3.1" xref="S4.Ex10.m1.11.11.1.1.1.3.3.1.cmml">+</mo><mn id="S4.Ex10.m1.11.11.1.1.1.3.3.3" xref="S4.Ex10.m1.11.11.1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></msub><mo id="S4.Ex10.m1.12.12.2.2a" xref="S4.Ex10.m1.12.12.2.3.cmml">⁡</mo><mrow id="S4.Ex10.m1.12.12.2.2.2" xref="S4.Ex10.m1.12.12.2.3.cmml"><mo id="S4.Ex10.m1.12.12.2.2.2.2" xref="S4.Ex10.m1.12.12.2.3.cmml">(</mo><mrow id="S4.Ex10.m1.12.12.2.2.2.1" xref="S4.Ex10.m1.12.12.2.2.2.1.cmml"><mfrac id="S4.Ex10.m1.12.12.2.2.2.1.2" xref="S4.Ex10.m1.12.12.2.2.2.1.2.cmml"><msup id="S4.Ex10.m1.12.12.2.2.2.1.2.2" xref="S4.Ex10.m1.12.12.2.2.2.1.2.2.cmml"><mn id="S4.Ex10.m1.12.12.2.2.2.1.2.2.2" xref="S4.Ex10.m1.12.12.2.2.2.1.2.2.2.cmml">2</mn><mi id="S4.Ex10.m1.12.12.2.2.2.1.2.2.3" xref="S4.Ex10.m1.12.12.2.2.2.1.2.2.3.cmml">d</mi></msup><mrow id="S4.Ex10.m1.12.12.2.2.2.1.2.3" xref="S4.Ex10.m1.12.12.2.2.2.1.2.3.cmml"><mi id="S4.Ex10.m1.12.12.2.2.2.1.2.3.2" xref="S4.Ex10.m1.12.12.2.2.2.1.2.3.2.cmml">d</mi><mo id="S4.Ex10.m1.12.12.2.2.2.1.2.3.1" xref="S4.Ex10.m1.12.12.2.2.2.1.2.3.1.cmml">!</mo></mrow></mfrac><mo id="S4.Ex10.m1.12.12.2.2.2.1.1" xref="S4.Ex10.m1.12.12.2.2.2.1.1.cmml">⁢</mo><msup id="S4.Ex10.m1.12.12.2.2.2.1.3" xref="S4.Ex10.m1.12.12.2.2.2.1.3.cmml"><mrow id="S4.Ex10.m1.12.12.2.2.2.1.3.2.2" xref="S4.Ex10.m1.10.10.cmml"><mo id="S4.Ex10.m1.12.12.2.2.2.1.3.2.2.1" xref="S4.Ex10.m1.10.10.cmml">(</mo><mfrac id="S4.Ex10.m1.10.10" xref="S4.Ex10.m1.10.10.cmml"><mrow id="S4.Ex10.m1.10.10.2" xref="S4.Ex10.m1.10.10.2.cmml"><mi id="S4.Ex10.m1.10.10.2.2" xref="S4.Ex10.m1.10.10.2.2.cmml">ε</mi><mo id="S4.Ex10.m1.10.10.2.1" xref="S4.Ex10.m1.10.10.2.1.cmml">−</mo><mrow id="S4.Ex10.m1.10.10.2.3" xref="S4.Ex10.m1.10.10.2.3.cmml"><mi id="S4.Ex10.m1.10.10.2.3.2" xref="S4.Ex10.m1.10.10.2.3.2.cmml">ε</mi><mo id="S4.Ex10.m1.10.10.2.3.1" xref="S4.Ex10.m1.10.10.2.3.1.cmml">⁢</mo><mi id="S4.Ex10.m1.10.10.2.3.3" xref="S4.Ex10.m1.10.10.2.3.3.cmml">λ</mi></mrow></mrow><mrow id="S4.Ex10.m1.10.10.3" xref="S4.Ex10.m1.10.10.3.cmml"><mn id="S4.Ex10.m1.10.10.3.2" xref="S4.Ex10.m1.10.10.3.2.cmml">2</mn><mo id="S4.Ex10.m1.10.10.3.1" xref="S4.Ex10.m1.10.10.3.1.cmml">+</mo><mrow id="S4.Ex10.m1.10.10.3.3" xref="S4.Ex10.m1.10.10.3.3.cmml"><mn id="S4.Ex10.m1.10.10.3.3.2" xref="S4.Ex10.m1.10.10.3.3.2.cmml">2</mn><mo id="S4.Ex10.m1.10.10.3.3.1" xref="S4.Ex10.m1.10.10.3.3.1.cmml">⁢</mo><mi id="S4.Ex10.m1.10.10.3.3.3" xref="S4.Ex10.m1.10.10.3.3.3.cmml">λ</mi></mrow></mrow></mfrac><mo id="S4.Ex10.m1.12.12.2.2.2.1.3.2.2.2" xref="S4.Ex10.m1.10.10.cmml">)</mo></mrow><mi id="S4.Ex10.m1.12.12.2.2.2.1.3.3" xref="S4.Ex10.m1.12.12.2.2.2.1.3.3.cmml">d</mi></msup></mrow><mo id="S4.Ex10.m1.12.12.2.2.2.3" xref="S4.Ex10.m1.12.12.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.12.12.6" xref="S4.Ex10.m1.12.12.6.cmml">=</mo><mfrac id="S4.Ex10.m1.5.5" xref="S4.Ex10.m1.5.5.cmml"><mrow id="S4.Ex10.m1.3.3.3.3" 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\frac{d}{d+1}\right)}=\frac{\log\left(\frac{d!}{2^{d}}(\frac{2}{\varepsilon}% \frac{1+\lambda}{1-\lambda})^{d}\right)}{\log\left(\frac{d+1}{d}\right)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.12d">italic_k ≤ roman_log start_POSTSUBSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG end_POSTSUBSCRIPT ( divide start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG italic_d ! end_ARG ( divide start_ARG italic_ε - italic_ε italic_λ end_ARG start_ARG 2 + 2 italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) = divide start_ARG roman_log ( divide start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG italic_d ! end_ARG ( divide start_ARG italic_ε - italic_ε italic_λ end_ARG start_ARG 2 + 2 italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) end_ARG start_ARG roman_log ( divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG ) end_ARG = divide start_ARG roman_log ( divide start_ARG italic_d ! end_ARG start_ARG 2 start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG ( divide start_ARG 2 end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) end_ARG start_ARG roman_log ( divide start_ARG italic_d + 1 end_ARG start_ARG italic_d end_ARG ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\leq\frac{\log d!-d+d\left(\log 4+\log(\frac{1}{\varepsilon})+\log(\frac{1}{1-% \lambda})\right)}{\log(\frac{d+1}{d})}" class="ltx_Math" display="block" id="S4.Ex11.m1.7"><semantics id="S4.Ex11.m1.7a"><mrow id="S4.Ex11.m1.7.8" 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id="S4.Ex11.m1.7.7.7.2.2.cmml" xref="S4.Ex11.m1.7.7.7.2.2"><plus id="S4.Ex11.m1.7.7.7.2.2.1.cmml" xref="S4.Ex11.m1.7.7.7.2.2.1"></plus><ci id="S4.Ex11.m1.7.7.7.2.2.2.cmml" xref="S4.Ex11.m1.7.7.7.2.2.2">𝑑</ci><cn id="S4.Ex11.m1.7.7.7.2.2.3.cmml" type="integer" xref="S4.Ex11.m1.7.7.7.2.2.3">1</cn></apply><ci id="S4.Ex11.m1.7.7.7.2.3.cmml" xref="S4.Ex11.m1.7.7.7.2.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.7c">\leq\frac{\log d!-d+d\left(\log 4+\log(\frac{1}{\varepsilon})+\log(\frac{1}{1-% \lambda})\right)}{\log(\frac{d+1}{d})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.7d">≤ divide start_ARG roman_log italic_d ! - italic_d + italic_d ( roman_log 4 + roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) + roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) ) end_ARG start_ARG roman_log ( divide start_ARG italic_d + 1 end_ARG start_ARG italic_d end_ARG ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\leq\frac{\mathcal{O}\left(d\left(\log d+\log(\frac{1}{\varepsilon})+\log(% \frac{1}{1-\lambda})\right)\right)}{\log(1+\frac{1}{d})}." class="ltx_Math" display="block" id="S4.Ex12.m1.8"><semantics id="S4.Ex12.m1.8a"><mrow id="S4.Ex12.m1.8.8.1" xref="S4.Ex12.m1.8.8.1.1.cmml"><mrow id="S4.Ex12.m1.8.8.1.1" xref="S4.Ex12.m1.8.8.1.1.cmml"><mi id="S4.Ex12.m1.8.8.1.1.2" xref="S4.Ex12.m1.8.8.1.1.2.cmml"></mi><mo id="S4.Ex12.m1.8.8.1.1.1" xref="S4.Ex12.m1.8.8.1.1.1.cmml">≤</mo><mfrac id="S4.Ex12.m1.7.7" xref="S4.Ex12.m1.7.7.cmml"><mrow id="S4.Ex12.m1.5.5.5" xref="S4.Ex12.m1.5.5.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex12.m1.5.5.5.7" 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id="S4.Ex12.m1.7.7.7.2.1.1.cmml" xref="S4.Ex12.m1.7.7.7.2.1.1"><plus id="S4.Ex12.m1.7.7.7.2.1.1.1.cmml" xref="S4.Ex12.m1.7.7.7.2.1.1.1"></plus><cn id="S4.Ex12.m1.7.7.7.2.1.1.2.cmml" type="integer" xref="S4.Ex12.m1.7.7.7.2.1.1.2">1</cn><apply id="S4.Ex12.m1.7.7.7.2.1.1.3.cmml" xref="S4.Ex12.m1.7.7.7.2.1.1.3"><divide id="S4.Ex12.m1.7.7.7.2.1.1.3.1.cmml" xref="S4.Ex12.m1.7.7.7.2.1.1.3"></divide><cn id="S4.Ex12.m1.7.7.7.2.1.1.3.2.cmml" type="integer" xref="S4.Ex12.m1.7.7.7.2.1.1.3.2">1</cn><ci id="S4.Ex12.m1.7.7.7.2.1.1.3.3.cmml" xref="S4.Ex12.m1.7.7.7.2.1.1.3.3">𝑑</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.8c">\leq\frac{\mathcal{O}\left(d\left(\log d+\log(\frac{1}{\varepsilon})+\log(% \frac{1}{1-\lambda})\right)\right)}{\log(1+\frac{1}{d})}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.8d">≤ divide start_ARG caligraphic_O ( italic_d ( roman_log italic_d + roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) + roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) ) ) end_ARG start_ARG roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.4.p2.28">For small <math alttext="\delta>0" class="ltx_Math" display="inline" id="S4.SS1.4.p2.8.m1.1"><semantics id="S4.SS1.4.p2.8.m1.1a"><mrow id="S4.SS1.4.p2.8.m1.1.1" xref="S4.SS1.4.p2.8.m1.1.1.cmml"><mi id="S4.SS1.4.p2.8.m1.1.1.2" xref="S4.SS1.4.p2.8.m1.1.1.2.cmml">δ</mi><mo id="S4.SS1.4.p2.8.m1.1.1.1" xref="S4.SS1.4.p2.8.m1.1.1.1.cmml">></mo><mn id="S4.SS1.4.p2.8.m1.1.1.3" xref="S4.SS1.4.p2.8.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.8.m1.1b"><apply id="S4.SS1.4.p2.8.m1.1.1.cmml" xref="S4.SS1.4.p2.8.m1.1.1"><gt id="S4.SS1.4.p2.8.m1.1.1.1.cmml" xref="S4.SS1.4.p2.8.m1.1.1.1"></gt><ci id="S4.SS1.4.p2.8.m1.1.1.2.cmml" xref="S4.SS1.4.p2.8.m1.1.1.2">𝛿</ci><cn id="S4.SS1.4.p2.8.m1.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.8.m1.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.8.m1.1d">italic_δ > 0</annotation></semantics></math>, we have <math alttext="\log(1+\delta)\approx\delta" class="ltx_Math" display="inline" id="S4.SS1.4.p2.9.m2.2"><semantics id="S4.SS1.4.p2.9.m2.2a"><mrow id="S4.SS1.4.p2.9.m2.2.2" xref="S4.SS1.4.p2.9.m2.2.2.cmml"><mrow id="S4.SS1.4.p2.9.m2.2.2.1.1" xref="S4.SS1.4.p2.9.m2.2.2.1.2.cmml"><mi id="S4.SS1.4.p2.9.m2.1.1" xref="S4.SS1.4.p2.9.m2.1.1.cmml">log</mi><mo id="S4.SS1.4.p2.9.m2.2.2.1.1a" xref="S4.SS1.4.p2.9.m2.2.2.1.2.cmml">⁡</mo><mrow id="S4.SS1.4.p2.9.m2.2.2.1.1.1" xref="S4.SS1.4.p2.9.m2.2.2.1.2.cmml"><mo id="S4.SS1.4.p2.9.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.9.m2.2.2.1.2.cmml">(</mo><mrow id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.cmml"><mn id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.2" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.2.cmml">1</mn><mo id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.1" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.1.cmml">+</mo><mi id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.3" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.3.cmml">δ</mi></mrow><mo id="S4.SS1.4.p2.9.m2.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.9.m2.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p2.9.m2.2.2.2" xref="S4.SS1.4.p2.9.m2.2.2.2.cmml">≈</mo><mi id="S4.SS1.4.p2.9.m2.2.2.3" xref="S4.SS1.4.p2.9.m2.2.2.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.9.m2.2b"><apply id="S4.SS1.4.p2.9.m2.2.2.cmml" xref="S4.SS1.4.p2.9.m2.2.2"><approx id="S4.SS1.4.p2.9.m2.2.2.2.cmml" xref="S4.SS1.4.p2.9.m2.2.2.2"></approx><apply id="S4.SS1.4.p2.9.m2.2.2.1.2.cmml" xref="S4.SS1.4.p2.9.m2.2.2.1.1"><log id="S4.SS1.4.p2.9.m2.1.1.cmml" xref="S4.SS1.4.p2.9.m2.1.1"></log><apply id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1"><plus id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.1"></plus><cn id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.2">1</cn><ci id="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.9.m2.2.2.1.1.1.1.3">𝛿</ci></apply></apply><ci id="S4.SS1.4.p2.9.m2.2.2.3.cmml" xref="S4.SS1.4.p2.9.m2.2.2.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.9.m2.2c">\log(1+\delta)\approx\delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.9.m2.2d">roman_log ( 1 + italic_δ ) ≈ italic_δ</annotation></semantics></math>, and thus we get <math alttext="k\leq\mathcal{O}(d^{2}(\log d+\log(\frac{1}{\varepsilon})+\log(\frac{1}{1-% \lambda}))" class="ltx_math_unparsed" display="inline" id="S4.SS1.4.p2.10.m3.4"><semantics id="S4.SS1.4.p2.10.m3.4a"><mrow id="S4.SS1.4.p2.10.m3.4b"><mi id="S4.SS1.4.p2.10.m3.4.5">k</mi><mo id="S4.SS1.4.p2.10.m3.4.6">≤</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p2.10.m3.4.7">𝒪</mi><mrow id="S4.SS1.4.p2.10.m3.4.8"><mo id="S4.SS1.4.p2.10.m3.4.8.1" stretchy="false">(</mo><msup id="S4.SS1.4.p2.10.m3.4.8.2"><mi id="S4.SS1.4.p2.10.m3.4.8.2.2">d</mi><mn id="S4.SS1.4.p2.10.m3.4.8.2.3">2</mn></msup><mrow id="S4.SS1.4.p2.10.m3.4.8.3"><mo id="S4.SS1.4.p2.10.m3.4.8.3.1" stretchy="false">(</mo><mi id="S4.SS1.4.p2.10.m3.4.8.3.2">log</mi><mi id="S4.SS1.4.p2.10.m3.4.8.3.3">d</mi><mo id="S4.SS1.4.p2.10.m3.4.8.3.4">+</mo><mi id="S4.SS1.4.p2.10.m3.1.1">log</mi><mrow id="S4.SS1.4.p2.10.m3.4.8.3.5"><mo id="S4.SS1.4.p2.10.m3.4.8.3.5.1" stretchy="false">(</mo><mfrac id="S4.SS1.4.p2.10.m3.2.2"><mn id="S4.SS1.4.p2.10.m3.2.2.2">1</mn><mi id="S4.SS1.4.p2.10.m3.2.2.3">ε</mi></mfrac><mo id="S4.SS1.4.p2.10.m3.4.8.3.5.2" stretchy="false">)</mo></mrow><mo id="S4.SS1.4.p2.10.m3.4.8.3.6">+</mo><mi id="S4.SS1.4.p2.10.m3.3.3">log</mi><mrow id="S4.SS1.4.p2.10.m3.4.8.3.7"><mo id="S4.SS1.4.p2.10.m3.4.8.3.7.1" stretchy="false">(</mo><mfrac id="S4.SS1.4.p2.10.m3.4.4"><mn id="S4.SS1.4.p2.10.m3.4.4.2">1</mn><mrow id="S4.SS1.4.p2.10.m3.4.4.3"><mn id="S4.SS1.4.p2.10.m3.4.4.3.2">1</mn><mo id="S4.SS1.4.p2.10.m3.4.4.3.1">−</mo><mi id="S4.SS1.4.p2.10.m3.4.4.3.3">λ</mi></mrow></mfrac><mo id="S4.SS1.4.p2.10.m3.4.8.3.7.2" stretchy="false">)</mo></mrow><mo id="S4.SS1.4.p2.10.m3.4.8.3.8" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS1.4.p2.10.m3.4c">k\leq\mathcal{O}(d^{2}(\log d+\log(\frac{1}{\varepsilon})+\log(\frac{1}{1-% \lambda}))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.10.m3.4d">italic_k ≤ caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log italic_d + roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) + roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math>. To get rid of the <math alttext="\log d" class="ltx_Math" display="inline" id="S4.SS1.4.p2.11.m4.1"><semantics id="S4.SS1.4.p2.11.m4.1a"><mrow id="S4.SS1.4.p2.11.m4.1.1" xref="S4.SS1.4.p2.11.m4.1.1.cmml"><mi id="S4.SS1.4.p2.11.m4.1.1.1" xref="S4.SS1.4.p2.11.m4.1.1.1.cmml">log</mi><mo id="S4.SS1.4.p2.11.m4.1.1a" lspace="0.167em" xref="S4.SS1.4.p2.11.m4.1.1.cmml">⁡</mo><mi id="S4.SS1.4.p2.11.m4.1.1.2" xref="S4.SS1.4.p2.11.m4.1.1.2.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.11.m4.1b"><apply id="S4.SS1.4.p2.11.m4.1.1.cmml" xref="S4.SS1.4.p2.11.m4.1.1"><log id="S4.SS1.4.p2.11.m4.1.1.1.cmml" xref="S4.SS1.4.p2.11.m4.1.1.1"></log><ci id="S4.SS1.4.p2.11.m4.1.1.2.cmml" xref="S4.SS1.4.p2.11.m4.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.11.m4.1c">\log d</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.11.m4.1d">roman_log italic_d</annotation></semantics></math> term, observe that if <math alttext="\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d" class="ltx_Math" display="inline" id="S4.SS1.4.p2.12.m5.3"><semantics id="S4.SS1.4.p2.12.m5.3a"><mrow id="S4.SS1.4.p2.12.m5.3.4" xref="S4.SS1.4.p2.12.m5.3.4.cmml"><mrow id="S4.SS1.4.p2.12.m5.3.4.2.2" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml"><mi id="S4.SS1.4.p2.12.m5.1.1" xref="S4.SS1.4.p2.12.m5.1.1.cmml">max</mi><mo id="S4.SS1.4.p2.12.m5.3.4.2.2a" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml">⁡</mo><mrow id="S4.SS1.4.p2.12.m5.3.4.2.2.1" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml"><mo id="S4.SS1.4.p2.12.m5.3.4.2.2.1.1" stretchy="false" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml">(</mo><mfrac id="S4.SS1.4.p2.12.m5.2.2" xref="S4.SS1.4.p2.12.m5.2.2.cmml"><mn id="S4.SS1.4.p2.12.m5.2.2.2" xref="S4.SS1.4.p2.12.m5.2.2.2.cmml">1</mn><mi id="S4.SS1.4.p2.12.m5.2.2.3" xref="S4.SS1.4.p2.12.m5.2.2.3.cmml">ε</mi></mfrac><mo id="S4.SS1.4.p2.12.m5.3.4.2.2.1.2" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml">,</mo><mfrac id="S4.SS1.4.p2.12.m5.3.3" xref="S4.SS1.4.p2.12.m5.3.3.cmml"><mn id="S4.SS1.4.p2.12.m5.3.3.2" xref="S4.SS1.4.p2.12.m5.3.3.2.cmml">1</mn><mrow id="S4.SS1.4.p2.12.m5.3.3.3" xref="S4.SS1.4.p2.12.m5.3.3.3.cmml"><mn id="S4.SS1.4.p2.12.m5.3.3.3.2" xref="S4.SS1.4.p2.12.m5.3.3.3.2.cmml">1</mn><mo id="S4.SS1.4.p2.12.m5.3.3.3.1" xref="S4.SS1.4.p2.12.m5.3.3.3.1.cmml">−</mo><mi id="S4.SS1.4.p2.12.m5.3.3.3.3" xref="S4.SS1.4.p2.12.m5.3.3.3.3.cmml">λ</mi></mrow></mfrac><mo id="S4.SS1.4.p2.12.m5.3.4.2.2.1.3" stretchy="false" xref="S4.SS1.4.p2.12.m5.3.4.2.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p2.12.m5.3.4.1" xref="S4.SS1.4.p2.12.m5.3.4.1.cmml"><</mo><mi id="S4.SS1.4.p2.12.m5.3.4.3" xref="S4.SS1.4.p2.12.m5.3.4.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.12.m5.3b"><apply id="S4.SS1.4.p2.12.m5.3.4.cmml" xref="S4.SS1.4.p2.12.m5.3.4"><lt id="S4.SS1.4.p2.12.m5.3.4.1.cmml" xref="S4.SS1.4.p2.12.m5.3.4.1"></lt><apply id="S4.SS1.4.p2.12.m5.3.4.2.1.cmml" xref="S4.SS1.4.p2.12.m5.3.4.2.2"><max id="S4.SS1.4.p2.12.m5.1.1.cmml" xref="S4.SS1.4.p2.12.m5.1.1"></max><apply id="S4.SS1.4.p2.12.m5.2.2.cmml" xref="S4.SS1.4.p2.12.m5.2.2"><divide id="S4.SS1.4.p2.12.m5.2.2.1.cmml" xref="S4.SS1.4.p2.12.m5.2.2"></divide><cn id="S4.SS1.4.p2.12.m5.2.2.2.cmml" type="integer" xref="S4.SS1.4.p2.12.m5.2.2.2">1</cn><ci id="S4.SS1.4.p2.12.m5.2.2.3.cmml" xref="S4.SS1.4.p2.12.m5.2.2.3">𝜀</ci></apply><apply id="S4.SS1.4.p2.12.m5.3.3.cmml" xref="S4.SS1.4.p2.12.m5.3.3"><divide id="S4.SS1.4.p2.12.m5.3.3.1.cmml" xref="S4.SS1.4.p2.12.m5.3.3"></divide><cn id="S4.SS1.4.p2.12.m5.3.3.2.cmml" type="integer" xref="S4.SS1.4.p2.12.m5.3.3.2">1</cn><apply id="S4.SS1.4.p2.12.m5.3.3.3.cmml" xref="S4.SS1.4.p2.12.m5.3.3.3"><minus id="S4.SS1.4.p2.12.m5.3.3.3.1.cmml" xref="S4.SS1.4.p2.12.m5.3.3.3.1"></minus><cn id="S4.SS1.4.p2.12.m5.3.3.3.2.cmml" type="integer" xref="S4.SS1.4.p2.12.m5.3.3.3.2">1</cn><ci id="S4.SS1.4.p2.12.m5.3.3.3.3.cmml" xref="S4.SS1.4.p2.12.m5.3.3.3.3">𝜆</ci></apply></apply></apply><ci id="S4.SS1.4.p2.12.m5.3.4.3.cmml" xref="S4.SS1.4.p2.12.m5.3.4.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.12.m5.3c">\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.12.m5.3d">roman_max ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG , divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) < italic_d</annotation></semantics></math> (i.e., whenever this term matters), a simple iteration algorithm can find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p2.13.m6.1"><semantics id="S4.SS1.4.p2.13.m6.1a"><mi id="S4.SS1.4.p2.13.m6.1.1" xref="S4.SS1.4.p2.13.m6.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.13.m6.1b"><ci id="S4.SS1.4.p2.13.m6.1.1.cmml" xref="S4.SS1.4.p2.13.m6.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.13.m6.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.13.m6.1d">italic_ε</annotation></semantics></math>-approximate fixpoint after at most <math alttext="\mathcal{O}(d\log d)" class="ltx_Math" display="inline" id="S4.SS1.4.p2.14.m7.1"><semantics id="S4.SS1.4.p2.14.m7.1a"><mrow id="S4.SS1.4.p2.14.m7.1.1" xref="S4.SS1.4.p2.14.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p2.14.m7.1.1.3" xref="S4.SS1.4.p2.14.m7.1.1.3.cmml">𝒪</mi><mo id="S4.SS1.4.p2.14.m7.1.1.2" xref="S4.SS1.4.p2.14.m7.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.14.m7.1.1.1.1" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.14.m7.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.4.p2.14.m7.1.1.1.1.1" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.14.m7.1.1.1.1.1.2" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.2.cmml">d</mi><mo id="S4.SS1.4.p2.14.m7.1.1.1.1.1.1" lspace="0.167em" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.1" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3a" lspace="0.167em" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.cmml">⁡</mo><mi id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.2" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.2.cmml">d</mi></mrow></mrow><mo id="S4.SS1.4.p2.14.m7.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.14.m7.1b"><apply id="S4.SS1.4.p2.14.m7.1.1.cmml" xref="S4.SS1.4.p2.14.m7.1.1"><times id="S4.SS1.4.p2.14.m7.1.1.2.cmml" xref="S4.SS1.4.p2.14.m7.1.1.2"></times><ci id="S4.SS1.4.p2.14.m7.1.1.3.cmml" xref="S4.SS1.4.p2.14.m7.1.1.3">𝒪</ci><apply id="S4.SS1.4.p2.14.m7.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1"><times id="S4.SS1.4.p2.14.m7.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.1"></times><ci id="S4.SS1.4.p2.14.m7.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.2">𝑑</ci><apply id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3"><log id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.1.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.1"></log><ci id="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.2.cmml" xref="S4.SS1.4.p2.14.m7.1.1.1.1.1.3.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.14.m7.1c">\mathcal{O}(d\log d)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.14.m7.1d">caligraphic_O ( italic_d roman_log italic_d )</annotation></semantics></math> queries. In particular, let <math alttext="x^{(0)}\in[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS1.4.p2.15.m8.3"><semantics id="S4.SS1.4.p2.15.m8.3a"><mrow id="S4.SS1.4.p2.15.m8.3.4" xref="S4.SS1.4.p2.15.m8.3.4.cmml"><msup id="S4.SS1.4.p2.15.m8.3.4.2" xref="S4.SS1.4.p2.15.m8.3.4.2.cmml"><mi id="S4.SS1.4.p2.15.m8.3.4.2.2" xref="S4.SS1.4.p2.15.m8.3.4.2.2.cmml">x</mi><mrow id="S4.SS1.4.p2.15.m8.1.1.1.3" xref="S4.SS1.4.p2.15.m8.3.4.2.cmml"><mo id="S4.SS1.4.p2.15.m8.1.1.1.3.1" stretchy="false" xref="S4.SS1.4.p2.15.m8.3.4.2.cmml">(</mo><mn id="S4.SS1.4.p2.15.m8.1.1.1.1" xref="S4.SS1.4.p2.15.m8.1.1.1.1.cmml">0</mn><mo id="S4.SS1.4.p2.15.m8.1.1.1.3.2" stretchy="false" xref="S4.SS1.4.p2.15.m8.3.4.2.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.15.m8.3.4.1" xref="S4.SS1.4.p2.15.m8.3.4.1.cmml">∈</mo><msup id="S4.SS1.4.p2.15.m8.3.4.3" xref="S4.SS1.4.p2.15.m8.3.4.3.cmml"><mrow id="S4.SS1.4.p2.15.m8.3.4.3.2.2" xref="S4.SS1.4.p2.15.m8.3.4.3.2.1.cmml"><mo id="S4.SS1.4.p2.15.m8.3.4.3.2.2.1" stretchy="false" xref="S4.SS1.4.p2.15.m8.3.4.3.2.1.cmml">[</mo><mn id="S4.SS1.4.p2.15.m8.2.2" xref="S4.SS1.4.p2.15.m8.2.2.cmml">0</mn><mo id="S4.SS1.4.p2.15.m8.3.4.3.2.2.2" xref="S4.SS1.4.p2.15.m8.3.4.3.2.1.cmml">,</mo><mn id="S4.SS1.4.p2.15.m8.3.3" xref="S4.SS1.4.p2.15.m8.3.3.cmml">1</mn><mo id="S4.SS1.4.p2.15.m8.3.4.3.2.2.3" stretchy="false" xref="S4.SS1.4.p2.15.m8.3.4.3.2.1.cmml">]</mo></mrow><mi id="S4.SS1.4.p2.15.m8.3.4.3.3" xref="S4.SS1.4.p2.15.m8.3.4.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.15.m8.3b"><apply id="S4.SS1.4.p2.15.m8.3.4.cmml" xref="S4.SS1.4.p2.15.m8.3.4"><in id="S4.SS1.4.p2.15.m8.3.4.1.cmml" xref="S4.SS1.4.p2.15.m8.3.4.1"></in><apply id="S4.SS1.4.p2.15.m8.3.4.2.cmml" xref="S4.SS1.4.p2.15.m8.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.4.p2.15.m8.3.4.2.1.cmml" xref="S4.SS1.4.p2.15.m8.3.4.2">superscript</csymbol><ci id="S4.SS1.4.p2.15.m8.3.4.2.2.cmml" xref="S4.SS1.4.p2.15.m8.3.4.2.2">𝑥</ci><cn id="S4.SS1.4.p2.15.m8.1.1.1.1.cmml" type="integer" xref="S4.SS1.4.p2.15.m8.1.1.1.1">0</cn></apply><apply id="S4.SS1.4.p2.15.m8.3.4.3.cmml" xref="S4.SS1.4.p2.15.m8.3.4.3"><csymbol cd="ambiguous" id="S4.SS1.4.p2.15.m8.3.4.3.1.cmml" xref="S4.SS1.4.p2.15.m8.3.4.3">superscript</csymbol><interval closure="closed" id="S4.SS1.4.p2.15.m8.3.4.3.2.1.cmml" xref="S4.SS1.4.p2.15.m8.3.4.3.2.2"><cn id="S4.SS1.4.p2.15.m8.2.2.cmml" type="integer" xref="S4.SS1.4.p2.15.m8.2.2">0</cn><cn id="S4.SS1.4.p2.15.m8.3.3.cmml" type="integer" xref="S4.SS1.4.p2.15.m8.3.3">1</cn></interval><ci id="S4.SS1.4.p2.15.m8.3.4.3.3.cmml" xref="S4.SS1.4.p2.15.m8.3.4.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.15.m8.3c">x^{(0)}\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.15.m8.3d">italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be arbitrary and consider the recursively defined iterates <math alttext="x^{(i)}\coloneqq f(x^{(i-1)})" class="ltx_Math" display="inline" id="S4.SS1.4.p2.16.m9.3"><semantics id="S4.SS1.4.p2.16.m9.3a"><mrow id="S4.SS1.4.p2.16.m9.3.3" xref="S4.SS1.4.p2.16.m9.3.3.cmml"><msup id="S4.SS1.4.p2.16.m9.3.3.3" xref="S4.SS1.4.p2.16.m9.3.3.3.cmml"><mi id="S4.SS1.4.p2.16.m9.3.3.3.2" xref="S4.SS1.4.p2.16.m9.3.3.3.2.cmml">x</mi><mrow id="S4.SS1.4.p2.16.m9.1.1.1.3" xref="S4.SS1.4.p2.16.m9.3.3.3.cmml"><mo id="S4.SS1.4.p2.16.m9.1.1.1.3.1" stretchy="false" xref="S4.SS1.4.p2.16.m9.3.3.3.cmml">(</mo><mi id="S4.SS1.4.p2.16.m9.1.1.1.1" xref="S4.SS1.4.p2.16.m9.1.1.1.1.cmml">i</mi><mo id="S4.SS1.4.p2.16.m9.1.1.1.3.2" stretchy="false" xref="S4.SS1.4.p2.16.m9.3.3.3.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.16.m9.3.3.2" xref="S4.SS1.4.p2.16.m9.3.3.2.cmml">≔</mo><mrow id="S4.SS1.4.p2.16.m9.3.3.1" xref="S4.SS1.4.p2.16.m9.3.3.1.cmml"><mi id="S4.SS1.4.p2.16.m9.3.3.1.3" xref="S4.SS1.4.p2.16.m9.3.3.1.3.cmml">f</mi><mo id="S4.SS1.4.p2.16.m9.3.3.1.2" xref="S4.SS1.4.p2.16.m9.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.16.m9.3.3.1.1.1" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.16.m9.3.3.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p2.16.m9.3.3.1.1.1.1" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.2" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.4.p2.16.m9.2.2.1.1" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.cmml"><mo id="S4.SS1.4.p2.16.m9.2.2.1.1.2" stretchy="false" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.4.p2.16.m9.2.2.1.1.1" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.cmml"><mi id="S4.SS1.4.p2.16.m9.2.2.1.1.1.2" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.16.m9.2.2.1.1.1.1" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.1.cmml">−</mo><mn id="S4.SS1.4.p2.16.m9.2.2.1.1.1.3" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.4.p2.16.m9.2.2.1.1.3" stretchy="false" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.16.m9.3.3.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.16.m9.3b"><apply id="S4.SS1.4.p2.16.m9.3.3.cmml" xref="S4.SS1.4.p2.16.m9.3.3"><ci id="S4.SS1.4.p2.16.m9.3.3.2.cmml" xref="S4.SS1.4.p2.16.m9.3.3.2">≔</ci><apply id="S4.SS1.4.p2.16.m9.3.3.3.cmml" xref="S4.SS1.4.p2.16.m9.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.4.p2.16.m9.3.3.3.1.cmml" xref="S4.SS1.4.p2.16.m9.3.3.3">superscript</csymbol><ci id="S4.SS1.4.p2.16.m9.3.3.3.2.cmml" xref="S4.SS1.4.p2.16.m9.3.3.3.2">𝑥</ci><ci id="S4.SS1.4.p2.16.m9.1.1.1.1.cmml" xref="S4.SS1.4.p2.16.m9.1.1.1.1">𝑖</ci></apply><apply id="S4.SS1.4.p2.16.m9.3.3.1.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1"><times id="S4.SS1.4.p2.16.m9.3.3.1.2.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1.2"></times><ci id="S4.SS1.4.p2.16.m9.3.3.1.3.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1.3">𝑓</ci><apply id="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.16.m9.3.3.1.1.1.1.2">𝑥</ci><apply id="S4.SS1.4.p2.16.m9.2.2.1.1.1.cmml" xref="S4.SS1.4.p2.16.m9.2.2.1.1"><minus id="S4.SS1.4.p2.16.m9.2.2.1.1.1.1.cmml" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.1"></minus><ci id="S4.SS1.4.p2.16.m9.2.2.1.1.1.2.cmml" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.2">𝑖</ci><cn id="S4.SS1.4.p2.16.m9.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.16.m9.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.16.m9.3c">x^{(i)}\coloneqq f(x^{(i-1)})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.16.m9.3d">italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ≔ italic_f ( italic_x start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math> for <math alttext="i\geq 1" class="ltx_Math" display="inline" id="S4.SS1.4.p2.17.m10.1"><semantics id="S4.SS1.4.p2.17.m10.1a"><mrow id="S4.SS1.4.p2.17.m10.1.1" xref="S4.SS1.4.p2.17.m10.1.1.cmml"><mi id="S4.SS1.4.p2.17.m10.1.1.2" xref="S4.SS1.4.p2.17.m10.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.17.m10.1.1.1" xref="S4.SS1.4.p2.17.m10.1.1.1.cmml">≥</mo><mn id="S4.SS1.4.p2.17.m10.1.1.3" xref="S4.SS1.4.p2.17.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.17.m10.1b"><apply id="S4.SS1.4.p2.17.m10.1.1.cmml" xref="S4.SS1.4.p2.17.m10.1.1"><geq id="S4.SS1.4.p2.17.m10.1.1.1.cmml" xref="S4.SS1.4.p2.17.m10.1.1.1"></geq><ci id="S4.SS1.4.p2.17.m10.1.1.2.cmml" xref="S4.SS1.4.p2.17.m10.1.1.2">𝑖</ci><cn id="S4.SS1.4.p2.17.m10.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.17.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.17.m10.1c">i\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.17.m10.1d">italic_i ≥ 1</annotation></semantics></math>. Since <math alttext="f" class="ltx_Math" display="inline" id="S4.SS1.4.p2.18.m11.1"><semantics id="S4.SS1.4.p2.18.m11.1a"><mi id="S4.SS1.4.p2.18.m11.1.1" xref="S4.SS1.4.p2.18.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.18.m11.1b"><ci id="S4.SS1.4.p2.18.m11.1.1.cmml" xref="S4.SS1.4.p2.18.m11.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.18.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.18.m11.1d">italic_f</annotation></semantics></math> is contracting, we have <math alttext="\lVert x^{(i)}-f(x^{(i)})\rVert_{p}\leq\lambda\lVert x^{(i-1)}-f(x^{(i-1)})% \rVert_{p}" class="ltx_Math" display="inline" id="S4.SS1.4.p2.19.m12.6"><semantics id="S4.SS1.4.p2.19.m12.6a"><mrow id="S4.SS1.4.p2.19.m12.6.6" xref="S4.SS1.4.p2.19.m12.6.6.cmml"><msub id="S4.SS1.4.p2.19.m12.5.5.1" xref="S4.SS1.4.p2.19.m12.5.5.1.cmml"><mrow id="S4.SS1.4.p2.19.m12.5.5.1.1.1" xref="S4.SS1.4.p2.19.m12.5.5.1.1.2.cmml"><mo fence="true" id="S4.SS1.4.p2.19.m12.5.5.1.1.1.2" rspace="0em" xref="S4.SS1.4.p2.19.m12.5.5.1.1.2.1.cmml">∥</mo><mrow id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.cmml"><msup id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.2" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.2.cmml">x</mi><mrow id="S4.SS1.4.p2.19.m12.1.1.1.3" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.cmml"><mo id="S4.SS1.4.p2.19.m12.1.1.1.3.1" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.cmml">(</mo><mi id="S4.SS1.4.p2.19.m12.1.1.1.1" xref="S4.SS1.4.p2.19.m12.1.1.1.1.cmml">i</mi><mo id="S4.SS1.4.p2.19.m12.1.1.1.3.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.2.cmml">−</mo><mrow id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.3" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.3.cmml">f</mi><mo id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.4.p2.19.m12.2.2.1.3" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.19.m12.2.2.1.3.1" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.SS1.4.p2.19.m12.2.2.1.1" xref="S4.SS1.4.p2.19.m12.2.2.1.1.cmml">i</mi><mo id="S4.SS1.4.p2.19.m12.2.2.1.3.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S4.SS1.4.p2.19.m12.5.5.1.1.1.3" lspace="0em" xref="S4.SS1.4.p2.19.m12.5.5.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS1.4.p2.19.m12.5.5.1.3" xref="S4.SS1.4.p2.19.m12.5.5.1.3.cmml">p</mi></msub><mo id="S4.SS1.4.p2.19.m12.6.6.3" xref="S4.SS1.4.p2.19.m12.6.6.3.cmml">≤</mo><mrow id="S4.SS1.4.p2.19.m12.6.6.2" xref="S4.SS1.4.p2.19.m12.6.6.2.cmml"><mi id="S4.SS1.4.p2.19.m12.6.6.2.3" xref="S4.SS1.4.p2.19.m12.6.6.2.3.cmml">λ</mi><mo id="S4.SS1.4.p2.19.m12.6.6.2.2" lspace="0em" xref="S4.SS1.4.p2.19.m12.6.6.2.2.cmml">⁢</mo><msub id="S4.SS1.4.p2.19.m12.6.6.2.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.cmml"><mrow id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.2.cmml"><mo fence="true" id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.2" rspace="0em" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.2.1.cmml">∥</mo><mrow id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.cmml"><msup id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.2" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.2.cmml">x</mi><mrow id="S4.SS1.4.p2.19.m12.3.3.1.1" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.cmml"><mo id="S4.SS1.4.p2.19.m12.3.3.1.1.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS1.4.p2.19.m12.3.3.1.1.1" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.3.3.1.1.1.2" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.19.m12.3.3.1.1.1.1" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.1.cmml">−</mo><mn id="S4.SS1.4.p2.19.m12.3.3.1.1.1.3" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.4.p2.19.m12.3.3.1.1.3" stretchy="false" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.2.cmml">−</mo><mrow id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.3" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.3.cmml">f</mi><mo id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.4.p2.19.m12.4.4.1.1" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.cmml"><mo id="S4.SS1.4.p2.19.m12.4.4.1.1.2" stretchy="false" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.cmml">(</mo><mrow id="S4.SS1.4.p2.19.m12.4.4.1.1.1" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.cmml"><mi id="S4.SS1.4.p2.19.m12.4.4.1.1.1.2" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.19.m12.4.4.1.1.1.1" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.1.cmml">−</mo><mn id="S4.SS1.4.p2.19.m12.4.4.1.1.1.3" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.4.p2.19.m12.4.4.1.1.3" stretchy="false" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.3" lspace="0em" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS1.4.p2.19.m12.6.6.2.1.3" xref="S4.SS1.4.p2.19.m12.6.6.2.1.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.19.m12.6b"><apply 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xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.3.2">𝑥</ci><ci id="S4.SS1.4.p2.19.m12.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.1.1.1.1">𝑖</ci></apply><apply id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1"><times id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.2"></times><ci id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.3">𝑓</ci><apply id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS1.4.p2.19.m12.2.2.1.1.cmml" xref="S4.SS1.4.p2.19.m12.2.2.1.1">𝑖</ci></apply></apply></apply></apply><ci id="S4.SS1.4.p2.19.m12.5.5.1.3.cmml" xref="S4.SS1.4.p2.19.m12.5.5.1.3">𝑝</ci></apply><apply id="S4.SS1.4.p2.19.m12.6.6.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2"><times id="S4.SS1.4.p2.19.m12.6.6.2.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.2"></times><ci id="S4.SS1.4.p2.19.m12.6.6.2.3.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.3">𝜆</ci><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.19.m12.6.6.2.1.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1">subscript</csymbol><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1"><csymbol cd="latexml" id="S4.SS1.4.p2.19.m12.6.6.2.1.1.2.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1"><minus id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.2"></minus><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3">superscript</csymbol><ci id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.3.2">𝑥</ci><apply id="S4.SS1.4.p2.19.m12.3.3.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.3.3.1.1"><minus id="S4.SS1.4.p2.19.m12.3.3.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.1"></minus><ci id="S4.SS1.4.p2.19.m12.3.3.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.2">𝑖</ci><cn id="S4.SS1.4.p2.19.m12.3.3.1.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.19.m12.3.3.1.1.1.3">1</cn></apply></apply><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1"><times id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.2"></times><ci id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.3">𝑓</ci><apply id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.1.1.1.1.1.1.1.2">𝑥</ci><apply id="S4.SS1.4.p2.19.m12.4.4.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.4.4.1.1"><minus id="S4.SS1.4.p2.19.m12.4.4.1.1.1.1.cmml" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.1"></minus><ci id="S4.SS1.4.p2.19.m12.4.4.1.1.1.2.cmml" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.2">𝑖</ci><cn id="S4.SS1.4.p2.19.m12.4.4.1.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.19.m12.4.4.1.1.1.3">1</cn></apply></apply></apply></apply></apply><ci id="S4.SS1.4.p2.19.m12.6.6.2.1.3.cmml" xref="S4.SS1.4.p2.19.m12.6.6.2.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.19.m12.6c">\lVert x^{(i)}-f(x^{(i)})\rVert_{p}\leq\lambda\lVert x^{(i-1)}-f(x^{(i-1)})% \rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.19.m12.6d">∥ italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT - italic_f ( italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_λ ∥ italic_x start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT - italic_f ( italic_x start_POSTSUPERSCRIPT ( italic_i - 1 ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="i\geq 1" class="ltx_Math" display="inline" id="S4.SS1.4.p2.20.m13.1"><semantics id="S4.SS1.4.p2.20.m13.1a"><mrow id="S4.SS1.4.p2.20.m13.1.1" xref="S4.SS1.4.p2.20.m13.1.1.cmml"><mi id="S4.SS1.4.p2.20.m13.1.1.2" xref="S4.SS1.4.p2.20.m13.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.20.m13.1.1.1" xref="S4.SS1.4.p2.20.m13.1.1.1.cmml">≥</mo><mn id="S4.SS1.4.p2.20.m13.1.1.3" xref="S4.SS1.4.p2.20.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.20.m13.1b"><apply id="S4.SS1.4.p2.20.m13.1.1.cmml" xref="S4.SS1.4.p2.20.m13.1.1"><geq id="S4.SS1.4.p2.20.m13.1.1.1.cmml" xref="S4.SS1.4.p2.20.m13.1.1.1"></geq><ci id="S4.SS1.4.p2.20.m13.1.1.2.cmml" xref="S4.SS1.4.p2.20.m13.1.1.2">𝑖</ci><cn id="S4.SS1.4.p2.20.m13.1.1.3.cmml" type="integer" xref="S4.SS1.4.p2.20.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.20.m13.1c">i\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.20.m13.1d">italic_i ≥ 1</annotation></semantics></math>. Together with the observation <math alttext="\lVert x^{(0)}-f(x^{(0)})\rVert_{p}\leq d" class="ltx_Math" display="inline" id="S4.SS1.4.p2.21.m14.3"><semantics id="S4.SS1.4.p2.21.m14.3a"><mrow id="S4.SS1.4.p2.21.m14.3.3" xref="S4.SS1.4.p2.21.m14.3.3.cmml"><msub id="S4.SS1.4.p2.21.m14.3.3.1" xref="S4.SS1.4.p2.21.m14.3.3.1.cmml"><mrow id="S4.SS1.4.p2.21.m14.3.3.1.1.1" xref="S4.SS1.4.p2.21.m14.3.3.1.1.2.cmml"><mo fence="true" id="S4.SS1.4.p2.21.m14.3.3.1.1.1.2" rspace="0em" xref="S4.SS1.4.p2.21.m14.3.3.1.1.2.1.cmml">∥</mo><mrow id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.cmml"><msup id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.2" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.2.cmml">x</mi><mrow id="S4.SS1.4.p2.21.m14.1.1.1.3" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.cmml"><mo id="S4.SS1.4.p2.21.m14.1.1.1.3.1" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.cmml">(</mo><mn id="S4.SS1.4.p2.21.m14.1.1.1.1" xref="S4.SS1.4.p2.21.m14.1.1.1.1.cmml">0</mn><mo id="S4.SS1.4.p2.21.m14.1.1.1.3.2" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.2" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.2.cmml">−</mo><mrow id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.3" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.3.cmml">f</mi><mo id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.2" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.4.p2.21.m14.2.2.1.3" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.21.m14.2.2.1.3.1" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S4.SS1.4.p2.21.m14.2.2.1.1" xref="S4.SS1.4.p2.21.m14.2.2.1.1.cmml">0</mn><mo id="S4.SS1.4.p2.21.m14.2.2.1.3.2" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S4.SS1.4.p2.21.m14.3.3.1.1.1.3" lspace="0em" xref="S4.SS1.4.p2.21.m14.3.3.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS1.4.p2.21.m14.3.3.1.3" xref="S4.SS1.4.p2.21.m14.3.3.1.3.cmml">p</mi></msub><mo id="S4.SS1.4.p2.21.m14.3.3.2" xref="S4.SS1.4.p2.21.m14.3.3.2.cmml">≤</mo><mi id="S4.SS1.4.p2.21.m14.3.3.3" xref="S4.SS1.4.p2.21.m14.3.3.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.21.m14.3b"><apply id="S4.SS1.4.p2.21.m14.3.3.cmml" xref="S4.SS1.4.p2.21.m14.3.3"><leq id="S4.SS1.4.p2.21.m14.3.3.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.2"></leq><apply id="S4.SS1.4.p2.21.m14.3.3.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.21.m14.3.3.1.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1">subscript</csymbol><apply id="S4.SS1.4.p2.21.m14.3.3.1.1.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1"><csymbol cd="latexml" id="S4.SS1.4.p2.21.m14.3.3.1.1.2.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1"><minus id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.2"></minus><apply id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3">superscript</csymbol><ci id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.3.2">𝑥</ci><cn id="S4.SS1.4.p2.21.m14.1.1.1.1.cmml" type="integer" xref="S4.SS1.4.p2.21.m14.1.1.1.1">0</cn></apply><apply id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1"><times id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.2"></times><ci id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.3">𝑓</ci><apply id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.1.1.1.1.1.1.1.2">𝑥</ci><cn id="S4.SS1.4.p2.21.m14.2.2.1.1.cmml" type="integer" xref="S4.SS1.4.p2.21.m14.2.2.1.1">0</cn></apply></apply></apply></apply><ci id="S4.SS1.4.p2.21.m14.3.3.1.3.cmml" xref="S4.SS1.4.p2.21.m14.3.3.1.3">𝑝</ci></apply><ci id="S4.SS1.4.p2.21.m14.3.3.3.cmml" xref="S4.SS1.4.p2.21.m14.3.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.21.m14.3c">\lVert x^{(0)}-f(x^{(0)})\rVert_{p}\leq d</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.21.m14.3d">∥ italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT - italic_f ( italic_x start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_d</annotation></semantics></math>, we get <math alttext="\lVert x^{(i)}-f(x^{(i)})\rVert_{p}\leq\lambda^{i}d" class="ltx_Math" display="inline" id="S4.SS1.4.p2.22.m15.3"><semantics id="S4.SS1.4.p2.22.m15.3a"><mrow id="S4.SS1.4.p2.22.m15.3.3" xref="S4.SS1.4.p2.22.m15.3.3.cmml"><msub id="S4.SS1.4.p2.22.m15.3.3.1" xref="S4.SS1.4.p2.22.m15.3.3.1.cmml"><mrow id="S4.SS1.4.p2.22.m15.3.3.1.1.1" xref="S4.SS1.4.p2.22.m15.3.3.1.1.2.cmml"><mo fence="true" id="S4.SS1.4.p2.22.m15.3.3.1.1.1.2" rspace="0em" xref="S4.SS1.4.p2.22.m15.3.3.1.1.2.1.cmml">∥</mo><mrow id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.cmml"><msup id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.2" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.2.cmml">x</mi><mrow id="S4.SS1.4.p2.22.m15.1.1.1.3" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.cmml"><mo id="S4.SS1.4.p2.22.m15.1.1.1.3.1" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.cmml">(</mo><mi id="S4.SS1.4.p2.22.m15.1.1.1.1" xref="S4.SS1.4.p2.22.m15.1.1.1.1.cmml">i</mi><mo id="S4.SS1.4.p2.22.m15.1.1.1.3.2" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.2" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.2.cmml">−</mo><mrow id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.3" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.3.cmml">f</mi><mo id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.2" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.4.p2.22.m15.2.2.1.3" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.22.m15.2.2.1.3.1" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.SS1.4.p2.22.m15.2.2.1.1" xref="S4.SS1.4.p2.22.m15.2.2.1.1.cmml">i</mi><mo id="S4.SS1.4.p2.22.m15.2.2.1.3.2" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S4.SS1.4.p2.22.m15.3.3.1.1.1.3" lspace="0em" xref="S4.SS1.4.p2.22.m15.3.3.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS1.4.p2.22.m15.3.3.1.3" xref="S4.SS1.4.p2.22.m15.3.3.1.3.cmml">p</mi></msub><mo id="S4.SS1.4.p2.22.m15.3.3.2" xref="S4.SS1.4.p2.22.m15.3.3.2.cmml">≤</mo><mrow id="S4.SS1.4.p2.22.m15.3.3.3" xref="S4.SS1.4.p2.22.m15.3.3.3.cmml"><msup id="S4.SS1.4.p2.22.m15.3.3.3.2" xref="S4.SS1.4.p2.22.m15.3.3.3.2.cmml"><mi id="S4.SS1.4.p2.22.m15.3.3.3.2.2" xref="S4.SS1.4.p2.22.m15.3.3.3.2.2.cmml">λ</mi><mi id="S4.SS1.4.p2.22.m15.3.3.3.2.3" xref="S4.SS1.4.p2.22.m15.3.3.3.2.3.cmml">i</mi></msup><mo id="S4.SS1.4.p2.22.m15.3.3.3.1" xref="S4.SS1.4.p2.22.m15.3.3.3.1.cmml">⁢</mo><mi id="S4.SS1.4.p2.22.m15.3.3.3.3" xref="S4.SS1.4.p2.22.m15.3.3.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.22.m15.3b"><apply id="S4.SS1.4.p2.22.m15.3.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3"><leq id="S4.SS1.4.p2.22.m15.3.3.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.2"></leq><apply id="S4.SS1.4.p2.22.m15.3.3.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.22.m15.3.3.1.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1">subscript</csymbol><apply id="S4.SS1.4.p2.22.m15.3.3.1.1.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1"><csymbol cd="latexml" id="S4.SS1.4.p2.22.m15.3.3.1.1.2.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1"><minus id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.2"></minus><apply id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3">superscript</csymbol><ci id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.3.2">𝑥</ci><ci id="S4.SS1.4.p2.22.m15.1.1.1.1.cmml" xref="S4.SS1.4.p2.22.m15.1.1.1.1">𝑖</ci></apply><apply id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1"><times id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.2"></times><ci id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.3">𝑓</ci><apply id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS1.4.p2.22.m15.2.2.1.1.cmml" xref="S4.SS1.4.p2.22.m15.2.2.1.1">𝑖</ci></apply></apply></apply></apply><ci id="S4.SS1.4.p2.22.m15.3.3.1.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.1.3">𝑝</ci></apply><apply id="S4.SS1.4.p2.22.m15.3.3.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3"><times id="S4.SS1.4.p2.22.m15.3.3.3.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.1"></times><apply id="S4.SS1.4.p2.22.m15.3.3.3.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.4.p2.22.m15.3.3.3.2.1.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.2">superscript</csymbol><ci id="S4.SS1.4.p2.22.m15.3.3.3.2.2.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.2.2">𝜆</ci><ci id="S4.SS1.4.p2.22.m15.3.3.3.2.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.2.3">𝑖</ci></apply><ci id="S4.SS1.4.p2.22.m15.3.3.3.3.cmml" xref="S4.SS1.4.p2.22.m15.3.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.22.m15.3c">\lVert x^{(i)}-f(x^{(i)})\rVert_{p}\leq\lambda^{i}d</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.22.m15.3d">∥ italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT - italic_f ( italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_λ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d</annotation></semantics></math> for all <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.4.p2.23.m16.1"><semantics id="S4.SS1.4.p2.23.m16.1a"><mi id="S4.SS1.4.p2.23.m16.1.1" xref="S4.SS1.4.p2.23.m16.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.23.m16.1b"><ci id="S4.SS1.4.p2.23.m16.1.1.cmml" xref="S4.SS1.4.p2.23.m16.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.23.m16.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.23.m16.1d">italic_i</annotation></semantics></math>. If <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.4.p2.24.m17.1"><semantics id="S4.SS1.4.p2.24.m17.1a"><mi id="S4.SS1.4.p2.24.m17.1.1" xref="S4.SS1.4.p2.24.m17.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.24.m17.1b"><ci id="S4.SS1.4.p2.24.m17.1.1.cmml" xref="S4.SS1.4.p2.24.m17.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.24.m17.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.24.m17.1d">italic_i</annotation></semantics></math> is large enough such that <math alttext="\lambda^{i}d<\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p2.25.m18.1"><semantics id="S4.SS1.4.p2.25.m18.1a"><mrow id="S4.SS1.4.p2.25.m18.1.1" xref="S4.SS1.4.p2.25.m18.1.1.cmml"><mrow id="S4.SS1.4.p2.25.m18.1.1.2" xref="S4.SS1.4.p2.25.m18.1.1.2.cmml"><msup id="S4.SS1.4.p2.25.m18.1.1.2.2" xref="S4.SS1.4.p2.25.m18.1.1.2.2.cmml"><mi id="S4.SS1.4.p2.25.m18.1.1.2.2.2" xref="S4.SS1.4.p2.25.m18.1.1.2.2.2.cmml">λ</mi><mi id="S4.SS1.4.p2.25.m18.1.1.2.2.3" xref="S4.SS1.4.p2.25.m18.1.1.2.2.3.cmml">i</mi></msup><mo id="S4.SS1.4.p2.25.m18.1.1.2.1" xref="S4.SS1.4.p2.25.m18.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.4.p2.25.m18.1.1.2.3" xref="S4.SS1.4.p2.25.m18.1.1.2.3.cmml">d</mi></mrow><mo id="S4.SS1.4.p2.25.m18.1.1.1" xref="S4.SS1.4.p2.25.m18.1.1.1.cmml"><</mo><mi id="S4.SS1.4.p2.25.m18.1.1.3" xref="S4.SS1.4.p2.25.m18.1.1.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.25.m18.1b"><apply id="S4.SS1.4.p2.25.m18.1.1.cmml" xref="S4.SS1.4.p2.25.m18.1.1"><lt id="S4.SS1.4.p2.25.m18.1.1.1.cmml" xref="S4.SS1.4.p2.25.m18.1.1.1"></lt><apply id="S4.SS1.4.p2.25.m18.1.1.2.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2"><times id="S4.SS1.4.p2.25.m18.1.1.2.1.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.1"></times><apply id="S4.SS1.4.p2.25.m18.1.1.2.2.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.4.p2.25.m18.1.1.2.2.1.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.2">superscript</csymbol><ci id="S4.SS1.4.p2.25.m18.1.1.2.2.2.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.2.2">𝜆</ci><ci id="S4.SS1.4.p2.25.m18.1.1.2.2.3.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS1.4.p2.25.m18.1.1.2.3.cmml" xref="S4.SS1.4.p2.25.m18.1.1.2.3">𝑑</ci></apply><ci id="S4.SS1.4.p2.25.m18.1.1.3.cmml" xref="S4.SS1.4.p2.25.m18.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.25.m18.1c">\lambda^{i}d<\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.25.m18.1d">italic_λ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d < italic_ε</annotation></semantics></math>, we must have found an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p2.26.m19.1"><semantics id="S4.SS1.4.p2.26.m19.1a"><mi id="S4.SS1.4.p2.26.m19.1.1" xref="S4.SS1.4.p2.26.m19.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.26.m19.1b"><ci id="S4.SS1.4.p2.26.m19.1.1.cmml" xref="S4.SS1.4.p2.26.m19.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.26.m19.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.26.m19.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. This is equivalent to <math alttext="i>\frac{\log d/\varepsilon}{\log\frac{1}{\lambda}}" class="ltx_Math" display="inline" id="S4.SS1.4.p2.27.m20.1"><semantics id="S4.SS1.4.p2.27.m20.1a"><mrow id="S4.SS1.4.p2.27.m20.1.1" xref="S4.SS1.4.p2.27.m20.1.1.cmml"><mi id="S4.SS1.4.p2.27.m20.1.1.2" xref="S4.SS1.4.p2.27.m20.1.1.2.cmml">i</mi><mo id="S4.SS1.4.p2.27.m20.1.1.1" xref="S4.SS1.4.p2.27.m20.1.1.1.cmml">></mo><mfrac id="S4.SS1.4.p2.27.m20.1.1.3" xref="S4.SS1.4.p2.27.m20.1.1.3.cmml"><mrow id="S4.SS1.4.p2.27.m20.1.1.3.2" xref="S4.SS1.4.p2.27.m20.1.1.3.2.cmml"><mi id="S4.SS1.4.p2.27.m20.1.1.3.2.1" xref="S4.SS1.4.p2.27.m20.1.1.3.2.1.cmml">log</mi><mo id="S4.SS1.4.p2.27.m20.1.1.3.2a" lspace="0.167em" xref="S4.SS1.4.p2.27.m20.1.1.3.2.cmml">⁡</mo><mrow id="S4.SS1.4.p2.27.m20.1.1.3.2.2" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.cmml"><mi id="S4.SS1.4.p2.27.m20.1.1.3.2.2.2" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.2.cmml">d</mi><mo id="S4.SS1.4.p2.27.m20.1.1.3.2.2.1" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.1.cmml">/</mo><mi id="S4.SS1.4.p2.27.m20.1.1.3.2.2.3" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.3.cmml">ε</mi></mrow></mrow><mrow id="S4.SS1.4.p2.27.m20.1.1.3.3" xref="S4.SS1.4.p2.27.m20.1.1.3.3.cmml"><mi id="S4.SS1.4.p2.27.m20.1.1.3.3.1" xref="S4.SS1.4.p2.27.m20.1.1.3.3.1.cmml">log</mi><mo id="S4.SS1.4.p2.27.m20.1.1.3.3a" lspace="0.167em" xref="S4.SS1.4.p2.27.m20.1.1.3.3.cmml">⁡</mo><mfrac id="S4.SS1.4.p2.27.m20.1.1.3.3.2" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2.cmml"><mn id="S4.SS1.4.p2.27.m20.1.1.3.3.2.2" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2.2.cmml">1</mn><mi id="S4.SS1.4.p2.27.m20.1.1.3.3.2.3" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2.3.cmml">λ</mi></mfrac></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.27.m20.1b"><apply id="S4.SS1.4.p2.27.m20.1.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1"><gt id="S4.SS1.4.p2.27.m20.1.1.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.1"></gt><ci id="S4.SS1.4.p2.27.m20.1.1.2.cmml" xref="S4.SS1.4.p2.27.m20.1.1.2">𝑖</ci><apply id="S4.SS1.4.p2.27.m20.1.1.3.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3"><divide id="S4.SS1.4.p2.27.m20.1.1.3.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3"></divide><apply id="S4.SS1.4.p2.27.m20.1.1.3.2.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2"><log id="S4.SS1.4.p2.27.m20.1.1.3.2.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2.1"></log><apply id="S4.SS1.4.p2.27.m20.1.1.3.2.2.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2"><divide id="S4.SS1.4.p2.27.m20.1.1.3.2.2.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.1"></divide><ci id="S4.SS1.4.p2.27.m20.1.1.3.2.2.2.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.2">𝑑</ci><ci id="S4.SS1.4.p2.27.m20.1.1.3.2.2.3.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.2.2.3">𝜀</ci></apply></apply><apply id="S4.SS1.4.p2.27.m20.1.1.3.3.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.3"><log id="S4.SS1.4.p2.27.m20.1.1.3.3.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.3.1"></log><apply id="S4.SS1.4.p2.27.m20.1.1.3.3.2.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2"><divide id="S4.SS1.4.p2.27.m20.1.1.3.3.2.1.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2"></divide><cn id="S4.SS1.4.p2.27.m20.1.1.3.3.2.2.cmml" type="integer" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2.2">1</cn><ci id="S4.SS1.4.p2.27.m20.1.1.3.3.2.3.cmml" xref="S4.SS1.4.p2.27.m20.1.1.3.3.2.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.27.m20.1c">i>\frac{\log d/\varepsilon}{\log\frac{1}{\lambda}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.27.m20.1d">italic_i > divide start_ARG roman_log italic_d / italic_ε end_ARG start_ARG roman_log divide start_ARG 1 end_ARG start_ARG italic_λ end_ARG end_ARG</annotation></semantics></math>. By using <math alttext="\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d" class="ltx_Math" display="inline" id="S4.SS1.4.p2.28.m21.3"><semantics id="S4.SS1.4.p2.28.m21.3a"><mrow id="S4.SS1.4.p2.28.m21.3.4" xref="S4.SS1.4.p2.28.m21.3.4.cmml"><mrow id="S4.SS1.4.p2.28.m21.3.4.2.2" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml"><mi id="S4.SS1.4.p2.28.m21.1.1" xref="S4.SS1.4.p2.28.m21.1.1.cmml">max</mi><mo id="S4.SS1.4.p2.28.m21.3.4.2.2a" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml">⁡</mo><mrow id="S4.SS1.4.p2.28.m21.3.4.2.2.1" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml"><mo id="S4.SS1.4.p2.28.m21.3.4.2.2.1.1" stretchy="false" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml">(</mo><mfrac id="S4.SS1.4.p2.28.m21.2.2" xref="S4.SS1.4.p2.28.m21.2.2.cmml"><mn id="S4.SS1.4.p2.28.m21.2.2.2" xref="S4.SS1.4.p2.28.m21.2.2.2.cmml">1</mn><mi id="S4.SS1.4.p2.28.m21.2.2.3" xref="S4.SS1.4.p2.28.m21.2.2.3.cmml">ε</mi></mfrac><mo id="S4.SS1.4.p2.28.m21.3.4.2.2.1.2" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml">,</mo><mfrac id="S4.SS1.4.p2.28.m21.3.3" xref="S4.SS1.4.p2.28.m21.3.3.cmml"><mn id="S4.SS1.4.p2.28.m21.3.3.2" xref="S4.SS1.4.p2.28.m21.3.3.2.cmml">1</mn><mrow id="S4.SS1.4.p2.28.m21.3.3.3" xref="S4.SS1.4.p2.28.m21.3.3.3.cmml"><mn id="S4.SS1.4.p2.28.m21.3.3.3.2" xref="S4.SS1.4.p2.28.m21.3.3.3.2.cmml">1</mn><mo id="S4.SS1.4.p2.28.m21.3.3.3.1" xref="S4.SS1.4.p2.28.m21.3.3.3.1.cmml">−</mo><mi id="S4.SS1.4.p2.28.m21.3.3.3.3" xref="S4.SS1.4.p2.28.m21.3.3.3.3.cmml">λ</mi></mrow></mfrac><mo id="S4.SS1.4.p2.28.m21.3.4.2.2.1.3" stretchy="false" xref="S4.SS1.4.p2.28.m21.3.4.2.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p2.28.m21.3.4.1" xref="S4.SS1.4.p2.28.m21.3.4.1.cmml"><</mo><mi id="S4.SS1.4.p2.28.m21.3.4.3" xref="S4.SS1.4.p2.28.m21.3.4.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.28.m21.3b"><apply id="S4.SS1.4.p2.28.m21.3.4.cmml" xref="S4.SS1.4.p2.28.m21.3.4"><lt id="S4.SS1.4.p2.28.m21.3.4.1.cmml" xref="S4.SS1.4.p2.28.m21.3.4.1"></lt><apply id="S4.SS1.4.p2.28.m21.3.4.2.1.cmml" xref="S4.SS1.4.p2.28.m21.3.4.2.2"><max id="S4.SS1.4.p2.28.m21.1.1.cmml" xref="S4.SS1.4.p2.28.m21.1.1"></max><apply id="S4.SS1.4.p2.28.m21.2.2.cmml" xref="S4.SS1.4.p2.28.m21.2.2"><divide id="S4.SS1.4.p2.28.m21.2.2.1.cmml" xref="S4.SS1.4.p2.28.m21.2.2"></divide><cn id="S4.SS1.4.p2.28.m21.2.2.2.cmml" type="integer" xref="S4.SS1.4.p2.28.m21.2.2.2">1</cn><ci id="S4.SS1.4.p2.28.m21.2.2.3.cmml" xref="S4.SS1.4.p2.28.m21.2.2.3">𝜀</ci></apply><apply id="S4.SS1.4.p2.28.m21.3.3.cmml" xref="S4.SS1.4.p2.28.m21.3.3"><divide id="S4.SS1.4.p2.28.m21.3.3.1.cmml" xref="S4.SS1.4.p2.28.m21.3.3"></divide><cn id="S4.SS1.4.p2.28.m21.3.3.2.cmml" type="integer" xref="S4.SS1.4.p2.28.m21.3.3.2">1</cn><apply id="S4.SS1.4.p2.28.m21.3.3.3.cmml" xref="S4.SS1.4.p2.28.m21.3.3.3"><minus id="S4.SS1.4.p2.28.m21.3.3.3.1.cmml" xref="S4.SS1.4.p2.28.m21.3.3.3.1"></minus><cn id="S4.SS1.4.p2.28.m21.3.3.3.2.cmml" type="integer" xref="S4.SS1.4.p2.28.m21.3.3.3.2">1</cn><ci id="S4.SS1.4.p2.28.m21.3.3.3.3.cmml" xref="S4.SS1.4.p2.28.m21.3.3.3.3">𝜆</ci></apply></apply></apply><ci id="S4.SS1.4.p2.28.m21.3.4.3.cmml" xref="S4.SS1.4.p2.28.m21.3.4.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.28.m21.3c">\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.28.m21.3d">roman_max ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG , divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) < italic_d</annotation></semantics></math>, we get</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\log(\frac{1}{\lambda})=\log(1+\frac{1-\lambda}{\lambda})\geq\log(1+\frac{1}{d% \lambda})\geq\log(1+\frac{1}{d})" class="ltx_Math" display="block" id="S4.Ex13.m1.8"><semantics id="S4.Ex13.m1.8a"><mrow id="S4.Ex13.m1.8.8" xref="S4.Ex13.m1.8.8.cmml"><mrow id="S4.Ex13.m1.8.8.5.2" xref="S4.Ex13.m1.8.8.5.1.cmml"><mi id="S4.Ex13.m1.1.1" xref="S4.Ex13.m1.1.1.cmml">log</mi><mo id="S4.Ex13.m1.8.8.5.2a" xref="S4.Ex13.m1.8.8.5.1.cmml">⁡</mo><mrow id="S4.Ex13.m1.8.8.5.2.1" xref="S4.Ex13.m1.8.8.5.1.cmml"><mo id="S4.Ex13.m1.8.8.5.2.1.1" stretchy="false" xref="S4.Ex13.m1.8.8.5.1.cmml">(</mo><mfrac id="S4.Ex13.m1.2.2" xref="S4.Ex13.m1.2.2.cmml"><mn id="S4.Ex13.m1.2.2.2" xref="S4.Ex13.m1.2.2.2.cmml">1</mn><mi id="S4.Ex13.m1.2.2.3" xref="S4.Ex13.m1.2.2.3.cmml">λ</mi></mfrac><mo id="S4.Ex13.m1.8.8.5.2.1.2" stretchy="false" xref="S4.Ex13.m1.8.8.5.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.8.8.6" xref="S4.Ex13.m1.8.8.6.cmml">=</mo><mrow id="S4.Ex13.m1.6.6.1.1" xref="S4.Ex13.m1.6.6.1.2.cmml"><mi id="S4.Ex13.m1.3.3" xref="S4.Ex13.m1.3.3.cmml">log</mi><mo id="S4.Ex13.m1.6.6.1.1a" xref="S4.Ex13.m1.6.6.1.2.cmml">⁡</mo><mrow id="S4.Ex13.m1.6.6.1.1.1" xref="S4.Ex13.m1.6.6.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.2.cmml">(</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.cmml"><mn id="S4.Ex13.m1.6.6.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.2.cmml">1</mn><mo id="S4.Ex13.m1.6.6.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.cmml">+</mo><mfrac id="S4.Ex13.m1.6.6.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.3.cmml"><mrow id="S4.Ex13.m1.6.6.1.1.1.1.3.2" xref="S4.Ex13.m1.6.6.1.1.1.1.3.2.cmml"><mn id="S4.Ex13.m1.6.6.1.1.1.1.3.2.2" xref="S4.Ex13.m1.6.6.1.1.1.1.3.2.2.cmml">1</mn><mo id="S4.Ex13.m1.6.6.1.1.1.1.3.2.1" xref="S4.Ex13.m1.6.6.1.1.1.1.3.2.1.cmml">−</mo><mi id="S4.Ex13.m1.6.6.1.1.1.1.3.2.3" xref="S4.Ex13.m1.6.6.1.1.1.1.3.2.3.cmml">λ</mi></mrow><mi id="S4.Ex13.m1.6.6.1.1.1.1.3.3" xref="S4.Ex13.m1.6.6.1.1.1.1.3.3.cmml">λ</mi></mfrac></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.8.8.7" xref="S4.Ex13.m1.8.8.7.cmml">≥</mo><mrow id="S4.Ex13.m1.7.7.2.1" xref="S4.Ex13.m1.7.7.2.2.cmml"><mi id="S4.Ex13.m1.4.4" xref="S4.Ex13.m1.4.4.cmml">log</mi><mo id="S4.Ex13.m1.7.7.2.1a" xref="S4.Ex13.m1.7.7.2.2.cmml">⁡</mo><mrow id="S4.Ex13.m1.7.7.2.1.1" xref="S4.Ex13.m1.7.7.2.2.cmml"><mo id="S4.Ex13.m1.7.7.2.1.1.2" stretchy="false" xref="S4.Ex13.m1.7.7.2.2.cmml">(</mo><mrow id="S4.Ex13.m1.7.7.2.1.1.1" xref="S4.Ex13.m1.7.7.2.1.1.1.cmml"><mn id="S4.Ex13.m1.7.7.2.1.1.1.2" xref="S4.Ex13.m1.7.7.2.1.1.1.2.cmml">1</mn><mo id="S4.Ex13.m1.7.7.2.1.1.1.1" xref="S4.Ex13.m1.7.7.2.1.1.1.1.cmml">+</mo><mfrac id="S4.Ex13.m1.7.7.2.1.1.1.3" xref="S4.Ex13.m1.7.7.2.1.1.1.3.cmml"><mn id="S4.Ex13.m1.7.7.2.1.1.1.3.2" xref="S4.Ex13.m1.7.7.2.1.1.1.3.2.cmml">1</mn><mrow id="S4.Ex13.m1.7.7.2.1.1.1.3.3" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.cmml"><mi id="S4.Ex13.m1.7.7.2.1.1.1.3.3.2" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.2.cmml">d</mi><mo id="S4.Ex13.m1.7.7.2.1.1.1.3.3.1" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.1.cmml">⁢</mo><mi id="S4.Ex13.m1.7.7.2.1.1.1.3.3.3" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.3.cmml">λ</mi></mrow></mfrac></mrow><mo id="S4.Ex13.m1.7.7.2.1.1.3" stretchy="false" xref="S4.Ex13.m1.7.7.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.8.8.8" xref="S4.Ex13.m1.8.8.8.cmml">≥</mo><mrow id="S4.Ex13.m1.8.8.3.1" xref="S4.Ex13.m1.8.8.3.2.cmml"><mi id="S4.Ex13.m1.5.5" xref="S4.Ex13.m1.5.5.cmml">log</mi><mo id="S4.Ex13.m1.8.8.3.1a" xref="S4.Ex13.m1.8.8.3.2.cmml">⁡</mo><mrow id="S4.Ex13.m1.8.8.3.1.1" xref="S4.Ex13.m1.8.8.3.2.cmml"><mo id="S4.Ex13.m1.8.8.3.1.1.2" stretchy="false" xref="S4.Ex13.m1.8.8.3.2.cmml">(</mo><mrow id="S4.Ex13.m1.8.8.3.1.1.1" xref="S4.Ex13.m1.8.8.3.1.1.1.cmml"><mn id="S4.Ex13.m1.8.8.3.1.1.1.2" xref="S4.Ex13.m1.8.8.3.1.1.1.2.cmml">1</mn><mo id="S4.Ex13.m1.8.8.3.1.1.1.1" xref="S4.Ex13.m1.8.8.3.1.1.1.1.cmml">+</mo><mfrac id="S4.Ex13.m1.8.8.3.1.1.1.3" xref="S4.Ex13.m1.8.8.3.1.1.1.3.cmml"><mn id="S4.Ex13.m1.8.8.3.1.1.1.3.2" xref="S4.Ex13.m1.8.8.3.1.1.1.3.2.cmml">1</mn><mi id="S4.Ex13.m1.8.8.3.1.1.1.3.3" xref="S4.Ex13.m1.8.8.3.1.1.1.3.3.cmml">d</mi></mfrac></mrow><mo id="S4.Ex13.m1.8.8.3.1.1.3" stretchy="false" xref="S4.Ex13.m1.8.8.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.8b"><apply id="S4.Ex13.m1.8.8.cmml" xref="S4.Ex13.m1.8.8"><and id="S4.Ex13.m1.8.8a.cmml" xref="S4.Ex13.m1.8.8"></and><apply id="S4.Ex13.m1.8.8b.cmml" xref="S4.Ex13.m1.8.8"><eq id="S4.Ex13.m1.8.8.6.cmml" xref="S4.Ex13.m1.8.8.6"></eq><apply id="S4.Ex13.m1.8.8.5.1.cmml" xref="S4.Ex13.m1.8.8.5.2"><log id="S4.Ex13.m1.1.1.cmml" xref="S4.Ex13.m1.1.1"></log><apply id="S4.Ex13.m1.2.2.cmml" xref="S4.Ex13.m1.2.2"><divide id="S4.Ex13.m1.2.2.1.cmml" xref="S4.Ex13.m1.2.2"></divide><cn id="S4.Ex13.m1.2.2.2.cmml" type="integer" xref="S4.Ex13.m1.2.2.2">1</cn><ci id="S4.Ex13.m1.2.2.3.cmml" xref="S4.Ex13.m1.2.2.3">𝜆</ci></apply></apply><apply id="S4.Ex13.m1.6.6.1.2.cmml" xref="S4.Ex13.m1.6.6.1.1"><log id="S4.Ex13.m1.3.3.cmml" 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xref="S4.Ex13.m1.8.8.7"></geq><share href="https://arxiv.org/html/2503.16089v1#S4.Ex13.m1.6.6.1.cmml" id="S4.Ex13.m1.8.8d.cmml" xref="S4.Ex13.m1.8.8"></share><apply id="S4.Ex13.m1.7.7.2.2.cmml" xref="S4.Ex13.m1.7.7.2.1"><log id="S4.Ex13.m1.4.4.cmml" xref="S4.Ex13.m1.4.4"></log><apply id="S4.Ex13.m1.7.7.2.1.1.1.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1"><plus id="S4.Ex13.m1.7.7.2.1.1.1.1.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.1"></plus><cn id="S4.Ex13.m1.7.7.2.1.1.1.2.cmml" type="integer" xref="S4.Ex13.m1.7.7.2.1.1.1.2">1</cn><apply id="S4.Ex13.m1.7.7.2.1.1.1.3.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3"><divide id="S4.Ex13.m1.7.7.2.1.1.1.3.1.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3"></divide><cn id="S4.Ex13.m1.7.7.2.1.1.1.3.2.cmml" type="integer" xref="S4.Ex13.m1.7.7.2.1.1.1.3.2">1</cn><apply id="S4.Ex13.m1.7.7.2.1.1.1.3.3.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3"><times id="S4.Ex13.m1.7.7.2.1.1.1.3.3.1.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.1"></times><ci id="S4.Ex13.m1.7.7.2.1.1.1.3.3.2.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.2">𝑑</ci><ci id="S4.Ex13.m1.7.7.2.1.1.1.3.3.3.cmml" xref="S4.Ex13.m1.7.7.2.1.1.1.3.3.3">𝜆</ci></apply></apply></apply></apply></apply><apply id="S4.Ex13.m1.8.8e.cmml" xref="S4.Ex13.m1.8.8"><geq id="S4.Ex13.m1.8.8.8.cmml" xref="S4.Ex13.m1.8.8.8"></geq><share href="https://arxiv.org/html/2503.16089v1#S4.Ex13.m1.7.7.2.cmml" id="S4.Ex13.m1.8.8f.cmml" xref="S4.Ex13.m1.8.8"></share><apply id="S4.Ex13.m1.8.8.3.2.cmml" xref="S4.Ex13.m1.8.8.3.1"><log id="S4.Ex13.m1.5.5.cmml" xref="S4.Ex13.m1.5.5"></log><apply id="S4.Ex13.m1.8.8.3.1.1.1.cmml" xref="S4.Ex13.m1.8.8.3.1.1.1"><plus id="S4.Ex13.m1.8.8.3.1.1.1.1.cmml" xref="S4.Ex13.m1.8.8.3.1.1.1.1"></plus><cn id="S4.Ex13.m1.8.8.3.1.1.1.2.cmml" type="integer" xref="S4.Ex13.m1.8.8.3.1.1.1.2">1</cn><apply id="S4.Ex13.m1.8.8.3.1.1.1.3.cmml" xref="S4.Ex13.m1.8.8.3.1.1.1.3"><divide id="S4.Ex13.m1.8.8.3.1.1.1.3.1.cmml" xref="S4.Ex13.m1.8.8.3.1.1.1.3"></divide><cn id="S4.Ex13.m1.8.8.3.1.1.1.3.2.cmml" type="integer" xref="S4.Ex13.m1.8.8.3.1.1.1.3.2">1</cn><ci id="S4.Ex13.m1.8.8.3.1.1.1.3.3.cmml" xref="S4.Ex13.m1.8.8.3.1.1.1.3.3">𝑑</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.8c">\log(\frac{1}{\lambda})=\log(1+\frac{1-\lambda}{\lambda})\geq\log(1+\frac{1}{d% \lambda})\geq\log(1+\frac{1}{d})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.8d">roman_log ( divide start_ARG 1 end_ARG start_ARG italic_λ end_ARG ) = roman_log ( 1 + divide start_ARG 1 - italic_λ end_ARG start_ARG italic_λ end_ARG ) ≥ roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG italic_d italic_λ end_ARG ) ≥ roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG italic_d end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.4.p2.29">and hence we obtain that <math alttext="\mathcal{O}(d\log d)" class="ltx_Math" display="inline" id="S4.SS1.4.p2.29.m1.1"><semantics id="S4.SS1.4.p2.29.m1.1a"><mrow id="S4.SS1.4.p2.29.m1.1.1" xref="S4.SS1.4.p2.29.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p2.29.m1.1.1.3" xref="S4.SS1.4.p2.29.m1.1.1.3.cmml">𝒪</mi><mo id="S4.SS1.4.p2.29.m1.1.1.2" xref="S4.SS1.4.p2.29.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p2.29.m1.1.1.1.1" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p2.29.m1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.4.p2.29.m1.1.1.1.1.1" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p2.29.m1.1.1.1.1.1.2" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.2.cmml">d</mi><mo id="S4.SS1.4.p2.29.m1.1.1.1.1.1.1" lspace="0.167em" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.cmml"><mi id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.1" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.2" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.2.cmml">d</mi></mrow></mrow><mo id="S4.SS1.4.p2.29.m1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p2.29.m1.1b"><apply id="S4.SS1.4.p2.29.m1.1.1.cmml" xref="S4.SS1.4.p2.29.m1.1.1"><times id="S4.SS1.4.p2.29.m1.1.1.2.cmml" xref="S4.SS1.4.p2.29.m1.1.1.2"></times><ci id="S4.SS1.4.p2.29.m1.1.1.3.cmml" xref="S4.SS1.4.p2.29.m1.1.1.3">𝒪</ci><apply id="S4.SS1.4.p2.29.m1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1"><times id="S4.SS1.4.p2.29.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.1"></times><ci id="S4.SS1.4.p2.29.m1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.2">𝑑</ci><apply id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3"><log id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.1.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.1"></log><ci id="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.2.cmml" xref="S4.SS1.4.p2.29.m1.1.1.1.1.1.3.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p2.29.m1.1c">\mathcal{O}(d\log d)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p2.29.m1.1d">caligraphic_O ( italic_d roman_log italic_d )</annotation></semantics></math> iterations suffice. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Rounding to the Grid in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.1.m1.1"><semantics id="S4.SS2.1.m1.1b"><msub id="S4.SS2.1.m1.1.1" xref="S4.SS2.1.m1.1.1.cmml"><mi id="S4.SS2.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.1.m1.1.1.3" xref="S4.SS2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.1.m1.1c"><apply id="S4.SS2.1.m1.1.1.cmml" xref="S4.SS2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.m1.1.1.1.cmml" xref="S4.SS2.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.1.m1.1.1.2.cmml" xref="S4.SS2.1.m1.1.1.2">ℓ</ci><cn id="S4.SS2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.m1.1d">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-Case</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">In this section, we adapt the algorithm from the previous section to also work in the discretized setting <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p1.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">ℓ</ci><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.SS2.p1.1.1">-GridContractionFixpoint</span>, proving the following theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Theorem 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.6.6">For every <math alttext="b\geq\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}\right)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.2"><semantics id="S4.Thmtheorem3.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.2.2.4" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.4.cmml">b</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.2.2.3" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.3.cmml">≥</mo><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.3.cmml"><msub id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.cmml"><mi 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xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2"><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><log id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.2"></log><cn id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.1.1.3">2</cn></apply><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1"><times id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.1"></times><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2"><divide id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2"></divide><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2"><times id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.1"></times><cn id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.2">2</cn><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.2.3">𝑑</ci></apply><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.2.3">𝜀</ci></apply><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3"><divide id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3"></divide><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2"><plus id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.1"></plus><cn id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.2">1</cn><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.2.3">𝜆</ci></apply><apply id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3"><minus id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.1"></minus><cn id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.2">1</cn><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.2.2.2.1.3.3.3">𝜆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.2c">b\geq\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.2d">italic_b ≥ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG 2 italic_d end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG )</annotation></semantics></math>, an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint of a <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.3.m3.1"><semantics id="S4.Thmtheorem3.p1.3.3.m3.1a"><mi id="S4.Thmtheorem3.p1.3.3.m3.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.3.m3.1b"><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.1d">italic_λ</annotation></semantics></math>-contracting (in <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.4.4.m4.1"><semantics id="S4.Thmtheorem3.p1.4.4.m4.1a"><msub id="S4.Thmtheorem3.p1.4.4.m4.1.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.1.2" mathvariant="normal" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">ℓ</mi><mn id="S4.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.4.4.m4.1b"><apply id="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2">ℓ</ci><cn id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.4.4.m4.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.4.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-norm) grid map <math alttext="f:G^{d}_{b}\rightarrow[0,1]^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.5.5.m5.2"><semantics id="S4.Thmtheorem3.p1.5.5.m5.2a"><mrow id="S4.Thmtheorem3.p1.5.5.m5.2.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.2.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.2.cmml">f</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.1.cmml">:</mo><mrow id="S4.Thmtheorem3.p1.5.5.m5.2.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.cmml"><msubsup id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.2.cmml">G</mi><mi id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.3.cmml">b</mi><mi id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.1" stretchy="false" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml">→</mo><msup id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.cmml"><mrow id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.1.cmml"><mo id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem3.p1.5.5.m5.1.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.1.cmml">,</mo><mn id="S4.Thmtheorem3.p1.5.5.m5.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.2.2.cmml">1</mn><mo id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.1.cmml">]</mo></mrow><mi id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.5.5.m5.2b"><apply id="S4.Thmtheorem3.p1.5.5.m5.2.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3"><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.1">:</ci><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.2">𝑓</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3"><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.1">→</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2">subscript</csymbol><apply id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.2">𝐺</ci><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.2.3">𝑑</ci></apply><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.2.3">𝑏</ci></apply><apply id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3">superscript</csymbol><interval closure="closed" id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.2.2"><cn id="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem3.p1.5.5.m5.1.1">0</cn><cn id="S4.Thmtheorem3.p1.5.5.m5.2.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.5.5.m5.2.2">1</cn></interval><ci id="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.2.3.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.5.5.m5.2c">f:G^{d}_{b}\rightarrow[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.5.5.m5.2d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> can be found using <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.6.6.m6.1"><semantics id="S4.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.6.6.m6.1.1.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.3.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.cmml"><msup id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.2.cmml">d</mi><mn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1.cmml">log</mi><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mfrac id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.cmml"><mn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2.cmml">1</mn><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml">ε</mi></mfrac></mrow><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mfrac id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mrow id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml"><mn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml">1</mn><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml">−</mo><mi id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mrow></mrow><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.3" stretchy="false" 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type="integer" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.3.3">2</cn></apply><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1"><plus id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.1"></plus><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2"><log id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.1"></log><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2"><divide id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2"></divide><cn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.2">1</cn><ci id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3"><log id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.1"></log><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2"><divide id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2"></divide><cn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3"><minus id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.Thmtheorem3.p1.6.6.m6.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.6.6.m6.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.6.6.m6.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math> queries.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.4">The main issue that we have to address is that we cannot always query the centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_c</annotation></semantics></math> of the remaining search space, since <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_c</annotation></semantics></math> is not guaranteed to lie on the grid <math alttext="G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><msubsup id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml"><mi id="S4.SS2.p2.3.m3.1.1.2.2" xref="S4.SS2.p2.3.m3.1.1.2.2.cmml">G</mi><mi id="S4.SS2.p2.3.m3.1.1.3" xref="S4.SS2.p2.3.m3.1.1.3.cmml">b</mi><mi id="S4.SS2.p2.3.m3.1.1.2.3" xref="S4.SS2.p2.3.m3.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">subscript</csymbol><apply id="S4.SS2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.1">superscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2.2">𝐺</ci><ci id="S4.SS2.p2.3.m3.1.1.2.3.cmml" xref="S4.SS2.p2.3.m3.1.1.2.3">𝑑</ci></apply><ci id="S4.SS2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.p2.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. In the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><msub id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><mi id="S4.SS2.p2.4.m4.1.1.2" mathvariant="normal" xref="S4.SS2.p2.4.m4.1.1.2.cmml">ℓ</mi><mi id="S4.SS2.p2.4.m4.1.1.3" mathvariant="normal" xref="S4.SS2.p2.4.m4.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2">ℓ</ci><infinity id="S4.SS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case, Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> solved this problem by choosing the measure for the remaining search space appropriately. Concretely, they design a special measure to determine the size of the remaining search space, and then show that rounding the centerpoint to the grid works for this measure. We do not repeat their argument here, but note that their technique also works if applied together with our (discrete) centerpoint theorem.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.3">Instead, we focus on the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><msub id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p3.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2">ℓ</ci><cn id="S4.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case. Fortunately, rounding in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><msub id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.2" mathvariant="normal" xref="S4.SS2.p3.2.m2.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.2">ℓ</ci><cn id="S4.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case turns out to be simpler than in the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><msub id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml"><mi id="S4.SS2.p3.3.m3.1.1.2" mathvariant="normal" xref="S4.SS2.p3.3.m3.1.1.2.cmml">ℓ</mi><mi id="S4.SS2.p3.3.m3.1.1.3" mathvariant="normal" xref="S4.SS2.p3.3.m3.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><apply id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p3.3.m3.1.1.2.cmml" xref="S4.SS2.p3.3.m3.1.1.2">ℓ</ci><infinity id="S4.SS2.p3.3.m3.1.1.3.cmml" xref="S4.SS2.p3.3.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-case. Indeed, we simply measure the size of the remaining search space as the number of grid points that still remain inside. It turns out that appropriately rounding the centerpoint to the grid does not change its centerpoint properties with respect to this measure of size. More formally, we get the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Lemma 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.4.4">Consider an arbitrary subset of points <math alttext="P\subseteq G^{d}_{b}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem4.p1.1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">P</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">⊆</mo><msubsup id="S4.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.2" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.2.cmml">G</mi><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml">b</mi><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.1.m1.1b"><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1"><subset id="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1"></subset><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2">𝑃</ci><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.2">𝐺</ci><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.1c">P\subseteq G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.1d">italic_P ⊆ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> on the grid. There exists a point <math alttext="c\in G^{d}_{b}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">∈</mo><msubsup id="S4.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.2.cmml">G</mi><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml">b</mi><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1"><in id="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2">𝑐</ci><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.2">𝐺</ci><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.3.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.3">𝑑</ci></apply><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">c\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_c ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="|\mathcal{H}^{1}_{c,v}\cap P|\geq\frac{|P|}{d+1}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.3.m3.4"><semantics id="S4.Thmtheorem4.p1.3.3.m3.4a"><mrow id="S4.Thmtheorem4.p1.3.3.m3.4.4" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.cmml"><mrow id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.2.cmml"><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.2.1.cmml">|</mo><mrow id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.cmml"><msubsup id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.2" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.2.cmml">ℋ</mi><mrow id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.4" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.3.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.1.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.1.1.cmml">c</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.4.1" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.2" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.2.cmml">v</mi></mrow><mn id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.3" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.3.cmml">1</mn></msubsup><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.1.cmml">∩</mo><mi id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.3.cmml">P</mi></mrow><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.3" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.2" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.2.cmml">≥</mo><mfrac id="S4.Thmtheorem4.p1.3.3.m3.3.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.cmml"><mrow id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.2.cmml"><mo id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.3.1" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.2.1.cmml">|</mo><mi id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.1.cmml">P</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.3.2" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.2.1.cmml">|</mo></mrow><mrow id="S4.Thmtheorem4.p1.3.3.m3.3.3.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.2" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.2.cmml">d</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.1" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.1.cmml">+</mo><mn id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.3.m3.4b"><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4"><geq id="S4.Thmtheorem4.p1.3.3.m3.4.4.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.2"></geq><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1"><abs id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.2.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.2"></abs><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1"><intersect id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.1"></intersect><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2">subscript</csymbol><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.2">ℋ</ci><cn id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.2.2.3">1</cn></apply><list id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.4"><ci id="S4.Thmtheorem4.p1.3.3.m3.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.1.1">𝑐</ci><ci id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.2">𝑣</ci></list></apply><ci id="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.1.1.1.3">𝑃</ci></apply></apply><apply id="S4.Thmtheorem4.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3"><divide id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3"></divide><apply id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.3"><abs id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.2.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.3.1"></abs><ci id="S4.Thmtheorem4.p1.3.3.m3.3.3.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.1.1">𝑃</ci></apply><apply id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3"><plus id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.1"></plus><ci id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.2">𝑑</ci><cn id="S4.Thmtheorem4.p1.3.3.m3.3.3.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.3.m3.4c">|\mathcal{H}^{1}_{c,v}\cap P|\geq\frac{|P|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.3.m3.4d">| caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT ∩ italic_P | ≥ divide start_ARG | italic_P | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math> for all <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.4.m4.1"><semantics id="S4.Thmtheorem4.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem4.p1.4.4.m4.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S4.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml">S</mi><mrow id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.2.cmml">d</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.3" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.4.m4.1b"><apply id="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1"><in id="S4.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.1"></in><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.2">𝑣</ci><apply id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.2">𝑆</ci><apply id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3"><minus id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.1"></minus><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.2">𝑑</ci><cn id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.4.m4.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.4.m4.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">We postpone the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a> and instead first use it to derive <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem3" title="Theorem 4.3. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>.</p> </div> <div class="ltx_proof" id="S4.SS2.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem3" title="Theorem 4.3. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>.</h6> <div class="ltx_para" id="S4.SS2.2.p1"> <p class="ltx_p" id="S4.SS2.2.p1.7">Overall, we use a similar strategy as in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem2" title="Theorem 4.2. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, but we use <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a> that is derived from our discrete centerpoint theorem. Concretely, let <math alttext="M\subseteq G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.1.m1.1"><semantics id="S4.SS2.2.p1.1.m1.1a"><mrow id="S4.SS2.2.p1.1.m1.1.1" xref="S4.SS2.2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.2.p1.1.m1.1.1.2" xref="S4.SS2.2.p1.1.m1.1.1.2.cmml">M</mi><mo id="S4.SS2.2.p1.1.m1.1.1.1" xref="S4.SS2.2.p1.1.m1.1.1.1.cmml">⊆</mo><msubsup id="S4.SS2.2.p1.1.m1.1.1.3" xref="S4.SS2.2.p1.1.m1.1.1.3.cmml"><mi id="S4.SS2.2.p1.1.m1.1.1.3.2.2" xref="S4.SS2.2.p1.1.m1.1.1.3.2.2.cmml">G</mi><mi id="S4.SS2.2.p1.1.m1.1.1.3.3" xref="S4.SS2.2.p1.1.m1.1.1.3.3.cmml">b</mi><mi id="S4.SS2.2.p1.1.m1.1.1.3.2.3" xref="S4.SS2.2.p1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.1.m1.1b"><apply id="S4.SS2.2.p1.1.m1.1.1.cmml" xref="S4.SS2.2.p1.1.m1.1.1"><subset id="S4.SS2.2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.2.p1.1.m1.1.1.1"></subset><ci id="S4.SS2.2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.2.p1.1.m1.1.1.2">𝑀</ci><apply id="S4.SS2.2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p1.1.m1.1.1.3.1.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3">subscript</csymbol><apply id="S4.SS2.2.p1.1.m1.1.1.3.2.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p1.1.m1.1.1.3.2.1.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.2.p1.1.m1.1.1.3.2.2.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3.2.2">𝐺</ci><ci id="S4.SS2.2.p1.1.m1.1.1.3.2.3.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S4.SS2.2.p1.1.m1.1.1.3.3.cmml" xref="S4.SS2.2.p1.1.m1.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.1.m1.1c">M\subseteq G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.1.m1.1d">italic_M ⊆ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> be the subset of grid points that could still be <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS2.2.p1.2.m2.1"><semantics id="S4.SS2.2.p1.2.m2.1a"><mi id="S4.SS2.2.p1.2.m2.1.1" xref="S4.SS2.2.p1.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.2.m2.1b"><ci id="S4.SS2.2.p1.2.m2.1.1.cmml" xref="S4.SS2.2.p1.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoints (they have not been ruled out yet). Applying <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a> to <math alttext="M" class="ltx_Math" display="inline" id="S4.SS2.2.p1.3.m3.1"><semantics id="S4.SS2.2.p1.3.m3.1a"><mi id="S4.SS2.2.p1.3.m3.1.1" xref="S4.SS2.2.p1.3.m3.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.3.m3.1b"><ci id="S4.SS2.2.p1.3.m3.1.1.cmml" xref="S4.SS2.2.p1.3.m3.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.3.m3.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.3.m3.1d">italic_M</annotation></semantics></math> yields a centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.2.p1.4.m4.1"><semantics id="S4.SS2.2.p1.4.m4.1a"><mi id="S4.SS2.2.p1.4.m4.1.1" xref="S4.SS2.2.p1.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.4.m4.1b"><ci id="S4.SS2.2.p1.4.m4.1.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.4.m4.1d">italic_c</annotation></semantics></math> on the grid <math alttext="G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.5.m5.1"><semantics id="S4.SS2.2.p1.5.m5.1a"><msubsup id="S4.SS2.2.p1.5.m5.1.1" xref="S4.SS2.2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.2.p1.5.m5.1.1.2.2" xref="S4.SS2.2.p1.5.m5.1.1.2.2.cmml">G</mi><mi id="S4.SS2.2.p1.5.m5.1.1.3" xref="S4.SS2.2.p1.5.m5.1.1.3.cmml">b</mi><mi id="S4.SS2.2.p1.5.m5.1.1.2.3" xref="S4.SS2.2.p1.5.m5.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.5.m5.1b"><apply id="S4.SS2.2.p1.5.m5.1.1.cmml" xref="S4.SS2.2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.2.p1.5.m5.1.1">subscript</csymbol><apply id="S4.SS2.2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.5.m5.1.1.2.1.cmml" xref="S4.SS2.2.p1.5.m5.1.1">superscript</csymbol><ci id="S4.SS2.2.p1.5.m5.1.1.2.2.cmml" xref="S4.SS2.2.p1.5.m5.1.1.2.2">𝐺</ci><ci id="S4.SS2.2.p1.5.m5.1.1.2.3.cmml" xref="S4.SS2.2.p1.5.m5.1.1.2.3">𝑑</ci></apply><ci id="S4.SS2.2.p1.5.m5.1.1.3.cmml" xref="S4.SS2.2.p1.5.m5.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.5.m5.1c">G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.5.m5.1d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Querying <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.2.p1.6.m6.1"><semantics id="S4.SS2.2.p1.6.m6.1a"><mi id="S4.SS2.2.p1.6.m6.1.1" xref="S4.SS2.2.p1.6.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.6.m6.1b"><ci id="S4.SS2.2.p1.6.m6.1.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.6.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.6.m6.1d">italic_c</annotation></semantics></math> guarantees that we can exclude at least <math alttext="\frac{|M|}{d+1}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.7.m7.1"><semantics id="S4.SS2.2.p1.7.m7.1a"><mfrac id="S4.SS2.2.p1.7.m7.1.1" xref="S4.SS2.2.p1.7.m7.1.1.cmml"><mrow id="S4.SS2.2.p1.7.m7.1.1.1.3" xref="S4.SS2.2.p1.7.m7.1.1.1.2.cmml"><mo id="S4.SS2.2.p1.7.m7.1.1.1.3.1" stretchy="false" xref="S4.SS2.2.p1.7.m7.1.1.1.2.1.cmml">|</mo><mi id="S4.SS2.2.p1.7.m7.1.1.1.1" xref="S4.SS2.2.p1.7.m7.1.1.1.1.cmml">M</mi><mo id="S4.SS2.2.p1.7.m7.1.1.1.3.2" stretchy="false" xref="S4.SS2.2.p1.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S4.SS2.2.p1.7.m7.1.1.3" xref="S4.SS2.2.p1.7.m7.1.1.3.cmml"><mi id="S4.SS2.2.p1.7.m7.1.1.3.2" xref="S4.SS2.2.p1.7.m7.1.1.3.2.cmml">d</mi><mo id="S4.SS2.2.p1.7.m7.1.1.3.1" xref="S4.SS2.2.p1.7.m7.1.1.3.1.cmml">+</mo><mn id="S4.SS2.2.p1.7.m7.1.1.3.3" xref="S4.SS2.2.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.7.m7.1b"><apply id="S4.SS2.2.p1.7.m7.1.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1"><divide id="S4.SS2.2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.2.p1.7.m7.1.1"></divide><apply id="S4.SS2.2.p1.7.m7.1.1.1.2.cmml" xref="S4.SS2.2.p1.7.m7.1.1.1.3"><abs id="S4.SS2.2.p1.7.m7.1.1.1.2.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1.1.3.1"></abs><ci id="S4.SS2.2.p1.7.m7.1.1.1.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1.1.1">𝑀</ci></apply><apply id="S4.SS2.2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.2.p1.7.m7.1.1.3"><plus id="S4.SS2.2.p1.7.m7.1.1.3.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1.3.1"></plus><ci id="S4.SS2.2.p1.7.m7.1.1.3.2.cmml" xref="S4.SS2.2.p1.7.m7.1.1.3.2">𝑑</ci><cn id="S4.SS2.2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S4.SS2.2.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.7.m7.1c">\frac{|M|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.7.m7.1d">divide start_ARG | italic_M | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math> points from our current search space in each non-terminating iteration.</p> </div> <div class="ltx_para" id="S4.SS2.3.p2"> <p class="ltx_p" id="S4.SS2.3.p2.5">Now <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> guarantees that as long as we have not queried an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS2.3.p2.1.m1.1"><semantics id="S4.SS2.3.p2.1.m1.1a"><mi id="S4.SS2.3.p2.1.m1.1.1" xref="S4.SS2.3.p2.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.1.m1.1b"><ci id="S4.SS2.3.p2.1.m1.1.1.cmml" xref="S4.SS2.3.p2.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.1.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint, no point in <math alttext="B^{1}(x^{\star},r_{\varepsilon,\lambda})\cap G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.3.p2.2.m2.4"><semantics id="S4.SS2.3.p2.2.m2.4a"><mrow id="S4.SS2.3.p2.2.m2.4.4" xref="S4.SS2.3.p2.2.m2.4.4.cmml"><mrow id="S4.SS2.3.p2.2.m2.4.4.2" xref="S4.SS2.3.p2.2.m2.4.4.2.cmml"><msup 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xref="S4.SS2.3.p2.2.m2.4.4.2.2.2.2.2.cmml">r</mi><mrow id="S4.SS2.3.p2.2.m2.2.2.2.4" xref="S4.SS2.3.p2.2.m2.2.2.2.3.cmml"><mi id="S4.SS2.3.p2.2.m2.1.1.1.1" xref="S4.SS2.3.p2.2.m2.1.1.1.1.cmml">ε</mi><mo id="S4.SS2.3.p2.2.m2.2.2.2.4.1" xref="S4.SS2.3.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.3.p2.2.m2.2.2.2.2" xref="S4.SS2.3.p2.2.m2.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS2.3.p2.2.m2.4.4.2.2.2.5" stretchy="false" xref="S4.SS2.3.p2.2.m2.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.3.p2.2.m2.4.4.3" xref="S4.SS2.3.p2.2.m2.4.4.3.cmml">∩</mo><msubsup id="S4.SS2.3.p2.2.m2.4.4.4" xref="S4.SS2.3.p2.2.m2.4.4.4.cmml"><mi id="S4.SS2.3.p2.2.m2.4.4.4.2.2" xref="S4.SS2.3.p2.2.m2.4.4.4.2.2.cmml">G</mi><mi id="S4.SS2.3.p2.2.m2.4.4.4.3" xref="S4.SS2.3.p2.2.m2.4.4.4.3.cmml">b</mi><mi id="S4.SS2.3.p2.2.m2.4.4.4.2.3" xref="S4.SS2.3.p2.2.m2.4.4.4.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.2.m2.4b"><apply id="S4.SS2.3.p2.2.m2.4.4.cmml" 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id="S4.SS2.3.p2.2.m2.4.4.4.2.1.cmml" xref="S4.SS2.3.p2.2.m2.4.4.4">superscript</csymbol><ci id="S4.SS2.3.p2.2.m2.4.4.4.2.2.cmml" xref="S4.SS2.3.p2.2.m2.4.4.4.2.2">𝐺</ci><ci id="S4.SS2.3.p2.2.m2.4.4.4.2.3.cmml" xref="S4.SS2.3.p2.2.m2.4.4.4.2.3">𝑑</ci></apply><ci id="S4.SS2.3.p2.2.m2.4.4.4.3.cmml" xref="S4.SS2.3.p2.2.m2.4.4.4.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.2.m2.4c">B^{1}(x^{\star},r_{\varepsilon,\lambda})\cap G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.2.m2.4d">italic_B start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT ) ∩ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> has been excluded from the search space yet (where <math alttext="r_{\varepsilon,\lambda}=\frac{\varepsilon-\varepsilon\lambda}{2+2\lambda}" class="ltx_Math" display="inline" id="S4.SS2.3.p2.3.m3.2"><semantics id="S4.SS2.3.p2.3.m3.2a"><mrow id="S4.SS2.3.p2.3.m3.2.3" xref="S4.SS2.3.p2.3.m3.2.3.cmml"><msub id="S4.SS2.3.p2.3.m3.2.3.2" xref="S4.SS2.3.p2.3.m3.2.3.2.cmml"><mi id="S4.SS2.3.p2.3.m3.2.3.2.2" xref="S4.SS2.3.p2.3.m3.2.3.2.2.cmml">r</mi><mrow id="S4.SS2.3.p2.3.m3.2.2.2.4" xref="S4.SS2.3.p2.3.m3.2.2.2.3.cmml"><mi id="S4.SS2.3.p2.3.m3.1.1.1.1" xref="S4.SS2.3.p2.3.m3.1.1.1.1.cmml">ε</mi><mo id="S4.SS2.3.p2.3.m3.2.2.2.4.1" xref="S4.SS2.3.p2.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS2.3.p2.3.m3.2.2.2.2" xref="S4.SS2.3.p2.3.m3.2.2.2.2.cmml">λ</mi></mrow></msub><mo id="S4.SS2.3.p2.3.m3.2.3.1" xref="S4.SS2.3.p2.3.m3.2.3.1.cmml">=</mo><mfrac id="S4.SS2.3.p2.3.m3.2.3.3" xref="S4.SS2.3.p2.3.m3.2.3.3.cmml"><mrow id="S4.SS2.3.p2.3.m3.2.3.3.2" xref="S4.SS2.3.p2.3.m3.2.3.3.2.cmml"><mi id="S4.SS2.3.p2.3.m3.2.3.3.2.2" xref="S4.SS2.3.p2.3.m3.2.3.3.2.2.cmml">ε</mi><mo id="S4.SS2.3.p2.3.m3.2.3.3.2.1" xref="S4.SS2.3.p2.3.m3.2.3.3.2.1.cmml">−</mo><mrow id="S4.SS2.3.p2.3.m3.2.3.3.2.3" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.cmml"><mi id="S4.SS2.3.p2.3.m3.2.3.3.2.3.2" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.2.cmml">ε</mi><mo id="S4.SS2.3.p2.3.m3.2.3.3.2.3.1" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.1.cmml">⁢</mo><mi id="S4.SS2.3.p2.3.m3.2.3.3.2.3.3" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.3.cmml">λ</mi></mrow></mrow><mrow id="S4.SS2.3.p2.3.m3.2.3.3.3" xref="S4.SS2.3.p2.3.m3.2.3.3.3.cmml"><mn id="S4.SS2.3.p2.3.m3.2.3.3.3.2" xref="S4.SS2.3.p2.3.m3.2.3.3.3.2.cmml">2</mn><mo id="S4.SS2.3.p2.3.m3.2.3.3.3.1" xref="S4.SS2.3.p2.3.m3.2.3.3.3.1.cmml">+</mo><mrow id="S4.SS2.3.p2.3.m3.2.3.3.3.3" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.cmml"><mn id="S4.SS2.3.p2.3.m3.2.3.3.3.3.2" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.2.cmml">2</mn><mo id="S4.SS2.3.p2.3.m3.2.3.3.3.3.1" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS2.3.p2.3.m3.2.3.3.3.3.3" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.3.cmml">λ</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.3.m3.2b"><apply id="S4.SS2.3.p2.3.m3.2.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3"><eq id="S4.SS2.3.p2.3.m3.2.3.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.1"></eq><apply id="S4.SS2.3.p2.3.m3.2.3.2.cmml" xref="S4.SS2.3.p2.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.3.p2.3.m3.2.3.2.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.2">subscript</csymbol><ci id="S4.SS2.3.p2.3.m3.2.3.2.2.cmml" xref="S4.SS2.3.p2.3.m3.2.3.2.2">𝑟</ci><list id="S4.SS2.3.p2.3.m3.2.2.2.3.cmml" xref="S4.SS2.3.p2.3.m3.2.2.2.4"><ci id="S4.SS2.3.p2.3.m3.1.1.1.1.cmml" xref="S4.SS2.3.p2.3.m3.1.1.1.1">𝜀</ci><ci id="S4.SS2.3.p2.3.m3.2.2.2.2.cmml" xref="S4.SS2.3.p2.3.m3.2.2.2.2">𝜆</ci></list></apply><apply id="S4.SS2.3.p2.3.m3.2.3.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3"><divide id="S4.SS2.3.p2.3.m3.2.3.3.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3"></divide><apply id="S4.SS2.3.p2.3.m3.2.3.3.2.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2"><minus id="S4.SS2.3.p2.3.m3.2.3.3.2.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.1"></minus><ci id="S4.SS2.3.p2.3.m3.2.3.3.2.2.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.2">𝜀</ci><apply id="S4.SS2.3.p2.3.m3.2.3.3.2.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3"><times id="S4.SS2.3.p2.3.m3.2.3.3.2.3.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.1"></times><ci id="S4.SS2.3.p2.3.m3.2.3.3.2.3.2.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.2">𝜀</ci><ci id="S4.SS2.3.p2.3.m3.2.3.3.2.3.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.2.3.3">𝜆</ci></apply></apply><apply id="S4.SS2.3.p2.3.m3.2.3.3.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.3"><plus id="S4.SS2.3.p2.3.m3.2.3.3.3.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.3.1"></plus><cn id="S4.SS2.3.p2.3.m3.2.3.3.3.2.cmml" type="integer" xref="S4.SS2.3.p2.3.m3.2.3.3.3.2">2</cn><apply id="S4.SS2.3.p2.3.m3.2.3.3.3.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3"><times id="S4.SS2.3.p2.3.m3.2.3.3.3.3.1.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.1"></times><cn id="S4.SS2.3.p2.3.m3.2.3.3.3.3.2.cmml" type="integer" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.2">2</cn><ci id="S4.SS2.3.p2.3.m3.2.3.3.3.3.3.cmml" xref="S4.SS2.3.p2.3.m3.2.3.3.3.3.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.3.m3.2c">r_{\varepsilon,\lambda}=\frac{\varepsilon-\varepsilon\lambda}{2+2\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.3.m3.2d">italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT = divide start_ARG italic_ε - italic_ε italic_λ end_ARG start_ARG 2 + 2 italic_λ end_ARG</annotation></semantics></math>). To guarantee that this intersection is non-empty, it suffices to have <math alttext="2^{-b}\leq\frac{r_{\varepsilon,\lambda}}{d}" class="ltx_Math" display="inline" id="S4.SS2.3.p2.4.m4.2"><semantics id="S4.SS2.3.p2.4.m4.2a"><mrow id="S4.SS2.3.p2.4.m4.2.3" xref="S4.SS2.3.p2.4.m4.2.3.cmml"><msup id="S4.SS2.3.p2.4.m4.2.3.2" xref="S4.SS2.3.p2.4.m4.2.3.2.cmml"><mn id="S4.SS2.3.p2.4.m4.2.3.2.2" xref="S4.SS2.3.p2.4.m4.2.3.2.2.cmml">2</mn><mrow id="S4.SS2.3.p2.4.m4.2.3.2.3" xref="S4.SS2.3.p2.4.m4.2.3.2.3.cmml"><mo id="S4.SS2.3.p2.4.m4.2.3.2.3a" xref="S4.SS2.3.p2.4.m4.2.3.2.3.cmml">−</mo><mi id="S4.SS2.3.p2.4.m4.2.3.2.3.2" xref="S4.SS2.3.p2.4.m4.2.3.2.3.2.cmml">b</mi></mrow></msup><mo id="S4.SS2.3.p2.4.m4.2.3.1" xref="S4.SS2.3.p2.4.m4.2.3.1.cmml">≤</mo><mfrac id="S4.SS2.3.p2.4.m4.2.2" xref="S4.SS2.3.p2.4.m4.2.2.cmml"><msub id="S4.SS2.3.p2.4.m4.2.2.2" xref="S4.SS2.3.p2.4.m4.2.2.2.cmml"><mi id="S4.SS2.3.p2.4.m4.2.2.2.4" xref="S4.SS2.3.p2.4.m4.2.2.2.4.cmml">r</mi><mrow id="S4.SS2.3.p2.4.m4.2.2.2.2.2.4" xref="S4.SS2.3.p2.4.m4.2.2.2.2.2.3.cmml"><mi id="S4.SS2.3.p2.4.m4.1.1.1.1.1.1" xref="S4.SS2.3.p2.4.m4.1.1.1.1.1.1.cmml">ε</mi><mo id="S4.SS2.3.p2.4.m4.2.2.2.2.2.4.1" xref="S4.SS2.3.p2.4.m4.2.2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.3.p2.4.m4.2.2.2.2.2.2" xref="S4.SS2.3.p2.4.m4.2.2.2.2.2.2.cmml">λ</mi></mrow></msub><mi id="S4.SS2.3.p2.4.m4.2.2.4" xref="S4.SS2.3.p2.4.m4.2.2.4.cmml">d</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.4.m4.2b"><apply id="S4.SS2.3.p2.4.m4.2.3.cmml" xref="S4.SS2.3.p2.4.m4.2.3"><leq id="S4.SS2.3.p2.4.m4.2.3.1.cmml" xref="S4.SS2.3.p2.4.m4.2.3.1"></leq><apply id="S4.SS2.3.p2.4.m4.2.3.2.cmml" xref="S4.SS2.3.p2.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.3.p2.4.m4.2.3.2.1.cmml" xref="S4.SS2.3.p2.4.m4.2.3.2">superscript</csymbol><cn id="S4.SS2.3.p2.4.m4.2.3.2.2.cmml" type="integer" xref="S4.SS2.3.p2.4.m4.2.3.2.2">2</cn><apply id="S4.SS2.3.p2.4.m4.2.3.2.3.cmml" xref="S4.SS2.3.p2.4.m4.2.3.2.3"><minus id="S4.SS2.3.p2.4.m4.2.3.2.3.1.cmml" xref="S4.SS2.3.p2.4.m4.2.3.2.3"></minus><ci id="S4.SS2.3.p2.4.m4.2.3.2.3.2.cmml" xref="S4.SS2.3.p2.4.m4.2.3.2.3.2">𝑏</ci></apply></apply><apply id="S4.SS2.3.p2.4.m4.2.2.cmml" xref="S4.SS2.3.p2.4.m4.2.2"><divide id="S4.SS2.3.p2.4.m4.2.2.3.cmml" xref="S4.SS2.3.p2.4.m4.2.2"></divide><apply id="S4.SS2.3.p2.4.m4.2.2.2.cmml" xref="S4.SS2.3.p2.4.m4.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.3.p2.4.m4.2.2.2.3.cmml" xref="S4.SS2.3.p2.4.m4.2.2.2">subscript</csymbol><ci id="S4.SS2.3.p2.4.m4.2.2.2.4.cmml" xref="S4.SS2.3.p2.4.m4.2.2.2.4">𝑟</ci><list id="S4.SS2.3.p2.4.m4.2.2.2.2.2.3.cmml" xref="S4.SS2.3.p2.4.m4.2.2.2.2.2.4"><ci id="S4.SS2.3.p2.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.3.p2.4.m4.1.1.1.1.1.1">𝜀</ci><ci id="S4.SS2.3.p2.4.m4.2.2.2.2.2.2.cmml" xref="S4.SS2.3.p2.4.m4.2.2.2.2.2.2">𝜆</ci></list></apply><ci id="S4.SS2.3.p2.4.m4.2.2.4.cmml" xref="S4.SS2.3.p2.4.m4.2.2.4">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.4.m4.2c">2^{-b}\leq\frac{r_{\varepsilon,\lambda}}{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.4.m4.2d">2 start_POSTSUPERSCRIPT - italic_b end_POSTSUPERSCRIPT ≤ divide start_ARG italic_r start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT end_ARG start_ARG italic_d end_ARG</annotation></semantics></math>, which translates to our assumption <math alttext="b\geq\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}\right)" class="ltx_Math" display="inline" id="S4.SS2.3.p2.5.m5.2"><semantics id="S4.SS2.3.p2.5.m5.2a"><mrow id="S4.SS2.3.p2.5.m5.2.2" xref="S4.SS2.3.p2.5.m5.2.2.cmml"><mi id="S4.SS2.3.p2.5.m5.2.2.4" xref="S4.SS2.3.p2.5.m5.2.2.4.cmml">b</mi><mo id="S4.SS2.3.p2.5.m5.2.2.3" xref="S4.SS2.3.p2.5.m5.2.2.3.cmml">≥</mo><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.3.cmml"><msub id="S4.SS2.3.p2.5.m5.1.1.1.1.1" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS2.3.p2.5.m5.1.1.1.1.1.2" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1.2.cmml">log</mi><mn id="S4.SS2.3.p2.5.m5.1.1.1.1.1.3" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.SS2.3.p2.5.m5.2.2.2.2a" xref="S4.SS2.3.p2.5.m5.2.2.2.3.cmml">⁡</mo><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.3.cmml"><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.3.cmml">(</mo><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.cmml"><mfrac id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.cmml"><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.cmml"><mn id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.2.cmml">2</mn><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.1" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.1.cmml">⁢</mo><mi id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.2.3.cmml">d</mi></mrow><mi id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.2.3.cmml">ε</mi></mfrac><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.1" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.1.cmml">⁢</mo><mfrac id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.cmml"><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.cmml"><mn id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.2.cmml">1</mn><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.1" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.1.cmml">+</mo><mi id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.3.cmml">λ</mi></mrow><mrow id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.cmml"><mn id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.2" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.2.cmml">1</mn><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.1" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.1.cmml">−</mo><mi id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.3" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.3.cmml">λ</mi></mrow></mfrac></mrow><mo id="S4.SS2.3.p2.5.m5.2.2.2.2.2.3" xref="S4.SS2.3.p2.5.m5.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.5.m5.2b"><apply id="S4.SS2.3.p2.5.m5.2.2.cmml" xref="S4.SS2.3.p2.5.m5.2.2"><geq id="S4.SS2.3.p2.5.m5.2.2.3.cmml" xref="S4.SS2.3.p2.5.m5.2.2.3"></geq><ci id="S4.SS2.3.p2.5.m5.2.2.4.cmml" xref="S4.SS2.3.p2.5.m5.2.2.4">𝑏</ci><apply id="S4.SS2.3.p2.5.m5.2.2.2.3.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2"><apply id="S4.SS2.3.p2.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p2.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1">subscript</csymbol><log id="S4.SS2.3.p2.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1.2"></log><cn id="S4.SS2.3.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.3.p2.5.m5.1.1.1.1.1.3">2</cn></apply><apply id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.cmml" 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xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3"></divide><apply id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2"><plus id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.1.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.1"></plus><cn id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.2.cmml" type="integer" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.2">1</cn><ci id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.3.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.2.3">𝜆</ci></apply><apply id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3"><minus id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.1.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.1"></minus><cn id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.2.cmml" type="integer" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.2">1</cn><ci id="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.3.cmml" xref="S4.SS2.3.p2.5.m5.2.2.2.2.2.1.3.3.3">𝜆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.5.m5.2c">b\geq\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.5.m5.2d">italic_b ≥ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG 2 italic_d end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG )</annotation></semantics></math>. In fact, we can assume</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="b=\left\lceil\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}% \right)\right\rceil" class="ltx_Math" display="block" id="S4.Ex14.m1.1"><semantics id="S4.Ex14.m1.1a"><mrow id="S4.Ex14.m1.1.1" xref="S4.Ex14.m1.1.1.cmml"><mi id="S4.Ex14.m1.1.1.3" xref="S4.Ex14.m1.1.1.3.cmml">b</mi><mo id="S4.Ex14.m1.1.1.2" xref="S4.Ex14.m1.1.1.2.cmml">=</mo><mrow id="S4.Ex14.m1.1.1.1.1" xref="S4.Ex14.m1.1.1.1.2.cmml"><mo id="S4.Ex14.m1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.2.1.cmml">⌈</mo><mrow id="S4.Ex14.m1.1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.1.1.3.cmml"><msub id="S4.Ex14.m1.1.1.1.1.1.1.1" xref="S4.Ex14.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S4.Ex14.m1.1.1.1.1.1.1.1.3" xref="S4.Ex14.m1.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.Ex14.m1.1.1.1.1.1.2a" xref="S4.Ex14.m1.1.1.1.1.1.3.cmml">⁡</mo><mrow id="S4.Ex14.m1.1.1.1.1.1.2.2" xref="S4.Ex14.m1.1.1.1.1.1.3.cmml"><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.2" xref="S4.Ex14.m1.1.1.1.1.1.3.cmml">(</mo><mrow id="S4.Ex14.m1.1.1.1.1.1.2.2.1" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.cmml"><mfrac id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.cmml"><mn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.2.cmml">2</mn><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.1" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.1.cmml">⁢</mo><mi id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.3.cmml">d</mi></mrow><mi id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.3.cmml">ε</mi></mfrac><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.1.1" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.1.cmml">⁢</mo><mfrac id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.cmml"><mn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.2.cmml">1</mn><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.1" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.1.cmml">+</mo><mi id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.3.cmml">λ</mi></mrow><mrow id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.cmml"><mn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.2" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.2.cmml">1</mn><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.1" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.1.cmml">−</mo><mi id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.3" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.3.cmml">λ</mi></mrow></mfrac></mrow><mo id="S4.Ex14.m1.1.1.1.1.1.2.2.3" xref="S4.Ex14.m1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo 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id="S4.Ex14.m1.1.1.1.1.1.2.2.1.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.1"></times><apply id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2"><divide id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2"></divide><apply id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2"><times id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.1"></times><cn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.2.cmml" type="integer" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.2">2</cn><ci id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.2.3">𝑑</ci></apply><ci id="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.2.3">𝜀</ci></apply><apply id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3"><divide id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3"></divide><apply id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2"><plus id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.1"></plus><cn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.2.cmml" type="integer" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.2">1</cn><ci id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.2.3">𝜆</ci></apply><apply id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3"><minus id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.1.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.1"></minus><cn id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.2.cmml" type="integer" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.2">1</cn><ci id="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.3.cmml" xref="S4.Ex14.m1.1.1.1.1.1.2.2.1.3.3.3">𝜆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.1c">b=\left\lceil\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}% \right)\right\rceil</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.1d">italic_b = ⌈ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG 2 italic_d end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG ) ⌉</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.3.p2.7">without loss of generality (as we can use a subgrid of this size, if the given grid is finer). For this choice of <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.3.p2.6.m1.1"><semantics id="S4.SS2.3.p2.6.m1.1a"><mi id="S4.SS2.3.p2.6.m1.1.1" xref="S4.SS2.3.p2.6.m1.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.6.m1.1b"><ci id="S4.SS2.3.p2.6.m1.1.1.cmml" xref="S4.SS2.3.p2.6.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.6.m1.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.6.m1.1d">italic_b</annotation></semantics></math>, we have <math alttext="|G^{d}_{b}|=(2^{b}+1)^{d}\leq 4^{db}" class="ltx_Math" display="inline" id="S4.SS2.3.p2.7.m2.2"><semantics id="S4.SS2.3.p2.7.m2.2a"><mrow id="S4.SS2.3.p2.7.m2.2.2" xref="S4.SS2.3.p2.7.m2.2.2.cmml"><mrow id="S4.SS2.3.p2.7.m2.1.1.1.1" xref="S4.SS2.3.p2.7.m2.1.1.1.2.cmml"><mo id="S4.SS2.3.p2.7.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.3.p2.7.m2.1.1.1.2.1.cmml">|</mo><msubsup id="S4.SS2.3.p2.7.m2.1.1.1.1.1" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.2" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.2.cmml">G</mi><mi id="S4.SS2.3.p2.7.m2.1.1.1.1.1.3" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.3.cmml">b</mi><mi id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.3" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.3.p2.7.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.3.p2.7.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.3.p2.7.m2.2.2.4" xref="S4.SS2.3.p2.7.m2.2.2.4.cmml">=</mo><msup id="S4.SS2.3.p2.7.m2.2.2.2" xref="S4.SS2.3.p2.7.m2.2.2.2.cmml"><mrow id="S4.SS2.3.p2.7.m2.2.2.2.1.1" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.cmml"><mo id="S4.SS2.3.p2.7.m2.2.2.2.1.1.2" stretchy="false" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.cmml"><msup id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.cmml"><mn id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.2" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.2.cmml">2</mn><mi id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.3" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.3.cmml">b</mi></msup><mo id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.1" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.1.cmml">+</mo><mn id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.3" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.3.p2.7.m2.2.2.2.1.1.3" stretchy="false" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="S4.SS2.3.p2.7.m2.2.2.2.3" xref="S4.SS2.3.p2.7.m2.2.2.2.3.cmml">d</mi></msup><mo id="S4.SS2.3.p2.7.m2.2.2.5" xref="S4.SS2.3.p2.7.m2.2.2.5.cmml">≤</mo><msup id="S4.SS2.3.p2.7.m2.2.2.6" xref="S4.SS2.3.p2.7.m2.2.2.6.cmml"><mn id="S4.SS2.3.p2.7.m2.2.2.6.2" xref="S4.SS2.3.p2.7.m2.2.2.6.2.cmml">4</mn><mrow id="S4.SS2.3.p2.7.m2.2.2.6.3" xref="S4.SS2.3.p2.7.m2.2.2.6.3.cmml"><mi id="S4.SS2.3.p2.7.m2.2.2.6.3.2" xref="S4.SS2.3.p2.7.m2.2.2.6.3.2.cmml">d</mi><mo id="S4.SS2.3.p2.7.m2.2.2.6.3.1" xref="S4.SS2.3.p2.7.m2.2.2.6.3.1.cmml">⁢</mo><mi id="S4.SS2.3.p2.7.m2.2.2.6.3.3" xref="S4.SS2.3.p2.7.m2.2.2.6.3.3.cmml">b</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.7.m2.2b"><apply id="S4.SS2.3.p2.7.m2.2.2.cmml" xref="S4.SS2.3.p2.7.m2.2.2"><and id="S4.SS2.3.p2.7.m2.2.2a.cmml" xref="S4.SS2.3.p2.7.m2.2.2"></and><apply id="S4.SS2.3.p2.7.m2.2.2b.cmml" xref="S4.SS2.3.p2.7.m2.2.2"><eq id="S4.SS2.3.p2.7.m2.2.2.4.cmml" xref="S4.SS2.3.p2.7.m2.2.2.4"></eq><apply id="S4.SS2.3.p2.7.m2.1.1.1.2.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1"><abs id="S4.SS2.3.p2.7.m2.1.1.1.2.1.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.2"></abs><apply id="S4.SS2.3.p2.7.m2.1.1.1.1.1.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p2.7.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1">subscript</csymbol><apply id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.1.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.2.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.2">𝐺</ci><ci id="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.3.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.2.3">𝑑</ci></apply><ci id="S4.SS2.3.p2.7.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.3.p2.7.m2.1.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.SS2.3.p2.7.m2.2.2.2.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.3.p2.7.m2.2.2.2.2.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2">superscript</csymbol><apply id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1"><plus id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.1.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.1"></plus><apply id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.1.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2">superscript</csymbol><cn id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.2.cmml" type="integer" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.2">2</cn><ci id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.3.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.2.3">𝑏</ci></apply><cn id="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.3.p2.7.m2.2.2.2.1.1.1.3">1</cn></apply><ci id="S4.SS2.3.p2.7.m2.2.2.2.3.cmml" xref="S4.SS2.3.p2.7.m2.2.2.2.3">𝑑</ci></apply></apply><apply id="S4.SS2.3.p2.7.m2.2.2c.cmml" xref="S4.SS2.3.p2.7.m2.2.2"><leq id="S4.SS2.3.p2.7.m2.2.2.5.cmml" xref="S4.SS2.3.p2.7.m2.2.2.5"></leq><share href="https://arxiv.org/html/2503.16089v1#S4.SS2.3.p2.7.m2.2.2.2.cmml" id="S4.SS2.3.p2.7.m2.2.2d.cmml" xref="S4.SS2.3.p2.7.m2.2.2"></share><apply id="S4.SS2.3.p2.7.m2.2.2.6.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6"><csymbol cd="ambiguous" id="S4.SS2.3.p2.7.m2.2.2.6.1.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6">superscript</csymbol><cn id="S4.SS2.3.p2.7.m2.2.2.6.2.cmml" type="integer" xref="S4.SS2.3.p2.7.m2.2.2.6.2">4</cn><apply id="S4.SS2.3.p2.7.m2.2.2.6.3.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6.3"><times id="S4.SS2.3.p2.7.m2.2.2.6.3.1.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6.3.1"></times><ci id="S4.SS2.3.p2.7.m2.2.2.6.3.2.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6.3.2">𝑑</ci><ci id="S4.SS2.3.p2.7.m2.2.2.6.3.3.cmml" xref="S4.SS2.3.p2.7.m2.2.2.6.3.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.7.m2.2c">|G^{d}_{b}|=(2^{b}+1)^{d}\leq 4^{db}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.7.m2.2d">| italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | = ( 2 start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT + 1 ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ≤ 4 start_POSTSUPERSCRIPT italic_d italic_b end_POSTSUPERSCRIPT</annotation></semantics></math>. With the calculation</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\log_{\frac{d}{d+1}}4^{-db}=\frac{\log 4^{-db}}{\log\frac{d}{d+1}}=\frac{db}{% \log(1+\frac{1}{d})}\leq\mathcal{O}(d^{2}b)\leq\mathcal{O}\left(d^{2}\left(% \log d+\log\varepsilon+\log\frac{1}{1-\lambda}\right)\right)" class="ltx_Math" display="block" id="S4.Ex15.m1.4"><semantics id="S4.Ex15.m1.4a"><mrow id="S4.Ex15.m1.4.4" xref="S4.Ex15.m1.4.4.cmml"><mrow id="S4.Ex15.m1.4.4.4" xref="S4.Ex15.m1.4.4.4.cmml"><msub id="S4.Ex15.m1.4.4.4.1" xref="S4.Ex15.m1.4.4.4.1.cmml"><mi id="S4.Ex15.m1.4.4.4.1.2" xref="S4.Ex15.m1.4.4.4.1.2.cmml">log</mi><mfrac id="S4.Ex15.m1.4.4.4.1.3" xref="S4.Ex15.m1.4.4.4.1.3.cmml"><mi id="S4.Ex15.m1.4.4.4.1.3.2" xref="S4.Ex15.m1.4.4.4.1.3.2.cmml">d</mi><mrow id="S4.Ex15.m1.4.4.4.1.3.3" xref="S4.Ex15.m1.4.4.4.1.3.3.cmml"><mi id="S4.Ex15.m1.4.4.4.1.3.3.2" xref="S4.Ex15.m1.4.4.4.1.3.3.2.cmml">d</mi><mo id="S4.Ex15.m1.4.4.4.1.3.3.1" xref="S4.Ex15.m1.4.4.4.1.3.3.1.cmml">+</mo><mn id="S4.Ex15.m1.4.4.4.1.3.3.3" xref="S4.Ex15.m1.4.4.4.1.3.3.3.cmml">1</mn></mrow></mfrac></msub><mo id="S4.Ex15.m1.4.4.4a" lspace="0.167em" xref="S4.Ex15.m1.4.4.4.cmml">⁡</mo><msup id="S4.Ex15.m1.4.4.4.2" xref="S4.Ex15.m1.4.4.4.2.cmml"><mn id="S4.Ex15.m1.4.4.4.2.2" xref="S4.Ex15.m1.4.4.4.2.2.cmml">4</mn><mrow id="S4.Ex15.m1.4.4.4.2.3" xref="S4.Ex15.m1.4.4.4.2.3.cmml"><mo id="S4.Ex15.m1.4.4.4.2.3a" xref="S4.Ex15.m1.4.4.4.2.3.cmml">−</mo><mrow id="S4.Ex15.m1.4.4.4.2.3.2" xref="S4.Ex15.m1.4.4.4.2.3.2.cmml"><mi id="S4.Ex15.m1.4.4.4.2.3.2.2" xref="S4.Ex15.m1.4.4.4.2.3.2.2.cmml">d</mi><mo id="S4.Ex15.m1.4.4.4.2.3.2.1" xref="S4.Ex15.m1.4.4.4.2.3.2.1.cmml">⁢</mo><mi id="S4.Ex15.m1.4.4.4.2.3.2.3" xref="S4.Ex15.m1.4.4.4.2.3.2.3.cmml">b</mi></mrow></mrow></msup></mrow><mo id="S4.Ex15.m1.4.4.5" xref="S4.Ex15.m1.4.4.5.cmml">=</mo><mfrac id="S4.Ex15.m1.4.4.6" xref="S4.Ex15.m1.4.4.6.cmml"><mrow id="S4.Ex15.m1.4.4.6.2" xref="S4.Ex15.m1.4.4.6.2.cmml"><mi id="S4.Ex15.m1.4.4.6.2.1" xref="S4.Ex15.m1.4.4.6.2.1.cmml">log</mi><mo id="S4.Ex15.m1.4.4.6.2a" lspace="0.167em" xref="S4.Ex15.m1.4.4.6.2.cmml">⁡</mo><msup id="S4.Ex15.m1.4.4.6.2.2" xref="S4.Ex15.m1.4.4.6.2.2.cmml"><mn id="S4.Ex15.m1.4.4.6.2.2.2" xref="S4.Ex15.m1.4.4.6.2.2.2.cmml">4</mn><mrow id="S4.Ex15.m1.4.4.6.2.2.3" xref="S4.Ex15.m1.4.4.6.2.2.3.cmml"><mo id="S4.Ex15.m1.4.4.6.2.2.3a" xref="S4.Ex15.m1.4.4.6.2.2.3.cmml">−</mo><mrow id="S4.Ex15.m1.4.4.6.2.2.3.2" xref="S4.Ex15.m1.4.4.6.2.2.3.2.cmml"><mi id="S4.Ex15.m1.4.4.6.2.2.3.2.2" xref="S4.Ex15.m1.4.4.6.2.2.3.2.2.cmml">d</mi><mo id="S4.Ex15.m1.4.4.6.2.2.3.2.1" xref="S4.Ex15.m1.4.4.6.2.2.3.2.1.cmml">⁢</mo><mi id="S4.Ex15.m1.4.4.6.2.2.3.2.3" xref="S4.Ex15.m1.4.4.6.2.2.3.2.3.cmml">b</mi></mrow></mrow></msup></mrow><mrow id="S4.Ex15.m1.4.4.6.3" xref="S4.Ex15.m1.4.4.6.3.cmml"><mi id="S4.Ex15.m1.4.4.6.3.1" xref="S4.Ex15.m1.4.4.6.3.1.cmml">log</mi><mo id="S4.Ex15.m1.4.4.6.3a" lspace="0.167em" xref="S4.Ex15.m1.4.4.6.3.cmml">⁡</mo><mfrac id="S4.Ex15.m1.4.4.6.3.2" xref="S4.Ex15.m1.4.4.6.3.2.cmml"><mi id="S4.Ex15.m1.4.4.6.3.2.2" xref="S4.Ex15.m1.4.4.6.3.2.2.cmml">d</mi><mrow id="S4.Ex15.m1.4.4.6.3.2.3" xref="S4.Ex15.m1.4.4.6.3.2.3.cmml"><mi id="S4.Ex15.m1.4.4.6.3.2.3.2" xref="S4.Ex15.m1.4.4.6.3.2.3.2.cmml">d</mi><mo id="S4.Ex15.m1.4.4.6.3.2.3.1" xref="S4.Ex15.m1.4.4.6.3.2.3.1.cmml">+</mo><mn id="S4.Ex15.m1.4.4.6.3.2.3.3" xref="S4.Ex15.m1.4.4.6.3.2.3.3.cmml">1</mn></mrow></mfrac></mrow></mfrac><mo id="S4.Ex15.m1.4.4.7" xref="S4.Ex15.m1.4.4.7.cmml">=</mo><mfrac id="S4.Ex15.m1.2.2" xref="S4.Ex15.m1.2.2.cmml"><mrow id="S4.Ex15.m1.2.2.4" xref="S4.Ex15.m1.2.2.4.cmml"><mi id="S4.Ex15.m1.2.2.4.2" xref="S4.Ex15.m1.2.2.4.2.cmml">d</mi><mo id="S4.Ex15.m1.2.2.4.1" xref="S4.Ex15.m1.2.2.4.1.cmml">⁢</mo><mi id="S4.Ex15.m1.2.2.4.3" xref="S4.Ex15.m1.2.2.4.3.cmml">b</mi></mrow><mrow id="S4.Ex15.m1.2.2.2.2" xref="S4.Ex15.m1.2.2.2.3.cmml"><mi id="S4.Ex15.m1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.cmml">log</mi><mo id="S4.Ex15.m1.2.2.2.2a" xref="S4.Ex15.m1.2.2.2.3.cmml">⁡</mo><mrow id="S4.Ex15.m1.2.2.2.2.1" xref="S4.Ex15.m1.2.2.2.3.cmml"><mo id="S4.Ex15.m1.2.2.2.2.1.2" stretchy="false" xref="S4.Ex15.m1.2.2.2.3.cmml">(</mo><mrow id="S4.Ex15.m1.2.2.2.2.1.1" xref="S4.Ex15.m1.2.2.2.2.1.1.cmml"><mn id="S4.Ex15.m1.2.2.2.2.1.1.2" xref="S4.Ex15.m1.2.2.2.2.1.1.2.cmml">1</mn><mo id="S4.Ex15.m1.2.2.2.2.1.1.1" xref="S4.Ex15.m1.2.2.2.2.1.1.1.cmml">+</mo><mfrac id="S4.Ex15.m1.2.2.2.2.1.1.3" xref="S4.Ex15.m1.2.2.2.2.1.1.3.cmml"><mn id="S4.Ex15.m1.2.2.2.2.1.1.3.2" xref="S4.Ex15.m1.2.2.2.2.1.1.3.2.cmml">1</mn><mi id="S4.Ex15.m1.2.2.2.2.1.1.3.3" xref="S4.Ex15.m1.2.2.2.2.1.1.3.3.cmml">d</mi></mfrac></mrow><mo id="S4.Ex15.m1.2.2.2.2.1.3" stretchy="false" xref="S4.Ex15.m1.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac><mo id="S4.Ex15.m1.4.4.8" xref="S4.Ex15.m1.4.4.8.cmml">≤</mo><mrow id="S4.Ex15.m1.3.3.1" xref="S4.Ex15.m1.3.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.3.3.1.3" xref="S4.Ex15.m1.3.3.1.3.cmml">𝒪</mi><mo id="S4.Ex15.m1.3.3.1.2" xref="S4.Ex15.m1.3.3.1.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.3.3.1.1.1" xref="S4.Ex15.m1.3.3.1.1.1.1.cmml"><mo id="S4.Ex15.m1.3.3.1.1.1.2" stretchy="false" xref="S4.Ex15.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.Ex15.m1.3.3.1.1.1.1" xref="S4.Ex15.m1.3.3.1.1.1.1.cmml"><msup id="S4.Ex15.m1.3.3.1.1.1.1.2" xref="S4.Ex15.m1.3.3.1.1.1.1.2.cmml"><mi id="S4.Ex15.m1.3.3.1.1.1.1.2.2" xref="S4.Ex15.m1.3.3.1.1.1.1.2.2.cmml">d</mi><mn id="S4.Ex15.m1.3.3.1.1.1.1.2.3" xref="S4.Ex15.m1.3.3.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S4.Ex15.m1.3.3.1.1.1.1.1" xref="S4.Ex15.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex15.m1.3.3.1.1.1.1.3" xref="S4.Ex15.m1.3.3.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Ex15.m1.3.3.1.1.1.3" stretchy="false" xref="S4.Ex15.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex15.m1.4.4.9" xref="S4.Ex15.m1.4.4.9.cmml">≤</mo><mrow id="S4.Ex15.m1.4.4.2" xref="S4.Ex15.m1.4.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.4.4.2.3" xref="S4.Ex15.m1.4.4.2.3.cmml">𝒪</mi><mo id="S4.Ex15.m1.4.4.2.2" xref="S4.Ex15.m1.4.4.2.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.4.4.2.1.1" xref="S4.Ex15.m1.4.4.2.1.1.1.cmml"><mo id="S4.Ex15.m1.4.4.2.1.1.2" xref="S4.Ex15.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="S4.Ex15.m1.4.4.2.1.1.1" xref="S4.Ex15.m1.4.4.2.1.1.1.cmml"><msup id="S4.Ex15.m1.4.4.2.1.1.1.3" xref="S4.Ex15.m1.4.4.2.1.1.1.3.cmml"><mi id="S4.Ex15.m1.4.4.2.1.1.1.3.2" xref="S4.Ex15.m1.4.4.2.1.1.1.3.2.cmml">d</mi><mn id="S4.Ex15.m1.4.4.2.1.1.1.3.3" xref="S4.Ex15.m1.4.4.2.1.1.1.3.3.cmml">2</mn></msup><mo id="S4.Ex15.m1.4.4.2.1.1.1.2" xref="S4.Ex15.m1.4.4.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.4.4.2.1.1.1.1.1" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex15.m1.4.4.2.1.1.1.1.1.2" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.cmml"><mrow id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.1" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.1.cmml">log</mi><mo id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2a" lspace="0.167em" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.cmml">⁡</mo><mi id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.2" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.2.2.cmml">d</mi></mrow><mo id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.1" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.1" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.2" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.3.2.cmml">ε</mi></mrow><mo id="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.1a" xref="S4.Ex15.m1.4.4.2.1.1.1.1.1.1.1.cmml">+</mo><mrow 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id="S4.Ex15.m1.4c">\log_{\frac{d}{d+1}}4^{-db}=\frac{\log 4^{-db}}{\log\frac{d}{d+1}}=\frac{db}{% \log(1+\frac{1}{d})}\leq\mathcal{O}(d^{2}b)\leq\mathcal{O}\left(d^{2}\left(% \log d+\log\varepsilon+\log\frac{1}{1-\lambda}\right)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.4d">roman_log start_POSTSUBSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG end_POSTSUBSCRIPT 4 start_POSTSUPERSCRIPT - italic_d italic_b end_POSTSUPERSCRIPT = divide start_ARG roman_log 4 start_POSTSUPERSCRIPT - italic_d italic_b end_POSTSUPERSCRIPT end_ARG start_ARG roman_log divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG end_ARG = divide start_ARG italic_d italic_b end_ARG start_ARG roman_log ( 1 + divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ) end_ARG ≤ caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_b ) ≤ caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log italic_d + roman_log italic_ε + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.3.p2.9">we therefore conclude that the algorithm must find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS2.3.p2.8.m1.1"><semantics id="S4.SS2.3.p2.8.m1.1a"><mi id="S4.SS2.3.p2.8.m1.1.1" xref="S4.SS2.3.p2.8.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p2.8.m1.1b"><ci id="S4.SS2.3.p2.8.m1.1.1.cmml" xref="S4.SS2.3.p2.8.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p2.8.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.8.m1.1d">italic_ε</annotation></semantics></math>-approximate fixpoint after at most <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d))" class="ltx_Math" 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id="S4.SS2.3.p2.9.m2.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p2.9.m2.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG + roman_log italic_d ) )</annotation></semantics></math> queries to grid points.</p> </div> <div class="ltx_para" id="S4.SS2.4.p3"> <p class="ltx_p" id="S4.SS2.4.p3.4">As in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem2" title="Theorem 4.2. ‣ 4.1 Solving ℓ_𝑝-ContractionFixpoint ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, we can get rid of the <math alttext="\log d" class="ltx_Math" display="inline" id="S4.SS2.4.p3.1.m1.1"><semantics id="S4.SS2.4.p3.1.m1.1a"><mrow id="S4.SS2.4.p3.1.m1.1.1" xref="S4.SS2.4.p3.1.m1.1.1.cmml"><mi id="S4.SS2.4.p3.1.m1.1.1.1" xref="S4.SS2.4.p3.1.m1.1.1.1.cmml">log</mi><mo id="S4.SS2.4.p3.1.m1.1.1a" lspace="0.167em" xref="S4.SS2.4.p3.1.m1.1.1.cmml">⁡</mo><mi id="S4.SS2.4.p3.1.m1.1.1.2" xref="S4.SS2.4.p3.1.m1.1.1.2.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p3.1.m1.1b"><apply id="S4.SS2.4.p3.1.m1.1.1.cmml" xref="S4.SS2.4.p3.1.m1.1.1"><log id="S4.SS2.4.p3.1.m1.1.1.1.cmml" xref="S4.SS2.4.p3.1.m1.1.1.1"></log><ci id="S4.SS2.4.p3.1.m1.1.1.2.cmml" xref="S4.SS2.4.p3.1.m1.1.1.2">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p3.1.m1.1c">\log d</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p3.1.m1.1d">roman_log italic_d</annotation></semantics></math> term in our query bound by observing that if <math alttext="\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d" 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id="S4.SS2.4.p3.2.m2.3c">\max(\frac{1}{\varepsilon},\frac{1}{1-\lambda})<d</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p3.2.m2.3d">roman_max ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG , divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) < italic_d</annotation></semantics></math>, a simple iteration algorithm (this time with rounding) can find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS2.4.p3.3.m3.1"><semantics id="S4.SS2.4.p3.3.m3.1a"><mi id="S4.SS2.4.p3.3.m3.1.1" xref="S4.SS2.4.p3.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p3.3.m3.1b"><ci id="S4.SS2.4.p3.3.m3.1.1.cmml" xref="S4.SS2.4.p3.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p3.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p3.3.m3.1d">italic_ε</annotation></semantics></math>-approximate fixpoint after at most <math 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xref="S4.SS2.4.p3.4.m4.1.1.1.1.1.3.1"></log><ci id="S4.SS2.4.p3.4.m4.1.1.1.1.1.3.2.cmml" xref="S4.SS2.4.p3.4.m4.1.1.1.1.1.3.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p3.4.m4.1c">\mathcal{O}(d\log d)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p3.4.m4.1d">caligraphic_O ( italic_d roman_log italic_d )</annotation></semantics></math> queries. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.1">It remains to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>. For this, we need some additional theory. In particular, we will proceed to give a characterization of containment in an <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><msub id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml"><mi id="S4.SS2.p5.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p5.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.p5.1.m1.1.1.3" xref="S4.SS2.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><apply id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p5.1.m1.1.1.2.cmml" xref="S4.SS2.p5.1.m1.1.1.2">ℓ</ci><cn id="S4.SS2.p5.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace based on tools from convex analysis. Once this characterization is established, it will be quite easy to derive <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>. Note that more details on this characterization can be found in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS1" title="A.1 Fundamentals of ℓ_𝑝-Halfspaces ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.1</span></a>.</p> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.4">The main tool that we need is the notion of subgradients of a convex function <math alttext="f:\mathbb{R}^{d}\rightarrow\mathbb{R}" class="ltx_Math" display="inline" id="S4.SS2.p6.1.m1.1"><semantics id="S4.SS2.p6.1.m1.1a"><mrow id="S4.SS2.p6.1.m1.1.1" xref="S4.SS2.p6.1.m1.1.1.cmml"><mi id="S4.SS2.p6.1.m1.1.1.2" xref="S4.SS2.p6.1.m1.1.1.2.cmml">f</mi><mo id="S4.SS2.p6.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.p6.1.m1.1.1.1.cmml">:</mo><mrow id="S4.SS2.p6.1.m1.1.1.3" xref="S4.SS2.p6.1.m1.1.1.3.cmml"><msup id="S4.SS2.p6.1.m1.1.1.3.2" 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xref="S4.SS2.p6.1.m1.1.1.3.2.2">ℝ</ci><ci id="S4.SS2.p6.1.m1.1.1.3.2.3.cmml" xref="S4.SS2.p6.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S4.SS2.p6.1.m1.1.1.3.3.cmml" xref="S4.SS2.p6.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.1.m1.1c">f:\mathbb{R}^{d}\rightarrow\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math>: a vector <math alttext="u\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS2.p6.2.m2.1"><semantics id="S4.SS2.p6.2.m2.1a"><mrow id="S4.SS2.p6.2.m2.1.1" xref="S4.SS2.p6.2.m2.1.1.cmml"><mi id="S4.SS2.p6.2.m2.1.1.2" xref="S4.SS2.p6.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS2.p6.2.m2.1.1.1" xref="S4.SS2.p6.2.m2.1.1.1.cmml">∈</mo><msup id="S4.SS2.p6.2.m2.1.1.3" xref="S4.SS2.p6.2.m2.1.1.3.cmml"><mi id="S4.SS2.p6.2.m2.1.1.3.2" xref="S4.SS2.p6.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S4.SS2.p6.2.m2.1.1.3.3" xref="S4.SS2.p6.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.2.m2.1b"><apply id="S4.SS2.p6.2.m2.1.1.cmml" xref="S4.SS2.p6.2.m2.1.1"><in id="S4.SS2.p6.2.m2.1.1.1.cmml" xref="S4.SS2.p6.2.m2.1.1.1"></in><ci id="S4.SS2.p6.2.m2.1.1.2.cmml" xref="S4.SS2.p6.2.m2.1.1.2">𝑢</ci><apply id="S4.SS2.p6.2.m2.1.1.3.cmml" xref="S4.SS2.p6.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p6.2.m2.1.1.3.1.cmml" xref="S4.SS2.p6.2.m2.1.1.3">superscript</csymbol><ci id="S4.SS2.p6.2.m2.1.1.3.2.cmml" xref="S4.SS2.p6.2.m2.1.1.3.2">ℝ</ci><ci id="S4.SS2.p6.2.m2.1.1.3.3.cmml" xref="S4.SS2.p6.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.2.m2.1c">u\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.2.m2.1d">italic_u ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is a subgradient of <math alttext="f" class="ltx_Math" display="inline" id="S4.SS2.p6.3.m3.1"><semantics id="S4.SS2.p6.3.m3.1a"><mi id="S4.SS2.p6.3.m3.1.1" xref="S4.SS2.p6.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.3.m3.1b"><ci id="S4.SS2.p6.3.m3.1.1.cmml" xref="S4.SS2.p6.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.3.m3.1d">italic_f</annotation></semantics></math> at <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS2.p6.4.m4.1"><semantics id="S4.SS2.p6.4.m4.1a"><mrow id="S4.SS2.p6.4.m4.1.1" xref="S4.SS2.p6.4.m4.1.1.cmml"><mi id="S4.SS2.p6.4.m4.1.1.2" xref="S4.SS2.p6.4.m4.1.1.2.cmml">x</mi><mo id="S4.SS2.p6.4.m4.1.1.1" xref="S4.SS2.p6.4.m4.1.1.1.cmml">∈</mo><msup id="S4.SS2.p6.4.m4.1.1.3" xref="S4.SS2.p6.4.m4.1.1.3.cmml"><mi id="S4.SS2.p6.4.m4.1.1.3.2" xref="S4.SS2.p6.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S4.SS2.p6.4.m4.1.1.3.3" xref="S4.SS2.p6.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.4.m4.1b"><apply id="S4.SS2.p6.4.m4.1.1.cmml" xref="S4.SS2.p6.4.m4.1.1"><in id="S4.SS2.p6.4.m4.1.1.1.cmml" xref="S4.SS2.p6.4.m4.1.1.1"></in><ci id="S4.SS2.p6.4.m4.1.1.2.cmml" xref="S4.SS2.p6.4.m4.1.1.2">𝑥</ci><apply id="S4.SS2.p6.4.m4.1.1.3.cmml" xref="S4.SS2.p6.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p6.4.m4.1.1.3.1.cmml" xref="S4.SS2.p6.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS2.p6.4.m4.1.1.3.2.cmml" xref="S4.SS2.p6.4.m4.1.1.3.2">ℝ</ci><ci id="S4.SS2.p6.4.m4.1.1.3.3.cmml" xref="S4.SS2.p6.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.4.m4.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.4.m4.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> if and only if</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(x^{\prime})-f(x)\geq\langle u,(x^{\prime}-x)\rangle" class="ltx_Math" display="block" id="S4.Ex16.m1.4"><semantics id="S4.Ex16.m1.4a"><mrow id="S4.Ex16.m1.4.4" xref="S4.Ex16.m1.4.4.cmml"><mrow id="S4.Ex16.m1.3.3.1" xref="S4.Ex16.m1.3.3.1.cmml"><mrow id="S4.Ex16.m1.3.3.1.1" xref="S4.Ex16.m1.3.3.1.1.cmml"><mi id="S4.Ex16.m1.3.3.1.1.3" xref="S4.Ex16.m1.3.3.1.1.3.cmml">f</mi><mo id="S4.Ex16.m1.3.3.1.1.2" xref="S4.Ex16.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.Ex16.m1.3.3.1.1.1.1" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml"><mo id="S4.Ex16.m1.3.3.1.1.1.1.2" stretchy="false" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex16.m1.3.3.1.1.1.1.1" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml"><mi id="S4.Ex16.m1.3.3.1.1.1.1.1.2" xref="S4.Ex16.m1.3.3.1.1.1.1.1.2.cmml">x</mi><mo id="S4.Ex16.m1.3.3.1.1.1.1.1.3" xref="S4.Ex16.m1.3.3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex16.m1.3.3.1.1.1.1.3" stretchy="false" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex16.m1.3.3.1.2" xref="S4.Ex16.m1.3.3.1.2.cmml">−</mo><mrow id="S4.Ex16.m1.3.3.1.3" xref="S4.Ex16.m1.3.3.1.3.cmml"><mi id="S4.Ex16.m1.3.3.1.3.2" xref="S4.Ex16.m1.3.3.1.3.2.cmml">f</mi><mo id="S4.Ex16.m1.3.3.1.3.1" xref="S4.Ex16.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="S4.Ex16.m1.3.3.1.3.3.2" xref="S4.Ex16.m1.3.3.1.3.cmml"><mo id="S4.Ex16.m1.3.3.1.3.3.2.1" stretchy="false" xref="S4.Ex16.m1.3.3.1.3.cmml">(</mo><mi id="S4.Ex16.m1.1.1" xref="S4.Ex16.m1.1.1.cmml">x</mi><mo id="S4.Ex16.m1.3.3.1.3.3.2.2" stretchy="false" xref="S4.Ex16.m1.3.3.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex16.m1.4.4.3" xref="S4.Ex16.m1.4.4.3.cmml">≥</mo><mrow id="S4.Ex16.m1.4.4.2.1" xref="S4.Ex16.m1.4.4.2.2.cmml"><mo id="S4.Ex16.m1.4.4.2.1.2" stretchy="false" xref="S4.Ex16.m1.4.4.2.2.cmml">⟨</mo><mi id="S4.Ex16.m1.2.2" xref="S4.Ex16.m1.2.2.cmml">u</mi><mo id="S4.Ex16.m1.4.4.2.1.3" xref="S4.Ex16.m1.4.4.2.2.cmml">,</mo><mrow id="S4.Ex16.m1.4.4.2.1.1.1" xref="S4.Ex16.m1.4.4.2.1.1.1.1.cmml"><mo id="S4.Ex16.m1.4.4.2.1.1.1.2" stretchy="false" xref="S4.Ex16.m1.4.4.2.1.1.1.1.cmml">(</mo><mrow id="S4.Ex16.m1.4.4.2.1.1.1.1" xref="S4.Ex16.m1.4.4.2.1.1.1.1.cmml"><msup id="S4.Ex16.m1.4.4.2.1.1.1.1.2" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2.cmml"><mi id="S4.Ex16.m1.4.4.2.1.1.1.1.2.2" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2.2.cmml">x</mi><mo id="S4.Ex16.m1.4.4.2.1.1.1.1.2.3" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S4.Ex16.m1.4.4.2.1.1.1.1.1" xref="S4.Ex16.m1.4.4.2.1.1.1.1.1.cmml">−</mo><mi id="S4.Ex16.m1.4.4.2.1.1.1.1.3" xref="S4.Ex16.m1.4.4.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.Ex16.m1.4.4.2.1.1.1.3" stretchy="false" xref="S4.Ex16.m1.4.4.2.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex16.m1.4.4.2.1.4" stretchy="false" xref="S4.Ex16.m1.4.4.2.2.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex16.m1.4b"><apply id="S4.Ex16.m1.4.4.cmml" xref="S4.Ex16.m1.4.4"><geq id="S4.Ex16.m1.4.4.3.cmml" xref="S4.Ex16.m1.4.4.3"></geq><apply id="S4.Ex16.m1.3.3.1.cmml" xref="S4.Ex16.m1.3.3.1"><minus id="S4.Ex16.m1.3.3.1.2.cmml" xref="S4.Ex16.m1.3.3.1.2"></minus><apply id="S4.Ex16.m1.3.3.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1"><times id="S4.Ex16.m1.3.3.1.1.2.cmml" xref="S4.Ex16.m1.3.3.1.1.2"></times><ci id="S4.Ex16.m1.3.3.1.1.3.cmml" xref="S4.Ex16.m1.3.3.1.1.3">𝑓</ci><apply id="S4.Ex16.m1.3.3.1.1.1.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S4.Ex16.m1.3.3.1.1.1.1.1.2.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1.1.2">𝑥</ci><ci id="S4.Ex16.m1.3.3.1.1.1.1.1.3.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.Ex16.m1.3.3.1.3.cmml" xref="S4.Ex16.m1.3.3.1.3"><times id="S4.Ex16.m1.3.3.1.3.1.cmml" xref="S4.Ex16.m1.3.3.1.3.1"></times><ci id="S4.Ex16.m1.3.3.1.3.2.cmml" xref="S4.Ex16.m1.3.3.1.3.2">𝑓</ci><ci id="S4.Ex16.m1.1.1.cmml" xref="S4.Ex16.m1.1.1">𝑥</ci></apply></apply><list id="S4.Ex16.m1.4.4.2.2.cmml" xref="S4.Ex16.m1.4.4.2.1"><ci id="S4.Ex16.m1.2.2.cmml" xref="S4.Ex16.m1.2.2">𝑢</ci><apply id="S4.Ex16.m1.4.4.2.1.1.1.1.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1"><minus id="S4.Ex16.m1.4.4.2.1.1.1.1.1.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.1"></minus><apply id="S4.Ex16.m1.4.4.2.1.1.1.1.2.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex16.m1.4.4.2.1.1.1.1.2.1.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2">superscript</csymbol><ci id="S4.Ex16.m1.4.4.2.1.1.1.1.2.2.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2.2">𝑥</ci><ci id="S4.Ex16.m1.4.4.2.1.1.1.1.2.3.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.2.3">′</ci></apply><ci id="S4.Ex16.m1.4.4.2.1.1.1.1.3.cmml" xref="S4.Ex16.m1.4.4.2.1.1.1.1.3">𝑥</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex16.m1.4c">f(x^{\prime})-f(x)\geq\langle u,(x^{\prime}-x)\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.4d">italic_f ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - italic_f ( italic_x ) ≥ ⟨ italic_u , ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_x ) ⟩</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p6.11">for all <math alttext="x^{\prime}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS2.p6.5.m1.1"><semantics id="S4.SS2.p6.5.m1.1a"><mrow id="S4.SS2.p6.5.m1.1.1" xref="S4.SS2.p6.5.m1.1.1.cmml"><msup id="S4.SS2.p6.5.m1.1.1.2" xref="S4.SS2.p6.5.m1.1.1.2.cmml"><mi id="S4.SS2.p6.5.m1.1.1.2.2" xref="S4.SS2.p6.5.m1.1.1.2.2.cmml">x</mi><mo id="S4.SS2.p6.5.m1.1.1.2.3" xref="S4.SS2.p6.5.m1.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.p6.5.m1.1.1.1" xref="S4.SS2.p6.5.m1.1.1.1.cmml">∈</mo><msup id="S4.SS2.p6.5.m1.1.1.3" xref="S4.SS2.p6.5.m1.1.1.3.cmml"><mi id="S4.SS2.p6.5.m1.1.1.3.2" xref="S4.SS2.p6.5.m1.1.1.3.2.cmml">ℝ</mi><mi id="S4.SS2.p6.5.m1.1.1.3.3" xref="S4.SS2.p6.5.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.5.m1.1b"><apply id="S4.SS2.p6.5.m1.1.1.cmml" xref="S4.SS2.p6.5.m1.1.1"><in id="S4.SS2.p6.5.m1.1.1.1.cmml" xref="S4.SS2.p6.5.m1.1.1.1"></in><apply id="S4.SS2.p6.5.m1.1.1.2.cmml" xref="S4.SS2.p6.5.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p6.5.m1.1.1.2.1.cmml" xref="S4.SS2.p6.5.m1.1.1.2">superscript</csymbol><ci id="S4.SS2.p6.5.m1.1.1.2.2.cmml" xref="S4.SS2.p6.5.m1.1.1.2.2">𝑥</ci><ci id="S4.SS2.p6.5.m1.1.1.2.3.cmml" xref="S4.SS2.p6.5.m1.1.1.2.3">′</ci></apply><apply id="S4.SS2.p6.5.m1.1.1.3.cmml" xref="S4.SS2.p6.5.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p6.5.m1.1.1.3.1.cmml" xref="S4.SS2.p6.5.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.p6.5.m1.1.1.3.2.cmml" xref="S4.SS2.p6.5.m1.1.1.3.2">ℝ</ci><ci id="S4.SS2.p6.5.m1.1.1.3.3.cmml" xref="S4.SS2.p6.5.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.5.m1.1c">x^{\prime}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.5.m1.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. The set <math alttext="\partial f(x)\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS2.p6.6.m2.1"><semantics id="S4.SS2.p6.6.m2.1a"><mrow id="S4.SS2.p6.6.m2.1.2" xref="S4.SS2.p6.6.m2.1.2.cmml"><mrow id="S4.SS2.p6.6.m2.1.2.2" xref="S4.SS2.p6.6.m2.1.2.2.cmml"><mo id="S4.SS2.p6.6.m2.1.2.2.1" rspace="0em" xref="S4.SS2.p6.6.m2.1.2.2.1.cmml">∂</mo><mrow id="S4.SS2.p6.6.m2.1.2.2.2" xref="S4.SS2.p6.6.m2.1.2.2.2.cmml"><mi id="S4.SS2.p6.6.m2.1.2.2.2.2" xref="S4.SS2.p6.6.m2.1.2.2.2.2.cmml">f</mi><mo id="S4.SS2.p6.6.m2.1.2.2.2.1" xref="S4.SS2.p6.6.m2.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p6.6.m2.1.2.2.2.3.2" xref="S4.SS2.p6.6.m2.1.2.2.2.cmml"><mo id="S4.SS2.p6.6.m2.1.2.2.2.3.2.1" stretchy="false" xref="S4.SS2.p6.6.m2.1.2.2.2.cmml">(</mo><mi id="S4.SS2.p6.6.m2.1.1" xref="S4.SS2.p6.6.m2.1.1.cmml">x</mi><mo id="S4.SS2.p6.6.m2.1.2.2.2.3.2.2" stretchy="false" xref="S4.SS2.p6.6.m2.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.p6.6.m2.1.2.1" xref="S4.SS2.p6.6.m2.1.2.1.cmml">⊆</mo><msup id="S4.SS2.p6.6.m2.1.2.3" xref="S4.SS2.p6.6.m2.1.2.3.cmml"><mi id="S4.SS2.p6.6.m2.1.2.3.2" xref="S4.SS2.p6.6.m2.1.2.3.2.cmml">ℝ</mi><mi id="S4.SS2.p6.6.m2.1.2.3.3" xref="S4.SS2.p6.6.m2.1.2.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.6.m2.1b"><apply id="S4.SS2.p6.6.m2.1.2.cmml" xref="S4.SS2.p6.6.m2.1.2"><subset id="S4.SS2.p6.6.m2.1.2.1.cmml" xref="S4.SS2.p6.6.m2.1.2.1"></subset><apply id="S4.SS2.p6.6.m2.1.2.2.cmml" xref="S4.SS2.p6.6.m2.1.2.2"><partialdiff id="S4.SS2.p6.6.m2.1.2.2.1.cmml" xref="S4.SS2.p6.6.m2.1.2.2.1"></partialdiff><apply id="S4.SS2.p6.6.m2.1.2.2.2.cmml" xref="S4.SS2.p6.6.m2.1.2.2.2"><times id="S4.SS2.p6.6.m2.1.2.2.2.1.cmml" xref="S4.SS2.p6.6.m2.1.2.2.2.1"></times><ci id="S4.SS2.p6.6.m2.1.2.2.2.2.cmml" xref="S4.SS2.p6.6.m2.1.2.2.2.2">𝑓</ci><ci id="S4.SS2.p6.6.m2.1.1.cmml" xref="S4.SS2.p6.6.m2.1.1">𝑥</ci></apply></apply><apply id="S4.SS2.p6.6.m2.1.2.3.cmml" xref="S4.SS2.p6.6.m2.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p6.6.m2.1.2.3.1.cmml" xref="S4.SS2.p6.6.m2.1.2.3">superscript</csymbol><ci id="S4.SS2.p6.6.m2.1.2.3.2.cmml" xref="S4.SS2.p6.6.m2.1.2.3.2">ℝ</ci><ci id="S4.SS2.p6.6.m2.1.2.3.3.cmml" xref="S4.SS2.p6.6.m2.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.6.m2.1c">\partial f(x)\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.6.m2.1d">∂ italic_f ( italic_x ) ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> of all subgradients of <math alttext="f" class="ltx_Math" display="inline" id="S4.SS2.p6.7.m3.1"><semantics id="S4.SS2.p6.7.m3.1a"><mi id="S4.SS2.p6.7.m3.1.1" xref="S4.SS2.p6.7.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.7.m3.1b"><ci id="S4.SS2.p6.7.m3.1.1.cmml" xref="S4.SS2.p6.7.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.7.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.7.m3.1d">italic_f</annotation></semantics></math> at <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.p6.8.m4.1"><semantics id="S4.SS2.p6.8.m4.1a"><mi id="S4.SS2.p6.8.m4.1.1" xref="S4.SS2.p6.8.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.8.m4.1b"><ci id="S4.SS2.p6.8.m4.1.1.cmml" xref="S4.SS2.p6.8.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.8.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.8.m4.1d">italic_x</annotation></semantics></math> is also called the subdifferential. If <math alttext="f" class="ltx_Math" display="inline" id="S4.SS2.p6.9.m5.1"><semantics id="S4.SS2.p6.9.m5.1a"><mi id="S4.SS2.p6.9.m5.1.1" xref="S4.SS2.p6.9.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.9.m5.1b"><ci id="S4.SS2.p6.9.m5.1.1.cmml" xref="S4.SS2.p6.9.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.9.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.9.m5.1d">italic_f</annotation></semantics></math> is differentiable at <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.p6.10.m6.1"><semantics id="S4.SS2.p6.10.m6.1a"><mi id="S4.SS2.p6.10.m6.1.1" xref="S4.SS2.p6.10.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.10.m6.1b"><ci id="S4.SS2.p6.10.m6.1.1.cmml" xref="S4.SS2.p6.10.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.10.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.10.m6.1d">italic_x</annotation></semantics></math>, then <math alttext="\partial f(x)=\{\nabla f(x)\}" class="ltx_Math" display="inline" id="S4.SS2.p6.11.m7.3"><semantics id="S4.SS2.p6.11.m7.3a"><mrow id="S4.SS2.p6.11.m7.3.3" xref="S4.SS2.p6.11.m7.3.3.cmml"><mrow id="S4.SS2.p6.11.m7.3.3.3" xref="S4.SS2.p6.11.m7.3.3.3.cmml"><mo id="S4.SS2.p6.11.m7.3.3.3.1" rspace="0em" xref="S4.SS2.p6.11.m7.3.3.3.1.cmml">∂</mo><mrow id="S4.SS2.p6.11.m7.3.3.3.2" xref="S4.SS2.p6.11.m7.3.3.3.2.cmml"><mi id="S4.SS2.p6.11.m7.3.3.3.2.2" xref="S4.SS2.p6.11.m7.3.3.3.2.2.cmml">f</mi><mo id="S4.SS2.p6.11.m7.3.3.3.2.1" xref="S4.SS2.p6.11.m7.3.3.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.p6.11.m7.3.3.3.2.3.2" xref="S4.SS2.p6.11.m7.3.3.3.2.cmml"><mo id="S4.SS2.p6.11.m7.3.3.3.2.3.2.1" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.3.2.cmml">(</mo><mi id="S4.SS2.p6.11.m7.1.1" xref="S4.SS2.p6.11.m7.1.1.cmml">x</mi><mo id="S4.SS2.p6.11.m7.3.3.3.2.3.2.2" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.3.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.p6.11.m7.3.3.2" xref="S4.SS2.p6.11.m7.3.3.2.cmml">=</mo><mrow id="S4.SS2.p6.11.m7.3.3.1.1" xref="S4.SS2.p6.11.m7.3.3.1.2.cmml"><mo id="S4.SS2.p6.11.m7.3.3.1.1.2" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.1.2.cmml">{</mo><mrow id="S4.SS2.p6.11.m7.3.3.1.1.1" xref="S4.SS2.p6.11.m7.3.3.1.1.1.cmml"><mrow id="S4.SS2.p6.11.m7.3.3.1.1.1.2" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2.cmml"><mo id="S4.SS2.p6.11.m7.3.3.1.1.1.2.1" rspace="0.167em" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2.1.cmml">∇</mo><mi id="S4.SS2.p6.11.m7.3.3.1.1.1.2.2" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2.2.cmml">f</mi></mrow><mo id="S4.SS2.p6.11.m7.3.3.1.1.1.1" xref="S4.SS2.p6.11.m7.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p6.11.m7.3.3.1.1.1.3.2" xref="S4.SS2.p6.11.m7.3.3.1.1.1.cmml"><mo id="S4.SS2.p6.11.m7.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.1.1.1.cmml">(</mo><mi id="S4.SS2.p6.11.m7.2.2" xref="S4.SS2.p6.11.m7.2.2.cmml">x</mi><mo id="S4.SS2.p6.11.m7.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p6.11.m7.3.3.1.1.3" stretchy="false" xref="S4.SS2.p6.11.m7.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.11.m7.3b"><apply id="S4.SS2.p6.11.m7.3.3.cmml" xref="S4.SS2.p6.11.m7.3.3"><eq id="S4.SS2.p6.11.m7.3.3.2.cmml" xref="S4.SS2.p6.11.m7.3.3.2"></eq><apply id="S4.SS2.p6.11.m7.3.3.3.cmml" xref="S4.SS2.p6.11.m7.3.3.3"><partialdiff id="S4.SS2.p6.11.m7.3.3.3.1.cmml" xref="S4.SS2.p6.11.m7.3.3.3.1"></partialdiff><apply id="S4.SS2.p6.11.m7.3.3.3.2.cmml" xref="S4.SS2.p6.11.m7.3.3.3.2"><times id="S4.SS2.p6.11.m7.3.3.3.2.1.cmml" xref="S4.SS2.p6.11.m7.3.3.3.2.1"></times><ci id="S4.SS2.p6.11.m7.3.3.3.2.2.cmml" xref="S4.SS2.p6.11.m7.3.3.3.2.2">𝑓</ci><ci id="S4.SS2.p6.11.m7.1.1.cmml" xref="S4.SS2.p6.11.m7.1.1">𝑥</ci></apply></apply><set id="S4.SS2.p6.11.m7.3.3.1.2.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1"><apply id="S4.SS2.p6.11.m7.3.3.1.1.1.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1.1"><times id="S4.SS2.p6.11.m7.3.3.1.1.1.1.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1.1.1"></times><apply id="S4.SS2.p6.11.m7.3.3.1.1.1.2.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2"><ci id="S4.SS2.p6.11.m7.3.3.1.1.1.2.1.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2.1">∇</ci><ci id="S4.SS2.p6.11.m7.3.3.1.1.1.2.2.cmml" xref="S4.SS2.p6.11.m7.3.3.1.1.1.2.2">𝑓</ci></apply><ci id="S4.SS2.p6.11.m7.2.2.cmml" xref="S4.SS2.p6.11.m7.2.2">𝑥</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.11.m7.3c">\partial f(x)=\{\nabla f(x)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.11.m7.3d">∂ italic_f ( italic_x ) = { ∇ italic_f ( italic_x ) }</annotation></semantics></math>. For more details on these concepts and convex analysis in general, we refer to the standard textbook by Rockafellar <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib34" title="">34</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.1">Subgradients are useful for us due to the following characterization of containment for <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS2.p7.1.m1.1"><semantics id="S4.SS2.p7.1.m1.1a"><msub id="S4.SS2.p7.1.m1.1.1" xref="S4.SS2.p7.1.m1.1.1.cmml"><mi id="S4.SS2.p7.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p7.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.SS2.p7.1.m1.1.1.3" xref="S4.SS2.p7.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.1.m1.1b"><apply id="S4.SS2.p7.1.m1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p7.1.m1.1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p7.1.m1.1.1.2.cmml" xref="S4.SS2.p7.1.m1.1.1.2">ℓ</ci><ci id="S4.SS2.p7.1.m1.1.1.3.cmml" xref="S4.SS2.p7.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces. The proof can be found in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS1" title="A.1 Fundamentals of ℓ_𝑝-Halfspaces ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Lemma 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.6.6">For any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.3"><semantics id="S4.Thmtheorem5.p1.1.1.m1.3a"><mrow id="S4.Thmtheorem5.p1.1.1.m1.3.4" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.3.4.2" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.1" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.1.1.m1.3.4.3" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.cmml"><mrow id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.1" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.3.3" mathvariant="normal" xref="S4.Thmtheorem5.p1.1.1.m1.3.3.cmml">∞</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.3b"><apply id="S4.Thmtheorem5.p1.1.1.m1.3.4.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4"><in id="S4.Thmtheorem5.p1.1.1.m1.3.4.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.1"></in><ci id="S4.Thmtheorem5.p1.1.1.m1.3.4.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.2">𝑝</ci><apply id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3"><union id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.2.2"><cn id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval><set id="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.4.3.3.2"><infinity id="S4.Thmtheorem5.p1.1.1.m1.3.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>, a point <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.1"><semantics id="S4.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">z</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S4.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.1b"><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1"><in id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2">𝑧</ci><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is contained in an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.1"><semantics id="S4.Thmtheorem5.p1.3.3.m3.1a"><msub id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.2" mathvariant="normal" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">ℓ</mi><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.1b"><apply id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2">ℓ</ci><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.4.m4.2"><semantics id="S4.Thmtheorem5.p1.4.4.m4.2a"><msubsup id="S4.Thmtheorem5.p1.4.4.m4.2.3" xref="S4.Thmtheorem5.p1.4.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.2" xref="S4.Thmtheorem5.p1.4.4.m4.2.3.2.2.cmml">ℋ</mi><mrow id="S4.Thmtheorem5.p1.4.4.m4.2.2.2.4" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml"><mi id="S4.Thmtheorem5.p1.4.4.m4.1.1.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml">x</mi><mo id="S4.Thmtheorem5.p1.4.4.m4.2.2.2.4.1" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem5.p1.4.4.m4.2.2.2.2" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.3" xref="S4.Thmtheorem5.p1.4.4.m4.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.4.4.m4.2b"><apply id="S4.Thmtheorem5.p1.4.4.m4.2.3.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.4.4.m4.2.3.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3">subscript</csymbol><apply id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3">superscript</csymbol><ci id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3.2.2">ℋ</ci><ci id="S4.Thmtheorem5.p1.4.4.m4.2.3.2.3.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.3.2.3">𝑝</ci></apply><list id="S4.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.2.4"><ci id="S4.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.1.1">𝑥</ci><ci id="S4.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.4.m4.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.4.m4.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> if and only if there exists a subgradient <math alttext="u\in\partial||z-x||_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.5.5.m5.1"><semantics id="S4.Thmtheorem5.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem5.p1.5.5.m5.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem5.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.3.cmml">u</mi><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.2" rspace="0.1389em" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.2.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.cmml"><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.2" lspace="0.1389em" rspace="0em" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.2.cmml">∂</mo><msub id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.cmml"><mrow id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.3" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.5.5.m5.1b"><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1"><in id="S4.Thmtheorem5.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.2"></in><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.3">𝑢</ci><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1"><partialdiff id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.2"></partialdiff><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.2">norm</csymbol><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1"><minus id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.1"></minus><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.2">𝑧</ci><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.5.5.m5.1c">u\in\partial||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.5.5.m5.1d">italic_u ∈ ∂ | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\langle u,v\rangle\geq 0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.6.6.m6.2"><semantics id="S4.Thmtheorem5.p1.6.6.m6.2a"><mrow id="S4.Thmtheorem5.p1.6.6.m6.2.3" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.cmml"><mrow id="S4.Thmtheorem5.p1.6.6.m6.2.3.2.2" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.2.1.cmml"><mo id="S4.Thmtheorem5.p1.6.6.m6.2.3.2.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.2.1.cmml">⟨</mo><mi id="S4.Thmtheorem5.p1.6.6.m6.1.1" xref="S4.Thmtheorem5.p1.6.6.m6.1.1.cmml">u</mi><mo id="S4.Thmtheorem5.p1.6.6.m6.2.3.2.2.2" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.6.6.m6.2.2" xref="S4.Thmtheorem5.p1.6.6.m6.2.2.cmml">v</mi><mo id="S4.Thmtheorem5.p1.6.6.m6.2.3.2.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.2.1.cmml">⟩</mo></mrow><mo id="S4.Thmtheorem5.p1.6.6.m6.2.3.1" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.1.cmml">≥</mo><mn id="S4.Thmtheorem5.p1.6.6.m6.2.3.3" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.6.6.m6.2b"><apply id="S4.Thmtheorem5.p1.6.6.m6.2.3.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.2.3"><geq id="S4.Thmtheorem5.p1.6.6.m6.2.3.1.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.1"></geq><list id="S4.Thmtheorem5.p1.6.6.m6.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.2.2"><ci id="S4.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.1.1">𝑢</ci><ci id="S4.Thmtheorem5.p1.6.6.m6.2.2.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.2.2">𝑣</ci></list><cn id="S4.Thmtheorem5.p1.6.6.m6.2.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.6.6.m6.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.6.6.m6.2c">\langle u,v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.6.6.m6.2d">⟨ italic_u , italic_v ⟩ ≥ 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.5">In this section, we are interested in applying <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.p8.1.m1.1"><semantics id="S4.SS2.p8.1.m1.1a"><msub id="S4.SS2.p8.1.m1.1.1" xref="S4.SS2.p8.1.m1.1.1.cmml"><mi id="S4.SS2.p8.1.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p8.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.p8.1.m1.1.1.3" xref="S4.SS2.p8.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.1.m1.1b"><apply id="S4.SS2.p8.1.m1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.1.m1.1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p8.1.m1.1.1.2.cmml" xref="S4.SS2.p8.1.m1.1.1.2">ℓ</ci><cn id="S4.SS2.p8.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case. Therefore, let us next describe the subgradients of <math alttext="||\cdot||_{1}" class="ltx_math_unparsed" display="inline" id="S4.SS2.p8.2.m2.1"><semantics id="S4.SS2.p8.2.m2.1a"><mrow id="S4.SS2.p8.2.m2.1b"><mo fence="false" id="S4.SS2.p8.2.m2.1.1" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="S4.SS2.p8.2.m2.1.2" stretchy="false">|</mo><mo id="S4.SS2.p8.2.m2.1.3" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.SS2.p8.2.m2.1.4" rspace="0.167em" stretchy="false">|</mo><msub id="S4.SS2.p8.2.m2.1.5"><mo fence="false" id="S4.SS2.p8.2.m2.1.5.2" stretchy="false">|</mo><mn id="S4.SS2.p8.2.m2.1.5.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S4.SS2.p8.2.m2.1c">||\cdot||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.2.m2.1d">| | ⋅ | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Due to <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a> (<math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS2.p8.3.m3.1"><semantics id="S4.SS2.p8.3.m3.1a"><msub id="S4.SS2.p8.3.m3.1.1" xref="S4.SS2.p8.3.m3.1.1.cmml"><mi id="S4.SS2.p8.3.m3.1.1.2" mathvariant="normal" xref="S4.SS2.p8.3.m3.1.1.2.cmml">ℓ</mi><mi id="S4.SS2.p8.3.m3.1.1.3" xref="S4.SS2.p8.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.3.m3.1b"><apply id="S4.SS2.p8.3.m3.1.1.cmml" xref="S4.SS2.p8.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.3.m3.1.1.1.cmml" xref="S4.SS2.p8.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p8.3.m3.1.1.2.cmml" xref="S4.SS2.p8.3.m3.1.1.2">ℓ</ci><ci id="S4.SS2.p8.3.m3.1.1.3.cmml" xref="S4.SS2.p8.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces are unions of rays), it will suffice to characterize the subgradients at points <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.p8.4.m4.1"><semantics id="S4.SS2.p8.4.m4.1a"><mi id="S4.SS2.p8.4.m4.1.1" xref="S4.SS2.p8.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.4.m4.1b"><ci id="S4.SS2.p8.4.m4.1.1.cmml" xref="S4.SS2.p8.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.4.m4.1d">italic_z</annotation></semantics></math> with <math alttext="||z||_{1}=1" class="ltx_Math" display="inline" id="S4.SS2.p8.5.m5.1"><semantics id="S4.SS2.p8.5.m5.1a"><mrow id="S4.SS2.p8.5.m5.1.2" xref="S4.SS2.p8.5.m5.1.2.cmml"><msub id="S4.SS2.p8.5.m5.1.2.2" xref="S4.SS2.p8.5.m5.1.2.2.cmml"><mrow id="S4.SS2.p8.5.m5.1.2.2.2.2" xref="S4.SS2.p8.5.m5.1.2.2.2.1.cmml"><mo id="S4.SS2.p8.5.m5.1.2.2.2.2.1" stretchy="false" xref="S4.SS2.p8.5.m5.1.2.2.2.1.1.cmml">‖</mo><mi id="S4.SS2.p8.5.m5.1.1" xref="S4.SS2.p8.5.m5.1.1.cmml">z</mi><mo id="S4.SS2.p8.5.m5.1.2.2.2.2.2" stretchy="false" xref="S4.SS2.p8.5.m5.1.2.2.2.1.1.cmml">‖</mo></mrow><mn id="S4.SS2.p8.5.m5.1.2.2.3" xref="S4.SS2.p8.5.m5.1.2.2.3.cmml">1</mn></msub><mo id="S4.SS2.p8.5.m5.1.2.1" xref="S4.SS2.p8.5.m5.1.2.1.cmml">=</mo><mn id="S4.SS2.p8.5.m5.1.2.3" xref="S4.SS2.p8.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.5.m5.1b"><apply id="S4.SS2.p8.5.m5.1.2.cmml" xref="S4.SS2.p8.5.m5.1.2"><eq id="S4.SS2.p8.5.m5.1.2.1.cmml" xref="S4.SS2.p8.5.m5.1.2.1"></eq><apply id="S4.SS2.p8.5.m5.1.2.2.cmml" xref="S4.SS2.p8.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p8.5.m5.1.2.2.1.cmml" xref="S4.SS2.p8.5.m5.1.2.2">subscript</csymbol><apply id="S4.SS2.p8.5.m5.1.2.2.2.1.cmml" xref="S4.SS2.p8.5.m5.1.2.2.2.2"><csymbol cd="latexml" id="S4.SS2.p8.5.m5.1.2.2.2.1.1.cmml" xref="S4.SS2.p8.5.m5.1.2.2.2.2.1">norm</csymbol><ci id="S4.SS2.p8.5.m5.1.1.cmml" xref="S4.SS2.p8.5.m5.1.1">𝑧</ci></apply><cn id="S4.SS2.p8.5.m5.1.2.2.3.cmml" type="integer" xref="S4.SS2.p8.5.m5.1.2.2.3">1</cn></apply><cn id="S4.SS2.p8.5.m5.1.2.3.cmml" type="integer" xref="S4.SS2.p8.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.5.m5.1c">||z||_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.5.m5.1d">| | italic_z | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_observation" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Observation 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.5.5">Consider arbitrary <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.1.m1.1"><semantics id="S4.Thmtheorem6.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem6.p1.1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">z</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S4.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.1.m1.1b"><apply id="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1"><in id="S4.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.2">𝑧</ci><apply id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{1}=1" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.1"><semantics id="S4.Thmtheorem6.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.cmml"><msub id="S4.Thmtheorem6.p1.2.2.m2.1.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.1.cmml"><mo id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.1.1.cmml">‖</mo><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml">z</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.1.1.cmml">‖</mo></mrow><mn id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.3.cmml">1</mn></msub><mo id="S4.Thmtheorem6.p1.2.2.m2.1.2.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.1.cmml">=</mo><mn id="S4.Thmtheorem6.p1.2.2.m2.1.2.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.2.m2.1b"><apply id="S4.Thmtheorem6.p1.2.2.m2.1.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2"><eq id="S4.Thmtheorem6.p1.2.2.m2.1.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.1"></eq><apply id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2">subscript</csymbol><apply id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.2"><csymbol cd="latexml" id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.2.2.1">norm</csymbol><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1">𝑧</ci></apply><cn id="S4.Thmtheorem6.p1.2.2.m2.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.2.3">1</cn></apply><cn id="S4.Thmtheorem6.p1.2.2.m2.1.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.1c">||z||_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.1d">| | italic_z | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>. A vector <math alttext="u\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.3.3.m3.1"><semantics id="S4.Thmtheorem6.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem6.p1.3.3.m3.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S4.Thmtheorem6.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.3.3.m3.1b"><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1"><in id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1"></in><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2">𝑢</ci><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.2">ℝ</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.1c">u\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.1d">italic_u ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is a subgradient of <math alttext="||\cdot||_{1}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem6.p1.4.4.m4.1"><semantics id="S4.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem6.p1.4.4.m4.1b"><mo fence="false" id="S4.Thmtheorem6.p1.4.4.m4.1.1" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="S4.Thmtheorem6.p1.4.4.m4.1.2" stretchy="false">|</mo><mo id="S4.Thmtheorem6.p1.4.4.m4.1.3" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.Thmtheorem6.p1.4.4.m4.1.4" rspace="0.167em" stretchy="false">|</mo><msub id="S4.Thmtheorem6.p1.4.4.m4.1.5"><mo fence="false" id="S4.Thmtheorem6.p1.4.4.m4.1.5.2" stretchy="false">|</mo><mn id="S4.Thmtheorem6.p1.4.4.m4.1.5.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.4.4.m4.1c">||\cdot||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.4.4.m4.1d">| | ⋅ | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> at <math alttext="z" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.5.5.m5.1"><semantics id="S4.Thmtheorem6.p1.5.5.m5.1a"><mi id="S4.Thmtheorem6.p1.5.5.m5.1.1" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.5.5.m5.1b"><ci id="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.5.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.5.5.m5.1d">italic_z</annotation></semantics></math> if and only if</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="u_{i}\in\begin{cases}\{1\}&\text{ if }z_{i}>0\\ \{-1\}&\text{ if }z_{i}<0\\ [-1,1]&\text{ if }z_{i}=0\end{cases}" class="ltx_Math" display="block" id="S4.Ex17.m1.6"><semantics id="S4.Ex17.m1.6a"><mrow id="S4.Ex17.m1.6.7" xref="S4.Ex17.m1.6.7.cmml"><msub id="S4.Ex17.m1.6.7.2" xref="S4.Ex17.m1.6.7.2.cmml"><mi id="S4.Ex17.m1.6.7.2.2" xref="S4.Ex17.m1.6.7.2.2.cmml">u</mi><mi id="S4.Ex17.m1.6.7.2.3" xref="S4.Ex17.m1.6.7.2.3.cmml">i</mi></msub><mo id="S4.Ex17.m1.6.7.1" xref="S4.Ex17.m1.6.7.1.cmml">∈</mo><mrow id="S4.Ex17.m1.6.6" xref="S4.Ex17.m1.6.7.3.1.cmml"><mo id="S4.Ex17.m1.6.6.7" xref="S4.Ex17.m1.6.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex17.m1.6.6.6" rowspacing="0pt" xref="S4.Ex17.m1.6.7.3.1.cmml"><mtr id="S4.Ex17.m1.6.6.6a" xref="S4.Ex17.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6b" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.1.1.1.1.1.1.3" xref="S4.Ex17.m1.1.1.1.1.1.1.2.cmml"><mo id="S4.Ex17.m1.1.1.1.1.1.1.3.1" stretchy="false" xref="S4.Ex17.m1.1.1.1.1.1.1.2.cmml">{</mo><mn id="S4.Ex17.m1.1.1.1.1.1.1.1" xref="S4.Ex17.m1.1.1.1.1.1.1.1.cmml">1</mn><mo id="S4.Ex17.m1.1.1.1.1.1.1.3.2" stretchy="false" xref="S4.Ex17.m1.1.1.1.1.1.1.2.cmml">}</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6c" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.2.2.2.2.2.1" xref="S4.Ex17.m1.2.2.2.2.2.1.cmml"><mrow id="S4.Ex17.m1.2.2.2.2.2.1.2" xref="S4.Ex17.m1.2.2.2.2.2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.2.2.2.2.2.1.2.2" xref="S4.Ex17.m1.2.2.2.2.2.1.2.2a.cmml"> if </mtext><mo id="S4.Ex17.m1.2.2.2.2.2.1.2.1" xref="S4.Ex17.m1.2.2.2.2.2.1.2.1.cmml">⁢</mo><msub id="S4.Ex17.m1.2.2.2.2.2.1.2.3" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3.cmml"><mi id="S4.Ex17.m1.2.2.2.2.2.1.2.3.2" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3.2.cmml">z</mi><mi id="S4.Ex17.m1.2.2.2.2.2.1.2.3.3" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S4.Ex17.m1.2.2.2.2.2.1.1" xref="S4.Ex17.m1.2.2.2.2.2.1.1.cmml">></mo><mn id="S4.Ex17.m1.2.2.2.2.2.1.3" xref="S4.Ex17.m1.2.2.2.2.2.1.3.cmml">0</mn></mrow></mtd></mtr><mtr id="S4.Ex17.m1.6.6.6d" xref="S4.Ex17.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6e" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.3.3.3.3.1.1.1" xref="S4.Ex17.m1.3.3.3.3.1.1.2.cmml"><mo id="S4.Ex17.m1.3.3.3.3.1.1.1.2" stretchy="false" xref="S4.Ex17.m1.3.3.3.3.1.1.2.cmml">{</mo><mrow id="S4.Ex17.m1.3.3.3.3.1.1.1.1" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1.cmml"><mo id="S4.Ex17.m1.3.3.3.3.1.1.1.1a" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1.cmml">−</mo><mn id="S4.Ex17.m1.3.3.3.3.1.1.1.1.2" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Ex17.m1.3.3.3.3.1.1.1.3" stretchy="false" xref="S4.Ex17.m1.3.3.3.3.1.1.2.cmml">}</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6f" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.4.4.4.4.2.1" xref="S4.Ex17.m1.4.4.4.4.2.1.cmml"><mrow id="S4.Ex17.m1.4.4.4.4.2.1.2" xref="S4.Ex17.m1.4.4.4.4.2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.4.4.4.4.2.1.2.2" xref="S4.Ex17.m1.4.4.4.4.2.1.2.2a.cmml"> if </mtext><mo id="S4.Ex17.m1.4.4.4.4.2.1.2.1" xref="S4.Ex17.m1.4.4.4.4.2.1.2.1.cmml">⁢</mo><msub id="S4.Ex17.m1.4.4.4.4.2.1.2.3" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3.cmml"><mi id="S4.Ex17.m1.4.4.4.4.2.1.2.3.2" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3.2.cmml">z</mi><mi id="S4.Ex17.m1.4.4.4.4.2.1.2.3.3" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S4.Ex17.m1.4.4.4.4.2.1.1" xref="S4.Ex17.m1.4.4.4.4.2.1.1.cmml"><</mo><mn id="S4.Ex17.m1.4.4.4.4.2.1.3" xref="S4.Ex17.m1.4.4.4.4.2.1.3.cmml">0</mn></mrow></mtd></mtr><mtr id="S4.Ex17.m1.6.6.6g" xref="S4.Ex17.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6h" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.5.5.5.5.1.1.2" xref="S4.Ex17.m1.5.5.5.5.1.1.3.cmml"><mo id="S4.Ex17.m1.5.5.5.5.1.1.2.2" stretchy="false" xref="S4.Ex17.m1.5.5.5.5.1.1.3.cmml">[</mo><mrow id="S4.Ex17.m1.5.5.5.5.1.1.2.1" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1.cmml"><mo id="S4.Ex17.m1.5.5.5.5.1.1.2.1a" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1.cmml">−</mo><mn id="S4.Ex17.m1.5.5.5.5.1.1.2.1.2" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1.2.cmml">1</mn></mrow><mo id="S4.Ex17.m1.5.5.5.5.1.1.2.3" xref="S4.Ex17.m1.5.5.5.5.1.1.3.cmml">,</mo><mn id="S4.Ex17.m1.5.5.5.5.1.1.1" xref="S4.Ex17.m1.5.5.5.5.1.1.1.cmml">1</mn><mo id="S4.Ex17.m1.5.5.5.5.1.1.2.4" stretchy="false" xref="S4.Ex17.m1.5.5.5.5.1.1.3.cmml">]</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex17.m1.6.6.6i" xref="S4.Ex17.m1.6.7.3.1.cmml"><mrow id="S4.Ex17.m1.6.6.6.6.2.1" xref="S4.Ex17.m1.6.6.6.6.2.1.cmml"><mrow id="S4.Ex17.m1.6.6.6.6.2.1.2" xref="S4.Ex17.m1.6.6.6.6.2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.6.6.6.6.2.1.2.2" xref="S4.Ex17.m1.6.6.6.6.2.1.2.2a.cmml"> if </mtext><mo id="S4.Ex17.m1.6.6.6.6.2.1.2.1" xref="S4.Ex17.m1.6.6.6.6.2.1.2.1.cmml">⁢</mo><msub id="S4.Ex17.m1.6.6.6.6.2.1.2.3" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3.cmml"><mi id="S4.Ex17.m1.6.6.6.6.2.1.2.3.2" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3.2.cmml">z</mi><mi id="S4.Ex17.m1.6.6.6.6.2.1.2.3.3" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S4.Ex17.m1.6.6.6.6.2.1.1" xref="S4.Ex17.m1.6.6.6.6.2.1.1.cmml">=</mo><mn id="S4.Ex17.m1.6.6.6.6.2.1.3" xref="S4.Ex17.m1.6.6.6.6.2.1.3.cmml">0</mn></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex17.m1.6b"><apply id="S4.Ex17.m1.6.7.cmml" xref="S4.Ex17.m1.6.7"><in id="S4.Ex17.m1.6.7.1.cmml" xref="S4.Ex17.m1.6.7.1"></in><apply id="S4.Ex17.m1.6.7.2.cmml" xref="S4.Ex17.m1.6.7.2"><csymbol cd="ambiguous" id="S4.Ex17.m1.6.7.2.1.cmml" xref="S4.Ex17.m1.6.7.2">subscript</csymbol><ci id="S4.Ex17.m1.6.7.2.2.cmml" xref="S4.Ex17.m1.6.7.2.2">𝑢</ci><ci id="S4.Ex17.m1.6.7.2.3.cmml" xref="S4.Ex17.m1.6.7.2.3">𝑖</ci></apply><apply id="S4.Ex17.m1.6.7.3.1.cmml" xref="S4.Ex17.m1.6.6"><csymbol cd="latexml" id="S4.Ex17.m1.6.7.3.1.1.cmml" xref="S4.Ex17.m1.6.6.7">cases</csymbol><set id="S4.Ex17.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex17.m1.1.1.1.1.1.1.3"><cn id="S4.Ex17.m1.1.1.1.1.1.1.1.cmml" type="integer" xref="S4.Ex17.m1.1.1.1.1.1.1.1">1</cn></set><apply id="S4.Ex17.m1.2.2.2.2.2.1.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1"><gt id="S4.Ex17.m1.2.2.2.2.2.1.1.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.1"></gt><apply id="S4.Ex17.m1.2.2.2.2.2.1.2.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2"><times id="S4.Ex17.m1.2.2.2.2.2.1.2.1.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.1"></times><ci id="S4.Ex17.m1.2.2.2.2.2.1.2.2a.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.2.2.2.2.2.1.2.2.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.2"> if </mtext></ci><apply id="S4.Ex17.m1.2.2.2.2.2.1.2.3.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3"><csymbol cd="ambiguous" id="S4.Ex17.m1.2.2.2.2.2.1.2.3.1.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3">subscript</csymbol><ci id="S4.Ex17.m1.2.2.2.2.2.1.2.3.2.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3.2">𝑧</ci><ci id="S4.Ex17.m1.2.2.2.2.2.1.2.3.3.cmml" xref="S4.Ex17.m1.2.2.2.2.2.1.2.3.3">𝑖</ci></apply></apply><cn id="S4.Ex17.m1.2.2.2.2.2.1.3.cmml" type="integer" xref="S4.Ex17.m1.2.2.2.2.2.1.3">0</cn></apply><set id="S4.Ex17.m1.3.3.3.3.1.1.2.cmml" xref="S4.Ex17.m1.3.3.3.3.1.1.1"><apply id="S4.Ex17.m1.3.3.3.3.1.1.1.1.cmml" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1"><minus id="S4.Ex17.m1.3.3.3.3.1.1.1.1.1.cmml" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1"></minus><cn id="S4.Ex17.m1.3.3.3.3.1.1.1.1.2.cmml" type="integer" xref="S4.Ex17.m1.3.3.3.3.1.1.1.1.2">1</cn></apply></set><apply id="S4.Ex17.m1.4.4.4.4.2.1.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1"><lt id="S4.Ex17.m1.4.4.4.4.2.1.1.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.1"></lt><apply id="S4.Ex17.m1.4.4.4.4.2.1.2.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2"><times id="S4.Ex17.m1.4.4.4.4.2.1.2.1.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.1"></times><ci id="S4.Ex17.m1.4.4.4.4.2.1.2.2a.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.4.4.4.4.2.1.2.2.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.2"> if </mtext></ci><apply id="S4.Ex17.m1.4.4.4.4.2.1.2.3.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3"><csymbol cd="ambiguous" id="S4.Ex17.m1.4.4.4.4.2.1.2.3.1.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3">subscript</csymbol><ci id="S4.Ex17.m1.4.4.4.4.2.1.2.3.2.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3.2">𝑧</ci><ci id="S4.Ex17.m1.4.4.4.4.2.1.2.3.3.cmml" xref="S4.Ex17.m1.4.4.4.4.2.1.2.3.3">𝑖</ci></apply></apply><cn id="S4.Ex17.m1.4.4.4.4.2.1.3.cmml" type="integer" xref="S4.Ex17.m1.4.4.4.4.2.1.3">0</cn></apply><interval closure="closed" id="S4.Ex17.m1.5.5.5.5.1.1.3.cmml" xref="S4.Ex17.m1.5.5.5.5.1.1.2"><apply id="S4.Ex17.m1.5.5.5.5.1.1.2.1.cmml" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1"><minus id="S4.Ex17.m1.5.5.5.5.1.1.2.1.1.cmml" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1"></minus><cn id="S4.Ex17.m1.5.5.5.5.1.1.2.1.2.cmml" type="integer" xref="S4.Ex17.m1.5.5.5.5.1.1.2.1.2">1</cn></apply><cn id="S4.Ex17.m1.5.5.5.5.1.1.1.cmml" type="integer" xref="S4.Ex17.m1.5.5.5.5.1.1.1">1</cn></interval><apply id="S4.Ex17.m1.6.6.6.6.2.1.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1"><eq id="S4.Ex17.m1.6.6.6.6.2.1.1.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.1"></eq><apply id="S4.Ex17.m1.6.6.6.6.2.1.2.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2"><times id="S4.Ex17.m1.6.6.6.6.2.1.2.1.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.1"></times><ci id="S4.Ex17.m1.6.6.6.6.2.1.2.2a.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex17.m1.6.6.6.6.2.1.2.2.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.2"> if </mtext></ci><apply id="S4.Ex17.m1.6.6.6.6.2.1.2.3.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3"><csymbol cd="ambiguous" id="S4.Ex17.m1.6.6.6.6.2.1.2.3.1.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3">subscript</csymbol><ci id="S4.Ex17.m1.6.6.6.6.2.1.2.3.2.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3.2">𝑧</ci><ci id="S4.Ex17.m1.6.6.6.6.2.1.2.3.3.cmml" xref="S4.Ex17.m1.6.6.6.6.2.1.2.3.3">𝑖</ci></apply></apply><cn id="S4.Ex17.m1.6.6.6.6.2.1.3.cmml" type="integer" xref="S4.Ex17.m1.6.6.6.6.2.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex17.m1.6c">u_{i}\in\begin{cases}\{1\}&\text{ if }z_{i}>0\\ \{-1\}&\text{ if }z_{i}<0\\ [-1,1]&\text{ if }z_{i}=0\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex17.m1.6d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { start_ROW start_CELL { 1 } end_CELL start_CELL if italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 end_CELL end_ROW start_ROW start_CELL { - 1 } end_CELL start_CELL if italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0 end_CELL end_ROW start_ROW start_CELL [ - 1 , 1 ] end_CELL start_CELL if italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem6.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.8.3">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.6.1.m1.1"><semantics id="S4.Thmtheorem6.p1.6.1.m1.1a"><mrow id="S4.Thmtheorem6.p1.6.1.m1.1.2" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.cmml"><mi id="S4.Thmtheorem6.p1.6.1.m1.1.2.2" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.2.cmml">i</mi><mo id="S4.Thmtheorem6.p1.6.1.m1.1.2.1" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.Thmtheorem6.p1.6.1.m1.1.2.3.2" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.3.1.cmml"><mo id="S4.Thmtheorem6.p1.6.1.m1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.3.1.1.cmml">[</mo><mi id="S4.Thmtheorem6.p1.6.1.m1.1.1" xref="S4.Thmtheorem6.p1.6.1.m1.1.1.cmml">d</mi><mo id="S4.Thmtheorem6.p1.6.1.m1.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.6.1.m1.1b"><apply id="S4.Thmtheorem6.p1.6.1.m1.1.2.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.2"><in id="S4.Thmtheorem6.p1.6.1.m1.1.2.1.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.1"></in><ci id="S4.Thmtheorem6.p1.6.1.m1.1.2.2.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.2">𝑖</ci><apply id="S4.Thmtheorem6.p1.6.1.m1.1.2.3.1.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.3.2"><csymbol cd="latexml" id="S4.Thmtheorem6.p1.6.1.m1.1.2.3.1.1.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.Thmtheorem6.p1.6.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.6.1.m1.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.6.1.m1.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.6.1.m1.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. In particular, we have <math alttext="\langle u,z\rangle=||u||_{\infty}=||z||_{1}=1" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.7.2.m2.4"><semantics id="S4.Thmtheorem6.p1.7.2.m2.4a"><mrow id="S4.Thmtheorem6.p1.7.2.m2.4.5" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.cmml"><mrow id="S4.Thmtheorem6.p1.7.2.m2.4.5.2.2" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.2.1.cmml"><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.2.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.2.1.cmml">⟨</mo><mi id="S4.Thmtheorem6.p1.7.2.m2.1.1" xref="S4.Thmtheorem6.p1.7.2.m2.1.1.cmml">u</mi><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.2.2.2" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.2.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.7.2.m2.2.2" xref="S4.Thmtheorem6.p1.7.2.m2.2.2.cmml">z</mi><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.2.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.2.1.cmml">⟩</mo></mrow><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.3" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.3.cmml">=</mo><msub id="S4.Thmtheorem6.p1.7.2.m2.4.5.4" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.cmml"><mrow id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.2" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.1.cmml"><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.1.1.cmml">‖</mo><mi id="S4.Thmtheorem6.p1.7.2.m2.3.3" xref="S4.Thmtheorem6.p1.7.2.m2.3.3.cmml">u</mi><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.1.1.cmml">‖</mo></mrow><mi id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.3" mathvariant="normal" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.3.cmml">∞</mi></msub><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.5" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.5.cmml">=</mo><msub id="S4.Thmtheorem6.p1.7.2.m2.4.5.6" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.cmml"><mrow id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.2" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.1.cmml"><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.1.1.cmml">‖</mo><mi id="S4.Thmtheorem6.p1.7.2.m2.4.4" xref="S4.Thmtheorem6.p1.7.2.m2.4.4.cmml">z</mi><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.1.1.cmml">‖</mo></mrow><mn id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.3" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.3.cmml">1</mn></msub><mo id="S4.Thmtheorem6.p1.7.2.m2.4.5.7" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.7.cmml">=</mo><mn id="S4.Thmtheorem6.p1.7.2.m2.4.5.8" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.8.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.7.2.m2.4b"><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"><and id="S4.Thmtheorem6.p1.7.2.m2.4.5a.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"></and><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5b.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"><eq id="S4.Thmtheorem6.p1.7.2.m2.4.5.3.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.3"></eq><list id="S4.Thmtheorem6.p1.7.2.m2.4.5.2.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.2.2"><ci id="S4.Thmtheorem6.p1.7.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.1.1">𝑢</ci><ci id="S4.Thmtheorem6.p1.7.2.m2.2.2.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.2.2">𝑧</ci></list><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4">subscript</csymbol><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.2"><csymbol cd="latexml" id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.1.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.2.2.1">norm</csymbol><ci id="S4.Thmtheorem6.p1.7.2.m2.3.3.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.3.3">𝑢</ci></apply><infinity id="S4.Thmtheorem6.p1.7.2.m2.4.5.4.3.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.4.3"></infinity></apply></apply><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5c.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"><eq id="S4.Thmtheorem6.p1.7.2.m2.4.5.5.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.5"></eq><share href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem6.p1.7.2.m2.4.5.4.cmml" id="S4.Thmtheorem6.p1.7.2.m2.4.5d.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"></share><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6">subscript</csymbol><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.2"><csymbol cd="latexml" id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.1.1.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.2.2.1">norm</csymbol><ci id="S4.Thmtheorem6.p1.7.2.m2.4.4.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.4">𝑧</ci></apply><cn id="S4.Thmtheorem6.p1.7.2.m2.4.5.6.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.6.3">1</cn></apply></apply><apply id="S4.Thmtheorem6.p1.7.2.m2.4.5e.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"><eq id="S4.Thmtheorem6.p1.7.2.m2.4.5.7.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.7"></eq><share href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem6.p1.7.2.m2.4.5.6.cmml" id="S4.Thmtheorem6.p1.7.2.m2.4.5f.cmml" xref="S4.Thmtheorem6.p1.7.2.m2.4.5"></share><cn id="S4.Thmtheorem6.p1.7.2.m2.4.5.8.cmml" type="integer" xref="S4.Thmtheorem6.p1.7.2.m2.4.5.8">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.7.2.m2.4c">\langle u,z\rangle=||u||_{\infty}=||z||_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.7.2.m2.4d">⟨ italic_u , italic_z ⟩ = | | italic_u | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = | | italic_z | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> for all <math alttext="u\in\partial||z||_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.8.3.m3.1"><semantics id="S4.Thmtheorem6.p1.8.3.m3.1a"><mrow id="S4.Thmtheorem6.p1.8.3.m3.1.2" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.cmml"><mi id="S4.Thmtheorem6.p1.8.3.m3.1.2.2" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.2.cmml">u</mi><mo id="S4.Thmtheorem6.p1.8.3.m3.1.2.1" rspace="0.1389em" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.1.cmml">∈</mo><mrow id="S4.Thmtheorem6.p1.8.3.m3.1.2.3" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.cmml"><mo id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.1" lspace="0.1389em" rspace="0em" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.1.cmml">∂</mo><msub id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.cmml"><mrow id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.2" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.1.cmml"><mo id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.1.1.cmml">‖</mo><mi id="S4.Thmtheorem6.p1.8.3.m3.1.1" xref="S4.Thmtheorem6.p1.8.3.m3.1.1.cmml">z</mi><mo id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.1.1.cmml">‖</mo></mrow><mn id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.3" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.8.3.m3.1b"><apply id="S4.Thmtheorem6.p1.8.3.m3.1.2.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2"><in id="S4.Thmtheorem6.p1.8.3.m3.1.2.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.1"></in><ci id="S4.Thmtheorem6.p1.8.3.m3.1.2.2.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.2">𝑢</ci><apply id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3"><partialdiff id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.1"></partialdiff><apply id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2">subscript</csymbol><apply id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.2"><csymbol cd="latexml" id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.1.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.2.2.1">norm</csymbol><ci id="S4.Thmtheorem6.p1.8.3.m3.1.1.cmml" xref="S4.Thmtheorem6.p1.8.3.m3.1.1">𝑧</ci></apply><cn id="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.8.3.m3.1.2.3.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.8.3.m3.1c">u\in\partial||z||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.8.3.m3.1d">italic_u ∈ ∂ | | italic_z | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p9"> <p class="ltx_p" id="S4.SS2.p9.1">Equipped with <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem6" title="Observation 4.6. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.6</span></a>, we are now ready to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>.</p> </div> <div class="ltx_proof" id="S4.SS2.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>.</h6> <div class="ltx_para" id="S4.SS2.5.p1"> <p class="ltx_p" id="S4.SS2.5.p1.11">Let <math alttext="P\subseteq G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.1.m1.1"><semantics id="S4.SS2.5.p1.1.m1.1a"><mrow id="S4.SS2.5.p1.1.m1.1.1" xref="S4.SS2.5.p1.1.m1.1.1.cmml"><mi id="S4.SS2.5.p1.1.m1.1.1.2" xref="S4.SS2.5.p1.1.m1.1.1.2.cmml">P</mi><mo id="S4.SS2.5.p1.1.m1.1.1.1" xref="S4.SS2.5.p1.1.m1.1.1.1.cmml">⊆</mo><msubsup id="S4.SS2.5.p1.1.m1.1.1.3" xref="S4.SS2.5.p1.1.m1.1.1.3.cmml"><mi id="S4.SS2.5.p1.1.m1.1.1.3.2.2" xref="S4.SS2.5.p1.1.m1.1.1.3.2.2.cmml">G</mi><mi id="S4.SS2.5.p1.1.m1.1.1.3.3" xref="S4.SS2.5.p1.1.m1.1.1.3.3.cmml">b</mi><mi id="S4.SS2.5.p1.1.m1.1.1.3.2.3" xref="S4.SS2.5.p1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.1.m1.1b"><apply id="S4.SS2.5.p1.1.m1.1.1.cmml" xref="S4.SS2.5.p1.1.m1.1.1"><subset id="S4.SS2.5.p1.1.m1.1.1.1.cmml" xref="S4.SS2.5.p1.1.m1.1.1.1"></subset><ci id="S4.SS2.5.p1.1.m1.1.1.2.cmml" xref="S4.SS2.5.p1.1.m1.1.1.2">𝑃</ci><apply id="S4.SS2.5.p1.1.m1.1.1.3.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.5.p1.1.m1.1.1.3.1.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3">subscript</csymbol><apply id="S4.SS2.5.p1.1.m1.1.1.3.2.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.5.p1.1.m1.1.1.3.2.1.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.5.p1.1.m1.1.1.3.2.2.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3.2.2">𝐺</ci><ci id="S4.SS2.5.p1.1.m1.1.1.3.2.3.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S4.SS2.5.p1.1.m1.1.1.3.3.cmml" xref="S4.SS2.5.p1.1.m1.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.1.m1.1c">P\subseteq G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.1.m1.1d">italic_P ⊆ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> be an arbitrary subset of the grid points, as given in the lemma. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a>, there exists a discrete <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.2.m2.1"><semantics id="S4.SS2.5.p1.2.m2.1a"><msub id="S4.SS2.5.p1.2.m2.1.1" xref="S4.SS2.5.p1.2.m2.1.1.cmml"><mi id="S4.SS2.5.p1.2.m2.1.1.2" mathvariant="normal" xref="S4.SS2.5.p1.2.m2.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.5.p1.2.m2.1.1.3" xref="S4.SS2.5.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.2.m2.1b"><apply id="S4.SS2.5.p1.2.m2.1.1.cmml" xref="S4.SS2.5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.2.m2.1.1.1.cmml" xref="S4.SS2.5.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.5.p1.2.m2.1.1.2.cmml" xref="S4.SS2.5.p1.2.m2.1.1.2">ℓ</ci><cn id="S4.SS2.5.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.5.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.5.p1.3.m3.1"><semantics id="S4.SS2.5.p1.3.m3.1a"><mi id="S4.SS2.5.p1.3.m3.1.1" xref="S4.SS2.5.p1.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.3.m3.1b"><ci id="S4.SS2.5.p1.3.m3.1.1.cmml" xref="S4.SS2.5.p1.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.3.m3.1d">italic_c</annotation></semantics></math> of <math alttext="P" class="ltx_Math" display="inline" id="S4.SS2.5.p1.4.m4.1"><semantics id="S4.SS2.5.p1.4.m4.1a"><mi id="S4.SS2.5.p1.4.m4.1.1" xref="S4.SS2.5.p1.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.4.m4.1b"><ci id="S4.SS2.5.p1.4.m4.1.1.cmml" xref="S4.SS2.5.p1.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.4.m4.1d">italic_P</annotation></semantics></math>. Moreover, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem19" title="Lemma 3.19. ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.19</span></a>, <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.5.p1.5.m5.1"><semantics id="S4.SS2.5.p1.5.m5.1a"><mi id="S4.SS2.5.p1.5.m5.1.1" xref="S4.SS2.5.p1.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.5.m5.1b"><ci id="S4.SS2.5.p1.5.m5.1.1.cmml" xref="S4.SS2.5.p1.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.5.m5.1d">italic_c</annotation></semantics></math> is contained in the convex hull <math alttext="\text{conv}(G^{d}_{b})=[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.6.m6.3"><semantics id="S4.SS2.5.p1.6.m6.3a"><mrow id="S4.SS2.5.p1.6.m6.3.3" xref="S4.SS2.5.p1.6.m6.3.3.cmml"><mrow id="S4.SS2.5.p1.6.m6.3.3.1" xref="S4.SS2.5.p1.6.m6.3.3.1.cmml"><mtext id="S4.SS2.5.p1.6.m6.3.3.1.3" xref="S4.SS2.5.p1.6.m6.3.3.1.3a.cmml">conv</mtext><mo id="S4.SS2.5.p1.6.m6.3.3.1.2" xref="S4.SS2.5.p1.6.m6.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.5.p1.6.m6.3.3.1.1.1" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.cmml"><mo id="S4.SS2.5.p1.6.m6.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.cmml">(</mo><msubsup id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.cmml"><mi id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.2" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.2.cmml">G</mi><mi id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.3" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.3.cmml">b</mi><mi id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.3" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.5.p1.6.m6.3.3.1.1.1.3" stretchy="false" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.5.p1.6.m6.3.3.2" xref="S4.SS2.5.p1.6.m6.3.3.2.cmml">=</mo><msup id="S4.SS2.5.p1.6.m6.3.3.3" xref="S4.SS2.5.p1.6.m6.3.3.3.cmml"><mrow id="S4.SS2.5.p1.6.m6.3.3.3.2.2" xref="S4.SS2.5.p1.6.m6.3.3.3.2.1.cmml"><mo id="S4.SS2.5.p1.6.m6.3.3.3.2.2.1" stretchy="false" xref="S4.SS2.5.p1.6.m6.3.3.3.2.1.cmml">[</mo><mn id="S4.SS2.5.p1.6.m6.1.1" xref="S4.SS2.5.p1.6.m6.1.1.cmml">0</mn><mo id="S4.SS2.5.p1.6.m6.3.3.3.2.2.2" xref="S4.SS2.5.p1.6.m6.3.3.3.2.1.cmml">,</mo><mn id="S4.SS2.5.p1.6.m6.2.2" xref="S4.SS2.5.p1.6.m6.2.2.cmml">1</mn><mo id="S4.SS2.5.p1.6.m6.3.3.3.2.2.3" stretchy="false" xref="S4.SS2.5.p1.6.m6.3.3.3.2.1.cmml">]</mo></mrow><mi id="S4.SS2.5.p1.6.m6.3.3.3.3" xref="S4.SS2.5.p1.6.m6.3.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.6.m6.3b"><apply id="S4.SS2.5.p1.6.m6.3.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3"><eq id="S4.SS2.5.p1.6.m6.3.3.2.cmml" xref="S4.SS2.5.p1.6.m6.3.3.2"></eq><apply id="S4.SS2.5.p1.6.m6.3.3.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1"><times id="S4.SS2.5.p1.6.m6.3.3.1.2.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.2"></times><ci id="S4.SS2.5.p1.6.m6.3.3.1.3a.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.3"><mtext id="S4.SS2.5.p1.6.m6.3.3.1.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.3">conv</mtext></ci><apply id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1">subscript</csymbol><apply id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1">superscript</csymbol><ci id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.2.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.2">𝐺</ci><ci id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.2.3">𝑑</ci></apply><ci id="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.SS2.5.p1.6.m6.3.3.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.5.p1.6.m6.3.3.3.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.3">superscript</csymbol><interval closure="closed" id="S4.SS2.5.p1.6.m6.3.3.3.2.1.cmml" xref="S4.SS2.5.p1.6.m6.3.3.3.2.2"><cn id="S4.SS2.5.p1.6.m6.1.1.cmml" type="integer" xref="S4.SS2.5.p1.6.m6.1.1">0</cn><cn id="S4.SS2.5.p1.6.m6.2.2.cmml" type="integer" xref="S4.SS2.5.p1.6.m6.2.2">1</cn></interval><ci id="S4.SS2.5.p1.6.m6.3.3.3.3.cmml" xref="S4.SS2.5.p1.6.m6.3.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.6.m6.3c">\text{conv}(G^{d}_{b})=[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.6.m6.3d">conv ( italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) = [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> of the grid. Let <math alttext="c^{\prime}\in G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.7.m7.1"><semantics id="S4.SS2.5.p1.7.m7.1a"><mrow id="S4.SS2.5.p1.7.m7.1.1" xref="S4.SS2.5.p1.7.m7.1.1.cmml"><msup id="S4.SS2.5.p1.7.m7.1.1.2" xref="S4.SS2.5.p1.7.m7.1.1.2.cmml"><mi id="S4.SS2.5.p1.7.m7.1.1.2.2" xref="S4.SS2.5.p1.7.m7.1.1.2.2.cmml">c</mi><mo id="S4.SS2.5.p1.7.m7.1.1.2.3" xref="S4.SS2.5.p1.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.5.p1.7.m7.1.1.1" xref="S4.SS2.5.p1.7.m7.1.1.1.cmml">∈</mo><msubsup id="S4.SS2.5.p1.7.m7.1.1.3" xref="S4.SS2.5.p1.7.m7.1.1.3.cmml"><mi id="S4.SS2.5.p1.7.m7.1.1.3.2.2" xref="S4.SS2.5.p1.7.m7.1.1.3.2.2.cmml">G</mi><mi id="S4.SS2.5.p1.7.m7.1.1.3.3" xref="S4.SS2.5.p1.7.m7.1.1.3.3.cmml">b</mi><mi id="S4.SS2.5.p1.7.m7.1.1.3.2.3" xref="S4.SS2.5.p1.7.m7.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.7.m7.1b"><apply id="S4.SS2.5.p1.7.m7.1.1.cmml" xref="S4.SS2.5.p1.7.m7.1.1"><in id="S4.SS2.5.p1.7.m7.1.1.1.cmml" xref="S4.SS2.5.p1.7.m7.1.1.1"></in><apply id="S4.SS2.5.p1.7.m7.1.1.2.cmml" xref="S4.SS2.5.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.5.p1.7.m7.1.1.2.1.cmml" xref="S4.SS2.5.p1.7.m7.1.1.2">superscript</csymbol><ci id="S4.SS2.5.p1.7.m7.1.1.2.2.cmml" xref="S4.SS2.5.p1.7.m7.1.1.2.2">𝑐</ci><ci id="S4.SS2.5.p1.7.m7.1.1.2.3.cmml" xref="S4.SS2.5.p1.7.m7.1.1.2.3">′</ci></apply><apply id="S4.SS2.5.p1.7.m7.1.1.3.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.5.p1.7.m7.1.1.3.1.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3">subscript</csymbol><apply id="S4.SS2.5.p1.7.m7.1.1.3.2.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.5.p1.7.m7.1.1.3.2.1.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3">superscript</csymbol><ci id="S4.SS2.5.p1.7.m7.1.1.3.2.2.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3.2.2">𝐺</ci><ci id="S4.SS2.5.p1.7.m7.1.1.3.2.3.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3.2.3">𝑑</ci></apply><ci id="S4.SS2.5.p1.7.m7.1.1.3.3.cmml" xref="S4.SS2.5.p1.7.m7.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.7.m7.1c">c^{\prime}\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.7.m7.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> be the grid point closest to <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.5.p1.8.m8.1"><semantics id="S4.SS2.5.p1.8.m8.1a"><mi id="S4.SS2.5.p1.8.m8.1.1" xref="S4.SS2.5.p1.8.m8.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.8.m8.1b"><ci id="S4.SS2.5.p1.8.m8.1.1.cmml" xref="S4.SS2.5.p1.8.m8.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.8.m8.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.8.m8.1d">italic_c</annotation></semantics></math> (in <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.9.m9.1"><semantics id="S4.SS2.5.p1.9.m9.1a"><msub id="S4.SS2.5.p1.9.m9.1.1" xref="S4.SS2.5.p1.9.m9.1.1.cmml"><mi id="S4.SS2.5.p1.9.m9.1.1.2" mathvariant="normal" xref="S4.SS2.5.p1.9.m9.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.5.p1.9.m9.1.1.3" xref="S4.SS2.5.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.9.m9.1b"><apply id="S4.SS2.5.p1.9.m9.1.1.cmml" xref="S4.SS2.5.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.9.m9.1.1.1.cmml" xref="S4.SS2.5.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.5.p1.9.m9.1.1.2.cmml" xref="S4.SS2.5.p1.9.m9.1.1.2">ℓ</ci><cn id="S4.SS2.5.p1.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.5.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.9.m9.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-distance). In other words, <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.10.m10.1"><semantics id="S4.SS2.5.p1.10.m10.1a"><msup id="S4.SS2.5.p1.10.m10.1.1" xref="S4.SS2.5.p1.10.m10.1.1.cmml"><mi id="S4.SS2.5.p1.10.m10.1.1.2" xref="S4.SS2.5.p1.10.m10.1.1.2.cmml">c</mi><mo id="S4.SS2.5.p1.10.m10.1.1.3" xref="S4.SS2.5.p1.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.10.m10.1b"><apply id="S4.SS2.5.p1.10.m10.1.1.cmml" xref="S4.SS2.5.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.10.m10.1.1.1.cmml" xref="S4.SS2.5.p1.10.m10.1.1">superscript</csymbol><ci id="S4.SS2.5.p1.10.m10.1.1.2.cmml" xref="S4.SS2.5.p1.10.m10.1.1.2">𝑐</ci><ci id="S4.SS2.5.p1.10.m10.1.1.3.cmml" xref="S4.SS2.5.p1.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.10.m10.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.10.m10.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is obtained from <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.5.p1.11.m11.1"><semantics id="S4.SS2.5.p1.11.m11.1a"><mi id="S4.SS2.5.p1.11.m11.1.1" xref="S4.SS2.5.p1.11.m11.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.11.m11.1b"><ci id="S4.SS2.5.p1.11.m11.1.1.cmml" xref="S4.SS2.5.p1.11.m11.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.11.m11.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.11.m11.1d">italic_c</annotation></semantics></math> by rounding each coordinate individually as little as possible.</p> </div> <div class="ltx_para" id="S4.SS2.6.p2"> <p class="ltx_p" id="S4.SS2.6.p2.13">We claim that <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.1.m1.1"><semantics id="S4.SS2.6.p2.1.m1.1a"><msup id="S4.SS2.6.p2.1.m1.1.1" xref="S4.SS2.6.p2.1.m1.1.1.cmml"><mi id="S4.SS2.6.p2.1.m1.1.1.2" xref="S4.SS2.6.p2.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS2.6.p2.1.m1.1.1.3" xref="S4.SS2.6.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.1.m1.1b"><apply id="S4.SS2.6.p2.1.m1.1.1.cmml" xref="S4.SS2.6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.1.m1.1.1.1.cmml" xref="S4.SS2.6.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.1.m1.1.1.2.cmml" xref="S4.SS2.6.p2.1.m1.1.1.2">𝑐</ci><ci id="S4.SS2.6.p2.1.m1.1.1.3.cmml" xref="S4.SS2.6.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.1.m1.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.1.m1.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is also a discrete <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.2.m2.1"><semantics id="S4.SS2.6.p2.2.m2.1a"><msub id="S4.SS2.6.p2.2.m2.1.1" xref="S4.SS2.6.p2.2.m2.1.1.cmml"><mi id="S4.SS2.6.p2.2.m2.1.1.2" mathvariant="normal" xref="S4.SS2.6.p2.2.m2.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.6.p2.2.m2.1.1.3" xref="S4.SS2.6.p2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.2.m2.1b"><apply id="S4.SS2.6.p2.2.m2.1.1.cmml" xref="S4.SS2.6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m2.1.1.1.cmml" xref="S4.SS2.6.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.6.p2.2.m2.1.1.2.cmml" xref="S4.SS2.6.p2.2.m2.1.1.2">ℓ</ci><cn id="S4.SS2.6.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of <math alttext="P" class="ltx_Math" display="inline" id="S4.SS2.6.p2.3.m3.1"><semantics id="S4.SS2.6.p2.3.m3.1a"><mi id="S4.SS2.6.p2.3.m3.1.1" xref="S4.SS2.6.p2.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.3.m3.1b"><ci id="S4.SS2.6.p2.3.m3.1.1.cmml" xref="S4.SS2.6.p2.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.3.m3.1d">italic_P</annotation></semantics></math>. To prove this, consider the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.4.m4.1"><semantics id="S4.SS2.6.p2.4.m4.1a"><msub id="S4.SS2.6.p2.4.m4.1.1" xref="S4.SS2.6.p2.4.m4.1.1.cmml"><mi id="S4.SS2.6.p2.4.m4.1.1.2" mathvariant="normal" xref="S4.SS2.6.p2.4.m4.1.1.2.cmml">ℓ</mi><mn id="S4.SS2.6.p2.4.m4.1.1.3" xref="S4.SS2.6.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.4.m4.1b"><apply id="S4.SS2.6.p2.4.m4.1.1.cmml" xref="S4.SS2.6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.4.m4.1.1.1.cmml" xref="S4.SS2.6.p2.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.6.p2.4.m4.1.1.2.cmml" xref="S4.SS2.6.p2.4.m4.1.1.2">ℓ</ci><cn id="S4.SS2.6.p2.4.m4.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.4.m4.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces <math alttext="\mathcal{H}^{1}_{c^{\prime},v}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.5.m5.2"><semantics id="S4.SS2.6.p2.5.m5.2a"><msubsup id="S4.SS2.6.p2.5.m5.2.3" xref="S4.SS2.6.p2.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.5.m5.2.3.2.2" xref="S4.SS2.6.p2.5.m5.2.3.2.2.cmml">ℋ</mi><mrow id="S4.SS2.6.p2.5.m5.2.2.2.2" xref="S4.SS2.6.p2.5.m5.2.2.2.3.cmml"><msup id="S4.SS2.6.p2.5.m5.2.2.2.2.1" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1.cmml"><mi id="S4.SS2.6.p2.5.m5.2.2.2.2.1.2" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1.2.cmml">c</mi><mo id="S4.SS2.6.p2.5.m5.2.2.2.2.1.3" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1.3.cmml">′</mo></msup><mo id="S4.SS2.6.p2.5.m5.2.2.2.2.2" xref="S4.SS2.6.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS2.6.p2.5.m5.1.1.1.1" xref="S4.SS2.6.p2.5.m5.1.1.1.1.cmml">v</mi></mrow><mn id="S4.SS2.6.p2.5.m5.2.3.2.3" xref="S4.SS2.6.p2.5.m5.2.3.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.5.m5.2b"><apply id="S4.SS2.6.p2.5.m5.2.3.cmml" xref="S4.SS2.6.p2.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.5.m5.2.3.1.cmml" xref="S4.SS2.6.p2.5.m5.2.3">subscript</csymbol><apply id="S4.SS2.6.p2.5.m5.2.3.2.cmml" xref="S4.SS2.6.p2.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.5.m5.2.3.2.1.cmml" xref="S4.SS2.6.p2.5.m5.2.3">superscript</csymbol><ci id="S4.SS2.6.p2.5.m5.2.3.2.2.cmml" xref="S4.SS2.6.p2.5.m5.2.3.2.2">ℋ</ci><cn id="S4.SS2.6.p2.5.m5.2.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.5.m5.2.3.2.3">1</cn></apply><list id="S4.SS2.6.p2.5.m5.2.2.2.3.cmml" xref="S4.SS2.6.p2.5.m5.2.2.2.2"><apply id="S4.SS2.6.p2.5.m5.2.2.2.2.1.cmml" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.5.m5.2.2.2.2.1.1.cmml" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1">superscript</csymbol><ci id="S4.SS2.6.p2.5.m5.2.2.2.2.1.2.cmml" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1.2">𝑐</ci><ci id="S4.SS2.6.p2.5.m5.2.2.2.2.1.3.cmml" xref="S4.SS2.6.p2.5.m5.2.2.2.2.1.3">′</ci></apply><ci id="S4.SS2.6.p2.5.m5.1.1.1.1.cmml" xref="S4.SS2.6.p2.5.m5.1.1.1.1">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.5.m5.2c">\mathcal{H}^{1}_{c^{\prime},v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.5.m5.2d">caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{H}^{1}_{c,v}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.6.m6.2"><semantics id="S4.SS2.6.p2.6.m6.2a"><msubsup id="S4.SS2.6.p2.6.m6.2.3" xref="S4.SS2.6.p2.6.m6.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.6.m6.2.3.2.2" xref="S4.SS2.6.p2.6.m6.2.3.2.2.cmml">ℋ</mi><mrow id="S4.SS2.6.p2.6.m6.2.2.2.4" xref="S4.SS2.6.p2.6.m6.2.2.2.3.cmml"><mi id="S4.SS2.6.p2.6.m6.1.1.1.1" xref="S4.SS2.6.p2.6.m6.1.1.1.1.cmml">c</mi><mo id="S4.SS2.6.p2.6.m6.2.2.2.4.1" xref="S4.SS2.6.p2.6.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS2.6.p2.6.m6.2.2.2.2" xref="S4.SS2.6.p2.6.m6.2.2.2.2.cmml">v</mi></mrow><mn id="S4.SS2.6.p2.6.m6.2.3.2.3" xref="S4.SS2.6.p2.6.m6.2.3.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.6.m6.2b"><apply id="S4.SS2.6.p2.6.m6.2.3.cmml" xref="S4.SS2.6.p2.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.6.m6.2.3.1.cmml" xref="S4.SS2.6.p2.6.m6.2.3">subscript</csymbol><apply id="S4.SS2.6.p2.6.m6.2.3.2.cmml" xref="S4.SS2.6.p2.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.6.m6.2.3.2.1.cmml" xref="S4.SS2.6.p2.6.m6.2.3">superscript</csymbol><ci id="S4.SS2.6.p2.6.m6.2.3.2.2.cmml" xref="S4.SS2.6.p2.6.m6.2.3.2.2">ℋ</ci><cn id="S4.SS2.6.p2.6.m6.2.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.6.m6.2.3.2.3">1</cn></apply><list id="S4.SS2.6.p2.6.m6.2.2.2.3.cmml" xref="S4.SS2.6.p2.6.m6.2.2.2.4"><ci id="S4.SS2.6.p2.6.m6.1.1.1.1.cmml" xref="S4.SS2.6.p2.6.m6.1.1.1.1">𝑐</ci><ci id="S4.SS2.6.p2.6.m6.2.2.2.2.cmml" xref="S4.SS2.6.p2.6.m6.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.6.m6.2c">\mathcal{H}^{1}_{c,v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.6.m6.2d">caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> for an arbitrary direction <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.7.m7.1"><semantics id="S4.SS2.6.p2.7.m7.1a"><mrow id="S4.SS2.6.p2.7.m7.1.1" xref="S4.SS2.6.p2.7.m7.1.1.cmml"><mi id="S4.SS2.6.p2.7.m7.1.1.2" xref="S4.SS2.6.p2.7.m7.1.1.2.cmml">v</mi><mo id="S4.SS2.6.p2.7.m7.1.1.1" xref="S4.SS2.6.p2.7.m7.1.1.1.cmml">∈</mo><msup id="S4.SS2.6.p2.7.m7.1.1.3" xref="S4.SS2.6.p2.7.m7.1.1.3.cmml"><mi id="S4.SS2.6.p2.7.m7.1.1.3.2" xref="S4.SS2.6.p2.7.m7.1.1.3.2.cmml">S</mi><mrow id="S4.SS2.6.p2.7.m7.1.1.3.3" xref="S4.SS2.6.p2.7.m7.1.1.3.3.cmml"><mi id="S4.SS2.6.p2.7.m7.1.1.3.3.2" xref="S4.SS2.6.p2.7.m7.1.1.3.3.2.cmml">d</mi><mo id="S4.SS2.6.p2.7.m7.1.1.3.3.1" xref="S4.SS2.6.p2.7.m7.1.1.3.3.1.cmml">−</mo><mn id="S4.SS2.6.p2.7.m7.1.1.3.3.3" xref="S4.SS2.6.p2.7.m7.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.7.m7.1b"><apply id="S4.SS2.6.p2.7.m7.1.1.cmml" xref="S4.SS2.6.p2.7.m7.1.1"><in id="S4.SS2.6.p2.7.m7.1.1.1.cmml" xref="S4.SS2.6.p2.7.m7.1.1.1"></in><ci id="S4.SS2.6.p2.7.m7.1.1.2.cmml" xref="S4.SS2.6.p2.7.m7.1.1.2">𝑣</ci><apply id="S4.SS2.6.p2.7.m7.1.1.3.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.7.m7.1.1.3.1.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3">superscript</csymbol><ci id="S4.SS2.6.p2.7.m7.1.1.3.2.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3.2">𝑆</ci><apply id="S4.SS2.6.p2.7.m7.1.1.3.3.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3.3"><minus id="S4.SS2.6.p2.7.m7.1.1.3.3.1.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3.3.1"></minus><ci id="S4.SS2.6.p2.7.m7.1.1.3.3.2.cmml" xref="S4.SS2.6.p2.7.m7.1.1.3.3.2">𝑑</ci><cn id="S4.SS2.6.p2.7.m7.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.6.p2.7.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.7.m7.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.7.m7.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. We prove that every point <math alttext="z\in P\cap\mathcal{H}^{1}_{c,v}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.8.m8.2"><semantics id="S4.SS2.6.p2.8.m8.2a"><mrow id="S4.SS2.6.p2.8.m8.2.3" xref="S4.SS2.6.p2.8.m8.2.3.cmml"><mi id="S4.SS2.6.p2.8.m8.2.3.2" xref="S4.SS2.6.p2.8.m8.2.3.2.cmml">z</mi><mo id="S4.SS2.6.p2.8.m8.2.3.1" xref="S4.SS2.6.p2.8.m8.2.3.1.cmml">∈</mo><mrow id="S4.SS2.6.p2.8.m8.2.3.3" xref="S4.SS2.6.p2.8.m8.2.3.3.cmml"><mi id="S4.SS2.6.p2.8.m8.2.3.3.2" xref="S4.SS2.6.p2.8.m8.2.3.3.2.cmml">P</mi><mo id="S4.SS2.6.p2.8.m8.2.3.3.1" xref="S4.SS2.6.p2.8.m8.2.3.3.1.cmml">∩</mo><msubsup id="S4.SS2.6.p2.8.m8.2.3.3.3" xref="S4.SS2.6.p2.8.m8.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.8.m8.2.3.3.3.2.2" xref="S4.SS2.6.p2.8.m8.2.3.3.3.2.2.cmml">ℋ</mi><mrow id="S4.SS2.6.p2.8.m8.2.2.2.4" xref="S4.SS2.6.p2.8.m8.2.2.2.3.cmml"><mi id="S4.SS2.6.p2.8.m8.1.1.1.1" xref="S4.SS2.6.p2.8.m8.1.1.1.1.cmml">c</mi><mo id="S4.SS2.6.p2.8.m8.2.2.2.4.1" xref="S4.SS2.6.p2.8.m8.2.2.2.3.cmml">,</mo><mi id="S4.SS2.6.p2.8.m8.2.2.2.2" xref="S4.SS2.6.p2.8.m8.2.2.2.2.cmml">v</mi></mrow><mn id="S4.SS2.6.p2.8.m8.2.3.3.3.2.3" xref="S4.SS2.6.p2.8.m8.2.3.3.3.2.3.cmml">1</mn></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.8.m8.2b"><apply id="S4.SS2.6.p2.8.m8.2.3.cmml" xref="S4.SS2.6.p2.8.m8.2.3"><in id="S4.SS2.6.p2.8.m8.2.3.1.cmml" xref="S4.SS2.6.p2.8.m8.2.3.1"></in><ci id="S4.SS2.6.p2.8.m8.2.3.2.cmml" xref="S4.SS2.6.p2.8.m8.2.3.2">𝑧</ci><apply id="S4.SS2.6.p2.8.m8.2.3.3.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3"><intersect id="S4.SS2.6.p2.8.m8.2.3.3.1.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.1"></intersect><ci id="S4.SS2.6.p2.8.m8.2.3.3.2.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.2">𝑃</ci><apply id="S4.SS2.6.p2.8.m8.2.3.3.3.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.8.m8.2.3.3.3.1.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.3">subscript</csymbol><apply id="S4.SS2.6.p2.8.m8.2.3.3.3.2.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.8.m8.2.3.3.3.2.1.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.3">superscript</csymbol><ci id="S4.SS2.6.p2.8.m8.2.3.3.3.2.2.cmml" xref="S4.SS2.6.p2.8.m8.2.3.3.3.2.2">ℋ</ci><cn id="S4.SS2.6.p2.8.m8.2.3.3.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.8.m8.2.3.3.3.2.3">1</cn></apply><list id="S4.SS2.6.p2.8.m8.2.2.2.3.cmml" xref="S4.SS2.6.p2.8.m8.2.2.2.4"><ci id="S4.SS2.6.p2.8.m8.1.1.1.1.cmml" xref="S4.SS2.6.p2.8.m8.1.1.1.1">𝑐</ci><ci id="S4.SS2.6.p2.8.m8.2.2.2.2.cmml" xref="S4.SS2.6.p2.8.m8.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.8.m8.2c">z\in P\cap\mathcal{H}^{1}_{c,v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.8.m8.2d">italic_z ∈ italic_P ∩ caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> must also be contained in <math alttext="\mathcal{H}^{1}_{c^{\prime},v}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.9.m9.2"><semantics id="S4.SS2.6.p2.9.m9.2a"><msubsup id="S4.SS2.6.p2.9.m9.2.3" xref="S4.SS2.6.p2.9.m9.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.9.m9.2.3.2.2" xref="S4.SS2.6.p2.9.m9.2.3.2.2.cmml">ℋ</mi><mrow id="S4.SS2.6.p2.9.m9.2.2.2.2" xref="S4.SS2.6.p2.9.m9.2.2.2.3.cmml"><msup id="S4.SS2.6.p2.9.m9.2.2.2.2.1" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1.cmml"><mi id="S4.SS2.6.p2.9.m9.2.2.2.2.1.2" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1.2.cmml">c</mi><mo id="S4.SS2.6.p2.9.m9.2.2.2.2.1.3" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1.3.cmml">′</mo></msup><mo id="S4.SS2.6.p2.9.m9.2.2.2.2.2" xref="S4.SS2.6.p2.9.m9.2.2.2.3.cmml">,</mo><mi id="S4.SS2.6.p2.9.m9.1.1.1.1" xref="S4.SS2.6.p2.9.m9.1.1.1.1.cmml">v</mi></mrow><mn id="S4.SS2.6.p2.9.m9.2.3.2.3" xref="S4.SS2.6.p2.9.m9.2.3.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.9.m9.2b"><apply id="S4.SS2.6.p2.9.m9.2.3.cmml" xref="S4.SS2.6.p2.9.m9.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.9.m9.2.3.1.cmml" xref="S4.SS2.6.p2.9.m9.2.3">subscript</csymbol><apply id="S4.SS2.6.p2.9.m9.2.3.2.cmml" xref="S4.SS2.6.p2.9.m9.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.9.m9.2.3.2.1.cmml" xref="S4.SS2.6.p2.9.m9.2.3">superscript</csymbol><ci id="S4.SS2.6.p2.9.m9.2.3.2.2.cmml" xref="S4.SS2.6.p2.9.m9.2.3.2.2">ℋ</ci><cn id="S4.SS2.6.p2.9.m9.2.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.9.m9.2.3.2.3">1</cn></apply><list id="S4.SS2.6.p2.9.m9.2.2.2.3.cmml" xref="S4.SS2.6.p2.9.m9.2.2.2.2"><apply id="S4.SS2.6.p2.9.m9.2.2.2.2.1.cmml" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.9.m9.2.2.2.2.1.1.cmml" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1">superscript</csymbol><ci id="S4.SS2.6.p2.9.m9.2.2.2.2.1.2.cmml" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1.2">𝑐</ci><ci id="S4.SS2.6.p2.9.m9.2.2.2.2.1.3.cmml" xref="S4.SS2.6.p2.9.m9.2.2.2.2.1.3">′</ci></apply><ci id="S4.SS2.6.p2.9.m9.1.1.1.1.cmml" xref="S4.SS2.6.p2.9.m9.1.1.1.1">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.9.m9.2c">\mathcal{H}^{1}_{c^{\prime},v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.9.m9.2d">caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Indeed, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>, there exists a subgradient <math alttext="u\in\partial||z-c||_{1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.10.m10.1"><semantics id="S4.SS2.6.p2.10.m10.1a"><mrow id="S4.SS2.6.p2.10.m10.1.1" xref="S4.SS2.6.p2.10.m10.1.1.cmml"><mi id="S4.SS2.6.p2.10.m10.1.1.3" xref="S4.SS2.6.p2.10.m10.1.1.3.cmml">u</mi><mo id="S4.SS2.6.p2.10.m10.1.1.2" rspace="0.1389em" xref="S4.SS2.6.p2.10.m10.1.1.2.cmml">∈</mo><mrow id="S4.SS2.6.p2.10.m10.1.1.1" xref="S4.SS2.6.p2.10.m10.1.1.1.cmml"><mo id="S4.SS2.6.p2.10.m10.1.1.1.2" lspace="0.1389em" rspace="0em" xref="S4.SS2.6.p2.10.m10.1.1.1.2.cmml">∂</mo><msub id="S4.SS2.6.p2.10.m10.1.1.1.1" xref="S4.SS2.6.p2.10.m10.1.1.1.1.cmml"><mrow id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.2.cmml"><mo id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.2" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.3" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.3.cmml">c</mi></mrow><mo id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.6.p2.10.m10.1.1.1.1.3" xref="S4.SS2.6.p2.10.m10.1.1.1.1.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.10.m10.1b"><apply id="S4.SS2.6.p2.10.m10.1.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1"><in id="S4.SS2.6.p2.10.m10.1.1.2.cmml" xref="S4.SS2.6.p2.10.m10.1.1.2"></in><ci id="S4.SS2.6.p2.10.m10.1.1.3.cmml" xref="S4.SS2.6.p2.10.m10.1.1.3">𝑢</ci><apply id="S4.SS2.6.p2.10.m10.1.1.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1"><partialdiff id="S4.SS2.6.p2.10.m10.1.1.1.2.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.2"></partialdiff><apply id="S4.SS2.6.p2.10.m10.1.1.1.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.10.m10.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1">subscript</csymbol><apply id="S4.SS2.6.p2.10.m10.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.6.p2.10.m10.1.1.1.1.1.2.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1"><minus id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.2">𝑧</ci><ci id="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.6.p2.10.m10.1.1.1.1.1.1.1.3">𝑐</ci></apply></apply><cn id="S4.SS2.6.p2.10.m10.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.10.m10.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.10.m10.1c">u\in\partial||z-c||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.10.m10.1d">italic_u ∈ ∂ | | italic_z - italic_c | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\langle u,v\rangle\geq 0" class="ltx_Math" display="inline" id="S4.SS2.6.p2.11.m11.2"><semantics id="S4.SS2.6.p2.11.m11.2a"><mrow id="S4.SS2.6.p2.11.m11.2.3" xref="S4.SS2.6.p2.11.m11.2.3.cmml"><mrow id="S4.SS2.6.p2.11.m11.2.3.2.2" xref="S4.SS2.6.p2.11.m11.2.3.2.1.cmml"><mo id="S4.SS2.6.p2.11.m11.2.3.2.2.1" stretchy="false" xref="S4.SS2.6.p2.11.m11.2.3.2.1.cmml">⟨</mo><mi id="S4.SS2.6.p2.11.m11.1.1" xref="S4.SS2.6.p2.11.m11.1.1.cmml">u</mi><mo id="S4.SS2.6.p2.11.m11.2.3.2.2.2" xref="S4.SS2.6.p2.11.m11.2.3.2.1.cmml">,</mo><mi id="S4.SS2.6.p2.11.m11.2.2" xref="S4.SS2.6.p2.11.m11.2.2.cmml">v</mi><mo id="S4.SS2.6.p2.11.m11.2.3.2.2.3" stretchy="false" xref="S4.SS2.6.p2.11.m11.2.3.2.1.cmml">⟩</mo></mrow><mo id="S4.SS2.6.p2.11.m11.2.3.1" xref="S4.SS2.6.p2.11.m11.2.3.1.cmml">≥</mo><mn id="S4.SS2.6.p2.11.m11.2.3.3" xref="S4.SS2.6.p2.11.m11.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.11.m11.2b"><apply id="S4.SS2.6.p2.11.m11.2.3.cmml" xref="S4.SS2.6.p2.11.m11.2.3"><geq id="S4.SS2.6.p2.11.m11.2.3.1.cmml" xref="S4.SS2.6.p2.11.m11.2.3.1"></geq><list id="S4.SS2.6.p2.11.m11.2.3.2.1.cmml" xref="S4.SS2.6.p2.11.m11.2.3.2.2"><ci id="S4.SS2.6.p2.11.m11.1.1.cmml" xref="S4.SS2.6.p2.11.m11.1.1">𝑢</ci><ci id="S4.SS2.6.p2.11.m11.2.2.cmml" xref="S4.SS2.6.p2.11.m11.2.2">𝑣</ci></list><cn id="S4.SS2.6.p2.11.m11.2.3.3.cmml" type="integer" xref="S4.SS2.6.p2.11.m11.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.11.m11.2c">\langle u,v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.11.m11.2d">⟨ italic_u , italic_v ⟩ ≥ 0</annotation></semantics></math>. We want to prove <math alttext="u\in\partial||z-c^{\prime}||_{1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.12.m12.1"><semantics id="S4.SS2.6.p2.12.m12.1a"><mrow id="S4.SS2.6.p2.12.m12.1.1" xref="S4.SS2.6.p2.12.m12.1.1.cmml"><mi id="S4.SS2.6.p2.12.m12.1.1.3" xref="S4.SS2.6.p2.12.m12.1.1.3.cmml">u</mi><mo id="S4.SS2.6.p2.12.m12.1.1.2" rspace="0.1389em" xref="S4.SS2.6.p2.12.m12.1.1.2.cmml">∈</mo><mrow id="S4.SS2.6.p2.12.m12.1.1.1" xref="S4.SS2.6.p2.12.m12.1.1.1.cmml"><mo id="S4.SS2.6.p2.12.m12.1.1.1.2" lspace="0.1389em" rspace="0em" xref="S4.SS2.6.p2.12.m12.1.1.1.2.cmml">∂</mo><msub id="S4.SS2.6.p2.12.m12.1.1.1.1" xref="S4.SS2.6.p2.12.m12.1.1.1.1.cmml"><mrow id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.2.cmml"><mo id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.2" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.2" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.2.cmml">c</mi><mo id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.3" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.6.p2.12.m12.1.1.1.1.3" xref="S4.SS2.6.p2.12.m12.1.1.1.1.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.12.m12.1b"><apply id="S4.SS2.6.p2.12.m12.1.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1"><in id="S4.SS2.6.p2.12.m12.1.1.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.2"></in><ci id="S4.SS2.6.p2.12.m12.1.1.3.cmml" xref="S4.SS2.6.p2.12.m12.1.1.3">𝑢</ci><apply id="S4.SS2.6.p2.12.m12.1.1.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1"><partialdiff id="S4.SS2.6.p2.12.m12.1.1.1.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.2"></partialdiff><apply id="S4.SS2.6.p2.12.m12.1.1.1.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.12.m12.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1">subscript</csymbol><apply id="S4.SS2.6.p2.12.m12.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.6.p2.12.m12.1.1.1.1.1.2.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1"><minus id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.2">𝑧</ci><apply id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.2">𝑐</ci><ci id="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.6.p2.12.m12.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><cn id="S4.SS2.6.p2.12.m12.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.12.m12.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.12.m12.1c">u\in\partial||z-c^{\prime}||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.12.m12.1d">italic_u ∈ ∂ | | italic_z - italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Observe that our choice of <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.13.m13.1"><semantics id="S4.SS2.6.p2.13.m13.1a"><msup id="S4.SS2.6.p2.13.m13.1.1" xref="S4.SS2.6.p2.13.m13.1.1.cmml"><mi id="S4.SS2.6.p2.13.m13.1.1.2" xref="S4.SS2.6.p2.13.m13.1.1.2.cmml">c</mi><mo id="S4.SS2.6.p2.13.m13.1.1.3" xref="S4.SS2.6.p2.13.m13.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.13.m13.1b"><apply id="S4.SS2.6.p2.13.m13.1.1.cmml" xref="S4.SS2.6.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.13.m13.1.1.1.cmml" xref="S4.SS2.6.p2.13.m13.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.13.m13.1.1.2.cmml" xref="S4.SS2.6.p2.13.m13.1.1.2">𝑐</ci><ci id="S4.SS2.6.p2.13.m13.1.1.3.cmml" xref="S4.SS2.6.p2.13.m13.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.13.m13.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.13.m13.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> guarantees</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z_{i}-c^{\prime}_{i}>0\implies z_{i}-c_{i}>0\text{ and }z_{i}-c^{\prime}_{i}<0% \implies z_{i}-c_{i}<0" class="ltx_Math" display="block" id="S4.Ex18.m1.1"><semantics id="S4.Ex18.m1.1a"><mrow id="S4.Ex18.m1.1.1" xref="S4.Ex18.m1.1.1.cmml"><mrow id="S4.Ex18.m1.1.1.2" xref="S4.Ex18.m1.1.1.2.cmml"><msub id="S4.Ex18.m1.1.1.2.2" xref="S4.Ex18.m1.1.1.2.2.cmml"><mi id="S4.Ex18.m1.1.1.2.2.2" xref="S4.Ex18.m1.1.1.2.2.2.cmml">z</mi><mi id="S4.Ex18.m1.1.1.2.2.3" xref="S4.Ex18.m1.1.1.2.2.3.cmml">i</mi></msub><mo id="S4.Ex18.m1.1.1.2.1" xref="S4.Ex18.m1.1.1.2.1.cmml">−</mo><msubsup id="S4.Ex18.m1.1.1.2.3" xref="S4.Ex18.m1.1.1.2.3.cmml"><mi id="S4.Ex18.m1.1.1.2.3.2.2" 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encoding="application/x-llamapun" id="S4.Ex18.m1.1d">italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 ⟹ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 and italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0 ⟹ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.6.p2.17">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="S4.SS2.6.p2.14.m1.1"><semantics id="S4.SS2.6.p2.14.m1.1a"><mrow id="S4.SS2.6.p2.14.m1.1.2" xref="S4.SS2.6.p2.14.m1.1.2.cmml"><mi id="S4.SS2.6.p2.14.m1.1.2.2" xref="S4.SS2.6.p2.14.m1.1.2.2.cmml">i</mi><mo id="S4.SS2.6.p2.14.m1.1.2.1" xref="S4.SS2.6.p2.14.m1.1.2.1.cmml">∈</mo><mrow id="S4.SS2.6.p2.14.m1.1.2.3.2" xref="S4.SS2.6.p2.14.m1.1.2.3.1.cmml"><mo id="S4.SS2.6.p2.14.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.6.p2.14.m1.1.2.3.1.1.cmml">[</mo><mi id="S4.SS2.6.p2.14.m1.1.1" xref="S4.SS2.6.p2.14.m1.1.1.cmml">d</mi><mo id="S4.SS2.6.p2.14.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.6.p2.14.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.14.m1.1b"><apply id="S4.SS2.6.p2.14.m1.1.2.cmml" xref="S4.SS2.6.p2.14.m1.1.2"><in id="S4.SS2.6.p2.14.m1.1.2.1.cmml" xref="S4.SS2.6.p2.14.m1.1.2.1"></in><ci id="S4.SS2.6.p2.14.m1.1.2.2.cmml" xref="S4.SS2.6.p2.14.m1.1.2.2">𝑖</ci><apply id="S4.SS2.6.p2.14.m1.1.2.3.1.cmml" xref="S4.SS2.6.p2.14.m1.1.2.3.2"><csymbol cd="latexml" id="S4.SS2.6.p2.14.m1.1.2.3.1.1.cmml" xref="S4.SS2.6.p2.14.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.SS2.6.p2.14.m1.1.1.cmml" xref="S4.SS2.6.p2.14.m1.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.14.m1.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.14.m1.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. With <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem6" title="Observation 4.6. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.6</span></a>, we therefore have <math alttext="u\in\partial||z-c^{\prime}||_{1}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.15.m2.1"><semantics id="S4.SS2.6.p2.15.m2.1a"><mrow id="S4.SS2.6.p2.15.m2.1.1" xref="S4.SS2.6.p2.15.m2.1.1.cmml"><mi id="S4.SS2.6.p2.15.m2.1.1.3" xref="S4.SS2.6.p2.15.m2.1.1.3.cmml">u</mi><mo id="S4.SS2.6.p2.15.m2.1.1.2" rspace="0.1389em" xref="S4.SS2.6.p2.15.m2.1.1.2.cmml">∈</mo><mrow id="S4.SS2.6.p2.15.m2.1.1.1" xref="S4.SS2.6.p2.15.m2.1.1.1.cmml"><mo id="S4.SS2.6.p2.15.m2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="S4.SS2.6.p2.15.m2.1.1.1.2.cmml">∂</mo><msub id="S4.SS2.6.p2.15.m2.1.1.1.1" xref="S4.SS2.6.p2.15.m2.1.1.1.1.cmml"><mrow id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.2.cmml"><mo id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.2" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.2" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.2.cmml">c</mi><mo id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.3" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.6.p2.15.m2.1.1.1.1.3" xref="S4.SS2.6.p2.15.m2.1.1.1.1.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.15.m2.1b"><apply id="S4.SS2.6.p2.15.m2.1.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1"><in id="S4.SS2.6.p2.15.m2.1.1.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.2"></in><ci id="S4.SS2.6.p2.15.m2.1.1.3.cmml" xref="S4.SS2.6.p2.15.m2.1.1.3">𝑢</ci><apply id="S4.SS2.6.p2.15.m2.1.1.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1"><partialdiff id="S4.SS2.6.p2.15.m2.1.1.1.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.2"></partialdiff><apply id="S4.SS2.6.p2.15.m2.1.1.1.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.15.m2.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1">subscript</csymbol><apply id="S4.SS2.6.p2.15.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.6.p2.15.m2.1.1.1.1.1.2.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1"><minus id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.2">𝑧</ci><apply id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.2">𝑐</ci><ci id="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.6.p2.15.m2.1.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply><cn id="S4.SS2.6.p2.15.m2.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.15.m2.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.15.m2.1c">u\in\partial||z-c^{\prime}||_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.15.m2.1d">italic_u ∈ ∂ | | italic_z - italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and conclude by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> that <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.6.p2.16.m3.1"><semantics id="S4.SS2.6.p2.16.m3.1a"><mi id="S4.SS2.6.p2.16.m3.1.1" xref="S4.SS2.6.p2.16.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.16.m3.1b"><ci id="S4.SS2.6.p2.16.m3.1.1.cmml" xref="S4.SS2.6.p2.16.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.16.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.16.m3.1d">italic_z</annotation></semantics></math> is contained in <math alttext="\mathcal{H}^{1}_{c^{\prime},v}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.17.m4.2"><semantics id="S4.SS2.6.p2.17.m4.2a"><msubsup id="S4.SS2.6.p2.17.m4.2.3" xref="S4.SS2.6.p2.17.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.17.m4.2.3.2.2" xref="S4.SS2.6.p2.17.m4.2.3.2.2.cmml">ℋ</mi><mrow id="S4.SS2.6.p2.17.m4.2.2.2.2" xref="S4.SS2.6.p2.17.m4.2.2.2.3.cmml"><msup id="S4.SS2.6.p2.17.m4.2.2.2.2.1" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1.cmml"><mi id="S4.SS2.6.p2.17.m4.2.2.2.2.1.2" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1.2.cmml">c</mi><mo id="S4.SS2.6.p2.17.m4.2.2.2.2.1.3" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1.3.cmml">′</mo></msup><mo id="S4.SS2.6.p2.17.m4.2.2.2.2.2" xref="S4.SS2.6.p2.17.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS2.6.p2.17.m4.1.1.1.1" xref="S4.SS2.6.p2.17.m4.1.1.1.1.cmml">v</mi></mrow><mn id="S4.SS2.6.p2.17.m4.2.3.2.3" xref="S4.SS2.6.p2.17.m4.2.3.2.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.17.m4.2b"><apply id="S4.SS2.6.p2.17.m4.2.3.cmml" xref="S4.SS2.6.p2.17.m4.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.17.m4.2.3.1.cmml" xref="S4.SS2.6.p2.17.m4.2.3">subscript</csymbol><apply id="S4.SS2.6.p2.17.m4.2.3.2.cmml" xref="S4.SS2.6.p2.17.m4.2.3"><csymbol cd="ambiguous" id="S4.SS2.6.p2.17.m4.2.3.2.1.cmml" xref="S4.SS2.6.p2.17.m4.2.3">superscript</csymbol><ci id="S4.SS2.6.p2.17.m4.2.3.2.2.cmml" xref="S4.SS2.6.p2.17.m4.2.3.2.2">ℋ</ci><cn id="S4.SS2.6.p2.17.m4.2.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.17.m4.2.3.2.3">1</cn></apply><list id="S4.SS2.6.p2.17.m4.2.2.2.3.cmml" xref="S4.SS2.6.p2.17.m4.2.2.2.2"><apply id="S4.SS2.6.p2.17.m4.2.2.2.2.1.cmml" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.17.m4.2.2.2.2.1.1.cmml" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1">superscript</csymbol><ci id="S4.SS2.6.p2.17.m4.2.2.2.2.1.2.cmml" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1.2">𝑐</ci><ci id="S4.SS2.6.p2.17.m4.2.2.2.2.1.3.cmml" xref="S4.SS2.6.p2.17.m4.2.2.2.2.1.3">′</ci></apply><ci id="S4.SS2.6.p2.17.m4.1.1.1.1.cmml" xref="S4.SS2.6.p2.17.m4.1.1.1.1">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.17.m4.2c">\mathcal{H}^{1}_{c^{\prime},v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.17.m4.2d">caligraphic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Total Search Version</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.4">The class <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><msup id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">superscript</csymbol><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p1.1.m1.1.1.3a.cmml" xref="S4.SS3.p1.1.m1.1.1.3"><mtext id="S4.SS3.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p1.1.m1.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math> (standing for <em class="ltx_emph ltx_font_italic" id="S4.SS3.p1.4.1">Total Function NP Search Problems, decision-tree view</em>), as defined by Göös et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib15" title="">15</a>]</cite>, captures total search problems (i.e., problems where every possible instance has a solution) that are specified by a long hidden bitstring accessible only through a bit querying oracle. To lie in <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><msup id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml"><mi id="S4.SS3.p1.2.m2.1.1.2" xref="S4.SS3.p1.2.m2.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p1.2.m2.1.1.3" xref="S4.SS3.p1.2.m2.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p1.2.m2.1.1.3a.cmml" xref="S4.SS3.p1.2.m2.1.1.3"><mtext id="S4.SS3.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p1.2.m2.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>, solutions must be efficiently verifiable (i.e., by decision trees of depth poly-logarithmic in the length of the bitstring). To lie in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><msup id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml"><mi id="S4.SS3.p1.3.m3.1.1.2" xref="S4.SS3.p1.3.m3.1.1.2.cmml">𝖥𝖯</mi><mtext id="S4.SS3.p1.3.m3.1.1.3" xref="S4.SS3.p1.3.m3.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><apply id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m3.1.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS3.p1.3.m3.1.1.2.cmml" xref="S4.SS3.p1.3.m3.1.1.2">𝖥𝖯</ci><ci id="S4.SS3.p1.3.m3.1.1.3a.cmml" xref="S4.SS3.p1.3.m3.1.1.3"><mtext id="S4.SS3.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p1.3.m3.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>, the subclass of efficiently solvable problems in <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><msup id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml"><mi id="S4.SS3.p1.4.m4.1.1.2" xref="S4.SS3.p1.4.m4.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p1.4.m4.1.1.3" xref="S4.SS3.p1.4.m4.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1">superscript</csymbol><ci id="S4.SS3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p1.4.m4.1.1.3a.cmml" xref="S4.SS3.p1.4.m4.1.1.3"><mtext id="S4.SS3.p1.4.m4.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p1.4.m4.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>, one must be able to efficiently <em class="ltx_emph ltx_font_italic" id="S4.SS3.p1.4.2">find</em> a solution as well, i.e., there has to be a decision tree of poly-logarithmic depth that always outputs a correct solution.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.1">To fit a <em class="ltx_emph ltx_font_italic" id="S4.SS3.p2.1.1">promise</em> search problem into <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><msup id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml"><mi id="S4.SS3.p2.1.m1.1.1.2" xref="S4.SS3.p2.1.m1.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p2.1.m1.1.1.3" xref="S4.SS3.p2.1.m1.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><apply id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS3.p2.1.m1.1.1.2.cmml" xref="S4.SS3.p2.1.m1.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p2.1.m1.1.1.3a.cmml" xref="S4.SS3.p2.1.m1.1.1.3"><mtext id="S4.SS3.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p2.1.m1.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>, we must introduce solution types, usually called <em class="ltx_emph ltx_font_italic" id="S4.SS3.p2.1.2">violations</em>, that are guaranteed to exist when the promise is violated.</p> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.24">Chen, Li, and Yannakakis <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib5" title="">5</a>]</cite> showed how the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><msub id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml"><mi id="S4.SS3.p3.1.m1.1.1.2" mathvariant="normal" xref="S4.SS3.p3.1.m1.1.1.2.cmml">ℓ</mi><mi id="S4.SS3.p3.1.m1.1.1.3" mathvariant="normal" xref="S4.SS3.p3.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><apply id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.1.m1.1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p3.1.m1.1.1.2.cmml" xref="S4.SS3.p3.1.m1.1.1.2">ℓ</ci><infinity 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, 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> extends to a <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS3.p3.4.m4.1"><semantics id="S4.SS3.p3.4.m4.1a"><mi id="S4.SS3.p3.4.m4.1.1" xref="S4.SS3.p3.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.4.m4.1b"><ci id="S4.SS3.p3.4.m4.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.4.m4.1d">italic_λ</annotation></semantics></math>-contraction <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p3.5.m5.1"><semantics id="S4.SS3.p3.5.m5.1a"><msup id="S4.SS3.p3.5.m5.1.1" xref="S4.SS3.p3.5.m5.1.1.cmml"><mi id="S4.SS3.p3.5.m5.1.1.2" xref="S4.SS3.p3.5.m5.1.1.2.cmml">f</mi><mo id="S4.SS3.p3.5.m5.1.1.3" xref="S4.SS3.p3.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.5.m5.1b"><apply id="S4.SS3.p3.5.m5.1.1.cmml" xref="S4.SS3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.5.m5.1.1.1.cmml" xref="S4.SS3.p3.5.m5.1.1">superscript</csymbol><ci id="S4.SS3.p3.5.m5.1.1.2.cmml" xref="S4.SS3.p3.5.m5.1.1.2">𝑓</ci><ci id="S4.SS3.p3.5.m5.1.1.3.cmml" xref="S4.SS3.p3.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.5.m5.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.5.m5.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS3.p3.6.m6.2"><semantics id="S4.SS3.p3.6.m6.2a"><msup id="S4.SS3.p3.6.m6.2.3" xref="S4.SS3.p3.6.m6.2.3.cmml"><mrow id="S4.SS3.p3.6.m6.2.3.2.2" xref="S4.SS3.p3.6.m6.2.3.2.1.cmml"><mo id="S4.SS3.p3.6.m6.2.3.2.2.1" stretchy="false" xref="S4.SS3.p3.6.m6.2.3.2.1.cmml">[</mo><mn id="S4.SS3.p3.6.m6.1.1" xref="S4.SS3.p3.6.m6.1.1.cmml">0</mn><mo id="S4.SS3.p3.6.m6.2.3.2.2.2" xref="S4.SS3.p3.6.m6.2.3.2.1.cmml">,</mo><mn id="S4.SS3.p3.6.m6.2.2" xref="S4.SS3.p3.6.m6.2.2.cmml">1</mn><mo id="S4.SS3.p3.6.m6.2.3.2.2.3" stretchy="false" xref="S4.SS3.p3.6.m6.2.3.2.1.cmml">]</mo></mrow><mi id="S4.SS3.p3.6.m6.2.3.3" xref="S4.SS3.p3.6.m6.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.6.m6.2b"><apply id="S4.SS3.p3.6.m6.2.3.cmml" xref="S4.SS3.p3.6.m6.2.3"><csymbol cd="ambiguous" id="S4.SS3.p3.6.m6.2.3.1.cmml" xref="S4.SS3.p3.6.m6.2.3">superscript</csymbol><interval closure="closed" id="S4.SS3.p3.6.m6.2.3.2.1.cmml" xref="S4.SS3.p3.6.m6.2.3.2.2"><cn id="S4.SS3.p3.6.m6.1.1.cmml" type="integer" xref="S4.SS3.p3.6.m6.1.1">0</cn><cn id="S4.SS3.p3.6.m6.2.2.cmml" type="integer" xref="S4.SS3.p3.6.m6.2.2">1</cn></interval><ci id="S4.SS3.p3.6.m6.2.3.3.cmml" xref="S4.SS3.p3.6.m6.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.6.m6.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.6.m6.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> if and only if <math alttext="f" class="ltx_Math" display="inline" id="S4.SS3.p3.7.m7.1"><semantics id="S4.SS3.p3.7.m7.1a"><mi id="S4.SS3.p3.7.m7.1.1" xref="S4.SS3.p3.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.7.m7.1b"><ci id="S4.SS3.p3.7.m7.1.1.cmml" xref="S4.SS3.p3.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.7.m7.1d">italic_f</annotation></semantics></math> is <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS3.p3.8.m8.1"><semantics id="S4.SS3.p3.8.m8.1a"><mi id="S4.SS3.p3.8.m8.1.1" xref="S4.SS3.p3.8.m8.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.8.m8.1b"><ci id="S4.SS3.p3.8.m8.1.1.cmml" xref="S4.SS3.p3.8.m8.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.8.m8.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.8.m8.1d">italic_λ</annotation></semantics></math>-contracting for all pairs of points in <math alttext="G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS3.p3.9.m9.1"><semantics id="S4.SS3.p3.9.m9.1a"><msubsup id="S4.SS3.p3.9.m9.1.1" xref="S4.SS3.p3.9.m9.1.1.cmml"><mi id="S4.SS3.p3.9.m9.1.1.2.2" xref="S4.SS3.p3.9.m9.1.1.2.2.cmml">G</mi><mi id="S4.SS3.p3.9.m9.1.1.3" xref="S4.SS3.p3.9.m9.1.1.3.cmml">b</mi><mi id="S4.SS3.p3.9.m9.1.1.2.3" xref="S4.SS3.p3.9.m9.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.9.m9.1b"><apply id="S4.SS3.p3.9.m9.1.1.cmml" xref="S4.SS3.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.9.m9.1.1.1.cmml" xref="S4.SS3.p3.9.m9.1.1">subscript</csymbol><apply id="S4.SS3.p3.9.m9.1.1.2.cmml" xref="S4.SS3.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.9.m9.1.1.2.1.cmml" xref="S4.SS3.p3.9.m9.1.1">superscript</csymbol><ci id="S4.SS3.p3.9.m9.1.1.2.2.cmml" xref="S4.SS3.p3.9.m9.1.1.2.2">𝐺</ci><ci id="S4.SS3.p3.9.m9.1.1.2.3.cmml" xref="S4.SS3.p3.9.m9.1.1.2.3">𝑑</ci></apply><ci id="S4.SS3.p3.9.m9.1.1.3.cmml" xref="S4.SS3.p3.9.m9.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.9.m9.1c">G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.9.m9.1d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, it suffices to introduce violations consisting of two points <math alttext="x,y\in G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS3.p3.10.m10.2"><semantics id="S4.SS3.p3.10.m10.2a"><mrow id="S4.SS3.p3.10.m10.2.3" xref="S4.SS3.p3.10.m10.2.3.cmml"><mrow id="S4.SS3.p3.10.m10.2.3.2.2" xref="S4.SS3.p3.10.m10.2.3.2.1.cmml"><mi id="S4.SS3.p3.10.m10.1.1" xref="S4.SS3.p3.10.m10.1.1.cmml">x</mi><mo id="S4.SS3.p3.10.m10.2.3.2.2.1" xref="S4.SS3.p3.10.m10.2.3.2.1.cmml">,</mo><mi id="S4.SS3.p3.10.m10.2.2" xref="S4.SS3.p3.10.m10.2.2.cmml">y</mi></mrow><mo id="S4.SS3.p3.10.m10.2.3.1" xref="S4.SS3.p3.10.m10.2.3.1.cmml">∈</mo><msubsup id="S4.SS3.p3.10.m10.2.3.3" xref="S4.SS3.p3.10.m10.2.3.3.cmml"><mi id="S4.SS3.p3.10.m10.2.3.3.2.2" xref="S4.SS3.p3.10.m10.2.3.3.2.2.cmml">G</mi><mi id="S4.SS3.p3.10.m10.2.3.3.3" xref="S4.SS3.p3.10.m10.2.3.3.3.cmml">b</mi><mi id="S4.SS3.p3.10.m10.2.3.3.2.3" xref="S4.SS3.p3.10.m10.2.3.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.10.m10.2b"><apply id="S4.SS3.p3.10.m10.2.3.cmml" xref="S4.SS3.p3.10.m10.2.3"><in id="S4.SS3.p3.10.m10.2.3.1.cmml" xref="S4.SS3.p3.10.m10.2.3.1"></in><list id="S4.SS3.p3.10.m10.2.3.2.1.cmml" xref="S4.SS3.p3.10.m10.2.3.2.2"><ci id="S4.SS3.p3.10.m10.1.1.cmml" xref="S4.SS3.p3.10.m10.1.1">𝑥</ci><ci id="S4.SS3.p3.10.m10.2.2.cmml" xref="S4.SS3.p3.10.m10.2.2">𝑦</ci></list><apply id="S4.SS3.p3.10.m10.2.3.3.cmml" xref="S4.SS3.p3.10.m10.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.10.m10.2.3.3.1.cmml" xref="S4.SS3.p3.10.m10.2.3.3">subscript</csymbol><apply id="S4.SS3.p3.10.m10.2.3.3.2.cmml" xref="S4.SS3.p3.10.m10.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.10.m10.2.3.3.2.1.cmml" xref="S4.SS3.p3.10.m10.2.3.3">superscript</csymbol><ci id="S4.SS3.p3.10.m10.2.3.3.2.2.cmml" xref="S4.SS3.p3.10.m10.2.3.3.2.2">𝐺</ci><ci id="S4.SS3.p3.10.m10.2.3.3.2.3.cmml" xref="S4.SS3.p3.10.m10.2.3.3.2.3">𝑑</ci></apply><ci id="S4.SS3.p3.10.m10.2.3.3.3.cmml" xref="S4.SS3.p3.10.m10.2.3.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.10.m10.2c">x,y\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.10.m10.2d">italic_x , italic_y ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> for which <math alttext="\lVert f(x)-f(y)\rVert_{\infty}>\lambda\cdot\lVert x-y\rVert_{\infty}" class="ltx_Math" display="inline" id="S4.SS3.p3.11.m11.4"><semantics id="S4.SS3.p3.11.m11.4a"><mrow id="S4.SS3.p3.11.m11.4.4" xref="S4.SS3.p3.11.m11.4.4.cmml"><msub id="S4.SS3.p3.11.m11.3.3.1" xref="S4.SS3.p3.11.m11.3.3.1.cmml"><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1" xref="S4.SS3.p3.11.m11.3.3.1.1.2.cmml"><mo fence="true" id="S4.SS3.p3.11.m11.3.3.1.1.1.2" rspace="0em" xref="S4.SS3.p3.11.m11.3.3.1.1.2.1.cmml">∥</mo><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1.1" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.cmml"><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.cmml"><mi id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.2" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.2.cmml">f</mi><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.1" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.3.2" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.cmml"><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.cmml">(</mo><mi id="S4.SS3.p3.11.m11.1.1" xref="S4.SS3.p3.11.m11.1.1.cmml">x</mi><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.1" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.cmml"><mi id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.2" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.2.cmml">f</mi><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.1" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.3.2" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.cmml"><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.3.2.1" stretchy="false" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.cmml">(</mo><mi id="S4.SS3.p3.11.m11.2.2" xref="S4.SS3.p3.11.m11.2.2.cmml">y</mi><mo id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.3.2.2" stretchy="false" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S4.SS3.p3.11.m11.3.3.1.1.1.3" lspace="0em" xref="S4.SS3.p3.11.m11.3.3.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS3.p3.11.m11.3.3.1.3" mathvariant="normal" xref="S4.SS3.p3.11.m11.3.3.1.3.cmml">∞</mi></msub><mo id="S4.SS3.p3.11.m11.4.4.3" xref="S4.SS3.p3.11.m11.4.4.3.cmml">></mo><mrow id="S4.SS3.p3.11.m11.4.4.2" xref="S4.SS3.p3.11.m11.4.4.2.cmml"><mi id="S4.SS3.p3.11.m11.4.4.2.3" xref="S4.SS3.p3.11.m11.4.4.2.3.cmml">λ</mi><mo id="S4.SS3.p3.11.m11.4.4.2.2" lspace="0.222em" xref="S4.SS3.p3.11.m11.4.4.2.2.cmml">⋅</mo><msub id="S4.SS3.p3.11.m11.4.4.2.1" xref="S4.SS3.p3.11.m11.4.4.2.1.cmml"><mrow id="S4.SS3.p3.11.m11.4.4.2.1.1.1" xref="S4.SS3.p3.11.m11.4.4.2.1.1.2.cmml"><mo fence="true" id="S4.SS3.p3.11.m11.4.4.2.1.1.1.2" lspace="0.055em" rspace="0em" xref="S4.SS3.p3.11.m11.4.4.2.1.1.2.1.cmml">∥</mo><mrow id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.cmml"><mi id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.2" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.2.cmml">x</mi><mo id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.1" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.1.cmml">−</mo><mi id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.3" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.3.cmml">y</mi></mrow><mo fence="true" id="S4.SS3.p3.11.m11.4.4.2.1.1.1.3" lspace="0em" xref="S4.SS3.p3.11.m11.4.4.2.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.SS3.p3.11.m11.4.4.2.1.3" mathvariant="normal" xref="S4.SS3.p3.11.m11.4.4.2.1.3.cmml">∞</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.11.m11.4b"><apply id="S4.SS3.p3.11.m11.4.4.cmml" xref="S4.SS3.p3.11.m11.4.4"><gt id="S4.SS3.p3.11.m11.4.4.3.cmml" xref="S4.SS3.p3.11.m11.4.4.3"></gt><apply id="S4.SS3.p3.11.m11.3.3.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1"><csymbol cd="ambiguous" id="S4.SS3.p3.11.m11.3.3.1.2.cmml" xref="S4.SS3.p3.11.m11.3.3.1">subscript</csymbol><apply id="S4.SS3.p3.11.m11.3.3.1.1.2.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1"><csymbol cd="latexml" id="S4.SS3.p3.11.m11.3.3.1.1.2.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.SS3.p3.11.m11.3.3.1.1.1.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1"><minus id="S4.SS3.p3.11.m11.3.3.1.1.1.1.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.1"></minus><apply id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2"><times id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.1"></times><ci id="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.2.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.2.2">𝑓</ci><ci id="S4.SS3.p3.11.m11.1.1.cmml" xref="S4.SS3.p3.11.m11.1.1">𝑥</ci></apply><apply id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3"><times id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.1.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.1"></times><ci id="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.2.cmml" xref="S4.SS3.p3.11.m11.3.3.1.1.1.1.3.2">𝑓</ci><ci id="S4.SS3.p3.11.m11.2.2.cmml" xref="S4.SS3.p3.11.m11.2.2">𝑦</ci></apply></apply></apply><infinity id="S4.SS3.p3.11.m11.3.3.1.3.cmml" xref="S4.SS3.p3.11.m11.3.3.1.3"></infinity></apply><apply id="S4.SS3.p3.11.m11.4.4.2.cmml" xref="S4.SS3.p3.11.m11.4.4.2"><ci id="S4.SS3.p3.11.m11.4.4.2.2.cmml" xref="S4.SS3.p3.11.m11.4.4.2.2">⋅</ci><ci id="S4.SS3.p3.11.m11.4.4.2.3.cmml" xref="S4.SS3.p3.11.m11.4.4.2.3">𝜆</ci><apply id="S4.SS3.p3.11.m11.4.4.2.1.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1"><csymbol cd="ambiguous" id="S4.SS3.p3.11.m11.4.4.2.1.2.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1">subscript</csymbol><apply id="S4.SS3.p3.11.m11.4.4.2.1.1.2.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1"><csymbol cd="latexml" id="S4.SS3.p3.11.m11.4.4.2.1.1.2.1.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1"><minus id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.1.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.1"></minus><ci id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.2.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.2">𝑥</ci><ci id="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.3.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.1.1.1.3">𝑦</ci></apply></apply><infinity id="S4.SS3.p3.11.m11.4.4.2.1.3.cmml" xref="S4.SS3.p3.11.m11.4.4.2.1.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.11.m11.4c">\lVert f(x)-f(y)\rVert_{\infty}>\lambda\cdot\lVert x-y\rVert_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.11.m11.4d">∥ italic_f ( italic_x ) - italic_f ( italic_y ) ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT > italic_λ ⋅ ∥ italic_x - italic_y ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>. If a function <math alttext="f:G^{d}_{b}\rightarrow G^{d}_{b^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p3.12.m12.1"><semantics id="S4.SS3.p3.12.m12.1a"><mrow id="S4.SS3.p3.12.m12.1.1" xref="S4.SS3.p3.12.m12.1.1.cmml"><mi id="S4.SS3.p3.12.m12.1.1.2" xref="S4.SS3.p3.12.m12.1.1.2.cmml">f</mi><mo id="S4.SS3.p3.12.m12.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p3.12.m12.1.1.1.cmml">:</mo><mrow id="S4.SS3.p3.12.m12.1.1.3" xref="S4.SS3.p3.12.m12.1.1.3.cmml"><msubsup id="S4.SS3.p3.12.m12.1.1.3.2" xref="S4.SS3.p3.12.m12.1.1.3.2.cmml"><mi id="S4.SS3.p3.12.m12.1.1.3.2.2.2" xref="S4.SS3.p3.12.m12.1.1.3.2.2.2.cmml">G</mi><mi id="S4.SS3.p3.12.m12.1.1.3.2.3" xref="S4.SS3.p3.12.m12.1.1.3.2.3.cmml">b</mi><mi id="S4.SS3.p3.12.m12.1.1.3.2.2.3" xref="S4.SS3.p3.12.m12.1.1.3.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS3.p3.12.m12.1.1.3.1" stretchy="false" xref="S4.SS3.p3.12.m12.1.1.3.1.cmml">→</mo><msubsup id="S4.SS3.p3.12.m12.1.1.3.3" xref="S4.SS3.p3.12.m12.1.1.3.3.cmml"><mi id="S4.SS3.p3.12.m12.1.1.3.3.2.2" xref="S4.SS3.p3.12.m12.1.1.3.3.2.2.cmml">G</mi><msup id="S4.SS3.p3.12.m12.1.1.3.3.3" xref="S4.SS3.p3.12.m12.1.1.3.3.3.cmml"><mi id="S4.SS3.p3.12.m12.1.1.3.3.3.2" xref="S4.SS3.p3.12.m12.1.1.3.3.3.2.cmml">b</mi><mo id="S4.SS3.p3.12.m12.1.1.3.3.3.3" xref="S4.SS3.p3.12.m12.1.1.3.3.3.3.cmml">′</mo></msup><mi id="S4.SS3.p3.12.m12.1.1.3.3.2.3" xref="S4.SS3.p3.12.m12.1.1.3.3.2.3.cmml">d</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.12.m12.1b"><apply id="S4.SS3.p3.12.m12.1.1.cmml" xref="S4.SS3.p3.12.m12.1.1"><ci id="S4.SS3.p3.12.m12.1.1.1.cmml" xref="S4.SS3.p3.12.m12.1.1.1">:</ci><ci id="S4.SS3.p3.12.m12.1.1.2.cmml" xref="S4.SS3.p3.12.m12.1.1.2">𝑓</ci><apply id="S4.SS3.p3.12.m12.1.1.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3"><ci id="S4.SS3.p3.12.m12.1.1.3.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.1">→</ci><apply id="S4.SS3.p3.12.m12.1.1.3.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m12.1.1.3.2.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2">subscript</csymbol><apply id="S4.SS3.p3.12.m12.1.1.3.2.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m12.1.1.3.2.2.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2">superscript</csymbol><ci id="S4.SS3.p3.12.m12.1.1.3.2.2.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2.2.2">𝐺</ci><ci id="S4.SS3.p3.12.m12.1.1.3.2.2.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2.2.3">𝑑</ci></apply><ci id="S4.SS3.p3.12.m12.1.1.3.2.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.2.3">𝑏</ci></apply><apply id="S4.SS3.p3.12.m12.1.1.3.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m12.1.1.3.3.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3">subscript</csymbol><apply id="S4.SS3.p3.12.m12.1.1.3.3.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m12.1.1.3.3.2.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p3.12.m12.1.1.3.3.2.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.2.2">𝐺</ci><ci id="S4.SS3.p3.12.m12.1.1.3.3.2.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.2.3">𝑑</ci></apply><apply id="S4.SS3.p3.12.m12.1.1.3.3.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.3"><csymbol cd="ambiguous" id="S4.SS3.p3.12.m12.1.1.3.3.3.1.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.3">superscript</csymbol><ci id="S4.SS3.p3.12.m12.1.1.3.3.3.2.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.3.2">𝑏</ci><ci id="S4.SS3.p3.12.m12.1.1.3.3.3.3.cmml" xref="S4.SS3.p3.12.m12.1.1.3.3.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.12.m12.1c">f:G^{d}_{b}\rightarrow G^{d}_{b^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.12.m12.1d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (for <math alttext="b^{\prime}\in\operatorname{poly}(b)" class="ltx_Math" display="inline" id="S4.SS3.p3.13.m13.2"><semantics id="S4.SS3.p3.13.m13.2a"><mrow id="S4.SS3.p3.13.m13.2.3" xref="S4.SS3.p3.13.m13.2.3.cmml"><msup id="S4.SS3.p3.13.m13.2.3.2" xref="S4.SS3.p3.13.m13.2.3.2.cmml"><mi id="S4.SS3.p3.13.m13.2.3.2.2" xref="S4.SS3.p3.13.m13.2.3.2.2.cmml">b</mi><mo id="S4.SS3.p3.13.m13.2.3.2.3" xref="S4.SS3.p3.13.m13.2.3.2.3.cmml">′</mo></msup><mo id="S4.SS3.p3.13.m13.2.3.1" xref="S4.SS3.p3.13.m13.2.3.1.cmml">∈</mo><mrow id="S4.SS3.p3.13.m13.2.3.3.2" xref="S4.SS3.p3.13.m13.2.3.3.1.cmml"><mi id="S4.SS3.p3.13.m13.1.1" xref="S4.SS3.p3.13.m13.1.1.cmml">poly</mi><mo id="S4.SS3.p3.13.m13.2.3.3.2a" xref="S4.SS3.p3.13.m13.2.3.3.1.cmml">⁡</mo><mrow id="S4.SS3.p3.13.m13.2.3.3.2.1" xref="S4.SS3.p3.13.m13.2.3.3.1.cmml"><mo id="S4.SS3.p3.13.m13.2.3.3.2.1.1" stretchy="false" xref="S4.SS3.p3.13.m13.2.3.3.1.cmml">(</mo><mi id="S4.SS3.p3.13.m13.2.2" xref="S4.SS3.p3.13.m13.2.2.cmml">b</mi><mo id="S4.SS3.p3.13.m13.2.3.3.2.1.2" stretchy="false" xref="S4.SS3.p3.13.m13.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.13.m13.2b"><apply id="S4.SS3.p3.13.m13.2.3.cmml" xref="S4.SS3.p3.13.m13.2.3"><in id="S4.SS3.p3.13.m13.2.3.1.cmml" xref="S4.SS3.p3.13.m13.2.3.1"></in><apply id="S4.SS3.p3.13.m13.2.3.2.cmml" xref="S4.SS3.p3.13.m13.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p3.13.m13.2.3.2.1.cmml" xref="S4.SS3.p3.13.m13.2.3.2">superscript</csymbol><ci id="S4.SS3.p3.13.m13.2.3.2.2.cmml" xref="S4.SS3.p3.13.m13.2.3.2.2">𝑏</ci><ci id="S4.SS3.p3.13.m13.2.3.2.3.cmml" xref="S4.SS3.p3.13.m13.2.3.2.3">′</ci></apply><apply id="S4.SS3.p3.13.m13.2.3.3.1.cmml" xref="S4.SS3.p3.13.m13.2.3.3.2"><ci id="S4.SS3.p3.13.m13.1.1.cmml" xref="S4.SS3.p3.13.m13.1.1">poly</ci><ci id="S4.SS3.p3.13.m13.2.2.cmml" xref="S4.SS3.p3.13.m13.2.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.13.m13.2c">b^{\prime}\in\operatorname{poly}(b)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.13.m13.2d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_poly ( italic_b )</annotation></semantics></math>) is now encoded as a bitstring of length <math alttext="2^{d\cdot b}\cdot b^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p3.14.m14.1"><semantics id="S4.SS3.p3.14.m14.1a"><mrow id="S4.SS3.p3.14.m14.1.1" xref="S4.SS3.p3.14.m14.1.1.cmml"><msup id="S4.SS3.p3.14.m14.1.1.2" xref="S4.SS3.p3.14.m14.1.1.2.cmml"><mn id="S4.SS3.p3.14.m14.1.1.2.2" xref="S4.SS3.p3.14.m14.1.1.2.2.cmml">2</mn><mrow id="S4.SS3.p3.14.m14.1.1.2.3" xref="S4.SS3.p3.14.m14.1.1.2.3.cmml"><mi id="S4.SS3.p3.14.m14.1.1.2.3.2" xref="S4.SS3.p3.14.m14.1.1.2.3.2.cmml">d</mi><mo id="S4.SS3.p3.14.m14.1.1.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.14.m14.1.1.2.3.1.cmml">⋅</mo><mi id="S4.SS3.p3.14.m14.1.1.2.3.3" xref="S4.SS3.p3.14.m14.1.1.2.3.3.cmml">b</mi></mrow></msup><mo id="S4.SS3.p3.14.m14.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.14.m14.1.1.1.cmml">⋅</mo><msup id="S4.SS3.p3.14.m14.1.1.3" xref="S4.SS3.p3.14.m14.1.1.3.cmml"><mi id="S4.SS3.p3.14.m14.1.1.3.2" xref="S4.SS3.p3.14.m14.1.1.3.2.cmml">b</mi><mo id="S4.SS3.p3.14.m14.1.1.3.3" xref="S4.SS3.p3.14.m14.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.14.m14.1b"><apply id="S4.SS3.p3.14.m14.1.1.cmml" xref="S4.SS3.p3.14.m14.1.1"><ci id="S4.SS3.p3.14.m14.1.1.1.cmml" xref="S4.SS3.p3.14.m14.1.1.1">⋅</ci><apply id="S4.SS3.p3.14.m14.1.1.2.cmml" xref="S4.SS3.p3.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.14.m14.1.1.2.1.cmml" xref="S4.SS3.p3.14.m14.1.1.2">superscript</csymbol><cn id="S4.SS3.p3.14.m14.1.1.2.2.cmml" type="integer" xref="S4.SS3.p3.14.m14.1.1.2.2">2</cn><apply id="S4.SS3.p3.14.m14.1.1.2.3.cmml" xref="S4.SS3.p3.14.m14.1.1.2.3"><ci id="S4.SS3.p3.14.m14.1.1.2.3.1.cmml" xref="S4.SS3.p3.14.m14.1.1.2.3.1">⋅</ci><ci id="S4.SS3.p3.14.m14.1.1.2.3.2.cmml" xref="S4.SS3.p3.14.m14.1.1.2.3.2">𝑑</ci><ci id="S4.SS3.p3.14.m14.1.1.2.3.3.cmml" xref="S4.SS3.p3.14.m14.1.1.2.3.3">𝑏</ci></apply></apply><apply id="S4.SS3.p3.14.m14.1.1.3.cmml" xref="S4.SS3.p3.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.14.m14.1.1.3.1.cmml" xref="S4.SS3.p3.14.m14.1.1.3">superscript</csymbol><ci id="S4.SS3.p3.14.m14.1.1.3.2.cmml" xref="S4.SS3.p3.14.m14.1.1.3.2">𝑏</ci><ci id="S4.SS3.p3.14.m14.1.1.3.3.cmml" xref="S4.SS3.p3.14.m14.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.14.m14.1c">2^{d\cdot b}\cdot b^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.14.m14.1d">2 start_POSTSUPERSCRIPT italic_d ⋅ italic_b end_POSTSUPERSCRIPT ⋅ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by simply concatenating the output values <math alttext="f(x)" class="ltx_Math" display="inline" id="S4.SS3.p3.15.m15.1"><semantics id="S4.SS3.p3.15.m15.1a"><mrow id="S4.SS3.p3.15.m15.1.2" xref="S4.SS3.p3.15.m15.1.2.cmml"><mi id="S4.SS3.p3.15.m15.1.2.2" xref="S4.SS3.p3.15.m15.1.2.2.cmml">f</mi><mo id="S4.SS3.p3.15.m15.1.2.1" xref="S4.SS3.p3.15.m15.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p3.15.m15.1.2.3.2" xref="S4.SS3.p3.15.m15.1.2.cmml"><mo id="S4.SS3.p3.15.m15.1.2.3.2.1" stretchy="false" xref="S4.SS3.p3.15.m15.1.2.cmml">(</mo><mi id="S4.SS3.p3.15.m15.1.1" xref="S4.SS3.p3.15.m15.1.1.cmml">x</mi><mo id="S4.SS3.p3.15.m15.1.2.3.2.2" stretchy="false" xref="S4.SS3.p3.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.15.m15.1b"><apply id="S4.SS3.p3.15.m15.1.2.cmml" xref="S4.SS3.p3.15.m15.1.2"><times id="S4.SS3.p3.15.m15.1.2.1.cmml" xref="S4.SS3.p3.15.m15.1.2.1"></times><ci id="S4.SS3.p3.15.m15.1.2.2.cmml" xref="S4.SS3.p3.15.m15.1.2.2">𝑓</ci><ci id="S4.SS3.p3.15.m15.1.1.cmml" xref="S4.SS3.p3.15.m15.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.15.m15.1c">f(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.15.m15.1d">italic_f ( italic_x )</annotation></semantics></math> for all <math alttext="x\in G^{d}_{b}" class="ltx_Math" display="inline" id="S4.SS3.p3.16.m16.1"><semantics id="S4.SS3.p3.16.m16.1a"><mrow id="S4.SS3.p3.16.m16.1.1" xref="S4.SS3.p3.16.m16.1.1.cmml"><mi id="S4.SS3.p3.16.m16.1.1.2" xref="S4.SS3.p3.16.m16.1.1.2.cmml">x</mi><mo id="S4.SS3.p3.16.m16.1.1.1" xref="S4.SS3.p3.16.m16.1.1.1.cmml">∈</mo><msubsup id="S4.SS3.p3.16.m16.1.1.3" xref="S4.SS3.p3.16.m16.1.1.3.cmml"><mi id="S4.SS3.p3.16.m16.1.1.3.2.2" xref="S4.SS3.p3.16.m16.1.1.3.2.2.cmml">G</mi><mi id="S4.SS3.p3.16.m16.1.1.3.3" xref="S4.SS3.p3.16.m16.1.1.3.3.cmml">b</mi><mi id="S4.SS3.p3.16.m16.1.1.3.2.3" xref="S4.SS3.p3.16.m16.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.16.m16.1b"><apply id="S4.SS3.p3.16.m16.1.1.cmml" xref="S4.SS3.p3.16.m16.1.1"><in id="S4.SS3.p3.16.m16.1.1.1.cmml" xref="S4.SS3.p3.16.m16.1.1.1"></in><ci id="S4.SS3.p3.16.m16.1.1.2.cmml" xref="S4.SS3.p3.16.m16.1.1.2">𝑥</ci><apply id="S4.SS3.p3.16.m16.1.1.3.cmml" xref="S4.SS3.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.16.m16.1.1.3.1.cmml" xref="S4.SS3.p3.16.m16.1.1.3">subscript</csymbol><apply id="S4.SS3.p3.16.m16.1.1.3.2.cmml" xref="S4.SS3.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.16.m16.1.1.3.2.1.cmml" xref="S4.SS3.p3.16.m16.1.1.3">superscript</csymbol><ci id="S4.SS3.p3.16.m16.1.1.3.2.2.cmml" xref="S4.SS3.p3.16.m16.1.1.3.2.2">𝐺</ci><ci id="S4.SS3.p3.16.m16.1.1.3.2.3.cmml" xref="S4.SS3.p3.16.m16.1.1.3.2.3">𝑑</ci></apply><ci id="S4.SS3.p3.16.m16.1.1.3.3.cmml" xref="S4.SS3.p3.16.m16.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.16.m16.1c">x\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.16.m16.1d">italic_x ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, both <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.p3.17.m17.1"><semantics id="S4.SS3.p3.17.m17.1a"><mi id="S4.SS3.p3.17.m17.1.1" xref="S4.SS3.p3.17.m17.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.17.m17.1b"><ci id="S4.SS3.p3.17.m17.1.1.cmml" xref="S4.SS3.p3.17.m17.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.17.m17.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.17.m17.1d">italic_ε</annotation></semantics></math>-approximate fixpoints as well as these violations can be verified by querying <math alttext="\mathcal{O}(b^{\prime})\in\operatorname{poly}\log(2^{d\cdot b}\cdot b^{\prime})" class="ltx_Math" display="inline" id="S4.SS3.p3.18.m18.3"><semantics id="S4.SS3.p3.18.m18.3a"><mrow id="S4.SS3.p3.18.m18.3.3" xref="S4.SS3.p3.18.m18.3.3.cmml"><mrow id="S4.SS3.p3.18.m18.2.2.1" xref="S4.SS3.p3.18.m18.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p3.18.m18.2.2.1.3" xref="S4.SS3.p3.18.m18.2.2.1.3.cmml">𝒪</mi><mo id="S4.SS3.p3.18.m18.2.2.1.2" xref="S4.SS3.p3.18.m18.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS3.p3.18.m18.2.2.1.1.1" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.cmml"><mo id="S4.SS3.p3.18.m18.2.2.1.1.1.2" stretchy="false" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.cmml">(</mo><msup id="S4.SS3.p3.18.m18.2.2.1.1.1.1" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.cmml"><mi id="S4.SS3.p3.18.m18.2.2.1.1.1.1.2" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.2.cmml">b</mi><mo id="S4.SS3.p3.18.m18.2.2.1.1.1.1.3" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS3.p3.18.m18.2.2.1.1.1.3" stretchy="false" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p3.18.m18.3.3.3" xref="S4.SS3.p3.18.m18.3.3.3.cmml">∈</mo><mrow id="S4.SS3.p3.18.m18.3.3.2" xref="S4.SS3.p3.18.m18.3.3.2.cmml"><mi id="S4.SS3.p3.18.m18.3.3.2.2" xref="S4.SS3.p3.18.m18.3.3.2.2.cmml">poly</mi><mo id="S4.SS3.p3.18.m18.3.3.2a" lspace="0.167em" xref="S4.SS3.p3.18.m18.3.3.2.cmml">⁡</mo><mrow id="S4.SS3.p3.18.m18.3.3.2.1.1" xref="S4.SS3.p3.18.m18.3.3.2.1.2.cmml"><mi id="S4.SS3.p3.18.m18.1.1" xref="S4.SS3.p3.18.m18.1.1.cmml">log</mi><mo id="S4.SS3.p3.18.m18.3.3.2.1.1a" xref="S4.SS3.p3.18.m18.3.3.2.1.2.cmml">⁡</mo><mrow id="S4.SS3.p3.18.m18.3.3.2.1.1.1" xref="S4.SS3.p3.18.m18.3.3.2.1.2.cmml"><mo id="S4.SS3.p3.18.m18.3.3.2.1.1.1.2" stretchy="false" xref="S4.SS3.p3.18.m18.3.3.2.1.2.cmml">(</mo><mrow id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.cmml"><msup id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.cmml"><mn id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.2" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.2.cmml">2</mn><mrow id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.cmml"><mi id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.2" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.2.cmml">d</mi><mo id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.1.cmml">⋅</mo><mi id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.3" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.3.cmml">b</mi></mrow></msup><mo id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.1.cmml">⋅</mo><msup id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.cmml"><mi id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.2" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.2.cmml">b</mi><mo id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.3" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.SS3.p3.18.m18.3.3.2.1.1.1.3" stretchy="false" xref="S4.SS3.p3.18.m18.3.3.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.18.m18.3b"><apply id="S4.SS3.p3.18.m18.3.3.cmml" xref="S4.SS3.p3.18.m18.3.3"><in id="S4.SS3.p3.18.m18.3.3.3.cmml" xref="S4.SS3.p3.18.m18.3.3.3"></in><apply id="S4.SS3.p3.18.m18.2.2.1.cmml" xref="S4.SS3.p3.18.m18.2.2.1"><times id="S4.SS3.p3.18.m18.2.2.1.2.cmml" xref="S4.SS3.p3.18.m18.2.2.1.2"></times><ci id="S4.SS3.p3.18.m18.2.2.1.3.cmml" xref="S4.SS3.p3.18.m18.2.2.1.3">𝒪</ci><apply id="S4.SS3.p3.18.m18.2.2.1.1.1.1.cmml" xref="S4.SS3.p3.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.18.m18.2.2.1.1.1.1.1.cmml" xref="S4.SS3.p3.18.m18.2.2.1.1.1">superscript</csymbol><ci id="S4.SS3.p3.18.m18.2.2.1.1.1.1.2.cmml" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.2">𝑏</ci><ci id="S4.SS3.p3.18.m18.2.2.1.1.1.1.3.cmml" xref="S4.SS3.p3.18.m18.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.SS3.p3.18.m18.3.3.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2"><ci id="S4.SS3.p3.18.m18.3.3.2.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2.2">poly</ci><apply id="S4.SS3.p3.18.m18.3.3.2.1.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1"><log id="S4.SS3.p3.18.m18.1.1.cmml" xref="S4.SS3.p3.18.m18.1.1"></log><apply id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1"><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.1.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.1">⋅</ci><apply id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.1.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2">superscript</csymbol><cn id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.2.cmml" type="integer" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.2">2</cn><apply id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3"><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.1.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.1">⋅</ci><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.2">𝑑</ci><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.3.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.2.3.3">𝑏</ci></apply></apply><apply id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.1.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3">superscript</csymbol><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.2.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.2">𝑏</ci><ci id="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.3.cmml" xref="S4.SS3.p3.18.m18.3.3.2.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.18.m18.3c">\mathcal{O}(b^{\prime})\in\operatorname{poly}\log(2^{d\cdot b}\cdot b^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.18.m18.3d">caligraphic_O ( italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ roman_poly roman_log ( 2 start_POSTSUPERSCRIPT italic_d ⋅ italic_b end_POSTSUPERSCRIPT ⋅ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> bits, thus placing the resulting problem in <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p3.19.m19.1"><semantics id="S4.SS3.p3.19.m19.1a"><msup id="S4.SS3.p3.19.m19.1.1" xref="S4.SS3.p3.19.m19.1.1.cmml"><mi id="S4.SS3.p3.19.m19.1.1.2" xref="S4.SS3.p3.19.m19.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p3.19.m19.1.1.3" xref="S4.SS3.p3.19.m19.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.19.m19.1b"><apply id="S4.SS3.p3.19.m19.1.1.cmml" xref="S4.SS3.p3.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.19.m19.1.1.1.cmml" xref="S4.SS3.p3.19.m19.1.1">superscript</csymbol><ci id="S4.SS3.p3.19.m19.1.1.2.cmml" xref="S4.SS3.p3.19.m19.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p3.19.m19.1.1.3a.cmml" xref="S4.SS3.p3.19.m19.1.1.3"><mtext id="S4.SS3.p3.19.m19.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p3.19.m19.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.19.m19.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.19.m19.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>. Furthermore, the algorithm of Chen, Li, and Yannakakis can also be used to solve this total version of the problem: if the function <math alttext="f" class="ltx_Math" display="inline" id="S4.SS3.p3.20.m20.1"><semantics id="S4.SS3.p3.20.m20.1a"><mi id="S4.SS3.p3.20.m20.1.1" xref="S4.SS3.p3.20.m20.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.20.m20.1b"><ci id="S4.SS3.p3.20.m20.1.1.cmml" xref="S4.SS3.p3.20.m20.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.20.m20.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.20.m20.1d">italic_f</annotation></semantics></math> is <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS3.p3.21.m21.1"><semantics id="S4.SS3.p3.21.m21.1a"><mi id="S4.SS3.p3.21.m21.1.1" xref="S4.SS3.p3.21.m21.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.21.m21.1b"><ci id="S4.SS3.p3.21.m21.1.1.cmml" xref="S4.SS3.p3.21.m21.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.21.m21.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.21.m21.1d">italic_λ</annotation></semantics></math>-contracting for all pairs of points queried by the algorithm, the algorithm must return an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.p3.22.m22.1"><semantics id="S4.SS3.p3.22.m22.1a"><mi id="S4.SS3.p3.22.m22.1.1" xref="S4.SS3.p3.22.m22.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.22.m22.1b"><ci id="S4.SS3.p3.22.m22.1.1.cmml" xref="S4.SS3.p3.22.m22.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.22.m22.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.22.m22.1d">italic_ε</annotation></semantics></math>-approximate fixpoint. Otherwise, the algorithm must have encountered a violation. Thus, <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.SS3.p3.23.m23.1"><semantics id="S4.SS3.p3.23.m23.1a"><msub id="S4.SS3.p3.23.m23.1.1" xref="S4.SS3.p3.23.m23.1.1.cmml"><mi id="S4.SS3.p3.23.m23.1.1.2" mathvariant="normal" xref="S4.SS3.p3.23.m23.1.1.2.cmml">ℓ</mi><mi id="S4.SS3.p3.23.m23.1.1.3" mathvariant="normal" xref="S4.SS3.p3.23.m23.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.23.m23.1b"><apply id="S4.SS3.p3.23.m23.1.1.cmml" xref="S4.SS3.p3.23.m23.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.23.m23.1.1.1.cmml" xref="S4.SS3.p3.23.m23.1.1">subscript</csymbol><ci id="S4.SS3.p3.23.m23.1.1.2.cmml" xref="S4.SS3.p3.23.m23.1.1.2">ℓ</ci><infinity id="S4.SS3.p3.23.m23.1.1.3.cmml" xref="S4.SS3.p3.23.m23.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.23.m23.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.23.m23.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.SS3.p3.24.2">-GridContractionFixpoint</span> is in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p3.24.m24.1"><semantics id="S4.SS3.p3.24.m24.1a"><msup id="S4.SS3.p3.24.m24.1.1" xref="S4.SS3.p3.24.m24.1.1.cmml"><mi id="S4.SS3.p3.24.m24.1.1.2" xref="S4.SS3.p3.24.m24.1.1.2.cmml">𝖥𝖯</mi><mtext id="S4.SS3.p3.24.m24.1.1.3" xref="S4.SS3.p3.24.m24.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.24.m24.1b"><apply id="S4.SS3.p3.24.m24.1.1.cmml" xref="S4.SS3.p3.24.m24.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.24.m24.1.1.1.cmml" xref="S4.SS3.p3.24.m24.1.1">superscript</csymbol><ci id="S4.SS3.p3.24.m24.1.1.2.cmml" xref="S4.SS3.p3.24.m24.1.1.2">𝖥𝖯</ci><ci id="S4.SS3.p3.24.m24.1.1.3a.cmml" xref="S4.SS3.p3.24.m24.1.1.3"><mtext id="S4.SS3.p3.24.m24.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p3.24.m24.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.24.m24.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.24.m24.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.p4"> <p class="ltx_p" id="S4.SS3.p4.15">The same strategy does not seem to work in the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS3.p4.1.m1.1"><semantics id="S4.SS3.p4.1.m1.1a"><msub id="S4.SS3.p4.1.m1.1.1" xref="S4.SS3.p4.1.m1.1.1.cmml"><mi id="S4.SS3.p4.1.m1.1.1.2" mathvariant="normal" xref="S4.SS3.p4.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS3.p4.1.m1.1.1.3" xref="S4.SS3.p4.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.1.m1.1b"><apply id="S4.SS3.p4.1.m1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.1.m1.1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p4.1.m1.1.1.2.cmml" xref="S4.SS3.p4.1.m1.1.1.2">ℓ</ci><cn id="S4.SS3.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-case. In fact, we suspect that the statement that a grid-map extends to a contraction map if and only if it is contracting for all grid points is not true for the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS3.p4.2.m2.1"><semantics id="S4.SS3.p4.2.m2.1a"><msub id="S4.SS3.p4.2.m2.1.1" xref="S4.SS3.p4.2.m2.1.1.cmml"><mi id="S4.SS3.p4.2.m2.1.1.2" mathvariant="normal" xref="S4.SS3.p4.2.m2.1.1.2.cmml">ℓ</mi><mn id="S4.SS3.p4.2.m2.1.1.3" xref="S4.SS3.p4.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.2.m2.1b"><apply id="S4.SS3.p4.2.m2.1.1.cmml" xref="S4.SS3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.2.m2.1.1.1.cmml" xref="S4.SS3.p4.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p4.2.m2.1.1.2.cmml" xref="S4.SS3.p4.2.m2.1.1.2">ℓ</ci><cn id="S4.SS3.p4.2.m2.1.1.3.cmml" type="integer" xref="S4.SS3.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.2.m2.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-norm and general <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="S4.SS3.p4.3.m3.1"><semantics id="S4.SS3.p4.3.m3.1a"><msub id="S4.SS3.p4.3.m3.1.1" xref="S4.SS3.p4.3.m3.1.1.cmml"><mi id="S4.SS3.p4.3.m3.1.1.2" mathvariant="normal" xref="S4.SS3.p4.3.m3.1.1.2.cmml">ℓ</mi><mi id="S4.SS3.p4.3.m3.1.1.3" xref="S4.SS3.p4.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.3.m3.1b"><apply id="S4.SS3.p4.3.m3.1.1.cmml" xref="S4.SS3.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.3.m3.1.1.1.cmml" xref="S4.SS3.p4.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p4.3.m3.1.1.2.cmml" xref="S4.SS3.p4.3.m3.1.1.2">ℓ</ci><ci id="S4.SS3.p4.3.m3.1.1.3.cmml" xref="S4.SS3.p4.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norms. In the case of <math alttext="p=\infty" class="ltx_Math" display="inline" id="S4.SS3.p4.4.m4.1"><semantics id="S4.SS3.p4.4.m4.1a"><mrow id="S4.SS3.p4.4.m4.1.1" xref="S4.SS3.p4.4.m4.1.1.cmml"><mi id="S4.SS3.p4.4.m4.1.1.2" xref="S4.SS3.p4.4.m4.1.1.2.cmml">p</mi><mo id="S4.SS3.p4.4.m4.1.1.1" xref="S4.SS3.p4.4.m4.1.1.1.cmml">=</mo><mi id="S4.SS3.p4.4.m4.1.1.3" mathvariant="normal" xref="S4.SS3.p4.4.m4.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.4.m4.1b"><apply id="S4.SS3.p4.4.m4.1.1.cmml" xref="S4.SS3.p4.4.m4.1.1"><eq id="S4.SS3.p4.4.m4.1.1.1.cmml" xref="S4.SS3.p4.4.m4.1.1.1"></eq><ci id="S4.SS3.p4.4.m4.1.1.2.cmml" xref="S4.SS3.p4.4.m4.1.1.2">𝑝</ci><infinity id="S4.SS3.p4.4.m4.1.1.3.cmml" xref="S4.SS3.p4.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.4.m4.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.4.m4.1d">italic_p = ∞</annotation></semantics></math>, Chen, Li, and Yannakakis constructed an extension explicitly: they extended the grid-function to <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="S4.SS3.p4.5.m5.2"><semantics id="S4.SS3.p4.5.m5.2a"><msup id="S4.SS3.p4.5.m5.2.3" xref="S4.SS3.p4.5.m5.2.3.cmml"><mrow id="S4.SS3.p4.5.m5.2.3.2.2" xref="S4.SS3.p4.5.m5.2.3.2.1.cmml"><mo id="S4.SS3.p4.5.m5.2.3.2.2.1" stretchy="false" xref="S4.SS3.p4.5.m5.2.3.2.1.cmml">[</mo><mn id="S4.SS3.p4.5.m5.1.1" xref="S4.SS3.p4.5.m5.1.1.cmml">0</mn><mo id="S4.SS3.p4.5.m5.2.3.2.2.2" xref="S4.SS3.p4.5.m5.2.3.2.1.cmml">,</mo><mn id="S4.SS3.p4.5.m5.2.2" xref="S4.SS3.p4.5.m5.2.2.cmml">1</mn><mo id="S4.SS3.p4.5.m5.2.3.2.2.3" stretchy="false" xref="S4.SS3.p4.5.m5.2.3.2.1.cmml">]</mo></mrow><mi id="S4.SS3.p4.5.m5.2.3.3" xref="S4.SS3.p4.5.m5.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.5.m5.2b"><apply id="S4.SS3.p4.5.m5.2.3.cmml" xref="S4.SS3.p4.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.5.m5.2.3.1.cmml" xref="S4.SS3.p4.5.m5.2.3">superscript</csymbol><interval closure="closed" id="S4.SS3.p4.5.m5.2.3.2.1.cmml" xref="S4.SS3.p4.5.m5.2.3.2.2"><cn id="S4.SS3.p4.5.m5.1.1.cmml" type="integer" xref="S4.SS3.p4.5.m5.1.1">0</cn><cn id="S4.SS3.p4.5.m5.2.2.cmml" type="integer" xref="S4.SS3.p4.5.m5.2.2">1</cn></interval><ci id="S4.SS3.p4.5.m5.2.3.3.cmml" xref="S4.SS3.p4.5.m5.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.5.m5.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.5.m5.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> using a formula that corresponds to applying McShane’s extension lemma <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib25" title="">25</a>]</cite> to every partial function <math alttext="f_{i}:G^{d}_{b}\rightarrow[0,1]" class="ltx_Math" display="inline" id="S4.SS3.p4.6.m6.2"><semantics id="S4.SS3.p4.6.m6.2a"><mrow id="S4.SS3.p4.6.m6.2.3" xref="S4.SS3.p4.6.m6.2.3.cmml"><msub id="S4.SS3.p4.6.m6.2.3.2" xref="S4.SS3.p4.6.m6.2.3.2.cmml"><mi id="S4.SS3.p4.6.m6.2.3.2.2" xref="S4.SS3.p4.6.m6.2.3.2.2.cmml">f</mi><mi id="S4.SS3.p4.6.m6.2.3.2.3" xref="S4.SS3.p4.6.m6.2.3.2.3.cmml">i</mi></msub><mo id="S4.SS3.p4.6.m6.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p4.6.m6.2.3.1.cmml">:</mo><mrow id="S4.SS3.p4.6.m6.2.3.3" xref="S4.SS3.p4.6.m6.2.3.3.cmml"><msubsup id="S4.SS3.p4.6.m6.2.3.3.2" xref="S4.SS3.p4.6.m6.2.3.3.2.cmml"><mi id="S4.SS3.p4.6.m6.2.3.3.2.2.2" xref="S4.SS3.p4.6.m6.2.3.3.2.2.2.cmml">G</mi><mi id="S4.SS3.p4.6.m6.2.3.3.2.3" xref="S4.SS3.p4.6.m6.2.3.3.2.3.cmml">b</mi><mi id="S4.SS3.p4.6.m6.2.3.3.2.2.3" xref="S4.SS3.p4.6.m6.2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS3.p4.6.m6.2.3.3.1" stretchy="false" xref="S4.SS3.p4.6.m6.2.3.3.1.cmml">→</mo><mrow id="S4.SS3.p4.6.m6.2.3.3.3.2" xref="S4.SS3.p4.6.m6.2.3.3.3.1.cmml"><mo id="S4.SS3.p4.6.m6.2.3.3.3.2.1" stretchy="false" xref="S4.SS3.p4.6.m6.2.3.3.3.1.cmml">[</mo><mn id="S4.SS3.p4.6.m6.1.1" xref="S4.SS3.p4.6.m6.1.1.cmml">0</mn><mo id="S4.SS3.p4.6.m6.2.3.3.3.2.2" xref="S4.SS3.p4.6.m6.2.3.3.3.1.cmml">,</mo><mn id="S4.SS3.p4.6.m6.2.2" xref="S4.SS3.p4.6.m6.2.2.cmml">1</mn><mo id="S4.SS3.p4.6.m6.2.3.3.3.2.3" stretchy="false" xref="S4.SS3.p4.6.m6.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.6.m6.2b"><apply id="S4.SS3.p4.6.m6.2.3.cmml" xref="S4.SS3.p4.6.m6.2.3"><ci id="S4.SS3.p4.6.m6.2.3.1.cmml" xref="S4.SS3.p4.6.m6.2.3.1">:</ci><apply id="S4.SS3.p4.6.m6.2.3.2.cmml" xref="S4.SS3.p4.6.m6.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p4.6.m6.2.3.2.1.cmml" xref="S4.SS3.p4.6.m6.2.3.2">subscript</csymbol><ci id="S4.SS3.p4.6.m6.2.3.2.2.cmml" xref="S4.SS3.p4.6.m6.2.3.2.2">𝑓</ci><ci id="S4.SS3.p4.6.m6.2.3.2.3.cmml" xref="S4.SS3.p4.6.m6.2.3.2.3">𝑖</ci></apply><apply id="S4.SS3.p4.6.m6.2.3.3.cmml" xref="S4.SS3.p4.6.m6.2.3.3"><ci id="S4.SS3.p4.6.m6.2.3.3.1.cmml" xref="S4.SS3.p4.6.m6.2.3.3.1">→</ci><apply id="S4.SS3.p4.6.m6.2.3.3.2.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS3.p4.6.m6.2.3.3.2.1.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2">subscript</csymbol><apply id="S4.SS3.p4.6.m6.2.3.3.2.2.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS3.p4.6.m6.2.3.3.2.2.1.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2">superscript</csymbol><ci id="S4.SS3.p4.6.m6.2.3.3.2.2.2.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2.2.2">𝐺</ci><ci id="S4.SS3.p4.6.m6.2.3.3.2.2.3.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2.2.3">𝑑</ci></apply><ci id="S4.SS3.p4.6.m6.2.3.3.2.3.cmml" xref="S4.SS3.p4.6.m6.2.3.3.2.3">𝑏</ci></apply><interval closure="closed" id="S4.SS3.p4.6.m6.2.3.3.3.1.cmml" xref="S4.SS3.p4.6.m6.2.3.3.3.2"><cn id="S4.SS3.p4.6.m6.1.1.cmml" type="integer" xref="S4.SS3.p4.6.m6.1.1">0</cn><cn id="S4.SS3.p4.6.m6.2.2.cmml" type="integer" xref="S4.SS3.p4.6.m6.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.6.m6.2c">f_{i}:G^{d}_{b}\rightarrow[0,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.6.m6.2d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → [ 0 , 1 ]</annotation></semantics></math> independently. McShane’s extension lemma guarantees that the partial functions <math alttext="f^{\prime}_{i}:[0,1]^{d}\rightarrow[0,1]" class="ltx_Math" display="inline" id="S4.SS3.p4.7.m7.4"><semantics id="S4.SS3.p4.7.m7.4a"><mrow id="S4.SS3.p4.7.m7.4.5" xref="S4.SS3.p4.7.m7.4.5.cmml"><msubsup id="S4.SS3.p4.7.m7.4.5.2" xref="S4.SS3.p4.7.m7.4.5.2.cmml"><mi id="S4.SS3.p4.7.m7.4.5.2.2.2" xref="S4.SS3.p4.7.m7.4.5.2.2.2.cmml">f</mi><mi id="S4.SS3.p4.7.m7.4.5.2.3" xref="S4.SS3.p4.7.m7.4.5.2.3.cmml">i</mi><mo id="S4.SS3.p4.7.m7.4.5.2.2.3" xref="S4.SS3.p4.7.m7.4.5.2.2.3.cmml">′</mo></msubsup><mo id="S4.SS3.p4.7.m7.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.SS3.p4.7.m7.4.5.1.cmml">:</mo><mrow id="S4.SS3.p4.7.m7.4.5.3" xref="S4.SS3.p4.7.m7.4.5.3.cmml"><msup id="S4.SS3.p4.7.m7.4.5.3.2" xref="S4.SS3.p4.7.m7.4.5.3.2.cmml"><mrow id="S4.SS3.p4.7.m7.4.5.3.2.2.2" xref="S4.SS3.p4.7.m7.4.5.3.2.2.1.cmml"><mo id="S4.SS3.p4.7.m7.4.5.3.2.2.2.1" stretchy="false" xref="S4.SS3.p4.7.m7.4.5.3.2.2.1.cmml">[</mo><mn id="S4.SS3.p4.7.m7.1.1" xref="S4.SS3.p4.7.m7.1.1.cmml">0</mn><mo id="S4.SS3.p4.7.m7.4.5.3.2.2.2.2" xref="S4.SS3.p4.7.m7.4.5.3.2.2.1.cmml">,</mo><mn id="S4.SS3.p4.7.m7.2.2" xref="S4.SS3.p4.7.m7.2.2.cmml">1</mn><mo id="S4.SS3.p4.7.m7.4.5.3.2.2.2.3" stretchy="false" xref="S4.SS3.p4.7.m7.4.5.3.2.2.1.cmml">]</mo></mrow><mi id="S4.SS3.p4.7.m7.4.5.3.2.3" xref="S4.SS3.p4.7.m7.4.5.3.2.3.cmml">d</mi></msup><mo id="S4.SS3.p4.7.m7.4.5.3.1" stretchy="false" xref="S4.SS3.p4.7.m7.4.5.3.1.cmml">→</mo><mrow id="S4.SS3.p4.7.m7.4.5.3.3.2" xref="S4.SS3.p4.7.m7.4.5.3.3.1.cmml"><mo id="S4.SS3.p4.7.m7.4.5.3.3.2.1" stretchy="false" xref="S4.SS3.p4.7.m7.4.5.3.3.1.cmml">[</mo><mn id="S4.SS3.p4.7.m7.3.3" xref="S4.SS3.p4.7.m7.3.3.cmml">0</mn><mo id="S4.SS3.p4.7.m7.4.5.3.3.2.2" xref="S4.SS3.p4.7.m7.4.5.3.3.1.cmml">,</mo><mn id="S4.SS3.p4.7.m7.4.4" xref="S4.SS3.p4.7.m7.4.4.cmml">1</mn><mo id="S4.SS3.p4.7.m7.4.5.3.3.2.3" stretchy="false" xref="S4.SS3.p4.7.m7.4.5.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.7.m7.4b"><apply id="S4.SS3.p4.7.m7.4.5.cmml" xref="S4.SS3.p4.7.m7.4.5"><ci id="S4.SS3.p4.7.m7.4.5.1.cmml" xref="S4.SS3.p4.7.m7.4.5.1">:</ci><apply id="S4.SS3.p4.7.m7.4.5.2.cmml" xref="S4.SS3.p4.7.m7.4.5.2"><csymbol cd="ambiguous" id="S4.SS3.p4.7.m7.4.5.2.1.cmml" xref="S4.SS3.p4.7.m7.4.5.2">subscript</csymbol><apply id="S4.SS3.p4.7.m7.4.5.2.2.cmml" xref="S4.SS3.p4.7.m7.4.5.2"><csymbol cd="ambiguous" id="S4.SS3.p4.7.m7.4.5.2.2.1.cmml" xref="S4.SS3.p4.7.m7.4.5.2">superscript</csymbol><ci id="S4.SS3.p4.7.m7.4.5.2.2.2.cmml" xref="S4.SS3.p4.7.m7.4.5.2.2.2">𝑓</ci><ci id="S4.SS3.p4.7.m7.4.5.2.2.3.cmml" xref="S4.SS3.p4.7.m7.4.5.2.2.3">′</ci></apply><ci id="S4.SS3.p4.7.m7.4.5.2.3.cmml" xref="S4.SS3.p4.7.m7.4.5.2.3">𝑖</ci></apply><apply id="S4.SS3.p4.7.m7.4.5.3.cmml" xref="S4.SS3.p4.7.m7.4.5.3"><ci id="S4.SS3.p4.7.m7.4.5.3.1.cmml" xref="S4.SS3.p4.7.m7.4.5.3.1">→</ci><apply id="S4.SS3.p4.7.m7.4.5.3.2.cmml" xref="S4.SS3.p4.7.m7.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS3.p4.7.m7.4.5.3.2.1.cmml" xref="S4.SS3.p4.7.m7.4.5.3.2">superscript</csymbol><interval closure="closed" id="S4.SS3.p4.7.m7.4.5.3.2.2.1.cmml" xref="S4.SS3.p4.7.m7.4.5.3.2.2.2"><cn id="S4.SS3.p4.7.m7.1.1.cmml" type="integer" xref="S4.SS3.p4.7.m7.1.1">0</cn><cn id="S4.SS3.p4.7.m7.2.2.cmml" type="integer" xref="S4.SS3.p4.7.m7.2.2">1</cn></interval><ci id="S4.SS3.p4.7.m7.4.5.3.2.3.cmml" xref="S4.SS3.p4.7.m7.4.5.3.2.3">𝑑</ci></apply><interval closure="closed" id="S4.SS3.p4.7.m7.4.5.3.3.1.cmml" xref="S4.SS3.p4.7.m7.4.5.3.3.2"><cn id="S4.SS3.p4.7.m7.3.3.cmml" type="integer" xref="S4.SS3.p4.7.m7.3.3">0</cn><cn id="S4.SS3.p4.7.m7.4.4.cmml" type="integer" xref="S4.SS3.p4.7.m7.4.4">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.7.m7.4c">f^{\prime}_{i}:[0,1]^{d}\rightarrow[0,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.7.m7.4d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → [ 0 , 1 ]</annotation></semantics></math> then fulfill the same contraction property as <math alttext="f_{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.8.m8.1"><semantics id="S4.SS3.p4.8.m8.1a"><msub id="S4.SS3.p4.8.m8.1.1" xref="S4.SS3.p4.8.m8.1.1.cmml"><mi id="S4.SS3.p4.8.m8.1.1.2" xref="S4.SS3.p4.8.m8.1.1.2.cmml">f</mi><mi id="S4.SS3.p4.8.m8.1.1.3" xref="S4.SS3.p4.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.8.m8.1b"><apply id="S4.SS3.p4.8.m8.1.1.cmml" xref="S4.SS3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.8.m8.1.1.1.cmml" xref="S4.SS3.p4.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.p4.8.m8.1.1.2.cmml" xref="S4.SS3.p4.8.m8.1.1.2">𝑓</ci><ci id="S4.SS3.p4.8.m8.1.1.3.cmml" xref="S4.SS3.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.8.m8.1c">f_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.8.m8.1d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which together with the definition of the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="S4.SS3.p4.9.m9.1"><semantics id="S4.SS3.p4.9.m9.1a"><msub id="S4.SS3.p4.9.m9.1.1" xref="S4.SS3.p4.9.m9.1.1.cmml"><mi id="S4.SS3.p4.9.m9.1.1.2" mathvariant="normal" xref="S4.SS3.p4.9.m9.1.1.2.cmml">ℓ</mi><mi id="S4.SS3.p4.9.m9.1.1.3" mathvariant="normal" xref="S4.SS3.p4.9.m9.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.9.m9.1b"><apply id="S4.SS3.p4.9.m9.1.1.cmml" xref="S4.SS3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.9.m9.1.1.1.cmml" xref="S4.SS3.p4.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p4.9.m9.1.1.2.cmml" xref="S4.SS3.p4.9.m9.1.1.2">ℓ</ci><infinity id="S4.SS3.p4.9.m9.1.1.3.cmml" xref="S4.SS3.p4.9.m9.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.9.m9.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-norm as a maximum over coordinates, yields the contraction property for <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p4.10.m10.1"><semantics id="S4.SS3.p4.10.m10.1a"><msup id="S4.SS3.p4.10.m10.1.1" xref="S4.SS3.p4.10.m10.1.1.cmml"><mi id="S4.SS3.p4.10.m10.1.1.2" xref="S4.SS3.p4.10.m10.1.1.2.cmml">f</mi><mo id="S4.SS3.p4.10.m10.1.1.3" xref="S4.SS3.p4.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.10.m10.1b"><apply id="S4.SS3.p4.10.m10.1.1.cmml" xref="S4.SS3.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.10.m10.1.1.1.cmml" xref="S4.SS3.p4.10.m10.1.1">superscript</csymbol><ci id="S4.SS3.p4.10.m10.1.1.2.cmml" xref="S4.SS3.p4.10.m10.1.1.2">𝑓</ci><ci id="S4.SS3.p4.10.m10.1.1.3.cmml" xref="S4.SS3.p4.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.10.m10.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.10.m10.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Trying this for <math alttext="p=1" class="ltx_Math" display="inline" id="S4.SS3.p4.11.m11.1"><semantics id="S4.SS3.p4.11.m11.1a"><mrow id="S4.SS3.p4.11.m11.1.1" xref="S4.SS3.p4.11.m11.1.1.cmml"><mi id="S4.SS3.p4.11.m11.1.1.2" xref="S4.SS3.p4.11.m11.1.1.2.cmml">p</mi><mo id="S4.SS3.p4.11.m11.1.1.1" xref="S4.SS3.p4.11.m11.1.1.1.cmml">=</mo><mn id="S4.SS3.p4.11.m11.1.1.3" xref="S4.SS3.p4.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.11.m11.1b"><apply id="S4.SS3.p4.11.m11.1.1.cmml" xref="S4.SS3.p4.11.m11.1.1"><eq id="S4.SS3.p4.11.m11.1.1.1.cmml" xref="S4.SS3.p4.11.m11.1.1.1"></eq><ci id="S4.SS3.p4.11.m11.1.1.2.cmml" xref="S4.SS3.p4.11.m11.1.1.2">𝑝</ci><cn id="S4.SS3.p4.11.m11.1.1.3.cmml" type="integer" xref="S4.SS3.p4.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.11.m11.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.11.m11.1d">italic_p = 1</annotation></semantics></math>, the analysis only yields that <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p4.12.m12.1"><semantics id="S4.SS3.p4.12.m12.1a"><msup id="S4.SS3.p4.12.m12.1.1" xref="S4.SS3.p4.12.m12.1.1.cmml"><mi id="S4.SS3.p4.12.m12.1.1.2" xref="S4.SS3.p4.12.m12.1.1.2.cmml">f</mi><mo id="S4.SS3.p4.12.m12.1.1.3" xref="S4.SS3.p4.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.12.m12.1b"><apply id="S4.SS3.p4.12.m12.1.1.cmml" xref="S4.SS3.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.12.m12.1.1.1.cmml" xref="S4.SS3.p4.12.m12.1.1">superscript</csymbol><ci id="S4.SS3.p4.12.m12.1.1.2.cmml" xref="S4.SS3.p4.12.m12.1.1.2">𝑓</ci><ci id="S4.SS3.p4.12.m12.1.1.3.cmml" xref="S4.SS3.p4.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.12.m12.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.12.m12.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="(d\lambda)" class="ltx_Math" display="inline" id="S4.SS3.p4.13.m13.1"><semantics id="S4.SS3.p4.13.m13.1a"><mrow id="S4.SS3.p4.13.m13.1.1.1" xref="S4.SS3.p4.13.m13.1.1.1.1.cmml"><mo id="S4.SS3.p4.13.m13.1.1.1.2" stretchy="false" xref="S4.SS3.p4.13.m13.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p4.13.m13.1.1.1.1" xref="S4.SS3.p4.13.m13.1.1.1.1.cmml"><mi id="S4.SS3.p4.13.m13.1.1.1.1.2" xref="S4.SS3.p4.13.m13.1.1.1.1.2.cmml">d</mi><mo id="S4.SS3.p4.13.m13.1.1.1.1.1" xref="S4.SS3.p4.13.m13.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.13.m13.1.1.1.1.3" xref="S4.SS3.p4.13.m13.1.1.1.1.3.cmml">λ</mi></mrow><mo id="S4.SS3.p4.13.m13.1.1.1.3" stretchy="false" xref="S4.SS3.p4.13.m13.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.13.m13.1b"><apply id="S4.SS3.p4.13.m13.1.1.1.1.cmml" xref="S4.SS3.p4.13.m13.1.1.1"><times id="S4.SS3.p4.13.m13.1.1.1.1.1.cmml" xref="S4.SS3.p4.13.m13.1.1.1.1.1"></times><ci id="S4.SS3.p4.13.m13.1.1.1.1.2.cmml" xref="S4.SS3.p4.13.m13.1.1.1.1.2">𝑑</ci><ci id="S4.SS3.p4.13.m13.1.1.1.1.3.cmml" xref="S4.SS3.p4.13.m13.1.1.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.13.m13.1c">(d\lambda)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.13.m13.1d">( italic_d italic_λ )</annotation></semantics></math>-Lipschitz, which is not enough to be contracting if we for example have <math alttext="d>1" class="ltx_Math" display="inline" id="S4.SS3.p4.14.m14.1"><semantics id="S4.SS3.p4.14.m14.1a"><mrow id="S4.SS3.p4.14.m14.1.1" xref="S4.SS3.p4.14.m14.1.1.cmml"><mi id="S4.SS3.p4.14.m14.1.1.2" xref="S4.SS3.p4.14.m14.1.1.2.cmml">d</mi><mo id="S4.SS3.p4.14.m14.1.1.1" xref="S4.SS3.p4.14.m14.1.1.1.cmml">></mo><mn id="S4.SS3.p4.14.m14.1.1.3" xref="S4.SS3.p4.14.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.14.m14.1b"><apply id="S4.SS3.p4.14.m14.1.1.cmml" xref="S4.SS3.p4.14.m14.1.1"><gt id="S4.SS3.p4.14.m14.1.1.1.cmml" xref="S4.SS3.p4.14.m14.1.1.1"></gt><ci id="S4.SS3.p4.14.m14.1.1.2.cmml" xref="S4.SS3.p4.14.m14.1.1.2">𝑑</ci><cn id="S4.SS3.p4.14.m14.1.1.3.cmml" type="integer" xref="S4.SS3.p4.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.14.m14.1c">d>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.14.m14.1d">italic_d > 1</annotation></semantics></math> and <math alttext="\lambda>0.5" class="ltx_Math" display="inline" id="S4.SS3.p4.15.m15.1"><semantics id="S4.SS3.p4.15.m15.1a"><mrow id="S4.SS3.p4.15.m15.1.1" xref="S4.SS3.p4.15.m15.1.1.cmml"><mi id="S4.SS3.p4.15.m15.1.1.2" xref="S4.SS3.p4.15.m15.1.1.2.cmml">λ</mi><mo id="S4.SS3.p4.15.m15.1.1.1" xref="S4.SS3.p4.15.m15.1.1.1.cmml">></mo><mn id="S4.SS3.p4.15.m15.1.1.3" xref="S4.SS3.p4.15.m15.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.15.m15.1b"><apply id="S4.SS3.p4.15.m15.1.1.cmml" xref="S4.SS3.p4.15.m15.1.1"><gt id="S4.SS3.p4.15.m15.1.1.1.cmml" xref="S4.SS3.p4.15.m15.1.1.1"></gt><ci id="S4.SS3.p4.15.m15.1.1.2.cmml" xref="S4.SS3.p4.15.m15.1.1.2">𝜆</ci><cn id="S4.SS3.p4.15.m15.1.1.3.cmml" type="float" xref="S4.SS3.p4.15.m15.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.15.m15.1c">\lambda>0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.15.m15.1d">italic_λ > 0.5</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.p5"> <p class="ltx_p" id="S4.SS3.p5.4">We get around this in a naive way by introducing a violation type based on the termination criterion of our algorithm for <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS3.p5.1.m1.1"><semantics id="S4.SS3.p5.1.m1.1a"><msub id="S4.SS3.p5.1.m1.1.1" xref="S4.SS3.p5.1.m1.1.1.cmml"><mi id="S4.SS3.p5.1.m1.1.1.2" mathvariant="normal" xref="S4.SS3.p5.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS3.p5.1.m1.1.1.3" xref="S4.SS3.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.1.m1.1b"><apply id="S4.SS3.p5.1.m1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p5.1.m1.1.1.2.cmml" xref="S4.SS3.p5.1.m1.1.1.2">ℓ</ci><cn id="S4.SS3.p5.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_smallcaps" id="S4.SS3.p5.4.1">-GridContractionFixpoint</span>. Concretely, if our algorithm does not find an <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.p5.2.m2.1"><semantics id="S4.SS3.p5.2.m2.1a"><mi id="S4.SS3.p5.2.m2.1.1" xref="S4.SS3.p5.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.2.m2.1b"><ci id="S4.SS3.p5.2.m2.1.1.cmml" xref="S4.SS3.p5.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoint after reaching its query bound, then the queried points <math alttext="x^{(1)},\dots,x^{(k)}" class="ltx_Math" display="inline" id="S4.SS3.p5.3.m3.5"><semantics id="S4.SS3.p5.3.m3.5a"><mrow id="S4.SS3.p5.3.m3.5.5.2" xref="S4.SS3.p5.3.m3.5.5.3.cmml"><msup id="S4.SS3.p5.3.m3.4.4.1.1" xref="S4.SS3.p5.3.m3.4.4.1.1.cmml"><mi id="S4.SS3.p5.3.m3.4.4.1.1.2" xref="S4.SS3.p5.3.m3.4.4.1.1.2.cmml">x</mi><mrow id="S4.SS3.p5.3.m3.1.1.1.3" xref="S4.SS3.p5.3.m3.4.4.1.1.cmml"><mo id="S4.SS3.p5.3.m3.1.1.1.3.1" stretchy="false" xref="S4.SS3.p5.3.m3.4.4.1.1.cmml">(</mo><mn id="S4.SS3.p5.3.m3.1.1.1.1" xref="S4.SS3.p5.3.m3.1.1.1.1.cmml">1</mn><mo id="S4.SS3.p5.3.m3.1.1.1.3.2" stretchy="false" xref="S4.SS3.p5.3.m3.4.4.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS3.p5.3.m3.5.5.2.3" xref="S4.SS3.p5.3.m3.5.5.3.cmml">,</mo><mi id="S4.SS3.p5.3.m3.3.3" mathvariant="normal" xref="S4.SS3.p5.3.m3.3.3.cmml">…</mi><mo id="S4.SS3.p5.3.m3.5.5.2.4" xref="S4.SS3.p5.3.m3.5.5.3.cmml">,</mo><msup id="S4.SS3.p5.3.m3.5.5.2.2" xref="S4.SS3.p5.3.m3.5.5.2.2.cmml"><mi id="S4.SS3.p5.3.m3.5.5.2.2.2" xref="S4.SS3.p5.3.m3.5.5.2.2.2.cmml">x</mi><mrow id="S4.SS3.p5.3.m3.2.2.1.3" xref="S4.SS3.p5.3.m3.5.5.2.2.cmml"><mo id="S4.SS3.p5.3.m3.2.2.1.3.1" stretchy="false" xref="S4.SS3.p5.3.m3.5.5.2.2.cmml">(</mo><mi id="S4.SS3.p5.3.m3.2.2.1.1" xref="S4.SS3.p5.3.m3.2.2.1.1.cmml">k</mi><mo id="S4.SS3.p5.3.m3.2.2.1.3.2" stretchy="false" xref="S4.SS3.p5.3.m3.5.5.2.2.cmml">)</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.3.m3.5b"><list id="S4.SS3.p5.3.m3.5.5.3.cmml" xref="S4.SS3.p5.3.m3.5.5.2"><apply id="S4.SS3.p5.3.m3.4.4.1.1.cmml" xref="S4.SS3.p5.3.m3.4.4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.4.4.1.1.1.cmml" xref="S4.SS3.p5.3.m3.4.4.1.1">superscript</csymbol><ci id="S4.SS3.p5.3.m3.4.4.1.1.2.cmml" xref="S4.SS3.p5.3.m3.4.4.1.1.2">𝑥</ci><cn id="S4.SS3.p5.3.m3.1.1.1.1.cmml" type="integer" xref="S4.SS3.p5.3.m3.1.1.1.1">1</cn></apply><ci id="S4.SS3.p5.3.m3.3.3.cmml" xref="S4.SS3.p5.3.m3.3.3">…</ci><apply id="S4.SS3.p5.3.m3.5.5.2.2.cmml" xref="S4.SS3.p5.3.m3.5.5.2.2"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.5.5.2.2.1.cmml" xref="S4.SS3.p5.3.m3.5.5.2.2">superscript</csymbol><ci id="S4.SS3.p5.3.m3.5.5.2.2.2.cmml" xref="S4.SS3.p5.3.m3.5.5.2.2.2">𝑥</ci><ci id="S4.SS3.p5.3.m3.2.2.1.1.cmml" xref="S4.SS3.p5.3.m3.2.2.1.1">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.3.m3.5c">x^{(1)},\dots,x^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.3.m3.5d">italic_x start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , … , italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> must satisfy <math alttext="G^{d}_{b}\setminus\bigcup_{i\in[k]}H^{1}_{x^{(i)},f(x^{(i)})}=\varnothing" class="ltx_Math" display="inline" id="S4.SS3.p5.4.m4.5"><semantics id="S4.SS3.p5.4.m4.5a"><mrow id="S4.SS3.p5.4.m4.5.6" xref="S4.SS3.p5.4.m4.5.6.cmml"><mrow id="S4.SS3.p5.4.m4.5.6.2" xref="S4.SS3.p5.4.m4.5.6.2.cmml"><msubsup id="S4.SS3.p5.4.m4.5.6.2.2" xref="S4.SS3.p5.4.m4.5.6.2.2.cmml"><mi id="S4.SS3.p5.4.m4.5.6.2.2.2.2" xref="S4.SS3.p5.4.m4.5.6.2.2.2.2.cmml">G</mi><mi id="S4.SS3.p5.4.m4.5.6.2.2.3" xref="S4.SS3.p5.4.m4.5.6.2.2.3.cmml">b</mi><mi id="S4.SS3.p5.4.m4.5.6.2.2.2.3" xref="S4.SS3.p5.4.m4.5.6.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS3.p5.4.m4.5.6.2.1" 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start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∖ ⋃ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT , italic_f ( italic_x start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = ∅</annotation></semantics></math>. In other words, we certify the existence of a violation of the contraction property with a set of points on the grid whose associated bisector halfspaces contain all grid points in their union.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Definition 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem7.p1.4.4">An instance of the <span class="ltx_text ltx_font_smallcaps" id="S4.Thmtheorem7.p1.1.1.1">Total-<math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.1.1.m1.1"><semantics id="S4.Thmtheorem7.p1.1.1.1.m1.1a"><msub id="S4.Thmtheorem7.p1.1.1.1.m1.1.1" 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end_POSTSUBSCRIPT</annotation></semantics></math>-ContractionFixpoint</span> problem consists of a bitstring encoding integers <math alttext="d,b,b^{\prime}\in\mathbb{N}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.2.m1.3"><semantics id="S4.Thmtheorem7.p1.2.2.m1.3a"><mrow id="S4.Thmtheorem7.p1.2.2.m1.3.3" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.cmml"><mrow id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.2.cmml"><mi id="S4.Thmtheorem7.p1.2.2.m1.1.1" xref="S4.Thmtheorem7.p1.2.2.m1.1.1.cmml">d</mi><mo id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.2" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.2.cmml">,</mo><mi id="S4.Thmtheorem7.p1.2.2.m1.2.2" xref="S4.Thmtheorem7.p1.2.2.m1.2.2.cmml">b</mi><mo id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.3" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.2.cmml">,</mo><msup id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.2" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.2.cmml">b</mi><mo id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.3" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S4.Thmtheorem7.p1.2.2.m1.3.3.2" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.2.cmml">∈</mo><mi id="S4.Thmtheorem7.p1.2.2.m1.3.3.3" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.2.m1.3b"><apply id="S4.Thmtheorem7.p1.2.2.m1.3.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3"><in id="S4.Thmtheorem7.p1.2.2.m1.3.3.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.2"></in><list id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1"><ci id="S4.Thmtheorem7.p1.2.2.m1.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.1.1">𝑑</ci><ci id="S4.Thmtheorem7.p1.2.2.m1.2.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.2.2">𝑏</ci><apply id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.2">𝑏</ci><ci id="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.1.1.1.3">′</ci></apply></list><ci id="S4.Thmtheorem7.p1.2.2.m1.3.3.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m1.3.3.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.2.m1.3c">d,b,b^{\prime}\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.2.m1.3d">italic_d , italic_b , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_N</annotation></semantics></math> and integral logarithms <math alttext="\log(\frac{1}{1-\lambda}),\log(\frac{1}{\varepsilon})" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.3.3.m2.6"><semantics id="S4.Thmtheorem7.p1.3.3.m2.6a"><mrow id="S4.Thmtheorem7.p1.3.3.m2.6.6.2" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.3.cmml"><mrow id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m2.1.1" xref="S4.Thmtheorem7.p1.3.3.m2.1.1.cmml">log</mi><mo id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2a" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml">⁡</mo><mrow id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2.1" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2.1.1" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml">(</mo><mfrac id="S4.Thmtheorem7.p1.3.3.m2.2.2" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.cmml"><mn id="S4.Thmtheorem7.p1.3.3.m2.2.2.2" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.2.cmml">1</mn><mrow id="S4.Thmtheorem7.p1.3.3.m2.2.2.3" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.cmml"><mn id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.2" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.2.cmml">1</mn><mo id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.1" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.1.cmml">−</mo><mi id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.3" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.3.cmml">λ</mi></mrow></mfrac><mo id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.3" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.3.cmml">,</mo><mrow id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m2.3.3" xref="S4.Thmtheorem7.p1.3.3.m2.3.3.cmml">log</mi><mo id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2a" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml">⁡</mo><mrow id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2.1" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml"><mo id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2.1.1" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml">(</mo><mfrac id="S4.Thmtheorem7.p1.3.3.m2.4.4" xref="S4.Thmtheorem7.p1.3.3.m2.4.4.cmml"><mn id="S4.Thmtheorem7.p1.3.3.m2.4.4.2" xref="S4.Thmtheorem7.p1.3.3.m2.4.4.2.cmml">1</mn><mi id="S4.Thmtheorem7.p1.3.3.m2.4.4.3" xref="S4.Thmtheorem7.p1.3.3.m2.4.4.3.cmml">ε</mi></mfrac><mo id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.3.3.m2.6b"><list id="S4.Thmtheorem7.p1.3.3.m2.6.6.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2"><apply id="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.5.5.1.1.2"><log id="S4.Thmtheorem7.p1.3.3.m2.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.1.1"></log><apply id="S4.Thmtheorem7.p1.3.3.m2.2.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.2.2"><divide id="S4.Thmtheorem7.p1.3.3.m2.2.2.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.2.2"></divide><cn id="S4.Thmtheorem7.p1.3.3.m2.2.2.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.2">1</cn><apply id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3"><minus id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.1"></minus><cn id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.2">1</cn><ci id="S4.Thmtheorem7.p1.3.3.m2.2.2.3.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.2.2.3.3">𝜆</ci></apply></apply></apply><apply id="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.6.6.2.2.2"><log id="S4.Thmtheorem7.p1.3.3.m2.3.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.3.3"></log><apply id="S4.Thmtheorem7.p1.3.3.m2.4.4.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.4.4"><divide id="S4.Thmtheorem7.p1.3.3.m2.4.4.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.4.4"></divide><cn id="S4.Thmtheorem7.p1.3.3.m2.4.4.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m2.4.4.2">1</cn><ci id="S4.Thmtheorem7.p1.3.3.m2.4.4.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m2.4.4.3">𝜀</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.3.3.m2.6c">\log(\frac{1}{1-\lambda}),\log(\frac{1}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.3.3.m2.6d">roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) , roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> in unary, as well as a function <math alttext="f:G^{d}_{b}\rightarrow G^{d}_{b^{\prime}}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.4.4.m3.1"><semantics id="S4.Thmtheorem7.p1.4.4.m3.1a"><mrow id="S4.Thmtheorem7.p1.4.4.m3.1.1" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem7.p1.4.4.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem7.p1.4.4.m3.1.1.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.cmml"><msubsup id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.2" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.2.cmml">G</mi><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.3.cmml">b</mi><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.3.cmml">d</mi></msubsup><mo id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.1" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.1.cmml">→</mo><msubsup id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.2" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.2.cmml">G</mi><msup id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.2" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.2.cmml">b</mi><mo id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.3.cmml">′</mo></msup><mi id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.3" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.3.cmml">d</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.4.4.m3.1b"><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1"><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.1">:</ci><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.2">𝑓</ci><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3"><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.1">→</ci><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2">subscript</csymbol><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2">superscript</csymbol><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.2">𝐺</ci><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.2.3">𝑑</ci></apply><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.2.3">𝑏</ci></apply><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3">subscript</csymbol><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3">superscript</csymbol><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.2">𝐺</ci><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.2.3">𝑑</ci></apply><apply id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3">superscript</csymbol><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.2">𝑏</ci><ci id="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m3.1.1.3.3.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.4.4.m3.1c">f:G^{d}_{b}\rightarrow G^{d}_{b^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.4.4.m3.1d">italic_f : italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT → italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> encoded as a concatenation of its values. The goal is to produce one of the following.</span></p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(S)</span> <div class="ltx_para" id="S4.I1.ix1.p1"> <p class="ltx_p" id="S4.I1.ix1.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I1.ix1.p1.2.1">A point </span><math alttext="x\in G^{d}_{b}" class="ltx_Math" display="inline" id="S4.I1.ix1.p1.1.m1.1"><semantics id="S4.I1.ix1.p1.1.m1.1a"><mrow id="S4.I1.ix1.p1.1.m1.1.1" xref="S4.I1.ix1.p1.1.m1.1.1.cmml"><mi id="S4.I1.ix1.p1.1.m1.1.1.2" xref="S4.I1.ix1.p1.1.m1.1.1.2.cmml">x</mi><mo id="S4.I1.ix1.p1.1.m1.1.1.1" xref="S4.I1.ix1.p1.1.m1.1.1.1.cmml">∈</mo><msubsup id="S4.I1.ix1.p1.1.m1.1.1.3" xref="S4.I1.ix1.p1.1.m1.1.1.3.cmml"><mi id="S4.I1.ix1.p1.1.m1.1.1.3.2.2" xref="S4.I1.ix1.p1.1.m1.1.1.3.2.2.cmml">G</mi><mi id="S4.I1.ix1.p1.1.m1.1.1.3.3" xref="S4.I1.ix1.p1.1.m1.1.1.3.3.cmml">b</mi><mi id="S4.I1.ix1.p1.1.m1.1.1.3.2.3" xref="S4.I1.ix1.p1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.ix1.p1.1.m1.1b"><apply id="S4.I1.ix1.p1.1.m1.1.1.cmml" xref="S4.I1.ix1.p1.1.m1.1.1"><in id="S4.I1.ix1.p1.1.m1.1.1.1.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.1"></in><ci id="S4.I1.ix1.p1.1.m1.1.1.2.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.2">𝑥</ci><apply id="S4.I1.ix1.p1.1.m1.1.1.3.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.ix1.p1.1.m1.1.1.3.1.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3">subscript</csymbol><apply id="S4.I1.ix1.p1.1.m1.1.1.3.2.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.ix1.p1.1.m1.1.1.3.2.1.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.I1.ix1.p1.1.m1.1.1.3.2.2.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3.2.2">𝐺</ci><ci id="S4.I1.ix1.p1.1.m1.1.1.3.2.3.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S4.I1.ix1.p1.1.m1.1.1.3.3.cmml" xref="S4.I1.ix1.p1.1.m1.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix1.p1.1.m1.1c">x\in G^{d}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix1.p1.1.m1.1d">italic_x ∈ italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.ix1.p1.2.2"> such that </span><math alttext="\lVert f(x)-x\rVert_{p}\leq\varepsilon" class="ltx_Math" display="inline" id="S4.I1.ix1.p1.2.m2.2"><semantics id="S4.I1.ix1.p1.2.m2.2a"><mrow id="S4.I1.ix1.p1.2.m2.2.2" xref="S4.I1.ix1.p1.2.m2.2.2.cmml"><msub id="S4.I1.ix1.p1.2.m2.2.2.1" xref="S4.I1.ix1.p1.2.m2.2.2.1.cmml"><mrow id="S4.I1.ix1.p1.2.m2.2.2.1.1.1" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.2.cmml"><mo fence="true" id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.2" rspace="0em" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.2.1.cmml">∥</mo><mrow id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.cmml"><mrow id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.cmml"><mi id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.2" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.2.cmml">f</mi><mo id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.1" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.3.2" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.cmml"><mo id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.cmml">(</mo><mi id="S4.I1.ix1.p1.2.m2.1.1" xref="S4.I1.ix1.p1.2.m2.1.1.cmml">x</mi><mo id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.1" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.3" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo fence="true" id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.3" lspace="0em" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="S4.I1.ix1.p1.2.m2.2.2.1.3" xref="S4.I1.ix1.p1.2.m2.2.2.1.3.cmml">p</mi></msub><mo id="S4.I1.ix1.p1.2.m2.2.2.2" xref="S4.I1.ix1.p1.2.m2.2.2.2.cmml">≤</mo><mi id="S4.I1.ix1.p1.2.m2.2.2.3" xref="S4.I1.ix1.p1.2.m2.2.2.3.cmml">ε</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.ix1.p1.2.m2.2b"><apply id="S4.I1.ix1.p1.2.m2.2.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2"><leq id="S4.I1.ix1.p1.2.m2.2.2.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.2"></leq><apply id="S4.I1.ix1.p1.2.m2.2.2.1.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1"><csymbol cd="ambiguous" id="S4.I1.ix1.p1.2.m2.2.2.1.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1">subscript</csymbol><apply id="S4.I1.ix1.p1.2.m2.2.2.1.1.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1"><csymbol cd="latexml" id="S4.I1.ix1.p1.2.m2.2.2.1.1.2.1.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1"><minus id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.1"></minus><apply id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2"><times id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.1.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.1"></times><ci id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.2.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.2.2">𝑓</ci><ci id="S4.I1.ix1.p1.2.m2.1.1.cmml" xref="S4.I1.ix1.p1.2.m2.1.1">𝑥</ci></apply><ci id="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S4.I1.ix1.p1.2.m2.2.2.1.3.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.1.3">𝑝</ci></apply><ci id="S4.I1.ix1.p1.2.m2.2.2.3.cmml" xref="S4.I1.ix1.p1.2.m2.2.2.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix1.p1.2.m2.2c">\lVert f(x)-x\rVert_{p}\leq\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix1.p1.2.m2.2d">∥ italic_f ( italic_x ) - italic_x ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_ε</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.ix1.p1.2.3">.</span></p> </div> </li> <li class="ltx_item" id="S4.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(V)</span> <div class="ltx_para" id="S4.I1.ix2.p1"> <p class="ltx_p" id="S4.I1.ix2.p1.4"><span class="ltx_text ltx_font_italic" id="S4.I1.ix2.p1.4.1">A set </span><math alttext="P" class="ltx_Math" display="inline" id="S4.I1.ix2.p1.1.m1.1"><semantics id="S4.I1.ix2.p1.1.m1.1a"><mi id="S4.I1.ix2.p1.1.m1.1.1" xref="S4.I1.ix2.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.I1.ix2.p1.1.m1.1b"><ci id="S4.I1.ix2.p1.1.m1.1.1.cmml" xref="S4.I1.ix2.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix2.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix2.p1.1.m1.1d">italic_P</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.ix2.p1.4.2"> of </span><math alttext="\operatorname{poly}(b,d,\log\left(\frac{1}{\varepsilon}\right),\log\left(\frac% {1}{1-\lambda}\right))" class="ltx_Math" display="inline" id="S4.I1.ix2.p1.2.m2.9"><semantics id="S4.I1.ix2.p1.2.m2.9a"><mrow id="S4.I1.ix2.p1.2.m2.9.9.2" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml"><mi id="S4.I1.ix2.p1.2.m2.5.5" xref="S4.I1.ix2.p1.2.m2.5.5.cmml">poly</mi><mo id="S4.I1.ix2.p1.2.m2.9.9.2a" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">⁡</mo><mrow id="S4.I1.ix2.p1.2.m2.9.9.2.2" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml"><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.3" stretchy="false" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">(</mo><mi id="S4.I1.ix2.p1.2.m2.6.6" xref="S4.I1.ix2.p1.2.m2.6.6.cmml">b</mi><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.4" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">,</mo><mi id="S4.I1.ix2.p1.2.m2.7.7" xref="S4.I1.ix2.p1.2.m2.7.7.cmml">d</mi><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.5" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">,</mo><mrow id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml"><mi id="S4.I1.ix2.p1.2.m2.1.1" xref="S4.I1.ix2.p1.2.m2.1.1.cmml">log</mi><mo id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2a" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml">⁡</mo><mrow id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2.1" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml"><mo id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2.1.1" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml">(</mo><mfrac id="S4.I1.ix2.p1.2.m2.2.2" xref="S4.I1.ix2.p1.2.m2.2.2.cmml"><mn id="S4.I1.ix2.p1.2.m2.2.2.2" xref="S4.I1.ix2.p1.2.m2.2.2.2.cmml">1</mn><mi id="S4.I1.ix2.p1.2.m2.2.2.3" xref="S4.I1.ix2.p1.2.m2.2.2.3.cmml">ε</mi></mfrac><mo id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2.1.2" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.6" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">,</mo><mrow id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml"><mi id="S4.I1.ix2.p1.2.m2.3.3" xref="S4.I1.ix2.p1.2.m2.3.3.cmml">log</mi><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2a" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml">⁡</mo><mrow id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2.1" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml"><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2.1.1" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml">(</mo><mfrac id="S4.I1.ix2.p1.2.m2.4.4" xref="S4.I1.ix2.p1.2.m2.4.4.cmml"><mn id="S4.I1.ix2.p1.2.m2.4.4.2" xref="S4.I1.ix2.p1.2.m2.4.4.2.cmml">1</mn><mrow id="S4.I1.ix2.p1.2.m2.4.4.3" xref="S4.I1.ix2.p1.2.m2.4.4.3.cmml"><mn id="S4.I1.ix2.p1.2.m2.4.4.3.2" xref="S4.I1.ix2.p1.2.m2.4.4.3.2.cmml">1</mn><mo id="S4.I1.ix2.p1.2.m2.4.4.3.1" xref="S4.I1.ix2.p1.2.m2.4.4.3.1.cmml">−</mo><mi id="S4.I1.ix2.p1.2.m2.4.4.3.3" xref="S4.I1.ix2.p1.2.m2.4.4.3.3.cmml">λ</mi></mrow></mfrac><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2.1.2" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.ix2.p1.2.m2.9.9.2.2.7" stretchy="false" xref="S4.I1.ix2.p1.2.m2.9.9.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.ix2.p1.2.m2.9b"><apply id="S4.I1.ix2.p1.2.m2.9.9.3.cmml" xref="S4.I1.ix2.p1.2.m2.9.9.2"><ci id="S4.I1.ix2.p1.2.m2.5.5.cmml" xref="S4.I1.ix2.p1.2.m2.5.5">poly</ci><ci id="S4.I1.ix2.p1.2.m2.6.6.cmml" xref="S4.I1.ix2.p1.2.m2.6.6">𝑏</ci><ci id="S4.I1.ix2.p1.2.m2.7.7.cmml" xref="S4.I1.ix2.p1.2.m2.7.7">𝑑</ci><apply id="S4.I1.ix2.p1.2.m2.8.8.1.1.1.1.cmml" xref="S4.I1.ix2.p1.2.m2.8.8.1.1.1.2"><log id="S4.I1.ix2.p1.2.m2.1.1.cmml" xref="S4.I1.ix2.p1.2.m2.1.1"></log><apply id="S4.I1.ix2.p1.2.m2.2.2.cmml" xref="S4.I1.ix2.p1.2.m2.2.2"><divide id="S4.I1.ix2.p1.2.m2.2.2.1.cmml" xref="S4.I1.ix2.p1.2.m2.2.2"></divide><cn id="S4.I1.ix2.p1.2.m2.2.2.2.cmml" type="integer" xref="S4.I1.ix2.p1.2.m2.2.2.2">1</cn><ci id="S4.I1.ix2.p1.2.m2.2.2.3.cmml" xref="S4.I1.ix2.p1.2.m2.2.2.3">𝜀</ci></apply></apply><apply id="S4.I1.ix2.p1.2.m2.9.9.2.2.2.1.cmml" xref="S4.I1.ix2.p1.2.m2.9.9.2.2.2.2"><log id="S4.I1.ix2.p1.2.m2.3.3.cmml" xref="S4.I1.ix2.p1.2.m2.3.3"></log><apply id="S4.I1.ix2.p1.2.m2.4.4.cmml" xref="S4.I1.ix2.p1.2.m2.4.4"><divide id="S4.I1.ix2.p1.2.m2.4.4.1.cmml" xref="S4.I1.ix2.p1.2.m2.4.4"></divide><cn id="S4.I1.ix2.p1.2.m2.4.4.2.cmml" type="integer" xref="S4.I1.ix2.p1.2.m2.4.4.2">1</cn><apply id="S4.I1.ix2.p1.2.m2.4.4.3.cmml" xref="S4.I1.ix2.p1.2.m2.4.4.3"><minus id="S4.I1.ix2.p1.2.m2.4.4.3.1.cmml" xref="S4.I1.ix2.p1.2.m2.4.4.3.1"></minus><cn id="S4.I1.ix2.p1.2.m2.4.4.3.2.cmml" type="integer" xref="S4.I1.ix2.p1.2.m2.4.4.3.2">1</cn><ci id="S4.I1.ix2.p1.2.m2.4.4.3.3.cmml" xref="S4.I1.ix2.p1.2.m2.4.4.3.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix2.p1.2.m2.9c">\operatorname{poly}(b,d,\log\left(\frac{1}{\varepsilon}\right),\log\left(\frac% {1}{1-\lambda}\right))</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix2.p1.2.m2.9d">roman_poly ( italic_b , italic_d , roman_log ( divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG ) , roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG ) )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.ix2.p1.4.3"> points in </span><math alttext="G^{d}_{b}" class="ltx_Math" display="inline" id="S4.I1.ix2.p1.3.m3.1"><semantics id="S4.I1.ix2.p1.3.m3.1a"><msubsup id="S4.I1.ix2.p1.3.m3.1.1" xref="S4.I1.ix2.p1.3.m3.1.1.cmml"><mi id="S4.I1.ix2.p1.3.m3.1.1.2.2" xref="S4.I1.ix2.p1.3.m3.1.1.2.2.cmml">G</mi><mi id="S4.I1.ix2.p1.3.m3.1.1.3" xref="S4.I1.ix2.p1.3.m3.1.1.3.cmml">b</mi><mi id="S4.I1.ix2.p1.3.m3.1.1.2.3" xref="S4.I1.ix2.p1.3.m3.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.I1.ix2.p1.3.m3.1b"><apply id="S4.I1.ix2.p1.3.m3.1.1.cmml" xref="S4.I1.ix2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.ix2.p1.3.m3.1.1.1.cmml" xref="S4.I1.ix2.p1.3.m3.1.1">subscript</csymbol><apply id="S4.I1.ix2.p1.3.m3.1.1.2.cmml" xref="S4.I1.ix2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.ix2.p1.3.m3.1.1.2.1.cmml" xref="S4.I1.ix2.p1.3.m3.1.1">superscript</csymbol><ci 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id="S4.I1.ix2.p1.4.m4.3d">italic_G start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊆ ⋃ start_POSTSUBSCRIPT italic_x ∈ italic_P end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_f ( italic_x ) end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.ix2.p1.4.5">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S4.SS3.p6"> <p class="ltx_p" id="S4.SS3.p6.3">Note that for our algorithm to work, we need to syntactically guarantee <math alttext="b\geq\log_{2}\left(\frac{2d}{\varepsilon}\frac{1+\lambda}{1-\lambda}\right)" class="ltx_Math" display="inline" id="S4.SS3.p6.1.m1.2"><semantics id="S4.SS3.p6.1.m1.2a"><mrow id="S4.SS3.p6.1.m1.2.2" xref="S4.SS3.p6.1.m1.2.2.cmml"><mi id="S4.SS3.p6.1.m1.2.2.4" xref="S4.SS3.p6.1.m1.2.2.4.cmml">b</mi><mo id="S4.SS3.p6.1.m1.2.2.3" xref="S4.SS3.p6.1.m1.2.2.3.cmml">≥</mo><mrow 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encoding="application/x-llamapun" id="S4.SS3.p6.1.m1.2d">italic_b ≥ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( divide start_ARG 2 italic_d end_ARG start_ARG italic_ε end_ARG divide start_ARG 1 + italic_λ end_ARG start_ARG 1 - italic_λ end_ARG )</annotation></semantics></math>. In fact, without this assumption, we would not even know if the problem is total. We also syntactically guarantee <math alttext="b^{\prime}\in\operatorname{poly}(b)" class="ltx_Math" display="inline" id="S4.SS3.p6.2.m2.2"><semantics id="S4.SS3.p6.2.m2.2a"><mrow id="S4.SS3.p6.2.m2.2.3" xref="S4.SS3.p6.2.m2.2.3.cmml"><msup id="S4.SS3.p6.2.m2.2.3.2" xref="S4.SS3.p6.2.m2.2.3.2.cmml"><mi id="S4.SS3.p6.2.m2.2.3.2.2" xref="S4.SS3.p6.2.m2.2.3.2.2.cmml">b</mi><mo id="S4.SS3.p6.2.m2.2.3.2.3" xref="S4.SS3.p6.2.m2.2.3.2.3.cmml">′</mo></msup><mo id="S4.SS3.p6.2.m2.2.3.1" xref="S4.SS3.p6.2.m2.2.3.1.cmml">∈</mo><mrow id="S4.SS3.p6.2.m2.2.3.3.2" xref="S4.SS3.p6.2.m2.2.3.3.1.cmml"><mi id="S4.SS3.p6.2.m2.1.1" xref="S4.SS3.p6.2.m2.1.1.cmml">poly</mi><mo id="S4.SS3.p6.2.m2.2.3.3.2a" xref="S4.SS3.p6.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S4.SS3.p6.2.m2.2.3.3.2.1" xref="S4.SS3.p6.2.m2.2.3.3.1.cmml"><mo id="S4.SS3.p6.2.m2.2.3.3.2.1.1" stretchy="false" xref="S4.SS3.p6.2.m2.2.3.3.1.cmml">(</mo><mi id="S4.SS3.p6.2.m2.2.2" xref="S4.SS3.p6.2.m2.2.2.cmml">b</mi><mo id="S4.SS3.p6.2.m2.2.3.3.2.1.2" stretchy="false" xref="S4.SS3.p6.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.2.m2.2b"><apply id="S4.SS3.p6.2.m2.2.3.cmml" xref="S4.SS3.p6.2.m2.2.3"><in id="S4.SS3.p6.2.m2.2.3.1.cmml" xref="S4.SS3.p6.2.m2.2.3.1"></in><apply id="S4.SS3.p6.2.m2.2.3.2.cmml" xref="S4.SS3.p6.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p6.2.m2.2.3.2.1.cmml" xref="S4.SS3.p6.2.m2.2.3.2">superscript</csymbol><ci id="S4.SS3.p6.2.m2.2.3.2.2.cmml" xref="S4.SS3.p6.2.m2.2.3.2.2">𝑏</ci><ci id="S4.SS3.p6.2.m2.2.3.2.3.cmml" xref="S4.SS3.p6.2.m2.2.3.2.3">′</ci></apply><apply id="S4.SS3.p6.2.m2.2.3.3.1.cmml" xref="S4.SS3.p6.2.m2.2.3.3.2"><ci id="S4.SS3.p6.2.m2.1.1.cmml" xref="S4.SS3.p6.2.m2.1.1">poly</ci><ci id="S4.SS3.p6.2.m2.2.2.cmml" xref="S4.SS3.p6.2.m2.2.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.2.m2.2c">b^{\prime}\in\operatorname{poly}(b)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.2.m2.2d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_poly ( italic_b )</annotation></semantics></math> to make sure that the number of bit queries that our algorithm makes to the encoding of <math alttext="f" class="ltx_Math" display="inline" id="S4.SS3.p6.3.m3.1"><semantics id="S4.SS3.p6.3.m3.1a"><mi id="S4.SS3.p6.3.m3.1.1" xref="S4.SS3.p6.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.3.m3.1b"><ci id="S4.SS3.p6.3.m3.1.1.cmml" xref="S4.SS3.p6.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.3.m3.1d">italic_f</annotation></semantics></math> is polynomially bounded. Such syntactic guarantees can be realized by adding trivial-to-verify additional violation types for the cases where the input parameters do not fulfill these assumptions.</p> </div> <div class="ltx_para" id="S4.SS3.p7"> <p class="ltx_p" id="S4.SS3.p7.5">We have shown in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem3" title="Theorem 4.3. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.3</span></a> that after <math alttext="\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d))" class="ltx_Math" display="inline" id="S4.SS3.p7.1.m1.1"><semantics id="S4.SS3.p7.1.m1.1a"><mrow id="S4.SS3.p7.1.m1.1.1" xref="S4.SS3.p7.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p7.1.m1.1.1.3" xref="S4.SS3.p7.1.m1.1.1.3.cmml">𝒪</mi><mo 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xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.1" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.1.cmml">log</mi><mo id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mfrac id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mn id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.2.cmml">1</mn><mi id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.3" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.3.cmml">ε</mi></mfrac></mrow><mo id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.1" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.1" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mfrac 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xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.2.2.3">𝜀</ci></apply></apply><apply id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3"><log id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.1"></log><apply id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2"><divide id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2"></divide><cn id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.2">1</cn><apply id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3"><minus id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.1.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.1"></minus><cn id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.2">1</cn><ci id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply><apply id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4"><log id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4.1"></log><ci id="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4.2.cmml" xref="S4.SS3.p7.1.m1.1.1.1.1.1.1.1.1.4.2">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.1.m1.1c">\mathcal{O}(d^{2}(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}+\log d))</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.1.m1.1d">caligraphic_O ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log divide start_ARG 1 end_ARG start_ARG italic_ε end_ARG + roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_λ end_ARG + roman_log italic_d ) )</annotation></semantics></math> queries to <math alttext="f" class="ltx_Math" display="inline" id="S4.SS3.p7.2.m2.1"><semantics id="S4.SS3.p7.2.m2.1a"><mi id="S4.SS3.p7.2.m2.1.1" xref="S4.SS3.p7.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p7.2.m2.1b"><ci id="S4.SS3.p7.2.m2.1.1.cmml" xref="S4.SS3.p7.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.2.m2.1d">italic_f</annotation></semantics></math>, the queries made by our algorithm must have produced either a solution (S), or a violation (V). Thus, our algorithm solves <span class="ltx_text ltx_font_smallcaps" id="S4.SS3.p7.3.1">Total-<math alttext="\ell_{1}" class="ltx_Math" display="inline" id="S4.SS3.p7.3.1.m1.1"><semantics id="S4.SS3.p7.3.1.m1.1a"><msub id="S4.SS3.p7.3.1.m1.1.1" xref="S4.SS3.p7.3.1.m1.1.1.cmml"><mi id="S4.SS3.p7.3.1.m1.1.1.2" mathvariant="normal" xref="S4.SS3.p7.3.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.SS3.p7.3.1.m1.1.1.3" xref="S4.SS3.p7.3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p7.3.1.m1.1b"><apply id="S4.SS3.p7.3.1.m1.1.1.cmml" xref="S4.SS3.p7.3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p7.3.1.m1.1.1.1.cmml" xref="S4.SS3.p7.3.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p7.3.1.m1.1.1.2.cmml" xref="S4.SS3.p7.3.1.m1.1.1.2">ℓ</ci><cn id="S4.SS3.p7.3.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p7.3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.3.1.m1.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.3.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-ContractionFixpoint</span> as well, placing it in <math alttext="\mathsf{TFNP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p7.4.m3.1"><semantics id="S4.SS3.p7.4.m3.1a"><msup id="S4.SS3.p7.4.m3.1.1" xref="S4.SS3.p7.4.m3.1.1.cmml"><mi id="S4.SS3.p7.4.m3.1.1.2" xref="S4.SS3.p7.4.m3.1.1.2.cmml">𝖳𝖥𝖭𝖯</mi><mtext id="S4.SS3.p7.4.m3.1.1.3" xref="S4.SS3.p7.4.m3.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p7.4.m3.1b"><apply id="S4.SS3.p7.4.m3.1.1.cmml" xref="S4.SS3.p7.4.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p7.4.m3.1.1.1.cmml" xref="S4.SS3.p7.4.m3.1.1">superscript</csymbol><ci id="S4.SS3.p7.4.m3.1.1.2.cmml" xref="S4.SS3.p7.4.m3.1.1.2">𝖳𝖥𝖭𝖯</ci><ci id="S4.SS3.p7.4.m3.1.1.3a.cmml" xref="S4.SS3.p7.4.m3.1.1.3"><mtext id="S4.SS3.p7.4.m3.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p7.4.m3.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.4.m3.1c">\mathsf{TFNP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.4.m3.1d">sansserif_TFNP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math> and even in <math alttext="\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.SS3.p7.5.m4.1"><semantics id="S4.SS3.p7.5.m4.1a"><msup id="S4.SS3.p7.5.m4.1.1" xref="S4.SS3.p7.5.m4.1.1.cmml"><mi id="S4.SS3.p7.5.m4.1.1.2" xref="S4.SS3.p7.5.m4.1.1.2.cmml">𝖥𝖯</mi><mtext id="S4.SS3.p7.5.m4.1.1.3" xref="S4.SS3.p7.5.m4.1.1.3a.cmml">dt</mtext></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p7.5.m4.1b"><apply id="S4.SS3.p7.5.m4.1.1.cmml" xref="S4.SS3.p7.5.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p7.5.m4.1.1.1.cmml" xref="S4.SS3.p7.5.m4.1.1">superscript</csymbol><ci id="S4.SS3.p7.5.m4.1.1.2.cmml" xref="S4.SS3.p7.5.m4.1.1.2">𝖥𝖯</ci><ci id="S4.SS3.p7.5.m4.1.1.3a.cmml" xref="S4.SS3.p7.5.m4.1.1.3"><mtext id="S4.SS3.p7.5.m4.1.1.3.cmml" mathsize="70%" xref="S4.SS3.p7.5.m4.1.1.3">dt</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.5.m4.1c">\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.5.m4.1d">sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Corollary 4.8</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.1"><math alttext="\textsc{Total-$\ell_{1}$-ContractionFixpoint}\in\mathsf{FP}^{\text{dt}}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.m1.1"><semantics id="S4.Thmtheorem8.p1.1.m1.1a"><mrow id="S4.Thmtheorem8.p1.1.m1.1.2" xref="S4.Thmtheorem8.p1.1.m1.1.2.cmml"><mrow id="S4.Thmtheorem8.p1.1.m1.1.1.1" xref="S4.Thmtheorem8.p1.1.m1.1.1.1c.cmml"><mtext class="ltx_font_smallcaps" id="S4.Thmtheorem8.p1.1.m1.1.1.1a" xref="S4.Thmtheorem8.p1.1.m1.1.1.1c.cmml">Total-</mtext><msub id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.2" mathvariant="normal" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.2.cmml">ℓ</mi><mn id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.3" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.3.cmml">1</mn></msub><mtext class="ltx_font_smallcaps" id="S4.Thmtheorem8.p1.1.m1.1.1.1b" xref="S4.Thmtheorem8.p1.1.m1.1.1.1c.cmml">-ContractionFixpoint</mtext></mrow><mo id="S4.Thmtheorem8.p1.1.m1.1.2.1" xref="S4.Thmtheorem8.p1.1.m1.1.2.1.cmml">∈</mo><msup id="S4.Thmtheorem8.p1.1.m1.1.2.2" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.cmml"><mi id="S4.Thmtheorem8.p1.1.m1.1.2.2.2" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.2.cmml">𝖥𝖯</mi><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem8.p1.1.m1.1.2.2.3" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.3a.cmml">dt</mtext></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.m1.1b"><apply id="S4.Thmtheorem8.p1.1.m1.1.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2"><in id="S4.Thmtheorem8.p1.1.m1.1.2.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2.1"></in><ci id="S4.Thmtheorem8.p1.1.m1.1.1.1c.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1"><mrow id="S4.Thmtheorem8.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1"><mtext class="ltx_font_smallcaps" id="S4.Thmtheorem8.p1.1.m1.1.1.1a.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1">Total-</mtext><msub id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1"><mi id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.2.cmml" mathvariant="normal" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.2">ℓ</mi><mn id="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1.m1.1.1.3">1</mn></msub><mtext class="ltx_font_smallcaps" id="S4.Thmtheorem8.p1.1.m1.1.1.1b.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.1">-ContractionFixpoint</mtext></mrow></ci><apply id="S4.Thmtheorem8.p1.1.m1.1.2.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.1.m1.1.2.2.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem8.p1.1.m1.1.2.2.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.2">𝖥𝖯</ci><ci id="S4.Thmtheorem8.p1.1.m1.1.2.2.3a.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem8.p1.1.m1.1.2.2.3.cmml" mathsize="70%" xref="S4.Thmtheorem8.p1.1.m1.1.2.2.3">dt</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.m1.1c">\textsc{Total-$\ell_{1}$-ContractionFixpoint}\in\mathsf{FP}^{\text{dt}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.m1.1d">Total- roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT -ContractionFixpoint ∈ sansserif_FP start_POSTSUPERSCRIPT dt end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem8.p1.1.1">.</span></p> </div> </div> <div class="ltx_para" id="S4.SS3.p8"> <p class="ltx_p" id="S4.SS3.p8.2">Note that for general <math alttext="p\notin\{1,2,\infty\}" class="ltx_Math" display="inline" id="S4.SS3.p8.1.m1.3"><semantics id="S4.SS3.p8.1.m1.3a"><mrow id="S4.SS3.p8.1.m1.3.4" xref="S4.SS3.p8.1.m1.3.4.cmml"><mi id="S4.SS3.p8.1.m1.3.4.2" xref="S4.SS3.p8.1.m1.3.4.2.cmml">p</mi><mo id="S4.SS3.p8.1.m1.3.4.1" xref="S4.SS3.p8.1.m1.3.4.1.cmml">∉</mo><mrow id="S4.SS3.p8.1.m1.3.4.3.2" xref="S4.SS3.p8.1.m1.3.4.3.1.cmml"><mo id="S4.SS3.p8.1.m1.3.4.3.2.1" stretchy="false" xref="S4.SS3.p8.1.m1.3.4.3.1.cmml">{</mo><mn id="S4.SS3.p8.1.m1.1.1" xref="S4.SS3.p8.1.m1.1.1.cmml">1</mn><mo id="S4.SS3.p8.1.m1.3.4.3.2.2" xref="S4.SS3.p8.1.m1.3.4.3.1.cmml">,</mo><mn id="S4.SS3.p8.1.m1.2.2" xref="S4.SS3.p8.1.m1.2.2.cmml">2</mn><mo id="S4.SS3.p8.1.m1.3.4.3.2.3" xref="S4.SS3.p8.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS3.p8.1.m1.3.3" mathvariant="normal" xref="S4.SS3.p8.1.m1.3.3.cmml">∞</mi><mo id="S4.SS3.p8.1.m1.3.4.3.2.4" stretchy="false" xref="S4.SS3.p8.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p8.1.m1.3b"><apply id="S4.SS3.p8.1.m1.3.4.cmml" xref="S4.SS3.p8.1.m1.3.4"><notin id="S4.SS3.p8.1.m1.3.4.1.cmml" xref="S4.SS3.p8.1.m1.3.4.1"></notin><ci id="S4.SS3.p8.1.m1.3.4.2.cmml" xref="S4.SS3.p8.1.m1.3.4.2">𝑝</ci><set id="S4.SS3.p8.1.m1.3.4.3.1.cmml" xref="S4.SS3.p8.1.m1.3.4.3.2"><cn id="S4.SS3.p8.1.m1.1.1.cmml" type="integer" xref="S4.SS3.p8.1.m1.1.1">1</cn><cn id="S4.SS3.p8.1.m1.2.2.cmml" type="integer" xref="S4.SS3.p8.1.m1.2.2">2</cn><infinity id="S4.SS3.p8.1.m1.3.3.cmml" xref="S4.SS3.p8.1.m1.3.3"></infinity></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.1.m1.3c">p\notin\{1,2,\infty\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.1.m1.3d">italic_p ∉ { 1 , 2 , ∞ }</annotation></semantics></math>, it is unclear whether an analogue of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem7" title="Definition 4.7. ‣ 4.3 Total Search Version ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">4.7</span></a> would even yield a total problem: we do not provide any rounding strategy for those values and hence it is unclear whether any grid-map without <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S4.SS3.p8.2.m2.1"><semantics id="S4.SS3.p8.2.m2.1a"><mi id="S4.SS3.p8.2.m2.1.1" xref="S4.SS3.p8.2.m2.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p8.2.m2.1b"><ci id="S4.SS3.p8.2.m2.1.1.cmml" xref="S4.SS3.p8.2.m2.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p8.2.m2.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p8.2.m2.1d">italic_ε</annotation></semantics></math>-approximate fixpoints on the grid must have a certificate for this in terms of a violation (V) as described above.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Stefan Banach. </span> <span class="ltx_bibblock">Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" 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CRC Press LLC, 3 edition, 2017. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>More on <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.1.m1.1"><semantics id="A1.1.m1.1b"><msub id="A1.1.m1.1.1" xref="A1.1.m1.1.1.cmml"><mi id="A1.1.m1.1.1.2" mathvariant="normal" xref="A1.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.1.m1.1.1.3" xref="A1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.1.m1.1c"><apply id="A1.1.m1.1.1.cmml" xref="A1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.1.m1.1.1.1.cmml" xref="A1.1.m1.1.1">subscript</csymbol><ci id="A1.1.m1.1.1.2.cmml" xref="A1.1.m1.1.1.2">ℓ</ci><ci id="A1.1.m1.1.1.3.cmml" xref="A1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.3">In this section, we will study limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><msub id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mi id="A1.p1.1.m1.1.1.2" mathvariant="normal" xref="A1.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.p1.1.m1.1.1.3" xref="A1.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><apply id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p1.1.m1.1.1.1.cmml" xref="A1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.p1.1.m1.1.1.2.cmml" xref="A1.p1.1.m1.1.1.2">ℓ</ci><ci id="A1.p1.1.m1.1.1.3.cmml" xref="A1.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces in more detail. The main goal is to provide proofs for the halfspace properties in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS2" title="3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.SS2" title="4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, but we will also develop some additional theory on the way. We split the exposition into two parts: fundamental properties of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><msub id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml"><mi id="A1.p1.2.m2.1.1.2" mathvariant="normal" xref="A1.p1.2.m2.1.1.2.cmml">ℓ</mi><mi id="A1.p1.2.m2.1.1.3" xref="A1.p1.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><apply id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p1.2.m2.1.1.1.cmml" xref="A1.p1.2.m2.1.1">subscript</csymbol><ci id="A1.p1.2.m2.1.1.2.cmml" xref="A1.p1.2.m2.1.1.2">ℓ</ci><ci id="A1.p1.2.m2.1.1.3.cmml" xref="A1.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces and insights about their shape are developed in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS1" title="A.1 Fundamentals of ℓ_𝑝-Halfspaces ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.1</span></a> while <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.SS2" title="A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.2</span></a> focuses on the interaction of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.p1.3.m3.1"><semantics id="A1.p1.3.m3.1a"><msub id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml"><mi id="A1.p1.3.m3.1.1.2" mathvariant="normal" xref="A1.p1.3.m3.1.1.2.cmml">ℓ</mi><mi id="A1.p1.3.m3.1.1.3" xref="A1.p1.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.1b"><apply id="A1.p1.3.m3.1.1.cmml" xref="A1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.p1.3.m3.1.1.1.cmml" xref="A1.p1.3.m3.1.1">subscript</csymbol><ci id="A1.p1.3.m3.1.1.2.cmml" xref="A1.p1.3.m3.1.1.2">ℓ</ci><ci id="A1.p1.3.m3.1.1.3.cmml" xref="A1.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces with mass distributions.</p> </div> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.1 </span>Fundamentals of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.1.m1.1"><semantics id="A1.SS1.1.m1.1b"><msub id="A1.SS1.1.m1.1.1" xref="A1.SS1.1.m1.1.1.cmml"><mi id="A1.SS1.1.m1.1.1.2" mathvariant="normal" xref="A1.SS1.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.1.m1.1.1.3" xref="A1.SS1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.1.m1.1c"><apply id="A1.SS1.1.m1.1.1.cmml" xref="A1.SS1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.1.m1.1.1.1.cmml" xref="A1.SS1.1.m1.1.1">subscript</csymbol><ci id="A1.SS1.1.m1.1.1.2.cmml" xref="A1.SS1.1.m1.1.1.2">ℓ</ci><ci id="A1.SS1.1.m1.1.1.3.cmml" xref="A1.SS1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces</h3> <div class="ltx_para" id="A1.SS1.p1"> <p class="ltx_p" id="A1.SS1.p1.2">For completeness sake, we recall the definition of limit <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p1.1.m1.1"><semantics id="A1.SS1.p1.1.m1.1a"><msub id="A1.SS1.p1.1.m1.1.1" xref="A1.SS1.p1.1.m1.1.1.cmml"><mi id="A1.SS1.p1.1.m1.1.1.2" mathvariant="normal" xref="A1.SS1.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p1.1.m1.1.1.3" xref="A1.SS1.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.1.m1.1b"><apply id="A1.SS1.p1.1.m1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.1.m1.1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.SS1.p1.1.m1.1.1.2.cmml" xref="A1.SS1.p1.1.m1.1.1.2">ℓ</ci><ci id="A1.SS1.p1.1.m1.1.1.3.cmml" xref="A1.SS1.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, which we will just refer to as <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p1.2.m2.1"><semantics id="A1.SS1.p1.2.m2.1a"><msub id="A1.SS1.p1.2.m2.1.1" xref="A1.SS1.p1.2.m2.1.1.cmml"><mi id="A1.SS1.p1.2.m2.1.1.2" mathvariant="normal" xref="A1.SS1.p1.2.m2.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p1.2.m2.1.1.3" xref="A1.SS1.p1.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.2.m2.1b"><apply id="A1.SS1.p1.2.m2.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.2.m2.1.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="A1.SS1.p1.2.m2.1.1.2.cmml" xref="A1.SS1.p1.2.m2.1.1.2">ℓ</ci><ci id="A1.SS1.p1.2.m2.1.1.3.cmml" xref="A1.SS1.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces in this appendix.</p> </div> <div class="ltx_para" id="A1.SS1.p2"> <p class="ltx_p" id="A1.SS1.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem2" title="Definition 3.2 (Limit ℓ_𝑝-Halfspace). ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.2</span></a></p> </div> <div class="ltx_para" id="A1.SS1.p3"> <p class="ltx_p" id="A1.SS1.p3.5">Recall that <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.p3.1.m1.2"><semantics id="A1.SS1.p3.1.m1.2a"><msubsup id="A1.SS1.p3.1.m1.2.3" xref="A1.SS1.p3.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.p3.1.m1.2.3.2.2" xref="A1.SS1.p3.1.m1.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.p3.1.m1.2.2.2.4" xref="A1.SS1.p3.1.m1.2.2.2.3.cmml"><mi id="A1.SS1.p3.1.m1.1.1.1.1" xref="A1.SS1.p3.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS1.p3.1.m1.2.2.2.4.1" xref="A1.SS1.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS1.p3.1.m1.2.2.2.2" xref="A1.SS1.p3.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.p3.1.m1.2.3.2.3" xref="A1.SS1.p3.1.m1.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.1.m1.2b"><apply id="A1.SS1.p3.1.m1.2.3.cmml" xref="A1.SS1.p3.1.m1.2.3"><csymbol cd="ambiguous" id="A1.SS1.p3.1.m1.2.3.1.cmml" xref="A1.SS1.p3.1.m1.2.3">subscript</csymbol><apply id="A1.SS1.p3.1.m1.2.3.2.cmml" xref="A1.SS1.p3.1.m1.2.3"><csymbol cd="ambiguous" id="A1.SS1.p3.1.m1.2.3.2.1.cmml" xref="A1.SS1.p3.1.m1.2.3">superscript</csymbol><ci id="A1.SS1.p3.1.m1.2.3.2.2.cmml" xref="A1.SS1.p3.1.m1.2.3.2.2">ℋ</ci><ci id="A1.SS1.p3.1.m1.2.3.2.3.cmml" xref="A1.SS1.p3.1.m1.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.p3.1.m1.2.2.2.3.cmml" xref="A1.SS1.p3.1.m1.2.2.2.4"><ci id="A1.SS1.p3.1.m1.1.1.1.1.cmml" xref="A1.SS1.p3.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS1.p3.1.m1.2.2.2.2.cmml" xref="A1.SS1.p3.1.m1.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.1.m1.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.1.m1.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is invariant under scaling <math alttext="v" class="ltx_Math" display="inline" id="A1.SS1.p3.2.m2.1"><semantics id="A1.SS1.p3.2.m2.1a"><mi id="A1.SS1.p3.2.m2.1.1" xref="A1.SS1.p3.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.2.m2.1b"><ci id="A1.SS1.p3.2.m2.1.1.cmml" xref="A1.SS1.p3.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.2.m2.1d">italic_v</annotation></semantics></math> with a positive scalar, and we therefore usually use <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="A1.SS1.p3.3.m3.1"><semantics id="A1.SS1.p3.3.m3.1a"><mrow id="A1.SS1.p3.3.m3.1.1" xref="A1.SS1.p3.3.m3.1.1.cmml"><mi id="A1.SS1.p3.3.m3.1.1.2" xref="A1.SS1.p3.3.m3.1.1.2.cmml">v</mi><mo id="A1.SS1.p3.3.m3.1.1.1" xref="A1.SS1.p3.3.m3.1.1.1.cmml">∈</mo><msup id="A1.SS1.p3.3.m3.1.1.3" xref="A1.SS1.p3.3.m3.1.1.3.cmml"><mi id="A1.SS1.p3.3.m3.1.1.3.2" xref="A1.SS1.p3.3.m3.1.1.3.2.cmml">S</mi><mrow id="A1.SS1.p3.3.m3.1.1.3.3" xref="A1.SS1.p3.3.m3.1.1.3.3.cmml"><mi id="A1.SS1.p3.3.m3.1.1.3.3.2" xref="A1.SS1.p3.3.m3.1.1.3.3.2.cmml">d</mi><mo id="A1.SS1.p3.3.m3.1.1.3.3.1" xref="A1.SS1.p3.3.m3.1.1.3.3.1.cmml">−</mo><mn id="A1.SS1.p3.3.m3.1.1.3.3.3" xref="A1.SS1.p3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.3.m3.1b"><apply id="A1.SS1.p3.3.m3.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1"><in id="A1.SS1.p3.3.m3.1.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1"></in><ci id="A1.SS1.p3.3.m3.1.1.2.cmml" xref="A1.SS1.p3.3.m3.1.1.2">𝑣</ci><apply id="A1.SS1.p3.3.m3.1.1.3.cmml" xref="A1.SS1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p3.3.m3.1.1.3.1.cmml" xref="A1.SS1.p3.3.m3.1.1.3">superscript</csymbol><ci id="A1.SS1.p3.3.m3.1.1.3.2.cmml" xref="A1.SS1.p3.3.m3.1.1.3.2">𝑆</ci><apply id="A1.SS1.p3.3.m3.1.1.3.3.cmml" xref="A1.SS1.p3.3.m3.1.1.3.3"><minus id="A1.SS1.p3.3.m3.1.1.3.3.1.cmml" xref="A1.SS1.p3.3.m3.1.1.3.3.1"></minus><ci id="A1.SS1.p3.3.m3.1.1.3.3.2.cmml" xref="A1.SS1.p3.3.m3.1.1.3.3.2">𝑑</ci><cn id="A1.SS1.p3.3.m3.1.1.3.3.3.cmml" type="integer" xref="A1.SS1.p3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.3.m3.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.3.m3.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. As a warm-up, we prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem5" title="Lemma 3.5. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.5</span></a>, which considers <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p3.4.m4.1"><semantics id="A1.SS1.p3.4.m4.1a"><msub id="A1.SS1.p3.4.m4.1.1" xref="A1.SS1.p3.4.m4.1.1.cmml"><mi id="A1.SS1.p3.4.m4.1.1.2" mathvariant="normal" xref="A1.SS1.p3.4.m4.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p3.4.m4.1.1.3" xref="A1.SS1.p3.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.4.m4.1b"><apply id="A1.SS1.p3.4.m4.1.1.cmml" xref="A1.SS1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS1.p3.4.m4.1.1.1.cmml" xref="A1.SS1.p3.4.m4.1.1">subscript</csymbol><ci id="A1.SS1.p3.4.m4.1.1.2.cmml" xref="A1.SS1.p3.4.m4.1.1.2">ℓ</ci><ci id="A1.SS1.p3.4.m4.1.1.3.cmml" xref="A1.SS1.p3.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces with axis-aligned directions <math alttext="v" class="ltx_Math" display="inline" id="A1.SS1.p3.5.m5.1"><semantics id="A1.SS1.p3.5.m5.1a"><mi id="A1.SS1.p3.5.m5.1.1" xref="A1.SS1.p3.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.5.m5.1b"><ci id="A1.SS1.p3.5.m5.1.1.cmml" xref="A1.SS1.p3.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.5.m5.1d">italic_v</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.SS1.p4"> <p class="ltx_p" id="A1.SS1.p4.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem5" title="Lemma 3.5. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.5</span></a></p> </div> <div class="ltx_proof" id="A1.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.2.p1"> <p class="ltx_p" id="A1.SS1.2.p1.9">Without loss of generality, assume that <math alttext="v=(1,0,\dots,0)\in S^{d-1}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.1.m1.4"><semantics id="A1.SS1.2.p1.1.m1.4a"><mrow id="A1.SS1.2.p1.1.m1.4.5" xref="A1.SS1.2.p1.1.m1.4.5.cmml"><mi id="A1.SS1.2.p1.1.m1.4.5.2" xref="A1.SS1.2.p1.1.m1.4.5.2.cmml">v</mi><mo id="A1.SS1.2.p1.1.m1.4.5.3" xref="A1.SS1.2.p1.1.m1.4.5.3.cmml">=</mo><mrow id="A1.SS1.2.p1.1.m1.4.5.4.2" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml"><mo id="A1.SS1.2.p1.1.m1.4.5.4.2.1" stretchy="false" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml">(</mo><mn id="A1.SS1.2.p1.1.m1.1.1" xref="A1.SS1.2.p1.1.m1.1.1.cmml">1</mn><mo id="A1.SS1.2.p1.1.m1.4.5.4.2.2" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml">,</mo><mn id="A1.SS1.2.p1.1.m1.2.2" xref="A1.SS1.2.p1.1.m1.2.2.cmml">0</mn><mo id="A1.SS1.2.p1.1.m1.4.5.4.2.3" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml">,</mo><mi id="A1.SS1.2.p1.1.m1.3.3" mathvariant="normal" xref="A1.SS1.2.p1.1.m1.3.3.cmml">…</mi><mo id="A1.SS1.2.p1.1.m1.4.5.4.2.4" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml">,</mo><mn id="A1.SS1.2.p1.1.m1.4.4" xref="A1.SS1.2.p1.1.m1.4.4.cmml">0</mn><mo id="A1.SS1.2.p1.1.m1.4.5.4.2.5" stretchy="false" xref="A1.SS1.2.p1.1.m1.4.5.4.1.cmml">)</mo></mrow><mo id="A1.SS1.2.p1.1.m1.4.5.5" xref="A1.SS1.2.p1.1.m1.4.5.5.cmml">∈</mo><msup id="A1.SS1.2.p1.1.m1.4.5.6" xref="A1.SS1.2.p1.1.m1.4.5.6.cmml"><mi id="A1.SS1.2.p1.1.m1.4.5.6.2" xref="A1.SS1.2.p1.1.m1.4.5.6.2.cmml">S</mi><mrow id="A1.SS1.2.p1.1.m1.4.5.6.3" xref="A1.SS1.2.p1.1.m1.4.5.6.3.cmml"><mi id="A1.SS1.2.p1.1.m1.4.5.6.3.2" xref="A1.SS1.2.p1.1.m1.4.5.6.3.2.cmml">d</mi><mo id="A1.SS1.2.p1.1.m1.4.5.6.3.1" xref="A1.SS1.2.p1.1.m1.4.5.6.3.1.cmml">−</mo><mn id="A1.SS1.2.p1.1.m1.4.5.6.3.3" xref="A1.SS1.2.p1.1.m1.4.5.6.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.1.m1.4b"><apply id="A1.SS1.2.p1.1.m1.4.5.cmml" xref="A1.SS1.2.p1.1.m1.4.5"><and id="A1.SS1.2.p1.1.m1.4.5a.cmml" xref="A1.SS1.2.p1.1.m1.4.5"></and><apply id="A1.SS1.2.p1.1.m1.4.5b.cmml" xref="A1.SS1.2.p1.1.m1.4.5"><eq id="A1.SS1.2.p1.1.m1.4.5.3.cmml" xref="A1.SS1.2.p1.1.m1.4.5.3"></eq><ci id="A1.SS1.2.p1.1.m1.4.5.2.cmml" xref="A1.SS1.2.p1.1.m1.4.5.2">𝑣</ci><vector id="A1.SS1.2.p1.1.m1.4.5.4.1.cmml" xref="A1.SS1.2.p1.1.m1.4.5.4.2"><cn id="A1.SS1.2.p1.1.m1.1.1.cmml" type="integer" xref="A1.SS1.2.p1.1.m1.1.1">1</cn><cn id="A1.SS1.2.p1.1.m1.2.2.cmml" type="integer" xref="A1.SS1.2.p1.1.m1.2.2">0</cn><ci id="A1.SS1.2.p1.1.m1.3.3.cmml" xref="A1.SS1.2.p1.1.m1.3.3">…</ci><cn id="A1.SS1.2.p1.1.m1.4.4.cmml" type="integer" xref="A1.SS1.2.p1.1.m1.4.4">0</cn></vector></apply><apply id="A1.SS1.2.p1.1.m1.4.5c.cmml" xref="A1.SS1.2.p1.1.m1.4.5"><in id="A1.SS1.2.p1.1.m1.4.5.5.cmml" xref="A1.SS1.2.p1.1.m1.4.5.5"></in><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.2.p1.1.m1.4.5.4.cmml" id="A1.SS1.2.p1.1.m1.4.5d.cmml" xref="A1.SS1.2.p1.1.m1.4.5"></share><apply id="A1.SS1.2.p1.1.m1.4.5.6.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6"><csymbol cd="ambiguous" id="A1.SS1.2.p1.1.m1.4.5.6.1.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6">superscript</csymbol><ci id="A1.SS1.2.p1.1.m1.4.5.6.2.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6.2">𝑆</ci><apply id="A1.SS1.2.p1.1.m1.4.5.6.3.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6.3"><minus id="A1.SS1.2.p1.1.m1.4.5.6.3.1.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6.3.1"></minus><ci id="A1.SS1.2.p1.1.m1.4.5.6.3.2.cmml" xref="A1.SS1.2.p1.1.m1.4.5.6.3.2">𝑑</ci><cn id="A1.SS1.2.p1.1.m1.4.5.6.3.3.cmml" type="integer" xref="A1.SS1.2.p1.1.m1.4.5.6.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.1.m1.4c">v=(1,0,\dots,0)\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.1.m1.4d">italic_v = ( 1 , 0 , … , 0 ) ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. By our definition of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.2.m2.1"><semantics id="A1.SS1.2.p1.2.m2.1a"><msub id="A1.SS1.2.p1.2.m2.1.1" xref="A1.SS1.2.p1.2.m2.1.1.cmml"><mi id="A1.SS1.2.p1.2.m2.1.1.2" mathvariant="normal" xref="A1.SS1.2.p1.2.m2.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.2.p1.2.m2.1.1.3" xref="A1.SS1.2.p1.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.2.m2.1b"><apply id="A1.SS1.2.p1.2.m2.1.1.cmml" xref="A1.SS1.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.2.m2.1.1.1.cmml" xref="A1.SS1.2.p1.2.m2.1.1">subscript</csymbol><ci id="A1.SS1.2.p1.2.m2.1.1.2.cmml" xref="A1.SS1.2.p1.2.m2.1.1.2">ℓ</ci><ci id="A1.SS1.2.p1.2.m2.1.1.3.cmml" xref="A1.SS1.2.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces, <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.2.p1.3.m3.1"><semantics id="A1.SS1.2.p1.3.m3.1a"><mi id="A1.SS1.2.p1.3.m3.1.1" xref="A1.SS1.2.p1.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.3.m3.1b"><ci id="A1.SS1.2.p1.3.m3.1.1.cmml" xref="A1.SS1.2.p1.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.3.m3.1d">italic_z</annotation></semantics></math> is contained in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.4.m4.2"><semantics id="A1.SS1.2.p1.4.m4.2a"><msubsup id="A1.SS1.2.p1.4.m4.2.3" xref="A1.SS1.2.p1.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.2.p1.4.m4.2.3.2.2" xref="A1.SS1.2.p1.4.m4.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.2.p1.4.m4.2.2.2.4" xref="A1.SS1.2.p1.4.m4.2.2.2.3.cmml"><mi id="A1.SS1.2.p1.4.m4.1.1.1.1" xref="A1.SS1.2.p1.4.m4.1.1.1.1.cmml">x</mi><mo id="A1.SS1.2.p1.4.m4.2.2.2.4.1" xref="A1.SS1.2.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="A1.SS1.2.p1.4.m4.2.2.2.2" xref="A1.SS1.2.p1.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.2.p1.4.m4.2.3.2.3" xref="A1.SS1.2.p1.4.m4.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.4.m4.2b"><apply id="A1.SS1.2.p1.4.m4.2.3.cmml" xref="A1.SS1.2.p1.4.m4.2.3"><csymbol cd="ambiguous" id="A1.SS1.2.p1.4.m4.2.3.1.cmml" xref="A1.SS1.2.p1.4.m4.2.3">subscript</csymbol><apply id="A1.SS1.2.p1.4.m4.2.3.2.cmml" xref="A1.SS1.2.p1.4.m4.2.3"><csymbol cd="ambiguous" id="A1.SS1.2.p1.4.m4.2.3.2.1.cmml" xref="A1.SS1.2.p1.4.m4.2.3">superscript</csymbol><ci id="A1.SS1.2.p1.4.m4.2.3.2.2.cmml" xref="A1.SS1.2.p1.4.m4.2.3.2.2">ℋ</ci><ci id="A1.SS1.2.p1.4.m4.2.3.2.3.cmml" xref="A1.SS1.2.p1.4.m4.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.2.p1.4.m4.2.2.2.3.cmml" xref="A1.SS1.2.p1.4.m4.2.2.2.4"><ci id="A1.SS1.2.p1.4.m4.1.1.1.1.cmml" xref="A1.SS1.2.p1.4.m4.1.1.1.1">𝑥</ci><ci id="A1.SS1.2.p1.4.m4.2.2.2.2.cmml" xref="A1.SS1.2.p1.4.m4.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.4.m4.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.4.m4.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\lVert x-z\rVert_{p}^{p}\leq\lVert x-\varepsilon v-z\rVert_{p}^{p}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.5.m5.2"><semantics id="A1.SS1.2.p1.5.m5.2a"><mrow id="A1.SS1.2.p1.5.m5.2.2" xref="A1.SS1.2.p1.5.m5.2.2.cmml"><msubsup id="A1.SS1.2.p1.5.m5.1.1.1" xref="A1.SS1.2.p1.5.m5.1.1.1.cmml"><mrow id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.2" rspace="0em" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.2" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.1" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.3" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.3" lspace="0em" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS1.2.p1.5.m5.1.1.1.1.3" xref="A1.SS1.2.p1.5.m5.1.1.1.1.3.cmml">p</mi><mi id="A1.SS1.2.p1.5.m5.1.1.1.3" xref="A1.SS1.2.p1.5.m5.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.SS1.2.p1.5.m5.2.2.3" rspace="0.1389em" xref="A1.SS1.2.p1.5.m5.2.2.3.cmml">≤</mo><msubsup id="A1.SS1.2.p1.5.m5.2.2.2" xref="A1.SS1.2.p1.5.m5.2.2.2.cmml"><mrow id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.2.cmml"><mo fence="true" 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id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.3" lspace="0em" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS1.2.p1.5.m5.2.2.2.1.3" xref="A1.SS1.2.p1.5.m5.2.2.2.1.3.cmml">p</mi><mi id="A1.SS1.2.p1.5.m5.2.2.2.3" xref="A1.SS1.2.p1.5.m5.2.2.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.5.m5.2b"><apply id="A1.SS1.2.p1.5.m5.2.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2"><leq id="A1.SS1.2.p1.5.m5.2.2.3.cmml" xref="A1.SS1.2.p1.5.m5.2.2.3"></leq><apply id="A1.SS1.2.p1.5.m5.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.5.m5.1.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1">superscript</csymbol><apply id="A1.SS1.2.p1.5.m5.1.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.5.m5.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1">subscript</csymbol><apply id="A1.SS1.2.p1.5.m5.1.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.2.p1.5.m5.1.1.1.1.1.2.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1"><minus id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.SS1.2.p1.5.m5.1.1.1.1.3.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.1.3">𝑝</ci></apply><ci id="A1.SS1.2.p1.5.m5.1.1.1.3.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1.3">𝑝</ci></apply><apply id="A1.SS1.2.p1.5.m5.2.2.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.5.m5.2.2.2.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2">superscript</csymbol><apply id="A1.SS1.2.p1.5.m5.2.2.2.1.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.5.m5.2.2.2.1.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2">subscript</csymbol><apply id="A1.SS1.2.p1.5.m5.2.2.2.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1"><csymbol cd="latexml" id="A1.SS1.2.p1.5.m5.2.2.2.1.1.2.1.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1"><minus id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.1"></minus><ci id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.2">𝑥</ci><apply id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3"><times id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.1.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.1"></times><ci id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.2.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.2">𝜀</ci><ci id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.3.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.4.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.SS1.2.p1.5.m5.2.2.2.1.3.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.1.3">𝑝</ci></apply><ci id="A1.SS1.2.p1.5.m5.2.2.2.3.cmml" xref="A1.SS1.2.p1.5.m5.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.5.m5.2c">\lVert x-z\rVert_{p}^{p}\leq\lVert x-\varepsilon v-z\rVert_{p}^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.5.m5.2d">∥ italic_x - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ≤ ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS1.2.p1.6.m6.1"><semantics id="A1.SS1.2.p1.6.m6.1a"><mrow id="A1.SS1.2.p1.6.m6.1.1" xref="A1.SS1.2.p1.6.m6.1.1.cmml"><mi id="A1.SS1.2.p1.6.m6.1.1.2" xref="A1.SS1.2.p1.6.m6.1.1.2.cmml">ε</mi><mo id="A1.SS1.2.p1.6.m6.1.1.1" xref="A1.SS1.2.p1.6.m6.1.1.1.cmml">></mo><mn id="A1.SS1.2.p1.6.m6.1.1.3" xref="A1.SS1.2.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.6.m6.1b"><apply id="A1.SS1.2.p1.6.m6.1.1.cmml" xref="A1.SS1.2.p1.6.m6.1.1"><gt id="A1.SS1.2.p1.6.m6.1.1.1.cmml" xref="A1.SS1.2.p1.6.m6.1.1.1"></gt><ci id="A1.SS1.2.p1.6.m6.1.1.2.cmml" xref="A1.SS1.2.p1.6.m6.1.1.2">𝜀</ci><cn id="A1.SS1.2.p1.6.m6.1.1.3.cmml" type="integer" xref="A1.SS1.2.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.6.m6.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.6.m6.1d">italic_ε > 0</annotation></semantics></math>, which happens if and only if <math alttext="|x_{1}-z_{1}|^{p}\leq|x_{1}-\varepsilon-z_{1}|^{p}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.7.m7.2"><semantics id="A1.SS1.2.p1.7.m7.2a"><mrow id="A1.SS1.2.p1.7.m7.2.2" xref="A1.SS1.2.p1.7.m7.2.2.cmml"><msup id="A1.SS1.2.p1.7.m7.1.1.1" xref="A1.SS1.2.p1.7.m7.1.1.1.cmml"><mrow id="A1.SS1.2.p1.7.m7.1.1.1.1.1" xref="A1.SS1.2.p1.7.m7.1.1.1.1.2.cmml"><mo id="A1.SS1.2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.2.p1.7.m7.1.1.1.1.2.1.cmml">|</mo><mrow id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml"><msub id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2.cmml"><mi id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2.2" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2.3" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.1" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.1.cmml">−</mo><msub id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.cmml"><mi id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.2" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.2.cmml">z</mi><mn id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.3" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="A1.SS1.2.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.2.p1.7.m7.1.1.1.1.2.1.cmml">|</mo></mrow><mi id="A1.SS1.2.p1.7.m7.1.1.1.3" xref="A1.SS1.2.p1.7.m7.1.1.1.3.cmml">p</mi></msup><mo id="A1.SS1.2.p1.7.m7.2.2.3" xref="A1.SS1.2.p1.7.m7.2.2.3.cmml">≤</mo><msup id="A1.SS1.2.p1.7.m7.2.2.2" xref="A1.SS1.2.p1.7.m7.2.2.2.cmml"><mrow id="A1.SS1.2.p1.7.m7.2.2.2.1.1" xref="A1.SS1.2.p1.7.m7.2.2.2.1.2.cmml"><mo id="A1.SS1.2.p1.7.m7.2.2.2.1.1.2" stretchy="false" xref="A1.SS1.2.p1.7.m7.2.2.2.1.2.1.cmml">|</mo><mrow id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.cmml"><msub id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.cmml"><mi id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.2" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.2.cmml">x</mi><mn id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.3" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.3.cmml">1</mn></msub><mo id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.1" 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id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.1.cmml" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.2.cmml" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.2">𝑧</ci><cn id="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A1.SS1.2.p1.7.m7.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><ci id="A1.SS1.2.p1.7.m7.1.1.1.3.cmml" xref="A1.SS1.2.p1.7.m7.1.1.1.3">𝑝</ci></apply><apply id="A1.SS1.2.p1.7.m7.2.2.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.7.m7.2.2.2.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2">superscript</csymbol><apply id="A1.SS1.2.p1.7.m7.2.2.2.1.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1"><abs id="A1.SS1.2.p1.7.m7.2.2.2.1.2.1.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.2"></abs><apply id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1"><minus id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.1.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.1"></minus><apply id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.1.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2">subscript</csymbol><ci id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.2">𝑥</ci><cn id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.3.cmml" type="integer" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.2.3">1</cn></apply><ci id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.3.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.3">𝜀</ci><apply id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4"><csymbol cd="ambiguous" id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.1.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4">subscript</csymbol><ci id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.2.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.2">𝑧</ci><cn id="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.3.cmml" type="integer" xref="A1.SS1.2.p1.7.m7.2.2.2.1.1.1.4.3">1</cn></apply></apply></apply><ci id="A1.SS1.2.p1.7.m7.2.2.2.3.cmml" xref="A1.SS1.2.p1.7.m7.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.7.m7.2c">|x_{1}-z_{1}|^{p}\leq|x_{1}-\varepsilon-z_{1}|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.7.m7.2d">| italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ≤ | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ε - italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>. This last inequality holds if and only if <math alttext="z_{1}\geq x_{1}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.8.m8.1"><semantics id="A1.SS1.2.p1.8.m8.1a"><mrow id="A1.SS1.2.p1.8.m8.1.1" xref="A1.SS1.2.p1.8.m8.1.1.cmml"><msub id="A1.SS1.2.p1.8.m8.1.1.2" xref="A1.SS1.2.p1.8.m8.1.1.2.cmml"><mi id="A1.SS1.2.p1.8.m8.1.1.2.2" xref="A1.SS1.2.p1.8.m8.1.1.2.2.cmml">z</mi><mn id="A1.SS1.2.p1.8.m8.1.1.2.3" xref="A1.SS1.2.p1.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="A1.SS1.2.p1.8.m8.1.1.1" xref="A1.SS1.2.p1.8.m8.1.1.1.cmml">≥</mo><msub id="A1.SS1.2.p1.8.m8.1.1.3" xref="A1.SS1.2.p1.8.m8.1.1.3.cmml"><mi id="A1.SS1.2.p1.8.m8.1.1.3.2" xref="A1.SS1.2.p1.8.m8.1.1.3.2.cmml">x</mi><mn id="A1.SS1.2.p1.8.m8.1.1.3.3" xref="A1.SS1.2.p1.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.8.m8.1b"><apply id="A1.SS1.2.p1.8.m8.1.1.cmml" xref="A1.SS1.2.p1.8.m8.1.1"><geq id="A1.SS1.2.p1.8.m8.1.1.1.cmml" xref="A1.SS1.2.p1.8.m8.1.1.1"></geq><apply id="A1.SS1.2.p1.8.m8.1.1.2.cmml" xref="A1.SS1.2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.8.m8.1.1.2.1.cmml" xref="A1.SS1.2.p1.8.m8.1.1.2">subscript</csymbol><ci id="A1.SS1.2.p1.8.m8.1.1.2.2.cmml" xref="A1.SS1.2.p1.8.m8.1.1.2.2">𝑧</ci><cn id="A1.SS1.2.p1.8.m8.1.1.2.3.cmml" type="integer" xref="A1.SS1.2.p1.8.m8.1.1.2.3">1</cn></apply><apply id="A1.SS1.2.p1.8.m8.1.1.3.cmml" xref="A1.SS1.2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.2.p1.8.m8.1.1.3.1.cmml" xref="A1.SS1.2.p1.8.m8.1.1.3">subscript</csymbol><ci id="A1.SS1.2.p1.8.m8.1.1.3.2.cmml" xref="A1.SS1.2.p1.8.m8.1.1.3.2">𝑥</ci><cn id="A1.SS1.2.p1.8.m8.1.1.3.3.cmml" type="integer" xref="A1.SS1.2.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.8.m8.1c">z_{1}\geq x_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.8.m8.1d">italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, independently of <math alttext="p" class="ltx_Math" display="inline" id="A1.SS1.2.p1.9.m9.1"><semantics id="A1.SS1.2.p1.9.m9.1a"><mi id="A1.SS1.2.p1.9.m9.1.1" xref="A1.SS1.2.p1.9.m9.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.9.m9.1b"><ci id="A1.SS1.2.p1.9.m9.1.1.cmml" xref="A1.SS1.2.p1.9.m9.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.9.m9.1c">p</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.9.m9.1d">italic_p</annotation></semantics></math>. This concludes the proof. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p5"> <p class="ltx_p" id="A1.SS1.p5.1">The proofs of the other properties require some more effort. In particular, for <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a>, we will need <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>, which we restate here for convenience.</p> </div> <div class="ltx_para" id="A1.SS1.p6"> <p class="ltx_p" id="A1.SS1.p6.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a></p> </div> <div class="ltx_para" id="A1.SS1.p7"> <p class="ltx_p" id="A1.SS1.p7.3">With this, we are ready to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a>, which states that <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p7.1.m1.1"><semantics id="A1.SS1.p7.1.m1.1a"><msub id="A1.SS1.p7.1.m1.1.1" xref="A1.SS1.p7.1.m1.1.1.cmml"><mi id="A1.SS1.p7.1.m1.1.1.2" mathvariant="normal" xref="A1.SS1.p7.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p7.1.m1.1.1.3" xref="A1.SS1.p7.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p7.1.m1.1b"><apply id="A1.SS1.p7.1.m1.1.1.cmml" xref="A1.SS1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.p7.1.m1.1.1.1.cmml" xref="A1.SS1.p7.1.m1.1.1">subscript</csymbol><ci id="A1.SS1.p7.1.m1.1.1.2.cmml" xref="A1.SS1.p7.1.m1.1.1.2">ℓ</ci><ci id="A1.SS1.p7.1.m1.1.1.3.cmml" xref="A1.SS1.p7.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p7.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p7.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces are unions of rays originating at <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.p7.2.m2.1"><semantics id="A1.SS1.p7.2.m2.1a"><mi id="A1.SS1.p7.2.m2.1.1" xref="A1.SS1.p7.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p7.2.m2.1b"><ci id="A1.SS1.p7.2.m2.1.1.cmml" xref="A1.SS1.p7.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p7.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p7.2.m2.1d">italic_x</annotation></semantics></math>. The proof mainly relies on convexity of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p7.3.m3.1"><semantics id="A1.SS1.p7.3.m3.1a"><msub id="A1.SS1.p7.3.m3.1.1" xref="A1.SS1.p7.3.m3.1.1.cmml"><mi id="A1.SS1.p7.3.m3.1.1.2" mathvariant="normal" xref="A1.SS1.p7.3.m3.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p7.3.m3.1.1.3" xref="A1.SS1.p7.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p7.3.m3.1b"><apply id="A1.SS1.p7.3.m3.1.1.cmml" xref="A1.SS1.p7.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS1.p7.3.m3.1.1.1.cmml" xref="A1.SS1.p7.3.m3.1.1">subscript</csymbol><ci id="A1.SS1.p7.3.m3.1.1.2.cmml" xref="A1.SS1.p7.3.m3.1.1.2">ℓ</ci><ci id="A1.SS1.p7.3.m3.1.1.3.cmml" xref="A1.SS1.p7.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p7.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p7.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-balls and <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>.</p> </div> <div class="ltx_para" id="A1.SS1.p8"> <p class="ltx_p" id="A1.SS1.p8.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.6</span></a></p> </div> <div class="ltx_proof" id="A1.SS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.3.p1"> <p class="ltx_p" id="A1.SS1.3.p1.11">We start with the first part of the statement for arbitrary <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.1.m1.3"><semantics id="A1.SS1.3.p1.1.m1.3a"><mrow id="A1.SS1.3.p1.1.m1.3.4" xref="A1.SS1.3.p1.1.m1.3.4.cmml"><mi id="A1.SS1.3.p1.1.m1.3.4.2" xref="A1.SS1.3.p1.1.m1.3.4.2.cmml">p</mi><mo id="A1.SS1.3.p1.1.m1.3.4.1" xref="A1.SS1.3.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="A1.SS1.3.p1.1.m1.3.4.3" xref="A1.SS1.3.p1.1.m1.3.4.3.cmml"><mrow id="A1.SS1.3.p1.1.m1.3.4.3.2.2" xref="A1.SS1.3.p1.1.m1.3.4.3.2.1.cmml"><mo id="A1.SS1.3.p1.1.m1.3.4.3.2.2.1" stretchy="false" xref="A1.SS1.3.p1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="A1.SS1.3.p1.1.m1.1.1" xref="A1.SS1.3.p1.1.m1.1.1.cmml">1</mn><mo id="A1.SS1.3.p1.1.m1.3.4.3.2.2.2" xref="A1.SS1.3.p1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="A1.SS1.3.p1.1.m1.2.2" mathvariant="normal" xref="A1.SS1.3.p1.1.m1.2.2.cmml">∞</mi><mo id="A1.SS1.3.p1.1.m1.3.4.3.2.2.3" stretchy="false" xref="A1.SS1.3.p1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.SS1.3.p1.1.m1.3.4.3.1" xref="A1.SS1.3.p1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="A1.SS1.3.p1.1.m1.3.4.3.3.2" xref="A1.SS1.3.p1.1.m1.3.4.3.3.1.cmml"><mo id="A1.SS1.3.p1.1.m1.3.4.3.3.2.1" stretchy="false" xref="A1.SS1.3.p1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="A1.SS1.3.p1.1.m1.3.3" mathvariant="normal" xref="A1.SS1.3.p1.1.m1.3.3.cmml">∞</mi><mo id="A1.SS1.3.p1.1.m1.3.4.3.3.2.2" stretchy="false" xref="A1.SS1.3.p1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.1.m1.3b"><apply id="A1.SS1.3.p1.1.m1.3.4.cmml" xref="A1.SS1.3.p1.1.m1.3.4"><in id="A1.SS1.3.p1.1.m1.3.4.1.cmml" xref="A1.SS1.3.p1.1.m1.3.4.1"></in><ci id="A1.SS1.3.p1.1.m1.3.4.2.cmml" xref="A1.SS1.3.p1.1.m1.3.4.2">𝑝</ci><apply id="A1.SS1.3.p1.1.m1.3.4.3.cmml" xref="A1.SS1.3.p1.1.m1.3.4.3"><union id="A1.SS1.3.p1.1.m1.3.4.3.1.cmml" xref="A1.SS1.3.p1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="A1.SS1.3.p1.1.m1.3.4.3.2.1.cmml" xref="A1.SS1.3.p1.1.m1.3.4.3.2.2"><cn id="A1.SS1.3.p1.1.m1.1.1.cmml" type="integer" xref="A1.SS1.3.p1.1.m1.1.1">1</cn><infinity id="A1.SS1.3.p1.1.m1.2.2.cmml" xref="A1.SS1.3.p1.1.m1.2.2"></infinity></interval><set id="A1.SS1.3.p1.1.m1.3.4.3.3.1.cmml" xref="A1.SS1.3.p1.1.m1.3.4.3.3.2"><infinity id="A1.SS1.3.p1.1.m1.3.3.cmml" xref="A1.SS1.3.p1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. For this, let <math alttext="L" class="ltx_Math" display="inline" id="A1.SS1.3.p1.2.m2.1"><semantics id="A1.SS1.3.p1.2.m2.1a"><mi id="A1.SS1.3.p1.2.m2.1.1" xref="A1.SS1.3.p1.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.2.m2.1b"><ci id="A1.SS1.3.p1.2.m2.1.1.cmml" xref="A1.SS1.3.p1.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.2.m2.1d">italic_L</annotation></semantics></math> be the line through <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.3.p1.3.m3.1"><semantics id="A1.SS1.3.p1.3.m3.1a"><mi id="A1.SS1.3.p1.3.m3.1.1" xref="A1.SS1.3.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.3.m3.1b"><ci id="A1.SS1.3.p1.3.m3.1.1.cmml" xref="A1.SS1.3.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.3.m3.1d">italic_x</annotation></semantics></math> in direction <math alttext="v" class="ltx_Math" display="inline" id="A1.SS1.3.p1.4.m4.1"><semantics id="A1.SS1.3.p1.4.m4.1a"><mi id="A1.SS1.3.p1.4.m4.1.1" xref="A1.SS1.3.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.4.m4.1b"><ci id="A1.SS1.3.p1.4.m4.1.1.cmml" xref="A1.SS1.3.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.4.m4.1d">italic_v</annotation></semantics></math>, and let <math alttext="R_{-}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.5.m5.1"><semantics id="A1.SS1.3.p1.5.m5.1a"><msub id="A1.SS1.3.p1.5.m5.1.1" xref="A1.SS1.3.p1.5.m5.1.1.cmml"><mi id="A1.SS1.3.p1.5.m5.1.1.2" xref="A1.SS1.3.p1.5.m5.1.1.2.cmml">R</mi><mo id="A1.SS1.3.p1.5.m5.1.1.3" xref="A1.SS1.3.p1.5.m5.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.5.m5.1b"><apply id="A1.SS1.3.p1.5.m5.1.1.cmml" xref="A1.SS1.3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.5.m5.1.1.1.cmml" xref="A1.SS1.3.p1.5.m5.1.1">subscript</csymbol><ci id="A1.SS1.3.p1.5.m5.1.1.2.cmml" xref="A1.SS1.3.p1.5.m5.1.1.2">𝑅</ci><minus id="A1.SS1.3.p1.5.m5.1.1.3.cmml" xref="A1.SS1.3.p1.5.m5.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.5.m5.1c">R_{-}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.5.m5.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> be the open ray from <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.3.p1.6.m6.1"><semantics id="A1.SS1.3.p1.6.m6.1a"><mi id="A1.SS1.3.p1.6.m6.1.1" xref="A1.SS1.3.p1.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.6.m6.1b"><ci id="A1.SS1.3.p1.6.m6.1.1.cmml" xref="A1.SS1.3.p1.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.6.m6.1d">italic_x</annotation></semantics></math> in direction <math alttext="-v" class="ltx_Math" display="inline" id="A1.SS1.3.p1.7.m7.1"><semantics id="A1.SS1.3.p1.7.m7.1a"><mrow id="A1.SS1.3.p1.7.m7.1.1" xref="A1.SS1.3.p1.7.m7.1.1.cmml"><mo id="A1.SS1.3.p1.7.m7.1.1a" xref="A1.SS1.3.p1.7.m7.1.1.cmml">−</mo><mi id="A1.SS1.3.p1.7.m7.1.1.2" xref="A1.SS1.3.p1.7.m7.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.7.m7.1b"><apply id="A1.SS1.3.p1.7.m7.1.1.cmml" xref="A1.SS1.3.p1.7.m7.1.1"><minus id="A1.SS1.3.p1.7.m7.1.1.1.cmml" xref="A1.SS1.3.p1.7.m7.1.1"></minus><ci id="A1.SS1.3.p1.7.m7.1.1.2.cmml" xref="A1.SS1.3.p1.7.m7.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.7.m7.1c">-v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.7.m7.1d">- italic_v</annotation></semantics></math>. Let <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.8.m8.1"><semantics id="A1.SS1.3.p1.8.m8.1a"><msub id="A1.SS1.3.p1.8.m8.1.1" xref="A1.SS1.3.p1.8.m8.1.1.cmml"><mi id="A1.SS1.3.p1.8.m8.1.1.2" xref="A1.SS1.3.p1.8.m8.1.1.2.cmml">B</mi><mi id="A1.SS1.3.p1.8.m8.1.1.3" xref="A1.SS1.3.p1.8.m8.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.8.m8.1b"><apply id="A1.SS1.3.p1.8.m8.1.1.cmml" xref="A1.SS1.3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.8.m8.1.1.1.cmml" xref="A1.SS1.3.p1.8.m8.1.1">subscript</csymbol><ci id="A1.SS1.3.p1.8.m8.1.1.2.cmml" xref="A1.SS1.3.p1.8.m8.1.1.2">𝐵</ci><ci id="A1.SS1.3.p1.8.m8.1.1.3.cmml" xref="A1.SS1.3.p1.8.m8.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.8.m8.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.8.m8.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> be the smallest <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.9.m9.1"><semantics id="A1.SS1.3.p1.9.m9.1a"><msub id="A1.SS1.3.p1.9.m9.1.1" xref="A1.SS1.3.p1.9.m9.1.1.cmml"><mi id="A1.SS1.3.p1.9.m9.1.1.2" mathvariant="normal" xref="A1.SS1.3.p1.9.m9.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.3.p1.9.m9.1.1.3" xref="A1.SS1.3.p1.9.m9.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.9.m9.1b"><apply id="A1.SS1.3.p1.9.m9.1.1.cmml" xref="A1.SS1.3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.9.m9.1.1.1.cmml" xref="A1.SS1.3.p1.9.m9.1.1">subscript</csymbol><ci id="A1.SS1.3.p1.9.m9.1.1.2.cmml" xref="A1.SS1.3.p1.9.m9.1.1.2">ℓ</ci><ci id="A1.SS1.3.p1.9.m9.1.1.3.cmml" xref="A1.SS1.3.p1.9.m9.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.9.m9.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-ball around a point <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.10.m10.1"><semantics id="A1.SS1.3.p1.10.m10.1a"><mrow id="A1.SS1.3.p1.10.m10.1.1" xref="A1.SS1.3.p1.10.m10.1.1.cmml"><mi id="A1.SS1.3.p1.10.m10.1.1.2" xref="A1.SS1.3.p1.10.m10.1.1.2.cmml">z</mi><mo id="A1.SS1.3.p1.10.m10.1.1.1" xref="A1.SS1.3.p1.10.m10.1.1.1.cmml">∈</mo><msup id="A1.SS1.3.p1.10.m10.1.1.3" xref="A1.SS1.3.p1.10.m10.1.1.3.cmml"><mi id="A1.SS1.3.p1.10.m10.1.1.3.2" xref="A1.SS1.3.p1.10.m10.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.3.p1.10.m10.1.1.3.3" xref="A1.SS1.3.p1.10.m10.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.10.m10.1b"><apply id="A1.SS1.3.p1.10.m10.1.1.cmml" xref="A1.SS1.3.p1.10.m10.1.1"><in id="A1.SS1.3.p1.10.m10.1.1.1.cmml" xref="A1.SS1.3.p1.10.m10.1.1.1"></in><ci id="A1.SS1.3.p1.10.m10.1.1.2.cmml" xref="A1.SS1.3.p1.10.m10.1.1.2">𝑧</ci><apply id="A1.SS1.3.p1.10.m10.1.1.3.cmml" xref="A1.SS1.3.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.3.p1.10.m10.1.1.3.1.cmml" xref="A1.SS1.3.p1.10.m10.1.1.3">superscript</csymbol><ci id="A1.SS1.3.p1.10.m10.1.1.3.2.cmml" xref="A1.SS1.3.p1.10.m10.1.1.3.2">ℝ</ci><ci id="A1.SS1.3.p1.10.m10.1.1.3.3.cmml" xref="A1.SS1.3.p1.10.m10.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.10.m10.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.10.m10.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> that contains <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.3.p1.11.m11.1"><semantics id="A1.SS1.3.p1.11.m11.1a"><mi id="A1.SS1.3.p1.11.m11.1.1" xref="A1.SS1.3.p1.11.m11.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.11.m11.1b"><ci id="A1.SS1.3.p1.11.m11.1.1.cmml" xref="A1.SS1.3.p1.11.m11.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.11.m11.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.11.m11.1d">italic_x</annotation></semantics></math>. From <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>, we know that</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z\in\mathcal{H}^{p}_{x,v}\iff R_{-}\cap B_{z}^{\circ}=\varnothing," class="ltx_Math" display="block" id="A1.Ex1.m1.3"><semantics id="A1.Ex1.m1.3a"><mrow id="A1.Ex1.m1.3.3.1" xref="A1.Ex1.m1.3.3.1.1.cmml"><mrow id="A1.Ex1.m1.3.3.1.1" xref="A1.Ex1.m1.3.3.1.1.cmml"><mrow id="A1.Ex1.m1.3.3.1.1.2" xref="A1.Ex1.m1.3.3.1.1.2.cmml"><mi id="A1.Ex1.m1.3.3.1.1.2.2" xref="A1.Ex1.m1.3.3.1.1.2.2.cmml">z</mi><mo id="A1.Ex1.m1.3.3.1.1.2.1" xref="A1.Ex1.m1.3.3.1.1.2.1.cmml">∈</mo><msubsup id="A1.Ex1.m1.3.3.1.1.2.3" xref="A1.Ex1.m1.3.3.1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Ex1.m1.3.3.1.1.2.3.2.2" xref="A1.Ex1.m1.3.3.1.1.2.3.2.2.cmml">ℋ</mi><mrow id="A1.Ex1.m1.2.2.2.4" xref="A1.Ex1.m1.2.2.2.3.cmml"><mi id="A1.Ex1.m1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.cmml">x</mi><mo id="A1.Ex1.m1.2.2.2.4.1" xref="A1.Ex1.m1.2.2.2.3.cmml">,</mo><mi id="A1.Ex1.m1.2.2.2.2" xref="A1.Ex1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Ex1.m1.3.3.1.1.2.3.2.3" xref="A1.Ex1.m1.3.3.1.1.2.3.2.3.cmml">p</mi></msubsup></mrow><mo id="A1.Ex1.m1.3.3.1.1.1" stretchy="false" xref="A1.Ex1.m1.3.3.1.1.1.cmml">⇔</mo><mrow id="A1.Ex1.m1.3.3.1.1.3" xref="A1.Ex1.m1.3.3.1.1.3.cmml"><mrow id="A1.Ex1.m1.3.3.1.1.3.2" xref="A1.Ex1.m1.3.3.1.1.3.2.cmml"><msub id="A1.Ex1.m1.3.3.1.1.3.2.2" xref="A1.Ex1.m1.3.3.1.1.3.2.2.cmml"><mi id="A1.Ex1.m1.3.3.1.1.3.2.2.2" xref="A1.Ex1.m1.3.3.1.1.3.2.2.2.cmml">R</mi><mo id="A1.Ex1.m1.3.3.1.1.3.2.2.3" xref="A1.Ex1.m1.3.3.1.1.3.2.2.3.cmml">−</mo></msub><mo id="A1.Ex1.m1.3.3.1.1.3.2.1" xref="A1.Ex1.m1.3.3.1.1.3.2.1.cmml">∩</mo><msubsup id="A1.Ex1.m1.3.3.1.1.3.2.3" xref="A1.Ex1.m1.3.3.1.1.3.2.3.cmml"><mi id="A1.Ex1.m1.3.3.1.1.3.2.3.2.2" xref="A1.Ex1.m1.3.3.1.1.3.2.3.2.2.cmml">B</mi><mi id="A1.Ex1.m1.3.3.1.1.3.2.3.2.3" xref="A1.Ex1.m1.3.3.1.1.3.2.3.2.3.cmml">z</mi><mo id="A1.Ex1.m1.3.3.1.1.3.2.3.3" xref="A1.Ex1.m1.3.3.1.1.3.2.3.3.cmml">∘</mo></msubsup></mrow><mo id="A1.Ex1.m1.3.3.1.1.3.1" xref="A1.Ex1.m1.3.3.1.1.3.1.cmml">=</mo><mi id="A1.Ex1.m1.3.3.1.1.3.3" mathvariant="normal" xref="A1.Ex1.m1.3.3.1.1.3.3.cmml">∅</mi></mrow></mrow><mo id="A1.Ex1.m1.3.3.1.2" xref="A1.Ex1.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex1.m1.3b"><apply id="A1.Ex1.m1.3.3.1.1.cmml" xref="A1.Ex1.m1.3.3.1"><csymbol cd="latexml" id="A1.Ex1.m1.3.3.1.1.1.cmml" xref="A1.Ex1.m1.3.3.1.1.1">iff</csymbol><apply id="A1.Ex1.m1.3.3.1.1.2.cmml" xref="A1.Ex1.m1.3.3.1.1.2"><in id="A1.Ex1.m1.3.3.1.1.2.1.cmml" xref="A1.Ex1.m1.3.3.1.1.2.1"></in><ci id="A1.Ex1.m1.3.3.1.1.2.2.cmml" xref="A1.Ex1.m1.3.3.1.1.2.2">𝑧</ci><apply id="A1.Ex1.m1.3.3.1.1.2.3.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="A1.Ex1.m1.3.3.1.1.2.3.1.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3">subscript</csymbol><apply id="A1.Ex1.m1.3.3.1.1.2.3.2.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="A1.Ex1.m1.3.3.1.1.2.3.2.1.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3">superscript</csymbol><ci id="A1.Ex1.m1.3.3.1.1.2.3.2.2.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3.2.2">ℋ</ci><ci id="A1.Ex1.m1.3.3.1.1.2.3.2.3.cmml" xref="A1.Ex1.m1.3.3.1.1.2.3.2.3">𝑝</ci></apply><list id="A1.Ex1.m1.2.2.2.3.cmml" xref="A1.Ex1.m1.2.2.2.4"><ci id="A1.Ex1.m1.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1">𝑥</ci><ci id="A1.Ex1.m1.2.2.2.2.cmml" xref="A1.Ex1.m1.2.2.2.2">𝑣</ci></list></apply></apply><apply id="A1.Ex1.m1.3.3.1.1.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3"><eq id="A1.Ex1.m1.3.3.1.1.3.1.cmml" xref="A1.Ex1.m1.3.3.1.1.3.1"></eq><apply id="A1.Ex1.m1.3.3.1.1.3.2.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2"><intersect id="A1.Ex1.m1.3.3.1.1.3.2.1.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.1"></intersect><apply id="A1.Ex1.m1.3.3.1.1.3.2.2.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.2"><csymbol cd="ambiguous" id="A1.Ex1.m1.3.3.1.1.3.2.2.1.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.2">subscript</csymbol><ci id="A1.Ex1.m1.3.3.1.1.3.2.2.2.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.2.2">𝑅</ci><minus id="A1.Ex1.m1.3.3.1.1.3.2.2.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.2.3"></minus></apply><apply id="A1.Ex1.m1.3.3.1.1.3.2.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="A1.Ex1.m1.3.3.1.1.3.2.3.1.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3">superscript</csymbol><apply id="A1.Ex1.m1.3.3.1.1.3.2.3.2.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="A1.Ex1.m1.3.3.1.1.3.2.3.2.1.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3">subscript</csymbol><ci id="A1.Ex1.m1.3.3.1.1.3.2.3.2.2.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3.2.2">𝐵</ci><ci id="A1.Ex1.m1.3.3.1.1.3.2.3.2.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3.2.3">𝑧</ci></apply><compose id="A1.Ex1.m1.3.3.1.1.3.2.3.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3.2.3.3"></compose></apply></apply><emptyset id="A1.Ex1.m1.3.3.1.1.3.3.cmml" xref="A1.Ex1.m1.3.3.1.1.3.3"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex1.m1.3c">z\in\mathcal{H}^{p}_{x,v}\iff R_{-}\cap B_{z}^{\circ}=\varnothing,</annotation><annotation encoding="application/x-llamapun" id="A1.Ex1.m1.3d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ⇔ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT = ∅ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS1.3.p1.13">where <math alttext="B_{z}^{\circ}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.12.m1.1"><semantics id="A1.SS1.3.p1.12.m1.1a"><msubsup id="A1.SS1.3.p1.12.m1.1.1" xref="A1.SS1.3.p1.12.m1.1.1.cmml"><mi id="A1.SS1.3.p1.12.m1.1.1.2.2" xref="A1.SS1.3.p1.12.m1.1.1.2.2.cmml">B</mi><mi id="A1.SS1.3.p1.12.m1.1.1.2.3" xref="A1.SS1.3.p1.12.m1.1.1.2.3.cmml">z</mi><mo id="A1.SS1.3.p1.12.m1.1.1.3" xref="A1.SS1.3.p1.12.m1.1.1.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.12.m1.1b"><apply id="A1.SS1.3.p1.12.m1.1.1.cmml" xref="A1.SS1.3.p1.12.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.12.m1.1.1.1.cmml" xref="A1.SS1.3.p1.12.m1.1.1">superscript</csymbol><apply id="A1.SS1.3.p1.12.m1.1.1.2.cmml" xref="A1.SS1.3.p1.12.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.12.m1.1.1.2.1.cmml" xref="A1.SS1.3.p1.12.m1.1.1">subscript</csymbol><ci id="A1.SS1.3.p1.12.m1.1.1.2.2.cmml" xref="A1.SS1.3.p1.12.m1.1.1.2.2">𝐵</ci><ci id="A1.SS1.3.p1.12.m1.1.1.2.3.cmml" xref="A1.SS1.3.p1.12.m1.1.1.2.3">𝑧</ci></apply><compose id="A1.SS1.3.p1.12.m1.1.1.3.cmml" xref="A1.SS1.3.p1.12.m1.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.12.m1.1c">B_{z}^{\circ}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.12.m1.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> denotes the interior of <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.3.p1.13.m2.1"><semantics id="A1.SS1.3.p1.13.m2.1a"><msub id="A1.SS1.3.p1.13.m2.1.1" xref="A1.SS1.3.p1.13.m2.1.1.cmml"><mi id="A1.SS1.3.p1.13.m2.1.1.2" xref="A1.SS1.3.p1.13.m2.1.1.2.cmml">B</mi><mi id="A1.SS1.3.p1.13.m2.1.1.3" xref="A1.SS1.3.p1.13.m2.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p1.13.m2.1b"><apply id="A1.SS1.3.p1.13.m2.1.1.cmml" xref="A1.SS1.3.p1.13.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.3.p1.13.m2.1.1.1.cmml" xref="A1.SS1.3.p1.13.m2.1.1">subscript</csymbol><ci id="A1.SS1.3.p1.13.m2.1.1.2.cmml" xref="A1.SS1.3.p1.13.m2.1.1.2">𝐵</ci><ci id="A1.SS1.3.p1.13.m2.1.1.3.cmml" xref="A1.SS1.3.p1.13.m2.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p1.13.m2.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p1.13.m2.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.SS1.4.p2"> <p class="ltx_p" id="A1.SS1.4.p2.16">Consider what happens when we move <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.4.p2.1.m1.1"><semantics id="A1.SS1.4.p2.1.m1.1a"><mi id="A1.SS1.4.p2.1.m1.1.1" xref="A1.SS1.4.p2.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.1.m1.1b"><ci id="A1.SS1.4.p2.1.m1.1.1.cmml" xref="A1.SS1.4.p2.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.1.m1.1d">italic_z</annotation></semantics></math> along the ray through <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.4.p2.2.m2.1"><semantics id="A1.SS1.4.p2.2.m2.1a"><mi id="A1.SS1.4.p2.2.m2.1.1" xref="A1.SS1.4.p2.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.2.m2.1b"><ci id="A1.SS1.4.p2.2.m2.1.1.cmml" xref="A1.SS1.4.p2.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.2.m2.1d">italic_x</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.4.p2.3.m3.1"><semantics id="A1.SS1.4.p2.3.m3.1a"><mi id="A1.SS1.4.p2.3.m3.1.1" xref="A1.SS1.4.p2.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.3.m3.1b"><ci id="A1.SS1.4.p2.3.m3.1.1.cmml" xref="A1.SS1.4.p2.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.3.m3.1d">italic_z</annotation></semantics></math>. In other words, consider a point <math alttext="z^{\prime}=x+\delta(z-x)" class="ltx_Math" display="inline" id="A1.SS1.4.p2.4.m4.1"><semantics id="A1.SS1.4.p2.4.m4.1a"><mrow id="A1.SS1.4.p2.4.m4.1.1" xref="A1.SS1.4.p2.4.m4.1.1.cmml"><msup id="A1.SS1.4.p2.4.m4.1.1.3" xref="A1.SS1.4.p2.4.m4.1.1.3.cmml"><mi id="A1.SS1.4.p2.4.m4.1.1.3.2" xref="A1.SS1.4.p2.4.m4.1.1.3.2.cmml">z</mi><mo id="A1.SS1.4.p2.4.m4.1.1.3.3" xref="A1.SS1.4.p2.4.m4.1.1.3.3.cmml">′</mo></msup><mo id="A1.SS1.4.p2.4.m4.1.1.2" xref="A1.SS1.4.p2.4.m4.1.1.2.cmml">=</mo><mrow id="A1.SS1.4.p2.4.m4.1.1.1" xref="A1.SS1.4.p2.4.m4.1.1.1.cmml"><mi id="A1.SS1.4.p2.4.m4.1.1.1.3" xref="A1.SS1.4.p2.4.m4.1.1.1.3.cmml">x</mi><mo id="A1.SS1.4.p2.4.m4.1.1.1.2" xref="A1.SS1.4.p2.4.m4.1.1.1.2.cmml">+</mo><mrow id="A1.SS1.4.p2.4.m4.1.1.1.1" xref="A1.SS1.4.p2.4.m4.1.1.1.1.cmml"><mi id="A1.SS1.4.p2.4.m4.1.1.1.1.3" xref="A1.SS1.4.p2.4.m4.1.1.1.1.3.cmml">δ</mi><mo id="A1.SS1.4.p2.4.m4.1.1.1.1.2" xref="A1.SS1.4.p2.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.cmml"><mo id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.2" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.1" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.3" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.4.m4.1b"><apply id="A1.SS1.4.p2.4.m4.1.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1"><eq id="A1.SS1.4.p2.4.m4.1.1.2.cmml" xref="A1.SS1.4.p2.4.m4.1.1.2"></eq><apply id="A1.SS1.4.p2.4.m4.1.1.3.cmml" xref="A1.SS1.4.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.4.m4.1.1.3.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1.3">superscript</csymbol><ci id="A1.SS1.4.p2.4.m4.1.1.3.2.cmml" xref="A1.SS1.4.p2.4.m4.1.1.3.2">𝑧</ci><ci id="A1.SS1.4.p2.4.m4.1.1.3.3.cmml" xref="A1.SS1.4.p2.4.m4.1.1.3.3">′</ci></apply><apply id="A1.SS1.4.p2.4.m4.1.1.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1"><plus id="A1.SS1.4.p2.4.m4.1.1.1.2.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.2"></plus><ci id="A1.SS1.4.p2.4.m4.1.1.1.3.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.3">𝑥</ci><apply id="A1.SS1.4.p2.4.m4.1.1.1.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1"><times id="A1.SS1.4.p2.4.m4.1.1.1.1.2.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.2"></times><ci id="A1.SS1.4.p2.4.m4.1.1.1.1.3.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.3">𝛿</ci><apply id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1"><minus id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.4.p2.4.m4.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.4.m4.1c">z^{\prime}=x+\delta(z-x)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.4.m4.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x + italic_δ ( italic_z - italic_x )</annotation></semantics></math> for some <math alttext="\delta>0" class="ltx_Math" display="inline" id="A1.SS1.4.p2.5.m5.1"><semantics id="A1.SS1.4.p2.5.m5.1a"><mrow id="A1.SS1.4.p2.5.m5.1.1" xref="A1.SS1.4.p2.5.m5.1.1.cmml"><mi id="A1.SS1.4.p2.5.m5.1.1.2" xref="A1.SS1.4.p2.5.m5.1.1.2.cmml">δ</mi><mo id="A1.SS1.4.p2.5.m5.1.1.1" xref="A1.SS1.4.p2.5.m5.1.1.1.cmml">></mo><mn id="A1.SS1.4.p2.5.m5.1.1.3" xref="A1.SS1.4.p2.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.5.m5.1b"><apply id="A1.SS1.4.p2.5.m5.1.1.cmml" xref="A1.SS1.4.p2.5.m5.1.1"><gt id="A1.SS1.4.p2.5.m5.1.1.1.cmml" xref="A1.SS1.4.p2.5.m5.1.1.1"></gt><ci id="A1.SS1.4.p2.5.m5.1.1.2.cmml" xref="A1.SS1.4.p2.5.m5.1.1.2">𝛿</ci><cn id="A1.SS1.4.p2.5.m5.1.1.3.cmml" type="integer" xref="A1.SS1.4.p2.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.5.m5.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.5.m5.1d">italic_δ > 0</annotation></semantics></math>. By definition, <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.4.p2.6.m6.1"><semantics id="A1.SS1.4.p2.6.m6.1a"><mi id="A1.SS1.4.p2.6.m6.1.1" xref="A1.SS1.4.p2.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.6.m6.1b"><ci id="A1.SS1.4.p2.6.m6.1.1.cmml" xref="A1.SS1.4.p2.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.6.m6.1d">italic_x</annotation></semantics></math> is on the boundary of both balls <math alttext="B_{z^{\prime}}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.7.m7.1"><semantics id="A1.SS1.4.p2.7.m7.1a"><msub id="A1.SS1.4.p2.7.m7.1.1" xref="A1.SS1.4.p2.7.m7.1.1.cmml"><mi id="A1.SS1.4.p2.7.m7.1.1.2" xref="A1.SS1.4.p2.7.m7.1.1.2.cmml">B</mi><msup id="A1.SS1.4.p2.7.m7.1.1.3" xref="A1.SS1.4.p2.7.m7.1.1.3.cmml"><mi id="A1.SS1.4.p2.7.m7.1.1.3.2" xref="A1.SS1.4.p2.7.m7.1.1.3.2.cmml">z</mi><mo id="A1.SS1.4.p2.7.m7.1.1.3.3" xref="A1.SS1.4.p2.7.m7.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.7.m7.1b"><apply id="A1.SS1.4.p2.7.m7.1.1.cmml" xref="A1.SS1.4.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.7.m7.1.1.1.cmml" xref="A1.SS1.4.p2.7.m7.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.7.m7.1.1.2.cmml" xref="A1.SS1.4.p2.7.m7.1.1.2">𝐵</ci><apply id="A1.SS1.4.p2.7.m7.1.1.3.cmml" xref="A1.SS1.4.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.7.m7.1.1.3.1.cmml" xref="A1.SS1.4.p2.7.m7.1.1.3">superscript</csymbol><ci id="A1.SS1.4.p2.7.m7.1.1.3.2.cmml" xref="A1.SS1.4.p2.7.m7.1.1.3.2">𝑧</ci><ci id="A1.SS1.4.p2.7.m7.1.1.3.3.cmml" xref="A1.SS1.4.p2.7.m7.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.7.m7.1c">B_{z^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.7.m7.1d">italic_B start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.8.m8.1"><semantics id="A1.SS1.4.p2.8.m8.1a"><msub id="A1.SS1.4.p2.8.m8.1.1" xref="A1.SS1.4.p2.8.m8.1.1.cmml"><mi id="A1.SS1.4.p2.8.m8.1.1.2" xref="A1.SS1.4.p2.8.m8.1.1.2.cmml">B</mi><mi id="A1.SS1.4.p2.8.m8.1.1.3" xref="A1.SS1.4.p2.8.m8.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.8.m8.1b"><apply id="A1.SS1.4.p2.8.m8.1.1.cmml" xref="A1.SS1.4.p2.8.m8.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.8.m8.1.1.1.cmml" xref="A1.SS1.4.p2.8.m8.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.8.m8.1.1.2.cmml" xref="A1.SS1.4.p2.8.m8.1.1.2">𝐵</ci><ci id="A1.SS1.4.p2.8.m8.1.1.3.cmml" xref="A1.SS1.4.p2.8.m8.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.8.m8.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.8.m8.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, observe that the ball <math alttext="B_{z^{\prime}}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.9.m9.1"><semantics id="A1.SS1.4.p2.9.m9.1a"><msub id="A1.SS1.4.p2.9.m9.1.1" xref="A1.SS1.4.p2.9.m9.1.1.cmml"><mi id="A1.SS1.4.p2.9.m9.1.1.2" xref="A1.SS1.4.p2.9.m9.1.1.2.cmml">B</mi><msup id="A1.SS1.4.p2.9.m9.1.1.3" xref="A1.SS1.4.p2.9.m9.1.1.3.cmml"><mi id="A1.SS1.4.p2.9.m9.1.1.3.2" xref="A1.SS1.4.p2.9.m9.1.1.3.2.cmml">z</mi><mo id="A1.SS1.4.p2.9.m9.1.1.3.3" xref="A1.SS1.4.p2.9.m9.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.9.m9.1b"><apply id="A1.SS1.4.p2.9.m9.1.1.cmml" xref="A1.SS1.4.p2.9.m9.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.9.m9.1.1.1.cmml" xref="A1.SS1.4.p2.9.m9.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.9.m9.1.1.2.cmml" xref="A1.SS1.4.p2.9.m9.1.1.2">𝐵</ci><apply id="A1.SS1.4.p2.9.m9.1.1.3.cmml" xref="A1.SS1.4.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.9.m9.1.1.3.1.cmml" xref="A1.SS1.4.p2.9.m9.1.1.3">superscript</csymbol><ci id="A1.SS1.4.p2.9.m9.1.1.3.2.cmml" xref="A1.SS1.4.p2.9.m9.1.1.3.2">𝑧</ci><ci id="A1.SS1.4.p2.9.m9.1.1.3.3.cmml" xref="A1.SS1.4.p2.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.9.m9.1c">B_{z^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.9.m9.1d">italic_B start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a translated (with center <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.10.m10.1"><semantics id="A1.SS1.4.p2.10.m10.1a"><msup id="A1.SS1.4.p2.10.m10.1.1" xref="A1.SS1.4.p2.10.m10.1.1.cmml"><mi id="A1.SS1.4.p2.10.m10.1.1.2" xref="A1.SS1.4.p2.10.m10.1.1.2.cmml">z</mi><mo id="A1.SS1.4.p2.10.m10.1.1.3" xref="A1.SS1.4.p2.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.10.m10.1b"><apply id="A1.SS1.4.p2.10.m10.1.1.cmml" xref="A1.SS1.4.p2.10.m10.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.10.m10.1.1.1.cmml" xref="A1.SS1.4.p2.10.m10.1.1">superscript</csymbol><ci id="A1.SS1.4.p2.10.m10.1.1.2.cmml" xref="A1.SS1.4.p2.10.m10.1.1.2">𝑧</ci><ci id="A1.SS1.4.p2.10.m10.1.1.3.cmml" xref="A1.SS1.4.p2.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.10.m10.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.10.m10.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> instead of <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.4.p2.11.m11.1"><semantics id="A1.SS1.4.p2.11.m11.1a"><mi id="A1.SS1.4.p2.11.m11.1.1" xref="A1.SS1.4.p2.11.m11.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.11.m11.1b"><ci id="A1.SS1.4.p2.11.m11.1.1.cmml" xref="A1.SS1.4.p2.11.m11.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.11.m11.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.11.m11.1d">italic_z</annotation></semantics></math>) and scaled (such that <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.4.p2.12.m12.1"><semantics id="A1.SS1.4.p2.12.m12.1a"><mi id="A1.SS1.4.p2.12.m12.1.1" xref="A1.SS1.4.p2.12.m12.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.12.m12.1b"><ci id="A1.SS1.4.p2.12.m12.1.1.cmml" xref="A1.SS1.4.p2.12.m12.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.12.m12.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.12.m12.1d">italic_x</annotation></semantics></math> stays on the boundary) version of <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.13.m13.1"><semantics id="A1.SS1.4.p2.13.m13.1a"><msub id="A1.SS1.4.p2.13.m13.1.1" xref="A1.SS1.4.p2.13.m13.1.1.cmml"><mi id="A1.SS1.4.p2.13.m13.1.1.2" xref="A1.SS1.4.p2.13.m13.1.1.2.cmml">B</mi><mi id="A1.SS1.4.p2.13.m13.1.1.3" xref="A1.SS1.4.p2.13.m13.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.13.m13.1b"><apply id="A1.SS1.4.p2.13.m13.1.1.cmml" xref="A1.SS1.4.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.13.m13.1.1.1.cmml" xref="A1.SS1.4.p2.13.m13.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.13.m13.1.1.2.cmml" xref="A1.SS1.4.p2.13.m13.1.1.2">𝐵</ci><ci id="A1.SS1.4.p2.13.m13.1.1.3.cmml" xref="A1.SS1.4.p2.13.m13.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.13.m13.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.13.m13.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, both <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.14.m14.1"><semantics id="A1.SS1.4.p2.14.m14.1a"><msub id="A1.SS1.4.p2.14.m14.1.1" xref="A1.SS1.4.p2.14.m14.1.1.cmml"><mi id="A1.SS1.4.p2.14.m14.1.1.2" xref="A1.SS1.4.p2.14.m14.1.1.2.cmml">B</mi><mi id="A1.SS1.4.p2.14.m14.1.1.3" xref="A1.SS1.4.p2.14.m14.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.14.m14.1b"><apply id="A1.SS1.4.p2.14.m14.1.1.cmml" xref="A1.SS1.4.p2.14.m14.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.14.m14.1.1.1.cmml" xref="A1.SS1.4.p2.14.m14.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.14.m14.1.1.2.cmml" xref="A1.SS1.4.p2.14.m14.1.1.2">𝐵</ci><ci id="A1.SS1.4.p2.14.m14.1.1.3.cmml" xref="A1.SS1.4.p2.14.m14.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.14.m14.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.14.m14.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="B_{z^{\prime}}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.15.m15.1"><semantics id="A1.SS1.4.p2.15.m15.1a"><msub id="A1.SS1.4.p2.15.m15.1.1" xref="A1.SS1.4.p2.15.m15.1.1.cmml"><mi id="A1.SS1.4.p2.15.m15.1.1.2" xref="A1.SS1.4.p2.15.m15.1.1.2.cmml">B</mi><msup id="A1.SS1.4.p2.15.m15.1.1.3" xref="A1.SS1.4.p2.15.m15.1.1.3.cmml"><mi id="A1.SS1.4.p2.15.m15.1.1.3.2" xref="A1.SS1.4.p2.15.m15.1.1.3.2.cmml">z</mi><mo id="A1.SS1.4.p2.15.m15.1.1.3.3" xref="A1.SS1.4.p2.15.m15.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.15.m15.1b"><apply id="A1.SS1.4.p2.15.m15.1.1.cmml" xref="A1.SS1.4.p2.15.m15.1.1"><csymbol cd="ambiguous" id="A1.SS1.4.p2.15.m15.1.1.1.cmml" xref="A1.SS1.4.p2.15.m15.1.1">subscript</csymbol><ci id="A1.SS1.4.p2.15.m15.1.1.2.cmml" xref="A1.SS1.4.p2.15.m15.1.1.2">𝐵</ci><apply id="A1.SS1.4.p2.15.m15.1.1.3.cmml" xref="A1.SS1.4.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.15.m15.1.1.3.1.cmml" xref="A1.SS1.4.p2.15.m15.1.1.3">superscript</csymbol><ci id="A1.SS1.4.p2.15.m15.1.1.3.2.cmml" xref="A1.SS1.4.p2.15.m15.1.1.3.2">𝑧</ci><ci id="A1.SS1.4.p2.15.m15.1.1.3.3.cmml" xref="A1.SS1.4.p2.15.m15.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.15.m15.1c">B_{z^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.15.m15.1d">italic_B start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> have the same tangent hyperplanes at the point <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.4.p2.16.m16.1"><semantics id="A1.SS1.4.p2.16.m16.1a"><mi id="A1.SS1.4.p2.16.m16.1.1" xref="A1.SS1.4.p2.16.m16.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.16.m16.1b"><ci id="A1.SS1.4.p2.16.m16.1.1.cmml" xref="A1.SS1.4.p2.16.m16.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.16.m16.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.16.m16.1d">italic_x</annotation></semantics></math>. Thus, we conclude</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="R_{-}\cap B_{z}^{\circ}=\varnothing\iff R_{-}\cap B_{z^{\prime}}^{\circ}=\varnothing," class="ltx_Math" display="block" id="A1.Ex2.m1.1"><semantics id="A1.Ex2.m1.1a"><mrow id="A1.Ex2.m1.1.1.1" xref="A1.Ex2.m1.1.1.1.1.cmml"><mrow id="A1.Ex2.m1.1.1.1.1" xref="A1.Ex2.m1.1.1.1.1.cmml"><mrow id="A1.Ex2.m1.1.1.1.1.2" xref="A1.Ex2.m1.1.1.1.1.2.cmml"><mrow id="A1.Ex2.m1.1.1.1.1.2.2" xref="A1.Ex2.m1.1.1.1.1.2.2.cmml"><msub id="A1.Ex2.m1.1.1.1.1.2.2.2" xref="A1.Ex2.m1.1.1.1.1.2.2.2.cmml"><mi id="A1.Ex2.m1.1.1.1.1.2.2.2.2" xref="A1.Ex2.m1.1.1.1.1.2.2.2.2.cmml">R</mi><mo id="A1.Ex2.m1.1.1.1.1.2.2.2.3" xref="A1.Ex2.m1.1.1.1.1.2.2.2.3.cmml">−</mo></msub><mo id="A1.Ex2.m1.1.1.1.1.2.2.1" xref="A1.Ex2.m1.1.1.1.1.2.2.1.cmml">∩</mo><msubsup id="A1.Ex2.m1.1.1.1.1.2.2.3" xref="A1.Ex2.m1.1.1.1.1.2.2.3.cmml"><mi id="A1.Ex2.m1.1.1.1.1.2.2.3.2.2" xref="A1.Ex2.m1.1.1.1.1.2.2.3.2.2.cmml">B</mi><mi id="A1.Ex2.m1.1.1.1.1.2.2.3.2.3" xref="A1.Ex2.m1.1.1.1.1.2.2.3.2.3.cmml">z</mi><mo id="A1.Ex2.m1.1.1.1.1.2.2.3.3" xref="A1.Ex2.m1.1.1.1.1.2.2.3.3.cmml">∘</mo></msubsup></mrow><mo id="A1.Ex2.m1.1.1.1.1.2.1" xref="A1.Ex2.m1.1.1.1.1.2.1.cmml">=</mo><mi id="A1.Ex2.m1.1.1.1.1.2.3" mathvariant="normal" xref="A1.Ex2.m1.1.1.1.1.2.3.cmml">∅</mi></mrow><mo id="A1.Ex2.m1.1.1.1.1.1" stretchy="false" xref="A1.Ex2.m1.1.1.1.1.1.cmml">⇔</mo><mrow id="A1.Ex2.m1.1.1.1.1.3" xref="A1.Ex2.m1.1.1.1.1.3.cmml"><mrow id="A1.Ex2.m1.1.1.1.1.3.2" xref="A1.Ex2.m1.1.1.1.1.3.2.cmml"><msub id="A1.Ex2.m1.1.1.1.1.3.2.2" xref="A1.Ex2.m1.1.1.1.1.3.2.2.cmml"><mi id="A1.Ex2.m1.1.1.1.1.3.2.2.2" xref="A1.Ex2.m1.1.1.1.1.3.2.2.2.cmml">R</mi><mo id="A1.Ex2.m1.1.1.1.1.3.2.2.3" xref="A1.Ex2.m1.1.1.1.1.3.2.2.3.cmml">−</mo></msub><mo id="A1.Ex2.m1.1.1.1.1.3.2.1" xref="A1.Ex2.m1.1.1.1.1.3.2.1.cmml">∩</mo><msubsup id="A1.Ex2.m1.1.1.1.1.3.2.3" xref="A1.Ex2.m1.1.1.1.1.3.2.3.cmml"><mi id="A1.Ex2.m1.1.1.1.1.3.2.3.2.2" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.2.cmml">B</mi><msup id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.cmml"><mi id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.2" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.2.cmml">z</mi><mo id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.3" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.3.cmml">′</mo></msup><mo id="A1.Ex2.m1.1.1.1.1.3.2.3.3" xref="A1.Ex2.m1.1.1.1.1.3.2.3.3.cmml">∘</mo></msubsup></mrow><mo id="A1.Ex2.m1.1.1.1.1.3.1" xref="A1.Ex2.m1.1.1.1.1.3.1.cmml">=</mo><mi id="A1.Ex2.m1.1.1.1.1.3.3" mathvariant="normal" xref="A1.Ex2.m1.1.1.1.1.3.3.cmml">∅</mi></mrow></mrow><mo id="A1.Ex2.m1.1.1.1.2" xref="A1.Ex2.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex2.m1.1b"><apply id="A1.Ex2.m1.1.1.1.1.cmml" xref="A1.Ex2.m1.1.1.1"><csymbol cd="latexml" id="A1.Ex2.m1.1.1.1.1.1.cmml" xref="A1.Ex2.m1.1.1.1.1.1">iff</csymbol><apply id="A1.Ex2.m1.1.1.1.1.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2"><eq id="A1.Ex2.m1.1.1.1.1.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.2.1"></eq><apply id="A1.Ex2.m1.1.1.1.1.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2"><intersect id="A1.Ex2.m1.1.1.1.1.2.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.1"></intersect><apply id="A1.Ex2.m1.1.1.1.1.2.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.2.2.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Ex2.m1.1.1.1.1.2.2.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.2.2">𝑅</ci><minus id="A1.Ex2.m1.1.1.1.1.2.2.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.2.3"></minus></apply><apply id="A1.Ex2.m1.1.1.1.1.2.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.2.2.3.1.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3">superscript</csymbol><apply id="A1.Ex2.m1.1.1.1.1.2.2.3.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.2.2.3.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3">subscript</csymbol><ci id="A1.Ex2.m1.1.1.1.1.2.2.3.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3.2.2">𝐵</ci><ci id="A1.Ex2.m1.1.1.1.1.2.2.3.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3.2.3">𝑧</ci></apply><compose id="A1.Ex2.m1.1.1.1.1.2.2.3.3.cmml" xref="A1.Ex2.m1.1.1.1.1.2.2.3.3"></compose></apply></apply><emptyset id="A1.Ex2.m1.1.1.1.1.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.2.3"></emptyset></apply><apply id="A1.Ex2.m1.1.1.1.1.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3"><eq id="A1.Ex2.m1.1.1.1.1.3.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.1"></eq><apply id="A1.Ex2.m1.1.1.1.1.3.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2"><intersect id="A1.Ex2.m1.1.1.1.1.3.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.1"></intersect><apply id="A1.Ex2.m1.1.1.1.1.3.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.3.2.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="A1.Ex2.m1.1.1.1.1.3.2.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.2.2">𝑅</ci><minus id="A1.Ex2.m1.1.1.1.1.3.2.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.2.3"></minus></apply><apply id="A1.Ex2.m1.1.1.1.1.3.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.3.2.3.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3">superscript</csymbol><apply id="A1.Ex2.m1.1.1.1.1.3.2.3.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.3.2.3.2.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3">subscript</csymbol><ci id="A1.Ex2.m1.1.1.1.1.3.2.3.2.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.2">𝐵</ci><apply id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.1.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3">superscript</csymbol><ci id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.2.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.2">𝑧</ci><ci id="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.2.3.3">′</ci></apply></apply><compose id="A1.Ex2.m1.1.1.1.1.3.2.3.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.2.3.3"></compose></apply></apply><emptyset id="A1.Ex2.m1.1.1.1.1.3.3.cmml" xref="A1.Ex2.m1.1.1.1.1.3.3"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex2.m1.1c">R_{-}\cap B_{z}^{\circ}=\varnothing\iff R_{-}\cap B_{z^{\prime}}^{\circ}=\varnothing,</annotation><annotation encoding="application/x-llamapun" id="A1.Ex2.m1.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT = ∅ ⇔ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT = ∅ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS1.4.p2.20">which means that <math alttext="z^{\prime}\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.17.m1.2"><semantics id="A1.SS1.4.p2.17.m1.2a"><mrow id="A1.SS1.4.p2.17.m1.2.3" xref="A1.SS1.4.p2.17.m1.2.3.cmml"><msup id="A1.SS1.4.p2.17.m1.2.3.2" xref="A1.SS1.4.p2.17.m1.2.3.2.cmml"><mi id="A1.SS1.4.p2.17.m1.2.3.2.2" xref="A1.SS1.4.p2.17.m1.2.3.2.2.cmml">z</mi><mo id="A1.SS1.4.p2.17.m1.2.3.2.3" xref="A1.SS1.4.p2.17.m1.2.3.2.3.cmml">′</mo></msup><mo id="A1.SS1.4.p2.17.m1.2.3.1" xref="A1.SS1.4.p2.17.m1.2.3.1.cmml">∈</mo><msubsup id="A1.SS1.4.p2.17.m1.2.3.3" xref="A1.SS1.4.p2.17.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.4.p2.17.m1.2.3.3.2.2" xref="A1.SS1.4.p2.17.m1.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.4.p2.17.m1.2.2.2.4" xref="A1.SS1.4.p2.17.m1.2.2.2.3.cmml"><mi id="A1.SS1.4.p2.17.m1.1.1.1.1" xref="A1.SS1.4.p2.17.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS1.4.p2.17.m1.2.2.2.4.1" xref="A1.SS1.4.p2.17.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS1.4.p2.17.m1.2.2.2.2" xref="A1.SS1.4.p2.17.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.4.p2.17.m1.2.3.3.2.3" xref="A1.SS1.4.p2.17.m1.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.17.m1.2b"><apply id="A1.SS1.4.p2.17.m1.2.3.cmml" xref="A1.SS1.4.p2.17.m1.2.3"><in id="A1.SS1.4.p2.17.m1.2.3.1.cmml" xref="A1.SS1.4.p2.17.m1.2.3.1"></in><apply id="A1.SS1.4.p2.17.m1.2.3.2.cmml" xref="A1.SS1.4.p2.17.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.4.p2.17.m1.2.3.2.1.cmml" xref="A1.SS1.4.p2.17.m1.2.3.2">superscript</csymbol><ci id="A1.SS1.4.p2.17.m1.2.3.2.2.cmml" xref="A1.SS1.4.p2.17.m1.2.3.2.2">𝑧</ci><ci id="A1.SS1.4.p2.17.m1.2.3.2.3.cmml" xref="A1.SS1.4.p2.17.m1.2.3.2.3">′</ci></apply><apply id="A1.SS1.4.p2.17.m1.2.3.3.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.17.m1.2.3.3.1.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3">subscript</csymbol><apply id="A1.SS1.4.p2.17.m1.2.3.3.2.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.17.m1.2.3.3.2.1.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3">superscript</csymbol><ci id="A1.SS1.4.p2.17.m1.2.3.3.2.2.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3.2.2">ℋ</ci><ci id="A1.SS1.4.p2.17.m1.2.3.3.2.3.cmml" xref="A1.SS1.4.p2.17.m1.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS1.4.p2.17.m1.2.2.2.3.cmml" xref="A1.SS1.4.p2.17.m1.2.2.2.4"><ci id="A1.SS1.4.p2.17.m1.1.1.1.1.cmml" xref="A1.SS1.4.p2.17.m1.1.1.1.1">𝑥</ci><ci id="A1.SS1.4.p2.17.m1.2.2.2.2.cmml" xref="A1.SS1.4.p2.17.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.17.m1.2c">z^{\prime}\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.17.m1.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.18.m2.2"><semantics id="A1.SS1.4.p2.18.m2.2a"><mrow id="A1.SS1.4.p2.18.m2.2.3" xref="A1.SS1.4.p2.18.m2.2.3.cmml"><mi id="A1.SS1.4.p2.18.m2.2.3.2" xref="A1.SS1.4.p2.18.m2.2.3.2.cmml">z</mi><mo id="A1.SS1.4.p2.18.m2.2.3.1" xref="A1.SS1.4.p2.18.m2.2.3.1.cmml">∈</mo><msubsup id="A1.SS1.4.p2.18.m2.2.3.3" xref="A1.SS1.4.p2.18.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.4.p2.18.m2.2.3.3.2.2" xref="A1.SS1.4.p2.18.m2.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.4.p2.18.m2.2.2.2.4" xref="A1.SS1.4.p2.18.m2.2.2.2.3.cmml"><mi id="A1.SS1.4.p2.18.m2.1.1.1.1" xref="A1.SS1.4.p2.18.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS1.4.p2.18.m2.2.2.2.4.1" xref="A1.SS1.4.p2.18.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS1.4.p2.18.m2.2.2.2.2" xref="A1.SS1.4.p2.18.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.4.p2.18.m2.2.3.3.2.3" xref="A1.SS1.4.p2.18.m2.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.18.m2.2b"><apply id="A1.SS1.4.p2.18.m2.2.3.cmml" xref="A1.SS1.4.p2.18.m2.2.3"><in id="A1.SS1.4.p2.18.m2.2.3.1.cmml" xref="A1.SS1.4.p2.18.m2.2.3.1"></in><ci id="A1.SS1.4.p2.18.m2.2.3.2.cmml" xref="A1.SS1.4.p2.18.m2.2.3.2">𝑧</ci><apply id="A1.SS1.4.p2.18.m2.2.3.3.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.18.m2.2.3.3.1.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3">subscript</csymbol><apply id="A1.SS1.4.p2.18.m2.2.3.3.2.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.18.m2.2.3.3.2.1.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3">superscript</csymbol><ci id="A1.SS1.4.p2.18.m2.2.3.3.2.2.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3.2.2">ℋ</ci><ci id="A1.SS1.4.p2.18.m2.2.3.3.2.3.cmml" xref="A1.SS1.4.p2.18.m2.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS1.4.p2.18.m2.2.2.2.3.cmml" xref="A1.SS1.4.p2.18.m2.2.2.2.4"><ci id="A1.SS1.4.p2.18.m2.1.1.1.1.cmml" xref="A1.SS1.4.p2.18.m2.1.1.1.1">𝑥</ci><ci id="A1.SS1.4.p2.18.m2.2.2.2.2.cmml" xref="A1.SS1.4.p2.18.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.18.m2.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.18.m2.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. This proves that <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.4.p2.19.m3.2"><semantics id="A1.SS1.4.p2.19.m3.2a"><msubsup id="A1.SS1.4.p2.19.m3.2.3" xref="A1.SS1.4.p2.19.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.4.p2.19.m3.2.3.2.2" xref="A1.SS1.4.p2.19.m3.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.4.p2.19.m3.2.2.2.4" xref="A1.SS1.4.p2.19.m3.2.2.2.3.cmml"><mi id="A1.SS1.4.p2.19.m3.1.1.1.1" xref="A1.SS1.4.p2.19.m3.1.1.1.1.cmml">x</mi><mo id="A1.SS1.4.p2.19.m3.2.2.2.4.1" xref="A1.SS1.4.p2.19.m3.2.2.2.3.cmml">,</mo><mi id="A1.SS1.4.p2.19.m3.2.2.2.2" xref="A1.SS1.4.p2.19.m3.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.4.p2.19.m3.2.3.2.3" xref="A1.SS1.4.p2.19.m3.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.19.m3.2b"><apply id="A1.SS1.4.p2.19.m3.2.3.cmml" xref="A1.SS1.4.p2.19.m3.2.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.19.m3.2.3.1.cmml" xref="A1.SS1.4.p2.19.m3.2.3">subscript</csymbol><apply id="A1.SS1.4.p2.19.m3.2.3.2.cmml" xref="A1.SS1.4.p2.19.m3.2.3"><csymbol cd="ambiguous" id="A1.SS1.4.p2.19.m3.2.3.2.1.cmml" xref="A1.SS1.4.p2.19.m3.2.3">superscript</csymbol><ci id="A1.SS1.4.p2.19.m3.2.3.2.2.cmml" xref="A1.SS1.4.p2.19.m3.2.3.2.2">ℋ</ci><ci id="A1.SS1.4.p2.19.m3.2.3.2.3.cmml" xref="A1.SS1.4.p2.19.m3.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.4.p2.19.m3.2.2.2.3.cmml" xref="A1.SS1.4.p2.19.m3.2.2.2.4"><ci id="A1.SS1.4.p2.19.m3.1.1.1.1.cmml" xref="A1.SS1.4.p2.19.m3.1.1.1.1">𝑥</ci><ci id="A1.SS1.4.p2.19.m3.2.2.2.2.cmml" xref="A1.SS1.4.p2.19.m3.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.19.m3.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.19.m3.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is a union of rays starting at <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.4.p2.20.m4.1"><semantics id="A1.SS1.4.p2.20.m4.1a"><mi id="A1.SS1.4.p2.20.m4.1.1" xref="A1.SS1.4.p2.20.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.4.p2.20.m4.1b"><ci id="A1.SS1.4.p2.20.m4.1.1.cmml" xref="A1.SS1.4.p2.20.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.4.p2.20.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.4.p2.20.m4.1d">italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.SS1.5.p3"> <p class="ltx_p" id="A1.SS1.5.p3.37">It remains to prove that the boundary <math alttext="\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.1.m1.2"><semantics id="A1.SS1.5.p3.1.m1.2a"><mrow id="A1.SS1.5.p3.1.m1.2.3" xref="A1.SS1.5.p3.1.m1.2.3.cmml"><mo id="A1.SS1.5.p3.1.m1.2.3.1" rspace="0em" xref="A1.SS1.5.p3.1.m1.2.3.1.cmml">∂</mo><msubsup id="A1.SS1.5.p3.1.m1.2.3.2" xref="A1.SS1.5.p3.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.5.p3.1.m1.2.3.2.2.2" xref="A1.SS1.5.p3.1.m1.2.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS1.5.p3.1.m1.2.2.2.4" xref="A1.SS1.5.p3.1.m1.2.2.2.3.cmml"><mi id="A1.SS1.5.p3.1.m1.1.1.1.1" xref="A1.SS1.5.p3.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.1.m1.2.2.2.4.1" xref="A1.SS1.5.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS1.5.p3.1.m1.2.2.2.2" xref="A1.SS1.5.p3.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.5.p3.1.m1.2.3.2.2.3" xref="A1.SS1.5.p3.1.m1.2.3.2.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.1.m1.2b"><apply id="A1.SS1.5.p3.1.m1.2.3.cmml" xref="A1.SS1.5.p3.1.m1.2.3"><partialdiff id="A1.SS1.5.p3.1.m1.2.3.1.cmml" xref="A1.SS1.5.p3.1.m1.2.3.1"></partialdiff><apply id="A1.SS1.5.p3.1.m1.2.3.2.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.1.m1.2.3.2.1.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2">subscript</csymbol><apply id="A1.SS1.5.p3.1.m1.2.3.2.2.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.1.m1.2.3.2.2.1.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2">superscript</csymbol><ci id="A1.SS1.5.p3.1.m1.2.3.2.2.2.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2.2.2">ℋ</ci><ci id="A1.SS1.5.p3.1.m1.2.3.2.2.3.cmml" xref="A1.SS1.5.p3.1.m1.2.3.2.2.3">𝑝</ci></apply><list id="A1.SS1.5.p3.1.m1.2.2.2.3.cmml" xref="A1.SS1.5.p3.1.m1.2.2.2.4"><ci id="A1.SS1.5.p3.1.m1.1.1.1.1.cmml" xref="A1.SS1.5.p3.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS1.5.p3.1.m1.2.2.2.2.cmml" xref="A1.SS1.5.p3.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.1.m1.2c">\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.1.m1.2d">∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.2.m2.2"><semantics id="A1.SS1.5.p3.2.m2.2a"><msubsup id="A1.SS1.5.p3.2.m2.2.3" xref="A1.SS1.5.p3.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.5.p3.2.m2.2.3.2.2" xref="A1.SS1.5.p3.2.m2.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.5.p3.2.m2.2.2.2.4" xref="A1.SS1.5.p3.2.m2.2.2.2.3.cmml"><mi id="A1.SS1.5.p3.2.m2.1.1.1.1" xref="A1.SS1.5.p3.2.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.2.m2.2.2.2.4.1" xref="A1.SS1.5.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS1.5.p3.2.m2.2.2.2.2" xref="A1.SS1.5.p3.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.5.p3.2.m2.2.3.2.3" xref="A1.SS1.5.p3.2.m2.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.2.m2.2b"><apply id="A1.SS1.5.p3.2.m2.2.3.cmml" xref="A1.SS1.5.p3.2.m2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.2.m2.2.3.1.cmml" xref="A1.SS1.5.p3.2.m2.2.3">subscript</csymbol><apply id="A1.SS1.5.p3.2.m2.2.3.2.cmml" xref="A1.SS1.5.p3.2.m2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.2.m2.2.3.2.1.cmml" xref="A1.SS1.5.p3.2.m2.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.2.m2.2.3.2.2.cmml" xref="A1.SS1.5.p3.2.m2.2.3.2.2">ℋ</ci><ci id="A1.SS1.5.p3.2.m2.2.3.2.3.cmml" xref="A1.SS1.5.p3.2.m2.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.5.p3.2.m2.2.2.2.3.cmml" xref="A1.SS1.5.p3.2.m2.2.2.2.4"><ci id="A1.SS1.5.p3.2.m2.1.1.1.1.cmml" xref="A1.SS1.5.p3.2.m2.1.1.1.1">𝑥</ci><ci id="A1.SS1.5.p3.2.m2.2.2.2.2.cmml" xref="A1.SS1.5.p3.2.m2.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.2.m2.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.2.m2.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is a union of lines through <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.5.p3.3.m3.1"><semantics id="A1.SS1.5.p3.3.m3.1a"><mi id="A1.SS1.5.p3.3.m3.1.1" xref="A1.SS1.5.p3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.3.m3.1b"><ci id="A1.SS1.5.p3.3.m3.1.1.cmml" xref="A1.SS1.5.p3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.3.m3.1d">italic_x</annotation></semantics></math>, assuming that <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="A1.SS1.5.p3.4.m4.2"><semantics id="A1.SS1.5.p3.4.m4.2a"><mrow id="A1.SS1.5.p3.4.m4.2.3" xref="A1.SS1.5.p3.4.m4.2.3.cmml"><mi id="A1.SS1.5.p3.4.m4.2.3.2" xref="A1.SS1.5.p3.4.m4.2.3.2.cmml">p</mi><mo id="A1.SS1.5.p3.4.m4.2.3.1" xref="A1.SS1.5.p3.4.m4.2.3.1.cmml">∈</mo><mrow id="A1.SS1.5.p3.4.m4.2.3.3.2" xref="A1.SS1.5.p3.4.m4.2.3.3.1.cmml"><mo id="A1.SS1.5.p3.4.m4.2.3.3.2.1" stretchy="false" xref="A1.SS1.5.p3.4.m4.2.3.3.1.cmml">(</mo><mn id="A1.SS1.5.p3.4.m4.1.1" xref="A1.SS1.5.p3.4.m4.1.1.cmml">1</mn><mo id="A1.SS1.5.p3.4.m4.2.3.3.2.2" xref="A1.SS1.5.p3.4.m4.2.3.3.1.cmml">,</mo><mi id="A1.SS1.5.p3.4.m4.2.2" mathvariant="normal" xref="A1.SS1.5.p3.4.m4.2.2.cmml">∞</mi><mo id="A1.SS1.5.p3.4.m4.2.3.3.2.3" stretchy="false" xref="A1.SS1.5.p3.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.4.m4.2b"><apply id="A1.SS1.5.p3.4.m4.2.3.cmml" xref="A1.SS1.5.p3.4.m4.2.3"><in id="A1.SS1.5.p3.4.m4.2.3.1.cmml" xref="A1.SS1.5.p3.4.m4.2.3.1"></in><ci id="A1.SS1.5.p3.4.m4.2.3.2.cmml" xref="A1.SS1.5.p3.4.m4.2.3.2">𝑝</ci><interval closure="open" id="A1.SS1.5.p3.4.m4.2.3.3.1.cmml" xref="A1.SS1.5.p3.4.m4.2.3.3.2"><cn id="A1.SS1.5.p3.4.m4.1.1.cmml" type="integer" xref="A1.SS1.5.p3.4.m4.1.1">1</cn><infinity id="A1.SS1.5.p3.4.m4.2.2.cmml" xref="A1.SS1.5.p3.4.m4.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.4.m4.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.4.m4.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>. For this, observe that the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.5.m5.1"><semantics id="A1.SS1.5.p3.5.m5.1a"><msub id="A1.SS1.5.p3.5.m5.1.1" xref="A1.SS1.5.p3.5.m5.1.1.cmml"><mi id="A1.SS1.5.p3.5.m5.1.1.2" mathvariant="normal" xref="A1.SS1.5.p3.5.m5.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.5.p3.5.m5.1.1.3" xref="A1.SS1.5.p3.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.5.m5.1b"><apply id="A1.SS1.5.p3.5.m5.1.1.cmml" xref="A1.SS1.5.p3.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.5.m5.1.1.1.cmml" xref="A1.SS1.5.p3.5.m5.1.1">subscript</csymbol><ci id="A1.SS1.5.p3.5.m5.1.1.2.cmml" xref="A1.SS1.5.p3.5.m5.1.1.2">ℓ</ci><ci id="A1.SS1.5.p3.5.m5.1.1.3.cmml" xref="A1.SS1.5.p3.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.5.m5.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-balls for <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="A1.SS1.5.p3.6.m6.2"><semantics id="A1.SS1.5.p3.6.m6.2a"><mrow id="A1.SS1.5.p3.6.m6.2.3" xref="A1.SS1.5.p3.6.m6.2.3.cmml"><mi id="A1.SS1.5.p3.6.m6.2.3.2" xref="A1.SS1.5.p3.6.m6.2.3.2.cmml">p</mi><mo id="A1.SS1.5.p3.6.m6.2.3.1" xref="A1.SS1.5.p3.6.m6.2.3.1.cmml">∈</mo><mrow id="A1.SS1.5.p3.6.m6.2.3.3.2" xref="A1.SS1.5.p3.6.m6.2.3.3.1.cmml"><mo id="A1.SS1.5.p3.6.m6.2.3.3.2.1" stretchy="false" xref="A1.SS1.5.p3.6.m6.2.3.3.1.cmml">(</mo><mn id="A1.SS1.5.p3.6.m6.1.1" xref="A1.SS1.5.p3.6.m6.1.1.cmml">1</mn><mo id="A1.SS1.5.p3.6.m6.2.3.3.2.2" xref="A1.SS1.5.p3.6.m6.2.3.3.1.cmml">,</mo><mi id="A1.SS1.5.p3.6.m6.2.2" mathvariant="normal" xref="A1.SS1.5.p3.6.m6.2.2.cmml">∞</mi><mo id="A1.SS1.5.p3.6.m6.2.3.3.2.3" stretchy="false" xref="A1.SS1.5.p3.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.6.m6.2b"><apply id="A1.SS1.5.p3.6.m6.2.3.cmml" xref="A1.SS1.5.p3.6.m6.2.3"><in id="A1.SS1.5.p3.6.m6.2.3.1.cmml" xref="A1.SS1.5.p3.6.m6.2.3.1"></in><ci id="A1.SS1.5.p3.6.m6.2.3.2.cmml" xref="A1.SS1.5.p3.6.m6.2.3.2">𝑝</ci><interval closure="open" id="A1.SS1.5.p3.6.m6.2.3.3.1.cmml" xref="A1.SS1.5.p3.6.m6.2.3.3.2"><cn id="A1.SS1.5.p3.6.m6.1.1.cmml" type="integer" xref="A1.SS1.5.p3.6.m6.1.1">1</cn><infinity id="A1.SS1.5.p3.6.m6.2.2.cmml" xref="A1.SS1.5.p3.6.m6.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.6.m6.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.6.m6.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math> are strictly convex. We claim that a point <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.7.m7.1"><semantics id="A1.SS1.5.p3.7.m7.1a"><mi id="A1.SS1.5.p3.7.m7.1.1" xref="A1.SS1.5.p3.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.7.m7.1b"><ci id="A1.SS1.5.p3.7.m7.1.1.cmml" xref="A1.SS1.5.p3.7.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.7.m7.1d">italic_z</annotation></semantics></math> is on the boundary of <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.8.m8.2"><semantics id="A1.SS1.5.p3.8.m8.2a"><msubsup id="A1.SS1.5.p3.8.m8.2.3" xref="A1.SS1.5.p3.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.5.p3.8.m8.2.3.2.2" xref="A1.SS1.5.p3.8.m8.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.5.p3.8.m8.2.2.2.4" xref="A1.SS1.5.p3.8.m8.2.2.2.3.cmml"><mi id="A1.SS1.5.p3.8.m8.1.1.1.1" xref="A1.SS1.5.p3.8.m8.1.1.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.8.m8.2.2.2.4.1" xref="A1.SS1.5.p3.8.m8.2.2.2.3.cmml">,</mo><mi id="A1.SS1.5.p3.8.m8.2.2.2.2" xref="A1.SS1.5.p3.8.m8.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.5.p3.8.m8.2.3.2.3" xref="A1.SS1.5.p3.8.m8.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.8.m8.2b"><apply id="A1.SS1.5.p3.8.m8.2.3.cmml" xref="A1.SS1.5.p3.8.m8.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.8.m8.2.3.1.cmml" xref="A1.SS1.5.p3.8.m8.2.3">subscript</csymbol><apply id="A1.SS1.5.p3.8.m8.2.3.2.cmml" xref="A1.SS1.5.p3.8.m8.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.8.m8.2.3.2.1.cmml" xref="A1.SS1.5.p3.8.m8.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.8.m8.2.3.2.2.cmml" xref="A1.SS1.5.p3.8.m8.2.3.2.2">ℋ</ci><ci id="A1.SS1.5.p3.8.m8.2.3.2.3.cmml" xref="A1.SS1.5.p3.8.m8.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.5.p3.8.m8.2.2.2.3.cmml" xref="A1.SS1.5.p3.8.m8.2.2.2.4"><ci id="A1.SS1.5.p3.8.m8.1.1.1.1.cmml" xref="A1.SS1.5.p3.8.m8.1.1.1.1">𝑥</ci><ci id="A1.SS1.5.p3.8.m8.2.2.2.2.cmml" xref="A1.SS1.5.p3.8.m8.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.8.m8.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.8.m8.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="B^{\circ}_{z}\cap L=\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.9.m9.1"><semantics id="A1.SS1.5.p3.9.m9.1a"><mrow id="A1.SS1.5.p3.9.m9.1.1" xref="A1.SS1.5.p3.9.m9.1.1.cmml"><mrow id="A1.SS1.5.p3.9.m9.1.1.2" xref="A1.SS1.5.p3.9.m9.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.9.m9.1.1.2.2" xref="A1.SS1.5.p3.9.m9.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.9.m9.1.1.2.2.2.2" xref="A1.SS1.5.p3.9.m9.1.1.2.2.2.2.cmml">B</mi><mi id="A1.SS1.5.p3.9.m9.1.1.2.2.3" xref="A1.SS1.5.p3.9.m9.1.1.2.2.3.cmml">z</mi><mo id="A1.SS1.5.p3.9.m9.1.1.2.2.2.3" xref="A1.SS1.5.p3.9.m9.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.9.m9.1.1.2.1" xref="A1.SS1.5.p3.9.m9.1.1.2.1.cmml">∩</mo><mi id="A1.SS1.5.p3.9.m9.1.1.2.3" xref="A1.SS1.5.p3.9.m9.1.1.2.3.cmml">L</mi></mrow><mo id="A1.SS1.5.p3.9.m9.1.1.1" xref="A1.SS1.5.p3.9.m9.1.1.1.cmml">=</mo><mi id="A1.SS1.5.p3.9.m9.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.9.m9.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.9.m9.1b"><apply id="A1.SS1.5.p3.9.m9.1.1.cmml" xref="A1.SS1.5.p3.9.m9.1.1"><eq id="A1.SS1.5.p3.9.m9.1.1.1.cmml" xref="A1.SS1.5.p3.9.m9.1.1.1"></eq><apply id="A1.SS1.5.p3.9.m9.1.1.2.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2"><intersect id="A1.SS1.5.p3.9.m9.1.1.2.1.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.9.m9.1.1.2.2.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.9.m9.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.9.m9.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.9.m9.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.9.m9.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.9.m9.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2.2.3"></compose></apply><ci id="A1.SS1.5.p3.9.m9.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.2.3">𝑧</ci></apply><ci id="A1.SS1.5.p3.9.m9.1.1.2.3.cmml" xref="A1.SS1.5.p3.9.m9.1.1.2.3">𝐿</ci></apply><emptyset id="A1.SS1.5.p3.9.m9.1.1.3.cmml" xref="A1.SS1.5.p3.9.m9.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.9.m9.1c">B^{\circ}_{z}\cap L=\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.9.m9.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∩ italic_L = ∅</annotation></semantics></math>. Indeed, any point <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.10.m10.1"><semantics id="A1.SS1.5.p3.10.m10.1a"><mi id="A1.SS1.5.p3.10.m10.1.1" xref="A1.SS1.5.p3.10.m10.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.10.m10.1b"><ci id="A1.SS1.5.p3.10.m10.1.1.cmml" xref="A1.SS1.5.p3.10.m10.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.10.m10.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.10.m10.1d">italic_z</annotation></semantics></math> with <math alttext="B^{\circ}_{z}\cap L=\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.11.m11.1"><semantics id="A1.SS1.5.p3.11.m11.1a"><mrow id="A1.SS1.5.p3.11.m11.1.1" xref="A1.SS1.5.p3.11.m11.1.1.cmml"><mrow id="A1.SS1.5.p3.11.m11.1.1.2" xref="A1.SS1.5.p3.11.m11.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.11.m11.1.1.2.2" xref="A1.SS1.5.p3.11.m11.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.11.m11.1.1.2.2.2.2" xref="A1.SS1.5.p3.11.m11.1.1.2.2.2.2.cmml">B</mi><mi id="A1.SS1.5.p3.11.m11.1.1.2.2.3" xref="A1.SS1.5.p3.11.m11.1.1.2.2.3.cmml">z</mi><mo id="A1.SS1.5.p3.11.m11.1.1.2.2.2.3" xref="A1.SS1.5.p3.11.m11.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.11.m11.1.1.2.1" xref="A1.SS1.5.p3.11.m11.1.1.2.1.cmml">∩</mo><mi id="A1.SS1.5.p3.11.m11.1.1.2.3" xref="A1.SS1.5.p3.11.m11.1.1.2.3.cmml">L</mi></mrow><mo id="A1.SS1.5.p3.11.m11.1.1.1" xref="A1.SS1.5.p3.11.m11.1.1.1.cmml">=</mo><mi id="A1.SS1.5.p3.11.m11.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.11.m11.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.11.m11.1b"><apply id="A1.SS1.5.p3.11.m11.1.1.cmml" xref="A1.SS1.5.p3.11.m11.1.1"><eq id="A1.SS1.5.p3.11.m11.1.1.1.cmml" xref="A1.SS1.5.p3.11.m11.1.1.1"></eq><apply id="A1.SS1.5.p3.11.m11.1.1.2.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2"><intersect id="A1.SS1.5.p3.11.m11.1.1.2.1.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.11.m11.1.1.2.2.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.11.m11.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.11.m11.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.11.m11.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.11.m11.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.11.m11.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2.2.3"></compose></apply><ci id="A1.SS1.5.p3.11.m11.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.2.3">𝑧</ci></apply><ci id="A1.SS1.5.p3.11.m11.1.1.2.3.cmml" xref="A1.SS1.5.p3.11.m11.1.1.2.3">𝐿</ci></apply><emptyset id="A1.SS1.5.p3.11.m11.1.1.3.cmml" xref="A1.SS1.5.p3.11.m11.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.11.m11.1c">B^{\circ}_{z}\cap L=\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.11.m11.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∩ italic_L = ∅</annotation></semantics></math> is clearly contained in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.12.m12.2"><semantics id="A1.SS1.5.p3.12.m12.2a"><msubsup id="A1.SS1.5.p3.12.m12.2.3" xref="A1.SS1.5.p3.12.m12.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.5.p3.12.m12.2.3.2.2" xref="A1.SS1.5.p3.12.m12.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.5.p3.12.m12.2.2.2.4" xref="A1.SS1.5.p3.12.m12.2.2.2.3.cmml"><mi id="A1.SS1.5.p3.12.m12.1.1.1.1" xref="A1.SS1.5.p3.12.m12.1.1.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.12.m12.2.2.2.4.1" xref="A1.SS1.5.p3.12.m12.2.2.2.3.cmml">,</mo><mi id="A1.SS1.5.p3.12.m12.2.2.2.2" xref="A1.SS1.5.p3.12.m12.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.5.p3.12.m12.2.3.2.3" xref="A1.SS1.5.p3.12.m12.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.12.m12.2b"><apply id="A1.SS1.5.p3.12.m12.2.3.cmml" xref="A1.SS1.5.p3.12.m12.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.12.m12.2.3.1.cmml" xref="A1.SS1.5.p3.12.m12.2.3">subscript</csymbol><apply id="A1.SS1.5.p3.12.m12.2.3.2.cmml" xref="A1.SS1.5.p3.12.m12.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.12.m12.2.3.2.1.cmml" xref="A1.SS1.5.p3.12.m12.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.12.m12.2.3.2.2.cmml" xref="A1.SS1.5.p3.12.m12.2.3.2.2">ℋ</ci><ci id="A1.SS1.5.p3.12.m12.2.3.2.3.cmml" xref="A1.SS1.5.p3.12.m12.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.5.p3.12.m12.2.2.2.3.cmml" xref="A1.SS1.5.p3.12.m12.2.2.2.4"><ci id="A1.SS1.5.p3.12.m12.1.1.1.1.cmml" xref="A1.SS1.5.p3.12.m12.1.1.1.1">𝑥</ci><ci id="A1.SS1.5.p3.12.m12.2.2.2.2.cmml" xref="A1.SS1.5.p3.12.m12.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.12.m12.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.12.m12.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> (by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>). Moreover, moving <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.13.m13.1"><semantics id="A1.SS1.5.p3.13.m13.1a"><mi id="A1.SS1.5.p3.13.m13.1.1" xref="A1.SS1.5.p3.13.m13.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.13.m13.1b"><ci id="A1.SS1.5.p3.13.m13.1.1.cmml" xref="A1.SS1.5.p3.13.m13.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.13.m13.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.13.m13.1d">italic_z</annotation></semantics></math> into the direction <math alttext="-v" class="ltx_Math" display="inline" id="A1.SS1.5.p3.14.m14.1"><semantics id="A1.SS1.5.p3.14.m14.1a"><mrow id="A1.SS1.5.p3.14.m14.1.1" xref="A1.SS1.5.p3.14.m14.1.1.cmml"><mo id="A1.SS1.5.p3.14.m14.1.1a" xref="A1.SS1.5.p3.14.m14.1.1.cmml">−</mo><mi id="A1.SS1.5.p3.14.m14.1.1.2" xref="A1.SS1.5.p3.14.m14.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.14.m14.1b"><apply id="A1.SS1.5.p3.14.m14.1.1.cmml" xref="A1.SS1.5.p3.14.m14.1.1"><minus id="A1.SS1.5.p3.14.m14.1.1.1.cmml" xref="A1.SS1.5.p3.14.m14.1.1"></minus><ci id="A1.SS1.5.p3.14.m14.1.1.2.cmml" xref="A1.SS1.5.p3.14.m14.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.14.m14.1c">-v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.14.m14.1d">- italic_v</annotation></semantics></math> yields a point <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.15.m15.1"><semantics id="A1.SS1.5.p3.15.m15.1a"><msup id="A1.SS1.5.p3.15.m15.1.1" xref="A1.SS1.5.p3.15.m15.1.1.cmml"><mi id="A1.SS1.5.p3.15.m15.1.1.2" xref="A1.SS1.5.p3.15.m15.1.1.2.cmml">z</mi><mo id="A1.SS1.5.p3.15.m15.1.1.3" xref="A1.SS1.5.p3.15.m15.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.15.m15.1b"><apply id="A1.SS1.5.p3.15.m15.1.1.cmml" xref="A1.SS1.5.p3.15.m15.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.15.m15.1.1.1.cmml" xref="A1.SS1.5.p3.15.m15.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.15.m15.1.1.2.cmml" xref="A1.SS1.5.p3.15.m15.1.1.2">𝑧</ci><ci id="A1.SS1.5.p3.15.m15.1.1.3.cmml" xref="A1.SS1.5.p3.15.m15.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.15.m15.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.15.m15.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="B^{\circ}_{z^{\prime}}\cap R_{-}\neq\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.16.m16.1"><semantics id="A1.SS1.5.p3.16.m16.1a"><mrow id="A1.SS1.5.p3.16.m16.1.1" xref="A1.SS1.5.p3.16.m16.1.1.cmml"><mrow id="A1.SS1.5.p3.16.m16.1.1.2" xref="A1.SS1.5.p3.16.m16.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.16.m16.1.1.2.2" xref="A1.SS1.5.p3.16.m16.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.16.m16.1.1.2.2.2.2" xref="A1.SS1.5.p3.16.m16.1.1.2.2.2.2.cmml">B</mi><msup id="A1.SS1.5.p3.16.m16.1.1.2.2.3" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3.cmml"><mi id="A1.SS1.5.p3.16.m16.1.1.2.2.3.2" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.16.m16.1.1.2.2.3.3" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3.3.cmml">′</mo></msup><mo id="A1.SS1.5.p3.16.m16.1.1.2.2.2.3" xref="A1.SS1.5.p3.16.m16.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.16.m16.1.1.2.1" xref="A1.SS1.5.p3.16.m16.1.1.2.1.cmml">∩</mo><msub id="A1.SS1.5.p3.16.m16.1.1.2.3" xref="A1.SS1.5.p3.16.m16.1.1.2.3.cmml"><mi id="A1.SS1.5.p3.16.m16.1.1.2.3.2" xref="A1.SS1.5.p3.16.m16.1.1.2.3.2.cmml">R</mi><mo id="A1.SS1.5.p3.16.m16.1.1.2.3.3" xref="A1.SS1.5.p3.16.m16.1.1.2.3.3.cmml">−</mo></msub></mrow><mo id="A1.SS1.5.p3.16.m16.1.1.1" xref="A1.SS1.5.p3.16.m16.1.1.1.cmml">≠</mo><mi id="A1.SS1.5.p3.16.m16.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.16.m16.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.16.m16.1b"><apply id="A1.SS1.5.p3.16.m16.1.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1"><neq id="A1.SS1.5.p3.16.m16.1.1.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.1"></neq><apply id="A1.SS1.5.p3.16.m16.1.1.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2"><intersect id="A1.SS1.5.p3.16.m16.1.1.2.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.16.m16.1.1.2.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.16.m16.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.16.m16.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.16.m16.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.16.m16.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.16.m16.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.2.3"></compose></apply><apply id="A1.SS1.5.p3.16.m16.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.16.m16.1.1.2.2.3.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.16.m16.1.1.2.2.3.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3.2">𝑧</ci><ci id="A1.SS1.5.p3.16.m16.1.1.2.2.3.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.2.3.3">′</ci></apply></apply><apply id="A1.SS1.5.p3.16.m16.1.1.2.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.16.m16.1.1.2.3.1.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.3">subscript</csymbol><ci id="A1.SS1.5.p3.16.m16.1.1.2.3.2.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.3.2">𝑅</ci><minus id="A1.SS1.5.p3.16.m16.1.1.2.3.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.2.3.3"></minus></apply></apply><emptyset id="A1.SS1.5.p3.16.m16.1.1.3.cmml" xref="A1.SS1.5.p3.16.m16.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.16.m16.1c">B^{\circ}_{z^{\prime}}\cap R_{-}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.16.m16.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∩ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math> (by strict convexity of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.17.m17.1"><semantics id="A1.SS1.5.p3.17.m17.1a"><msub id="A1.SS1.5.p3.17.m17.1.1" xref="A1.SS1.5.p3.17.m17.1.1.cmml"><mi id="A1.SS1.5.p3.17.m17.1.1.2" mathvariant="normal" xref="A1.SS1.5.p3.17.m17.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.5.p3.17.m17.1.1.3" xref="A1.SS1.5.p3.17.m17.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.17.m17.1b"><apply id="A1.SS1.5.p3.17.m17.1.1.cmml" xref="A1.SS1.5.p3.17.m17.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.17.m17.1.1.1.cmml" xref="A1.SS1.5.p3.17.m17.1.1">subscript</csymbol><ci id="A1.SS1.5.p3.17.m17.1.1.2.cmml" xref="A1.SS1.5.p3.17.m17.1.1.2">ℓ</ci><ci id="A1.SS1.5.p3.17.m17.1.1.3.cmml" xref="A1.SS1.5.p3.17.m17.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.17.m17.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.17.m17.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-balls). We conclude <math alttext="z\in\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.18.m18.2"><semantics id="A1.SS1.5.p3.18.m18.2a"><mrow id="A1.SS1.5.p3.18.m18.2.3" xref="A1.SS1.5.p3.18.m18.2.3.cmml"><mi id="A1.SS1.5.p3.18.m18.2.3.2" xref="A1.SS1.5.p3.18.m18.2.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.18.m18.2.3.1" rspace="0.1389em" xref="A1.SS1.5.p3.18.m18.2.3.1.cmml">∈</mo><mrow id="A1.SS1.5.p3.18.m18.2.3.3" xref="A1.SS1.5.p3.18.m18.2.3.3.cmml"><mo id="A1.SS1.5.p3.18.m18.2.3.3.1" lspace="0.1389em" rspace="0em" xref="A1.SS1.5.p3.18.m18.2.3.3.1.cmml">∂</mo><msubsup id="A1.SS1.5.p3.18.m18.2.3.3.2" xref="A1.SS1.5.p3.18.m18.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.5.p3.18.m18.2.3.3.2.2.2" xref="A1.SS1.5.p3.18.m18.2.3.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS1.5.p3.18.m18.2.2.2.4" xref="A1.SS1.5.p3.18.m18.2.2.2.3.cmml"><mi id="A1.SS1.5.p3.18.m18.1.1.1.1" xref="A1.SS1.5.p3.18.m18.1.1.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.18.m18.2.2.2.4.1" xref="A1.SS1.5.p3.18.m18.2.2.2.3.cmml">,</mo><mi id="A1.SS1.5.p3.18.m18.2.2.2.2" xref="A1.SS1.5.p3.18.m18.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.5.p3.18.m18.2.3.3.2.2.3" xref="A1.SS1.5.p3.18.m18.2.3.3.2.2.3.cmml">p</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.18.m18.2b"><apply id="A1.SS1.5.p3.18.m18.2.3.cmml" xref="A1.SS1.5.p3.18.m18.2.3"><in id="A1.SS1.5.p3.18.m18.2.3.1.cmml" xref="A1.SS1.5.p3.18.m18.2.3.1"></in><ci id="A1.SS1.5.p3.18.m18.2.3.2.cmml" xref="A1.SS1.5.p3.18.m18.2.3.2">𝑧</ci><apply id="A1.SS1.5.p3.18.m18.2.3.3.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3"><partialdiff id="A1.SS1.5.p3.18.m18.2.3.3.1.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.1"></partialdiff><apply id="A1.SS1.5.p3.18.m18.2.3.3.2.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.18.m18.2.3.3.2.1.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2">subscript</csymbol><apply id="A1.SS1.5.p3.18.m18.2.3.3.2.2.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.18.m18.2.3.3.2.2.1.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2">superscript</csymbol><ci id="A1.SS1.5.p3.18.m18.2.3.3.2.2.2.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2.2.2">ℋ</ci><ci id="A1.SS1.5.p3.18.m18.2.3.3.2.2.3.cmml" xref="A1.SS1.5.p3.18.m18.2.3.3.2.2.3">𝑝</ci></apply><list id="A1.SS1.5.p3.18.m18.2.2.2.3.cmml" xref="A1.SS1.5.p3.18.m18.2.2.2.4"><ci id="A1.SS1.5.p3.18.m18.1.1.1.1.cmml" xref="A1.SS1.5.p3.18.m18.1.1.1.1">𝑥</ci><ci id="A1.SS1.5.p3.18.m18.2.2.2.2.cmml" xref="A1.SS1.5.p3.18.m18.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.18.m18.2c">z\in\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.18.m18.2d">italic_z ∈ ∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Conversely, consider an arbitrary point <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.19.m19.1"><semantics id="A1.SS1.5.p3.19.m19.1a"><mi id="A1.SS1.5.p3.19.m19.1.1" xref="A1.SS1.5.p3.19.m19.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.19.m19.1b"><ci id="A1.SS1.5.p3.19.m19.1.1.cmml" xref="A1.SS1.5.p3.19.m19.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.19.m19.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.19.m19.1d">italic_z</annotation></semantics></math> with <math alttext="B^{\circ}_{z}\cap L\neq\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.20.m20.1"><semantics id="A1.SS1.5.p3.20.m20.1a"><mrow id="A1.SS1.5.p3.20.m20.1.1" xref="A1.SS1.5.p3.20.m20.1.1.cmml"><mrow id="A1.SS1.5.p3.20.m20.1.1.2" xref="A1.SS1.5.p3.20.m20.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.20.m20.1.1.2.2" xref="A1.SS1.5.p3.20.m20.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.20.m20.1.1.2.2.2.2" xref="A1.SS1.5.p3.20.m20.1.1.2.2.2.2.cmml">B</mi><mi id="A1.SS1.5.p3.20.m20.1.1.2.2.3" xref="A1.SS1.5.p3.20.m20.1.1.2.2.3.cmml">z</mi><mo id="A1.SS1.5.p3.20.m20.1.1.2.2.2.3" xref="A1.SS1.5.p3.20.m20.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.20.m20.1.1.2.1" xref="A1.SS1.5.p3.20.m20.1.1.2.1.cmml">∩</mo><mi id="A1.SS1.5.p3.20.m20.1.1.2.3" xref="A1.SS1.5.p3.20.m20.1.1.2.3.cmml">L</mi></mrow><mo id="A1.SS1.5.p3.20.m20.1.1.1" xref="A1.SS1.5.p3.20.m20.1.1.1.cmml">≠</mo><mi id="A1.SS1.5.p3.20.m20.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.20.m20.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.20.m20.1b"><apply id="A1.SS1.5.p3.20.m20.1.1.cmml" xref="A1.SS1.5.p3.20.m20.1.1"><neq id="A1.SS1.5.p3.20.m20.1.1.1.cmml" xref="A1.SS1.5.p3.20.m20.1.1.1"></neq><apply id="A1.SS1.5.p3.20.m20.1.1.2.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2"><intersect id="A1.SS1.5.p3.20.m20.1.1.2.1.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.20.m20.1.1.2.2.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.20.m20.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.20.m20.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.20.m20.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.20.m20.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.20.m20.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2.2.3"></compose></apply><ci id="A1.SS1.5.p3.20.m20.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.2.3">𝑧</ci></apply><ci id="A1.SS1.5.p3.20.m20.1.1.2.3.cmml" xref="A1.SS1.5.p3.20.m20.1.1.2.3">𝐿</ci></apply><emptyset id="A1.SS1.5.p3.20.m20.1.1.3.cmml" xref="A1.SS1.5.p3.20.m20.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.20.m20.1c">B^{\circ}_{z}\cap L\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.20.m20.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∩ italic_L ≠ ∅</annotation></semantics></math>. If <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.21.m21.1"><semantics id="A1.SS1.5.p3.21.m21.1a"><msubsup id="A1.SS1.5.p3.21.m21.1.1" xref="A1.SS1.5.p3.21.m21.1.1.cmml"><mi id="A1.SS1.5.p3.21.m21.1.1.2.2" xref="A1.SS1.5.p3.21.m21.1.1.2.2.cmml">B</mi><mi id="A1.SS1.5.p3.21.m21.1.1.3" xref="A1.SS1.5.p3.21.m21.1.1.3.cmml">z</mi><mo id="A1.SS1.5.p3.21.m21.1.1.2.3" xref="A1.SS1.5.p3.21.m21.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.21.m21.1b"><apply id="A1.SS1.5.p3.21.m21.1.1.cmml" xref="A1.SS1.5.p3.21.m21.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.21.m21.1.1.1.cmml" xref="A1.SS1.5.p3.21.m21.1.1">subscript</csymbol><apply id="A1.SS1.5.p3.21.m21.1.1.2.cmml" xref="A1.SS1.5.p3.21.m21.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.21.m21.1.1.2.1.cmml" xref="A1.SS1.5.p3.21.m21.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.21.m21.1.1.2.2.cmml" xref="A1.SS1.5.p3.21.m21.1.1.2.2">𝐵</ci><compose id="A1.SS1.5.p3.21.m21.1.1.2.3.cmml" xref="A1.SS1.5.p3.21.m21.1.1.2.3"></compose></apply><ci id="A1.SS1.5.p3.21.m21.1.1.3.cmml" xref="A1.SS1.5.p3.21.m21.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.21.m21.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.21.m21.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> intersects <math alttext="R_{-}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.22.m22.1"><semantics id="A1.SS1.5.p3.22.m22.1a"><msub id="A1.SS1.5.p3.22.m22.1.1" xref="A1.SS1.5.p3.22.m22.1.1.cmml"><mi id="A1.SS1.5.p3.22.m22.1.1.2" xref="A1.SS1.5.p3.22.m22.1.1.2.cmml">R</mi><mo id="A1.SS1.5.p3.22.m22.1.1.3" xref="A1.SS1.5.p3.22.m22.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.22.m22.1b"><apply id="A1.SS1.5.p3.22.m22.1.1.cmml" xref="A1.SS1.5.p3.22.m22.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.22.m22.1.1.1.cmml" xref="A1.SS1.5.p3.22.m22.1.1">subscript</csymbol><ci id="A1.SS1.5.p3.22.m22.1.1.2.cmml" xref="A1.SS1.5.p3.22.m22.1.1.2">𝑅</ci><minus id="A1.SS1.5.p3.22.m22.1.1.3.cmml" xref="A1.SS1.5.p3.22.m22.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.22.m22.1c">R_{-}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.22.m22.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math>, slightly moving <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.23.m23.1"><semantics id="A1.SS1.5.p3.23.m23.1a"><mi id="A1.SS1.5.p3.23.m23.1.1" xref="A1.SS1.5.p3.23.m23.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.23.m23.1b"><ci id="A1.SS1.5.p3.23.m23.1.1.cmml" xref="A1.SS1.5.p3.23.m23.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.23.m23.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.23.m23.1d">italic_z</annotation></semantics></math> does not change this. Concretely, any point <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.24.m24.1"><semantics id="A1.SS1.5.p3.24.m24.1a"><msup id="A1.SS1.5.p3.24.m24.1.1" xref="A1.SS1.5.p3.24.m24.1.1.cmml"><mi id="A1.SS1.5.p3.24.m24.1.1.2" xref="A1.SS1.5.p3.24.m24.1.1.2.cmml">z</mi><mo id="A1.SS1.5.p3.24.m24.1.1.3" xref="A1.SS1.5.p3.24.m24.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.24.m24.1b"><apply id="A1.SS1.5.p3.24.m24.1.1.cmml" xref="A1.SS1.5.p3.24.m24.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.24.m24.1.1.1.cmml" xref="A1.SS1.5.p3.24.m24.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.24.m24.1.1.2.cmml" xref="A1.SS1.5.p3.24.m24.1.1.2">𝑧</ci><ci id="A1.SS1.5.p3.24.m24.1.1.3.cmml" xref="A1.SS1.5.p3.24.m24.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.24.m24.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.24.m24.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in a small neighborhood around <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.25.m25.1"><semantics id="A1.SS1.5.p3.25.m25.1a"><mi id="A1.SS1.5.p3.25.m25.1.1" xref="A1.SS1.5.p3.25.m25.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.25.m25.1b"><ci id="A1.SS1.5.p3.25.m25.1.1.cmml" xref="A1.SS1.5.p3.25.m25.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.25.m25.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.25.m25.1d">italic_z</annotation></semantics></math> satisfies <math alttext="B^{\circ}_{z^{\prime}}\cap R_{-}\neq\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.26.m26.1"><semantics id="A1.SS1.5.p3.26.m26.1a"><mrow id="A1.SS1.5.p3.26.m26.1.1" xref="A1.SS1.5.p3.26.m26.1.1.cmml"><mrow id="A1.SS1.5.p3.26.m26.1.1.2" xref="A1.SS1.5.p3.26.m26.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.26.m26.1.1.2.2" xref="A1.SS1.5.p3.26.m26.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.26.m26.1.1.2.2.2.2" xref="A1.SS1.5.p3.26.m26.1.1.2.2.2.2.cmml">B</mi><msup id="A1.SS1.5.p3.26.m26.1.1.2.2.3" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3.cmml"><mi id="A1.SS1.5.p3.26.m26.1.1.2.2.3.2" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.26.m26.1.1.2.2.3.3" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3.3.cmml">′</mo></msup><mo id="A1.SS1.5.p3.26.m26.1.1.2.2.2.3" xref="A1.SS1.5.p3.26.m26.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.26.m26.1.1.2.1" xref="A1.SS1.5.p3.26.m26.1.1.2.1.cmml">∩</mo><msub id="A1.SS1.5.p3.26.m26.1.1.2.3" xref="A1.SS1.5.p3.26.m26.1.1.2.3.cmml"><mi id="A1.SS1.5.p3.26.m26.1.1.2.3.2" xref="A1.SS1.5.p3.26.m26.1.1.2.3.2.cmml">R</mi><mo id="A1.SS1.5.p3.26.m26.1.1.2.3.3" xref="A1.SS1.5.p3.26.m26.1.1.2.3.3.cmml">−</mo></msub></mrow><mo id="A1.SS1.5.p3.26.m26.1.1.1" xref="A1.SS1.5.p3.26.m26.1.1.1.cmml">≠</mo><mi id="A1.SS1.5.p3.26.m26.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.26.m26.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.26.m26.1b"><apply id="A1.SS1.5.p3.26.m26.1.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1"><neq id="A1.SS1.5.p3.26.m26.1.1.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.1"></neq><apply id="A1.SS1.5.p3.26.m26.1.1.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2"><intersect id="A1.SS1.5.p3.26.m26.1.1.2.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.26.m26.1.1.2.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.26.m26.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.26.m26.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.26.m26.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.26.m26.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.26.m26.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.2.3"></compose></apply><apply id="A1.SS1.5.p3.26.m26.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.26.m26.1.1.2.2.3.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.26.m26.1.1.2.2.3.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3.2">𝑧</ci><ci id="A1.SS1.5.p3.26.m26.1.1.2.2.3.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.2.3.3">′</ci></apply></apply><apply id="A1.SS1.5.p3.26.m26.1.1.2.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.26.m26.1.1.2.3.1.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.3">subscript</csymbol><ci id="A1.SS1.5.p3.26.m26.1.1.2.3.2.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.3.2">𝑅</ci><minus id="A1.SS1.5.p3.26.m26.1.1.2.3.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.2.3.3"></minus></apply></apply><emptyset id="A1.SS1.5.p3.26.m26.1.1.3.cmml" xref="A1.SS1.5.p3.26.m26.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.26.m26.1c">B^{\circ}_{z^{\prime}}\cap R_{-}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.26.m26.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∩ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math>, and thus none of these points is contained in the halfspace. We conclude that <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.27.m27.1"><semantics id="A1.SS1.5.p3.27.m27.1a"><mi id="A1.SS1.5.p3.27.m27.1.1" xref="A1.SS1.5.p3.27.m27.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.27.m27.1b"><ci id="A1.SS1.5.p3.27.m27.1.1.cmml" xref="A1.SS1.5.p3.27.m27.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.27.m27.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.27.m27.1d">italic_z</annotation></semantics></math> cannot be on the boundary. Otherwise, <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.28.m28.1"><semantics id="A1.SS1.5.p3.28.m28.1a"><msubsup id="A1.SS1.5.p3.28.m28.1.1" xref="A1.SS1.5.p3.28.m28.1.1.cmml"><mi id="A1.SS1.5.p3.28.m28.1.1.2.2" xref="A1.SS1.5.p3.28.m28.1.1.2.2.cmml">B</mi><mi id="A1.SS1.5.p3.28.m28.1.1.3" xref="A1.SS1.5.p3.28.m28.1.1.3.cmml">z</mi><mo id="A1.SS1.5.p3.28.m28.1.1.2.3" xref="A1.SS1.5.p3.28.m28.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.28.m28.1b"><apply id="A1.SS1.5.p3.28.m28.1.1.cmml" xref="A1.SS1.5.p3.28.m28.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.28.m28.1.1.1.cmml" xref="A1.SS1.5.p3.28.m28.1.1">subscript</csymbol><apply id="A1.SS1.5.p3.28.m28.1.1.2.cmml" xref="A1.SS1.5.p3.28.m28.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.28.m28.1.1.2.1.cmml" xref="A1.SS1.5.p3.28.m28.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.28.m28.1.1.2.2.cmml" xref="A1.SS1.5.p3.28.m28.1.1.2.2">𝐵</ci><compose id="A1.SS1.5.p3.28.m28.1.1.2.3.cmml" xref="A1.SS1.5.p3.28.m28.1.1.2.3"></compose></apply><ci id="A1.SS1.5.p3.28.m28.1.1.3.cmml" xref="A1.SS1.5.p3.28.m28.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.28.m28.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.28.m28.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> intersects <math alttext="R_{+}\coloneqq L\setminus(R_{-}\cup\{x\})" class="ltx_Math" display="inline" id="A1.SS1.5.p3.29.m29.2"><semantics id="A1.SS1.5.p3.29.m29.2a"><mrow id="A1.SS1.5.p3.29.m29.2.2" xref="A1.SS1.5.p3.29.m29.2.2.cmml"><msub id="A1.SS1.5.p3.29.m29.2.2.3" xref="A1.SS1.5.p3.29.m29.2.2.3.cmml"><mi id="A1.SS1.5.p3.29.m29.2.2.3.2" xref="A1.SS1.5.p3.29.m29.2.2.3.2.cmml">R</mi><mo id="A1.SS1.5.p3.29.m29.2.2.3.3" xref="A1.SS1.5.p3.29.m29.2.2.3.3.cmml">+</mo></msub><mo id="A1.SS1.5.p3.29.m29.2.2.2" xref="A1.SS1.5.p3.29.m29.2.2.2.cmml">≔</mo><mrow id="A1.SS1.5.p3.29.m29.2.2.1" xref="A1.SS1.5.p3.29.m29.2.2.1.cmml"><mi id="A1.SS1.5.p3.29.m29.2.2.1.3" xref="A1.SS1.5.p3.29.m29.2.2.1.3.cmml">L</mi><mo id="A1.SS1.5.p3.29.m29.2.2.1.2" xref="A1.SS1.5.p3.29.m29.2.2.1.2.cmml">∖</mo><mrow id="A1.SS1.5.p3.29.m29.2.2.1.1.1" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.cmml"><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.2" stretchy="false" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.cmml"><msub id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.cmml"><mi id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.2" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.2.cmml">R</mi><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.3" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.3.cmml">−</mo></msub><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.1" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.1.cmml">∪</mo><mrow id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.2" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.1.cmml"><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.2.1" stretchy="false" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.1.cmml">{</mo><mi id="A1.SS1.5.p3.29.m29.1.1" xref="A1.SS1.5.p3.29.m29.1.1.cmml">x</mi><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.2.2" stretchy="false" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="A1.SS1.5.p3.29.m29.2.2.1.1.1.3" stretchy="false" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.29.m29.2b"><apply id="A1.SS1.5.p3.29.m29.2.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2"><ci id="A1.SS1.5.p3.29.m29.2.2.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2.2">≔</ci><apply id="A1.SS1.5.p3.29.m29.2.2.3.cmml" xref="A1.SS1.5.p3.29.m29.2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.29.m29.2.2.3.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.3">subscript</csymbol><ci id="A1.SS1.5.p3.29.m29.2.2.3.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2.3.2">𝑅</ci><plus id="A1.SS1.5.p3.29.m29.2.2.3.3.cmml" xref="A1.SS1.5.p3.29.m29.2.2.3.3"></plus></apply><apply id="A1.SS1.5.p3.29.m29.2.2.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1"><setdiff id="A1.SS1.5.p3.29.m29.2.2.1.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.2"></setdiff><ci id="A1.SS1.5.p3.29.m29.2.2.1.3.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.3">𝐿</ci><apply id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1"><union id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.1"></union><apply id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2">subscript</csymbol><ci id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.2.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.2">𝑅</ci><minus id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.3.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.2.3"></minus></apply><set id="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.1.cmml" xref="A1.SS1.5.p3.29.m29.2.2.1.1.1.1.3.2"><ci id="A1.SS1.5.p3.29.m29.1.1.cmml" xref="A1.SS1.5.p3.29.m29.1.1">𝑥</ci></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.29.m29.2c">R_{+}\coloneqq L\setminus(R_{-}\cup\{x\})</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.29.m29.2d">italic_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ≔ italic_L ∖ ( italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∪ { italic_x } )</annotation></semantics></math>, but then slightly moving <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.30.m30.1"><semantics id="A1.SS1.5.p3.30.m30.1a"><mi id="A1.SS1.5.p3.30.m30.1.1" xref="A1.SS1.5.p3.30.m30.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.30.m30.1b"><ci id="A1.SS1.5.p3.30.m30.1.1.cmml" xref="A1.SS1.5.p3.30.m30.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.30.m30.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.30.m30.1d">italic_z</annotation></semantics></math> does not change this. In other words, any point <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.31.m31.1"><semantics id="A1.SS1.5.p3.31.m31.1a"><msup id="A1.SS1.5.p3.31.m31.1.1" xref="A1.SS1.5.p3.31.m31.1.1.cmml"><mi id="A1.SS1.5.p3.31.m31.1.1.2" xref="A1.SS1.5.p3.31.m31.1.1.2.cmml">z</mi><mo id="A1.SS1.5.p3.31.m31.1.1.3" xref="A1.SS1.5.p3.31.m31.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.31.m31.1b"><apply id="A1.SS1.5.p3.31.m31.1.1.cmml" xref="A1.SS1.5.p3.31.m31.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.31.m31.1.1.1.cmml" xref="A1.SS1.5.p3.31.m31.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.31.m31.1.1.2.cmml" xref="A1.SS1.5.p3.31.m31.1.1.2">𝑧</ci><ci id="A1.SS1.5.p3.31.m31.1.1.3.cmml" xref="A1.SS1.5.p3.31.m31.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.31.m31.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.31.m31.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in a small neighborhood around <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.32.m32.1"><semantics id="A1.SS1.5.p3.32.m32.1a"><mi id="A1.SS1.5.p3.32.m32.1.1" xref="A1.SS1.5.p3.32.m32.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.32.m32.1b"><ci id="A1.SS1.5.p3.32.m32.1.1.cmml" xref="A1.SS1.5.p3.32.m32.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.32.m32.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.32.m32.1d">italic_z</annotation></semantics></math> satisfies <math alttext="B^{\circ}_{z^{\prime}}\cap R_{+}\neq\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.33.m33.1"><semantics id="A1.SS1.5.p3.33.m33.1a"><mrow id="A1.SS1.5.p3.33.m33.1.1" xref="A1.SS1.5.p3.33.m33.1.1.cmml"><mrow id="A1.SS1.5.p3.33.m33.1.1.2" xref="A1.SS1.5.p3.33.m33.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.33.m33.1.1.2.2" xref="A1.SS1.5.p3.33.m33.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.33.m33.1.1.2.2.2.2" xref="A1.SS1.5.p3.33.m33.1.1.2.2.2.2.cmml">B</mi><msup id="A1.SS1.5.p3.33.m33.1.1.2.2.3" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3.cmml"><mi id="A1.SS1.5.p3.33.m33.1.1.2.2.3.2" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.33.m33.1.1.2.2.3.3" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3.3.cmml">′</mo></msup><mo id="A1.SS1.5.p3.33.m33.1.1.2.2.2.3" xref="A1.SS1.5.p3.33.m33.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.33.m33.1.1.2.1" xref="A1.SS1.5.p3.33.m33.1.1.2.1.cmml">∩</mo><msub id="A1.SS1.5.p3.33.m33.1.1.2.3" xref="A1.SS1.5.p3.33.m33.1.1.2.3.cmml"><mi id="A1.SS1.5.p3.33.m33.1.1.2.3.2" xref="A1.SS1.5.p3.33.m33.1.1.2.3.2.cmml">R</mi><mo id="A1.SS1.5.p3.33.m33.1.1.2.3.3" xref="A1.SS1.5.p3.33.m33.1.1.2.3.3.cmml">+</mo></msub></mrow><mo id="A1.SS1.5.p3.33.m33.1.1.1" xref="A1.SS1.5.p3.33.m33.1.1.1.cmml">≠</mo><mi id="A1.SS1.5.p3.33.m33.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.33.m33.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.33.m33.1b"><apply id="A1.SS1.5.p3.33.m33.1.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1"><neq id="A1.SS1.5.p3.33.m33.1.1.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.1"></neq><apply id="A1.SS1.5.p3.33.m33.1.1.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2"><intersect id="A1.SS1.5.p3.33.m33.1.1.2.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.33.m33.1.1.2.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.33.m33.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.33.m33.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.33.m33.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.33.m33.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.33.m33.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.2.3"></compose></apply><apply id="A1.SS1.5.p3.33.m33.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.33.m33.1.1.2.2.3.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.33.m33.1.1.2.2.3.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3.2">𝑧</ci><ci id="A1.SS1.5.p3.33.m33.1.1.2.2.3.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.2.3.3">′</ci></apply></apply><apply id="A1.SS1.5.p3.33.m33.1.1.2.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.33.m33.1.1.2.3.1.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.3">subscript</csymbol><ci id="A1.SS1.5.p3.33.m33.1.1.2.3.2.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.3.2">𝑅</ci><plus id="A1.SS1.5.p3.33.m33.1.1.2.3.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.2.3.3"></plus></apply></apply><emptyset id="A1.SS1.5.p3.33.m33.1.1.3.cmml" xref="A1.SS1.5.p3.33.m33.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.33.m33.1c">B^{\circ}_{z^{\prime}}\cap R_{+}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.33.m33.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∩ italic_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math>, which also implies <math alttext="B^{\circ}_{z^{\prime}}\cap R_{-}=\varnothing" class="ltx_Math" display="inline" id="A1.SS1.5.p3.34.m34.1"><semantics id="A1.SS1.5.p3.34.m34.1a"><mrow id="A1.SS1.5.p3.34.m34.1.1" xref="A1.SS1.5.p3.34.m34.1.1.cmml"><mrow id="A1.SS1.5.p3.34.m34.1.1.2" xref="A1.SS1.5.p3.34.m34.1.1.2.cmml"><msubsup id="A1.SS1.5.p3.34.m34.1.1.2.2" xref="A1.SS1.5.p3.34.m34.1.1.2.2.cmml"><mi id="A1.SS1.5.p3.34.m34.1.1.2.2.2.2" xref="A1.SS1.5.p3.34.m34.1.1.2.2.2.2.cmml">B</mi><msup id="A1.SS1.5.p3.34.m34.1.1.2.2.3" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3.cmml"><mi id="A1.SS1.5.p3.34.m34.1.1.2.2.3.2" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.34.m34.1.1.2.2.3.3" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3.3.cmml">′</mo></msup><mo id="A1.SS1.5.p3.34.m34.1.1.2.2.2.3" xref="A1.SS1.5.p3.34.m34.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.5.p3.34.m34.1.1.2.1" xref="A1.SS1.5.p3.34.m34.1.1.2.1.cmml">∩</mo><msub id="A1.SS1.5.p3.34.m34.1.1.2.3" xref="A1.SS1.5.p3.34.m34.1.1.2.3.cmml"><mi id="A1.SS1.5.p3.34.m34.1.1.2.3.2" xref="A1.SS1.5.p3.34.m34.1.1.2.3.2.cmml">R</mi><mo id="A1.SS1.5.p3.34.m34.1.1.2.3.3" xref="A1.SS1.5.p3.34.m34.1.1.2.3.3.cmml">−</mo></msub></mrow><mo id="A1.SS1.5.p3.34.m34.1.1.1" xref="A1.SS1.5.p3.34.m34.1.1.1.cmml">=</mo><mi id="A1.SS1.5.p3.34.m34.1.1.3" mathvariant="normal" xref="A1.SS1.5.p3.34.m34.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.34.m34.1b"><apply id="A1.SS1.5.p3.34.m34.1.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1"><eq id="A1.SS1.5.p3.34.m34.1.1.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.1"></eq><apply id="A1.SS1.5.p3.34.m34.1.1.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2"><intersect id="A1.SS1.5.p3.34.m34.1.1.2.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.1"></intersect><apply id="A1.SS1.5.p3.34.m34.1.1.2.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.34.m34.1.1.2.2.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2">subscript</csymbol><apply id="A1.SS1.5.p3.34.m34.1.1.2.2.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.5.p3.34.m34.1.1.2.2.2.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2">superscript</csymbol><ci id="A1.SS1.5.p3.34.m34.1.1.2.2.2.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.5.p3.34.m34.1.1.2.2.2.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.2.3"></compose></apply><apply id="A1.SS1.5.p3.34.m34.1.1.2.2.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.34.m34.1.1.2.2.3.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3">superscript</csymbol><ci id="A1.SS1.5.p3.34.m34.1.1.2.2.3.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3.2">𝑧</ci><ci id="A1.SS1.5.p3.34.m34.1.1.2.2.3.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.2.3.3">′</ci></apply></apply><apply id="A1.SS1.5.p3.34.m34.1.1.2.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.34.m34.1.1.2.3.1.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.3">subscript</csymbol><ci id="A1.SS1.5.p3.34.m34.1.1.2.3.2.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.3.2">𝑅</ci><minus id="A1.SS1.5.p3.34.m34.1.1.2.3.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.2.3.3"></minus></apply></apply><emptyset id="A1.SS1.5.p3.34.m34.1.1.3.cmml" xref="A1.SS1.5.p3.34.m34.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.34.m34.1c">B^{\circ}_{z^{\prime}}\cap R_{-}=\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.34.m34.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∩ italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT = ∅</annotation></semantics></math> (as otherwise, <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.5.p3.35.m35.1"><semantics id="A1.SS1.5.p3.35.m35.1a"><mi id="A1.SS1.5.p3.35.m35.1.1" xref="A1.SS1.5.p3.35.m35.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.35.m35.1b"><ci id="A1.SS1.5.p3.35.m35.1.1.cmml" xref="A1.SS1.5.p3.35.m35.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.35.m35.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.35.m35.1d">italic_x</annotation></semantics></math> would be contained in <math alttext="B^{\circ}_{z^{\prime}}" class="ltx_Math" display="inline" id="A1.SS1.5.p3.36.m36.1"><semantics id="A1.SS1.5.p3.36.m36.1a"><msubsup id="A1.SS1.5.p3.36.m36.1.1" xref="A1.SS1.5.p3.36.m36.1.1.cmml"><mi id="A1.SS1.5.p3.36.m36.1.1.2.2" xref="A1.SS1.5.p3.36.m36.1.1.2.2.cmml">B</mi><msup id="A1.SS1.5.p3.36.m36.1.1.3" xref="A1.SS1.5.p3.36.m36.1.1.3.cmml"><mi id="A1.SS1.5.p3.36.m36.1.1.3.2" xref="A1.SS1.5.p3.36.m36.1.1.3.2.cmml">z</mi><mo id="A1.SS1.5.p3.36.m36.1.1.3.3" xref="A1.SS1.5.p3.36.m36.1.1.3.3.cmml">′</mo></msup><mo id="A1.SS1.5.p3.36.m36.1.1.2.3" xref="A1.SS1.5.p3.36.m36.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.36.m36.1b"><apply id="A1.SS1.5.p3.36.m36.1.1.cmml" xref="A1.SS1.5.p3.36.m36.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.36.m36.1.1.1.cmml" xref="A1.SS1.5.p3.36.m36.1.1">subscript</csymbol><apply id="A1.SS1.5.p3.36.m36.1.1.2.cmml" xref="A1.SS1.5.p3.36.m36.1.1"><csymbol cd="ambiguous" id="A1.SS1.5.p3.36.m36.1.1.2.1.cmml" xref="A1.SS1.5.p3.36.m36.1.1">superscript</csymbol><ci id="A1.SS1.5.p3.36.m36.1.1.2.2.cmml" xref="A1.SS1.5.p3.36.m36.1.1.2.2">𝐵</ci><compose id="A1.SS1.5.p3.36.m36.1.1.2.3.cmml" xref="A1.SS1.5.p3.36.m36.1.1.2.3"></compose></apply><apply id="A1.SS1.5.p3.36.m36.1.1.3.cmml" xref="A1.SS1.5.p3.36.m36.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.5.p3.36.m36.1.1.3.1.cmml" xref="A1.SS1.5.p3.36.m36.1.1.3">superscript</csymbol><ci id="A1.SS1.5.p3.36.m36.1.1.3.2.cmml" xref="A1.SS1.5.p3.36.m36.1.1.3.2">𝑧</ci><ci id="A1.SS1.5.p3.36.m36.1.1.3.3.cmml" xref="A1.SS1.5.p3.36.m36.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.36.m36.1c">B^{\circ}_{z^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.36.m36.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> by convexity). We conclude that <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.5.p3.37.m37.1"><semantics id="A1.SS1.5.p3.37.m37.1a"><mi id="A1.SS1.5.p3.37.m37.1.1" xref="A1.SS1.5.p3.37.m37.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.5.p3.37.m37.1b"><ci id="A1.SS1.5.p3.37.m37.1.1.cmml" xref="A1.SS1.5.p3.37.m37.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.5.p3.37.m37.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.5.p3.37.m37.1d">italic_z</annotation></semantics></math> is not on the boundary.</p> </div> <div class="ltx_para" id="A1.SS1.6.p4"> <p class="ltx_p" id="A1.SS1.6.p4.12">With this characterization of the boundary points, we can now finish the proof. Consider an arbitrary <math alttext="z\in\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.1.m1.2"><semantics id="A1.SS1.6.p4.1.m1.2a"><mrow id="A1.SS1.6.p4.1.m1.2.3" xref="A1.SS1.6.p4.1.m1.2.3.cmml"><mi id="A1.SS1.6.p4.1.m1.2.3.2" xref="A1.SS1.6.p4.1.m1.2.3.2.cmml">z</mi><mo id="A1.SS1.6.p4.1.m1.2.3.1" rspace="0.1389em" xref="A1.SS1.6.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="A1.SS1.6.p4.1.m1.2.3.3" xref="A1.SS1.6.p4.1.m1.2.3.3.cmml"><mo id="A1.SS1.6.p4.1.m1.2.3.3.1" lspace="0.1389em" rspace="0em" xref="A1.SS1.6.p4.1.m1.2.3.3.1.cmml">∂</mo><msubsup id="A1.SS1.6.p4.1.m1.2.3.3.2" xref="A1.SS1.6.p4.1.m1.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.6.p4.1.m1.2.3.3.2.2.2" xref="A1.SS1.6.p4.1.m1.2.3.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS1.6.p4.1.m1.2.2.2.4" xref="A1.SS1.6.p4.1.m1.2.2.2.3.cmml"><mi id="A1.SS1.6.p4.1.m1.1.1.1.1" xref="A1.SS1.6.p4.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS1.6.p4.1.m1.2.2.2.4.1" xref="A1.SS1.6.p4.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS1.6.p4.1.m1.2.2.2.2" xref="A1.SS1.6.p4.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.6.p4.1.m1.2.3.3.2.2.3" xref="A1.SS1.6.p4.1.m1.2.3.3.2.2.3.cmml">p</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.1.m1.2b"><apply id="A1.SS1.6.p4.1.m1.2.3.cmml" xref="A1.SS1.6.p4.1.m1.2.3"><in id="A1.SS1.6.p4.1.m1.2.3.1.cmml" xref="A1.SS1.6.p4.1.m1.2.3.1"></in><ci id="A1.SS1.6.p4.1.m1.2.3.2.cmml" xref="A1.SS1.6.p4.1.m1.2.3.2">𝑧</ci><apply id="A1.SS1.6.p4.1.m1.2.3.3.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3"><partialdiff id="A1.SS1.6.p4.1.m1.2.3.3.1.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.1"></partialdiff><apply id="A1.SS1.6.p4.1.m1.2.3.3.2.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.1.m1.2.3.3.2.1.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2">subscript</csymbol><apply id="A1.SS1.6.p4.1.m1.2.3.3.2.2.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.1.m1.2.3.3.2.2.1.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2">superscript</csymbol><ci id="A1.SS1.6.p4.1.m1.2.3.3.2.2.2.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2.2.2">ℋ</ci><ci id="A1.SS1.6.p4.1.m1.2.3.3.2.2.3.cmml" xref="A1.SS1.6.p4.1.m1.2.3.3.2.2.3">𝑝</ci></apply><list id="A1.SS1.6.p4.1.m1.2.2.2.3.cmml" xref="A1.SS1.6.p4.1.m1.2.2.2.4"><ci id="A1.SS1.6.p4.1.m1.1.1.1.1.cmml" xref="A1.SS1.6.p4.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS1.6.p4.1.m1.2.2.2.2.cmml" xref="A1.SS1.6.p4.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.1.m1.2c">z\in\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.1.m1.2d">italic_z ∈ ∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. We must have <math alttext="B^{\circ}_{z}\cap L=\varnothing" class="ltx_Math" display="inline" id="A1.SS1.6.p4.2.m2.1"><semantics id="A1.SS1.6.p4.2.m2.1a"><mrow id="A1.SS1.6.p4.2.m2.1.1" xref="A1.SS1.6.p4.2.m2.1.1.cmml"><mrow id="A1.SS1.6.p4.2.m2.1.1.2" xref="A1.SS1.6.p4.2.m2.1.1.2.cmml"><msubsup id="A1.SS1.6.p4.2.m2.1.1.2.2" xref="A1.SS1.6.p4.2.m2.1.1.2.2.cmml"><mi id="A1.SS1.6.p4.2.m2.1.1.2.2.2.2" xref="A1.SS1.6.p4.2.m2.1.1.2.2.2.2.cmml">B</mi><mi id="A1.SS1.6.p4.2.m2.1.1.2.2.3" xref="A1.SS1.6.p4.2.m2.1.1.2.2.3.cmml">z</mi><mo id="A1.SS1.6.p4.2.m2.1.1.2.2.2.3" xref="A1.SS1.6.p4.2.m2.1.1.2.2.2.3.cmml">∘</mo></msubsup><mo id="A1.SS1.6.p4.2.m2.1.1.2.1" xref="A1.SS1.6.p4.2.m2.1.1.2.1.cmml">∩</mo><mi id="A1.SS1.6.p4.2.m2.1.1.2.3" xref="A1.SS1.6.p4.2.m2.1.1.2.3.cmml">L</mi></mrow><mo id="A1.SS1.6.p4.2.m2.1.1.1" xref="A1.SS1.6.p4.2.m2.1.1.1.cmml">=</mo><mi id="A1.SS1.6.p4.2.m2.1.1.3" mathvariant="normal" xref="A1.SS1.6.p4.2.m2.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.2.m2.1b"><apply id="A1.SS1.6.p4.2.m2.1.1.cmml" xref="A1.SS1.6.p4.2.m2.1.1"><eq id="A1.SS1.6.p4.2.m2.1.1.1.cmml" xref="A1.SS1.6.p4.2.m2.1.1.1"></eq><apply id="A1.SS1.6.p4.2.m2.1.1.2.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2"><intersect id="A1.SS1.6.p4.2.m2.1.1.2.1.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.1"></intersect><apply id="A1.SS1.6.p4.2.m2.1.1.2.2.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.2.m2.1.1.2.2.1.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2">subscript</csymbol><apply id="A1.SS1.6.p4.2.m2.1.1.2.2.2.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.2.m2.1.1.2.2.2.1.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2">superscript</csymbol><ci id="A1.SS1.6.p4.2.m2.1.1.2.2.2.2.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2.2.2">𝐵</ci><compose id="A1.SS1.6.p4.2.m2.1.1.2.2.2.3.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2.2.3"></compose></apply><ci id="A1.SS1.6.p4.2.m2.1.1.2.2.3.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.2.3">𝑧</ci></apply><ci id="A1.SS1.6.p4.2.m2.1.1.2.3.cmml" xref="A1.SS1.6.p4.2.m2.1.1.2.3">𝐿</ci></apply><emptyset id="A1.SS1.6.p4.2.m2.1.1.3.cmml" xref="A1.SS1.6.p4.2.m2.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.2.m2.1c">B^{\circ}_{z}\cap L=\varnothing</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.2.m2.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∩ italic_L = ∅</annotation></semantics></math>. In other words, <math alttext="L" class="ltx_Math" display="inline" id="A1.SS1.6.p4.3.m3.1"><semantics id="A1.SS1.6.p4.3.m3.1a"><mi id="A1.SS1.6.p4.3.m3.1.1" xref="A1.SS1.6.p4.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.3.m3.1b"><ci id="A1.SS1.6.p4.3.m3.1.1.cmml" xref="A1.SS1.6.p4.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.3.m3.1d">italic_L</annotation></semantics></math> is tangential to <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.4.m4.1"><semantics id="A1.SS1.6.p4.4.m4.1a"><msub id="A1.SS1.6.p4.4.m4.1.1" xref="A1.SS1.6.p4.4.m4.1.1.cmml"><mi id="A1.SS1.6.p4.4.m4.1.1.2" xref="A1.SS1.6.p4.4.m4.1.1.2.cmml">B</mi><mi id="A1.SS1.6.p4.4.m4.1.1.3" xref="A1.SS1.6.p4.4.m4.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.4.m4.1b"><apply id="A1.SS1.6.p4.4.m4.1.1.cmml" xref="A1.SS1.6.p4.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS1.6.p4.4.m4.1.1.1.cmml" xref="A1.SS1.6.p4.4.m4.1.1">subscript</csymbol><ci id="A1.SS1.6.p4.4.m4.1.1.2.cmml" xref="A1.SS1.6.p4.4.m4.1.1.2">𝐵</ci><ci id="A1.SS1.6.p4.4.m4.1.1.3.cmml" xref="A1.SS1.6.p4.4.m4.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.4.m4.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.4.m4.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>. This remains true for any other point <math alttext="z^{\prime}=x+\delta(z-x)" class="ltx_Math" display="inline" id="A1.SS1.6.p4.5.m5.1"><semantics id="A1.SS1.6.p4.5.m5.1a"><mrow id="A1.SS1.6.p4.5.m5.1.1" xref="A1.SS1.6.p4.5.m5.1.1.cmml"><msup id="A1.SS1.6.p4.5.m5.1.1.3" xref="A1.SS1.6.p4.5.m5.1.1.3.cmml"><mi id="A1.SS1.6.p4.5.m5.1.1.3.2" xref="A1.SS1.6.p4.5.m5.1.1.3.2.cmml">z</mi><mo id="A1.SS1.6.p4.5.m5.1.1.3.3" xref="A1.SS1.6.p4.5.m5.1.1.3.3.cmml">′</mo></msup><mo id="A1.SS1.6.p4.5.m5.1.1.2" xref="A1.SS1.6.p4.5.m5.1.1.2.cmml">=</mo><mrow id="A1.SS1.6.p4.5.m5.1.1.1" xref="A1.SS1.6.p4.5.m5.1.1.1.cmml"><mi id="A1.SS1.6.p4.5.m5.1.1.1.3" xref="A1.SS1.6.p4.5.m5.1.1.1.3.cmml">x</mi><mo id="A1.SS1.6.p4.5.m5.1.1.1.2" xref="A1.SS1.6.p4.5.m5.1.1.1.2.cmml">+</mo><mrow id="A1.SS1.6.p4.5.m5.1.1.1.1" xref="A1.SS1.6.p4.5.m5.1.1.1.1.cmml"><mi id="A1.SS1.6.p4.5.m5.1.1.1.1.3" xref="A1.SS1.6.p4.5.m5.1.1.1.1.3.cmml">δ</mi><mo id="A1.SS1.6.p4.5.m5.1.1.1.1.2" xref="A1.SS1.6.p4.5.m5.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.cmml"><mo id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.2" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.1" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.3" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.5.m5.1b"><apply id="A1.SS1.6.p4.5.m5.1.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1"><eq id="A1.SS1.6.p4.5.m5.1.1.2.cmml" xref="A1.SS1.6.p4.5.m5.1.1.2"></eq><apply id="A1.SS1.6.p4.5.m5.1.1.3.cmml" xref="A1.SS1.6.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.6.p4.5.m5.1.1.3.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1.3">superscript</csymbol><ci id="A1.SS1.6.p4.5.m5.1.1.3.2.cmml" xref="A1.SS1.6.p4.5.m5.1.1.3.2">𝑧</ci><ci id="A1.SS1.6.p4.5.m5.1.1.3.3.cmml" xref="A1.SS1.6.p4.5.m5.1.1.3.3">′</ci></apply><apply id="A1.SS1.6.p4.5.m5.1.1.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1"><plus id="A1.SS1.6.p4.5.m5.1.1.1.2.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.2"></plus><ci id="A1.SS1.6.p4.5.m5.1.1.1.3.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.3">𝑥</ci><apply id="A1.SS1.6.p4.5.m5.1.1.1.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1"><times id="A1.SS1.6.p4.5.m5.1.1.1.1.2.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.2"></times><ci id="A1.SS1.6.p4.5.m5.1.1.1.1.3.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.3">𝛿</ci><apply id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1"><minus id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.6.p4.5.m5.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.5.m5.1c">z^{\prime}=x+\delta(z-x)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.5.m5.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x + italic_δ ( italic_z - italic_x )</annotation></semantics></math> for <math alttext="\delta\in\mathbb{R}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.6.m6.1"><semantics id="A1.SS1.6.p4.6.m6.1a"><mrow id="A1.SS1.6.p4.6.m6.1.1" xref="A1.SS1.6.p4.6.m6.1.1.cmml"><mi id="A1.SS1.6.p4.6.m6.1.1.2" xref="A1.SS1.6.p4.6.m6.1.1.2.cmml">δ</mi><mo id="A1.SS1.6.p4.6.m6.1.1.1" xref="A1.SS1.6.p4.6.m6.1.1.1.cmml">∈</mo><mi id="A1.SS1.6.p4.6.m6.1.1.3" xref="A1.SS1.6.p4.6.m6.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.6.m6.1b"><apply id="A1.SS1.6.p4.6.m6.1.1.cmml" xref="A1.SS1.6.p4.6.m6.1.1"><in id="A1.SS1.6.p4.6.m6.1.1.1.cmml" xref="A1.SS1.6.p4.6.m6.1.1.1"></in><ci id="A1.SS1.6.p4.6.m6.1.1.2.cmml" xref="A1.SS1.6.p4.6.m6.1.1.2">𝛿</ci><ci id="A1.SS1.6.p4.6.m6.1.1.3.cmml" xref="A1.SS1.6.p4.6.m6.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.6.m6.1c">\delta\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.6.m6.1d">italic_δ ∈ blackboard_R</annotation></semantics></math> on the line through <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.6.p4.7.m7.1"><semantics id="A1.SS1.6.p4.7.m7.1a"><mi id="A1.SS1.6.p4.7.m7.1.1" xref="A1.SS1.6.p4.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.7.m7.1b"><ci id="A1.SS1.6.p4.7.m7.1.1.cmml" xref="A1.SS1.6.p4.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.7.m7.1d">italic_x</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.6.p4.8.m8.1"><semantics id="A1.SS1.6.p4.8.m8.1a"><mi id="A1.SS1.6.p4.8.m8.1.1" xref="A1.SS1.6.p4.8.m8.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.8.m8.1b"><ci id="A1.SS1.6.p4.8.m8.1.1.cmml" xref="A1.SS1.6.p4.8.m8.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.8.m8.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.8.m8.1d">italic_z</annotation></semantics></math> with ball <math alttext="B_{z^{\prime}}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.9.m9.1"><semantics id="A1.SS1.6.p4.9.m9.1a"><msub id="A1.SS1.6.p4.9.m9.1.1" xref="A1.SS1.6.p4.9.m9.1.1.cmml"><mi id="A1.SS1.6.p4.9.m9.1.1.2" xref="A1.SS1.6.p4.9.m9.1.1.2.cmml">B</mi><msup id="A1.SS1.6.p4.9.m9.1.1.3" xref="A1.SS1.6.p4.9.m9.1.1.3.cmml"><mi id="A1.SS1.6.p4.9.m9.1.1.3.2" xref="A1.SS1.6.p4.9.m9.1.1.3.2.cmml">z</mi><mo id="A1.SS1.6.p4.9.m9.1.1.3.3" xref="A1.SS1.6.p4.9.m9.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.9.m9.1b"><apply id="A1.SS1.6.p4.9.m9.1.1.cmml" xref="A1.SS1.6.p4.9.m9.1.1"><csymbol cd="ambiguous" id="A1.SS1.6.p4.9.m9.1.1.1.cmml" xref="A1.SS1.6.p4.9.m9.1.1">subscript</csymbol><ci id="A1.SS1.6.p4.9.m9.1.1.2.cmml" xref="A1.SS1.6.p4.9.m9.1.1.2">𝐵</ci><apply id="A1.SS1.6.p4.9.m9.1.1.3.cmml" xref="A1.SS1.6.p4.9.m9.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.6.p4.9.m9.1.1.3.1.cmml" xref="A1.SS1.6.p4.9.m9.1.1.3">superscript</csymbol><ci id="A1.SS1.6.p4.9.m9.1.1.3.2.cmml" xref="A1.SS1.6.p4.9.m9.1.1.3.2">𝑧</ci><ci id="A1.SS1.6.p4.9.m9.1.1.3.3.cmml" xref="A1.SS1.6.p4.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.9.m9.1c">B_{z^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.9.m9.1d">italic_B start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (using point-symmetry of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.10.m10.1"><semantics id="A1.SS1.6.p4.10.m10.1a"><msub id="A1.SS1.6.p4.10.m10.1.1" xref="A1.SS1.6.p4.10.m10.1.1.cmml"><mi id="A1.SS1.6.p4.10.m10.1.1.2" mathvariant="normal" xref="A1.SS1.6.p4.10.m10.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.6.p4.10.m10.1.1.3" xref="A1.SS1.6.p4.10.m10.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.10.m10.1b"><apply id="A1.SS1.6.p4.10.m10.1.1.cmml" xref="A1.SS1.6.p4.10.m10.1.1"><csymbol cd="ambiguous" id="A1.SS1.6.p4.10.m10.1.1.1.cmml" xref="A1.SS1.6.p4.10.m10.1.1">subscript</csymbol><ci id="A1.SS1.6.p4.10.m10.1.1.2.cmml" xref="A1.SS1.6.p4.10.m10.1.1.2">ℓ</ci><ci id="A1.SS1.6.p4.10.m10.1.1.3.cmml" xref="A1.SS1.6.p4.10.m10.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.10.m10.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.10.m10.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-balls around their center for <math alttext="\delta<0" class="ltx_Math" display="inline" id="A1.SS1.6.p4.11.m11.1"><semantics id="A1.SS1.6.p4.11.m11.1a"><mrow id="A1.SS1.6.p4.11.m11.1.1" xref="A1.SS1.6.p4.11.m11.1.1.cmml"><mi id="A1.SS1.6.p4.11.m11.1.1.2" xref="A1.SS1.6.p4.11.m11.1.1.2.cmml">δ</mi><mo id="A1.SS1.6.p4.11.m11.1.1.1" xref="A1.SS1.6.p4.11.m11.1.1.1.cmml"><</mo><mn id="A1.SS1.6.p4.11.m11.1.1.3" xref="A1.SS1.6.p4.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.11.m11.1b"><apply id="A1.SS1.6.p4.11.m11.1.1.cmml" xref="A1.SS1.6.p4.11.m11.1.1"><lt id="A1.SS1.6.p4.11.m11.1.1.1.cmml" xref="A1.SS1.6.p4.11.m11.1.1.1"></lt><ci id="A1.SS1.6.p4.11.m11.1.1.2.cmml" xref="A1.SS1.6.p4.11.m11.1.1.2">𝛿</ci><cn id="A1.SS1.6.p4.11.m11.1.1.3.cmml" type="integer" xref="A1.SS1.6.p4.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.11.m11.1c">\delta<0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.11.m11.1d">italic_δ < 0</annotation></semantics></math>). This again implies <math alttext="z^{\prime}\in\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.6.p4.12.m12.2"><semantics id="A1.SS1.6.p4.12.m12.2a"><mrow id="A1.SS1.6.p4.12.m12.2.3" xref="A1.SS1.6.p4.12.m12.2.3.cmml"><msup id="A1.SS1.6.p4.12.m12.2.3.2" xref="A1.SS1.6.p4.12.m12.2.3.2.cmml"><mi id="A1.SS1.6.p4.12.m12.2.3.2.2" xref="A1.SS1.6.p4.12.m12.2.3.2.2.cmml">z</mi><mo id="A1.SS1.6.p4.12.m12.2.3.2.3" xref="A1.SS1.6.p4.12.m12.2.3.2.3.cmml">′</mo></msup><mo id="A1.SS1.6.p4.12.m12.2.3.1" rspace="0.1389em" xref="A1.SS1.6.p4.12.m12.2.3.1.cmml">∈</mo><mrow id="A1.SS1.6.p4.12.m12.2.3.3" xref="A1.SS1.6.p4.12.m12.2.3.3.cmml"><mo id="A1.SS1.6.p4.12.m12.2.3.3.1" lspace="0.1389em" rspace="0em" xref="A1.SS1.6.p4.12.m12.2.3.3.1.cmml">∂</mo><msubsup id="A1.SS1.6.p4.12.m12.2.3.3.2" xref="A1.SS1.6.p4.12.m12.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.6.p4.12.m12.2.3.3.2.2.2" xref="A1.SS1.6.p4.12.m12.2.3.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS1.6.p4.12.m12.2.2.2.4" xref="A1.SS1.6.p4.12.m12.2.2.2.3.cmml"><mi id="A1.SS1.6.p4.12.m12.1.1.1.1" xref="A1.SS1.6.p4.12.m12.1.1.1.1.cmml">x</mi><mo id="A1.SS1.6.p4.12.m12.2.2.2.4.1" xref="A1.SS1.6.p4.12.m12.2.2.2.3.cmml">,</mo><mi id="A1.SS1.6.p4.12.m12.2.2.2.2" xref="A1.SS1.6.p4.12.m12.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.6.p4.12.m12.2.3.3.2.2.3" xref="A1.SS1.6.p4.12.m12.2.3.3.2.2.3.cmml">p</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.6.p4.12.m12.2b"><apply id="A1.SS1.6.p4.12.m12.2.3.cmml" xref="A1.SS1.6.p4.12.m12.2.3"><in id="A1.SS1.6.p4.12.m12.2.3.1.cmml" xref="A1.SS1.6.p4.12.m12.2.3.1"></in><apply id="A1.SS1.6.p4.12.m12.2.3.2.cmml" xref="A1.SS1.6.p4.12.m12.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.12.m12.2.3.2.1.cmml" xref="A1.SS1.6.p4.12.m12.2.3.2">superscript</csymbol><ci id="A1.SS1.6.p4.12.m12.2.3.2.2.cmml" xref="A1.SS1.6.p4.12.m12.2.3.2.2">𝑧</ci><ci id="A1.SS1.6.p4.12.m12.2.3.2.3.cmml" xref="A1.SS1.6.p4.12.m12.2.3.2.3">′</ci></apply><apply id="A1.SS1.6.p4.12.m12.2.3.3.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3"><partialdiff id="A1.SS1.6.p4.12.m12.2.3.3.1.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.1"></partialdiff><apply id="A1.SS1.6.p4.12.m12.2.3.3.2.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.12.m12.2.3.3.2.1.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2">subscript</csymbol><apply id="A1.SS1.6.p4.12.m12.2.3.3.2.2.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS1.6.p4.12.m12.2.3.3.2.2.1.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2">superscript</csymbol><ci id="A1.SS1.6.p4.12.m12.2.3.3.2.2.2.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2.2.2">ℋ</ci><ci id="A1.SS1.6.p4.12.m12.2.3.3.2.2.3.cmml" xref="A1.SS1.6.p4.12.m12.2.3.3.2.2.3">𝑝</ci></apply><list id="A1.SS1.6.p4.12.m12.2.2.2.3.cmml" xref="A1.SS1.6.p4.12.m12.2.2.2.4"><ci id="A1.SS1.6.p4.12.m12.1.1.1.1.cmml" xref="A1.SS1.6.p4.12.m12.1.1.1.1">𝑥</ci><ci id="A1.SS1.6.p4.12.m12.2.2.2.2.cmml" xref="A1.SS1.6.p4.12.m12.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.6.p4.12.m12.2c">z^{\prime}\in\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.6.p4.12.m12.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ ∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p9"> <p class="ltx_p" id="A1.SS1.p9.4">With <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a> at our disposal, we now want to tackle <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a> next. However, before getting there, we will need to recall some tools from convex analysis. Concretely, we will need the notion of subgradients of a convex function <math alttext="f:\mathbb{R}^{d}\rightarrow\mathbb{R}" class="ltx_Math" display="inline" id="A1.SS1.p9.1.m1.1"><semantics id="A1.SS1.p9.1.m1.1a"><mrow id="A1.SS1.p9.1.m1.1.1" xref="A1.SS1.p9.1.m1.1.1.cmml"><mi id="A1.SS1.p9.1.m1.1.1.2" xref="A1.SS1.p9.1.m1.1.1.2.cmml">f</mi><mo id="A1.SS1.p9.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="A1.SS1.p9.1.m1.1.1.1.cmml">:</mo><mrow id="A1.SS1.p9.1.m1.1.1.3" xref="A1.SS1.p9.1.m1.1.1.3.cmml"><msup id="A1.SS1.p9.1.m1.1.1.3.2" xref="A1.SS1.p9.1.m1.1.1.3.2.cmml"><mi id="A1.SS1.p9.1.m1.1.1.3.2.2" xref="A1.SS1.p9.1.m1.1.1.3.2.2.cmml">ℝ</mi><mi id="A1.SS1.p9.1.m1.1.1.3.2.3" xref="A1.SS1.p9.1.m1.1.1.3.2.3.cmml">d</mi></msup><mo id="A1.SS1.p9.1.m1.1.1.3.1" stretchy="false" xref="A1.SS1.p9.1.m1.1.1.3.1.cmml">→</mo><mi id="A1.SS1.p9.1.m1.1.1.3.3" xref="A1.SS1.p9.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.1.m1.1b"><apply id="A1.SS1.p9.1.m1.1.1.cmml" xref="A1.SS1.p9.1.m1.1.1"><ci id="A1.SS1.p9.1.m1.1.1.1.cmml" xref="A1.SS1.p9.1.m1.1.1.1">:</ci><ci id="A1.SS1.p9.1.m1.1.1.2.cmml" xref="A1.SS1.p9.1.m1.1.1.2">𝑓</ci><apply id="A1.SS1.p9.1.m1.1.1.3.cmml" xref="A1.SS1.p9.1.m1.1.1.3"><ci id="A1.SS1.p9.1.m1.1.1.3.1.cmml" xref="A1.SS1.p9.1.m1.1.1.3.1">→</ci><apply id="A1.SS1.p9.1.m1.1.1.3.2.cmml" xref="A1.SS1.p9.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS1.p9.1.m1.1.1.3.2.1.cmml" xref="A1.SS1.p9.1.m1.1.1.3.2">superscript</csymbol><ci id="A1.SS1.p9.1.m1.1.1.3.2.2.cmml" xref="A1.SS1.p9.1.m1.1.1.3.2.2">ℝ</ci><ci id="A1.SS1.p9.1.m1.1.1.3.2.3.cmml" xref="A1.SS1.p9.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="A1.SS1.p9.1.m1.1.1.3.3.cmml" xref="A1.SS1.p9.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.1.m1.1c">f:\mathbb{R}^{d}\rightarrow\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math>: a vector <math alttext="u\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.p9.2.m2.1"><semantics id="A1.SS1.p9.2.m2.1a"><mrow id="A1.SS1.p9.2.m2.1.1" xref="A1.SS1.p9.2.m2.1.1.cmml"><mi id="A1.SS1.p9.2.m2.1.1.2" xref="A1.SS1.p9.2.m2.1.1.2.cmml">u</mi><mo id="A1.SS1.p9.2.m2.1.1.1" xref="A1.SS1.p9.2.m2.1.1.1.cmml">∈</mo><msup id="A1.SS1.p9.2.m2.1.1.3" xref="A1.SS1.p9.2.m2.1.1.3.cmml"><mi id="A1.SS1.p9.2.m2.1.1.3.2" xref="A1.SS1.p9.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.p9.2.m2.1.1.3.3" xref="A1.SS1.p9.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.2.m2.1b"><apply id="A1.SS1.p9.2.m2.1.1.cmml" xref="A1.SS1.p9.2.m2.1.1"><in id="A1.SS1.p9.2.m2.1.1.1.cmml" xref="A1.SS1.p9.2.m2.1.1.1"></in><ci id="A1.SS1.p9.2.m2.1.1.2.cmml" xref="A1.SS1.p9.2.m2.1.1.2">𝑢</ci><apply id="A1.SS1.p9.2.m2.1.1.3.cmml" xref="A1.SS1.p9.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p9.2.m2.1.1.3.1.cmml" xref="A1.SS1.p9.2.m2.1.1.3">superscript</csymbol><ci id="A1.SS1.p9.2.m2.1.1.3.2.cmml" xref="A1.SS1.p9.2.m2.1.1.3.2">ℝ</ci><ci id="A1.SS1.p9.2.m2.1.1.3.3.cmml" xref="A1.SS1.p9.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.2.m2.1c">u\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.2.m2.1d">italic_u ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> is a subgradient of <math alttext="f" class="ltx_Math" display="inline" id="A1.SS1.p9.3.m3.1"><semantics id="A1.SS1.p9.3.m3.1a"><mi id="A1.SS1.p9.3.m3.1.1" xref="A1.SS1.p9.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.3.m3.1b"><ci id="A1.SS1.p9.3.m3.1.1.cmml" xref="A1.SS1.p9.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.3.m3.1d">italic_f</annotation></semantics></math> at <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.p9.4.m4.1"><semantics id="A1.SS1.p9.4.m4.1a"><mrow id="A1.SS1.p9.4.m4.1.1" xref="A1.SS1.p9.4.m4.1.1.cmml"><mi id="A1.SS1.p9.4.m4.1.1.2" xref="A1.SS1.p9.4.m4.1.1.2.cmml">x</mi><mo id="A1.SS1.p9.4.m4.1.1.1" xref="A1.SS1.p9.4.m4.1.1.1.cmml">∈</mo><msup id="A1.SS1.p9.4.m4.1.1.3" xref="A1.SS1.p9.4.m4.1.1.3.cmml"><mi id="A1.SS1.p9.4.m4.1.1.3.2" xref="A1.SS1.p9.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.p9.4.m4.1.1.3.3" xref="A1.SS1.p9.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.4.m4.1b"><apply id="A1.SS1.p9.4.m4.1.1.cmml" xref="A1.SS1.p9.4.m4.1.1"><in id="A1.SS1.p9.4.m4.1.1.1.cmml" xref="A1.SS1.p9.4.m4.1.1.1"></in><ci id="A1.SS1.p9.4.m4.1.1.2.cmml" xref="A1.SS1.p9.4.m4.1.1.2">𝑥</ci><apply id="A1.SS1.p9.4.m4.1.1.3.cmml" xref="A1.SS1.p9.4.m4.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p9.4.m4.1.1.3.1.cmml" xref="A1.SS1.p9.4.m4.1.1.3">superscript</csymbol><ci id="A1.SS1.p9.4.m4.1.1.3.2.cmml" xref="A1.SS1.p9.4.m4.1.1.3.2">ℝ</ci><ci id="A1.SS1.p9.4.m4.1.1.3.3.cmml" xref="A1.SS1.p9.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.4.m4.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.4.m4.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> if and only if</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(x^{\prime})-f(x)\geq\langle u,(x^{\prime}-x)\rangle" class="ltx_Math" display="block" id="A1.Ex3.m1.4"><semantics id="A1.Ex3.m1.4a"><mrow id="A1.Ex3.m1.4.4" xref="A1.Ex3.m1.4.4.cmml"><mrow id="A1.Ex3.m1.3.3.1" xref="A1.Ex3.m1.3.3.1.cmml"><mrow id="A1.Ex3.m1.3.3.1.1" xref="A1.Ex3.m1.3.3.1.1.cmml"><mi id="A1.Ex3.m1.3.3.1.1.3" xref="A1.Ex3.m1.3.3.1.1.3.cmml">f</mi><mo id="A1.Ex3.m1.3.3.1.1.2" xref="A1.Ex3.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="A1.Ex3.m1.3.3.1.1.1.1" xref="A1.Ex3.m1.3.3.1.1.1.1.1.cmml"><mo id="A1.Ex3.m1.3.3.1.1.1.1.2" stretchy="false" xref="A1.Ex3.m1.3.3.1.1.1.1.1.cmml">(</mo><msup id="A1.Ex3.m1.3.3.1.1.1.1.1" xref="A1.Ex3.m1.3.3.1.1.1.1.1.cmml"><mi id="A1.Ex3.m1.3.3.1.1.1.1.1.2" xref="A1.Ex3.m1.3.3.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex3.m1.3.3.1.1.1.1.1.3" xref="A1.Ex3.m1.3.3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="A1.Ex3.m1.3.3.1.1.1.1.3" stretchy="false" xref="A1.Ex3.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex3.m1.3.3.1.2" xref="A1.Ex3.m1.3.3.1.2.cmml">−</mo><mrow id="A1.Ex3.m1.3.3.1.3" xref="A1.Ex3.m1.3.3.1.3.cmml"><mi id="A1.Ex3.m1.3.3.1.3.2" xref="A1.Ex3.m1.3.3.1.3.2.cmml">f</mi><mo id="A1.Ex3.m1.3.3.1.3.1" xref="A1.Ex3.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="A1.Ex3.m1.3.3.1.3.3.2" xref="A1.Ex3.m1.3.3.1.3.cmml"><mo id="A1.Ex3.m1.3.3.1.3.3.2.1" stretchy="false" xref="A1.Ex3.m1.3.3.1.3.cmml">(</mo><mi id="A1.Ex3.m1.1.1" xref="A1.Ex3.m1.1.1.cmml">x</mi><mo id="A1.Ex3.m1.3.3.1.3.3.2.2" stretchy="false" xref="A1.Ex3.m1.3.3.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A1.Ex3.m1.4.4.3" xref="A1.Ex3.m1.4.4.3.cmml">≥</mo><mrow id="A1.Ex3.m1.4.4.2.1" xref="A1.Ex3.m1.4.4.2.2.cmml"><mo id="A1.Ex3.m1.4.4.2.1.2" stretchy="false" xref="A1.Ex3.m1.4.4.2.2.cmml">⟨</mo><mi id="A1.Ex3.m1.2.2" xref="A1.Ex3.m1.2.2.cmml">u</mi><mo id="A1.Ex3.m1.4.4.2.1.3" xref="A1.Ex3.m1.4.4.2.2.cmml">,</mo><mrow id="A1.Ex3.m1.4.4.2.1.1.1" xref="A1.Ex3.m1.4.4.2.1.1.1.1.cmml"><mo id="A1.Ex3.m1.4.4.2.1.1.1.2" stretchy="false" xref="A1.Ex3.m1.4.4.2.1.1.1.1.cmml">(</mo><mrow id="A1.Ex3.m1.4.4.2.1.1.1.1" xref="A1.Ex3.m1.4.4.2.1.1.1.1.cmml"><msup id="A1.Ex3.m1.4.4.2.1.1.1.1.2" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2.cmml"><mi id="A1.Ex3.m1.4.4.2.1.1.1.1.2.2" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2.2.cmml">x</mi><mo id="A1.Ex3.m1.4.4.2.1.1.1.1.2.3" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A1.Ex3.m1.4.4.2.1.1.1.1.1" xref="A1.Ex3.m1.4.4.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex3.m1.4.4.2.1.1.1.1.3" xref="A1.Ex3.m1.4.4.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.Ex3.m1.4.4.2.1.1.1.3" stretchy="false" xref="A1.Ex3.m1.4.4.2.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Ex3.m1.4.4.2.1.4" stretchy="false" xref="A1.Ex3.m1.4.4.2.2.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex3.m1.4b"><apply id="A1.Ex3.m1.4.4.cmml" xref="A1.Ex3.m1.4.4"><geq id="A1.Ex3.m1.4.4.3.cmml" xref="A1.Ex3.m1.4.4.3"></geq><apply id="A1.Ex3.m1.3.3.1.cmml" xref="A1.Ex3.m1.3.3.1"><minus id="A1.Ex3.m1.3.3.1.2.cmml" xref="A1.Ex3.m1.3.3.1.2"></minus><apply id="A1.Ex3.m1.3.3.1.1.cmml" xref="A1.Ex3.m1.3.3.1.1"><times id="A1.Ex3.m1.3.3.1.1.2.cmml" xref="A1.Ex3.m1.3.3.1.1.2"></times><ci id="A1.Ex3.m1.3.3.1.1.3.cmml" xref="A1.Ex3.m1.3.3.1.1.3">𝑓</ci><apply id="A1.Ex3.m1.3.3.1.1.1.1.1.cmml" xref="A1.Ex3.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex3.m1.3.3.1.1.1.1.1.1.cmml" xref="A1.Ex3.m1.3.3.1.1.1.1">superscript</csymbol><ci id="A1.Ex3.m1.3.3.1.1.1.1.1.2.cmml" xref="A1.Ex3.m1.3.3.1.1.1.1.1.2">𝑥</ci><ci id="A1.Ex3.m1.3.3.1.1.1.1.1.3.cmml" xref="A1.Ex3.m1.3.3.1.1.1.1.1.3">′</ci></apply></apply><apply id="A1.Ex3.m1.3.3.1.3.cmml" xref="A1.Ex3.m1.3.3.1.3"><times id="A1.Ex3.m1.3.3.1.3.1.cmml" xref="A1.Ex3.m1.3.3.1.3.1"></times><ci id="A1.Ex3.m1.3.3.1.3.2.cmml" xref="A1.Ex3.m1.3.3.1.3.2">𝑓</ci><ci id="A1.Ex3.m1.1.1.cmml" xref="A1.Ex3.m1.1.1">𝑥</ci></apply></apply><list id="A1.Ex3.m1.4.4.2.2.cmml" xref="A1.Ex3.m1.4.4.2.1"><ci id="A1.Ex3.m1.2.2.cmml" xref="A1.Ex3.m1.2.2">𝑢</ci><apply id="A1.Ex3.m1.4.4.2.1.1.1.1.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1"><minus id="A1.Ex3.m1.4.4.2.1.1.1.1.1.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.1"></minus><apply id="A1.Ex3.m1.4.4.2.1.1.1.1.2.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex3.m1.4.4.2.1.1.1.1.2.1.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2">superscript</csymbol><ci id="A1.Ex3.m1.4.4.2.1.1.1.1.2.2.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2.2">𝑥</ci><ci id="A1.Ex3.m1.4.4.2.1.1.1.1.2.3.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.2.3">′</ci></apply><ci id="A1.Ex3.m1.4.4.2.1.1.1.1.3.cmml" xref="A1.Ex3.m1.4.4.2.1.1.1.1.3">𝑥</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex3.m1.4c">f(x^{\prime})-f(x)\geq\langle u,(x^{\prime}-x)\rangle</annotation><annotation encoding="application/x-llamapun" id="A1.Ex3.m1.4d">italic_f ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - italic_f ( italic_x ) ≥ ⟨ italic_u , ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_x ) ⟩</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS1.p9.11">for all <math alttext="x^{\prime}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.p9.5.m1.1"><semantics id="A1.SS1.p9.5.m1.1a"><mrow id="A1.SS1.p9.5.m1.1.1" xref="A1.SS1.p9.5.m1.1.1.cmml"><msup id="A1.SS1.p9.5.m1.1.1.2" xref="A1.SS1.p9.5.m1.1.1.2.cmml"><mi id="A1.SS1.p9.5.m1.1.1.2.2" xref="A1.SS1.p9.5.m1.1.1.2.2.cmml">x</mi><mo id="A1.SS1.p9.5.m1.1.1.2.3" xref="A1.SS1.p9.5.m1.1.1.2.3.cmml">′</mo></msup><mo id="A1.SS1.p9.5.m1.1.1.1" xref="A1.SS1.p9.5.m1.1.1.1.cmml">∈</mo><msup id="A1.SS1.p9.5.m1.1.1.3" xref="A1.SS1.p9.5.m1.1.1.3.cmml"><mi id="A1.SS1.p9.5.m1.1.1.3.2" xref="A1.SS1.p9.5.m1.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.p9.5.m1.1.1.3.3" xref="A1.SS1.p9.5.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.5.m1.1b"><apply id="A1.SS1.p9.5.m1.1.1.cmml" xref="A1.SS1.p9.5.m1.1.1"><in id="A1.SS1.p9.5.m1.1.1.1.cmml" xref="A1.SS1.p9.5.m1.1.1.1"></in><apply id="A1.SS1.p9.5.m1.1.1.2.cmml" xref="A1.SS1.p9.5.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.p9.5.m1.1.1.2.1.cmml" xref="A1.SS1.p9.5.m1.1.1.2">superscript</csymbol><ci id="A1.SS1.p9.5.m1.1.1.2.2.cmml" xref="A1.SS1.p9.5.m1.1.1.2.2">𝑥</ci><ci id="A1.SS1.p9.5.m1.1.1.2.3.cmml" xref="A1.SS1.p9.5.m1.1.1.2.3">′</ci></apply><apply id="A1.SS1.p9.5.m1.1.1.3.cmml" xref="A1.SS1.p9.5.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p9.5.m1.1.1.3.1.cmml" xref="A1.SS1.p9.5.m1.1.1.3">superscript</csymbol><ci id="A1.SS1.p9.5.m1.1.1.3.2.cmml" xref="A1.SS1.p9.5.m1.1.1.3.2">ℝ</ci><ci id="A1.SS1.p9.5.m1.1.1.3.3.cmml" xref="A1.SS1.p9.5.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.5.m1.1c">x^{\prime}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.5.m1.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. The set <math alttext="\partial f(x)\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.p9.6.m2.1"><semantics id="A1.SS1.p9.6.m2.1a"><mrow id="A1.SS1.p9.6.m2.1.2" xref="A1.SS1.p9.6.m2.1.2.cmml"><mrow id="A1.SS1.p9.6.m2.1.2.2" xref="A1.SS1.p9.6.m2.1.2.2.cmml"><mo id="A1.SS1.p9.6.m2.1.2.2.1" rspace="0em" xref="A1.SS1.p9.6.m2.1.2.2.1.cmml">∂</mo><mrow id="A1.SS1.p9.6.m2.1.2.2.2" xref="A1.SS1.p9.6.m2.1.2.2.2.cmml"><mi id="A1.SS1.p9.6.m2.1.2.2.2.2" xref="A1.SS1.p9.6.m2.1.2.2.2.2.cmml">f</mi><mo id="A1.SS1.p9.6.m2.1.2.2.2.1" xref="A1.SS1.p9.6.m2.1.2.2.2.1.cmml">⁢</mo><mrow id="A1.SS1.p9.6.m2.1.2.2.2.3.2" xref="A1.SS1.p9.6.m2.1.2.2.2.cmml"><mo id="A1.SS1.p9.6.m2.1.2.2.2.3.2.1" stretchy="false" xref="A1.SS1.p9.6.m2.1.2.2.2.cmml">(</mo><mi id="A1.SS1.p9.6.m2.1.1" xref="A1.SS1.p9.6.m2.1.1.cmml">x</mi><mo id="A1.SS1.p9.6.m2.1.2.2.2.3.2.2" stretchy="false" xref="A1.SS1.p9.6.m2.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A1.SS1.p9.6.m2.1.2.1" xref="A1.SS1.p9.6.m2.1.2.1.cmml">⊆</mo><msup id="A1.SS1.p9.6.m2.1.2.3" xref="A1.SS1.p9.6.m2.1.2.3.cmml"><mi id="A1.SS1.p9.6.m2.1.2.3.2" xref="A1.SS1.p9.6.m2.1.2.3.2.cmml">ℝ</mi><mi id="A1.SS1.p9.6.m2.1.2.3.3" xref="A1.SS1.p9.6.m2.1.2.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.6.m2.1b"><apply id="A1.SS1.p9.6.m2.1.2.cmml" xref="A1.SS1.p9.6.m2.1.2"><subset id="A1.SS1.p9.6.m2.1.2.1.cmml" xref="A1.SS1.p9.6.m2.1.2.1"></subset><apply id="A1.SS1.p9.6.m2.1.2.2.cmml" xref="A1.SS1.p9.6.m2.1.2.2"><partialdiff id="A1.SS1.p9.6.m2.1.2.2.1.cmml" xref="A1.SS1.p9.6.m2.1.2.2.1"></partialdiff><apply id="A1.SS1.p9.6.m2.1.2.2.2.cmml" xref="A1.SS1.p9.6.m2.1.2.2.2"><times id="A1.SS1.p9.6.m2.1.2.2.2.1.cmml" xref="A1.SS1.p9.6.m2.1.2.2.2.1"></times><ci id="A1.SS1.p9.6.m2.1.2.2.2.2.cmml" xref="A1.SS1.p9.6.m2.1.2.2.2.2">𝑓</ci><ci id="A1.SS1.p9.6.m2.1.1.cmml" xref="A1.SS1.p9.6.m2.1.1">𝑥</ci></apply></apply><apply id="A1.SS1.p9.6.m2.1.2.3.cmml" xref="A1.SS1.p9.6.m2.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.p9.6.m2.1.2.3.1.cmml" xref="A1.SS1.p9.6.m2.1.2.3">superscript</csymbol><ci id="A1.SS1.p9.6.m2.1.2.3.2.cmml" xref="A1.SS1.p9.6.m2.1.2.3.2">ℝ</ci><ci id="A1.SS1.p9.6.m2.1.2.3.3.cmml" xref="A1.SS1.p9.6.m2.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.6.m2.1c">\partial f(x)\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.6.m2.1d">∂ italic_f ( italic_x ) ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> of all subgradients of <math alttext="f" class="ltx_Math" display="inline" id="A1.SS1.p9.7.m3.1"><semantics id="A1.SS1.p9.7.m3.1a"><mi id="A1.SS1.p9.7.m3.1.1" xref="A1.SS1.p9.7.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.7.m3.1b"><ci id="A1.SS1.p9.7.m3.1.1.cmml" xref="A1.SS1.p9.7.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.7.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.7.m3.1d">italic_f</annotation></semantics></math> at <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.p9.8.m4.1"><semantics id="A1.SS1.p9.8.m4.1a"><mi id="A1.SS1.p9.8.m4.1.1" xref="A1.SS1.p9.8.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.8.m4.1b"><ci id="A1.SS1.p9.8.m4.1.1.cmml" xref="A1.SS1.p9.8.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.8.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.8.m4.1d">italic_x</annotation></semantics></math> is also called the subdifferential. If <math alttext="f" class="ltx_Math" display="inline" id="A1.SS1.p9.9.m5.1"><semantics id="A1.SS1.p9.9.m5.1a"><mi id="A1.SS1.p9.9.m5.1.1" xref="A1.SS1.p9.9.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.9.m5.1b"><ci id="A1.SS1.p9.9.m5.1.1.cmml" xref="A1.SS1.p9.9.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.9.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.9.m5.1d">italic_f</annotation></semantics></math> is differentiable at <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.p9.10.m6.1"><semantics id="A1.SS1.p9.10.m6.1a"><mi id="A1.SS1.p9.10.m6.1.1" xref="A1.SS1.p9.10.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.10.m6.1b"><ci id="A1.SS1.p9.10.m6.1.1.cmml" xref="A1.SS1.p9.10.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.10.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.10.m6.1d">italic_x</annotation></semantics></math>, then <math alttext="\partial f(x)=\{\nabla f(x)\}" class="ltx_Math" display="inline" id="A1.SS1.p9.11.m7.3"><semantics id="A1.SS1.p9.11.m7.3a"><mrow id="A1.SS1.p9.11.m7.3.3" xref="A1.SS1.p9.11.m7.3.3.cmml"><mrow id="A1.SS1.p9.11.m7.3.3.3" xref="A1.SS1.p9.11.m7.3.3.3.cmml"><mo id="A1.SS1.p9.11.m7.3.3.3.1" rspace="0em" xref="A1.SS1.p9.11.m7.3.3.3.1.cmml">∂</mo><mrow id="A1.SS1.p9.11.m7.3.3.3.2" xref="A1.SS1.p9.11.m7.3.3.3.2.cmml"><mi id="A1.SS1.p9.11.m7.3.3.3.2.2" xref="A1.SS1.p9.11.m7.3.3.3.2.2.cmml">f</mi><mo id="A1.SS1.p9.11.m7.3.3.3.2.1" xref="A1.SS1.p9.11.m7.3.3.3.2.1.cmml">⁢</mo><mrow id="A1.SS1.p9.11.m7.3.3.3.2.3.2" xref="A1.SS1.p9.11.m7.3.3.3.2.cmml"><mo id="A1.SS1.p9.11.m7.3.3.3.2.3.2.1" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.3.2.cmml">(</mo><mi id="A1.SS1.p9.11.m7.1.1" xref="A1.SS1.p9.11.m7.1.1.cmml">x</mi><mo id="A1.SS1.p9.11.m7.3.3.3.2.3.2.2" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.3.2.cmml">)</mo></mrow></mrow></mrow><mo id="A1.SS1.p9.11.m7.3.3.2" xref="A1.SS1.p9.11.m7.3.3.2.cmml">=</mo><mrow id="A1.SS1.p9.11.m7.3.3.1.1" xref="A1.SS1.p9.11.m7.3.3.1.2.cmml"><mo id="A1.SS1.p9.11.m7.3.3.1.1.2" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.1.2.cmml">{</mo><mrow id="A1.SS1.p9.11.m7.3.3.1.1.1" xref="A1.SS1.p9.11.m7.3.3.1.1.1.cmml"><mrow id="A1.SS1.p9.11.m7.3.3.1.1.1.2" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2.cmml"><mo id="A1.SS1.p9.11.m7.3.3.1.1.1.2.1" rspace="0.167em" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2.1.cmml">∇</mo><mi id="A1.SS1.p9.11.m7.3.3.1.1.1.2.2" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2.2.cmml">f</mi></mrow><mo id="A1.SS1.p9.11.m7.3.3.1.1.1.1" xref="A1.SS1.p9.11.m7.3.3.1.1.1.1.cmml">⁢</mo><mrow id="A1.SS1.p9.11.m7.3.3.1.1.1.3.2" xref="A1.SS1.p9.11.m7.3.3.1.1.1.cmml"><mo id="A1.SS1.p9.11.m7.3.3.1.1.1.3.2.1" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.1.1.1.cmml">(</mo><mi id="A1.SS1.p9.11.m7.2.2" xref="A1.SS1.p9.11.m7.2.2.cmml">x</mi><mo id="A1.SS1.p9.11.m7.3.3.1.1.1.3.2.2" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.p9.11.m7.3.3.1.1.3" stretchy="false" xref="A1.SS1.p9.11.m7.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p9.11.m7.3b"><apply id="A1.SS1.p9.11.m7.3.3.cmml" xref="A1.SS1.p9.11.m7.3.3"><eq id="A1.SS1.p9.11.m7.3.3.2.cmml" xref="A1.SS1.p9.11.m7.3.3.2"></eq><apply id="A1.SS1.p9.11.m7.3.3.3.cmml" xref="A1.SS1.p9.11.m7.3.3.3"><partialdiff id="A1.SS1.p9.11.m7.3.3.3.1.cmml" xref="A1.SS1.p9.11.m7.3.3.3.1"></partialdiff><apply id="A1.SS1.p9.11.m7.3.3.3.2.cmml" xref="A1.SS1.p9.11.m7.3.3.3.2"><times id="A1.SS1.p9.11.m7.3.3.3.2.1.cmml" xref="A1.SS1.p9.11.m7.3.3.3.2.1"></times><ci id="A1.SS1.p9.11.m7.3.3.3.2.2.cmml" xref="A1.SS1.p9.11.m7.3.3.3.2.2">𝑓</ci><ci id="A1.SS1.p9.11.m7.1.1.cmml" xref="A1.SS1.p9.11.m7.1.1">𝑥</ci></apply></apply><set id="A1.SS1.p9.11.m7.3.3.1.2.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1"><apply id="A1.SS1.p9.11.m7.3.3.1.1.1.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1.1"><times id="A1.SS1.p9.11.m7.3.3.1.1.1.1.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1.1.1"></times><apply id="A1.SS1.p9.11.m7.3.3.1.1.1.2.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2"><ci id="A1.SS1.p9.11.m7.3.3.1.1.1.2.1.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2.1">∇</ci><ci id="A1.SS1.p9.11.m7.3.3.1.1.1.2.2.cmml" xref="A1.SS1.p9.11.m7.3.3.1.1.1.2.2">𝑓</ci></apply><ci id="A1.SS1.p9.11.m7.2.2.cmml" xref="A1.SS1.p9.11.m7.2.2">𝑥</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p9.11.m7.3c">\partial f(x)=\{\nabla f(x)\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p9.11.m7.3d">∂ italic_f ( italic_x ) = { ∇ italic_f ( italic_x ) }</annotation></semantics></math>. For more details on convex analysis, we refer to the standard textbook by Rockafellar <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib34" title="">34</a>]</cite>.</p> </div> <div class="ltx_para" id="A1.SS1.p10"> <p class="ltx_p" id="A1.SS1.p10.3">Subgradients are useful because they allow us to further characterize containment of a given point <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.p10.1.m1.1"><semantics id="A1.SS1.p10.1.m1.1a"><mi id="A1.SS1.p10.1.m1.1.1" xref="A1.SS1.p10.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p10.1.m1.1b"><ci id="A1.SS1.p10.1.m1.1.1.cmml" xref="A1.SS1.p10.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p10.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p10.1.m1.1d">italic_z</annotation></semantics></math> in an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p10.2.m2.1"><semantics id="A1.SS1.p10.2.m2.1a"><msub id="A1.SS1.p10.2.m2.1.1" xref="A1.SS1.p10.2.m2.1.1.cmml"><mi id="A1.SS1.p10.2.m2.1.1.2" mathvariant="normal" xref="A1.SS1.p10.2.m2.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p10.2.m2.1.1.3" xref="A1.SS1.p10.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p10.2.m2.1b"><apply id="A1.SS1.p10.2.m2.1.1.cmml" xref="A1.SS1.p10.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.p10.2.m2.1.1.1.cmml" xref="A1.SS1.p10.2.m2.1.1">subscript</csymbol><ci id="A1.SS1.p10.2.m2.1.1.2.cmml" xref="A1.SS1.p10.2.m2.1.1.2">ℓ</ci><ci id="A1.SS1.p10.2.m2.1.1.3.cmml" xref="A1.SS1.p10.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p10.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p10.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.p10.3.m3.2"><semantics id="A1.SS1.p10.3.m3.2a"><msubsup id="A1.SS1.p10.3.m3.2.3" xref="A1.SS1.p10.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.p10.3.m3.2.3.2.2" xref="A1.SS1.p10.3.m3.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.p10.3.m3.2.2.2.4" xref="A1.SS1.p10.3.m3.2.2.2.3.cmml"><mi id="A1.SS1.p10.3.m3.1.1.1.1" xref="A1.SS1.p10.3.m3.1.1.1.1.cmml">x</mi><mo id="A1.SS1.p10.3.m3.2.2.2.4.1" xref="A1.SS1.p10.3.m3.2.2.2.3.cmml">,</mo><mi id="A1.SS1.p10.3.m3.2.2.2.2" xref="A1.SS1.p10.3.m3.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.p10.3.m3.2.3.2.3" xref="A1.SS1.p10.3.m3.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.p10.3.m3.2b"><apply id="A1.SS1.p10.3.m3.2.3.cmml" xref="A1.SS1.p10.3.m3.2.3"><csymbol cd="ambiguous" id="A1.SS1.p10.3.m3.2.3.1.cmml" xref="A1.SS1.p10.3.m3.2.3">subscript</csymbol><apply id="A1.SS1.p10.3.m3.2.3.2.cmml" xref="A1.SS1.p10.3.m3.2.3"><csymbol cd="ambiguous" id="A1.SS1.p10.3.m3.2.3.2.1.cmml" xref="A1.SS1.p10.3.m3.2.3">superscript</csymbol><ci id="A1.SS1.p10.3.m3.2.3.2.2.cmml" xref="A1.SS1.p10.3.m3.2.3.2.2">ℋ</ci><ci id="A1.SS1.p10.3.m3.2.3.2.3.cmml" xref="A1.SS1.p10.3.m3.2.3.2.3">𝑝</ci></apply><list id="A1.SS1.p10.3.m3.2.2.2.3.cmml" xref="A1.SS1.p10.3.m3.2.2.2.4"><ci id="A1.SS1.p10.3.m3.1.1.1.1.cmml" xref="A1.SS1.p10.3.m3.1.1.1.1">𝑥</ci><ci id="A1.SS1.p10.3.m3.2.2.2.2.cmml" xref="A1.SS1.p10.3.m3.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p10.3.m3.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p10.3.m3.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> as follows.</p> </div> <div class="ltx_para" id="A1.SS1.p11"> <p class="ltx_p" id="A1.SS1.p11.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.5</span></a></p> </div> <div class="ltx_proof" id="A1.SS1.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.7.p1"> <p class="ltx_p" id="A1.SS1.7.p1.24">Assume first that there exists such a subgradient <math alttext="u\in\partial||z-x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.1.m1.1"><semantics id="A1.SS1.7.p1.1.m1.1a"><mrow id="A1.SS1.7.p1.1.m1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.cmml"><mi id="A1.SS1.7.p1.1.m1.1.1.3" xref="A1.SS1.7.p1.1.m1.1.1.3.cmml">u</mi><mo id="A1.SS1.7.p1.1.m1.1.1.2" rspace="0.1389em" xref="A1.SS1.7.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="A1.SS1.7.p1.1.m1.1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.1.cmml"><mo id="A1.SS1.7.p1.1.m1.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS1.7.p1.1.m1.1.1.1.2.cmml">∂</mo><msub id="A1.SS1.7.p1.1.m1.1.1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.1.1.cmml"><mrow id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.2.cmml"><mo id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.2" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.1" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.3" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.7.p1.1.m1.1.1.1.1.3" xref="A1.SS1.7.p1.1.m1.1.1.1.1.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.1.m1.1b"><apply id="A1.SS1.7.p1.1.m1.1.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1"><in id="A1.SS1.7.p1.1.m1.1.1.2.cmml" xref="A1.SS1.7.p1.1.m1.1.1.2"></in><ci id="A1.SS1.7.p1.1.m1.1.1.3.cmml" xref="A1.SS1.7.p1.1.m1.1.1.3">𝑢</ci><apply id="A1.SS1.7.p1.1.m1.1.1.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1"><partialdiff id="A1.SS1.7.p1.1.m1.1.1.1.2.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.2"></partialdiff><apply id="A1.SS1.7.p1.1.m1.1.1.1.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.1.m1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.1.m1.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.7.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1"><minus id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.7.p1.1.m1.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.1.m1.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.1.m1.1c">u\in\partial||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.1.m1.1d">italic_u ∈ ∂ | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\langle u,v\rangle\geq 0" class="ltx_Math" display="inline" id="A1.SS1.7.p1.2.m2.2"><semantics id="A1.SS1.7.p1.2.m2.2a"><mrow id="A1.SS1.7.p1.2.m2.2.3" xref="A1.SS1.7.p1.2.m2.2.3.cmml"><mrow id="A1.SS1.7.p1.2.m2.2.3.2.2" xref="A1.SS1.7.p1.2.m2.2.3.2.1.cmml"><mo id="A1.SS1.7.p1.2.m2.2.3.2.2.1" stretchy="false" xref="A1.SS1.7.p1.2.m2.2.3.2.1.cmml">⟨</mo><mi id="A1.SS1.7.p1.2.m2.1.1" xref="A1.SS1.7.p1.2.m2.1.1.cmml">u</mi><mo id="A1.SS1.7.p1.2.m2.2.3.2.2.2" xref="A1.SS1.7.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="A1.SS1.7.p1.2.m2.2.2" xref="A1.SS1.7.p1.2.m2.2.2.cmml">v</mi><mo id="A1.SS1.7.p1.2.m2.2.3.2.2.3" stretchy="false" xref="A1.SS1.7.p1.2.m2.2.3.2.1.cmml">⟩</mo></mrow><mo id="A1.SS1.7.p1.2.m2.2.3.1" xref="A1.SS1.7.p1.2.m2.2.3.1.cmml">≥</mo><mn id="A1.SS1.7.p1.2.m2.2.3.3" xref="A1.SS1.7.p1.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.2.m2.2b"><apply id="A1.SS1.7.p1.2.m2.2.3.cmml" xref="A1.SS1.7.p1.2.m2.2.3"><geq id="A1.SS1.7.p1.2.m2.2.3.1.cmml" xref="A1.SS1.7.p1.2.m2.2.3.1"></geq><list id="A1.SS1.7.p1.2.m2.2.3.2.1.cmml" xref="A1.SS1.7.p1.2.m2.2.3.2.2"><ci id="A1.SS1.7.p1.2.m2.1.1.cmml" xref="A1.SS1.7.p1.2.m2.1.1">𝑢</ci><ci id="A1.SS1.7.p1.2.m2.2.2.cmml" xref="A1.SS1.7.p1.2.m2.2.2">𝑣</ci></list><cn id="A1.SS1.7.p1.2.m2.2.3.3.cmml" type="integer" xref="A1.SS1.7.p1.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.2.m2.2c">\langle u,v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.2.m2.2d">⟨ italic_u , italic_v ⟩ ≥ 0</annotation></semantics></math>. Choosing <math alttext="x^{\prime}=z-x+\varepsilon v" class="ltx_Math" display="inline" id="A1.SS1.7.p1.3.m3.1"><semantics id="A1.SS1.7.p1.3.m3.1a"><mrow id="A1.SS1.7.p1.3.m3.1.1" xref="A1.SS1.7.p1.3.m3.1.1.cmml"><msup id="A1.SS1.7.p1.3.m3.1.1.2" xref="A1.SS1.7.p1.3.m3.1.1.2.cmml"><mi id="A1.SS1.7.p1.3.m3.1.1.2.2" xref="A1.SS1.7.p1.3.m3.1.1.2.2.cmml">x</mi><mo id="A1.SS1.7.p1.3.m3.1.1.2.3" xref="A1.SS1.7.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="A1.SS1.7.p1.3.m3.1.1.1" xref="A1.SS1.7.p1.3.m3.1.1.1.cmml">=</mo><mrow id="A1.SS1.7.p1.3.m3.1.1.3" xref="A1.SS1.7.p1.3.m3.1.1.3.cmml"><mrow id="A1.SS1.7.p1.3.m3.1.1.3.2" xref="A1.SS1.7.p1.3.m3.1.1.3.2.cmml"><mi id="A1.SS1.7.p1.3.m3.1.1.3.2.2" xref="A1.SS1.7.p1.3.m3.1.1.3.2.2.cmml">z</mi><mo id="A1.SS1.7.p1.3.m3.1.1.3.2.1" xref="A1.SS1.7.p1.3.m3.1.1.3.2.1.cmml">−</mo><mi id="A1.SS1.7.p1.3.m3.1.1.3.2.3" xref="A1.SS1.7.p1.3.m3.1.1.3.2.3.cmml">x</mi></mrow><mo id="A1.SS1.7.p1.3.m3.1.1.3.1" xref="A1.SS1.7.p1.3.m3.1.1.3.1.cmml">+</mo><mrow id="A1.SS1.7.p1.3.m3.1.1.3.3" xref="A1.SS1.7.p1.3.m3.1.1.3.3.cmml"><mi id="A1.SS1.7.p1.3.m3.1.1.3.3.2" xref="A1.SS1.7.p1.3.m3.1.1.3.3.2.cmml">ε</mi><mo id="A1.SS1.7.p1.3.m3.1.1.3.3.1" xref="A1.SS1.7.p1.3.m3.1.1.3.3.1.cmml">⁢</mo><mi id="A1.SS1.7.p1.3.m3.1.1.3.3.3" xref="A1.SS1.7.p1.3.m3.1.1.3.3.3.cmml">v</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.3.m3.1b"><apply id="A1.SS1.7.p1.3.m3.1.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1"><eq id="A1.SS1.7.p1.3.m3.1.1.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1.1"></eq><apply id="A1.SS1.7.p1.3.m3.1.1.2.cmml" xref="A1.SS1.7.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.7.p1.3.m3.1.1.2.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1.2">superscript</csymbol><ci id="A1.SS1.7.p1.3.m3.1.1.2.2.cmml" xref="A1.SS1.7.p1.3.m3.1.1.2.2">𝑥</ci><ci id="A1.SS1.7.p1.3.m3.1.1.2.3.cmml" xref="A1.SS1.7.p1.3.m3.1.1.2.3">′</ci></apply><apply id="A1.SS1.7.p1.3.m3.1.1.3.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3"><plus id="A1.SS1.7.p1.3.m3.1.1.3.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.1"></plus><apply id="A1.SS1.7.p1.3.m3.1.1.3.2.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.2"><minus id="A1.SS1.7.p1.3.m3.1.1.3.2.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.2.1"></minus><ci id="A1.SS1.7.p1.3.m3.1.1.3.2.2.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.2.2">𝑧</ci><ci id="A1.SS1.7.p1.3.m3.1.1.3.2.3.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.2.3">𝑥</ci></apply><apply id="A1.SS1.7.p1.3.m3.1.1.3.3.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.3"><times id="A1.SS1.7.p1.3.m3.1.1.3.3.1.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.3.1"></times><ci id="A1.SS1.7.p1.3.m3.1.1.3.3.2.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.3.2">𝜀</ci><ci id="A1.SS1.7.p1.3.m3.1.1.3.3.3.cmml" xref="A1.SS1.7.p1.3.m3.1.1.3.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.3.m3.1c">x^{\prime}=z-x+\varepsilon v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.3.m3.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_z - italic_x + italic_ε italic_v</annotation></semantics></math> in the definition of subgradients, we conclude <math alttext="||z-x+\varepsilon v||_{p}-||z-x||_{p}\geq\varepsilon\langle u,v\rangle\geq 0" class="ltx_Math" display="inline" id="A1.SS1.7.p1.4.m4.4"><semantics id="A1.SS1.7.p1.4.m4.4a"><mrow id="A1.SS1.7.p1.4.m4.4.4" xref="A1.SS1.7.p1.4.m4.4.4.cmml"><mrow id="A1.SS1.7.p1.4.m4.4.4.2" xref="A1.SS1.7.p1.4.m4.4.4.2.cmml"><msub id="A1.SS1.7.p1.4.m4.3.3.1.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.cmml"><mrow id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.2.cmml"><mo id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.2" stretchy="false" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.cmml"><mrow id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.cmml"><mi id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.2" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.2.cmml">z</mi><mo id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.1.cmml">−</mo><mi id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.3" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.3.cmml">x</mi></mrow><mo id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.1.cmml">+</mo><mrow id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.cmml"><mi id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.2" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.1" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.3" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.3.cmml">v</mi></mrow></mrow><mo id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.3" stretchy="false" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.7.p1.4.m4.3.3.1.1.3" xref="A1.SS1.7.p1.4.m4.3.3.1.1.3.cmml">p</mi></msub><mo id="A1.SS1.7.p1.4.m4.4.4.2.3" xref="A1.SS1.7.p1.4.m4.4.4.2.3.cmml">−</mo><msub id="A1.SS1.7.p1.4.m4.4.4.2.2" xref="A1.SS1.7.p1.4.m4.4.4.2.2.cmml"><mrow id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.2.cmml"><mo id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.2" stretchy="false" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.2.1.cmml">‖</mo><mrow id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.cmml"><mi id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.2" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.2.cmml">z</mi><mo id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.1" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.3" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.3" stretchy="false" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.7.p1.4.m4.4.4.2.2.3" xref="A1.SS1.7.p1.4.m4.4.4.2.2.3.cmml">p</mi></msub></mrow><mo id="A1.SS1.7.p1.4.m4.4.4.4" xref="A1.SS1.7.p1.4.m4.4.4.4.cmml">≥</mo><mrow id="A1.SS1.7.p1.4.m4.4.4.5" xref="A1.SS1.7.p1.4.m4.4.4.5.cmml"><mi id="A1.SS1.7.p1.4.m4.4.4.5.2" xref="A1.SS1.7.p1.4.m4.4.4.5.2.cmml">ε</mi><mo id="A1.SS1.7.p1.4.m4.4.4.5.1" xref="A1.SS1.7.p1.4.m4.4.4.5.1.cmml">⁢</mo><mrow id="A1.SS1.7.p1.4.m4.4.4.5.3.2" xref="A1.SS1.7.p1.4.m4.4.4.5.3.1.cmml"><mo id="A1.SS1.7.p1.4.m4.4.4.5.3.2.1" stretchy="false" xref="A1.SS1.7.p1.4.m4.4.4.5.3.1.cmml">⟨</mo><mi id="A1.SS1.7.p1.4.m4.1.1" xref="A1.SS1.7.p1.4.m4.1.1.cmml">u</mi><mo id="A1.SS1.7.p1.4.m4.4.4.5.3.2.2" xref="A1.SS1.7.p1.4.m4.4.4.5.3.1.cmml">,</mo><mi id="A1.SS1.7.p1.4.m4.2.2" xref="A1.SS1.7.p1.4.m4.2.2.cmml">v</mi><mo id="A1.SS1.7.p1.4.m4.4.4.5.3.2.3" stretchy="false" xref="A1.SS1.7.p1.4.m4.4.4.5.3.1.cmml">⟩</mo></mrow></mrow><mo id="A1.SS1.7.p1.4.m4.4.4.6" xref="A1.SS1.7.p1.4.m4.4.4.6.cmml">≥</mo><mn id="A1.SS1.7.p1.4.m4.4.4.7" xref="A1.SS1.7.p1.4.m4.4.4.7.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.4.m4.4b"><apply id="A1.SS1.7.p1.4.m4.4.4.cmml" xref="A1.SS1.7.p1.4.m4.4.4"><and id="A1.SS1.7.p1.4.m4.4.4a.cmml" xref="A1.SS1.7.p1.4.m4.4.4"></and><apply id="A1.SS1.7.p1.4.m4.4.4b.cmml" xref="A1.SS1.7.p1.4.m4.4.4"><geq id="A1.SS1.7.p1.4.m4.4.4.4.cmml" xref="A1.SS1.7.p1.4.m4.4.4.4"></geq><apply id="A1.SS1.7.p1.4.m4.4.4.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2"><minus id="A1.SS1.7.p1.4.m4.4.4.2.3.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.3"></minus><apply id="A1.SS1.7.p1.4.m4.3.3.1.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.4.m4.3.3.1.1.2.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.4.m4.3.3.1.1.1.2.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.7.p1.4.m4.3.3.1.1.1.2.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1"><plus id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.1"></plus><apply id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2"><minus id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.1"></minus><ci id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.2.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.2">𝑧</ci><ci id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.3.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.2.3">𝑥</ci></apply><apply id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3"><times id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.1.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.1"></times><ci id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.2.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.2">𝜀</ci><ci id="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.3.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.1.1.1.3.3">𝑣</ci></apply></apply></apply><ci id="A1.SS1.7.p1.4.m4.3.3.1.1.3.cmml" xref="A1.SS1.7.p1.4.m4.3.3.1.1.3">𝑝</ci></apply><apply id="A1.SS1.7.p1.4.m4.4.4.2.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2"><csymbol cd="ambiguous" id="A1.SS1.7.p1.4.m4.4.4.2.2.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2">subscript</csymbol><apply id="A1.SS1.7.p1.4.m4.4.4.2.2.1.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1"><csymbol cd="latexml" id="A1.SS1.7.p1.4.m4.4.4.2.2.1.2.1.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.2">norm</csymbol><apply id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1"><minus id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.1.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.1"></minus><ci id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.2">𝑧</ci><ci id="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.3.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.7.p1.4.m4.4.4.2.2.3.cmml" xref="A1.SS1.7.p1.4.m4.4.4.2.2.3">𝑝</ci></apply></apply><apply id="A1.SS1.7.p1.4.m4.4.4.5.cmml" xref="A1.SS1.7.p1.4.m4.4.4.5"><times id="A1.SS1.7.p1.4.m4.4.4.5.1.cmml" xref="A1.SS1.7.p1.4.m4.4.4.5.1"></times><ci id="A1.SS1.7.p1.4.m4.4.4.5.2.cmml" xref="A1.SS1.7.p1.4.m4.4.4.5.2">𝜀</ci><list id="A1.SS1.7.p1.4.m4.4.4.5.3.1.cmml" xref="A1.SS1.7.p1.4.m4.4.4.5.3.2"><ci id="A1.SS1.7.p1.4.m4.1.1.cmml" xref="A1.SS1.7.p1.4.m4.1.1">𝑢</ci><ci id="A1.SS1.7.p1.4.m4.2.2.cmml" xref="A1.SS1.7.p1.4.m4.2.2">𝑣</ci></list></apply></apply><apply id="A1.SS1.7.p1.4.m4.4.4c.cmml" xref="A1.SS1.7.p1.4.m4.4.4"><geq id="A1.SS1.7.p1.4.m4.4.4.6.cmml" xref="A1.SS1.7.p1.4.m4.4.4.6"></geq><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.7.p1.4.m4.4.4.5.cmml" id="A1.SS1.7.p1.4.m4.4.4d.cmml" xref="A1.SS1.7.p1.4.m4.4.4"></share><cn id="A1.SS1.7.p1.4.m4.4.4.7.cmml" type="integer" xref="A1.SS1.7.p1.4.m4.4.4.7">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.4.m4.4c">||z-x+\varepsilon v||_{p}-||z-x||_{p}\geq\varepsilon\langle u,v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.4.m4.4d">| | italic_z - italic_x + italic_ε italic_v | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ italic_ε ⟨ italic_u , italic_v ⟩ ≥ 0</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS1.7.p1.5.m5.1"><semantics id="A1.SS1.7.p1.5.m5.1a"><mrow id="A1.SS1.7.p1.5.m5.1.1" xref="A1.SS1.7.p1.5.m5.1.1.cmml"><mi id="A1.SS1.7.p1.5.m5.1.1.2" xref="A1.SS1.7.p1.5.m5.1.1.2.cmml">ε</mi><mo id="A1.SS1.7.p1.5.m5.1.1.1" xref="A1.SS1.7.p1.5.m5.1.1.1.cmml">></mo><mn id="A1.SS1.7.p1.5.m5.1.1.3" xref="A1.SS1.7.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.5.m5.1b"><apply id="A1.SS1.7.p1.5.m5.1.1.cmml" xref="A1.SS1.7.p1.5.m5.1.1"><gt id="A1.SS1.7.p1.5.m5.1.1.1.cmml" xref="A1.SS1.7.p1.5.m5.1.1.1"></gt><ci id="A1.SS1.7.p1.5.m5.1.1.2.cmml" xref="A1.SS1.7.p1.5.m5.1.1.2">𝜀</ci><cn id="A1.SS1.7.p1.5.m5.1.1.3.cmml" type="integer" xref="A1.SS1.7.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.5.m5.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.5.m5.1d">italic_ε > 0</annotation></semantics></math>. Thus, <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.7.p1.6.m6.1"><semantics id="A1.SS1.7.p1.6.m6.1a"><mi id="A1.SS1.7.p1.6.m6.1.1" xref="A1.SS1.7.p1.6.m6.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.6.m6.1b"><ci id="A1.SS1.7.p1.6.m6.1.1.cmml" xref="A1.SS1.7.p1.6.m6.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.6.m6.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.6.m6.1d">italic_z</annotation></semantics></math> is contained in the halfspace. Conversely, assume <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.7.m7.2"><semantics id="A1.SS1.7.p1.7.m7.2a"><mrow id="A1.SS1.7.p1.7.m7.2.3" xref="A1.SS1.7.p1.7.m7.2.3.cmml"><mi id="A1.SS1.7.p1.7.m7.2.3.2" xref="A1.SS1.7.p1.7.m7.2.3.2.cmml">z</mi><mo id="A1.SS1.7.p1.7.m7.2.3.1" xref="A1.SS1.7.p1.7.m7.2.3.1.cmml">∈</mo><msubsup id="A1.SS1.7.p1.7.m7.2.3.3" xref="A1.SS1.7.p1.7.m7.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.7.p1.7.m7.2.3.3.2.2" xref="A1.SS1.7.p1.7.m7.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.7.p1.7.m7.2.2.2.4" xref="A1.SS1.7.p1.7.m7.2.2.2.3.cmml"><mi id="A1.SS1.7.p1.7.m7.1.1.1.1" xref="A1.SS1.7.p1.7.m7.1.1.1.1.cmml">x</mi><mo id="A1.SS1.7.p1.7.m7.2.2.2.4.1" xref="A1.SS1.7.p1.7.m7.2.2.2.3.cmml">,</mo><mi id="A1.SS1.7.p1.7.m7.2.2.2.2" xref="A1.SS1.7.p1.7.m7.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.7.p1.7.m7.2.3.3.2.3" xref="A1.SS1.7.p1.7.m7.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.7.m7.2b"><apply id="A1.SS1.7.p1.7.m7.2.3.cmml" xref="A1.SS1.7.p1.7.m7.2.3"><in id="A1.SS1.7.p1.7.m7.2.3.1.cmml" xref="A1.SS1.7.p1.7.m7.2.3.1"></in><ci id="A1.SS1.7.p1.7.m7.2.3.2.cmml" xref="A1.SS1.7.p1.7.m7.2.3.2">𝑧</ci><apply id="A1.SS1.7.p1.7.m7.2.3.3.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.7.p1.7.m7.2.3.3.1.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3">subscript</csymbol><apply id="A1.SS1.7.p1.7.m7.2.3.3.2.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.7.p1.7.m7.2.3.3.2.1.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3">superscript</csymbol><ci id="A1.SS1.7.p1.7.m7.2.3.3.2.2.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3.2.2">ℋ</ci><ci id="A1.SS1.7.p1.7.m7.2.3.3.2.3.cmml" xref="A1.SS1.7.p1.7.m7.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS1.7.p1.7.m7.2.2.2.3.cmml" xref="A1.SS1.7.p1.7.m7.2.2.2.4"><ci id="A1.SS1.7.p1.7.m7.1.1.1.1.cmml" xref="A1.SS1.7.p1.7.m7.1.1.1.1">𝑥</ci><ci id="A1.SS1.7.p1.7.m7.2.2.2.2.cmml" xref="A1.SS1.7.p1.7.m7.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.7.m7.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.7.m7.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem3" title="Observation 3.3. ‣ 3.1 ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.3</span></a>, this implies that the intersection of the open ray <math alttext="R_{-}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.8.m8.1"><semantics id="A1.SS1.7.p1.8.m8.1a"><msub id="A1.SS1.7.p1.8.m8.1.1" xref="A1.SS1.7.p1.8.m8.1.1.cmml"><mi id="A1.SS1.7.p1.8.m8.1.1.2" xref="A1.SS1.7.p1.8.m8.1.1.2.cmml">R</mi><mo id="A1.SS1.7.p1.8.m8.1.1.3" xref="A1.SS1.7.p1.8.m8.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.8.m8.1b"><apply id="A1.SS1.7.p1.8.m8.1.1.cmml" xref="A1.SS1.7.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.8.m8.1.1.1.cmml" xref="A1.SS1.7.p1.8.m8.1.1">subscript</csymbol><ci id="A1.SS1.7.p1.8.m8.1.1.2.cmml" xref="A1.SS1.7.p1.8.m8.1.1.2">𝑅</ci><minus id="A1.SS1.7.p1.8.m8.1.1.3.cmml" xref="A1.SS1.7.p1.8.m8.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.8.m8.1c">R_{-}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.8.m8.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.7.p1.9.m9.1"><semantics id="A1.SS1.7.p1.9.m9.1a"><mi id="A1.SS1.7.p1.9.m9.1.1" xref="A1.SS1.7.p1.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.9.m9.1b"><ci id="A1.SS1.7.p1.9.m9.1.1.cmml" xref="A1.SS1.7.p1.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.9.m9.1d">italic_x</annotation></semantics></math> in direction <math alttext="-v" class="ltx_Math" display="inline" id="A1.SS1.7.p1.10.m10.1"><semantics id="A1.SS1.7.p1.10.m10.1a"><mrow id="A1.SS1.7.p1.10.m10.1.1" xref="A1.SS1.7.p1.10.m10.1.1.cmml"><mo id="A1.SS1.7.p1.10.m10.1.1a" xref="A1.SS1.7.p1.10.m10.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.10.m10.1.1.2" xref="A1.SS1.7.p1.10.m10.1.1.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.10.m10.1b"><apply id="A1.SS1.7.p1.10.m10.1.1.cmml" xref="A1.SS1.7.p1.10.m10.1.1"><minus id="A1.SS1.7.p1.10.m10.1.1.1.cmml" xref="A1.SS1.7.p1.10.m10.1.1"></minus><ci id="A1.SS1.7.p1.10.m10.1.1.2.cmml" xref="A1.SS1.7.p1.10.m10.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.10.m10.1c">-v</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.10.m10.1d">- italic_v</annotation></semantics></math> with the interior <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.11.m11.1"><semantics id="A1.SS1.7.p1.11.m11.1a"><msubsup id="A1.SS1.7.p1.11.m11.1.1" xref="A1.SS1.7.p1.11.m11.1.1.cmml"><mi id="A1.SS1.7.p1.11.m11.1.1.2.2" xref="A1.SS1.7.p1.11.m11.1.1.2.2.cmml">B</mi><mi id="A1.SS1.7.p1.11.m11.1.1.3" xref="A1.SS1.7.p1.11.m11.1.1.3.cmml">z</mi><mo id="A1.SS1.7.p1.11.m11.1.1.2.3" xref="A1.SS1.7.p1.11.m11.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.11.m11.1b"><apply id="A1.SS1.7.p1.11.m11.1.1.cmml" xref="A1.SS1.7.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.11.m11.1.1.1.cmml" xref="A1.SS1.7.p1.11.m11.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.11.m11.1.1.2.cmml" xref="A1.SS1.7.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.11.m11.1.1.2.1.cmml" xref="A1.SS1.7.p1.11.m11.1.1">superscript</csymbol><ci id="A1.SS1.7.p1.11.m11.1.1.2.2.cmml" xref="A1.SS1.7.p1.11.m11.1.1.2.2">𝐵</ci><compose id="A1.SS1.7.p1.11.m11.1.1.2.3.cmml" xref="A1.SS1.7.p1.11.m11.1.1.2.3"></compose></apply><ci id="A1.SS1.7.p1.11.m11.1.1.3.cmml" xref="A1.SS1.7.p1.11.m11.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.11.m11.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.11.m11.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> of the smallest ball <math alttext="B_{z}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.12.m12.1"><semantics id="A1.SS1.7.p1.12.m12.1a"><msub id="A1.SS1.7.p1.12.m12.1.1" xref="A1.SS1.7.p1.12.m12.1.1.cmml"><mi id="A1.SS1.7.p1.12.m12.1.1.2" xref="A1.SS1.7.p1.12.m12.1.1.2.cmml">B</mi><mi id="A1.SS1.7.p1.12.m12.1.1.3" xref="A1.SS1.7.p1.12.m12.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.12.m12.1b"><apply id="A1.SS1.7.p1.12.m12.1.1.cmml" xref="A1.SS1.7.p1.12.m12.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.12.m12.1.1.1.cmml" xref="A1.SS1.7.p1.12.m12.1.1">subscript</csymbol><ci id="A1.SS1.7.p1.12.m12.1.1.2.cmml" xref="A1.SS1.7.p1.12.m12.1.1.2">𝐵</ci><ci id="A1.SS1.7.p1.12.m12.1.1.3.cmml" xref="A1.SS1.7.p1.12.m12.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.12.m12.1c">B_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.12.m12.1d">italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> containing <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.7.p1.13.m13.1"><semantics id="A1.SS1.7.p1.13.m13.1a"><mi id="A1.SS1.7.p1.13.m13.1.1" xref="A1.SS1.7.p1.13.m13.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.13.m13.1b"><ci id="A1.SS1.7.p1.13.m13.1.1.cmml" xref="A1.SS1.7.p1.13.m13.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.13.m13.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.13.m13.1d">italic_x</annotation></semantics></math> around <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.7.p1.14.m14.1"><semantics id="A1.SS1.7.p1.14.m14.1a"><mi id="A1.SS1.7.p1.14.m14.1.1" xref="A1.SS1.7.p1.14.m14.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.14.m14.1b"><ci id="A1.SS1.7.p1.14.m14.1.1.cmml" xref="A1.SS1.7.p1.14.m14.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.14.m14.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.14.m14.1d">italic_z</annotation></semantics></math> is empty. Both of these objects are convex. Thus, there exists a hyperplane separating <math alttext="R_{-}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.15.m15.1"><semantics id="A1.SS1.7.p1.15.m15.1a"><msub id="A1.SS1.7.p1.15.m15.1.1" xref="A1.SS1.7.p1.15.m15.1.1.cmml"><mi id="A1.SS1.7.p1.15.m15.1.1.2" xref="A1.SS1.7.p1.15.m15.1.1.2.cmml">R</mi><mo id="A1.SS1.7.p1.15.m15.1.1.3" xref="A1.SS1.7.p1.15.m15.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.15.m15.1b"><apply id="A1.SS1.7.p1.15.m15.1.1.cmml" xref="A1.SS1.7.p1.15.m15.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.15.m15.1.1.1.cmml" xref="A1.SS1.7.p1.15.m15.1.1">subscript</csymbol><ci id="A1.SS1.7.p1.15.m15.1.1.2.cmml" xref="A1.SS1.7.p1.15.m15.1.1.2">𝑅</ci><minus id="A1.SS1.7.p1.15.m15.1.1.3.cmml" xref="A1.SS1.7.p1.15.m15.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.15.m15.1c">R_{-}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.15.m15.1d">italic_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.16.m16.1"><semantics id="A1.SS1.7.p1.16.m16.1a"><msubsup id="A1.SS1.7.p1.16.m16.1.1" xref="A1.SS1.7.p1.16.m16.1.1.cmml"><mi id="A1.SS1.7.p1.16.m16.1.1.2.2" xref="A1.SS1.7.p1.16.m16.1.1.2.2.cmml">B</mi><mi id="A1.SS1.7.p1.16.m16.1.1.3" xref="A1.SS1.7.p1.16.m16.1.1.3.cmml">z</mi><mo id="A1.SS1.7.p1.16.m16.1.1.2.3" xref="A1.SS1.7.p1.16.m16.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.16.m16.1b"><apply id="A1.SS1.7.p1.16.m16.1.1.cmml" xref="A1.SS1.7.p1.16.m16.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.16.m16.1.1.1.cmml" xref="A1.SS1.7.p1.16.m16.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.16.m16.1.1.2.cmml" xref="A1.SS1.7.p1.16.m16.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.16.m16.1.1.2.1.cmml" xref="A1.SS1.7.p1.16.m16.1.1">superscript</csymbol><ci id="A1.SS1.7.p1.16.m16.1.1.2.2.cmml" xref="A1.SS1.7.p1.16.m16.1.1.2.2">𝐵</ci><compose id="A1.SS1.7.p1.16.m16.1.1.2.3.cmml" xref="A1.SS1.7.p1.16.m16.1.1.2.3"></compose></apply><ci id="A1.SS1.7.p1.16.m16.1.1.3.cmml" xref="A1.SS1.7.p1.16.m16.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.16.m16.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.16.m16.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>. Note that this hyperplane must go through <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.7.p1.17.m17.1"><semantics id="A1.SS1.7.p1.17.m17.1a"><mi id="A1.SS1.7.p1.17.m17.1.1" xref="A1.SS1.7.p1.17.m17.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.17.m17.1b"><ci id="A1.SS1.7.p1.17.m17.1.1.cmml" xref="A1.SS1.7.p1.17.m17.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.17.m17.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.17.m17.1d">italic_x</annotation></semantics></math>, and a scaling of its normal vector must be a subgradient in <math alttext="\partial||x-z||_{p}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.18.m18.1"><semantics id="A1.SS1.7.p1.18.m18.1a"><mrow id="A1.SS1.7.p1.18.m18.1.1" xref="A1.SS1.7.p1.18.m18.1.1.cmml"><mo id="A1.SS1.7.p1.18.m18.1.1.2" rspace="0em" xref="A1.SS1.7.p1.18.m18.1.1.2.cmml">∂</mo><msub id="A1.SS1.7.p1.18.m18.1.1.1" xref="A1.SS1.7.p1.18.m18.1.1.1.cmml"><mrow id="A1.SS1.7.p1.18.m18.1.1.1.1.1" xref="A1.SS1.7.p1.18.m18.1.1.1.1.2.cmml"><mo id="A1.SS1.7.p1.18.m18.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.7.p1.18.m18.1.1.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.cmml"><mi id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.2" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.1" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.3" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="A1.SS1.7.p1.18.m18.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.7.p1.18.m18.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.7.p1.18.m18.1.1.1.3" xref="A1.SS1.7.p1.18.m18.1.1.1.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.18.m18.1b"><apply id="A1.SS1.7.p1.18.m18.1.1.cmml" xref="A1.SS1.7.p1.18.m18.1.1"><partialdiff id="A1.SS1.7.p1.18.m18.1.1.2.cmml" xref="A1.SS1.7.p1.18.m18.1.1.2"></partialdiff><apply id="A1.SS1.7.p1.18.m18.1.1.1.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.18.m18.1.1.1.2.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.18.m18.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.7.p1.18.m18.1.1.1.1.2.1.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1"><minus id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.SS1.7.p1.18.m18.1.1.1.3.cmml" xref="A1.SS1.7.p1.18.m18.1.1.1.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.18.m18.1c">\partial||x-z||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.18.m18.1d">∂ | | italic_x - italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. Naturally, this subgradient <math alttext="u" class="ltx_Math" display="inline" id="A1.SS1.7.p1.19.m19.1"><semantics id="A1.SS1.7.p1.19.m19.1a"><mi id="A1.SS1.7.p1.19.m19.1.1" xref="A1.SS1.7.p1.19.m19.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.19.m19.1b"><ci id="A1.SS1.7.p1.19.m19.1.1.cmml" xref="A1.SS1.7.p1.19.m19.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.19.m19.1c">u</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.19.m19.1d">italic_u</annotation></semantics></math> at <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.7.p1.20.m20.1"><semantics id="A1.SS1.7.p1.20.m20.1a"><mi id="A1.SS1.7.p1.20.m20.1.1" xref="A1.SS1.7.p1.20.m20.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.20.m20.1b"><ci id="A1.SS1.7.p1.20.m20.1.1.cmml" xref="A1.SS1.7.p1.20.m20.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.20.m20.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.20.m20.1d">italic_x</annotation></semantics></math> points away from <math alttext="B^{\circ}_{z}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.21.m21.1"><semantics id="A1.SS1.7.p1.21.m21.1a"><msubsup id="A1.SS1.7.p1.21.m21.1.1" xref="A1.SS1.7.p1.21.m21.1.1.cmml"><mi id="A1.SS1.7.p1.21.m21.1.1.2.2" xref="A1.SS1.7.p1.21.m21.1.1.2.2.cmml">B</mi><mi id="A1.SS1.7.p1.21.m21.1.1.3" xref="A1.SS1.7.p1.21.m21.1.1.3.cmml">z</mi><mo id="A1.SS1.7.p1.21.m21.1.1.2.3" xref="A1.SS1.7.p1.21.m21.1.1.2.3.cmml">∘</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.21.m21.1b"><apply id="A1.SS1.7.p1.21.m21.1.1.cmml" xref="A1.SS1.7.p1.21.m21.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.21.m21.1.1.1.cmml" xref="A1.SS1.7.p1.21.m21.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.21.m21.1.1.2.cmml" xref="A1.SS1.7.p1.21.m21.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.21.m21.1.1.2.1.cmml" xref="A1.SS1.7.p1.21.m21.1.1">superscript</csymbol><ci id="A1.SS1.7.p1.21.m21.1.1.2.2.cmml" xref="A1.SS1.7.p1.21.m21.1.1.2.2">𝐵</ci><compose id="A1.SS1.7.p1.21.m21.1.1.2.3.cmml" xref="A1.SS1.7.p1.21.m21.1.1.2.3"></compose></apply><ci id="A1.SS1.7.p1.21.m21.1.1.3.cmml" xref="A1.SS1.7.p1.21.m21.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.21.m21.1c">B^{\circ}_{z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.21.m21.1d">italic_B start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math>, and we have <math alttext="\langle u,-v\rangle\geq 0" class="ltx_Math" display="inline" id="A1.SS1.7.p1.22.m22.2"><semantics id="A1.SS1.7.p1.22.m22.2a"><mrow id="A1.SS1.7.p1.22.m22.2.2" xref="A1.SS1.7.p1.22.m22.2.2.cmml"><mrow id="A1.SS1.7.p1.22.m22.2.2.1.1" xref="A1.SS1.7.p1.22.m22.2.2.1.2.cmml"><mo id="A1.SS1.7.p1.22.m22.2.2.1.1.2" stretchy="false" xref="A1.SS1.7.p1.22.m22.2.2.1.2.cmml">⟨</mo><mi id="A1.SS1.7.p1.22.m22.1.1" xref="A1.SS1.7.p1.22.m22.1.1.cmml">u</mi><mo id="A1.SS1.7.p1.22.m22.2.2.1.1.3" xref="A1.SS1.7.p1.22.m22.2.2.1.2.cmml">,</mo><mrow id="A1.SS1.7.p1.22.m22.2.2.1.1.1" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1.cmml"><mo id="A1.SS1.7.p1.22.m22.2.2.1.1.1a" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.22.m22.2.2.1.1.1.2" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1.2.cmml">v</mi></mrow><mo id="A1.SS1.7.p1.22.m22.2.2.1.1.4" stretchy="false" xref="A1.SS1.7.p1.22.m22.2.2.1.2.cmml">⟩</mo></mrow><mo id="A1.SS1.7.p1.22.m22.2.2.2" xref="A1.SS1.7.p1.22.m22.2.2.2.cmml">≥</mo><mn id="A1.SS1.7.p1.22.m22.2.2.3" xref="A1.SS1.7.p1.22.m22.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.22.m22.2b"><apply id="A1.SS1.7.p1.22.m22.2.2.cmml" xref="A1.SS1.7.p1.22.m22.2.2"><geq id="A1.SS1.7.p1.22.m22.2.2.2.cmml" xref="A1.SS1.7.p1.22.m22.2.2.2"></geq><list id="A1.SS1.7.p1.22.m22.2.2.1.2.cmml" xref="A1.SS1.7.p1.22.m22.2.2.1.1"><ci id="A1.SS1.7.p1.22.m22.1.1.cmml" xref="A1.SS1.7.p1.22.m22.1.1">𝑢</ci><apply id="A1.SS1.7.p1.22.m22.2.2.1.1.1.cmml" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1"><minus id="A1.SS1.7.p1.22.m22.2.2.1.1.1.1.cmml" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1"></minus><ci id="A1.SS1.7.p1.22.m22.2.2.1.1.1.2.cmml" xref="A1.SS1.7.p1.22.m22.2.2.1.1.1.2">𝑣</ci></apply></list><cn id="A1.SS1.7.p1.22.m22.2.2.3.cmml" type="integer" xref="A1.SS1.7.p1.22.m22.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.22.m22.2c">\langle u,-v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.22.m22.2d">⟨ italic_u , - italic_v ⟩ ≥ 0</annotation></semantics></math>. It remains to observe <math alttext="-u\in\partial||z-x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.7.p1.23.m23.1"><semantics id="A1.SS1.7.p1.23.m23.1a"><mrow id="A1.SS1.7.p1.23.m23.1.1" xref="A1.SS1.7.p1.23.m23.1.1.cmml"><mrow id="A1.SS1.7.p1.23.m23.1.1.3" xref="A1.SS1.7.p1.23.m23.1.1.3.cmml"><mo id="A1.SS1.7.p1.23.m23.1.1.3a" xref="A1.SS1.7.p1.23.m23.1.1.3.cmml">−</mo><mi id="A1.SS1.7.p1.23.m23.1.1.3.2" xref="A1.SS1.7.p1.23.m23.1.1.3.2.cmml">u</mi></mrow><mo id="A1.SS1.7.p1.23.m23.1.1.2" rspace="0.1389em" xref="A1.SS1.7.p1.23.m23.1.1.2.cmml">∈</mo><mrow id="A1.SS1.7.p1.23.m23.1.1.1" xref="A1.SS1.7.p1.23.m23.1.1.1.cmml"><mo id="A1.SS1.7.p1.23.m23.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS1.7.p1.23.m23.1.1.1.2.cmml">∂</mo><msub id="A1.SS1.7.p1.23.m23.1.1.1.1" xref="A1.SS1.7.p1.23.m23.1.1.1.1.cmml"><mrow id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.2.cmml"><mo id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.2" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.1" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.3" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.7.p1.23.m23.1.1.1.1.3" xref="A1.SS1.7.p1.23.m23.1.1.1.1.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.23.m23.1b"><apply id="A1.SS1.7.p1.23.m23.1.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1"><in id="A1.SS1.7.p1.23.m23.1.1.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.2"></in><apply id="A1.SS1.7.p1.23.m23.1.1.3.cmml" xref="A1.SS1.7.p1.23.m23.1.1.3"><minus id="A1.SS1.7.p1.23.m23.1.1.3.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.3"></minus><ci id="A1.SS1.7.p1.23.m23.1.1.3.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.3.2">𝑢</ci></apply><apply id="A1.SS1.7.p1.23.m23.1.1.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1"><partialdiff id="A1.SS1.7.p1.23.m23.1.1.1.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.2"></partialdiff><apply id="A1.SS1.7.p1.23.m23.1.1.1.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.7.p1.23.m23.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1">subscript</csymbol><apply id="A1.SS1.7.p1.23.m23.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.7.p1.23.m23.1.1.1.1.1.2.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1"><minus id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.7.p1.23.m23.1.1.1.1.3.cmml" xref="A1.SS1.7.p1.23.m23.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.23.m23.1c">-u\in\partial||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.23.m23.1d">- italic_u ∈ ∂ | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="-u" class="ltx_Math" display="inline" id="A1.SS1.7.p1.24.m24.1"><semantics id="A1.SS1.7.p1.24.m24.1a"><mrow id="A1.SS1.7.p1.24.m24.1.1" xref="A1.SS1.7.p1.24.m24.1.1.cmml"><mo id="A1.SS1.7.p1.24.m24.1.1a" xref="A1.SS1.7.p1.24.m24.1.1.cmml">−</mo><mi id="A1.SS1.7.p1.24.m24.1.1.2" xref="A1.SS1.7.p1.24.m24.1.1.2.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.7.p1.24.m24.1b"><apply id="A1.SS1.7.p1.24.m24.1.1.cmml" xref="A1.SS1.7.p1.24.m24.1.1"><minus id="A1.SS1.7.p1.24.m24.1.1.1.cmml" xref="A1.SS1.7.p1.24.m24.1.1"></minus><ci id="A1.SS1.7.p1.24.m24.1.1.2.cmml" xref="A1.SS1.7.p1.24.m24.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.7.p1.24.m24.1c">-u</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.7.p1.24.m24.1d">- italic_u</annotation></semantics></math> is the desired subgradient). ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p12"> <p class="ltx_p" id="A1.SS1.p12.6">In order to use <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>, we need to determine the subgradients of the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p12.1.m1.1"><semantics id="A1.SS1.p12.1.m1.1a"><msub id="A1.SS1.p12.1.m1.1.1" xref="A1.SS1.p12.1.m1.1.1.cmml"><mi id="A1.SS1.p12.1.m1.1.1.2" mathvariant="normal" xref="A1.SS1.p12.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p12.1.m1.1.1.3" xref="A1.SS1.p12.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.1.m1.1b"><apply id="A1.SS1.p12.1.m1.1.1.cmml" xref="A1.SS1.p12.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.p12.1.m1.1.1.1.cmml" xref="A1.SS1.p12.1.m1.1.1">subscript</csymbol><ci id="A1.SS1.p12.1.m1.1.1.2.cmml" xref="A1.SS1.p12.1.m1.1.1.2">ℓ</ci><ci id="A1.SS1.p12.1.m1.1.1.3.cmml" xref="A1.SS1.p12.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm for all <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.SS1.p12.2.m2.3"><semantics id="A1.SS1.p12.2.m2.3a"><mrow id="A1.SS1.p12.2.m2.3.4" xref="A1.SS1.p12.2.m2.3.4.cmml"><mi id="A1.SS1.p12.2.m2.3.4.2" xref="A1.SS1.p12.2.m2.3.4.2.cmml">p</mi><mo id="A1.SS1.p12.2.m2.3.4.1" xref="A1.SS1.p12.2.m2.3.4.1.cmml">∈</mo><mrow id="A1.SS1.p12.2.m2.3.4.3" xref="A1.SS1.p12.2.m2.3.4.3.cmml"><mrow id="A1.SS1.p12.2.m2.3.4.3.2.2" xref="A1.SS1.p12.2.m2.3.4.3.2.1.cmml"><mo id="A1.SS1.p12.2.m2.3.4.3.2.2.1" stretchy="false" xref="A1.SS1.p12.2.m2.3.4.3.2.1.cmml">[</mo><mn id="A1.SS1.p12.2.m2.1.1" xref="A1.SS1.p12.2.m2.1.1.cmml">1</mn><mo id="A1.SS1.p12.2.m2.3.4.3.2.2.2" xref="A1.SS1.p12.2.m2.3.4.3.2.1.cmml">,</mo><mi id="A1.SS1.p12.2.m2.2.2" mathvariant="normal" xref="A1.SS1.p12.2.m2.2.2.cmml">∞</mi><mo id="A1.SS1.p12.2.m2.3.4.3.2.2.3" stretchy="false" xref="A1.SS1.p12.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.SS1.p12.2.m2.3.4.3.1" xref="A1.SS1.p12.2.m2.3.4.3.1.cmml">∪</mo><mrow id="A1.SS1.p12.2.m2.3.4.3.3.2" xref="A1.SS1.p12.2.m2.3.4.3.3.1.cmml"><mo id="A1.SS1.p12.2.m2.3.4.3.3.2.1" stretchy="false" xref="A1.SS1.p12.2.m2.3.4.3.3.1.cmml">{</mo><mi id="A1.SS1.p12.2.m2.3.3" mathvariant="normal" xref="A1.SS1.p12.2.m2.3.3.cmml">∞</mi><mo id="A1.SS1.p12.2.m2.3.4.3.3.2.2" stretchy="false" xref="A1.SS1.p12.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.2.m2.3b"><apply id="A1.SS1.p12.2.m2.3.4.cmml" xref="A1.SS1.p12.2.m2.3.4"><in id="A1.SS1.p12.2.m2.3.4.1.cmml" xref="A1.SS1.p12.2.m2.3.4.1"></in><ci id="A1.SS1.p12.2.m2.3.4.2.cmml" xref="A1.SS1.p12.2.m2.3.4.2">𝑝</ci><apply id="A1.SS1.p12.2.m2.3.4.3.cmml" xref="A1.SS1.p12.2.m2.3.4.3"><union id="A1.SS1.p12.2.m2.3.4.3.1.cmml" xref="A1.SS1.p12.2.m2.3.4.3.1"></union><interval closure="closed-open" id="A1.SS1.p12.2.m2.3.4.3.2.1.cmml" xref="A1.SS1.p12.2.m2.3.4.3.2.2"><cn id="A1.SS1.p12.2.m2.1.1.cmml" type="integer" xref="A1.SS1.p12.2.m2.1.1">1</cn><infinity id="A1.SS1.p12.2.m2.2.2.cmml" xref="A1.SS1.p12.2.m2.2.2"></infinity></interval><set id="A1.SS1.p12.2.m2.3.4.3.3.1.cmml" xref="A1.SS1.p12.2.m2.3.4.3.3.2"><infinity id="A1.SS1.p12.2.m2.3.3.cmml" xref="A1.SS1.p12.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. We start with the cases <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="A1.SS1.p12.3.m3.2"><semantics id="A1.SS1.p12.3.m3.2a"><mrow id="A1.SS1.p12.3.m3.2.3" xref="A1.SS1.p12.3.m3.2.3.cmml"><mi id="A1.SS1.p12.3.m3.2.3.2" xref="A1.SS1.p12.3.m3.2.3.2.cmml">p</mi><mo id="A1.SS1.p12.3.m3.2.3.1" xref="A1.SS1.p12.3.m3.2.3.1.cmml">∈</mo><mrow id="A1.SS1.p12.3.m3.2.3.3.2" xref="A1.SS1.p12.3.m3.2.3.3.1.cmml"><mo id="A1.SS1.p12.3.m3.2.3.3.2.1" stretchy="false" xref="A1.SS1.p12.3.m3.2.3.3.1.cmml">(</mo><mn id="A1.SS1.p12.3.m3.1.1" xref="A1.SS1.p12.3.m3.1.1.cmml">1</mn><mo id="A1.SS1.p12.3.m3.2.3.3.2.2" xref="A1.SS1.p12.3.m3.2.3.3.1.cmml">,</mo><mi id="A1.SS1.p12.3.m3.2.2" mathvariant="normal" xref="A1.SS1.p12.3.m3.2.2.cmml">∞</mi><mo id="A1.SS1.p12.3.m3.2.3.3.2.3" stretchy="false" xref="A1.SS1.p12.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.3.m3.2b"><apply id="A1.SS1.p12.3.m3.2.3.cmml" xref="A1.SS1.p12.3.m3.2.3"><in id="A1.SS1.p12.3.m3.2.3.1.cmml" xref="A1.SS1.p12.3.m3.2.3.1"></in><ci id="A1.SS1.p12.3.m3.2.3.2.cmml" xref="A1.SS1.p12.3.m3.2.3.2">𝑝</ci><interval closure="open" id="A1.SS1.p12.3.m3.2.3.3.1.cmml" xref="A1.SS1.p12.3.m3.2.3.3.2"><cn id="A1.SS1.p12.3.m3.1.1.cmml" type="integer" xref="A1.SS1.p12.3.m3.1.1">1</cn><infinity id="A1.SS1.p12.3.m3.2.2.cmml" xref="A1.SS1.p12.3.m3.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.3.m3.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.3.m3.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, and focus on those points <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.p12.4.m4.1"><semantics id="A1.SS1.p12.4.m4.1a"><mrow id="A1.SS1.p12.4.m4.1.1" xref="A1.SS1.p12.4.m4.1.1.cmml"><mi id="A1.SS1.p12.4.m4.1.1.2" xref="A1.SS1.p12.4.m4.1.1.2.cmml">z</mi><mo id="A1.SS1.p12.4.m4.1.1.1" xref="A1.SS1.p12.4.m4.1.1.1.cmml">∈</mo><msup id="A1.SS1.p12.4.m4.1.1.3" xref="A1.SS1.p12.4.m4.1.1.3.cmml"><mi id="A1.SS1.p12.4.m4.1.1.3.2" xref="A1.SS1.p12.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.p12.4.m4.1.1.3.3" xref="A1.SS1.p12.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.4.m4.1b"><apply id="A1.SS1.p12.4.m4.1.1.cmml" xref="A1.SS1.p12.4.m4.1.1"><in id="A1.SS1.p12.4.m4.1.1.1.cmml" xref="A1.SS1.p12.4.m4.1.1.1"></in><ci id="A1.SS1.p12.4.m4.1.1.2.cmml" xref="A1.SS1.p12.4.m4.1.1.2">𝑧</ci><apply id="A1.SS1.p12.4.m4.1.1.3.cmml" xref="A1.SS1.p12.4.m4.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p12.4.m4.1.1.3.1.cmml" xref="A1.SS1.p12.4.m4.1.1.3">superscript</csymbol><ci id="A1.SS1.p12.4.m4.1.1.3.2.cmml" xref="A1.SS1.p12.4.m4.1.1.3.2">ℝ</ci><ci id="A1.SS1.p12.4.m4.1.1.3.3.cmml" xref="A1.SS1.p12.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.4.m4.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.4.m4.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{p}=1" class="ltx_Math" display="inline" id="A1.SS1.p12.5.m5.1"><semantics id="A1.SS1.p12.5.m5.1a"><mrow id="A1.SS1.p12.5.m5.1.2" xref="A1.SS1.p12.5.m5.1.2.cmml"><msub id="A1.SS1.p12.5.m5.1.2.2" xref="A1.SS1.p12.5.m5.1.2.2.cmml"><mrow id="A1.SS1.p12.5.m5.1.2.2.2.2" xref="A1.SS1.p12.5.m5.1.2.2.2.1.cmml"><mo id="A1.SS1.p12.5.m5.1.2.2.2.2.1" stretchy="false" xref="A1.SS1.p12.5.m5.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.p12.5.m5.1.1" xref="A1.SS1.p12.5.m5.1.1.cmml">z</mi><mo id="A1.SS1.p12.5.m5.1.2.2.2.2.2" stretchy="false" xref="A1.SS1.p12.5.m5.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.p12.5.m5.1.2.2.3" xref="A1.SS1.p12.5.m5.1.2.2.3.cmml">p</mi></msub><mo id="A1.SS1.p12.5.m5.1.2.1" xref="A1.SS1.p12.5.m5.1.2.1.cmml">=</mo><mn id="A1.SS1.p12.5.m5.1.2.3" xref="A1.SS1.p12.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.5.m5.1b"><apply id="A1.SS1.p12.5.m5.1.2.cmml" xref="A1.SS1.p12.5.m5.1.2"><eq id="A1.SS1.p12.5.m5.1.2.1.cmml" xref="A1.SS1.p12.5.m5.1.2.1"></eq><apply id="A1.SS1.p12.5.m5.1.2.2.cmml" xref="A1.SS1.p12.5.m5.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.p12.5.m5.1.2.2.1.cmml" xref="A1.SS1.p12.5.m5.1.2.2">subscript</csymbol><apply id="A1.SS1.p12.5.m5.1.2.2.2.1.cmml" xref="A1.SS1.p12.5.m5.1.2.2.2.2"><csymbol cd="latexml" id="A1.SS1.p12.5.m5.1.2.2.2.1.1.cmml" xref="A1.SS1.p12.5.m5.1.2.2.2.2.1">norm</csymbol><ci id="A1.SS1.p12.5.m5.1.1.cmml" xref="A1.SS1.p12.5.m5.1.1">𝑧</ci></apply><ci id="A1.SS1.p12.5.m5.1.2.2.3.cmml" xref="A1.SS1.p12.5.m5.1.2.2.3">𝑝</ci></apply><cn id="A1.SS1.p12.5.m5.1.2.3.cmml" type="integer" xref="A1.SS1.p12.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.5.m5.1c">||z||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.5.m5.1d">| | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Note that by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a>, knowledge about the cases with <math alttext="||z||_{p}=1" class="ltx_Math" display="inline" id="A1.SS1.p12.6.m6.1"><semantics id="A1.SS1.p12.6.m6.1a"><mrow id="A1.SS1.p12.6.m6.1.2" xref="A1.SS1.p12.6.m6.1.2.cmml"><msub id="A1.SS1.p12.6.m6.1.2.2" xref="A1.SS1.p12.6.m6.1.2.2.cmml"><mrow id="A1.SS1.p12.6.m6.1.2.2.2.2" xref="A1.SS1.p12.6.m6.1.2.2.2.1.cmml"><mo id="A1.SS1.p12.6.m6.1.2.2.2.2.1" stretchy="false" xref="A1.SS1.p12.6.m6.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.p12.6.m6.1.1" xref="A1.SS1.p12.6.m6.1.1.cmml">z</mi><mo id="A1.SS1.p12.6.m6.1.2.2.2.2.2" stretchy="false" xref="A1.SS1.p12.6.m6.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.p12.6.m6.1.2.2.3" xref="A1.SS1.p12.6.m6.1.2.2.3.cmml">p</mi></msub><mo id="A1.SS1.p12.6.m6.1.2.1" xref="A1.SS1.p12.6.m6.1.2.1.cmml">=</mo><mn id="A1.SS1.p12.6.m6.1.2.3" xref="A1.SS1.p12.6.m6.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p12.6.m6.1b"><apply id="A1.SS1.p12.6.m6.1.2.cmml" xref="A1.SS1.p12.6.m6.1.2"><eq id="A1.SS1.p12.6.m6.1.2.1.cmml" xref="A1.SS1.p12.6.m6.1.2.1"></eq><apply id="A1.SS1.p12.6.m6.1.2.2.cmml" xref="A1.SS1.p12.6.m6.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.p12.6.m6.1.2.2.1.cmml" xref="A1.SS1.p12.6.m6.1.2.2">subscript</csymbol><apply id="A1.SS1.p12.6.m6.1.2.2.2.1.cmml" xref="A1.SS1.p12.6.m6.1.2.2.2.2"><csymbol cd="latexml" id="A1.SS1.p12.6.m6.1.2.2.2.1.1.cmml" xref="A1.SS1.p12.6.m6.1.2.2.2.2.1">norm</csymbol><ci id="A1.SS1.p12.6.m6.1.1.cmml" xref="A1.SS1.p12.6.m6.1.1">𝑧</ci></apply><ci id="A1.SS1.p12.6.m6.1.2.2.3.cmml" xref="A1.SS1.p12.6.m6.1.2.2.3">𝑝</ci></apply><cn id="A1.SS1.p12.6.m6.1.2.3.cmml" type="integer" xref="A1.SS1.p12.6.m6.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p12.6.m6.1c">||z||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p12.6.m6.1d">| | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math> already suffices to make full use of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_observation" id="A1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem1.1.1.1">Observation A.1</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem1.p1"> <p class="ltx_p" id="A1.Thmtheorem1.p1.6"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem1.p1.6.6">For <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.1.1.m1.2"><semantics id="A1.Thmtheorem1.p1.1.1.m1.2a"><mrow id="A1.Thmtheorem1.p1.1.1.m1.2.3" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="A1.Thmtheorem1.p1.1.1.m1.2.3.2" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="A1.Thmtheorem1.p1.1.1.m1.2.3.1" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="A1.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="A1.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="A1.Thmtheorem1.p1.1.1.m1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="A1.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="A1.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="A1.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="A1.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.1.1.m1.2b"><apply id="A1.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.2.3"><in id="A1.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.1"></in><ci id="A1.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="A1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.2.3.3.2"><cn id="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="A1.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math> and arbitrary <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.2.2.m2.1"><semantics id="A1.Thmtheorem1.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem1.p1.2.2.m2.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">z</mi><mo id="A1.Thmtheorem1.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem1.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.2.2.m2.1b"><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1"><in id="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.1"></in><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2">𝑧</ci><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.2.2.m2.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.2.2.m2.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{p}=1" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.3.3.m3.1"><semantics id="A1.Thmtheorem1.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.cmml"><msub id="A1.Thmtheorem1.p1.3.3.m3.1.2.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.cmml"><mrow id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.2" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.1.cmml"><mo id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.2.1" stretchy="false" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml">z</mi><mo id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.2.2" stretchy="false" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.3.cmml">p</mi></msub><mo id="A1.Thmtheorem1.p1.3.3.m3.1.2.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.1.cmml">=</mo><mn id="A1.Thmtheorem1.p1.3.3.m3.1.2.3" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.3.3.m3.1b"><apply id="A1.Thmtheorem1.p1.3.3.m3.1.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2"><eq id="A1.Thmtheorem1.p1.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.1"></eq><apply id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2">subscript</csymbol><apply id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1">𝑧</ci></apply><ci id="A1.Thmtheorem1.p1.3.3.m3.1.2.2.3.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.2.3">𝑝</ci></apply><cn id="A1.Thmtheorem1.p1.3.3.m3.1.2.3.cmml" type="integer" xref="A1.Thmtheorem1.p1.3.3.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.3.3.m3.1c">||z||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.3.3.m3.1d">| | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math>, <math alttext="||\cdot||_{p}" class="ltx_math_unparsed" display="inline" id="A1.Thmtheorem1.p1.4.4.m4.1"><semantics id="A1.Thmtheorem1.p1.4.4.m4.1a"><mrow id="A1.Thmtheorem1.p1.4.4.m4.1b"><mo fence="false" id="A1.Thmtheorem1.p1.4.4.m4.1.1" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="A1.Thmtheorem1.p1.4.4.m4.1.2" stretchy="false">|</mo><mo id="A1.Thmtheorem1.p1.4.4.m4.1.3" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A1.Thmtheorem1.p1.4.4.m4.1.4" rspace="0.167em" stretchy="false">|</mo><msub id="A1.Thmtheorem1.p1.4.4.m4.1.5"><mo fence="false" id="A1.Thmtheorem1.p1.4.4.m4.1.5.2" stretchy="false">|</mo><mi id="A1.Thmtheorem1.p1.4.4.m4.1.5.3">p</mi></msub></mrow><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.4.4.m4.1c">||\cdot||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.4.4.m4.1d">| | ⋅ | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is differentiable at <math alttext="z" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.5.5.m5.1"><semantics id="A1.Thmtheorem1.p1.5.5.m5.1a"><mi id="A1.Thmtheorem1.p1.5.5.m5.1.1" xref="A1.Thmtheorem1.p1.5.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.5.5.m5.1b"><ci id="A1.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem1.p1.5.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.5.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.5.5.m5.1d">italic_z</annotation></semantics></math> and the gradient <math alttext="\nabla||z||_{p}" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.6.6.m6.1"><semantics id="A1.Thmtheorem1.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem1.p1.6.6.m6.1.2" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.cmml"><mo id="A1.Thmtheorem1.p1.6.6.m6.1.2.2" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.2.cmml">∇</mo><mo id="A1.Thmtheorem1.p1.6.6.m6.1.2.1" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.1.cmml">⁢</mo><msub id="A1.Thmtheorem1.p1.6.6.m6.1.2.3" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.cmml"><mrow id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.2" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.1.cmml"><mo id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.2.1" stretchy="false" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem1.p1.6.6.m6.1.1" xref="A1.Thmtheorem1.p1.6.6.m6.1.1.cmml">z</mi><mo id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.2.2" stretchy="false" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.3" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.6.6.m6.1b"><apply id="A1.Thmtheorem1.p1.6.6.m6.1.2.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2"><times id="A1.Thmtheorem1.p1.6.6.m6.1.2.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.1"></times><ci id="A1.Thmtheorem1.p1.6.6.m6.1.2.2.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.2">∇</ci><apply id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3">subscript</csymbol><apply id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.2"><csymbol cd="latexml" id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.1.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.2.2.1">norm</csymbol><ci id="A1.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.1">𝑧</ci></apply><ci id="A1.Thmtheorem1.p1.6.6.m6.1.2.3.3.cmml" xref="A1.Thmtheorem1.p1.6.6.m6.1.2.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.6.6.m6.1c">\nabla||z||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.6.6.m6.1d">∇ | | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is given by</span></p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\nabla||z||_{p}=\begin{bmatrix}|z_{1}|^{p-1}sign(z_{1})\\ \vdots\\ |z_{d}|^{p-1}sign(z_{d})\end{bmatrix}." class="ltx_Math" display="block" id="A1.Ex4.m1.3"><semantics id="A1.Ex4.m1.3a"><mrow id="A1.Ex4.m1.3.3.1" xref="A1.Ex4.m1.3.3.1.1.cmml"><mrow id="A1.Ex4.m1.3.3.1.1" xref="A1.Ex4.m1.3.3.1.1.cmml"><mrow id="A1.Ex4.m1.3.3.1.1.2" xref="A1.Ex4.m1.3.3.1.1.2.cmml"><mo id="A1.Ex4.m1.3.3.1.1.2.2" xref="A1.Ex4.m1.3.3.1.1.2.2.cmml">∇</mo><mo id="A1.Ex4.m1.3.3.1.1.2.1" xref="A1.Ex4.m1.3.3.1.1.2.1.cmml">⁢</mo><msub id="A1.Ex4.m1.3.3.1.1.2.3" xref="A1.Ex4.m1.3.3.1.1.2.3.cmml"><mrow id="A1.Ex4.m1.3.3.1.1.2.3.2.2" xref="A1.Ex4.m1.3.3.1.1.2.3.2.1.cmml"><mo id="A1.Ex4.m1.3.3.1.1.2.3.2.2.1" stretchy="false" xref="A1.Ex4.m1.3.3.1.1.2.3.2.1.1.cmml">‖</mo><mi id="A1.Ex4.m1.2.2" xref="A1.Ex4.m1.2.2.cmml">z</mi><mo id="A1.Ex4.m1.3.3.1.1.2.3.2.2.2" stretchy="false" xref="A1.Ex4.m1.3.3.1.1.2.3.2.1.1.cmml">‖</mo></mrow><mi id="A1.Ex4.m1.3.3.1.1.2.3.3" xref="A1.Ex4.m1.3.3.1.1.2.3.3.cmml">p</mi></msub></mrow><mo id="A1.Ex4.m1.3.3.1.1.1" xref="A1.Ex4.m1.3.3.1.1.1.cmml">=</mo><mrow id="A1.Ex4.m1.1.1.3" xref="A1.Ex4.m1.1.1.2.cmml"><mo id="A1.Ex4.m1.1.1.3.1" xref="A1.Ex4.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="A1.Ex4.m1.1.1.1.1" rowspacing="0pt" xref="A1.Ex4.m1.1.1.1.1.cmml"><mtr id="A1.Ex4.m1.1.1.1.1a" xref="A1.Ex4.m1.1.1.1.1.cmml"><mtd id="A1.Ex4.m1.1.1.1.1b" xref="A1.Ex4.m1.1.1.1.1.cmml"><mrow id="A1.Ex4.m1.1.1.1.1.2.2.2.2" 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start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p - 1 end_POSTSUPERSCRIPT italic_s italic_i italic_g italic_n ( italic_z start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.Thmtheorem1.p1.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem1.p1.7.1">In particular, we have <math alttext="\langle\nabla||z||_{p},z\rangle=\left\lVert\nabla\left(||z||_{p}\right)\right% \rVert_{\frac{p}{p-1}}=||z||_{p}=1" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.7.1.m1.7"><semantics id="A1.Thmtheorem1.p1.7.1.m1.7a"><mrow id="A1.Thmtheorem1.p1.7.1.m1.7.7" xref="A1.Thmtheorem1.p1.7.1.m1.7.7.cmml"><mrow id="A1.Thmtheorem1.p1.7.1.m1.6.6.1.1" xref="A1.Thmtheorem1.p1.7.1.m1.6.6.1.2.cmml"><mo id="A1.Thmtheorem1.p1.7.1.m1.6.6.1.1.2" stretchy="false" 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xref="A1.SS1.p13.3.m3.1.1.1.cmml">∈</mo><msup id="A1.SS1.p13.3.m3.1.1.3" xref="A1.SS1.p13.3.m3.1.1.3.cmml"><mi id="A1.SS1.p13.3.m3.1.1.3.2" xref="A1.SS1.p13.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.p13.3.m3.1.1.3.3" xref="A1.SS1.p13.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p13.3.m3.1b"><apply id="A1.SS1.p13.3.m3.1.1.cmml" xref="A1.SS1.p13.3.m3.1.1"><in id="A1.SS1.p13.3.m3.1.1.1.cmml" xref="A1.SS1.p13.3.m3.1.1.1"></in><ci id="A1.SS1.p13.3.m3.1.1.2.cmml" xref="A1.SS1.p13.3.m3.1.1.2">𝑧</ci><apply id="A1.SS1.p13.3.m3.1.1.3.cmml" xref="A1.SS1.p13.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p13.3.m3.1.1.3.1.cmml" xref="A1.SS1.p13.3.m3.1.1.3">superscript</csymbol><ci id="A1.SS1.p13.3.m3.1.1.3.2.cmml" xref="A1.SS1.p13.3.m3.1.1.3.2">ℝ</ci><ci id="A1.SS1.p13.3.m3.1.1.3.3.cmml" xref="A1.SS1.p13.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p13.3.m3.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p13.3.m3.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{\infty}=1" class="ltx_Math" display="inline" id="A1.SS1.p13.4.m4.1"><semantics id="A1.SS1.p13.4.m4.1a"><mrow id="A1.SS1.p13.4.m4.1.2" xref="A1.SS1.p13.4.m4.1.2.cmml"><msub id="A1.SS1.p13.4.m4.1.2.2" xref="A1.SS1.p13.4.m4.1.2.2.cmml"><mrow id="A1.SS1.p13.4.m4.1.2.2.2.2" xref="A1.SS1.p13.4.m4.1.2.2.2.1.cmml"><mo id="A1.SS1.p13.4.m4.1.2.2.2.2.1" stretchy="false" xref="A1.SS1.p13.4.m4.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.p13.4.m4.1.1" xref="A1.SS1.p13.4.m4.1.1.cmml">z</mi><mo id="A1.SS1.p13.4.m4.1.2.2.2.2.2" stretchy="false" xref="A1.SS1.p13.4.m4.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.p13.4.m4.1.2.2.3" mathvariant="normal" xref="A1.SS1.p13.4.m4.1.2.2.3.cmml">∞</mi></msub><mo id="A1.SS1.p13.4.m4.1.2.1" xref="A1.SS1.p13.4.m4.1.2.1.cmml">=</mo><mn id="A1.SS1.p13.4.m4.1.2.3" xref="A1.SS1.p13.4.m4.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p13.4.m4.1b"><apply id="A1.SS1.p13.4.m4.1.2.cmml" xref="A1.SS1.p13.4.m4.1.2"><eq id="A1.SS1.p13.4.m4.1.2.1.cmml" xref="A1.SS1.p13.4.m4.1.2.1"></eq><apply id="A1.SS1.p13.4.m4.1.2.2.cmml" xref="A1.SS1.p13.4.m4.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.p13.4.m4.1.2.2.1.cmml" xref="A1.SS1.p13.4.m4.1.2.2">subscript</csymbol><apply id="A1.SS1.p13.4.m4.1.2.2.2.1.cmml" xref="A1.SS1.p13.4.m4.1.2.2.2.2"><csymbol cd="latexml" id="A1.SS1.p13.4.m4.1.2.2.2.1.1.cmml" xref="A1.SS1.p13.4.m4.1.2.2.2.2.1">norm</csymbol><ci id="A1.SS1.p13.4.m4.1.1.cmml" xref="A1.SS1.p13.4.m4.1.1">𝑧</ci></apply><infinity id="A1.SS1.p13.4.m4.1.2.2.3.cmml" xref="A1.SS1.p13.4.m4.1.2.2.3"></infinity></apply><cn id="A1.SS1.p13.4.m4.1.2.3.cmml" type="integer" xref="A1.SS1.p13.4.m4.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p13.4.m4.1c">||z||_{\infty}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p13.4.m4.1d">| | italic_z | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = 1</annotation></semantics></math> (<math alttext="||z||_{1}=1" class="ltx_Math" display="inline" id="A1.SS1.p13.5.m5.1"><semantics id="A1.SS1.p13.5.m5.1a"><mrow id="A1.SS1.p13.5.m5.1.2" xref="A1.SS1.p13.5.m5.1.2.cmml"><msub id="A1.SS1.p13.5.m5.1.2.2" xref="A1.SS1.p13.5.m5.1.2.2.cmml"><mrow id="A1.SS1.p13.5.m5.1.2.2.2.2" xref="A1.SS1.p13.5.m5.1.2.2.2.1.cmml"><mo id="A1.SS1.p13.5.m5.1.2.2.2.2.1" stretchy="false" xref="A1.SS1.p13.5.m5.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.p13.5.m5.1.1" xref="A1.SS1.p13.5.m5.1.1.cmml">z</mi><mo id="A1.SS1.p13.5.m5.1.2.2.2.2.2" stretchy="false" xref="A1.SS1.p13.5.m5.1.2.2.2.1.1.cmml">‖</mo></mrow><mn id="A1.SS1.p13.5.m5.1.2.2.3" xref="A1.SS1.p13.5.m5.1.2.2.3.cmml">1</mn></msub><mo id="A1.SS1.p13.5.m5.1.2.1" xref="A1.SS1.p13.5.m5.1.2.1.cmml">=</mo><mn id="A1.SS1.p13.5.m5.1.2.3" xref="A1.SS1.p13.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p13.5.m5.1b"><apply id="A1.SS1.p13.5.m5.1.2.cmml" xref="A1.SS1.p13.5.m5.1.2"><eq id="A1.SS1.p13.5.m5.1.2.1.cmml" xref="A1.SS1.p13.5.m5.1.2.1"></eq><apply id="A1.SS1.p13.5.m5.1.2.2.cmml" xref="A1.SS1.p13.5.m5.1.2.2"><csymbol cd="ambiguous" id="A1.SS1.p13.5.m5.1.2.2.1.cmml" xref="A1.SS1.p13.5.m5.1.2.2">subscript</csymbol><apply id="A1.SS1.p13.5.m5.1.2.2.2.1.cmml" xref="A1.SS1.p13.5.m5.1.2.2.2.2"><csymbol cd="latexml" id="A1.SS1.p13.5.m5.1.2.2.2.1.1.cmml" xref="A1.SS1.p13.5.m5.1.2.2.2.2.1">norm</csymbol><ci id="A1.SS1.p13.5.m5.1.1.cmml" xref="A1.SS1.p13.5.m5.1.1">𝑧</ci></apply><cn id="A1.SS1.p13.5.m5.1.2.2.3.cmml" type="integer" xref="A1.SS1.p13.5.m5.1.2.2.3">1</cn></apply><cn id="A1.SS1.p13.5.m5.1.2.3.cmml" type="integer" xref="A1.SS1.p13.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p13.5.m5.1c">||z||_{1}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p13.5.m5.1d">| | italic_z | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, respectively). Note that these subdifferentials can also be found in the literature (see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#bib.bib34" title="">34</a>]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_observation" id="A1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem2.1.1.1">Observation A.2</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem2.p1"> <p class="ltx_p" id="A1.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem2.p1.3.3">Consider arbitrary <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.1.1.m1.1"><semantics id="A1.Thmtheorem2.p1.1.1.m1.1a"><mrow id="A1.Thmtheorem2.p1.1.1.m1.1.1" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1.2" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">z</mi><mo id="A1.Thmtheorem2.p1.1.1.m1.1.1.1" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem2.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.1.1.m1.1b"><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1"><in id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1"></in><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.2">𝑧</ci><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.1.1.m1.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.1.1.m1.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{\infty}=1" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.2.2.m2.1"><semantics id="A1.Thmtheorem2.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem2.p1.2.2.m2.1.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.cmml"><msub id="A1.Thmtheorem2.p1.2.2.m2.1.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.cmml"><mrow id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.1.cmml"><mo id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml">z</mi><mo id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.3" mathvariant="normal" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.3.cmml">∞</mi></msub><mo id="A1.Thmtheorem2.p1.2.2.m2.1.2.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.1.cmml">=</mo><mn id="A1.Thmtheorem2.p1.2.2.m2.1.2.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.2.2.m2.1b"><apply id="A1.Thmtheorem2.p1.2.2.m2.1.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2"><eq id="A1.Thmtheorem2.p1.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.1"></eq><apply id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2">subscript</csymbol><apply id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1">𝑧</ci></apply><infinity id="A1.Thmtheorem2.p1.2.2.m2.1.2.2.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.2.3"></infinity></apply><cn id="A1.Thmtheorem2.p1.2.2.m2.1.2.3.cmml" type="integer" xref="A1.Thmtheorem2.p1.2.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.2.2.m2.1c">||z||_{\infty}=1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.2.2.m2.1d">| | italic_z | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = 1</annotation></semantics></math>. The set of subgradients <math alttext="\partial||z||_{\infty}" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.3.3.m3.1"><semantics id="A1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem2.p1.3.3.m3.1.2" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mo id="A1.Thmtheorem2.p1.3.3.m3.1.2.1" rspace="0em" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.1.cmml">∂</mo><msub id="A1.Thmtheorem2.p1.3.3.m3.1.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.cmml"><mrow id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.1.cmml"><mo id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem2.p1.3.3.m3.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.1.1.cmml">z</mi><mo id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.3" mathvariant="normal" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.3.cmml">∞</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.3.3.m3.1b"><apply id="A1.Thmtheorem2.p1.3.3.m3.1.2.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2"><partialdiff id="A1.Thmtheorem2.p1.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.1"></partialdiff><apply id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2">subscript</csymbol><apply id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.1.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.1">𝑧</ci></apply><infinity id="A1.Thmtheorem2.p1.3.3.m3.1.2.2.3.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.2.2.3"></infinity></apply></apply></annotation-xml><annotation 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id="A1.Ex5.m1.2.2.1.1.1.1.1.1.3.1.1.2.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.3.1.1.2">𝑒</ci><ci id="A1.Ex5.m1.2.2.1.1.1.1.1.1.3.1.1.3.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.3.1.1.3">𝑖</ci></apply><apply id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2"><eq id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.1.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.1"></eq><apply id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2"><csymbol cd="ambiguous" id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.1.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2">subscript</csymbol><ci id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.2.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.2">𝑧</ci><ci id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.3.cmml" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.2.3">𝑖</ci></apply><cn id="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.3.cmml" type="integer" xref="A1.Ex5.m1.2.2.1.1.1.1.1.1.4.2.2.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex5.m1.2c">\partial||z||_{\infty}=\text{conv}\left(\{-e_{i}\mid z_{i}=-1\}\cup\{e_{i}\mid z% _{i}=1\}\right),</annotation><annotation encoding="application/x-llamapun" id="A1.Ex5.m1.2d">∂ | | italic_z | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = conv ( { - italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 1 } ∪ { italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 } ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.Thmtheorem2.p1.8"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem2.p1.8.5">where <math alttext="e_{i}" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.4.1.m1.1"><semantics id="A1.Thmtheorem2.p1.4.1.m1.1a"><msub id="A1.Thmtheorem2.p1.4.1.m1.1.1" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.cmml"><mi id="A1.Thmtheorem2.p1.4.1.m1.1.1.2" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.2.cmml">e</mi><mi id="A1.Thmtheorem2.p1.4.1.m1.1.1.3" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.4.1.m1.1b"><apply id="A1.Thmtheorem2.p1.4.1.m1.1.1.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.4.1.m1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem2.p1.4.1.m1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.2">𝑒</ci><ci id="A1.Thmtheorem2.p1.4.1.m1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.4.1.m1.1c">e_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.4.1.m1.1d">italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> denotes the <math alttext="i" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.5.2.m2.1"><semantics id="A1.Thmtheorem2.p1.5.2.m2.1a"><mi id="A1.Thmtheorem2.p1.5.2.m2.1.1" xref="A1.Thmtheorem2.p1.5.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.5.2.m2.1b"><ci id="A1.Thmtheorem2.p1.5.2.m2.1.1.cmml" xref="A1.Thmtheorem2.p1.5.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.5.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.5.2.m2.1d">italic_i</annotation></semantics></math>-th standard unit vector in <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.6.3.m3.1"><semantics id="A1.Thmtheorem2.p1.6.3.m3.1a"><msup id="A1.Thmtheorem2.p1.6.3.m3.1.1" xref="A1.Thmtheorem2.p1.6.3.m3.1.1.cmml"><mi id="A1.Thmtheorem2.p1.6.3.m3.1.1.2" xref="A1.Thmtheorem2.p1.6.3.m3.1.1.2.cmml">ℝ</mi><mi id="A1.Thmtheorem2.p1.6.3.m3.1.1.3" xref="A1.Thmtheorem2.p1.6.3.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.6.3.m3.1b"><apply id="A1.Thmtheorem2.p1.6.3.m3.1.1.cmml" xref="A1.Thmtheorem2.p1.6.3.m3.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.6.3.m3.1.1.1.cmml" xref="A1.Thmtheorem2.p1.6.3.m3.1.1">superscript</csymbol><ci id="A1.Thmtheorem2.p1.6.3.m3.1.1.2.cmml" xref="A1.Thmtheorem2.p1.6.3.m3.1.1.2">ℝ</ci><ci id="A1.Thmtheorem2.p1.6.3.m3.1.1.3.cmml" xref="A1.Thmtheorem2.p1.6.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.6.3.m3.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.6.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. In particular, we have <math alttext="\langle u,z\rangle=||u||_{1}=||z||_{\infty}=1" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.7.4.m4.4"><semantics id="A1.Thmtheorem2.p1.7.4.m4.4a"><mrow id="A1.Thmtheorem2.p1.7.4.m4.4.5" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.cmml"><mrow id="A1.Thmtheorem2.p1.7.4.m4.4.5.2.2" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.2.1.cmml"><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.2.1.cmml">⟨</mo><mi id="A1.Thmtheorem2.p1.7.4.m4.1.1" xref="A1.Thmtheorem2.p1.7.4.m4.1.1.cmml">u</mi><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.2.2.2" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.2.1.cmml">,</mo><mi id="A1.Thmtheorem2.p1.7.4.m4.2.2" xref="A1.Thmtheorem2.p1.7.4.m4.2.2.cmml">z</mi><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.2.2.3" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.2.1.cmml">⟩</mo></mrow><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.3" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.3.cmml">=</mo><msub id="A1.Thmtheorem2.p1.7.4.m4.4.5.4" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.cmml"><mrow id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.2" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.1.cmml"><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem2.p1.7.4.m4.3.3" xref="A1.Thmtheorem2.p1.7.4.m4.3.3.cmml">u</mi><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.1.1.cmml">‖</mo></mrow><mn id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.3" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.3.cmml">1</mn></msub><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.5" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.5.cmml">=</mo><msub id="A1.Thmtheorem2.p1.7.4.m4.4.5.6" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.cmml"><mrow id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.2" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.1.cmml"><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem2.p1.7.4.m4.4.4" xref="A1.Thmtheorem2.p1.7.4.m4.4.4.cmml">z</mi><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.3" mathvariant="normal" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.3.cmml">∞</mi></msub><mo id="A1.Thmtheorem2.p1.7.4.m4.4.5.7" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.7.cmml">=</mo><mn id="A1.Thmtheorem2.p1.7.4.m4.4.5.8" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.8.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.7.4.m4.4b"><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"><and id="A1.Thmtheorem2.p1.7.4.m4.4.5a.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"></and><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5b.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"><eq id="A1.Thmtheorem2.p1.7.4.m4.4.5.3.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.3"></eq><list id="A1.Thmtheorem2.p1.7.4.m4.4.5.2.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.2.2"><ci id="A1.Thmtheorem2.p1.7.4.m4.1.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.1.1">𝑢</ci><ci id="A1.Thmtheorem2.p1.7.4.m4.2.2.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.2.2">𝑧</ci></list><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4">subscript</csymbol><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.2"><csymbol cd="latexml" id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.1.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.2.2.1">norm</csymbol><ci id="A1.Thmtheorem2.p1.7.4.m4.3.3.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.3.3">𝑢</ci></apply><cn id="A1.Thmtheorem2.p1.7.4.m4.4.5.4.3.cmml" type="integer" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.4.3">1</cn></apply></apply><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5c.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"><eq id="A1.Thmtheorem2.p1.7.4.m4.4.5.5.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.5"></eq><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem2.p1.7.4.m4.4.5.4.cmml" id="A1.Thmtheorem2.p1.7.4.m4.4.5d.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"></share><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6">subscript</csymbol><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.2"><csymbol cd="latexml" id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.1.1.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.2.2.1">norm</csymbol><ci id="A1.Thmtheorem2.p1.7.4.m4.4.4.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.4">𝑧</ci></apply><infinity id="A1.Thmtheorem2.p1.7.4.m4.4.5.6.3.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.6.3"></infinity></apply></apply><apply id="A1.Thmtheorem2.p1.7.4.m4.4.5e.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"><eq id="A1.Thmtheorem2.p1.7.4.m4.4.5.7.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.7"></eq><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem2.p1.7.4.m4.4.5.6.cmml" id="A1.Thmtheorem2.p1.7.4.m4.4.5f.cmml" xref="A1.Thmtheorem2.p1.7.4.m4.4.5"></share><cn id="A1.Thmtheorem2.p1.7.4.m4.4.5.8.cmml" type="integer" xref="A1.Thmtheorem2.p1.7.4.m4.4.5.8">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.7.4.m4.4c">\langle u,z\rangle=||u||_{1}=||z||_{\infty}=1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.7.4.m4.4d">⟨ italic_u , italic_z ⟩ = | | italic_u | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = | | italic_z | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = 1</annotation></semantics></math> for all <math alttext="u\in\partial||z||_{\infty}" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.8.5.m5.1"><semantics id="A1.Thmtheorem2.p1.8.5.m5.1a"><mrow id="A1.Thmtheorem2.p1.8.5.m5.1.2" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.cmml"><mi id="A1.Thmtheorem2.p1.8.5.m5.1.2.2" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.2.cmml">u</mi><mo id="A1.Thmtheorem2.p1.8.5.m5.1.2.1" rspace="0.1389em" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.1.cmml">∈</mo><mrow id="A1.Thmtheorem2.p1.8.5.m5.1.2.3" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.cmml"><mo id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.1" lspace="0.1389em" rspace="0em" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.1.cmml">∂</mo><msub id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.cmml"><mrow id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.2" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.1.cmml"><mo id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem2.p1.8.5.m5.1.1" xref="A1.Thmtheorem2.p1.8.5.m5.1.1.cmml">z</mi><mo id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.3" mathvariant="normal" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.3.cmml">∞</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.8.5.m5.1b"><apply id="A1.Thmtheorem2.p1.8.5.m5.1.2.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2"><in id="A1.Thmtheorem2.p1.8.5.m5.1.2.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.1"></in><ci id="A1.Thmtheorem2.p1.8.5.m5.1.2.2.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.2">𝑢</ci><apply id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3"><partialdiff id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.1"></partialdiff><apply id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2">subscript</csymbol><apply id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.1.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem2.p1.8.5.m5.1.1.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.1">𝑧</ci></apply><infinity id="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.3.cmml" xref="A1.Thmtheorem2.p1.8.5.m5.1.2.3.2.3"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.8.5.m5.1c">u\in\partial||z||_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.8.5.m5.1d">italic_u ∈ ∂ | | italic_z | | start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="A1.SS1.p14"> <p class="ltx_p" id="A1.SS1.p14.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem6" title="Observation 4.6. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">4.6</span></a></p> </div> <div class="ltx_para" id="A1.SS1.p15"> <p class="ltx_p" id="A1.SS1.p15.6">Consider a subgradient <math alttext="u" class="ltx_Math" display="inline" id="A1.SS1.p15.1.m1.1"><semantics id="A1.SS1.p15.1.m1.1a"><mi id="A1.SS1.p15.1.m1.1.1" xref="A1.SS1.p15.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p15.1.m1.1b"><ci id="A1.SS1.p15.1.m1.1.1.cmml" xref="A1.SS1.p15.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p15.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.1.m1.1d">italic_u</annotation></semantics></math> of <math alttext="||\cdot||_{p}" class="ltx_math_unparsed" display="inline" id="A1.SS1.p15.2.m2.1"><semantics id="A1.SS1.p15.2.m2.1a"><mrow id="A1.SS1.p15.2.m2.1b"><mo fence="false" id="A1.SS1.p15.2.m2.1.1" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="A1.SS1.p15.2.m2.1.2" stretchy="false">|</mo><mo id="A1.SS1.p15.2.m2.1.3" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A1.SS1.p15.2.m2.1.4" rspace="0.167em" stretchy="false">|</mo><msub id="A1.SS1.p15.2.m2.1.5"><mo fence="false" id="A1.SS1.p15.2.m2.1.5.2" stretchy="false">|</mo><mi id="A1.SS1.p15.2.m2.1.5.3">p</mi></msub></mrow><annotation encoding="application/x-tex" id="A1.SS1.p15.2.m2.1c">||\cdot||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.2.m2.1d">| | ⋅ | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> at point <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.p15.3.m3.1"><semantics id="A1.SS1.p15.3.m3.1a"><mi id="A1.SS1.p15.3.m3.1.1" xref="A1.SS1.p15.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p15.3.m3.1b"><ci id="A1.SS1.p15.3.m3.1.1.cmml" xref="A1.SS1.p15.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p15.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.3.m3.1d">italic_z</annotation></semantics></math>. Given the characterization of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.p15.4.m4.1"><semantics id="A1.SS1.p15.4.m4.1a"><msub id="A1.SS1.p15.4.m4.1.1" xref="A1.SS1.p15.4.m4.1.1.cmml"><mi id="A1.SS1.p15.4.m4.1.1.2" mathvariant="normal" xref="A1.SS1.p15.4.m4.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.p15.4.m4.1.1.3" xref="A1.SS1.p15.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p15.4.m4.1b"><apply id="A1.SS1.p15.4.m4.1.1.cmml" xref="A1.SS1.p15.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS1.p15.4.m4.1.1.1.cmml" xref="A1.SS1.p15.4.m4.1.1">subscript</csymbol><ci id="A1.SS1.p15.4.m4.1.1.2.cmml" xref="A1.SS1.p15.4.m4.1.1.2">ℓ</ci><ci id="A1.SS1.p15.4.m4.1.1.3.cmml" xref="A1.SS1.p15.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p15.4.m4.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.4.m4.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>, it will be useful to prove an upper bound on the angle between <math alttext="u" class="ltx_Math" display="inline" id="A1.SS1.p15.5.m5.1"><semantics id="A1.SS1.p15.5.m5.1a"><mi id="A1.SS1.p15.5.m5.1.1" xref="A1.SS1.p15.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p15.5.m5.1b"><ci id="A1.SS1.p15.5.m5.1.1.cmml" xref="A1.SS1.p15.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p15.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.5.m5.1d">italic_u</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.p15.6.m6.1"><semantics id="A1.SS1.p15.6.m6.1a"><mi id="A1.SS1.p15.6.m6.1.1" xref="A1.SS1.p15.6.m6.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p15.6.m6.1b"><ci id="A1.SS1.p15.6.m6.1.1.cmml" xref="A1.SS1.p15.6.m6.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p15.6.m6.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p15.6.m6.1d">italic_z</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="A1.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem3.1.1.1">Lemma A.3</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem3.p1"> <p class="ltx_p" id="A1.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem3.p1.5.5">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.1.1.m1.3"><semantics id="A1.Thmtheorem3.p1.1.1.m1.3a"><mrow id="A1.Thmtheorem3.p1.1.1.m1.3.4" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.cmml"><mi id="A1.Thmtheorem3.p1.1.1.m1.3.4.2" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.1" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="A1.Thmtheorem3.p1.1.1.m1.3.4.3" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.cmml"><mrow id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.2" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="A1.Thmtheorem3.p1.1.1.m1.1.1" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.cmml">1</mn><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.2.2" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="A1.Thmtheorem3.p1.1.1.m1.2.2" mathvariant="normal" xref="A1.Thmtheorem3.p1.1.1.m1.2.2.cmml">∞</mi><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.1" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.2" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="A1.Thmtheorem3.p1.1.1.m1.3.3" mathvariant="normal" xref="A1.Thmtheorem3.p1.1.1.m1.3.3.cmml">∞</mi><mo id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.1.1.m1.3b"><apply id="A1.Thmtheorem3.p1.1.1.m1.3.4.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4"><in id="A1.Thmtheorem3.p1.1.1.m1.3.4.1.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.1"></in><ci id="A1.Thmtheorem3.p1.1.1.m1.3.4.2.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.2">𝑝</ci><apply id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3"><union id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.1.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.1.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.2.2"><cn id="A1.Thmtheorem3.p1.1.1.m1.1.1.cmml" type="integer" xref="A1.Thmtheorem3.p1.1.1.m1.1.1">1</cn><infinity id="A1.Thmtheorem3.p1.1.1.m1.2.2.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.2.2"></infinity></interval><set id="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.1.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.4.3.3.2"><infinity id="A1.Thmtheorem3.p1.1.1.m1.3.3.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and consider arbitrary <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.2.2.m2.1"><semantics id="A1.Thmtheorem3.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem3.p1.2.2.m2.1.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">z</mi><mo id="A1.Thmtheorem3.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem3.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml"><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.2" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.3" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.2.2.m2.1b"><apply id="A1.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1"><in id="A1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.1"></in><ci id="A1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.2">𝑧</ci><apply id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.2">ℝ</ci><ci id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.2.2.m2.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.2.2.m2.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="||z||_{p}=1" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.3.3.m3.1"><semantics id="A1.Thmtheorem3.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.cmml"><msub id="A1.Thmtheorem3.p1.3.3.m3.1.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.cmml"><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.1.cmml"><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem3.p1.3.3.m3.1.1" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.cmml">z</mi><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.3" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.3.cmml">p</mi></msub><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.1" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.1.cmml">=</mo><mn id="A1.Thmtheorem3.p1.3.3.m3.1.2.3" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.3.3.m3.1b"><apply id="A1.Thmtheorem3.p1.3.3.m3.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2"><eq id="A1.Thmtheorem3.p1.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.1"></eq><apply id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2">subscript</csymbol><apply id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1">𝑧</ci></apply><ci id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.3">𝑝</ci></apply><cn id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.cmml" type="integer" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.3.3.m3.1c">||z||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.3.3.m3.1d">| | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Then we have <math alttext="\measuredangle(u,z)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.4.4.m4.2"><semantics id="A1.Thmtheorem3.p1.4.4.m4.2a"><mrow id="A1.Thmtheorem3.p1.4.4.m4.2.3" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.cmml"><mrow id="A1.Thmtheorem3.p1.4.4.m4.2.3.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.cmml"><mi id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.2" mathvariant="normal" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.2.cmml">∡</mi><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.1" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.1.cmml"><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.1.cmml">(</mo><mi id="A1.Thmtheorem3.p1.4.4.m4.1.1" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml">u</mi><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.2.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.1.cmml">,</mo><mi id="A1.Thmtheorem3.p1.4.4.m4.2.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.2.cmml">z</mi><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.2.3" stretchy="false" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.1" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.1.cmml">≤</mo><mrow id="A1.Thmtheorem3.p1.4.4.m4.2.3.3" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.cmml"><mfrac id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.cmml"><mi id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.2.cmml">π</mi><mn id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.3" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.3.cmml">2</mn></mfrac><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.1" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml">−</mo><msqrt id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.cmml"><mrow id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.cmml"><mpadded id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2" voffset="0.3em" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2.cmml"><mn id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2a" mathsize="70%" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1.cmml"><mo id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1a" stretchy="true" symmetric="true" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1.cmml">/</mo></mpadded><mi id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.3" mathsize="70%" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.3.cmml">d</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.4.4.m4.2b"><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3"><leq id="A1.Thmtheorem3.p1.4.4.m4.2.3.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.1"></leq><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2"><times id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.1"></times><ci id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.2">∡</ci><interval closure="open" id="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.2.3.2"><ci id="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1">𝑢</ci><ci id="A1.Thmtheorem3.p1.4.4.m4.2.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.2">𝑧</ci></interval></apply><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3"><minus id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.1"></minus><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2"><divide id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2"></divide><ci id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.2">𝜋</ci><cn id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.3.cmml" type="integer" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.2.3">2</cn></apply><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3"><root id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3a.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3"></root><apply id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2"><divide id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.1"></divide><cn id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2.cmml" type="integer" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.2">1</cn><ci id="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.2.3.3.3.2.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.4.4.m4.2c">\measuredangle(u,z)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.4.4.m4.2d">∡ ( italic_u , italic_z ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG - square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math> for all subgradients <math alttext="u\in\partial||z||_{p}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.5.5.m5.1"><semantics id="A1.Thmtheorem3.p1.5.5.m5.1a"><mrow id="A1.Thmtheorem3.p1.5.5.m5.1.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.cmml"><mi id="A1.Thmtheorem3.p1.5.5.m5.1.2.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.2.cmml">u</mi><mo id="A1.Thmtheorem3.p1.5.5.m5.1.2.1" rspace="0.1389em" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.1.cmml">∈</mo><mrow id="A1.Thmtheorem3.p1.5.5.m5.1.2.3" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.cmml"><mo id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.1" lspace="0.1389em" rspace="0em" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.1.cmml">∂</mo><msub id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.cmml"><mrow id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.1.cmml"><mo id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.1.1.cmml">‖</mo><mi id="A1.Thmtheorem3.p1.5.5.m5.1.1" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml">z</mi><mo id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.3" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.5.5.m5.1b"><apply id="A1.Thmtheorem3.p1.5.5.m5.1.2.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2"><in id="A1.Thmtheorem3.p1.5.5.m5.1.2.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.1"></in><ci id="A1.Thmtheorem3.p1.5.5.m5.1.2.2.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.2">𝑢</ci><apply id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3"><partialdiff id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.1"></partialdiff><apply id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2">subscript</csymbol><apply id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.1.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.2.2.1">norm</csymbol><ci id="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1">𝑧</ci></apply><ci id="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.3.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.2.3.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.5.5.m5.1c">u\in\partial||z||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.5.5.m5.1d">italic_u ∈ ∂ | | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="A1.SS1.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.8.p1"> <p class="ltx_p" id="A1.SS1.8.p1.7">We heavily use the previous observations. In particular, recall that we have <math alttext="\langle u,z\rangle=||u||_{\frac{p}{p-1}}=||z||_{p}=1" class="ltx_Math" display="inline" id="A1.SS1.8.p1.1.m1.4"><semantics id="A1.SS1.8.p1.1.m1.4a"><mrow id="A1.SS1.8.p1.1.m1.4.5" xref="A1.SS1.8.p1.1.m1.4.5.cmml"><mrow id="A1.SS1.8.p1.1.m1.4.5.2.2" xref="A1.SS1.8.p1.1.m1.4.5.2.1.cmml"><mo id="A1.SS1.8.p1.1.m1.4.5.2.2.1" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.2.1.cmml">⟨</mo><mi id="A1.SS1.8.p1.1.m1.1.1" xref="A1.SS1.8.p1.1.m1.1.1.cmml">u</mi><mo id="A1.SS1.8.p1.1.m1.4.5.2.2.2" xref="A1.SS1.8.p1.1.m1.4.5.2.1.cmml">,</mo><mi id="A1.SS1.8.p1.1.m1.2.2" xref="A1.SS1.8.p1.1.m1.2.2.cmml">z</mi><mo id="A1.SS1.8.p1.1.m1.4.5.2.2.3" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.2.1.cmml">⟩</mo></mrow><mo id="A1.SS1.8.p1.1.m1.4.5.3" xref="A1.SS1.8.p1.1.m1.4.5.3.cmml">=</mo><msub id="A1.SS1.8.p1.1.m1.4.5.4" xref="A1.SS1.8.p1.1.m1.4.5.4.cmml"><mrow id="A1.SS1.8.p1.1.m1.4.5.4.2.2" xref="A1.SS1.8.p1.1.m1.4.5.4.2.1.cmml"><mo id="A1.SS1.8.p1.1.m1.4.5.4.2.2.1" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.4.2.1.1.cmml">‖</mo><mi id="A1.SS1.8.p1.1.m1.3.3" xref="A1.SS1.8.p1.1.m1.3.3.cmml">u</mi><mo id="A1.SS1.8.p1.1.m1.4.5.4.2.2.2" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.4.2.1.1.cmml">‖</mo></mrow><mfrac id="A1.SS1.8.p1.1.m1.4.5.4.3" xref="A1.SS1.8.p1.1.m1.4.5.4.3.cmml"><mi id="A1.SS1.8.p1.1.m1.4.5.4.3.2" xref="A1.SS1.8.p1.1.m1.4.5.4.3.2.cmml">p</mi><mrow id="A1.SS1.8.p1.1.m1.4.5.4.3.3" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.cmml"><mi id="A1.SS1.8.p1.1.m1.4.5.4.3.3.2" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.2.cmml">p</mi><mo id="A1.SS1.8.p1.1.m1.4.5.4.3.3.1" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.1.cmml">−</mo><mn id="A1.SS1.8.p1.1.m1.4.5.4.3.3.3" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.3.cmml">1</mn></mrow></mfrac></msub><mo id="A1.SS1.8.p1.1.m1.4.5.5" xref="A1.SS1.8.p1.1.m1.4.5.5.cmml">=</mo><msub id="A1.SS1.8.p1.1.m1.4.5.6" xref="A1.SS1.8.p1.1.m1.4.5.6.cmml"><mrow id="A1.SS1.8.p1.1.m1.4.5.6.2.2" xref="A1.SS1.8.p1.1.m1.4.5.6.2.1.cmml"><mo id="A1.SS1.8.p1.1.m1.4.5.6.2.2.1" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.6.2.1.1.cmml">‖</mo><mi id="A1.SS1.8.p1.1.m1.4.4" xref="A1.SS1.8.p1.1.m1.4.4.cmml">z</mi><mo id="A1.SS1.8.p1.1.m1.4.5.6.2.2.2" stretchy="false" xref="A1.SS1.8.p1.1.m1.4.5.6.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.8.p1.1.m1.4.5.6.3" xref="A1.SS1.8.p1.1.m1.4.5.6.3.cmml">p</mi></msub><mo id="A1.SS1.8.p1.1.m1.4.5.7" xref="A1.SS1.8.p1.1.m1.4.5.7.cmml">=</mo><mn id="A1.SS1.8.p1.1.m1.4.5.8" xref="A1.SS1.8.p1.1.m1.4.5.8.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.1.m1.4b"><apply id="A1.SS1.8.p1.1.m1.4.5.cmml" xref="A1.SS1.8.p1.1.m1.4.5"><and id="A1.SS1.8.p1.1.m1.4.5a.cmml" xref="A1.SS1.8.p1.1.m1.4.5"></and><apply id="A1.SS1.8.p1.1.m1.4.5b.cmml" xref="A1.SS1.8.p1.1.m1.4.5"><eq id="A1.SS1.8.p1.1.m1.4.5.3.cmml" xref="A1.SS1.8.p1.1.m1.4.5.3"></eq><list id="A1.SS1.8.p1.1.m1.4.5.2.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.2.2"><ci id="A1.SS1.8.p1.1.m1.1.1.cmml" xref="A1.SS1.8.p1.1.m1.1.1">𝑢</ci><ci id="A1.SS1.8.p1.1.m1.2.2.cmml" xref="A1.SS1.8.p1.1.m1.2.2">𝑧</ci></list><apply id="A1.SS1.8.p1.1.m1.4.5.4.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4"><csymbol cd="ambiguous" id="A1.SS1.8.p1.1.m1.4.5.4.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4">subscript</csymbol><apply id="A1.SS1.8.p1.1.m1.4.5.4.2.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.1.m1.4.5.4.2.1.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.1.m1.3.3.cmml" xref="A1.SS1.8.p1.1.m1.3.3">𝑢</ci></apply><apply id="A1.SS1.8.p1.1.m1.4.5.4.3.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3"><divide id="A1.SS1.8.p1.1.m1.4.5.4.3.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3"></divide><ci id="A1.SS1.8.p1.1.m1.4.5.4.3.2.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3.2">𝑝</ci><apply id="A1.SS1.8.p1.1.m1.4.5.4.3.3.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3"><minus id="A1.SS1.8.p1.1.m1.4.5.4.3.3.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.1"></minus><ci id="A1.SS1.8.p1.1.m1.4.5.4.3.3.2.cmml" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.2">𝑝</ci><cn id="A1.SS1.8.p1.1.m1.4.5.4.3.3.3.cmml" type="integer" xref="A1.SS1.8.p1.1.m1.4.5.4.3.3.3">1</cn></apply></apply></apply></apply><apply id="A1.SS1.8.p1.1.m1.4.5c.cmml" xref="A1.SS1.8.p1.1.m1.4.5"><eq id="A1.SS1.8.p1.1.m1.4.5.5.cmml" xref="A1.SS1.8.p1.1.m1.4.5.5"></eq><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.8.p1.1.m1.4.5.4.cmml" id="A1.SS1.8.p1.1.m1.4.5d.cmml" xref="A1.SS1.8.p1.1.m1.4.5"></share><apply id="A1.SS1.8.p1.1.m1.4.5.6.cmml" xref="A1.SS1.8.p1.1.m1.4.5.6"><csymbol cd="ambiguous" id="A1.SS1.8.p1.1.m1.4.5.6.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.6">subscript</csymbol><apply id="A1.SS1.8.p1.1.m1.4.5.6.2.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.6.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.1.m1.4.5.6.2.1.1.cmml" xref="A1.SS1.8.p1.1.m1.4.5.6.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.1.m1.4.4.cmml" xref="A1.SS1.8.p1.1.m1.4.4">𝑧</ci></apply><ci id="A1.SS1.8.p1.1.m1.4.5.6.3.cmml" xref="A1.SS1.8.p1.1.m1.4.5.6.3">𝑝</ci></apply></apply><apply id="A1.SS1.8.p1.1.m1.4.5e.cmml" xref="A1.SS1.8.p1.1.m1.4.5"><eq id="A1.SS1.8.p1.1.m1.4.5.7.cmml" xref="A1.SS1.8.p1.1.m1.4.5.7"></eq><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.8.p1.1.m1.4.5.6.cmml" id="A1.SS1.8.p1.1.m1.4.5f.cmml" xref="A1.SS1.8.p1.1.m1.4.5"></share><cn id="A1.SS1.8.p1.1.m1.4.5.8.cmml" type="integer" xref="A1.SS1.8.p1.1.m1.4.5.8">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.1.m1.4c">\langle u,z\rangle=||u||_{\frac{p}{p-1}}=||z||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.1.m1.4d">⟨ italic_u , italic_z ⟩ = | | italic_u | | start_POSTSUBSCRIPT divide start_ARG italic_p end_ARG start_ARG italic_p - 1 end_ARG end_POSTSUBSCRIPT = | | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math> for all subgradients <math alttext="u\in\partial||z||_{p}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.2.m2.1"><semantics id="A1.SS1.8.p1.2.m2.1a"><mrow id="A1.SS1.8.p1.2.m2.1.2" xref="A1.SS1.8.p1.2.m2.1.2.cmml"><mi id="A1.SS1.8.p1.2.m2.1.2.2" xref="A1.SS1.8.p1.2.m2.1.2.2.cmml">u</mi><mo id="A1.SS1.8.p1.2.m2.1.2.1" rspace="0.1389em" xref="A1.SS1.8.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="A1.SS1.8.p1.2.m2.1.2.3" xref="A1.SS1.8.p1.2.m2.1.2.3.cmml"><mo id="A1.SS1.8.p1.2.m2.1.2.3.1" lspace="0.1389em" rspace="0em" xref="A1.SS1.8.p1.2.m2.1.2.3.1.cmml">∂</mo><msub id="A1.SS1.8.p1.2.m2.1.2.3.2" xref="A1.SS1.8.p1.2.m2.1.2.3.2.cmml"><mrow id="A1.SS1.8.p1.2.m2.1.2.3.2.2.2" xref="A1.SS1.8.p1.2.m2.1.2.3.2.2.1.cmml"><mo id="A1.SS1.8.p1.2.m2.1.2.3.2.2.2.1" stretchy="false" xref="A1.SS1.8.p1.2.m2.1.2.3.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.8.p1.2.m2.1.1" xref="A1.SS1.8.p1.2.m2.1.1.cmml">z</mi><mo id="A1.SS1.8.p1.2.m2.1.2.3.2.2.2.2" stretchy="false" xref="A1.SS1.8.p1.2.m2.1.2.3.2.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.8.p1.2.m2.1.2.3.2.3" xref="A1.SS1.8.p1.2.m2.1.2.3.2.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.2.m2.1b"><apply id="A1.SS1.8.p1.2.m2.1.2.cmml" xref="A1.SS1.8.p1.2.m2.1.2"><in id="A1.SS1.8.p1.2.m2.1.2.1.cmml" xref="A1.SS1.8.p1.2.m2.1.2.1"></in><ci id="A1.SS1.8.p1.2.m2.1.2.2.cmml" xref="A1.SS1.8.p1.2.m2.1.2.2">𝑢</ci><apply id="A1.SS1.8.p1.2.m2.1.2.3.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3"><partialdiff id="A1.SS1.8.p1.2.m2.1.2.3.1.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.1"></partialdiff><apply id="A1.SS1.8.p1.2.m2.1.2.3.2.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.8.p1.2.m2.1.2.3.2.1.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.2">subscript</csymbol><apply id="A1.SS1.8.p1.2.m2.1.2.3.2.2.1.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.2.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.2.m2.1.2.3.2.2.1.1.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.2.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.2.m2.1.1.cmml" xref="A1.SS1.8.p1.2.m2.1.1">𝑧</ci></apply><ci id="A1.SS1.8.p1.2.m2.1.2.3.2.3.cmml" xref="A1.SS1.8.p1.2.m2.1.2.3.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.2.m2.1c">u\in\partial||z||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.2.m2.1d">italic_u ∈ ∂ | | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> (where <math alttext="\frac{p}{p-1}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.3.m3.1"><semantics id="A1.SS1.8.p1.3.m3.1a"><mfrac id="A1.SS1.8.p1.3.m3.1.1" xref="A1.SS1.8.p1.3.m3.1.1.cmml"><mi id="A1.SS1.8.p1.3.m3.1.1.2" xref="A1.SS1.8.p1.3.m3.1.1.2.cmml">p</mi><mrow id="A1.SS1.8.p1.3.m3.1.1.3" xref="A1.SS1.8.p1.3.m3.1.1.3.cmml"><mi id="A1.SS1.8.p1.3.m3.1.1.3.2" xref="A1.SS1.8.p1.3.m3.1.1.3.2.cmml">p</mi><mo id="A1.SS1.8.p1.3.m3.1.1.3.1" xref="A1.SS1.8.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="A1.SS1.8.p1.3.m3.1.1.3.3" xref="A1.SS1.8.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></mfrac><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.3.m3.1b"><apply id="A1.SS1.8.p1.3.m3.1.1.cmml" xref="A1.SS1.8.p1.3.m3.1.1"><divide id="A1.SS1.8.p1.3.m3.1.1.1.cmml" xref="A1.SS1.8.p1.3.m3.1.1"></divide><ci id="A1.SS1.8.p1.3.m3.1.1.2.cmml" xref="A1.SS1.8.p1.3.m3.1.1.2">𝑝</ci><apply id="A1.SS1.8.p1.3.m3.1.1.3.cmml" xref="A1.SS1.8.p1.3.m3.1.1.3"><minus id="A1.SS1.8.p1.3.m3.1.1.3.1.cmml" xref="A1.SS1.8.p1.3.m3.1.1.3.1"></minus><ci id="A1.SS1.8.p1.3.m3.1.1.3.2.cmml" xref="A1.SS1.8.p1.3.m3.1.1.3.2">𝑝</ci><cn id="A1.SS1.8.p1.3.m3.1.1.3.3.cmml" type="integer" xref="A1.SS1.8.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.3.m3.1c">\frac{p}{p-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.3.m3.1d">divide start_ARG italic_p end_ARG start_ARG italic_p - 1 end_ARG</annotation></semantics></math> is interpreted as <math alttext="1" class="ltx_Math" display="inline" id="A1.SS1.8.p1.4.m4.1"><semantics id="A1.SS1.8.p1.4.m4.1a"><mn id="A1.SS1.8.p1.4.m4.1.1" xref="A1.SS1.8.p1.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.4.m4.1b"><cn id="A1.SS1.8.p1.4.m4.1.1.cmml" type="integer" xref="A1.SS1.8.p1.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.4.m4.1d">1</annotation></semantics></math> for <math alttext="p=\infty" class="ltx_Math" display="inline" id="A1.SS1.8.p1.5.m5.1"><semantics id="A1.SS1.8.p1.5.m5.1a"><mrow id="A1.SS1.8.p1.5.m5.1.1" xref="A1.SS1.8.p1.5.m5.1.1.cmml"><mi id="A1.SS1.8.p1.5.m5.1.1.2" xref="A1.SS1.8.p1.5.m5.1.1.2.cmml">p</mi><mo id="A1.SS1.8.p1.5.m5.1.1.1" xref="A1.SS1.8.p1.5.m5.1.1.1.cmml">=</mo><mi id="A1.SS1.8.p1.5.m5.1.1.3" mathvariant="normal" xref="A1.SS1.8.p1.5.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.5.m5.1b"><apply id="A1.SS1.8.p1.5.m5.1.1.cmml" xref="A1.SS1.8.p1.5.m5.1.1"><eq id="A1.SS1.8.p1.5.m5.1.1.1.cmml" xref="A1.SS1.8.p1.5.m5.1.1.1"></eq><ci id="A1.SS1.8.p1.5.m5.1.1.2.cmml" xref="A1.SS1.8.p1.5.m5.1.1.2">𝑝</ci><infinity id="A1.SS1.8.p1.5.m5.1.1.3.cmml" xref="A1.SS1.8.p1.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.5.m5.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.5.m5.1d">italic_p = ∞</annotation></semantics></math> and as <math alttext="\infty" class="ltx_Math" display="inline" id="A1.SS1.8.p1.6.m6.1"><semantics id="A1.SS1.8.p1.6.m6.1a"><mi id="A1.SS1.8.p1.6.m6.1.1" mathvariant="normal" xref="A1.SS1.8.p1.6.m6.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.6.m6.1b"><infinity id="A1.SS1.8.p1.6.m6.1.1.cmml" xref="A1.SS1.8.p1.6.m6.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.6.m6.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.6.m6.1d">∞</annotation></semantics></math> for <math alttext="p=1" class="ltx_Math" display="inline" id="A1.SS1.8.p1.7.m7.1"><semantics id="A1.SS1.8.p1.7.m7.1a"><mrow id="A1.SS1.8.p1.7.m7.1.1" xref="A1.SS1.8.p1.7.m7.1.1.cmml"><mi id="A1.SS1.8.p1.7.m7.1.1.2" xref="A1.SS1.8.p1.7.m7.1.1.2.cmml">p</mi><mo id="A1.SS1.8.p1.7.m7.1.1.1" xref="A1.SS1.8.p1.7.m7.1.1.1.cmml">=</mo><mn id="A1.SS1.8.p1.7.m7.1.1.3" xref="A1.SS1.8.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.7.m7.1b"><apply id="A1.SS1.8.p1.7.m7.1.1.cmml" xref="A1.SS1.8.p1.7.m7.1.1"><eq id="A1.SS1.8.p1.7.m7.1.1.1.cmml" xref="A1.SS1.8.p1.7.m7.1.1.1"></eq><ci id="A1.SS1.8.p1.7.m7.1.1.2.cmml" xref="A1.SS1.8.p1.7.m7.1.1.2">𝑝</ci><cn id="A1.SS1.8.p1.7.m7.1.1.3.cmml" type="integer" xref="A1.SS1.8.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.7.m7.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.7.m7.1d">italic_p = 1</annotation></semantics></math>, respectively). We thus get</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\cos(\measuredangle(u,z))=\frac{\langle u,z\rangle}{||u||_{2}||z||_{2}}=\frac{% 1}{||u||_{2}||z||_{2}}." class="ltx_Math" display="block" id="A1.Ex6.m1.10"><semantics id="A1.Ex6.m1.10a"><mrow id="A1.Ex6.m1.10.10.1" xref="A1.Ex6.m1.10.10.1.1.cmml"><mrow id="A1.Ex6.m1.10.10.1.1" xref="A1.Ex6.m1.10.10.1.1.cmml"><mrow id="A1.Ex6.m1.10.10.1.1.1.1" xref="A1.Ex6.m1.10.10.1.1.1.2.cmml"><mi id="A1.Ex6.m1.9.9" xref="A1.Ex6.m1.9.9.cmml">cos</mi><mo id="A1.Ex6.m1.10.10.1.1.1.1a" xref="A1.Ex6.m1.10.10.1.1.1.2.cmml">⁡</mo><mrow id="A1.Ex6.m1.10.10.1.1.1.1.1" xref="A1.Ex6.m1.10.10.1.1.1.2.cmml"><mo id="A1.Ex6.m1.10.10.1.1.1.1.1.2" stretchy="false" xref="A1.Ex6.m1.10.10.1.1.1.2.cmml">(</mo><mrow id="A1.Ex6.m1.10.10.1.1.1.1.1.1" 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1}{||u||_{2}||z||_{2}}.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex6.m1.10d">roman_cos ( ∡ ( italic_u , italic_z ) ) = divide start_ARG ⟨ italic_u , italic_z ⟩ end_ARG start_ARG | | italic_u | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | | italic_z | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG | | italic_u | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | | italic_z | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS1.8.p1.14">Recall the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.8.m1.1"><semantics id="A1.SS1.8.p1.8.m1.1a"><msub id="A1.SS1.8.p1.8.m1.1.1" xref="A1.SS1.8.p1.8.m1.1.1.cmml"><mi id="A1.SS1.8.p1.8.m1.1.1.2" mathvariant="normal" xref="A1.SS1.8.p1.8.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS1.8.p1.8.m1.1.1.3" xref="A1.SS1.8.p1.8.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.8.m1.1b"><apply id="A1.SS1.8.p1.8.m1.1.1.cmml" xref="A1.SS1.8.p1.8.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.8.p1.8.m1.1.1.1.cmml" xref="A1.SS1.8.p1.8.m1.1.1">subscript</csymbol><ci id="A1.SS1.8.p1.8.m1.1.1.2.cmml" xref="A1.SS1.8.p1.8.m1.1.1.2">ℓ</ci><ci id="A1.SS1.8.p1.8.m1.1.1.3.cmml" xref="A1.SS1.8.p1.8.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.8.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.8.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm inequalities <math alttext="||x||_{2}\leq||x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.9.m2.2"><semantics id="A1.SS1.8.p1.9.m2.2a"><mrow id="A1.SS1.8.p1.9.m2.2.3" xref="A1.SS1.8.p1.9.m2.2.3.cmml"><msub id="A1.SS1.8.p1.9.m2.2.3.2" xref="A1.SS1.8.p1.9.m2.2.3.2.cmml"><mrow 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xref="A1.SS1.8.p1.9.m2.2.3.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.9.m2.2b"><apply id="A1.SS1.8.p1.9.m2.2.3.cmml" xref="A1.SS1.8.p1.9.m2.2.3"><leq id="A1.SS1.8.p1.9.m2.2.3.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.1"></leq><apply id="A1.SS1.8.p1.9.m2.2.3.2.cmml" xref="A1.SS1.8.p1.9.m2.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.8.p1.9.m2.2.3.2.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.2">subscript</csymbol><apply id="A1.SS1.8.p1.9.m2.2.3.2.2.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.2.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.9.m2.2.3.2.2.1.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.2.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.9.m2.1.1.cmml" xref="A1.SS1.8.p1.9.m2.1.1">𝑥</ci></apply><cn id="A1.SS1.8.p1.9.m2.2.3.2.3.cmml" type="integer" xref="A1.SS1.8.p1.9.m2.2.3.2.3">2</cn></apply><apply id="A1.SS1.8.p1.9.m2.2.3.3.cmml" xref="A1.SS1.8.p1.9.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.8.p1.9.m2.2.3.3.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.3">subscript</csymbol><apply id="A1.SS1.8.p1.9.m2.2.3.3.2.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.3.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.9.m2.2.3.3.2.1.1.cmml" xref="A1.SS1.8.p1.9.m2.2.3.3.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.9.m2.2.2.cmml" xref="A1.SS1.8.p1.9.m2.2.2">𝑥</ci></apply><ci id="A1.SS1.8.p1.9.m2.2.3.3.3.cmml" xref="A1.SS1.8.p1.9.m2.2.3.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.9.m2.2c">||x||_{2}\leq||x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.9.m2.2d">| | italic_x | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ | | italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="p\leq 2" class="ltx_Math" display="inline" id="A1.SS1.8.p1.10.m3.1"><semantics id="A1.SS1.8.p1.10.m3.1a"><mrow id="A1.SS1.8.p1.10.m3.1.1" xref="A1.SS1.8.p1.10.m3.1.1.cmml"><mi id="A1.SS1.8.p1.10.m3.1.1.2" xref="A1.SS1.8.p1.10.m3.1.1.2.cmml">p</mi><mo id="A1.SS1.8.p1.10.m3.1.1.1" xref="A1.SS1.8.p1.10.m3.1.1.1.cmml">≤</mo><mn id="A1.SS1.8.p1.10.m3.1.1.3" xref="A1.SS1.8.p1.10.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.10.m3.1b"><apply id="A1.SS1.8.p1.10.m3.1.1.cmml" xref="A1.SS1.8.p1.10.m3.1.1"><leq id="A1.SS1.8.p1.10.m3.1.1.1.cmml" xref="A1.SS1.8.p1.10.m3.1.1.1"></leq><ci id="A1.SS1.8.p1.10.m3.1.1.2.cmml" xref="A1.SS1.8.p1.10.m3.1.1.2">𝑝</ci><cn id="A1.SS1.8.p1.10.m3.1.1.3.cmml" type="integer" xref="A1.SS1.8.p1.10.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.10.m3.1c">p\leq 2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.10.m3.1d">italic_p ≤ 2</annotation></semantics></math> and <math alttext="||x||_{2}\leq\sqrt{d}||x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.11.m4.2"><semantics id="A1.SS1.8.p1.11.m4.2a"><mrow id="A1.SS1.8.p1.11.m4.2.3" xref="A1.SS1.8.p1.11.m4.2.3.cmml"><msub id="A1.SS1.8.p1.11.m4.2.3.2" xref="A1.SS1.8.p1.11.m4.2.3.2.cmml"><mrow id="A1.SS1.8.p1.11.m4.2.3.2.2.2" xref="A1.SS1.8.p1.11.m4.2.3.2.2.1.cmml"><mo id="A1.SS1.8.p1.11.m4.2.3.2.2.2.1" stretchy="false" xref="A1.SS1.8.p1.11.m4.2.3.2.2.1.1.cmml">‖</mo><mi id="A1.SS1.8.p1.11.m4.1.1" xref="A1.SS1.8.p1.11.m4.1.1.cmml">x</mi><mo id="A1.SS1.8.p1.11.m4.2.3.2.2.2.2" stretchy="false" xref="A1.SS1.8.p1.11.m4.2.3.2.2.1.1.cmml">‖</mo></mrow><mn id="A1.SS1.8.p1.11.m4.2.3.2.3" xref="A1.SS1.8.p1.11.m4.2.3.2.3.cmml">2</mn></msub><mo id="A1.SS1.8.p1.11.m4.2.3.1" xref="A1.SS1.8.p1.11.m4.2.3.1.cmml">≤</mo><mrow id="A1.SS1.8.p1.11.m4.2.3.3" xref="A1.SS1.8.p1.11.m4.2.3.3.cmml"><msqrt id="A1.SS1.8.p1.11.m4.2.3.3.2" xref="A1.SS1.8.p1.11.m4.2.3.3.2.cmml"><mi id="A1.SS1.8.p1.11.m4.2.3.3.2.2" xref="A1.SS1.8.p1.11.m4.2.3.3.2.2.cmml">d</mi></msqrt><mo id="A1.SS1.8.p1.11.m4.2.3.3.1" xref="A1.SS1.8.p1.11.m4.2.3.3.1.cmml">⁢</mo><msub id="A1.SS1.8.p1.11.m4.2.3.3.3" xref="A1.SS1.8.p1.11.m4.2.3.3.3.cmml"><mrow id="A1.SS1.8.p1.11.m4.2.3.3.3.2.2" xref="A1.SS1.8.p1.11.m4.2.3.3.3.2.1.cmml"><mo id="A1.SS1.8.p1.11.m4.2.3.3.3.2.2.1" stretchy="false" xref="A1.SS1.8.p1.11.m4.2.3.3.3.2.1.1.cmml">‖</mo><mi id="A1.SS1.8.p1.11.m4.2.2" xref="A1.SS1.8.p1.11.m4.2.2.cmml">x</mi><mo id="A1.SS1.8.p1.11.m4.2.3.3.3.2.2.2" stretchy="false" xref="A1.SS1.8.p1.11.m4.2.3.3.3.2.1.1.cmml">‖</mo></mrow><mi id="A1.SS1.8.p1.11.m4.2.3.3.3.3" xref="A1.SS1.8.p1.11.m4.2.3.3.3.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.11.m4.2b"><apply id="A1.SS1.8.p1.11.m4.2.3.cmml" xref="A1.SS1.8.p1.11.m4.2.3"><leq id="A1.SS1.8.p1.11.m4.2.3.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.1"></leq><apply id="A1.SS1.8.p1.11.m4.2.3.2.cmml" xref="A1.SS1.8.p1.11.m4.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.8.p1.11.m4.2.3.2.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.2">subscript</csymbol><apply id="A1.SS1.8.p1.11.m4.2.3.2.2.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.2.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.11.m4.2.3.2.2.1.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.2.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.11.m4.1.1.cmml" xref="A1.SS1.8.p1.11.m4.1.1">𝑥</ci></apply><cn id="A1.SS1.8.p1.11.m4.2.3.2.3.cmml" type="integer" xref="A1.SS1.8.p1.11.m4.2.3.2.3">2</cn></apply><apply id="A1.SS1.8.p1.11.m4.2.3.3.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3"><times id="A1.SS1.8.p1.11.m4.2.3.3.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.1"></times><apply id="A1.SS1.8.p1.11.m4.2.3.3.2.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.2"><root id="A1.SS1.8.p1.11.m4.2.3.3.2a.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.2"></root><ci id="A1.SS1.8.p1.11.m4.2.3.3.2.2.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.2.2">𝑑</ci></apply><apply id="A1.SS1.8.p1.11.m4.2.3.3.3.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.3"><csymbol cd="ambiguous" id="A1.SS1.8.p1.11.m4.2.3.3.3.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.3">subscript</csymbol><apply id="A1.SS1.8.p1.11.m4.2.3.3.3.2.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.3.2.2"><csymbol cd="latexml" id="A1.SS1.8.p1.11.m4.2.3.3.3.2.1.1.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.3.2.2.1">norm</csymbol><ci id="A1.SS1.8.p1.11.m4.2.2.cmml" xref="A1.SS1.8.p1.11.m4.2.2">𝑥</ci></apply><ci id="A1.SS1.8.p1.11.m4.2.3.3.3.3.cmml" xref="A1.SS1.8.p1.11.m4.2.3.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.11.m4.2c">||x||_{2}\leq\sqrt{d}||x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.11.m4.2d">| | italic_x | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ square-root start_ARG italic_d end_ARG | | italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="2\leq p" class="ltx_Math" display="inline" id="A1.SS1.8.p1.12.m5.1"><semantics id="A1.SS1.8.p1.12.m5.1a"><mrow id="A1.SS1.8.p1.12.m5.1.1" xref="A1.SS1.8.p1.12.m5.1.1.cmml"><mn id="A1.SS1.8.p1.12.m5.1.1.2" xref="A1.SS1.8.p1.12.m5.1.1.2.cmml">2</mn><mo id="A1.SS1.8.p1.12.m5.1.1.1" xref="A1.SS1.8.p1.12.m5.1.1.1.cmml">≤</mo><mi id="A1.SS1.8.p1.12.m5.1.1.3" xref="A1.SS1.8.p1.12.m5.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.12.m5.1b"><apply id="A1.SS1.8.p1.12.m5.1.1.cmml" xref="A1.SS1.8.p1.12.m5.1.1"><leq id="A1.SS1.8.p1.12.m5.1.1.1.cmml" xref="A1.SS1.8.p1.12.m5.1.1.1"></leq><cn id="A1.SS1.8.p1.12.m5.1.1.2.cmml" type="integer" xref="A1.SS1.8.p1.12.m5.1.1.2">2</cn><ci id="A1.SS1.8.p1.12.m5.1.1.3.cmml" xref="A1.SS1.8.p1.12.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.12.m5.1c">2\leq p</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.12.m5.1d">2 ≤ italic_p</annotation></semantics></math>. Further, for any <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.13.m6.3"><semantics id="A1.SS1.8.p1.13.m6.3a"><mrow id="A1.SS1.8.p1.13.m6.3.4" xref="A1.SS1.8.p1.13.m6.3.4.cmml"><mi id="A1.SS1.8.p1.13.m6.3.4.2" xref="A1.SS1.8.p1.13.m6.3.4.2.cmml">p</mi><mo id="A1.SS1.8.p1.13.m6.3.4.1" xref="A1.SS1.8.p1.13.m6.3.4.1.cmml">∈</mo><mrow id="A1.SS1.8.p1.13.m6.3.4.3" xref="A1.SS1.8.p1.13.m6.3.4.3.cmml"><mrow id="A1.SS1.8.p1.13.m6.3.4.3.2.2" xref="A1.SS1.8.p1.13.m6.3.4.3.2.1.cmml"><mo id="A1.SS1.8.p1.13.m6.3.4.3.2.2.1" stretchy="false" xref="A1.SS1.8.p1.13.m6.3.4.3.2.1.cmml">[</mo><mn id="A1.SS1.8.p1.13.m6.1.1" xref="A1.SS1.8.p1.13.m6.1.1.cmml">1</mn><mo id="A1.SS1.8.p1.13.m6.3.4.3.2.2.2" xref="A1.SS1.8.p1.13.m6.3.4.3.2.1.cmml">,</mo><mi id="A1.SS1.8.p1.13.m6.2.2" mathvariant="normal" xref="A1.SS1.8.p1.13.m6.2.2.cmml">∞</mi><mo id="A1.SS1.8.p1.13.m6.3.4.3.2.2.3" stretchy="false" xref="A1.SS1.8.p1.13.m6.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.SS1.8.p1.13.m6.3.4.3.1" xref="A1.SS1.8.p1.13.m6.3.4.3.1.cmml">∪</mo><mrow id="A1.SS1.8.p1.13.m6.3.4.3.3.2" xref="A1.SS1.8.p1.13.m6.3.4.3.3.1.cmml"><mo id="A1.SS1.8.p1.13.m6.3.4.3.3.2.1" stretchy="false" xref="A1.SS1.8.p1.13.m6.3.4.3.3.1.cmml">{</mo><mi id="A1.SS1.8.p1.13.m6.3.3" mathvariant="normal" xref="A1.SS1.8.p1.13.m6.3.3.cmml">∞</mi><mo id="A1.SS1.8.p1.13.m6.3.4.3.3.2.2" stretchy="false" xref="A1.SS1.8.p1.13.m6.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.13.m6.3b"><apply id="A1.SS1.8.p1.13.m6.3.4.cmml" xref="A1.SS1.8.p1.13.m6.3.4"><in id="A1.SS1.8.p1.13.m6.3.4.1.cmml" xref="A1.SS1.8.p1.13.m6.3.4.1"></in><ci id="A1.SS1.8.p1.13.m6.3.4.2.cmml" xref="A1.SS1.8.p1.13.m6.3.4.2">𝑝</ci><apply id="A1.SS1.8.p1.13.m6.3.4.3.cmml" xref="A1.SS1.8.p1.13.m6.3.4.3"><union id="A1.SS1.8.p1.13.m6.3.4.3.1.cmml" xref="A1.SS1.8.p1.13.m6.3.4.3.1"></union><interval closure="closed-open" id="A1.SS1.8.p1.13.m6.3.4.3.2.1.cmml" xref="A1.SS1.8.p1.13.m6.3.4.3.2.2"><cn id="A1.SS1.8.p1.13.m6.1.1.cmml" type="integer" xref="A1.SS1.8.p1.13.m6.1.1">1</cn><infinity id="A1.SS1.8.p1.13.m6.2.2.cmml" xref="A1.SS1.8.p1.13.m6.2.2"></infinity></interval><set id="A1.SS1.8.p1.13.m6.3.4.3.3.1.cmml" xref="A1.SS1.8.p1.13.m6.3.4.3.3.2"><infinity id="A1.SS1.8.p1.13.m6.3.3.cmml" xref="A1.SS1.8.p1.13.m6.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.13.m6.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.13.m6.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> we have <math alttext="p<2\iff\frac{p}{p-1}>2" class="ltx_Math" display="inline" id="A1.SS1.8.p1.14.m7.1"><semantics id="A1.SS1.8.p1.14.m7.1a"><mrow id="A1.SS1.8.p1.14.m7.1.1" xref="A1.SS1.8.p1.14.m7.1.1.cmml"><mrow id="A1.SS1.8.p1.14.m7.1.1.2" xref="A1.SS1.8.p1.14.m7.1.1.2.cmml"><mi id="A1.SS1.8.p1.14.m7.1.1.2.2" xref="A1.SS1.8.p1.14.m7.1.1.2.2.cmml">p</mi><mo id="A1.SS1.8.p1.14.m7.1.1.2.1" xref="A1.SS1.8.p1.14.m7.1.1.2.1.cmml"><</mo><mn id="A1.SS1.8.p1.14.m7.1.1.2.3" xref="A1.SS1.8.p1.14.m7.1.1.2.3.cmml">2</mn></mrow><mo id="A1.SS1.8.p1.14.m7.1.1.1" stretchy="false" xref="A1.SS1.8.p1.14.m7.1.1.1.cmml">⇔</mo><mrow id="A1.SS1.8.p1.14.m7.1.1.3" xref="A1.SS1.8.p1.14.m7.1.1.3.cmml"><mfrac id="A1.SS1.8.p1.14.m7.1.1.3.2" xref="A1.SS1.8.p1.14.m7.1.1.3.2.cmml"><mi id="A1.SS1.8.p1.14.m7.1.1.3.2.2" xref="A1.SS1.8.p1.14.m7.1.1.3.2.2.cmml">p</mi><mrow id="A1.SS1.8.p1.14.m7.1.1.3.2.3" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.cmml"><mi id="A1.SS1.8.p1.14.m7.1.1.3.2.3.2" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.2.cmml">p</mi><mo id="A1.SS1.8.p1.14.m7.1.1.3.2.3.1" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.1.cmml">−</mo><mn id="A1.SS1.8.p1.14.m7.1.1.3.2.3.3" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.3.cmml">1</mn></mrow></mfrac><mo id="A1.SS1.8.p1.14.m7.1.1.3.1" xref="A1.SS1.8.p1.14.m7.1.1.3.1.cmml">></mo><mn id="A1.SS1.8.p1.14.m7.1.1.3.3" xref="A1.SS1.8.p1.14.m7.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.8.p1.14.m7.1b"><apply id="A1.SS1.8.p1.14.m7.1.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1"><csymbol cd="latexml" id="A1.SS1.8.p1.14.m7.1.1.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1.1">iff</csymbol><apply id="A1.SS1.8.p1.14.m7.1.1.2.cmml" xref="A1.SS1.8.p1.14.m7.1.1.2"><lt id="A1.SS1.8.p1.14.m7.1.1.2.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1.2.1"></lt><ci id="A1.SS1.8.p1.14.m7.1.1.2.2.cmml" xref="A1.SS1.8.p1.14.m7.1.1.2.2">𝑝</ci><cn id="A1.SS1.8.p1.14.m7.1.1.2.3.cmml" type="integer" xref="A1.SS1.8.p1.14.m7.1.1.2.3">2</cn></apply><apply id="A1.SS1.8.p1.14.m7.1.1.3.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3"><gt id="A1.SS1.8.p1.14.m7.1.1.3.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.1"></gt><apply id="A1.SS1.8.p1.14.m7.1.1.3.2.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2"><divide id="A1.SS1.8.p1.14.m7.1.1.3.2.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2"></divide><ci id="A1.SS1.8.p1.14.m7.1.1.3.2.2.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2.2">𝑝</ci><apply id="A1.SS1.8.p1.14.m7.1.1.3.2.3.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3"><minus id="A1.SS1.8.p1.14.m7.1.1.3.2.3.1.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.1"></minus><ci id="A1.SS1.8.p1.14.m7.1.1.3.2.3.2.cmml" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.2">𝑝</ci><cn id="A1.SS1.8.p1.14.m7.1.1.3.2.3.3.cmml" type="integer" xref="A1.SS1.8.p1.14.m7.1.1.3.2.3.3">1</cn></apply></apply><cn id="A1.SS1.8.p1.14.m7.1.1.3.3.cmml" type="integer" xref="A1.SS1.8.p1.14.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.8.p1.14.m7.1c">p<2\iff\frac{p}{p-1}>2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.14.m7.1d">italic_p < 2 ⇔ divide start_ARG italic_p end_ARG start_ARG italic_p - 1 end_ARG > 2</annotation></semantics></math>. Thus, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{1}{||u||_{2}||z||_{2}}\geq\frac{1}{\sqrt{d}||u||_{\frac{p}{p-1}}||z||_{p% }}=\frac{1}{\sqrt{d}}." class="ltx_Math" display="block" id="A1.Ex7.m1.5"><semantics id="A1.Ex7.m1.5a"><mrow id="A1.Ex7.m1.5.5.1" xref="A1.Ex7.m1.5.5.1.1.cmml"><mrow id="A1.Ex7.m1.5.5.1.1" xref="A1.Ex7.m1.5.5.1.1.cmml"><mfrac id="A1.Ex7.m1.2.2" xref="A1.Ex7.m1.2.2.cmml"><mn id="A1.Ex7.m1.2.2.4" xref="A1.Ex7.m1.2.2.4.cmml">1</mn><mrow id="A1.Ex7.m1.2.2.2" xref="A1.Ex7.m1.2.2.2.cmml"><msub id="A1.Ex7.m1.2.2.2.4" xref="A1.Ex7.m1.2.2.2.4.cmml"><mrow id="A1.Ex7.m1.2.2.2.4.2.2" xref="A1.Ex7.m1.2.2.2.4.2.1.cmml"><mo id="A1.Ex7.m1.2.2.2.4.2.2.1" stretchy="false" xref="A1.Ex7.m1.2.2.2.4.2.1.1.cmml">‖</mo><mi id="A1.Ex7.m1.1.1.1.1" xref="A1.Ex7.m1.1.1.1.1.cmml">u</mi><mo id="A1.Ex7.m1.2.2.2.4.2.2.2" stretchy="false" xref="A1.Ex7.m1.2.2.2.4.2.1.1.cmml">‖</mo></mrow><mn id="A1.Ex7.m1.2.2.2.4.3" xref="A1.Ex7.m1.2.2.2.4.3.cmml">2</mn></msub><mo id="A1.Ex7.m1.2.2.2.3" xref="A1.Ex7.m1.2.2.2.3.cmml">⁢</mo><msub id="A1.Ex7.m1.2.2.2.5" xref="A1.Ex7.m1.2.2.2.5.cmml"><mrow id="A1.Ex7.m1.2.2.2.5.2.2" xref="A1.Ex7.m1.2.2.2.5.2.1.cmml"><mo id="A1.Ex7.m1.2.2.2.5.2.2.1" stretchy="false" xref="A1.Ex7.m1.2.2.2.5.2.1.1.cmml">‖</mo><mi id="A1.Ex7.m1.2.2.2.2" xref="A1.Ex7.m1.2.2.2.2.cmml">z</mi><mo id="A1.Ex7.m1.2.2.2.5.2.2.2" stretchy="false" xref="A1.Ex7.m1.2.2.2.5.2.1.1.cmml">‖</mo></mrow><mn id="A1.Ex7.m1.2.2.2.5.3" xref="A1.Ex7.m1.2.2.2.5.3.cmml">2</mn></msub></mrow></mfrac><mo id="A1.Ex7.m1.5.5.1.1.2" xref="A1.Ex7.m1.5.5.1.1.2.cmml">≥</mo><mfrac id="A1.Ex7.m1.4.4" xref="A1.Ex7.m1.4.4.cmml"><mn id="A1.Ex7.m1.4.4.4" xref="A1.Ex7.m1.4.4.4.cmml">1</mn><mrow id="A1.Ex7.m1.4.4.2" xref="A1.Ex7.m1.4.4.2.cmml"><msqrt id="A1.Ex7.m1.4.4.2.4" xref="A1.Ex7.m1.4.4.2.4.cmml"><mi id="A1.Ex7.m1.4.4.2.4.2" xref="A1.Ex7.m1.4.4.2.4.2.cmml">d</mi></msqrt><mo id="A1.Ex7.m1.4.4.2.3" xref="A1.Ex7.m1.4.4.2.3.cmml">⁢</mo><msub id="A1.Ex7.m1.4.4.2.5" xref="A1.Ex7.m1.4.4.2.5.cmml"><mrow id="A1.Ex7.m1.4.4.2.5.2.2" xref="A1.Ex7.m1.4.4.2.5.2.1.cmml"><mo id="A1.Ex7.m1.4.4.2.5.2.2.1" stretchy="false" xref="A1.Ex7.m1.4.4.2.5.2.1.1.cmml">‖</mo><mi id="A1.Ex7.m1.3.3.1.1" xref="A1.Ex7.m1.3.3.1.1.cmml">u</mi><mo id="A1.Ex7.m1.4.4.2.5.2.2.2" stretchy="false" xref="A1.Ex7.m1.4.4.2.5.2.1.1.cmml">‖</mo></mrow><mfrac id="A1.Ex7.m1.4.4.2.5.3" xref="A1.Ex7.m1.4.4.2.5.3.cmml"><mi id="A1.Ex7.m1.4.4.2.5.3.2" xref="A1.Ex7.m1.4.4.2.5.3.2.cmml">p</mi><mrow id="A1.Ex7.m1.4.4.2.5.3.3" xref="A1.Ex7.m1.4.4.2.5.3.3.cmml"><mi id="A1.Ex7.m1.4.4.2.5.3.3.2" xref="A1.Ex7.m1.4.4.2.5.3.3.2.cmml">p</mi><mo id="A1.Ex7.m1.4.4.2.5.3.3.1" xref="A1.Ex7.m1.4.4.2.5.3.3.1.cmml">−</mo><mn id="A1.Ex7.m1.4.4.2.5.3.3.3" xref="A1.Ex7.m1.4.4.2.5.3.3.3.cmml">1</mn></mrow></mfrac></msub><mo id="A1.Ex7.m1.4.4.2.3a" xref="A1.Ex7.m1.4.4.2.3.cmml">⁢</mo><msub id="A1.Ex7.m1.4.4.2.6" xref="A1.Ex7.m1.4.4.2.6.cmml"><mrow id="A1.Ex7.m1.4.4.2.6.2.2" xref="A1.Ex7.m1.4.4.2.6.2.1.cmml"><mo id="A1.Ex7.m1.4.4.2.6.2.2.1" stretchy="false" xref="A1.Ex7.m1.4.4.2.6.2.1.1.cmml">‖</mo><mi id="A1.Ex7.m1.4.4.2.2" xref="A1.Ex7.m1.4.4.2.2.cmml">z</mi><mo id="A1.Ex7.m1.4.4.2.6.2.2.2" stretchy="false" xref="A1.Ex7.m1.4.4.2.6.2.1.1.cmml">‖</mo></mrow><mi id="A1.Ex7.m1.4.4.2.6.3" xref="A1.Ex7.m1.4.4.2.6.3.cmml">p</mi></msub></mrow></mfrac><mo id="A1.Ex7.m1.5.5.1.1.3" xref="A1.Ex7.m1.5.5.1.1.3.cmml">=</mo><mfrac id="A1.Ex7.m1.5.5.1.1.4" xref="A1.Ex7.m1.5.5.1.1.4.cmml"><mn id="A1.Ex7.m1.5.5.1.1.4.2" xref="A1.Ex7.m1.5.5.1.1.4.2.cmml">1</mn><msqrt id="A1.Ex7.m1.5.5.1.1.4.3" xref="A1.Ex7.m1.5.5.1.1.4.3.cmml"><mi id="A1.Ex7.m1.5.5.1.1.4.3.2" xref="A1.Ex7.m1.5.5.1.1.4.3.2.cmml">d</mi></msqrt></mfrac></mrow><mo id="A1.Ex7.m1.5.5.1.2" lspace="0em" xref="A1.Ex7.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex7.m1.5b"><apply id="A1.Ex7.m1.5.5.1.1.cmml" xref="A1.Ex7.m1.5.5.1"><and id="A1.Ex7.m1.5.5.1.1a.cmml" xref="A1.Ex7.m1.5.5.1"></and><apply id="A1.Ex7.m1.5.5.1.1b.cmml" xref="A1.Ex7.m1.5.5.1"><geq id="A1.Ex7.m1.5.5.1.1.2.cmml" xref="A1.Ex7.m1.5.5.1.1.2"></geq><apply id="A1.Ex7.m1.2.2.cmml" xref="A1.Ex7.m1.2.2"><divide id="A1.Ex7.m1.2.2.3.cmml" xref="A1.Ex7.m1.2.2"></divide><cn id="A1.Ex7.m1.2.2.4.cmml" type="integer" xref="A1.Ex7.m1.2.2.4">1</cn><apply id="A1.Ex7.m1.2.2.2.cmml" xref="A1.Ex7.m1.2.2.2"><times id="A1.Ex7.m1.2.2.2.3.cmml" xref="A1.Ex7.m1.2.2.2.3"></times><apply id="A1.Ex7.m1.2.2.2.4.cmml" xref="A1.Ex7.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.Ex7.m1.2.2.2.4.1.cmml" xref="A1.Ex7.m1.2.2.2.4">subscript</csymbol><apply id="A1.Ex7.m1.2.2.2.4.2.1.cmml" xref="A1.Ex7.m1.2.2.2.4.2.2"><csymbol cd="latexml" id="A1.Ex7.m1.2.2.2.4.2.1.1.cmml" xref="A1.Ex7.m1.2.2.2.4.2.2.1">norm</csymbol><ci id="A1.Ex7.m1.1.1.1.1.cmml" xref="A1.Ex7.m1.1.1.1.1">𝑢</ci></apply><cn 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xref="A1.Ex7.m1.4.4.2.5.3.3.3">1</cn></apply></apply></apply><apply id="A1.Ex7.m1.4.4.2.6.cmml" xref="A1.Ex7.m1.4.4.2.6"><csymbol cd="ambiguous" id="A1.Ex7.m1.4.4.2.6.1.cmml" xref="A1.Ex7.m1.4.4.2.6">subscript</csymbol><apply id="A1.Ex7.m1.4.4.2.6.2.1.cmml" xref="A1.Ex7.m1.4.4.2.6.2.2"><csymbol cd="latexml" id="A1.Ex7.m1.4.4.2.6.2.1.1.cmml" xref="A1.Ex7.m1.4.4.2.6.2.2.1">norm</csymbol><ci id="A1.Ex7.m1.4.4.2.2.cmml" xref="A1.Ex7.m1.4.4.2.2">𝑧</ci></apply><ci id="A1.Ex7.m1.4.4.2.6.3.cmml" xref="A1.Ex7.m1.4.4.2.6.3">𝑝</ci></apply></apply></apply></apply><apply id="A1.Ex7.m1.5.5.1.1c.cmml" xref="A1.Ex7.m1.5.5.1"><eq id="A1.Ex7.m1.5.5.1.1.3.cmml" xref="A1.Ex7.m1.5.5.1.1.3"></eq><share href="https://arxiv.org/html/2503.16089v1#A1.Ex7.m1.4.4.cmml" id="A1.Ex7.m1.5.5.1.1d.cmml" xref="A1.Ex7.m1.5.5.1"></share><apply id="A1.Ex7.m1.5.5.1.1.4.cmml" xref="A1.Ex7.m1.5.5.1.1.4"><divide id="A1.Ex7.m1.5.5.1.1.4.1.cmml" xref="A1.Ex7.m1.5.5.1.1.4"></divide><cn id="A1.Ex7.m1.5.5.1.1.4.2.cmml" type="integer" xref="A1.Ex7.m1.5.5.1.1.4.2">1</cn><apply id="A1.Ex7.m1.5.5.1.1.4.3.cmml" xref="A1.Ex7.m1.5.5.1.1.4.3"><root id="A1.Ex7.m1.5.5.1.1.4.3a.cmml" xref="A1.Ex7.m1.5.5.1.1.4.3"></root><ci id="A1.Ex7.m1.5.5.1.1.4.3.2.cmml" xref="A1.Ex7.m1.5.5.1.1.4.3.2">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex7.m1.5c">\frac{1}{||u||_{2}||z||_{2}}\geq\frac{1}{\sqrt{d}||u||_{\frac{p}{p-1}}||z||_{p% }}=\frac{1}{\sqrt{d}}.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex7.m1.5d">divide start_ARG 1 end_ARG start_ARG | | italic_u | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | | italic_z | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ≥ divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG | | italic_u | | start_POSTSUBSCRIPT divide start_ARG italic_p end_ARG start_ARG italic_p - 1 end_ARG end_POSTSUBSCRIPT | | italic_z | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS1.8.p1.15">Thus, we conclude <math alttext="\measuredangle(u,z)\leq\frac{\pi}{2}-\frac{1}{\sqrt{d}}" class="ltx_Math" display="inline" id="A1.SS1.8.p1.15.m1.2"><semantics id="A1.SS1.8.p1.15.m1.2a"><mrow id="A1.SS1.8.p1.15.m1.2.3" xref="A1.SS1.8.p1.15.m1.2.3.cmml"><mrow id="A1.SS1.8.p1.15.m1.2.3.2" xref="A1.SS1.8.p1.15.m1.2.3.2.cmml"><mi id="A1.SS1.8.p1.15.m1.2.3.2.2" mathvariant="normal" xref="A1.SS1.8.p1.15.m1.2.3.2.2.cmml">∡</mi><mo id="A1.SS1.8.p1.15.m1.2.3.2.1" xref="A1.SS1.8.p1.15.m1.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS1.8.p1.15.m1.2.3.2.3.2" xref="A1.SS1.8.p1.15.m1.2.3.2.3.1.cmml"><mo id="A1.SS1.8.p1.15.m1.2.3.2.3.2.1" stretchy="false" xref="A1.SS1.8.p1.15.m1.2.3.2.3.1.cmml">(</mo><mi id="A1.SS1.8.p1.15.m1.1.1" xref="A1.SS1.8.p1.15.m1.1.1.cmml">u</mi><mo id="A1.SS1.8.p1.15.m1.2.3.2.3.2.2" xref="A1.SS1.8.p1.15.m1.2.3.2.3.1.cmml">,</mo><mi id="A1.SS1.8.p1.15.m1.2.2" xref="A1.SS1.8.p1.15.m1.2.2.cmml">z</mi><mo id="A1.SS1.8.p1.15.m1.2.3.2.3.2.3" stretchy="false" xref="A1.SS1.8.p1.15.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.8.p1.15.m1.2.3.1" xref="A1.SS1.8.p1.15.m1.2.3.1.cmml">≤</mo><mrow id="A1.SS1.8.p1.15.m1.2.3.3" xref="A1.SS1.8.p1.15.m1.2.3.3.cmml"><mfrac id="A1.SS1.8.p1.15.m1.2.3.3.2" xref="A1.SS1.8.p1.15.m1.2.3.3.2.cmml"><mi id="A1.SS1.8.p1.15.m1.2.3.3.2.2" xref="A1.SS1.8.p1.15.m1.2.3.3.2.2.cmml">π</mi><mn id="A1.SS1.8.p1.15.m1.2.3.3.2.3" xref="A1.SS1.8.p1.15.m1.2.3.3.2.3.cmml">2</mn></mfrac><mo id="A1.SS1.8.p1.15.m1.2.3.3.1" xref="A1.SS1.8.p1.15.m1.2.3.3.1.cmml">−</mo><mfrac id="A1.SS1.8.p1.15.m1.2.3.3.3" xref="A1.SS1.8.p1.15.m1.2.3.3.3.cmml"><mn id="A1.SS1.8.p1.15.m1.2.3.3.3.2" xref="A1.SS1.8.p1.15.m1.2.3.3.3.2.cmml">1</mn><msqrt id="A1.SS1.8.p1.15.m1.2.3.3.3.3" xref="A1.SS1.8.p1.15.m1.2.3.3.3.3.cmml"><mi 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id="A1.SS1.8.p1.15.m1.2c">\measuredangle(u,z)\leq\frac{\pi}{2}-\frac{1}{\sqrt{d}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.8.p1.15.m1.2d">∡ ( italic_u , italic_z ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG - divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>, as desired. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p16"> <p class="ltx_p" id="A1.SS1.p16.1">With this theory in place, we are now ready to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a>.</p> </div> <div class="ltx_para" id="A1.SS1.p17"> <p class="ltx_p" id="A1.SS1.p17.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.7</span></a></p> </div> <div class="ltx_proof" id="A1.SS1.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.9.p1"> <p class="ltx_p" id="A1.SS1.9.p1.7">By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.6</span></a> (every halfspace is a union of rays), we can assume <math alttext="||z-x||_{p}=1" class="ltx_Math" display="inline" id="A1.SS1.9.p1.1.m1.1"><semantics id="A1.SS1.9.p1.1.m1.1a"><mrow id="A1.SS1.9.p1.1.m1.1.1" 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encoding="MathML-Content" id="A1.SS1.9.p1.1.m1.1b"><apply id="A1.SS1.9.p1.1.m1.1.1.cmml" xref="A1.SS1.9.p1.1.m1.1.1"><eq id="A1.SS1.9.p1.1.m1.1.1.2.cmml" xref="A1.SS1.9.p1.1.m1.1.1.2"></eq><apply id="A1.SS1.9.p1.1.m1.1.1.1.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.9.p1.1.m1.1.1.1.2.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1">subscript</csymbol><apply id="A1.SS1.9.p1.1.m1.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.9.p1.1.m1.1.1.1.1.2.1.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1.1"><minus id="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.9.p1.1.m1.1.1.1.3.cmml" xref="A1.SS1.9.p1.1.m1.1.1.1.3">𝑝</ci></apply><cn id="A1.SS1.9.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.SS1.9.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.1.m1.1c">||z-x||_{p}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.1.m1.1d">| | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1</annotation></semantics></math>, without loss of generality. Now consider the first statement of the lemma. <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> says that <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.9.p1.2.m2.2"><semantics id="A1.SS1.9.p1.2.m2.2a"><mrow id="A1.SS1.9.p1.2.m2.2.3" xref="A1.SS1.9.p1.2.m2.2.3.cmml"><mi id="A1.SS1.9.p1.2.m2.2.3.2" xref="A1.SS1.9.p1.2.m2.2.3.2.cmml">z</mi><mo id="A1.SS1.9.p1.2.m2.2.3.1" xref="A1.SS1.9.p1.2.m2.2.3.1.cmml">∈</mo><msubsup id="A1.SS1.9.p1.2.m2.2.3.3" xref="A1.SS1.9.p1.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.9.p1.2.m2.2.3.3.2.2" xref="A1.SS1.9.p1.2.m2.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.9.p1.2.m2.2.2.2.4" xref="A1.SS1.9.p1.2.m2.2.2.2.3.cmml"><mi id="A1.SS1.9.p1.2.m2.1.1.1.1" xref="A1.SS1.9.p1.2.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS1.9.p1.2.m2.2.2.2.4.1" xref="A1.SS1.9.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS1.9.p1.2.m2.2.2.2.2" xref="A1.SS1.9.p1.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.9.p1.2.m2.2.3.3.2.3" xref="A1.SS1.9.p1.2.m2.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.9.p1.2.m2.2b"><apply id="A1.SS1.9.p1.2.m2.2.3.cmml" xref="A1.SS1.9.p1.2.m2.2.3"><in id="A1.SS1.9.p1.2.m2.2.3.1.cmml" xref="A1.SS1.9.p1.2.m2.2.3.1"></in><ci id="A1.SS1.9.p1.2.m2.2.3.2.cmml" xref="A1.SS1.9.p1.2.m2.2.3.2">𝑧</ci><apply id="A1.SS1.9.p1.2.m2.2.3.3.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.9.p1.2.m2.2.3.3.1.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3">subscript</csymbol><apply id="A1.SS1.9.p1.2.m2.2.3.3.2.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.9.p1.2.m2.2.3.3.2.1.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3">superscript</csymbol><ci id="A1.SS1.9.p1.2.m2.2.3.3.2.2.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3.2.2">ℋ</ci><ci id="A1.SS1.9.p1.2.m2.2.3.3.2.3.cmml" xref="A1.SS1.9.p1.2.m2.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS1.9.p1.2.m2.2.2.2.3.cmml" xref="A1.SS1.9.p1.2.m2.2.2.2.4"><ci id="A1.SS1.9.p1.2.m2.1.1.1.1.cmml" xref="A1.SS1.9.p1.2.m2.1.1.1.1">𝑥</ci><ci id="A1.SS1.9.p1.2.m2.2.2.2.2.cmml" xref="A1.SS1.9.p1.2.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.2.m2.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.2.m2.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> implies <math alttext="\langle u,v\rangle\geq 0" class="ltx_Math" display="inline" id="A1.SS1.9.p1.3.m3.2"><semantics id="A1.SS1.9.p1.3.m3.2a"><mrow id="A1.SS1.9.p1.3.m3.2.3" xref="A1.SS1.9.p1.3.m3.2.3.cmml"><mrow id="A1.SS1.9.p1.3.m3.2.3.2.2" xref="A1.SS1.9.p1.3.m3.2.3.2.1.cmml"><mo id="A1.SS1.9.p1.3.m3.2.3.2.2.1" stretchy="false" xref="A1.SS1.9.p1.3.m3.2.3.2.1.cmml">⟨</mo><mi id="A1.SS1.9.p1.3.m3.1.1" xref="A1.SS1.9.p1.3.m3.1.1.cmml">u</mi><mo id="A1.SS1.9.p1.3.m3.2.3.2.2.2" xref="A1.SS1.9.p1.3.m3.2.3.2.1.cmml">,</mo><mi id="A1.SS1.9.p1.3.m3.2.2" xref="A1.SS1.9.p1.3.m3.2.2.cmml">v</mi><mo id="A1.SS1.9.p1.3.m3.2.3.2.2.3" stretchy="false" xref="A1.SS1.9.p1.3.m3.2.3.2.1.cmml">⟩</mo></mrow><mo id="A1.SS1.9.p1.3.m3.2.3.1" xref="A1.SS1.9.p1.3.m3.2.3.1.cmml">≥</mo><mn id="A1.SS1.9.p1.3.m3.2.3.3" xref="A1.SS1.9.p1.3.m3.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.9.p1.3.m3.2b"><apply id="A1.SS1.9.p1.3.m3.2.3.cmml" xref="A1.SS1.9.p1.3.m3.2.3"><geq id="A1.SS1.9.p1.3.m3.2.3.1.cmml" xref="A1.SS1.9.p1.3.m3.2.3.1"></geq><list id="A1.SS1.9.p1.3.m3.2.3.2.1.cmml" xref="A1.SS1.9.p1.3.m3.2.3.2.2"><ci id="A1.SS1.9.p1.3.m3.1.1.cmml" xref="A1.SS1.9.p1.3.m3.1.1">𝑢</ci><ci id="A1.SS1.9.p1.3.m3.2.2.cmml" xref="A1.SS1.9.p1.3.m3.2.2">𝑣</ci></list><cn id="A1.SS1.9.p1.3.m3.2.3.3.cmml" type="integer" xref="A1.SS1.9.p1.3.m3.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.3.m3.2c">\langle u,v\rangle\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.3.m3.2d">⟨ italic_u , italic_v ⟩ ≥ 0</annotation></semantics></math> for some subgradient <math alttext="u\in\partial||z-x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.9.p1.4.m4.1"><semantics id="A1.SS1.9.p1.4.m4.1a"><mrow id="A1.SS1.9.p1.4.m4.1.1" xref="A1.SS1.9.p1.4.m4.1.1.cmml"><mi id="A1.SS1.9.p1.4.m4.1.1.3" xref="A1.SS1.9.p1.4.m4.1.1.3.cmml">u</mi><mo id="A1.SS1.9.p1.4.m4.1.1.2" rspace="0.1389em" xref="A1.SS1.9.p1.4.m4.1.1.2.cmml">∈</mo><mrow id="A1.SS1.9.p1.4.m4.1.1.1" xref="A1.SS1.9.p1.4.m4.1.1.1.cmml"><mo id="A1.SS1.9.p1.4.m4.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS1.9.p1.4.m4.1.1.1.2.cmml">∂</mo><msub 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id="A1.SS1.9.p1.4.m4.1.1.2.cmml" xref="A1.SS1.9.p1.4.m4.1.1.2"></in><ci id="A1.SS1.9.p1.4.m4.1.1.3.cmml" xref="A1.SS1.9.p1.4.m4.1.1.3">𝑢</ci><apply id="A1.SS1.9.p1.4.m4.1.1.1.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1"><partialdiff id="A1.SS1.9.p1.4.m4.1.1.1.2.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.2"></partialdiff><apply id="A1.SS1.9.p1.4.m4.1.1.1.1.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.9.p1.4.m4.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1">subscript</csymbol><apply id="A1.SS1.9.p1.4.m4.1.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.9.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1"><minus id="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.9.p1.4.m4.1.1.1.1.3.cmml" xref="A1.SS1.9.p1.4.m4.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.4.m4.1c">u\in\partial||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.4.m4.1d">italic_u ∈ ∂ | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. Concretely, this means that <math alttext="\measuredangle(u,v)\leq\frac{\pi}{2}" class="ltx_Math" display="inline" id="A1.SS1.9.p1.5.m5.2"><semantics id="A1.SS1.9.p1.5.m5.2a"><mrow id="A1.SS1.9.p1.5.m5.2.3" xref="A1.SS1.9.p1.5.m5.2.3.cmml"><mrow id="A1.SS1.9.p1.5.m5.2.3.2" xref="A1.SS1.9.p1.5.m5.2.3.2.cmml"><mi id="A1.SS1.9.p1.5.m5.2.3.2.2" mathvariant="normal" xref="A1.SS1.9.p1.5.m5.2.3.2.2.cmml">∡</mi><mo id="A1.SS1.9.p1.5.m5.2.3.2.1" xref="A1.SS1.9.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS1.9.p1.5.m5.2.3.2.3.2" xref="A1.SS1.9.p1.5.m5.2.3.2.3.1.cmml"><mo id="A1.SS1.9.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="A1.SS1.9.p1.5.m5.2.3.2.3.1.cmml">(</mo><mi id="A1.SS1.9.p1.5.m5.1.1" xref="A1.SS1.9.p1.5.m5.1.1.cmml">u</mi><mo id="A1.SS1.9.p1.5.m5.2.3.2.3.2.2" xref="A1.SS1.9.p1.5.m5.2.3.2.3.1.cmml">,</mo><mi id="A1.SS1.9.p1.5.m5.2.2" xref="A1.SS1.9.p1.5.m5.2.2.cmml">v</mi><mo id="A1.SS1.9.p1.5.m5.2.3.2.3.2.3" stretchy="false" xref="A1.SS1.9.p1.5.m5.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.9.p1.5.m5.2.3.1" xref="A1.SS1.9.p1.5.m5.2.3.1.cmml">≤</mo><mfrac id="A1.SS1.9.p1.5.m5.2.3.3" xref="A1.SS1.9.p1.5.m5.2.3.3.cmml"><mi id="A1.SS1.9.p1.5.m5.2.3.3.2" xref="A1.SS1.9.p1.5.m5.2.3.3.2.cmml">π</mi><mn id="A1.SS1.9.p1.5.m5.2.3.3.3" xref="A1.SS1.9.p1.5.m5.2.3.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.9.p1.5.m5.2b"><apply id="A1.SS1.9.p1.5.m5.2.3.cmml" xref="A1.SS1.9.p1.5.m5.2.3"><leq id="A1.SS1.9.p1.5.m5.2.3.1.cmml" xref="A1.SS1.9.p1.5.m5.2.3.1"></leq><apply id="A1.SS1.9.p1.5.m5.2.3.2.cmml" xref="A1.SS1.9.p1.5.m5.2.3.2"><times id="A1.SS1.9.p1.5.m5.2.3.2.1.cmml" xref="A1.SS1.9.p1.5.m5.2.3.2.1"></times><ci id="A1.SS1.9.p1.5.m5.2.3.2.2.cmml" xref="A1.SS1.9.p1.5.m5.2.3.2.2">∡</ci><interval closure="open" id="A1.SS1.9.p1.5.m5.2.3.2.3.1.cmml" xref="A1.SS1.9.p1.5.m5.2.3.2.3.2"><ci id="A1.SS1.9.p1.5.m5.1.1.cmml" xref="A1.SS1.9.p1.5.m5.1.1">𝑢</ci><ci id="A1.SS1.9.p1.5.m5.2.2.cmml" xref="A1.SS1.9.p1.5.m5.2.2">𝑣</ci></interval></apply><apply id="A1.SS1.9.p1.5.m5.2.3.3.cmml" xref="A1.SS1.9.p1.5.m5.2.3.3"><divide id="A1.SS1.9.p1.5.m5.2.3.3.1.cmml" xref="A1.SS1.9.p1.5.m5.2.3.3"></divide><ci id="A1.SS1.9.p1.5.m5.2.3.3.2.cmml" xref="A1.SS1.9.p1.5.m5.2.3.3.2">𝜋</ci><cn id="A1.SS1.9.p1.5.m5.2.3.3.3.cmml" type="integer" xref="A1.SS1.9.p1.5.m5.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.5.m5.2c">\measuredangle(u,v)\leq\frac{\pi}{2}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.5.m5.2d">∡ ( italic_u , italic_v ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. Moreover, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem3" title="Lemma A.3. ‣ A.1 Fundamentals of ℓ_𝑝-Halfspaces ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">A.3</span></a>, we have <math alttext="\measuredangle(u,z-x)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="A1.SS1.9.p1.6.m6.2"><semantics id="A1.SS1.9.p1.6.m6.2a"><mrow id="A1.SS1.9.p1.6.m6.2.2" xref="A1.SS1.9.p1.6.m6.2.2.cmml"><mrow id="A1.SS1.9.p1.6.m6.2.2.1" xref="A1.SS1.9.p1.6.m6.2.2.1.cmml"><mi id="A1.SS1.9.p1.6.m6.2.2.1.3" mathvariant="normal" xref="A1.SS1.9.p1.6.m6.2.2.1.3.cmml">∡</mi><mo id="A1.SS1.9.p1.6.m6.2.2.1.2" xref="A1.SS1.9.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="A1.SS1.9.p1.6.m6.2.2.1.1.1" xref="A1.SS1.9.p1.6.m6.2.2.1.1.2.cmml"><mo id="A1.SS1.9.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="A1.SS1.9.p1.6.m6.2.2.1.1.2.cmml">(</mo><mi id="A1.SS1.9.p1.6.m6.1.1" xref="A1.SS1.9.p1.6.m6.1.1.cmml">u</mi><mo id="A1.SS1.9.p1.6.m6.2.2.1.1.1.3" xref="A1.SS1.9.p1.6.m6.2.2.1.1.2.cmml">,</mo><mrow id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.2" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.1" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.3" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.9.p1.6.m6.2.2.1.1.1.4" stretchy="false" xref="A1.SS1.9.p1.6.m6.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.9.p1.6.m6.2.2.2" xref="A1.SS1.9.p1.6.m6.2.2.2.cmml">≤</mo><mrow id="A1.SS1.9.p1.6.m6.2.2.3" xref="A1.SS1.9.p1.6.m6.2.2.3.cmml"><mfrac id="A1.SS1.9.p1.6.m6.2.2.3.2" xref="A1.SS1.9.p1.6.m6.2.2.3.2.cmml"><mi id="A1.SS1.9.p1.6.m6.2.2.3.2.2" xref="A1.SS1.9.p1.6.m6.2.2.3.2.2.cmml">π</mi><mn id="A1.SS1.9.p1.6.m6.2.2.3.2.3" xref="A1.SS1.9.p1.6.m6.2.2.3.2.3.cmml">2</mn></mfrac><mo id="A1.SS1.9.p1.6.m6.2.2.3.1" xref="A1.SS1.9.p1.6.m6.2.2.3.1.cmml">−</mo><msqrt id="A1.SS1.9.p1.6.m6.2.2.3.3" xref="A1.SS1.9.p1.6.m6.2.2.3.3.cmml"><mrow id="A1.SS1.9.p1.6.m6.2.2.3.3.2" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.cmml"><mpadded id="A1.SS1.9.p1.6.m6.2.2.3.3.2.2" voffset="0.3em" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.2.cmml"><mn id="A1.SS1.9.p1.6.m6.2.2.3.3.2.2a" mathsize="70%" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.SS1.9.p1.6.m6.2.2.3.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.1.cmml"><mo id="A1.SS1.9.p1.6.m6.2.2.3.3.2.1a" stretchy="true" symmetric="true" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.1.cmml">/</mo></mpadded><mi id="A1.SS1.9.p1.6.m6.2.2.3.3.2.3" mathsize="70%" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.3.cmml">d</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.9.p1.6.m6.2b"><apply id="A1.SS1.9.p1.6.m6.2.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2"><leq id="A1.SS1.9.p1.6.m6.2.2.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.2"></leq><apply id="A1.SS1.9.p1.6.m6.2.2.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1"><times id="A1.SS1.9.p1.6.m6.2.2.1.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.2"></times><ci id="A1.SS1.9.p1.6.m6.2.2.1.3.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.3">∡</ci><interval closure="open" id="A1.SS1.9.p1.6.m6.2.2.1.1.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1"><ci id="A1.SS1.9.p1.6.m6.1.1.cmml" xref="A1.SS1.9.p1.6.m6.1.1">𝑢</ci><apply id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1"><minus id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.1"></minus><ci id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="A1.SS1.9.p1.6.m6.2.2.1.1.1.1.3">𝑥</ci></apply></interval></apply><apply id="A1.SS1.9.p1.6.m6.2.2.3.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3"><minus id="A1.SS1.9.p1.6.m6.2.2.3.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.1"></minus><apply id="A1.SS1.9.p1.6.m6.2.2.3.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.2"><divide id="A1.SS1.9.p1.6.m6.2.2.3.2.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.2"></divide><ci id="A1.SS1.9.p1.6.m6.2.2.3.2.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.2.2">𝜋</ci><cn id="A1.SS1.9.p1.6.m6.2.2.3.2.3.cmml" type="integer" xref="A1.SS1.9.p1.6.m6.2.2.3.2.3">2</cn></apply><apply id="A1.SS1.9.p1.6.m6.2.2.3.3.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.3"><root id="A1.SS1.9.p1.6.m6.2.2.3.3a.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.3"></root><apply id="A1.SS1.9.p1.6.m6.2.2.3.3.2.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2"><divide id="A1.SS1.9.p1.6.m6.2.2.3.3.2.1.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.1"></divide><cn id="A1.SS1.9.p1.6.m6.2.2.3.3.2.2.cmml" type="integer" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.2">1</cn><ci id="A1.SS1.9.p1.6.m6.2.2.3.3.2.3.cmml" xref="A1.SS1.9.p1.6.m6.2.2.3.3.2.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.6.m6.2c">\measuredangle(u,z-x)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.6.m6.2d">∡ ( italic_u , italic_z - italic_x ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG - square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. Combining the two bounds, we conclude <math alttext="\measuredangle(\overrightarrow{xz},v)=\measuredangle(z-x,v)\leq\pi-\sqrt{% \nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="A1.SS1.9.p1.7.m7.4"><semantics id="A1.SS1.9.p1.7.m7.4a"><mrow id="A1.SS1.9.p1.7.m7.4.4" xref="A1.SS1.9.p1.7.m7.4.4.cmml"><mrow id="A1.SS1.9.p1.7.m7.4.4.3" xref="A1.SS1.9.p1.7.m7.4.4.3.cmml"><mi id="A1.SS1.9.p1.7.m7.4.4.3.2" mathvariant="normal" xref="A1.SS1.9.p1.7.m7.4.4.3.2.cmml">∡</mi><mo id="A1.SS1.9.p1.7.m7.4.4.3.1" xref="A1.SS1.9.p1.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="A1.SS1.9.p1.7.m7.4.4.3.3.2" xref="A1.SS1.9.p1.7.m7.4.4.3.3.1.cmml"><mo id="A1.SS1.9.p1.7.m7.4.4.3.3.2.1" stretchy="false" xref="A1.SS1.9.p1.7.m7.4.4.3.3.1.cmml">(</mo><mover accent="true" id="A1.SS1.9.p1.7.m7.1.1" xref="A1.SS1.9.p1.7.m7.1.1.cmml"><mrow id="A1.SS1.9.p1.7.m7.1.1.2" xref="A1.SS1.9.p1.7.m7.1.1.2.cmml"><mi id="A1.SS1.9.p1.7.m7.1.1.2.2" xref="A1.SS1.9.p1.7.m7.1.1.2.2.cmml">x</mi><mo id="A1.SS1.9.p1.7.m7.1.1.2.1" xref="A1.SS1.9.p1.7.m7.1.1.2.1.cmml">⁢</mo><mi id="A1.SS1.9.p1.7.m7.1.1.2.3" xref="A1.SS1.9.p1.7.m7.1.1.2.3.cmml">z</mi></mrow><mo id="A1.SS1.9.p1.7.m7.1.1.1" stretchy="false" xref="A1.SS1.9.p1.7.m7.1.1.1.cmml">→</mo></mover><mo id="A1.SS1.9.p1.7.m7.4.4.3.3.2.2" xref="A1.SS1.9.p1.7.m7.4.4.3.3.1.cmml">,</mo><mi id="A1.SS1.9.p1.7.m7.2.2" xref="A1.SS1.9.p1.7.m7.2.2.cmml">v</mi><mo id="A1.SS1.9.p1.7.m7.4.4.3.3.2.3" stretchy="false" xref="A1.SS1.9.p1.7.m7.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.9.p1.7.m7.4.4.4" xref="A1.SS1.9.p1.7.m7.4.4.4.cmml">=</mo><mrow id="A1.SS1.9.p1.7.m7.4.4.1" xref="A1.SS1.9.p1.7.m7.4.4.1.cmml"><mi id="A1.SS1.9.p1.7.m7.4.4.1.3" mathvariant="normal" xref="A1.SS1.9.p1.7.m7.4.4.1.3.cmml">∡</mi><mo id="A1.SS1.9.p1.7.m7.4.4.1.2" xref="A1.SS1.9.p1.7.m7.4.4.1.2.cmml">⁢</mo><mrow id="A1.SS1.9.p1.7.m7.4.4.1.1.1" xref="A1.SS1.9.p1.7.m7.4.4.1.1.2.cmml"><mo id="A1.SS1.9.p1.7.m7.4.4.1.1.1.2" stretchy="false" xref="A1.SS1.9.p1.7.m7.4.4.1.1.2.cmml">(</mo><mrow id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.cmml"><mi id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.2" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.1" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.3" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.9.p1.7.m7.4.4.1.1.1.3" xref="A1.SS1.9.p1.7.m7.4.4.1.1.2.cmml">,</mo><mi id="A1.SS1.9.p1.7.m7.3.3" xref="A1.SS1.9.p1.7.m7.3.3.cmml">v</mi><mo id="A1.SS1.9.p1.7.m7.4.4.1.1.1.4" stretchy="false" xref="A1.SS1.9.p1.7.m7.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.9.p1.7.m7.4.4.5" xref="A1.SS1.9.p1.7.m7.4.4.5.cmml">≤</mo><mrow id="A1.SS1.9.p1.7.m7.4.4.6" xref="A1.SS1.9.p1.7.m7.4.4.6.cmml"><mi id="A1.SS1.9.p1.7.m7.4.4.6.2" xref="A1.SS1.9.p1.7.m7.4.4.6.2.cmml">π</mi><mo id="A1.SS1.9.p1.7.m7.4.4.6.1" xref="A1.SS1.9.p1.7.m7.4.4.6.1.cmml">−</mo><msqrt id="A1.SS1.9.p1.7.m7.4.4.6.3" xref="A1.SS1.9.p1.7.m7.4.4.6.3.cmml"><mrow id="A1.SS1.9.p1.7.m7.4.4.6.3.2" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.cmml"><mpadded id="A1.SS1.9.p1.7.m7.4.4.6.3.2.2" voffset="0.3em" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.2.cmml"><mn id="A1.SS1.9.p1.7.m7.4.4.6.3.2.2a" mathsize="70%" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.SS1.9.p1.7.m7.4.4.6.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.1.cmml"><mo id="A1.SS1.9.p1.7.m7.4.4.6.3.2.1a" stretchy="true" symmetric="true" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.1.cmml">/</mo></mpadded><mi id="A1.SS1.9.p1.7.m7.4.4.6.3.2.3" mathsize="70%" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.3.cmml">d</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.9.p1.7.m7.4b"><apply id="A1.SS1.9.p1.7.m7.4.4.cmml" xref="A1.SS1.9.p1.7.m7.4.4"><and id="A1.SS1.9.p1.7.m7.4.4a.cmml" xref="A1.SS1.9.p1.7.m7.4.4"></and><apply id="A1.SS1.9.p1.7.m7.4.4b.cmml" xref="A1.SS1.9.p1.7.m7.4.4"><eq id="A1.SS1.9.p1.7.m7.4.4.4.cmml" xref="A1.SS1.9.p1.7.m7.4.4.4"></eq><apply id="A1.SS1.9.p1.7.m7.4.4.3.cmml" xref="A1.SS1.9.p1.7.m7.4.4.3"><times id="A1.SS1.9.p1.7.m7.4.4.3.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.3.1"></times><ci id="A1.SS1.9.p1.7.m7.4.4.3.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.3.2">∡</ci><interval closure="open" id="A1.SS1.9.p1.7.m7.4.4.3.3.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.3.3.2"><apply id="A1.SS1.9.p1.7.m7.1.1.cmml" xref="A1.SS1.9.p1.7.m7.1.1"><ci id="A1.SS1.9.p1.7.m7.1.1.1.cmml" xref="A1.SS1.9.p1.7.m7.1.1.1">→</ci><apply id="A1.SS1.9.p1.7.m7.1.1.2.cmml" xref="A1.SS1.9.p1.7.m7.1.1.2"><times id="A1.SS1.9.p1.7.m7.1.1.2.1.cmml" xref="A1.SS1.9.p1.7.m7.1.1.2.1"></times><ci id="A1.SS1.9.p1.7.m7.1.1.2.2.cmml" xref="A1.SS1.9.p1.7.m7.1.1.2.2">𝑥</ci><ci id="A1.SS1.9.p1.7.m7.1.1.2.3.cmml" xref="A1.SS1.9.p1.7.m7.1.1.2.3">𝑧</ci></apply></apply><ci id="A1.SS1.9.p1.7.m7.2.2.cmml" xref="A1.SS1.9.p1.7.m7.2.2">𝑣</ci></interval></apply><apply id="A1.SS1.9.p1.7.m7.4.4.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1"><times id="A1.SS1.9.p1.7.m7.4.4.1.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.2"></times><ci id="A1.SS1.9.p1.7.m7.4.4.1.3.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.3">∡</ci><interval closure="open" id="A1.SS1.9.p1.7.m7.4.4.1.1.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1"><apply id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1"><minus id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.1"></minus><ci id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.3.cmml" xref="A1.SS1.9.p1.7.m7.4.4.1.1.1.1.3">𝑥</ci></apply><ci id="A1.SS1.9.p1.7.m7.3.3.cmml" xref="A1.SS1.9.p1.7.m7.3.3">𝑣</ci></interval></apply></apply><apply id="A1.SS1.9.p1.7.m7.4.4c.cmml" xref="A1.SS1.9.p1.7.m7.4.4"><leq id="A1.SS1.9.p1.7.m7.4.4.5.cmml" xref="A1.SS1.9.p1.7.m7.4.4.5"></leq><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.9.p1.7.m7.4.4.1.cmml" id="A1.SS1.9.p1.7.m7.4.4d.cmml" xref="A1.SS1.9.p1.7.m7.4.4"></share><apply id="A1.SS1.9.p1.7.m7.4.4.6.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6"><minus id="A1.SS1.9.p1.7.m7.4.4.6.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.1"></minus><ci id="A1.SS1.9.p1.7.m7.4.4.6.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.2">𝜋</ci><apply id="A1.SS1.9.p1.7.m7.4.4.6.3.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.3"><root id="A1.SS1.9.p1.7.m7.4.4.6.3a.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.3"></root><apply id="A1.SS1.9.p1.7.m7.4.4.6.3.2.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2"><divide id="A1.SS1.9.p1.7.m7.4.4.6.3.2.1.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.1"></divide><cn id="A1.SS1.9.p1.7.m7.4.4.6.3.2.2.cmml" type="integer" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.2">1</cn><ci id="A1.SS1.9.p1.7.m7.4.4.6.3.2.3.cmml" xref="A1.SS1.9.p1.7.m7.4.4.6.3.2.3">𝑑</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.9.p1.7.m7.4c">\measuredangle(\overrightarrow{xz},v)=\measuredangle(z-x,v)\leq\pi-\sqrt{% \nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.9.p1.7.m7.4d">∡ ( over→ start_ARG italic_x italic_z end_ARG , italic_v ) = ∡ ( italic_z - italic_x , italic_v ) ≤ italic_π - square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>, as desired.</p> </div> <div class="ltx_para" id="A1.SS1.10.p2"> <p class="ltx_p" id="A1.SS1.10.p2.6">For the second statement, assume that <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.10.p2.1.m1.1"><semantics id="A1.SS1.10.p2.1.m1.1a"><mi id="A1.SS1.10.p2.1.m1.1.1" xref="A1.SS1.10.p2.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.1.m1.1b"><ci id="A1.SS1.10.p2.1.m1.1.1.cmml" xref="A1.SS1.10.p2.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.1.m1.1d">italic_z</annotation></semantics></math> satisfies <math alttext="\measuredangle(\overrightarrow{xz},v)\leq\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="A1.SS1.10.p2.2.m2.2"><semantics id="A1.SS1.10.p2.2.m2.2a"><mrow id="A1.SS1.10.p2.2.m2.2.3" xref="A1.SS1.10.p2.2.m2.2.3.cmml"><mrow id="A1.SS1.10.p2.2.m2.2.3.2" xref="A1.SS1.10.p2.2.m2.2.3.2.cmml"><mi id="A1.SS1.10.p2.2.m2.2.3.2.2" mathvariant="normal" xref="A1.SS1.10.p2.2.m2.2.3.2.2.cmml">∡</mi><mo id="A1.SS1.10.p2.2.m2.2.3.2.1" xref="A1.SS1.10.p2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS1.10.p2.2.m2.2.3.2.3.2" xref="A1.SS1.10.p2.2.m2.2.3.2.3.1.cmml"><mo id="A1.SS1.10.p2.2.m2.2.3.2.3.2.1" stretchy="false" xref="A1.SS1.10.p2.2.m2.2.3.2.3.1.cmml">(</mo><mover accent="true" id="A1.SS1.10.p2.2.m2.1.1" xref="A1.SS1.10.p2.2.m2.1.1.cmml"><mrow id="A1.SS1.10.p2.2.m2.1.1.2" xref="A1.SS1.10.p2.2.m2.1.1.2.cmml"><mi id="A1.SS1.10.p2.2.m2.1.1.2.2" xref="A1.SS1.10.p2.2.m2.1.1.2.2.cmml">x</mi><mo id="A1.SS1.10.p2.2.m2.1.1.2.1" xref="A1.SS1.10.p2.2.m2.1.1.2.1.cmml">⁢</mo><mi id="A1.SS1.10.p2.2.m2.1.1.2.3" xref="A1.SS1.10.p2.2.m2.1.1.2.3.cmml">z</mi></mrow><mo id="A1.SS1.10.p2.2.m2.1.1.1" stretchy="false" xref="A1.SS1.10.p2.2.m2.1.1.1.cmml">→</mo></mover><mo id="A1.SS1.10.p2.2.m2.2.3.2.3.2.2" xref="A1.SS1.10.p2.2.m2.2.3.2.3.1.cmml">,</mo><mi id="A1.SS1.10.p2.2.m2.2.2" xref="A1.SS1.10.p2.2.m2.2.2.cmml">v</mi><mo id="A1.SS1.10.p2.2.m2.2.3.2.3.2.3" stretchy="false" xref="A1.SS1.10.p2.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.10.p2.2.m2.2.3.1" xref="A1.SS1.10.p2.2.m2.2.3.1.cmml">≤</mo><msqrt id="A1.SS1.10.p2.2.m2.2.3.3" xref="A1.SS1.10.p2.2.m2.2.3.3.cmml"><mrow id="A1.SS1.10.p2.2.m2.2.3.3.2" xref="A1.SS1.10.p2.2.m2.2.3.3.2.cmml"><mpadded id="A1.SS1.10.p2.2.m2.2.3.3.2.2" voffset="0.3em" xref="A1.SS1.10.p2.2.m2.2.3.3.2.2.cmml"><mn id="A1.SS1.10.p2.2.m2.2.3.3.2.2a" mathsize="70%" xref="A1.SS1.10.p2.2.m2.2.3.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.SS1.10.p2.2.m2.2.3.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.SS1.10.p2.2.m2.2.3.3.2.1.cmml"><mo id="A1.SS1.10.p2.2.m2.2.3.3.2.1a" stretchy="true" symmetric="true" xref="A1.SS1.10.p2.2.m2.2.3.3.2.1.cmml">/</mo></mpadded><mi id="A1.SS1.10.p2.2.m2.2.3.3.2.3" mathsize="70%" xref="A1.SS1.10.p2.2.m2.2.3.3.2.3.cmml">d</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.2.m2.2b"><apply id="A1.SS1.10.p2.2.m2.2.3.cmml" xref="A1.SS1.10.p2.2.m2.2.3"><leq id="A1.SS1.10.p2.2.m2.2.3.1.cmml" xref="A1.SS1.10.p2.2.m2.2.3.1"></leq><apply id="A1.SS1.10.p2.2.m2.2.3.2.cmml" xref="A1.SS1.10.p2.2.m2.2.3.2"><times id="A1.SS1.10.p2.2.m2.2.3.2.1.cmml" xref="A1.SS1.10.p2.2.m2.2.3.2.1"></times><ci id="A1.SS1.10.p2.2.m2.2.3.2.2.cmml" xref="A1.SS1.10.p2.2.m2.2.3.2.2">∡</ci><interval closure="open" id="A1.SS1.10.p2.2.m2.2.3.2.3.1.cmml" xref="A1.SS1.10.p2.2.m2.2.3.2.3.2"><apply id="A1.SS1.10.p2.2.m2.1.1.cmml" xref="A1.SS1.10.p2.2.m2.1.1"><ci id="A1.SS1.10.p2.2.m2.1.1.1.cmml" xref="A1.SS1.10.p2.2.m2.1.1.1">→</ci><apply id="A1.SS1.10.p2.2.m2.1.1.2.cmml" xref="A1.SS1.10.p2.2.m2.1.1.2"><times id="A1.SS1.10.p2.2.m2.1.1.2.1.cmml" xref="A1.SS1.10.p2.2.m2.1.1.2.1"></times><ci id="A1.SS1.10.p2.2.m2.1.1.2.2.cmml" xref="A1.SS1.10.p2.2.m2.1.1.2.2">𝑥</ci><ci id="A1.SS1.10.p2.2.m2.1.1.2.3.cmml" xref="A1.SS1.10.p2.2.m2.1.1.2.3">𝑧</ci></apply></apply><ci id="A1.SS1.10.p2.2.m2.2.2.cmml" xref="A1.SS1.10.p2.2.m2.2.2">𝑣</ci></interval></apply><apply id="A1.SS1.10.p2.2.m2.2.3.3.cmml" xref="A1.SS1.10.p2.2.m2.2.3.3"><root id="A1.SS1.10.p2.2.m2.2.3.3a.cmml" xref="A1.SS1.10.p2.2.m2.2.3.3"></root><apply id="A1.SS1.10.p2.2.m2.2.3.3.2.cmml" xref="A1.SS1.10.p2.2.m2.2.3.3.2"><divide id="A1.SS1.10.p2.2.m2.2.3.3.2.1.cmml" xref="A1.SS1.10.p2.2.m2.2.3.3.2.1"></divide><cn id="A1.SS1.10.p2.2.m2.2.3.3.2.2.cmml" type="integer" xref="A1.SS1.10.p2.2.m2.2.3.3.2.2">1</cn><ci id="A1.SS1.10.p2.2.m2.2.3.3.2.3.cmml" xref="A1.SS1.10.p2.2.m2.2.3.3.2.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.2.m2.2c">\measuredangle(\overrightarrow{xz},v)\leq\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.2.m2.2d">∡ ( over→ start_ARG italic_x italic_z end_ARG , italic_v ) ≤ square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem3" title="Lemma A.3. ‣ A.1 Fundamentals of ℓ_𝑝-Halfspaces ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">A.3</span></a>, there exists a subgradient <math alttext="u\in\partial||z-x||_{p}" class="ltx_Math" display="inline" id="A1.SS1.10.p2.3.m3.1"><semantics id="A1.SS1.10.p2.3.m3.1a"><mrow id="A1.SS1.10.p2.3.m3.1.1" xref="A1.SS1.10.p2.3.m3.1.1.cmml"><mi id="A1.SS1.10.p2.3.m3.1.1.3" xref="A1.SS1.10.p2.3.m3.1.1.3.cmml">u</mi><mo id="A1.SS1.10.p2.3.m3.1.1.2" rspace="0.1389em" xref="A1.SS1.10.p2.3.m3.1.1.2.cmml">∈</mo><mrow id="A1.SS1.10.p2.3.m3.1.1.1" xref="A1.SS1.10.p2.3.m3.1.1.1.cmml"><mo id="A1.SS1.10.p2.3.m3.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS1.10.p2.3.m3.1.1.1.2.cmml">∂</mo><msub id="A1.SS1.10.p2.3.m3.1.1.1.1" xref="A1.SS1.10.p2.3.m3.1.1.1.1.cmml"><mrow id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.2.cmml"><mo id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.cmml"><mi id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.2" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.1" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.3" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi id="A1.SS1.10.p2.3.m3.1.1.1.1.3" xref="A1.SS1.10.p2.3.m3.1.1.1.1.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.3.m3.1b"><apply id="A1.SS1.10.p2.3.m3.1.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1"><in id="A1.SS1.10.p2.3.m3.1.1.2.cmml" xref="A1.SS1.10.p2.3.m3.1.1.2"></in><ci id="A1.SS1.10.p2.3.m3.1.1.3.cmml" xref="A1.SS1.10.p2.3.m3.1.1.3">𝑢</ci><apply id="A1.SS1.10.p2.3.m3.1.1.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1"><partialdiff id="A1.SS1.10.p2.3.m3.1.1.1.2.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.2"></partialdiff><apply id="A1.SS1.10.p2.3.m3.1.1.1.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.10.p2.3.m3.1.1.1.1.2.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1">subscript</csymbol><apply id="A1.SS1.10.p2.3.m3.1.1.1.1.1.2.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.10.p2.3.m3.1.1.1.1.1.2.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.2">norm</csymbol><apply id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1"><minus id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.1.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.2.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.3.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.SS1.10.p2.3.m3.1.1.1.1.3.cmml" xref="A1.SS1.10.p2.3.m3.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.3.m3.1c">u\in\partial||z-x||_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.3.m3.1d">italic_u ∈ ∂ | | italic_z - italic_x | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\measuredangle(u,z-x)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}" class="ltx_Math" display="inline" id="A1.SS1.10.p2.4.m4.2"><semantics id="A1.SS1.10.p2.4.m4.2a"><mrow id="A1.SS1.10.p2.4.m4.2.2" xref="A1.SS1.10.p2.4.m4.2.2.cmml"><mrow id="A1.SS1.10.p2.4.m4.2.2.1" xref="A1.SS1.10.p2.4.m4.2.2.1.cmml"><mi id="A1.SS1.10.p2.4.m4.2.2.1.3" mathvariant="normal" xref="A1.SS1.10.p2.4.m4.2.2.1.3.cmml">∡</mi><mo id="A1.SS1.10.p2.4.m4.2.2.1.2" xref="A1.SS1.10.p2.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="A1.SS1.10.p2.4.m4.2.2.1.1.1" xref="A1.SS1.10.p2.4.m4.2.2.1.1.2.cmml"><mo id="A1.SS1.10.p2.4.m4.2.2.1.1.1.2" stretchy="false" xref="A1.SS1.10.p2.4.m4.2.2.1.1.2.cmml">(</mo><mi id="A1.SS1.10.p2.4.m4.1.1" xref="A1.SS1.10.p2.4.m4.1.1.cmml">u</mi><mo id="A1.SS1.10.p2.4.m4.2.2.1.1.1.3" xref="A1.SS1.10.p2.4.m4.2.2.1.1.2.cmml">,</mo><mrow id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.cmml"><mi id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.2" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.1" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.3" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS1.10.p2.4.m4.2.2.1.1.1.4" stretchy="false" xref="A1.SS1.10.p2.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.10.p2.4.m4.2.2.2" xref="A1.SS1.10.p2.4.m4.2.2.2.cmml">≤</mo><mrow id="A1.SS1.10.p2.4.m4.2.2.3" xref="A1.SS1.10.p2.4.m4.2.2.3.cmml"><mfrac id="A1.SS1.10.p2.4.m4.2.2.3.2" xref="A1.SS1.10.p2.4.m4.2.2.3.2.cmml"><mi id="A1.SS1.10.p2.4.m4.2.2.3.2.2" xref="A1.SS1.10.p2.4.m4.2.2.3.2.2.cmml">π</mi><mn id="A1.SS1.10.p2.4.m4.2.2.3.2.3" xref="A1.SS1.10.p2.4.m4.2.2.3.2.3.cmml">2</mn></mfrac><mo id="A1.SS1.10.p2.4.m4.2.2.3.1" xref="A1.SS1.10.p2.4.m4.2.2.3.1.cmml">−</mo><msqrt id="A1.SS1.10.p2.4.m4.2.2.3.3" xref="A1.SS1.10.p2.4.m4.2.2.3.3.cmml"><mrow id="A1.SS1.10.p2.4.m4.2.2.3.3.2" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.cmml"><mpadded id="A1.SS1.10.p2.4.m4.2.2.3.3.2.2" voffset="0.3em" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.2.cmml"><mn id="A1.SS1.10.p2.4.m4.2.2.3.3.2.2a" mathsize="70%" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.SS1.10.p2.4.m4.2.2.3.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.1.cmml"><mo id="A1.SS1.10.p2.4.m4.2.2.3.3.2.1a" stretchy="true" symmetric="true" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.1.cmml">/</mo></mpadded><mi id="A1.SS1.10.p2.4.m4.2.2.3.3.2.3" mathsize="70%" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.3.cmml">d</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.4.m4.2b"><apply id="A1.SS1.10.p2.4.m4.2.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2"><leq id="A1.SS1.10.p2.4.m4.2.2.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.2"></leq><apply id="A1.SS1.10.p2.4.m4.2.2.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1"><times id="A1.SS1.10.p2.4.m4.2.2.1.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.2"></times><ci id="A1.SS1.10.p2.4.m4.2.2.1.3.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.3">∡</ci><interval closure="open" id="A1.SS1.10.p2.4.m4.2.2.1.1.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1"><ci id="A1.SS1.10.p2.4.m4.1.1.cmml" xref="A1.SS1.10.p2.4.m4.1.1">𝑢</ci><apply id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1"><minus id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.1"></minus><ci id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.2">𝑧</ci><ci id="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.3.cmml" xref="A1.SS1.10.p2.4.m4.2.2.1.1.1.1.3">𝑥</ci></apply></interval></apply><apply id="A1.SS1.10.p2.4.m4.2.2.3.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3"><minus id="A1.SS1.10.p2.4.m4.2.2.3.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.1"></minus><apply id="A1.SS1.10.p2.4.m4.2.2.3.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.2"><divide id="A1.SS1.10.p2.4.m4.2.2.3.2.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.2"></divide><ci id="A1.SS1.10.p2.4.m4.2.2.3.2.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.2.2">𝜋</ci><cn id="A1.SS1.10.p2.4.m4.2.2.3.2.3.cmml" type="integer" xref="A1.SS1.10.p2.4.m4.2.2.3.2.3">2</cn></apply><apply id="A1.SS1.10.p2.4.m4.2.2.3.3.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.3"><root id="A1.SS1.10.p2.4.m4.2.2.3.3a.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.3"></root><apply id="A1.SS1.10.p2.4.m4.2.2.3.3.2.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2"><divide id="A1.SS1.10.p2.4.m4.2.2.3.3.2.1.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.1"></divide><cn id="A1.SS1.10.p2.4.m4.2.2.3.3.2.2.cmml" type="integer" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.2">1</cn><ci id="A1.SS1.10.p2.4.m4.2.2.3.3.2.3.cmml" xref="A1.SS1.10.p2.4.m4.2.2.3.3.2.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.4.m4.2c">\measuredangle(u,z-x)\leq\frac{\pi}{2}-\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.4.m4.2d">∡ ( italic_u , italic_z - italic_x ) ≤ divide start_ARG italic_π end_ARG start_ARG 2 end_ARG - square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. We conclude <math alttext="\measuredangle(u,v)\leq\pi" class="ltx_Math" display="inline" id="A1.SS1.10.p2.5.m5.2"><semantics id="A1.SS1.10.p2.5.m5.2a"><mrow id="A1.SS1.10.p2.5.m5.2.3" xref="A1.SS1.10.p2.5.m5.2.3.cmml"><mrow id="A1.SS1.10.p2.5.m5.2.3.2" xref="A1.SS1.10.p2.5.m5.2.3.2.cmml"><mi id="A1.SS1.10.p2.5.m5.2.3.2.2" mathvariant="normal" xref="A1.SS1.10.p2.5.m5.2.3.2.2.cmml">∡</mi><mo id="A1.SS1.10.p2.5.m5.2.3.2.1" xref="A1.SS1.10.p2.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS1.10.p2.5.m5.2.3.2.3.2" xref="A1.SS1.10.p2.5.m5.2.3.2.3.1.cmml"><mo id="A1.SS1.10.p2.5.m5.2.3.2.3.2.1" stretchy="false" xref="A1.SS1.10.p2.5.m5.2.3.2.3.1.cmml">(</mo><mi id="A1.SS1.10.p2.5.m5.1.1" xref="A1.SS1.10.p2.5.m5.1.1.cmml">u</mi><mo id="A1.SS1.10.p2.5.m5.2.3.2.3.2.2" xref="A1.SS1.10.p2.5.m5.2.3.2.3.1.cmml">,</mo><mi id="A1.SS1.10.p2.5.m5.2.2" xref="A1.SS1.10.p2.5.m5.2.2.cmml">v</mi><mo id="A1.SS1.10.p2.5.m5.2.3.2.3.2.3" stretchy="false" xref="A1.SS1.10.p2.5.m5.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.10.p2.5.m5.2.3.1" xref="A1.SS1.10.p2.5.m5.2.3.1.cmml">≤</mo><mi id="A1.SS1.10.p2.5.m5.2.3.3" xref="A1.SS1.10.p2.5.m5.2.3.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.5.m5.2b"><apply id="A1.SS1.10.p2.5.m5.2.3.cmml" xref="A1.SS1.10.p2.5.m5.2.3"><leq id="A1.SS1.10.p2.5.m5.2.3.1.cmml" xref="A1.SS1.10.p2.5.m5.2.3.1"></leq><apply id="A1.SS1.10.p2.5.m5.2.3.2.cmml" xref="A1.SS1.10.p2.5.m5.2.3.2"><times id="A1.SS1.10.p2.5.m5.2.3.2.1.cmml" xref="A1.SS1.10.p2.5.m5.2.3.2.1"></times><ci id="A1.SS1.10.p2.5.m5.2.3.2.2.cmml" xref="A1.SS1.10.p2.5.m5.2.3.2.2">∡</ci><interval closure="open" id="A1.SS1.10.p2.5.m5.2.3.2.3.1.cmml" xref="A1.SS1.10.p2.5.m5.2.3.2.3.2"><ci id="A1.SS1.10.p2.5.m5.1.1.cmml" xref="A1.SS1.10.p2.5.m5.1.1">𝑢</ci><ci id="A1.SS1.10.p2.5.m5.2.2.cmml" xref="A1.SS1.10.p2.5.m5.2.2">𝑣</ci></interval></apply><ci id="A1.SS1.10.p2.5.m5.2.3.3.cmml" xref="A1.SS1.10.p2.5.m5.2.3.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.5.m5.2c">\measuredangle(u,v)\leq\pi</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.5.m5.2d">∡ ( italic_u , italic_v ) ≤ italic_π</annotation></semantics></math>, which implies <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS1.10.p2.6.m6.2"><semantics id="A1.SS1.10.p2.6.m6.2a"><mrow id="A1.SS1.10.p2.6.m6.2.3" xref="A1.SS1.10.p2.6.m6.2.3.cmml"><mi id="A1.SS1.10.p2.6.m6.2.3.2" xref="A1.SS1.10.p2.6.m6.2.3.2.cmml">z</mi><mo id="A1.SS1.10.p2.6.m6.2.3.1" xref="A1.SS1.10.p2.6.m6.2.3.1.cmml">∈</mo><msubsup id="A1.SS1.10.p2.6.m6.2.3.3" xref="A1.SS1.10.p2.6.m6.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS1.10.p2.6.m6.2.3.3.2.2" xref="A1.SS1.10.p2.6.m6.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS1.10.p2.6.m6.2.2.2.4" xref="A1.SS1.10.p2.6.m6.2.2.2.3.cmml"><mi id="A1.SS1.10.p2.6.m6.1.1.1.1" xref="A1.SS1.10.p2.6.m6.1.1.1.1.cmml">x</mi><mo id="A1.SS1.10.p2.6.m6.2.2.2.4.1" xref="A1.SS1.10.p2.6.m6.2.2.2.3.cmml">,</mo><mi id="A1.SS1.10.p2.6.m6.2.2.2.2" xref="A1.SS1.10.p2.6.m6.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS1.10.p2.6.m6.2.3.3.2.3" xref="A1.SS1.10.p2.6.m6.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.10.p2.6.m6.2b"><apply id="A1.SS1.10.p2.6.m6.2.3.cmml" xref="A1.SS1.10.p2.6.m6.2.3"><in id="A1.SS1.10.p2.6.m6.2.3.1.cmml" xref="A1.SS1.10.p2.6.m6.2.3.1"></in><ci id="A1.SS1.10.p2.6.m6.2.3.2.cmml" xref="A1.SS1.10.p2.6.m6.2.3.2">𝑧</ci><apply id="A1.SS1.10.p2.6.m6.2.3.3.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.10.p2.6.m6.2.3.3.1.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3">subscript</csymbol><apply id="A1.SS1.10.p2.6.m6.2.3.3.2.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.10.p2.6.m6.2.3.3.2.1.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3">superscript</csymbol><ci id="A1.SS1.10.p2.6.m6.2.3.3.2.2.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3.2.2">ℋ</ci><ci id="A1.SS1.10.p2.6.m6.2.3.3.2.3.cmml" xref="A1.SS1.10.p2.6.m6.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS1.10.p2.6.m6.2.2.2.3.cmml" xref="A1.SS1.10.p2.6.m6.2.2.2.4"><ci id="A1.SS1.10.p2.6.m6.1.1.1.1.cmml" xref="A1.SS1.10.p2.6.m6.1.1.1.1">𝑥</ci><ci id="A1.SS1.10.p2.6.m6.2.2.2.2.cmml" xref="A1.SS1.10.p2.6.m6.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.10.p2.6.m6.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.10.p2.6.m6.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Rounding to the Grid in the ℓ₁-Case ‣ 4 Finding Fixpoints of ℓ_𝑝-Contraction Maps ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p18"> <p class="ltx_p" id="A1.SS1.p18.1">Finally, we can prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.8</span></a> using <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a>.</p> </div> <div class="ltx_para" id="A1.SS1.p19"> <p class="ltx_p" id="A1.SS1.p19.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.8</span></a></p> </div> <div class="ltx_proof" id="A1.SS1.11"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.11.p1"> <p class="ltx_p" id="A1.SS1.11.p1.7">By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a>, it suffices to prove that all points <math alttext="z\in[0,1]^{d}" class="ltx_Math" display="inline" id="A1.SS1.11.p1.1.m1.2"><semantics id="A1.SS1.11.p1.1.m1.2a"><mrow id="A1.SS1.11.p1.1.m1.2.3" xref="A1.SS1.11.p1.1.m1.2.3.cmml"><mi id="A1.SS1.11.p1.1.m1.2.3.2" xref="A1.SS1.11.p1.1.m1.2.3.2.cmml">z</mi><mo id="A1.SS1.11.p1.1.m1.2.3.1" xref="A1.SS1.11.p1.1.m1.2.3.1.cmml">∈</mo><msup id="A1.SS1.11.p1.1.m1.2.3.3" xref="A1.SS1.11.p1.1.m1.2.3.3.cmml"><mrow id="A1.SS1.11.p1.1.m1.2.3.3.2.2" xref="A1.SS1.11.p1.1.m1.2.3.3.2.1.cmml"><mo id="A1.SS1.11.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="A1.SS1.11.p1.1.m1.2.3.3.2.1.cmml">[</mo><mn id="A1.SS1.11.p1.1.m1.1.1" xref="A1.SS1.11.p1.1.m1.1.1.cmml">0</mn><mo id="A1.SS1.11.p1.1.m1.2.3.3.2.2.2" xref="A1.SS1.11.p1.1.m1.2.3.3.2.1.cmml">,</mo><mn id="A1.SS1.11.p1.1.m1.2.2" xref="A1.SS1.11.p1.1.m1.2.2.cmml">1</mn><mo id="A1.SS1.11.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="A1.SS1.11.p1.1.m1.2.3.3.2.1.cmml">]</mo></mrow><mi id="A1.SS1.11.p1.1.m1.2.3.3.3" xref="A1.SS1.11.p1.1.m1.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.1.m1.2b"><apply id="A1.SS1.11.p1.1.m1.2.3.cmml" xref="A1.SS1.11.p1.1.m1.2.3"><in id="A1.SS1.11.p1.1.m1.2.3.1.cmml" xref="A1.SS1.11.p1.1.m1.2.3.1"></in><ci id="A1.SS1.11.p1.1.m1.2.3.2.cmml" xref="A1.SS1.11.p1.1.m1.2.3.2">𝑧</ci><apply id="A1.SS1.11.p1.1.m1.2.3.3.cmml" xref="A1.SS1.11.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS1.11.p1.1.m1.2.3.3.1.cmml" xref="A1.SS1.11.p1.1.m1.2.3.3">superscript</csymbol><interval closure="closed" id="A1.SS1.11.p1.1.m1.2.3.3.2.1.cmml" xref="A1.SS1.11.p1.1.m1.2.3.3.2.2"><cn id="A1.SS1.11.p1.1.m1.1.1.cmml" type="integer" xref="A1.SS1.11.p1.1.m1.1.1">0</cn><cn id="A1.SS1.11.p1.1.m1.2.2.cmml" type="integer" xref="A1.SS1.11.p1.1.m1.2.2">1</cn></interval><ci id="A1.SS1.11.p1.1.m1.2.3.3.3.cmml" xref="A1.SS1.11.p1.1.m1.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.1.m1.2c">z\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.1.m1.2d">italic_z ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> satisfy <math 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id="A1.SS1.11.p1.2.m2.1.1.2.3" xref="A1.SS1.11.p1.2.m2.1.1.2.3.cmml">z</mi></mrow><mo id="A1.SS1.11.p1.2.m2.1.1.1" stretchy="false" xref="A1.SS1.11.p1.2.m2.1.1.1.cmml">→</mo></mover><mo id="A1.SS1.11.p1.2.m2.2.2.1.1.1.3" xref="A1.SS1.11.p1.2.m2.2.2.1.1.2.cmml">,</mo><mrow id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1a" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.2" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.2.cmml">x</mi></mrow><mo id="A1.SS1.11.p1.2.m2.2.2.1.1.1.4" stretchy="false" xref="A1.SS1.11.p1.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.11.p1.2.m2.2.2.2" xref="A1.SS1.11.p1.2.m2.2.2.2.cmml">≤</mo><msqrt id="A1.SS1.11.p1.2.m2.2.2.3" xref="A1.SS1.11.p1.2.m2.2.2.3.cmml"><mrow id="A1.SS1.11.p1.2.m2.2.2.3.2" xref="A1.SS1.11.p1.2.m2.2.2.3.2.cmml"><mpadded id="A1.SS1.11.p1.2.m2.2.2.3.2.2" voffset="0.3em" xref="A1.SS1.11.p1.2.m2.2.2.3.2.2.cmml"><mn id="A1.SS1.11.p1.2.m2.2.2.3.2.2a" mathsize="70%" xref="A1.SS1.11.p1.2.m2.2.2.3.2.2.cmml">1</mn></mpadded><mpadded id="A1.SS1.11.p1.2.m2.2.2.3.2.1" lspace="-0.1em" width="-0.15em" xref="A1.SS1.11.p1.2.m2.2.2.3.2.1.cmml"><mo id="A1.SS1.11.p1.2.m2.2.2.3.2.1a" stretchy="true" symmetric="true" xref="A1.SS1.11.p1.2.m2.2.2.3.2.1.cmml">/</mo></mpadded><mi id="A1.SS1.11.p1.2.m2.2.2.3.2.3" mathsize="70%" xref="A1.SS1.11.p1.2.m2.2.2.3.2.3.cmml">d</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.2.m2.2b"><apply id="A1.SS1.11.p1.2.m2.2.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2"><leq id="A1.SS1.11.p1.2.m2.2.2.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2.2"></leq><apply id="A1.SS1.11.p1.2.m2.2.2.1.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1"><times id="A1.SS1.11.p1.2.m2.2.2.1.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.2"></times><ci id="A1.SS1.11.p1.2.m2.2.2.1.3.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.3">∡</ci><interval closure="open" id="A1.SS1.11.p1.2.m2.2.2.1.1.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1"><apply id="A1.SS1.11.p1.2.m2.1.1.cmml" xref="A1.SS1.11.p1.2.m2.1.1"><ci id="A1.SS1.11.p1.2.m2.1.1.1.cmml" xref="A1.SS1.11.p1.2.m2.1.1.1">→</ci><apply id="A1.SS1.11.p1.2.m2.1.1.2.cmml" xref="A1.SS1.11.p1.2.m2.1.1.2"><times id="A1.SS1.11.p1.2.m2.1.1.2.1.cmml" xref="A1.SS1.11.p1.2.m2.1.1.2.1"></times><ci id="A1.SS1.11.p1.2.m2.1.1.2.2.cmml" xref="A1.SS1.11.p1.2.m2.1.1.2.2">𝑥</ci><ci id="A1.SS1.11.p1.2.m2.1.1.2.3.cmml" xref="A1.SS1.11.p1.2.m2.1.1.2.3">𝑧</ci></apply></apply><apply id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1"><minus id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1"></minus><ci id="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2.1.1.1.1.2">𝑥</ci></apply></interval></apply><apply id="A1.SS1.11.p1.2.m2.2.2.3.cmml" xref="A1.SS1.11.p1.2.m2.2.2.3"><root id="A1.SS1.11.p1.2.m2.2.2.3a.cmml" xref="A1.SS1.11.p1.2.m2.2.2.3"></root><apply id="A1.SS1.11.p1.2.m2.2.2.3.2.cmml" xref="A1.SS1.11.p1.2.m2.2.2.3.2"><divide id="A1.SS1.11.p1.2.m2.2.2.3.2.1.cmml" xref="A1.SS1.11.p1.2.m2.2.2.3.2.1"></divide><cn id="A1.SS1.11.p1.2.m2.2.2.3.2.2.cmml" type="integer" xref="A1.SS1.11.p1.2.m2.2.2.3.2.2">1</cn><ci id="A1.SS1.11.p1.2.m2.2.2.3.2.3.cmml" xref="A1.SS1.11.p1.2.m2.2.2.3.2.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.2.m2.2c">\measuredangle(\overrightarrow{xz},-x)\leq\sqrt{\nicefrac{{1}}{{d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.2.m2.2d">∡ ( over→ start_ARG italic_x italic_z end_ARG , - italic_x ) ≤ square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. To see this, consider the triangle spanned by the origin <math alttext="0\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS1.11.p1.3.m3.1"><semantics id="A1.SS1.11.p1.3.m3.1a"><mrow id="A1.SS1.11.p1.3.m3.1.1" xref="A1.SS1.11.p1.3.m3.1.1.cmml"><mn id="A1.SS1.11.p1.3.m3.1.1.2" xref="A1.SS1.11.p1.3.m3.1.1.2.cmml">0</mn><mo id="A1.SS1.11.p1.3.m3.1.1.1" xref="A1.SS1.11.p1.3.m3.1.1.1.cmml">∈</mo><msup id="A1.SS1.11.p1.3.m3.1.1.3" xref="A1.SS1.11.p1.3.m3.1.1.3.cmml"><mi id="A1.SS1.11.p1.3.m3.1.1.3.2" xref="A1.SS1.11.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS1.11.p1.3.m3.1.1.3.3" xref="A1.SS1.11.p1.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.3.m3.1b"><apply id="A1.SS1.11.p1.3.m3.1.1.cmml" xref="A1.SS1.11.p1.3.m3.1.1"><in id="A1.SS1.11.p1.3.m3.1.1.1.cmml" xref="A1.SS1.11.p1.3.m3.1.1.1"></in><cn id="A1.SS1.11.p1.3.m3.1.1.2.cmml" type="integer" xref="A1.SS1.11.p1.3.m3.1.1.2">0</cn><apply id="A1.SS1.11.p1.3.m3.1.1.3.cmml" xref="A1.SS1.11.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.11.p1.3.m3.1.1.3.1.cmml" xref="A1.SS1.11.p1.3.m3.1.1.3">superscript</csymbol><ci id="A1.SS1.11.p1.3.m3.1.1.3.2.cmml" xref="A1.SS1.11.p1.3.m3.1.1.3.2">ℝ</ci><ci id="A1.SS1.11.p1.3.m3.1.1.3.3.cmml" xref="A1.SS1.11.p1.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.3.m3.1c">0\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.3.m3.1d">0 ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and the two points <math alttext="z" class="ltx_Math" display="inline" id="A1.SS1.11.p1.4.m4.1"><semantics id="A1.SS1.11.p1.4.m4.1a"><mi id="A1.SS1.11.p1.4.m4.1.1" xref="A1.SS1.11.p1.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.4.m4.1b"><ci id="A1.SS1.11.p1.4.m4.1.1.cmml" xref="A1.SS1.11.p1.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.4.m4.1d">italic_z</annotation></semantics></math> and <math alttext="x" class="ltx_Math" display="inline" id="A1.SS1.11.p1.5.m5.1"><semantics id="A1.SS1.11.p1.5.m5.1a"><mi id="A1.SS1.11.p1.5.m5.1.1" xref="A1.SS1.11.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.5.m5.1b"><ci id="A1.SS1.11.p1.5.m5.1.1.cmml" xref="A1.SS1.11.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.5.m5.1d">italic_x</annotation></semantics></math>. We know that <math alttext="||z-0||_{2}=||z||_{2}\leq\sqrt{d}" class="ltx_Math" display="inline" id="A1.SS1.11.p1.6.m6.2"><semantics id="A1.SS1.11.p1.6.m6.2a"><mrow id="A1.SS1.11.p1.6.m6.2.2" xref="A1.SS1.11.p1.6.m6.2.2.cmml"><msub id="A1.SS1.11.p1.6.m6.2.2.1" xref="A1.SS1.11.p1.6.m6.2.2.1.cmml"><mrow id="A1.SS1.11.p1.6.m6.2.2.1.1.1" xref="A1.SS1.11.p1.6.m6.2.2.1.1.2.cmml"><mo id="A1.SS1.11.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="A1.SS1.11.p1.6.m6.2.2.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.2" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.2.cmml">z</mi><mo id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.1" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.1.cmml">−</mo><mn id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.3" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.3.cmml">0</mn></mrow><mo id="A1.SS1.11.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="A1.SS1.11.p1.6.m6.2.2.1.1.2.1.cmml">‖</mo></mrow><mn id="A1.SS1.11.p1.6.m6.2.2.1.3" xref="A1.SS1.11.p1.6.m6.2.2.1.3.cmml">2</mn></msub><mo id="A1.SS1.11.p1.6.m6.2.2.3" xref="A1.SS1.11.p1.6.m6.2.2.3.cmml">=</mo><msub id="A1.SS1.11.p1.6.m6.2.2.4" xref="A1.SS1.11.p1.6.m6.2.2.4.cmml"><mrow id="A1.SS1.11.p1.6.m6.2.2.4.2.2" xref="A1.SS1.11.p1.6.m6.2.2.4.2.1.cmml"><mo id="A1.SS1.11.p1.6.m6.2.2.4.2.2.1" stretchy="false" xref="A1.SS1.11.p1.6.m6.2.2.4.2.1.1.cmml">‖</mo><mi id="A1.SS1.11.p1.6.m6.1.1" xref="A1.SS1.11.p1.6.m6.1.1.cmml">z</mi><mo id="A1.SS1.11.p1.6.m6.2.2.4.2.2.2" stretchy="false" xref="A1.SS1.11.p1.6.m6.2.2.4.2.1.1.cmml">‖</mo></mrow><mn id="A1.SS1.11.p1.6.m6.2.2.4.3" xref="A1.SS1.11.p1.6.m6.2.2.4.3.cmml">2</mn></msub><mo id="A1.SS1.11.p1.6.m6.2.2.5" xref="A1.SS1.11.p1.6.m6.2.2.5.cmml">≤</mo><msqrt id="A1.SS1.11.p1.6.m6.2.2.6" xref="A1.SS1.11.p1.6.m6.2.2.6.cmml"><mi id="A1.SS1.11.p1.6.m6.2.2.6.2" xref="A1.SS1.11.p1.6.m6.2.2.6.2.cmml">d</mi></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.6.m6.2b"><apply id="A1.SS1.11.p1.6.m6.2.2.cmml" xref="A1.SS1.11.p1.6.m6.2.2"><and id="A1.SS1.11.p1.6.m6.2.2a.cmml" xref="A1.SS1.11.p1.6.m6.2.2"></and><apply id="A1.SS1.11.p1.6.m6.2.2b.cmml" xref="A1.SS1.11.p1.6.m6.2.2"><eq id="A1.SS1.11.p1.6.m6.2.2.3.cmml" xref="A1.SS1.11.p1.6.m6.2.2.3"></eq><apply id="A1.SS1.11.p1.6.m6.2.2.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1"><csymbol cd="ambiguous" id="A1.SS1.11.p1.6.m6.2.2.1.2.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1">subscript</csymbol><apply id="A1.SS1.11.p1.6.m6.2.2.1.1.2.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1"><csymbol cd="latexml" id="A1.SS1.11.p1.6.m6.2.2.1.1.2.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.2">norm</csymbol><apply id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1"><minus id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.1"></minus><ci id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.2">𝑧</ci><cn id="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.3.cmml" type="integer" xref="A1.SS1.11.p1.6.m6.2.2.1.1.1.1.3">0</cn></apply></apply><cn id="A1.SS1.11.p1.6.m6.2.2.1.3.cmml" type="integer" xref="A1.SS1.11.p1.6.m6.2.2.1.3">2</cn></apply><apply id="A1.SS1.11.p1.6.m6.2.2.4.cmml" xref="A1.SS1.11.p1.6.m6.2.2.4"><csymbol cd="ambiguous" id="A1.SS1.11.p1.6.m6.2.2.4.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.4">subscript</csymbol><apply id="A1.SS1.11.p1.6.m6.2.2.4.2.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.4.2.2"><csymbol cd="latexml" id="A1.SS1.11.p1.6.m6.2.2.4.2.1.1.cmml" xref="A1.SS1.11.p1.6.m6.2.2.4.2.2.1">norm</csymbol><ci id="A1.SS1.11.p1.6.m6.1.1.cmml" xref="A1.SS1.11.p1.6.m6.1.1">𝑧</ci></apply><cn id="A1.SS1.11.p1.6.m6.2.2.4.3.cmml" type="integer" xref="A1.SS1.11.p1.6.m6.2.2.4.3">2</cn></apply></apply><apply id="A1.SS1.11.p1.6.m6.2.2c.cmml" xref="A1.SS1.11.p1.6.m6.2.2"><leq id="A1.SS1.11.p1.6.m6.2.2.5.cmml" xref="A1.SS1.11.p1.6.m6.2.2.5"></leq><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.11.p1.6.m6.2.2.4.cmml" id="A1.SS1.11.p1.6.m6.2.2d.cmml" xref="A1.SS1.11.p1.6.m6.2.2"></share><apply id="A1.SS1.11.p1.6.m6.2.2.6.cmml" xref="A1.SS1.11.p1.6.m6.2.2.6"><root id="A1.SS1.11.p1.6.m6.2.2.6a.cmml" xref="A1.SS1.11.p1.6.m6.2.2.6"></root><ci id="A1.SS1.11.p1.6.m6.2.2.6.2.cmml" xref="A1.SS1.11.p1.6.m6.2.2.6.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.6.m6.2c">||z-0||_{2}=||z||_{2}\leq\sqrt{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.6.m6.2d">| | italic_z - 0 | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = | | italic_z | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ square-root start_ARG italic_d end_ARG</annotation></semantics></math>, and <math alttext="||x-0||_{2}=||x||_{2}>2d" class="ltx_Math" display="inline" id="A1.SS1.11.p1.7.m7.2"><semantics id="A1.SS1.11.p1.7.m7.2a"><mrow id="A1.SS1.11.p1.7.m7.2.2" xref="A1.SS1.11.p1.7.m7.2.2.cmml"><msub id="A1.SS1.11.p1.7.m7.2.2.1" xref="A1.SS1.11.p1.7.m7.2.2.1.cmml"><mrow id="A1.SS1.11.p1.7.m7.2.2.1.1.1" xref="A1.SS1.11.p1.7.m7.2.2.1.1.2.cmml"><mo id="A1.SS1.11.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="A1.SS1.11.p1.7.m7.2.2.1.1.2.1.cmml">‖</mo><mrow id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.2" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.1" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.1.cmml">−</mo><mn id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.3" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.3.cmml">0</mn></mrow><mo id="A1.SS1.11.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="A1.SS1.11.p1.7.m7.2.2.1.1.2.1.cmml">‖</mo></mrow><mn id="A1.SS1.11.p1.7.m7.2.2.1.3" xref="A1.SS1.11.p1.7.m7.2.2.1.3.cmml">2</mn></msub><mo id="A1.SS1.11.p1.7.m7.2.2.3" xref="A1.SS1.11.p1.7.m7.2.2.3.cmml">=</mo><msub id="A1.SS1.11.p1.7.m7.2.2.4" xref="A1.SS1.11.p1.7.m7.2.2.4.cmml"><mrow id="A1.SS1.11.p1.7.m7.2.2.4.2.2" xref="A1.SS1.11.p1.7.m7.2.2.4.2.1.cmml"><mo id="A1.SS1.11.p1.7.m7.2.2.4.2.2.1" stretchy="false" xref="A1.SS1.11.p1.7.m7.2.2.4.2.1.1.cmml">‖</mo><mi id="A1.SS1.11.p1.7.m7.1.1" xref="A1.SS1.11.p1.7.m7.1.1.cmml">x</mi><mo id="A1.SS1.11.p1.7.m7.2.2.4.2.2.2" stretchy="false" xref="A1.SS1.11.p1.7.m7.2.2.4.2.1.1.cmml">‖</mo></mrow><mn id="A1.SS1.11.p1.7.m7.2.2.4.3" xref="A1.SS1.11.p1.7.m7.2.2.4.3.cmml">2</mn></msub><mo id="A1.SS1.11.p1.7.m7.2.2.5" xref="A1.SS1.11.p1.7.m7.2.2.5.cmml">></mo><mrow id="A1.SS1.11.p1.7.m7.2.2.6" xref="A1.SS1.11.p1.7.m7.2.2.6.cmml"><mn id="A1.SS1.11.p1.7.m7.2.2.6.2" xref="A1.SS1.11.p1.7.m7.2.2.6.2.cmml">2</mn><mo id="A1.SS1.11.p1.7.m7.2.2.6.1" xref="A1.SS1.11.p1.7.m7.2.2.6.1.cmml">⁢</mo><mi id="A1.SS1.11.p1.7.m7.2.2.6.3" xref="A1.SS1.11.p1.7.m7.2.2.6.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.11.p1.7.m7.2b"><apply id="A1.SS1.11.p1.7.m7.2.2.cmml" xref="A1.SS1.11.p1.7.m7.2.2"><and id="A1.SS1.11.p1.7.m7.2.2a.cmml" xref="A1.SS1.11.p1.7.m7.2.2"></and><apply id="A1.SS1.11.p1.7.m7.2.2b.cmml" xref="A1.SS1.11.p1.7.m7.2.2"><eq id="A1.SS1.11.p1.7.m7.2.2.3.cmml" xref="A1.SS1.11.p1.7.m7.2.2.3"></eq><apply id="A1.SS1.11.p1.7.m7.2.2.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1"><csymbol cd="ambiguous" id="A1.SS1.11.p1.7.m7.2.2.1.2.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1">subscript</csymbol><apply id="A1.SS1.11.p1.7.m7.2.2.1.1.2.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1"><csymbol cd="latexml" id="A1.SS1.11.p1.7.m7.2.2.1.1.2.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.2">norm</csymbol><apply id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1"><minus id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.1"></minus><ci id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.2">𝑥</ci><cn id="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.3.cmml" type="integer" xref="A1.SS1.11.p1.7.m7.2.2.1.1.1.1.3">0</cn></apply></apply><cn id="A1.SS1.11.p1.7.m7.2.2.1.3.cmml" type="integer" xref="A1.SS1.11.p1.7.m7.2.2.1.3">2</cn></apply><apply id="A1.SS1.11.p1.7.m7.2.2.4.cmml" xref="A1.SS1.11.p1.7.m7.2.2.4"><csymbol cd="ambiguous" id="A1.SS1.11.p1.7.m7.2.2.4.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.4">subscript</csymbol><apply id="A1.SS1.11.p1.7.m7.2.2.4.2.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.4.2.2"><csymbol cd="latexml" id="A1.SS1.11.p1.7.m7.2.2.4.2.1.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.4.2.2.1">norm</csymbol><ci id="A1.SS1.11.p1.7.m7.1.1.cmml" xref="A1.SS1.11.p1.7.m7.1.1">𝑥</ci></apply><cn id="A1.SS1.11.p1.7.m7.2.2.4.3.cmml" type="integer" xref="A1.SS1.11.p1.7.m7.2.2.4.3">2</cn></apply></apply><apply id="A1.SS1.11.p1.7.m7.2.2c.cmml" xref="A1.SS1.11.p1.7.m7.2.2"><gt id="A1.SS1.11.p1.7.m7.2.2.5.cmml" xref="A1.SS1.11.p1.7.m7.2.2.5"></gt><share href="https://arxiv.org/html/2503.16089v1#A1.SS1.11.p1.7.m7.2.2.4.cmml" id="A1.SS1.11.p1.7.m7.2.2d.cmml" xref="A1.SS1.11.p1.7.m7.2.2"></share><apply id="A1.SS1.11.p1.7.m7.2.2.6.cmml" xref="A1.SS1.11.p1.7.m7.2.2.6"><times id="A1.SS1.11.p1.7.m7.2.2.6.1.cmml" xref="A1.SS1.11.p1.7.m7.2.2.6.1"></times><cn id="A1.SS1.11.p1.7.m7.2.2.6.2.cmml" type="integer" xref="A1.SS1.11.p1.7.m7.2.2.6.2">2</cn><ci id="A1.SS1.11.p1.7.m7.2.2.6.3.cmml" xref="A1.SS1.11.p1.7.m7.2.2.6.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.7.m7.2c">||x-0||_{2}=||x||_{2}>2d</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.7.m7.2d">| | italic_x - 0 | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = | | italic_x | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 2 italic_d</annotation></semantics></math>. By the law of sines, we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx5"> <tbody id="A1.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sin(\measuredangle(\overrightarrow{xz},\overrightarrow{x0}))" class="ltx_Math" display="inline" id="A1.Ex8.m1.4"><semantics id="A1.Ex8.m1.4a"><mrow id="A1.Ex8.m1.4.4.1" xref="A1.Ex8.m1.4.4.2.cmml"><mi id="A1.Ex8.m1.3.3" xref="A1.Ex8.m1.3.3.cmml">sin</mi><mo id="A1.Ex8.m1.4.4.1a" xref="A1.Ex8.m1.4.4.2.cmml">⁡</mo><mrow id="A1.Ex8.m1.4.4.1.1" xref="A1.Ex8.m1.4.4.2.cmml"><mo id="A1.Ex8.m1.4.4.1.1.2" stretchy="false" xref="A1.Ex8.m1.4.4.2.cmml">(</mo><mrow id="A1.Ex8.m1.4.4.1.1.1" xref="A1.Ex8.m1.4.4.1.1.1.cmml"><mi id="A1.Ex8.m1.4.4.1.1.1.2" mathvariant="normal" xref="A1.Ex8.m1.4.4.1.1.1.2.cmml">∡</mi><mo id="A1.Ex8.m1.4.4.1.1.1.1" xref="A1.Ex8.m1.4.4.1.1.1.1.cmml">⁢</mo><mrow id="A1.Ex8.m1.4.4.1.1.1.3.2" xref="A1.Ex8.m1.4.4.1.1.1.3.1.cmml"><mo id="A1.Ex8.m1.4.4.1.1.1.3.2.1" stretchy="false" xref="A1.Ex8.m1.4.4.1.1.1.3.1.cmml">(</mo><mover accent="true" id="A1.Ex8.m1.1.1" xref="A1.Ex8.m1.1.1.cmml"><mrow id="A1.Ex8.m1.1.1.2" xref="A1.Ex8.m1.1.1.2.cmml"><mi id="A1.Ex8.m1.1.1.2.2" xref="A1.Ex8.m1.1.1.2.2.cmml">x</mi><mo id="A1.Ex8.m1.1.1.2.1" xref="A1.Ex8.m1.1.1.2.1.cmml">⁢</mo><mi id="A1.Ex8.m1.1.1.2.3" xref="A1.Ex8.m1.1.1.2.3.cmml">z</mi></mrow><mo id="A1.Ex8.m1.1.1.1" stretchy="false" xref="A1.Ex8.m1.1.1.1.cmml">→</mo></mover><mo id="A1.Ex8.m1.4.4.1.1.1.3.2.2" xref="A1.Ex8.m1.4.4.1.1.1.3.1.cmml">,</mo><mover accent="true" id="A1.Ex8.m1.2.2" xref="A1.Ex8.m1.2.2.cmml"><mrow id="A1.Ex8.m1.2.2.2" xref="A1.Ex8.m1.2.2.2.cmml"><mi id="A1.Ex8.m1.2.2.2.2" xref="A1.Ex8.m1.2.2.2.2.cmml">x</mi><mo id="A1.Ex8.m1.2.2.2.1" xref="A1.Ex8.m1.2.2.2.1.cmml">⁢</mo><mn id="A1.Ex8.m1.2.2.2.3" xref="A1.Ex8.m1.2.2.2.3.cmml">0</mn></mrow><mo id="A1.Ex8.m1.2.2.1" stretchy="false" xref="A1.Ex8.m1.2.2.1.cmml">→</mo></mover><mo id="A1.Ex8.m1.4.4.1.1.1.3.2.3" stretchy="false" xref="A1.Ex8.m1.4.4.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex8.m1.4.4.1.1.3" stretchy="false" xref="A1.Ex8.m1.4.4.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex8.m1.4b"><apply id="A1.Ex8.m1.4.4.2.cmml" xref="A1.Ex8.m1.4.4.1"><sin id="A1.Ex8.m1.3.3.cmml" xref="A1.Ex8.m1.3.3"></sin><apply id="A1.Ex8.m1.4.4.1.1.1.cmml" xref="A1.Ex8.m1.4.4.1.1.1"><times id="A1.Ex8.m1.4.4.1.1.1.1.cmml" xref="A1.Ex8.m1.4.4.1.1.1.1"></times><ci id="A1.Ex8.m1.4.4.1.1.1.2.cmml" xref="A1.Ex8.m1.4.4.1.1.1.2">∡</ci><interval closure="open" id="A1.Ex8.m1.4.4.1.1.1.3.1.cmml" xref="A1.Ex8.m1.4.4.1.1.1.3.2"><apply id="A1.Ex8.m1.1.1.cmml" xref="A1.Ex8.m1.1.1"><ci id="A1.Ex8.m1.1.1.1.cmml" xref="A1.Ex8.m1.1.1.1">→</ci><apply id="A1.Ex8.m1.1.1.2.cmml" xref="A1.Ex8.m1.1.1.2"><times id="A1.Ex8.m1.1.1.2.1.cmml" xref="A1.Ex8.m1.1.1.2.1"></times><ci id="A1.Ex8.m1.1.1.2.2.cmml" xref="A1.Ex8.m1.1.1.2.2">𝑥</ci><ci id="A1.Ex8.m1.1.1.2.3.cmml" xref="A1.Ex8.m1.1.1.2.3">𝑧</ci></apply></apply><apply id="A1.Ex8.m1.2.2.cmml" xref="A1.Ex8.m1.2.2"><ci id="A1.Ex8.m1.2.2.1.cmml" xref="A1.Ex8.m1.2.2.1">→</ci><apply id="A1.Ex8.m1.2.2.2.cmml" xref="A1.Ex8.m1.2.2.2"><times id="A1.Ex8.m1.2.2.2.1.cmml" xref="A1.Ex8.m1.2.2.2.1"></times><ci id="A1.Ex8.m1.2.2.2.2.cmml" xref="A1.Ex8.m1.2.2.2.2">𝑥</ci><cn id="A1.Ex8.m1.2.2.2.3.cmml" type="integer" xref="A1.Ex8.m1.2.2.2.3">0</cn></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex8.m1.4c">\displaystyle\sin(\measuredangle(\overrightarrow{xz},\overrightarrow{x0}))</annotation><annotation encoding="application/x-llamapun" id="A1.Ex8.m1.4d">roman_sin ( ∡ ( over→ start_ARG italic_x italic_z end_ARG , over→ start_ARG italic_x 0 end_ARG ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=||z-0||_{2}\frac{\sin(\measuredangle(\overrightarrow{zx},% \overrightarrow{z0}))}{||x-0||_{2}}" class="ltx_Math" display="inline" id="A1.Ex8.m2.6"><semantics id="A1.Ex8.m2.6a"><mrow id="A1.Ex8.m2.6.6" xref="A1.Ex8.m2.6.6.cmml"><mi id="A1.Ex8.m2.6.6.3" xref="A1.Ex8.m2.6.6.3.cmml"></mi><mo id="A1.Ex8.m2.6.6.2" xref="A1.Ex8.m2.6.6.2.cmml">=</mo><mrow id="A1.Ex8.m2.6.6.1" xref="A1.Ex8.m2.6.6.1.cmml"><msub id="A1.Ex8.m2.6.6.1.1" xref="A1.Ex8.m2.6.6.1.1.cmml"><mrow id="A1.Ex8.m2.6.6.1.1.1.1" xref="A1.Ex8.m2.6.6.1.1.1.2.cmml"><mo id="A1.Ex8.m2.6.6.1.1.1.1.2" stretchy="false" xref="A1.Ex8.m2.6.6.1.1.1.2.1.cmml">‖</mo><mrow id="A1.Ex8.m2.6.6.1.1.1.1.1" xref="A1.Ex8.m2.6.6.1.1.1.1.1.cmml"><mi id="A1.Ex8.m2.6.6.1.1.1.1.1.2" xref="A1.Ex8.m2.6.6.1.1.1.1.1.2.cmml">z</mi><mo id="A1.Ex8.m2.6.6.1.1.1.1.1.1" xref="A1.Ex8.m2.6.6.1.1.1.1.1.1.cmml">−</mo><mn id="A1.Ex8.m2.6.6.1.1.1.1.1.3" xref="A1.Ex8.m2.6.6.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="A1.Ex8.m2.6.6.1.1.1.1.3" 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id="A1.Ex9.m1.5.5.5.1.2.cmml" xref="A1.Ex9.m1.5.5.5.1.1"><csymbol cd="latexml" id="A1.Ex9.m1.5.5.5.1.2.1.cmml" xref="A1.Ex9.m1.5.5.5.1.1.2">norm</csymbol><apply id="A1.Ex9.m1.5.5.5.1.1.1.cmml" xref="A1.Ex9.m1.5.5.5.1.1.1"><minus id="A1.Ex9.m1.5.5.5.1.1.1.1.cmml" xref="A1.Ex9.m1.5.5.5.1.1.1.1"></minus><ci id="A1.Ex9.m1.5.5.5.1.1.1.2.cmml" xref="A1.Ex9.m1.5.5.5.1.1.1.2">𝑥</ci><cn id="A1.Ex9.m1.5.5.5.1.1.1.3.cmml" type="integer" xref="A1.Ex9.m1.5.5.5.1.1.1.3">0</cn></apply></apply><cn id="A1.Ex9.m1.5.5.5.3.cmml" type="integer" xref="A1.Ex9.m1.5.5.5.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex9.m1.5c">\displaystyle\leq\sqrt{d}\frac{\sin(\measuredangle(\overrightarrow{zx},% \overrightarrow{z0}))}{||x-0||_{2}}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex9.m1.5d">≤ square-root start_ARG italic_d end_ARG divide start_ARG roman_sin ( ∡ ( over→ start_ARG italic_z italic_x end_ARG , over→ start_ARG italic_z 0 end_ARG ) ) end_ARG start_ARG | | italic_x - 0 | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle<\sqrt{d}\frac{1}{2d}" class="ltx_Math" display="inline" id="A1.Ex10.m1.1"><semantics id="A1.Ex10.m1.1a"><mrow id="A1.Ex10.m1.1.1" xref="A1.Ex10.m1.1.1.cmml"><mi id="A1.Ex10.m1.1.1.2" xref="A1.Ex10.m1.1.1.2.cmml"></mi><mo id="A1.Ex10.m1.1.1.1" xref="A1.Ex10.m1.1.1.1.cmml"><</mo><mrow id="A1.Ex10.m1.1.1.3" xref="A1.Ex10.m1.1.1.3.cmml"><msqrt id="A1.Ex10.m1.1.1.3.2" xref="A1.Ex10.m1.1.1.3.2.cmml"><mi id="A1.Ex10.m1.1.1.3.2.2" xref="A1.Ex10.m1.1.1.3.2.2.cmml">d</mi></msqrt><mo id="A1.Ex10.m1.1.1.3.1" xref="A1.Ex10.m1.1.1.3.1.cmml">⁢</mo><mstyle displaystyle="true" id="A1.Ex10.m1.1.1.3.3" xref="A1.Ex10.m1.1.1.3.3.cmml"><mfrac id="A1.Ex10.m1.1.1.3.3a" xref="A1.Ex10.m1.1.1.3.3.cmml"><mn id="A1.Ex10.m1.1.1.3.3.2" xref="A1.Ex10.m1.1.1.3.3.2.cmml">1</mn><mrow id="A1.Ex10.m1.1.1.3.3.3" xref="A1.Ex10.m1.1.1.3.3.3.cmml"><mn id="A1.Ex10.m1.1.1.3.3.3.2" xref="A1.Ex10.m1.1.1.3.3.3.2.cmml">2</mn><mo id="A1.Ex10.m1.1.1.3.3.3.1" xref="A1.Ex10.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="A1.Ex10.m1.1.1.3.3.3.3" xref="A1.Ex10.m1.1.1.3.3.3.3.cmml">d</mi></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex10.m1.1b"><apply id="A1.Ex10.m1.1.1.cmml" xref="A1.Ex10.m1.1.1"><lt id="A1.Ex10.m1.1.1.1.cmml" xref="A1.Ex10.m1.1.1.1"></lt><csymbol cd="latexml" id="A1.Ex10.m1.1.1.2.cmml" xref="A1.Ex10.m1.1.1.2">absent</csymbol><apply id="A1.Ex10.m1.1.1.3.cmml" xref="A1.Ex10.m1.1.1.3"><times id="A1.Ex10.m1.1.1.3.1.cmml" xref="A1.Ex10.m1.1.1.3.1"></times><apply id="A1.Ex10.m1.1.1.3.2.cmml" xref="A1.Ex10.m1.1.1.3.2"><root id="A1.Ex10.m1.1.1.3.2a.cmml" xref="A1.Ex10.m1.1.1.3.2"></root><ci id="A1.Ex10.m1.1.1.3.2.2.cmml" xref="A1.Ex10.m1.1.1.3.2.2">𝑑</ci></apply><apply id="A1.Ex10.m1.1.1.3.3.cmml" xref="A1.Ex10.m1.1.1.3.3"><divide id="A1.Ex10.m1.1.1.3.3.1.cmml" xref="A1.Ex10.m1.1.1.3.3"></divide><cn id="A1.Ex10.m1.1.1.3.3.2.cmml" type="integer" xref="A1.Ex10.m1.1.1.3.3.2">1</cn><apply id="A1.Ex10.m1.1.1.3.3.3.cmml" xref="A1.Ex10.m1.1.1.3.3.3"><times id="A1.Ex10.m1.1.1.3.3.3.1.cmml" xref="A1.Ex10.m1.1.1.3.3.3.1"></times><cn id="A1.Ex10.m1.1.1.3.3.3.2.cmml" type="integer" xref="A1.Ex10.m1.1.1.3.3.3.2">2</cn><ci id="A1.Ex10.m1.1.1.3.3.3.3.cmml" xref="A1.Ex10.m1.1.1.3.3.3.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex10.m1.1c">\displaystyle<\sqrt{d}\frac{1}{2d}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex10.m1.1d">< square-root start_ARG italic_d end_ARG divide start_ARG 1 end_ARG start_ARG 2 italic_d end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A1.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{2\sqrt{d}}" class="ltx_Math" display="inline" id="A1.Ex11.m1.1"><semantics id="A1.Ex11.m1.1a"><mrow id="A1.Ex11.m1.1.1" xref="A1.Ex11.m1.1.1.cmml"><mi id="A1.Ex11.m1.1.1.2" xref="A1.Ex11.m1.1.1.2.cmml"></mi><mo id="A1.Ex11.m1.1.1.1" xref="A1.Ex11.m1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="A1.Ex11.m1.1.1.3" xref="A1.Ex11.m1.1.1.3.cmml"><mfrac id="A1.Ex11.m1.1.1.3a" xref="A1.Ex11.m1.1.1.3.cmml"><mn id="A1.Ex11.m1.1.1.3.2" xref="A1.Ex11.m1.1.1.3.2.cmml">1</mn><mrow id="A1.Ex11.m1.1.1.3.3" xref="A1.Ex11.m1.1.1.3.3.cmml"><mn id="A1.Ex11.m1.1.1.3.3.2" xref="A1.Ex11.m1.1.1.3.3.2.cmml">2</mn><mo id="A1.Ex11.m1.1.1.3.3.1" xref="A1.Ex11.m1.1.1.3.3.1.cmml">⁢</mo><msqrt id="A1.Ex11.m1.1.1.3.3.3" xref="A1.Ex11.m1.1.1.3.3.3.cmml"><mi id="A1.Ex11.m1.1.1.3.3.3.2" xref="A1.Ex11.m1.1.1.3.3.3.2.cmml">d</mi></msqrt></mrow></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex11.m1.1b"><apply id="A1.Ex11.m1.1.1.cmml" xref="A1.Ex11.m1.1.1"><eq id="A1.Ex11.m1.1.1.1.cmml" xref="A1.Ex11.m1.1.1.1"></eq><csymbol cd="latexml" id="A1.Ex11.m1.1.1.2.cmml" xref="A1.Ex11.m1.1.1.2">absent</csymbol><apply id="A1.Ex11.m1.1.1.3.cmml" xref="A1.Ex11.m1.1.1.3"><divide id="A1.Ex11.m1.1.1.3.1.cmml" xref="A1.Ex11.m1.1.1.3"></divide><cn id="A1.Ex11.m1.1.1.3.2.cmml" type="integer" xref="A1.Ex11.m1.1.1.3.2">1</cn><apply id="A1.Ex11.m1.1.1.3.3.cmml" xref="A1.Ex11.m1.1.1.3.3"><times id="A1.Ex11.m1.1.1.3.3.1.cmml" xref="A1.Ex11.m1.1.1.3.3.1"></times><cn id="A1.Ex11.m1.1.1.3.3.2.cmml" type="integer" xref="A1.Ex11.m1.1.1.3.3.2">2</cn><apply id="A1.Ex11.m1.1.1.3.3.3.cmml" xref="A1.Ex11.m1.1.1.3.3.3"><root 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xref="A1.SS1.11.p1.8.m1.2.3.3"></root><apply id="A1.SS1.11.p1.8.m1.2.3.3.2.cmml" xref="A1.SS1.11.p1.8.m1.2.3.3.2"><divide id="A1.SS1.11.p1.8.m1.2.3.3.2.1.cmml" xref="A1.SS1.11.p1.8.m1.2.3.3.2.1"></divide><cn id="A1.SS1.11.p1.8.m1.2.3.3.2.2.cmml" type="integer" xref="A1.SS1.11.p1.8.m1.2.3.3.2.2">1</cn><ci id="A1.SS1.11.p1.8.m1.2.3.3.2.3.cmml" xref="A1.SS1.11.p1.8.m1.2.3.3.2.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.11.p1.8.m1.2c">\measuredangle(\overrightarrow{xz},\overrightarrow{x0})<\sqrt{\nicefrac{{1}}{{% d}}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.11.p1.8.m1.2d">∡ ( over→ start_ARG italic_x italic_z end_ARG , over→ start_ARG italic_x 0 end_ARG ) < square-root start_ARG / start_ARG 1 end_ARG start_ARG italic_d end_ARG end_ARG</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="A1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.2 </span><math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS2.1.m1.1"><semantics id="A1.SS2.1.m1.1b"><msub id="A1.SS2.1.m1.1.1" xref="A1.SS2.1.m1.1.1.cmml"><mi id="A1.SS2.1.m1.1.1.2" mathvariant="normal" xref="A1.SS2.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS2.1.m1.1.1.3" xref="A1.SS2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.1.m1.1c"><apply id="A1.SS2.1.m1.1.1.cmml" xref="A1.SS2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.m1.1.1.1.cmml" xref="A1.SS2.1.m1.1.1">subscript</csymbol><ci id="A1.SS2.1.m1.1.1.2.cmml" xref="A1.SS2.1.m1.1.1.2">ℓ</ci><ci id="A1.SS2.1.m1.1.1.3.cmml" xref="A1.SS2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.m1.1d">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.m1.1e">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-Halfspaces and Mass Distributions</h3> <div class="ltx_para" id="A1.SS2.p1"> <p class="ltx_p" id="A1.SS2.p1.3">We now move on to the interaction of <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS2.p1.1.m1.1"><semantics id="A1.SS2.p1.1.m1.1a"><msub id="A1.SS2.p1.1.m1.1.1" xref="A1.SS2.p1.1.m1.1.1.cmml"><mi id="A1.SS2.p1.1.m1.1.1.2" mathvariant="normal" xref="A1.SS2.p1.1.m1.1.1.2.cmml">ℓ</mi><mi id="A1.SS2.p1.1.m1.1.1.3" xref="A1.SS2.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.1.m1.1b"><apply id="A1.SS2.p1.1.m1.1.1.cmml" xref="A1.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.1.m1.1.1.1.cmml" xref="A1.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="A1.SS2.p1.1.m1.1.1.2.cmml" xref="A1.SS2.p1.1.m1.1.1.2">ℓ</ci><ci id="A1.SS2.p1.1.m1.1.1.3.cmml" xref="A1.SS2.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.1.m1.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.1.m1.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspaces with mass distributions. We briefly recall that mass distributions are finite absolutely continuous (w.r.t. the Lebesgue measure) measures on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.p1.2.m2.1"><semantics id="A1.SS2.p1.2.m2.1a"><msup id="A1.SS2.p1.2.m2.1.1" xref="A1.SS2.p1.2.m2.1.1.cmml"><mi id="A1.SS2.p1.2.m2.1.1.2" xref="A1.SS2.p1.2.m2.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.p1.2.m2.1.1.3" xref="A1.SS2.p1.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.2.m2.1b"><apply id="A1.SS2.p1.2.m2.1.1.cmml" xref="A1.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.2.m2.1.1.1.cmml" xref="A1.SS2.p1.2.m2.1.1">superscript</csymbol><ci id="A1.SS2.p1.2.m2.1.1.2.cmml" xref="A1.SS2.p1.2.m2.1.1.2">ℝ</ci><ci id="A1.SS2.p1.2.m2.1.1.3.cmml" xref="A1.SS2.p1.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. As mentioned in <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.SS2" title="3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3.2</span></a>, by the Radon-Nikodym theorem, we can therefore think of mass distributions as probability distributions with a density (by assuming <math alttext="\mu(\mathbb{R}^{d})=1" class="ltx_Math" display="inline" id="A1.SS2.p1.3.m3.1"><semantics id="A1.SS2.p1.3.m3.1a"><mrow id="A1.SS2.p1.3.m3.1.1" xref="A1.SS2.p1.3.m3.1.1.cmml"><mrow id="A1.SS2.p1.3.m3.1.1.1" xref="A1.SS2.p1.3.m3.1.1.1.cmml"><mi id="A1.SS2.p1.3.m3.1.1.1.3" xref="A1.SS2.p1.3.m3.1.1.1.3.cmml">μ</mi><mo id="A1.SS2.p1.3.m3.1.1.1.2" xref="A1.SS2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.3.m3.1.1.1.1.1" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="A1.SS2.p1.3.m3.1.1.1.1.1.1" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.3.m3.1.1.1.1.1.1.2" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.p1.3.m3.1.1.1.1.1.1.3" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="A1.SS2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.p1.3.m3.1.1.2" xref="A1.SS2.p1.3.m3.1.1.2.cmml">=</mo><mn id="A1.SS2.p1.3.m3.1.1.3" xref="A1.SS2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.3.m3.1b"><apply id="A1.SS2.p1.3.m3.1.1.cmml" xref="A1.SS2.p1.3.m3.1.1"><eq id="A1.SS2.p1.3.m3.1.1.2.cmml" xref="A1.SS2.p1.3.m3.1.1.2"></eq><apply id="A1.SS2.p1.3.m3.1.1.1.cmml" xref="A1.SS2.p1.3.m3.1.1.1"><times id="A1.SS2.p1.3.m3.1.1.1.2.cmml" xref="A1.SS2.p1.3.m3.1.1.1.2"></times><ci id="A1.SS2.p1.3.m3.1.1.1.3.cmml" xref="A1.SS2.p1.3.m3.1.1.1.3">𝜇</ci><apply id="A1.SS2.p1.3.m3.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.3.m3.1.1.1.1.1">superscript</csymbol><ci id="A1.SS2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.2">ℝ</ci><ci id="A1.SS2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.3.m3.1.1.1.1.1.1.3">𝑑</ci></apply></apply><cn id="A1.SS2.p1.3.m3.1.1.3.cmml" type="integer" xref="A1.SS2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.3.m3.1c">\mu(\mathbb{R}^{d})=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.3.m3.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>, without loss of generality). We will use this point of view repeatedly in this section: it allows us to use the language of probability theory (instead of the language of measure theory).</p> </div> <div class="ltx_para" id="A1.SS2.p2"> <p class="ltx_p" id="A1.SS2.p2.4">Before we prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.10</span></a>, i.e., that <math alttext="\mu(\mathcal{H}_{x,v}^{p})" class="ltx_Math" display="inline" id="A1.SS2.p2.1.m1.3"><semantics id="A1.SS2.p2.1.m1.3a"><mrow id="A1.SS2.p2.1.m1.3.3" xref="A1.SS2.p2.1.m1.3.3.cmml"><mi id="A1.SS2.p2.1.m1.3.3.3" xref="A1.SS2.p2.1.m1.3.3.3.cmml">μ</mi><mo id="A1.SS2.p2.1.m1.3.3.2" xref="A1.SS2.p2.1.m1.3.3.2.cmml">⁢</mo><mrow id="A1.SS2.p2.1.m1.3.3.1.1" xref="A1.SS2.p2.1.m1.3.3.1.1.1.cmml"><mo id="A1.SS2.p2.1.m1.3.3.1.1.2" stretchy="false" xref="A1.SS2.p2.1.m1.3.3.1.1.1.cmml">(</mo><msubsup id="A1.SS2.p2.1.m1.3.3.1.1.1" xref="A1.SS2.p2.1.m1.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.p2.1.m1.3.3.1.1.1.2.2" xref="A1.SS2.p2.1.m1.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.SS2.p2.1.m1.2.2.2.4" xref="A1.SS2.p2.1.m1.2.2.2.3.cmml"><mi id="A1.SS2.p2.1.m1.1.1.1.1" xref="A1.SS2.p2.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.p2.1.m1.2.2.2.4.1" xref="A1.SS2.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.p2.1.m1.2.2.2.2" xref="A1.SS2.p2.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.p2.1.m1.3.3.1.1.1.3" xref="A1.SS2.p2.1.m1.3.3.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.SS2.p2.1.m1.3.3.1.1.3" stretchy="false" xref="A1.SS2.p2.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.1.m1.3b"><apply id="A1.SS2.p2.1.m1.3.3.cmml" xref="A1.SS2.p2.1.m1.3.3"><times id="A1.SS2.p2.1.m1.3.3.2.cmml" xref="A1.SS2.p2.1.m1.3.3.2"></times><ci id="A1.SS2.p2.1.m1.3.3.3.cmml" xref="A1.SS2.p2.1.m1.3.3.3">𝜇</ci><apply id="A1.SS2.p2.1.m1.3.3.1.1.1.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p2.1.m1.3.3.1.1.1.1.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1">superscript</csymbol><apply id="A1.SS2.p2.1.m1.3.3.1.1.1.2.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1">subscript</csymbol><ci id="A1.SS2.p2.1.m1.3.3.1.1.1.2.2.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1.1.2.2">ℋ</ci><list id="A1.SS2.p2.1.m1.2.2.2.3.cmml" xref="A1.SS2.p2.1.m1.2.2.2.4"><ci id="A1.SS2.p2.1.m1.1.1.1.1.cmml" xref="A1.SS2.p2.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.p2.1.m1.2.2.2.2.cmml" xref="A1.SS2.p2.1.m1.2.2.2.2">𝑣</ci></list></apply><ci id="A1.SS2.p2.1.m1.3.3.1.1.1.3.cmml" xref="A1.SS2.p2.1.m1.3.3.1.1.1.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.1.m1.3c">\mu(\mathcal{H}_{x,v}^{p})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.1.m1.3d">italic_μ ( caligraphic_H start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT )</annotation></semantics></math> is continuous in <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.p2.2.m2.1"><semantics id="A1.SS2.p2.2.m2.1a"><mi id="A1.SS2.p2.2.m2.1.1" xref="A1.SS2.p2.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.2.m2.1b"><ci id="A1.SS2.p2.2.m2.1.1.cmml" xref="A1.SS2.p2.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.2.m2.1d">italic_x</annotation></semantics></math>, we first want to prove that the boundary of an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.SS2.p2.3.m3.1"><semantics id="A1.SS2.p2.3.m3.1a"><msub id="A1.SS2.p2.3.m3.1.1" xref="A1.SS2.p2.3.m3.1.1.cmml"><mi id="A1.SS2.p2.3.m3.1.1.2" mathvariant="normal" xref="A1.SS2.p2.3.m3.1.1.2.cmml">ℓ</mi><mi id="A1.SS2.p2.3.m3.1.1.3" xref="A1.SS2.p2.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.3.m3.1b"><apply id="A1.SS2.p2.3.m3.1.1.cmml" xref="A1.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p2.3.m3.1.1.1.cmml" xref="A1.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="A1.SS2.p2.3.m3.1.1.2.cmml" xref="A1.SS2.p2.3.m3.1.1.2">ℓ</ci><ci id="A1.SS2.p2.3.m3.1.1.3.cmml" xref="A1.SS2.p2.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace has measure <math alttext="0" class="ltx_Math" display="inline" id="A1.SS2.p2.4.m4.1"><semantics id="A1.SS2.p2.4.m4.1a"><mn id="A1.SS2.p2.4.m4.1.1" xref="A1.SS2.p2.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.4.m4.1b"><cn id="A1.SS2.p2.4.m4.1.1.cmml" type="integer" xref="A1.SS2.p2.4.m4.1.1">0</cn></annotation-xml></semantics></math>. The following lemma will help with this.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="A1.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem4.1.1.1">Lemma A.4</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem4.2.2"> </span>(Inside and Outside Orthant)<span class="ltx_text ltx_font_bold" id="A1.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem4.p1"> <p class="ltx_p" id="A1.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem4.p1.4.4">Let <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.1.1.m1.2"><semantics id="A1.Thmtheorem4.p1.1.1.m1.2a"><msubsup id="A1.Thmtheorem4.p1.1.1.m1.2.3" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.2" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem4.p1.1.1.m1.2.2.2.4" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.1.1.m1.1.1.1.1" xref="A1.Thmtheorem4.p1.1.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.1.1.m1.2.2.2.4.1" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem4.p1.1.1.m1.2.2.2.2" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.3" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.1.1.m1.2b"><apply id="A1.Thmtheorem4.p1.1.1.m1.2.3.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3">subscript</csymbol><apply id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.1.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.2.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.2.2">ℋ</ci><ci id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.3.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem4.p1.1.1.m1.2.2.2.3.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.2.4"><ci id="A1.Thmtheorem4.p1.1.1.m1.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem4.p1.1.1.m1.2.2.2.2.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.1.1.m1.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.1.1.m1.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be an arbitrary <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.2.2.m2.1"><semantics id="A1.Thmtheorem4.p1.2.2.m2.1a"><msub id="A1.Thmtheorem4.p1.2.2.m2.1.1" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem4.p1.2.2.m2.1.1.2" mathvariant="normal" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">ℓ</mi><mi id="A1.Thmtheorem4.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.2.2.m2.1b"><apply id="A1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.2">ℓ</ci><ci id="A1.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.2.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.2.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace for some <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.3.3.m3.3"><semantics id="A1.Thmtheorem4.p1.3.3.m3.3a"><mrow id="A1.Thmtheorem4.p1.3.3.m3.3.4" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.cmml"><mi id="A1.Thmtheorem4.p1.3.3.m3.3.4.2" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.2.cmml">p</mi><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.1" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.1.cmml">∈</mo><mrow id="A1.Thmtheorem4.p1.3.3.m3.3.4.3" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.cmml"><mrow id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.2" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.1.cmml"><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.1.cmml">[</mo><mn id="A1.Thmtheorem4.p1.3.3.m3.1.1" xref="A1.Thmtheorem4.p1.3.3.m3.1.1.cmml">1</mn><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.2.2" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.1.cmml">,</mo><mi id="A1.Thmtheorem4.p1.3.3.m3.2.2" mathvariant="normal" xref="A1.Thmtheorem4.p1.3.3.m3.2.2.cmml">∞</mi><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.2.3" stretchy="false" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.1" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.1.cmml">∪</mo><mrow id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.2" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.1.cmml"><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.1.cmml">{</mo><mi id="A1.Thmtheorem4.p1.3.3.m3.3.3" mathvariant="normal" xref="A1.Thmtheorem4.p1.3.3.m3.3.3.cmml">∞</mi><mo id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.2.2" stretchy="false" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.3.3.m3.3b"><apply id="A1.Thmtheorem4.p1.3.3.m3.3.4.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4"><in id="A1.Thmtheorem4.p1.3.3.m3.3.4.1.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.1"></in><ci id="A1.Thmtheorem4.p1.3.3.m3.3.4.2.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.2">𝑝</ci><apply id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3"><union id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.1.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.1"></union><interval closure="closed-open" id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.1.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.2.2"><cn id="A1.Thmtheorem4.p1.3.3.m3.1.1.cmml" type="integer" xref="A1.Thmtheorem4.p1.3.3.m3.1.1">1</cn><infinity id="A1.Thmtheorem4.p1.3.3.m3.2.2.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.2.2"></infinity></interval><set id="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.1.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.4.3.3.2"><infinity id="A1.Thmtheorem4.p1.3.3.m3.3.3.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.3.3.m3.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.3.3.m3.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. Consider arbitrary <math alttext="z,z^{\prime}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.4.4.m4.2"><semantics id="A1.Thmtheorem4.p1.4.4.m4.2a"><mrow id="A1.Thmtheorem4.p1.4.4.m4.2.2" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.cmml"><mrow id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.2.cmml"><mi id="A1.Thmtheorem4.p1.4.4.m4.1.1" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.cmml">z</mi><mo id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.2" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.2.cmml">,</mo><msup id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.cmml"><mi id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.2" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.2.cmml">z</mi><mo id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.3" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="A1.Thmtheorem4.p1.4.4.m4.2.2.2" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.2.cmml">∈</mo><msup id="A1.Thmtheorem4.p1.4.4.m4.2.2.3" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.2" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.3" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.4.4.m4.2b"><apply id="A1.Thmtheorem4.p1.4.4.m4.2.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2"><in id="A1.Thmtheorem4.p1.4.4.m4.2.2.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.2"></in><list id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1"><ci id="A1.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.1.1">𝑧</ci><apply id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1">superscript</csymbol><ci id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.3">′</ci></apply></list><apply id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3.2">ℝ</ci><ci id="A1.Thmtheorem4.p1.4.4.m4.2.2.3.3.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.2.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.4.4.m4.2c">z,z^{\prime}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.4.4.m4.2d">italic_z , italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> satisfying both</span></p> <ul class="ltx_itemize" id="A1.I1"> <li class="ltx_item" id="A1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A1.I1.i1.p1"> <p class="ltx_p" id="A1.I1.i1.p1.1"><math alttext="v_{i}>0\implies z^{\prime}_{i}-z_{i}\geq 0" class="ltx_Math" display="inline" id="A1.I1.i1.p1.1.m1.1"><semantics id="A1.I1.i1.p1.1.m1.1a"><mrow id="A1.I1.i1.p1.1.m1.1.1" xref="A1.I1.i1.p1.1.m1.1.1.cmml"><msub id="A1.I1.i1.p1.1.m1.1.1.2" xref="A1.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="A1.I1.i1.p1.1.m1.1.1.2.2" xref="A1.I1.i1.p1.1.m1.1.1.2.2.cmml">v</mi><mi id="A1.I1.i1.p1.1.m1.1.1.2.3" xref="A1.I1.i1.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.I1.i1.p1.1.m1.1.1.3" xref="A1.I1.i1.p1.1.m1.1.1.3.cmml">></mo><mn id="A1.I1.i1.p1.1.m1.1.1.4" xref="A1.I1.i1.p1.1.m1.1.1.4.cmml">0</mn><mo id="A1.I1.i1.p1.1.m1.1.1.5" stretchy="false" xref="A1.I1.i1.p1.1.m1.1.1.5.cmml">⟹</mo><mrow id="A1.I1.i1.p1.1.m1.1.1.6" xref="A1.I1.i1.p1.1.m1.1.1.6.cmml"><msubsup id="A1.I1.i1.p1.1.m1.1.1.6.2" xref="A1.I1.i1.p1.1.m1.1.1.6.2.cmml"><mi id="A1.I1.i1.p1.1.m1.1.1.6.2.2.2" xref="A1.I1.i1.p1.1.m1.1.1.6.2.2.2.cmml">z</mi><mi id="A1.I1.i1.p1.1.m1.1.1.6.2.3" xref="A1.I1.i1.p1.1.m1.1.1.6.2.3.cmml">i</mi><mo id="A1.I1.i1.p1.1.m1.1.1.6.2.2.3" xref="A1.I1.i1.p1.1.m1.1.1.6.2.2.3.cmml">′</mo></msubsup><mo id="A1.I1.i1.p1.1.m1.1.1.6.1" xref="A1.I1.i1.p1.1.m1.1.1.6.1.cmml">−</mo><msub id="A1.I1.i1.p1.1.m1.1.1.6.3" xref="A1.I1.i1.p1.1.m1.1.1.6.3.cmml"><mi id="A1.I1.i1.p1.1.m1.1.1.6.3.2" xref="A1.I1.i1.p1.1.m1.1.1.6.3.2.cmml">z</mi><mi id="A1.I1.i1.p1.1.m1.1.1.6.3.3" xref="A1.I1.i1.p1.1.m1.1.1.6.3.3.cmml">i</mi></msub></mrow><mo id="A1.I1.i1.p1.1.m1.1.1.7" xref="A1.I1.i1.p1.1.m1.1.1.7.cmml">≥</mo><mn id="A1.I1.i1.p1.1.m1.1.1.8" xref="A1.I1.i1.p1.1.m1.1.1.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.1.m1.1b"><apply id="A1.I1.i1.p1.1.m1.1.1.cmml" xref="A1.I1.i1.p1.1.m1.1.1"><and id="A1.I1.i1.p1.1.m1.1.1a.cmml" xref="A1.I1.i1.p1.1.m1.1.1"></and><apply id="A1.I1.i1.p1.1.m1.1.1b.cmml" 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xref="A1.I1.i1.p1.1.m1.1.1.6.3.3">𝑖</ci></apply></apply></apply><apply id="A1.I1.i1.p1.1.m1.1.1e.cmml" xref="A1.I1.i1.p1.1.m1.1.1"><geq id="A1.I1.i1.p1.1.m1.1.1.7.cmml" xref="A1.I1.i1.p1.1.m1.1.1.7"></geq><share href="https://arxiv.org/html/2503.16089v1#A1.I1.i1.p1.1.m1.1.1.6.cmml" id="A1.I1.i1.p1.1.m1.1.1f.cmml" xref="A1.I1.i1.p1.1.m1.1.1"></share><cn id="A1.I1.i1.p1.1.m1.1.1.8.cmml" type="integer" xref="A1.I1.i1.p1.1.m1.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.1.m1.1c">v_{i}>0\implies z^{\prime}_{i}-z_{i}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.1.m1.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 ⟹ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i1.p1.1.1"></span></p> </div> </li> <li class="ltx_item" id="A1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A1.I1.i2.p1"> <p class="ltx_p" id="A1.I1.i2.p1.1"><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.1.1">and </span><math alttext="v_{i}<0\implies z^{\prime}_{i}-z_{i}\leq 0" class="ltx_Math" display="inline" id="A1.I1.i2.p1.1.m1.1"><semantics id="A1.I1.i2.p1.1.m1.1a"><mrow id="A1.I1.i2.p1.1.m1.1.1" xref="A1.I1.i2.p1.1.m1.1.1.cmml"><msub id="A1.I1.i2.p1.1.m1.1.1.2" xref="A1.I1.i2.p1.1.m1.1.1.2.cmml"><mi id="A1.I1.i2.p1.1.m1.1.1.2.2" xref="A1.I1.i2.p1.1.m1.1.1.2.2.cmml">v</mi><mi id="A1.I1.i2.p1.1.m1.1.1.2.3" xref="A1.I1.i2.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.I1.i2.p1.1.m1.1.1.3" xref="A1.I1.i2.p1.1.m1.1.1.3.cmml"><</mo><mn id="A1.I1.i2.p1.1.m1.1.1.4" xref="A1.I1.i2.p1.1.m1.1.1.4.cmml">0</mn><mo id="A1.I1.i2.p1.1.m1.1.1.5" stretchy="false" xref="A1.I1.i2.p1.1.m1.1.1.5.cmml">⟹</mo><mrow id="A1.I1.i2.p1.1.m1.1.1.6" 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id="A1.I1.i2.p1.1.m1.1.1a.cmml" xref="A1.I1.i2.p1.1.m1.1.1"></and><apply id="A1.I1.i2.p1.1.m1.1.1b.cmml" xref="A1.I1.i2.p1.1.m1.1.1"><lt id="A1.I1.i2.p1.1.m1.1.1.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.3"></lt><apply id="A1.I1.i2.p1.1.m1.1.1.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="A1.I1.i2.p1.1.m1.1.1.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.2.2">𝑣</ci><ci id="A1.I1.i2.p1.1.m1.1.1.2.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="A1.I1.i2.p1.1.m1.1.1.4.cmml" type="integer" xref="A1.I1.i2.p1.1.m1.1.1.4">0</cn></apply><apply id="A1.I1.i2.p1.1.m1.1.1c.cmml" xref="A1.I1.i2.p1.1.m1.1.1"><implies id="A1.I1.i2.p1.1.m1.1.1.5.cmml" xref="A1.I1.i2.p1.1.m1.1.1.5"></implies><share href="https://arxiv.org/html/2503.16089v1#A1.I1.i2.p1.1.m1.1.1.4.cmml" id="A1.I1.i2.p1.1.m1.1.1d.cmml" xref="A1.I1.i2.p1.1.m1.1.1"></share><apply id="A1.I1.i2.p1.1.m1.1.1.6.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6"><minus id="A1.I1.i2.p1.1.m1.1.1.6.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.1"></minus><apply id="A1.I1.i2.p1.1.m1.1.1.6.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.1.6.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2">subscript</csymbol><apply id="A1.I1.i2.p1.1.m1.1.1.6.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.1.6.2.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2">superscript</csymbol><ci id="A1.I1.i2.p1.1.m1.1.1.6.2.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2.2.2">𝑧</ci><ci id="A1.I1.i2.p1.1.m1.1.1.6.2.2.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2.2.3">′</ci></apply><ci id="A1.I1.i2.p1.1.m1.1.1.6.2.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.2.3">𝑖</ci></apply><apply id="A1.I1.i2.p1.1.m1.1.1.6.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.3"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.1.6.3.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.3">subscript</csymbol><ci id="A1.I1.i2.p1.1.m1.1.1.6.3.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.3.2">𝑧</ci><ci id="A1.I1.i2.p1.1.m1.1.1.6.3.3.cmml" xref="A1.I1.i2.p1.1.m1.1.1.6.3.3">𝑖</ci></apply></apply></apply><apply id="A1.I1.i2.p1.1.m1.1.1e.cmml" xref="A1.I1.i2.p1.1.m1.1.1"><leq id="A1.I1.i2.p1.1.m1.1.1.7.cmml" xref="A1.I1.i2.p1.1.m1.1.1.7"></leq><share href="https://arxiv.org/html/2503.16089v1#A1.I1.i2.p1.1.m1.1.1.6.cmml" id="A1.I1.i2.p1.1.m1.1.1f.cmml" xref="A1.I1.i2.p1.1.m1.1.1"></share><cn id="A1.I1.i2.p1.1.m1.1.1.8.cmml" type="integer" xref="A1.I1.i2.p1.1.m1.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.1.m1.1c">v_{i}<0\implies z^{\prime}_{i}-z_{i}\leq 0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.1.m1.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0 ⟹ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.1.2"></span></p> </div> </li> </ul> <p class="ltx_p" id="A1.Thmtheorem4.p1.13"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem4.p1.13.9">for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.5.1.m1.1"><semantics id="A1.Thmtheorem4.p1.5.1.m1.1a"><mrow id="A1.Thmtheorem4.p1.5.1.m1.1.2" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.cmml"><mi id="A1.Thmtheorem4.p1.5.1.m1.1.2.2" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.2.cmml">i</mi><mo id="A1.Thmtheorem4.p1.5.1.m1.1.2.1" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.1.cmml">∈</mo><mrow id="A1.Thmtheorem4.p1.5.1.m1.1.2.3.2" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.3.1.cmml"><mo id="A1.Thmtheorem4.p1.5.1.m1.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.3.1.1.cmml">[</mo><mi id="A1.Thmtheorem4.p1.5.1.m1.1.1" xref="A1.Thmtheorem4.p1.5.1.m1.1.1.cmml">d</mi><mo id="A1.Thmtheorem4.p1.5.1.m1.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.5.1.m1.1b"><apply id="A1.Thmtheorem4.p1.5.1.m1.1.2.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.2"><in id="A1.Thmtheorem4.p1.5.1.m1.1.2.1.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.1"></in><ci id="A1.Thmtheorem4.p1.5.1.m1.1.2.2.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.2">𝑖</ci><apply id="A1.Thmtheorem4.p1.5.1.m1.1.2.3.1.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.3.2"><csymbol cd="latexml" id="A1.Thmtheorem4.p1.5.1.m1.1.2.3.1.1.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="A1.Thmtheorem4.p1.5.1.m1.1.1.cmml" xref="A1.Thmtheorem4.p1.5.1.m1.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.5.1.m1.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.5.1.m1.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math>. Then <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.6.2.m2.2"><semantics id="A1.Thmtheorem4.p1.6.2.m2.2a"><mrow id="A1.Thmtheorem4.p1.6.2.m2.2.3" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.6.2.m2.2.3.2" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.2.cmml">z</mi><mo id="A1.Thmtheorem4.p1.6.2.m2.2.3.1" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.1.cmml">∈</mo><msubsup id="A1.Thmtheorem4.p1.6.2.m2.2.3.3" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.2" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem4.p1.6.2.m2.2.2.2.4" xref="A1.Thmtheorem4.p1.6.2.m2.2.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.6.2.m2.1.1.1.1" xref="A1.Thmtheorem4.p1.6.2.m2.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.6.2.m2.2.2.2.4.1" xref="A1.Thmtheorem4.p1.6.2.m2.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem4.p1.6.2.m2.2.2.2.2" xref="A1.Thmtheorem4.p1.6.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.3" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.6.2.m2.2b"><apply id="A1.Thmtheorem4.p1.6.2.m2.2.3.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3"><in id="A1.Thmtheorem4.p1.6.2.m2.2.3.1.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.1"></in><ci id="A1.Thmtheorem4.p1.6.2.m2.2.3.2.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.2">𝑧</ci><apply id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.1.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3">subscript</csymbol><apply id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.1.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.2.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.2">ℋ</ci><ci id="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.3.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.3.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem4.p1.6.2.m2.2.2.2.3.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.2.2.4"><ci id="A1.Thmtheorem4.p1.6.2.m2.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem4.p1.6.2.m2.2.2.2.2.cmml" xref="A1.Thmtheorem4.p1.6.2.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.6.2.m2.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.6.2.m2.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> implies <math alttext="z^{\prime}\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.7.3.m3.2"><semantics id="A1.Thmtheorem4.p1.7.3.m3.2a"><mrow id="A1.Thmtheorem4.p1.7.3.m3.2.3" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.cmml"><msup id="A1.Thmtheorem4.p1.7.3.m3.2.3.2" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2.cmml"><mi id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.2" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2.2.cmml">z</mi><mo id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.3" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2.3.cmml">′</mo></msup><mo id="A1.Thmtheorem4.p1.7.3.m3.2.3.1" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.1.cmml">∈</mo><msubsup id="A1.Thmtheorem4.p1.7.3.m3.2.3.3" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.2" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem4.p1.7.3.m3.2.2.2.4" xref="A1.Thmtheorem4.p1.7.3.m3.2.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.7.3.m3.1.1.1.1" xref="A1.Thmtheorem4.p1.7.3.m3.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.7.3.m3.2.2.2.4.1" xref="A1.Thmtheorem4.p1.7.3.m3.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem4.p1.7.3.m3.2.2.2.2" xref="A1.Thmtheorem4.p1.7.3.m3.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.3" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.7.3.m3.2b"><apply id="A1.Thmtheorem4.p1.7.3.m3.2.3.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3"><in id="A1.Thmtheorem4.p1.7.3.m3.2.3.1.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.1"></in><apply id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.1.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2">superscript</csymbol><ci id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.2.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.7.3.m3.2.3.2.3.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.2.3">′</ci></apply><apply id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.1.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3">subscript</csymbol><apply id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.1.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.2.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.2">ℋ</ci><ci id="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.3.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.3.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem4.p1.7.3.m3.2.2.2.3.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.2.2.4"><ci id="A1.Thmtheorem4.p1.7.3.m3.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem4.p1.7.3.m3.2.2.2.2.cmml" xref="A1.Thmtheorem4.p1.7.3.m3.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.7.3.m3.2c">z^{\prime}\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.7.3.m3.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, for any <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.8.4.m4.2"><semantics id="A1.Thmtheorem4.p1.8.4.m4.2a"><mrow id="A1.Thmtheorem4.p1.8.4.m4.2.3" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.cmml"><mi id="A1.Thmtheorem4.p1.8.4.m4.2.3.2" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.2.cmml">z</mi><mo id="A1.Thmtheorem4.p1.8.4.m4.2.3.1" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.1.cmml">∉</mo><msubsup id="A1.Thmtheorem4.p1.8.4.m4.2.3.3" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.2" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem4.p1.8.4.m4.2.2.2.4" xref="A1.Thmtheorem4.p1.8.4.m4.2.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.8.4.m4.1.1.1.1" xref="A1.Thmtheorem4.p1.8.4.m4.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.8.4.m4.2.2.2.4.1" xref="A1.Thmtheorem4.p1.8.4.m4.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem4.p1.8.4.m4.2.2.2.2" xref="A1.Thmtheorem4.p1.8.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.3" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.8.4.m4.2b"><apply id="A1.Thmtheorem4.p1.8.4.m4.2.3.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3"><notin id="A1.Thmtheorem4.p1.8.4.m4.2.3.1.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.1"></notin><ci id="A1.Thmtheorem4.p1.8.4.m4.2.3.2.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.2">𝑧</ci><apply id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.1.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3">subscript</csymbol><apply id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.1.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.2.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.2">ℋ</ci><ci id="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.3.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.3.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem4.p1.8.4.m4.2.2.2.3.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.2.2.4"><ci id="A1.Thmtheorem4.p1.8.4.m4.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem4.p1.8.4.m4.2.2.2.2.cmml" xref="A1.Thmtheorem4.p1.8.4.m4.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.8.4.m4.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.8.4.m4.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>, all <math alttext="z^{\prime}\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.9.5.m5.1"><semantics id="A1.Thmtheorem4.p1.9.5.m5.1a"><mrow id="A1.Thmtheorem4.p1.9.5.m5.1.1" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.cmml"><msup id="A1.Thmtheorem4.p1.9.5.m5.1.1.2" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2.cmml"><mi id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.2" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2.2.cmml">z</mi><mo id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.3" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="A1.Thmtheorem4.p1.9.5.m5.1.1.1" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem4.p1.9.5.m5.1.1.3" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3.cmml"><mi id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.2" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.3" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.9.5.m5.1b"><apply id="A1.Thmtheorem4.p1.9.5.m5.1.1.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1"><in id="A1.Thmtheorem4.p1.9.5.m5.1.1.1.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.1"></in><apply id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.1.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2">superscript</csymbol><ci id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.2.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.9.5.m5.1.1.2.3.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.2.3">′</ci></apply><apply id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.1.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.2.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3.2">ℝ</ci><ci id="A1.Thmtheorem4.p1.9.5.m5.1.1.3.3.cmml" xref="A1.Thmtheorem4.p1.9.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.9.5.m5.1c">z^{\prime}\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.9.5.m5.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> satisfying <math alttext="v_{i}>0\implies z^{\prime}_{i}-z_{i}\leq 0" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.10.6.m6.1"><semantics id="A1.Thmtheorem4.p1.10.6.m6.1a"><mrow id="A1.Thmtheorem4.p1.10.6.m6.1.1" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.cmml"><msub id="A1.Thmtheorem4.p1.10.6.m6.1.1.2" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2.cmml"><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.2" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2.2.cmml">v</mi><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="A1.Thmtheorem4.p1.10.6.m6.1.1.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.3.cmml">></mo><mn id="A1.Thmtheorem4.p1.10.6.m6.1.1.4" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.4.cmml">0</mn><mo id="A1.Thmtheorem4.p1.10.6.m6.1.1.5" stretchy="false" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.5.cmml">⟹</mo><mrow id="A1.Thmtheorem4.p1.10.6.m6.1.1.6" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.cmml"><msubsup id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.cmml"><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.2" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.2.cmml">z</mi><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.3.cmml">i</mi><mo id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.3.cmml">′</mo></msubsup><mo id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.1" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.1.cmml">−</mo><msub id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.cmml"><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.2" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.2.cmml">z</mi><mi id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.3" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.3.cmml">i</mi></msub></mrow><mo id="A1.Thmtheorem4.p1.10.6.m6.1.1.7" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.7.cmml">≤</mo><mn id="A1.Thmtheorem4.p1.10.6.m6.1.1.8" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.10.6.m6.1b"><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"><and id="A1.Thmtheorem4.p1.10.6.m6.1.1a.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"></and><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1b.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"><gt id="A1.Thmtheorem4.p1.10.6.m6.1.1.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.3"></gt><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2">subscript</csymbol><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2.2">𝑣</ci><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.2.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.2.3">𝑖</ci></apply><cn id="A1.Thmtheorem4.p1.10.6.m6.1.1.4.cmml" type="integer" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.4">0</cn></apply><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1c.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"><implies id="A1.Thmtheorem4.p1.10.6.m6.1.1.5.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.5"></implies><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem4.p1.10.6.m6.1.1.4.cmml" id="A1.Thmtheorem4.p1.10.6.m6.1.1d.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"></share><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6"><minus id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.1"></minus><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2">subscript</csymbol><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2">superscript</csymbol><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.2.3">′</ci></apply><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.2.3">𝑖</ci></apply><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.1.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3">subscript</csymbol><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.2.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.3.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.6.3.3">𝑖</ci></apply></apply></apply><apply id="A1.Thmtheorem4.p1.10.6.m6.1.1e.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"><leq id="A1.Thmtheorem4.p1.10.6.m6.1.1.7.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.7"></leq><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem4.p1.10.6.m6.1.1.6.cmml" id="A1.Thmtheorem4.p1.10.6.m6.1.1f.cmml" xref="A1.Thmtheorem4.p1.10.6.m6.1.1"></share><cn id="A1.Thmtheorem4.p1.10.6.m6.1.1.8.cmml" type="integer" xref="A1.Thmtheorem4.p1.10.6.m6.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.10.6.m6.1c">v_{i}>0\implies z^{\prime}_{i}-z_{i}\leq 0</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.10.6.m6.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 ⟹ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 0</annotation></semantics></math> and <math alttext="v_{i}<0\implies z^{\prime}_{i}-z_{i}\geq 0" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.11.7.m7.1"><semantics id="A1.Thmtheorem4.p1.11.7.m7.1a"><mrow id="A1.Thmtheorem4.p1.11.7.m7.1.1" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.cmml"><msub id="A1.Thmtheorem4.p1.11.7.m7.1.1.2" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2.cmml"><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.2" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2.2.cmml">v</mi><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="A1.Thmtheorem4.p1.11.7.m7.1.1.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.3.cmml"><</mo><mn id="A1.Thmtheorem4.p1.11.7.m7.1.1.4" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.4.cmml">0</mn><mo id="A1.Thmtheorem4.p1.11.7.m7.1.1.5" stretchy="false" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.5.cmml">⟹</mo><mrow id="A1.Thmtheorem4.p1.11.7.m7.1.1.6" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.cmml"><msubsup id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.cmml"><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.2" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.2.cmml">z</mi><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.3.cmml">i</mi><mo id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.3.cmml">′</mo></msubsup><mo id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.1" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.1.cmml">−</mo><msub id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.cmml"><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.2" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.2.cmml">z</mi><mi id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.3" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.3.cmml">i</mi></msub></mrow><mo id="A1.Thmtheorem4.p1.11.7.m7.1.1.7" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.7.cmml">≥</mo><mn id="A1.Thmtheorem4.p1.11.7.m7.1.1.8" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.11.7.m7.1b"><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"><and id="A1.Thmtheorem4.p1.11.7.m7.1.1a.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"></and><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1b.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"><lt id="A1.Thmtheorem4.p1.11.7.m7.1.1.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.3"></lt><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2">subscript</csymbol><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2.2">𝑣</ci><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.2.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.2.3">𝑖</ci></apply><cn id="A1.Thmtheorem4.p1.11.7.m7.1.1.4.cmml" type="integer" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.4">0</cn></apply><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1c.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"><implies id="A1.Thmtheorem4.p1.11.7.m7.1.1.5.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.5"></implies><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem4.p1.11.7.m7.1.1.4.cmml" id="A1.Thmtheorem4.p1.11.7.m7.1.1d.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"></share><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6"><minus id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.1"></minus><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2">subscript</csymbol><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2">superscript</csymbol><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.2.3">′</ci></apply><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.2.3">𝑖</ci></apply><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.1.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3">subscript</csymbol><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.2.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.2">𝑧</ci><ci id="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.3.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.6.3.3">𝑖</ci></apply></apply></apply><apply id="A1.Thmtheorem4.p1.11.7.m7.1.1e.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"><geq id="A1.Thmtheorem4.p1.11.7.m7.1.1.7.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.7"></geq><share href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem4.p1.11.7.m7.1.1.6.cmml" id="A1.Thmtheorem4.p1.11.7.m7.1.1f.cmml" xref="A1.Thmtheorem4.p1.11.7.m7.1.1"></share><cn id="A1.Thmtheorem4.p1.11.7.m7.1.1.8.cmml" type="integer" xref="A1.Thmtheorem4.p1.11.7.m7.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.11.7.m7.1c">v_{i}<0\implies z^{\prime}_{i}-z_{i}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.11.7.m7.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0 ⟹ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> for all <math alttext="i\in[d]" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.12.8.m8.1"><semantics id="A1.Thmtheorem4.p1.12.8.m8.1a"><mrow id="A1.Thmtheorem4.p1.12.8.m8.1.2" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.cmml"><mi id="A1.Thmtheorem4.p1.12.8.m8.1.2.2" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.2.cmml">i</mi><mo id="A1.Thmtheorem4.p1.12.8.m8.1.2.1" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.1.cmml">∈</mo><mrow id="A1.Thmtheorem4.p1.12.8.m8.1.2.3.2" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.3.1.cmml"><mo id="A1.Thmtheorem4.p1.12.8.m8.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.3.1.1.cmml">[</mo><mi id="A1.Thmtheorem4.p1.12.8.m8.1.1" xref="A1.Thmtheorem4.p1.12.8.m8.1.1.cmml">d</mi><mo id="A1.Thmtheorem4.p1.12.8.m8.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.12.8.m8.1b"><apply id="A1.Thmtheorem4.p1.12.8.m8.1.2.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.2"><in id="A1.Thmtheorem4.p1.12.8.m8.1.2.1.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.1"></in><ci id="A1.Thmtheorem4.p1.12.8.m8.1.2.2.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.2">𝑖</ci><apply id="A1.Thmtheorem4.p1.12.8.m8.1.2.3.1.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.3.2"><csymbol cd="latexml" id="A1.Thmtheorem4.p1.12.8.m8.1.2.3.1.1.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="A1.Thmtheorem4.p1.12.8.m8.1.1.cmml" xref="A1.Thmtheorem4.p1.12.8.m8.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.12.8.m8.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.12.8.m8.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math> are not contained in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.13.9.m9.2"><semantics id="A1.Thmtheorem4.p1.13.9.m9.2a"><msubsup id="A1.Thmtheorem4.p1.13.9.m9.2.3" xref="A1.Thmtheorem4.p1.13.9.m9.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.2" xref="A1.Thmtheorem4.p1.13.9.m9.2.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem4.p1.13.9.m9.2.2.2.4" xref="A1.Thmtheorem4.p1.13.9.m9.2.2.2.3.cmml"><mi id="A1.Thmtheorem4.p1.13.9.m9.1.1.1.1" xref="A1.Thmtheorem4.p1.13.9.m9.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.13.9.m9.2.2.2.4.1" xref="A1.Thmtheorem4.p1.13.9.m9.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem4.p1.13.9.m9.2.2.2.2" xref="A1.Thmtheorem4.p1.13.9.m9.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.3" xref="A1.Thmtheorem4.p1.13.9.m9.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.13.9.m9.2b"><apply id="A1.Thmtheorem4.p1.13.9.m9.2.3.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.13.9.m9.2.3.1.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3">subscript</csymbol><apply id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.1.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3">superscript</csymbol><ci id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.2.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3.2.2">ℋ</ci><ci id="A1.Thmtheorem4.p1.13.9.m9.2.3.2.3.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem4.p1.13.9.m9.2.2.2.3.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.2.2.4"><ci id="A1.Thmtheorem4.p1.13.9.m9.1.1.1.1.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem4.p1.13.9.m9.2.2.2.2.cmml" xref="A1.Thmtheorem4.p1.13.9.m9.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.13.9.m9.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.13.9.m9.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="A1.SS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS2.2.p1"> <p class="ltx_p" id="A1.SS2.2.p1.9">Let <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.1.m1.2"><semantics id="A1.SS2.2.p1.1.m1.2a"><mrow id="A1.SS2.2.p1.1.m1.2.3" xref="A1.SS2.2.p1.1.m1.2.3.cmml"><mi id="A1.SS2.2.p1.1.m1.2.3.2" xref="A1.SS2.2.p1.1.m1.2.3.2.cmml">z</mi><mo id="A1.SS2.2.p1.1.m1.2.3.1" xref="A1.SS2.2.p1.1.m1.2.3.1.cmml">∈</mo><msubsup id="A1.SS2.2.p1.1.m1.2.3.3" xref="A1.SS2.2.p1.1.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.2.p1.1.m1.2.3.3.2.2" xref="A1.SS2.2.p1.1.m1.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.2.p1.1.m1.2.2.2.4" xref="A1.SS2.2.p1.1.m1.2.2.2.3.cmml"><mi id="A1.SS2.2.p1.1.m1.1.1.1.1" xref="A1.SS2.2.p1.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.2.p1.1.m1.2.2.2.4.1" xref="A1.SS2.2.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.2.p1.1.m1.2.2.2.2" xref="A1.SS2.2.p1.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.2.p1.1.m1.2.3.3.2.3" xref="A1.SS2.2.p1.1.m1.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.1.m1.2b"><apply id="A1.SS2.2.p1.1.m1.2.3.cmml" xref="A1.SS2.2.p1.1.m1.2.3"><in id="A1.SS2.2.p1.1.m1.2.3.1.cmml" xref="A1.SS2.2.p1.1.m1.2.3.1"></in><ci id="A1.SS2.2.p1.1.m1.2.3.2.cmml" xref="A1.SS2.2.p1.1.m1.2.3.2">𝑧</ci><apply id="A1.SS2.2.p1.1.m1.2.3.3.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.1.m1.2.3.3.1.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3">subscript</csymbol><apply id="A1.SS2.2.p1.1.m1.2.3.3.2.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.1.m1.2.3.3.2.1.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3">superscript</csymbol><ci id="A1.SS2.2.p1.1.m1.2.3.3.2.2.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.2.p1.1.m1.2.3.3.2.3.cmml" xref="A1.SS2.2.p1.1.m1.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.2.p1.1.m1.2.2.2.3.cmml" xref="A1.SS2.2.p1.1.m1.2.2.2.4"><ci id="A1.SS2.2.p1.1.m1.1.1.1.1.cmml" xref="A1.SS2.2.p1.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.2.p1.1.m1.2.2.2.2.cmml" xref="A1.SS2.2.p1.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.1.m1.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.1.m1.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be arbitrary and consider first the case <math alttext="p\neq\infty" class="ltx_Math" display="inline" id="A1.SS2.2.p1.2.m2.1"><semantics id="A1.SS2.2.p1.2.m2.1a"><mrow id="A1.SS2.2.p1.2.m2.1.1" xref="A1.SS2.2.p1.2.m2.1.1.cmml"><mi id="A1.SS2.2.p1.2.m2.1.1.2" xref="A1.SS2.2.p1.2.m2.1.1.2.cmml">p</mi><mo id="A1.SS2.2.p1.2.m2.1.1.1" xref="A1.SS2.2.p1.2.m2.1.1.1.cmml">≠</mo><mi id="A1.SS2.2.p1.2.m2.1.1.3" mathvariant="normal" xref="A1.SS2.2.p1.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.2.m2.1b"><apply id="A1.SS2.2.p1.2.m2.1.1.cmml" xref="A1.SS2.2.p1.2.m2.1.1"><neq id="A1.SS2.2.p1.2.m2.1.1.1.cmml" xref="A1.SS2.2.p1.2.m2.1.1.1"></neq><ci id="A1.SS2.2.p1.2.m2.1.1.2.cmml" xref="A1.SS2.2.p1.2.m2.1.1.2">𝑝</ci><infinity id="A1.SS2.2.p1.2.m2.1.1.3.cmml" xref="A1.SS2.2.p1.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.2.m2.1c">p\neq\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.2.m2.1d">italic_p ≠ ∞</annotation></semantics></math>. Without loss of generality, we can consider the case where <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.3.m3.1"><semantics id="A1.SS2.2.p1.3.m3.1a"><msup id="A1.SS2.2.p1.3.m3.1.1" xref="A1.SS2.2.p1.3.m3.1.1.cmml"><mi id="A1.SS2.2.p1.3.m3.1.1.2" xref="A1.SS2.2.p1.3.m3.1.1.2.cmml">z</mi><mo id="A1.SS2.2.p1.3.m3.1.1.3" xref="A1.SS2.2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.3.m3.1b"><apply id="A1.SS2.2.p1.3.m3.1.1.cmml" xref="A1.SS2.2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.3.m3.1.1.1.cmml" xref="A1.SS2.2.p1.3.m3.1.1">superscript</csymbol><ci id="A1.SS2.2.p1.3.m3.1.1.2.cmml" xref="A1.SS2.2.p1.3.m3.1.1.2">𝑧</ci><ci id="A1.SS2.2.p1.3.m3.1.1.3.cmml" xref="A1.SS2.2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.3.m3.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.3.m3.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="A1.SS2.2.p1.4.m4.1"><semantics id="A1.SS2.2.p1.4.m4.1a"><mi id="A1.SS2.2.p1.4.m4.1.1" xref="A1.SS2.2.p1.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.4.m4.1b"><ci id="A1.SS2.2.p1.4.m4.1.1.cmml" xref="A1.SS2.2.p1.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.4.m4.1d">italic_z</annotation></semantics></math> only differ in one coordinate, i.e., <math alttext="z^{\prime}_{i}=z_{i}+\alpha v_{i}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.5.m5.1"><semantics id="A1.SS2.2.p1.5.m5.1a"><mrow id="A1.SS2.2.p1.5.m5.1.1" xref="A1.SS2.2.p1.5.m5.1.1.cmml"><msubsup id="A1.SS2.2.p1.5.m5.1.1.2" xref="A1.SS2.2.p1.5.m5.1.1.2.cmml"><mi id="A1.SS2.2.p1.5.m5.1.1.2.2.2" xref="A1.SS2.2.p1.5.m5.1.1.2.2.2.cmml">z</mi><mi id="A1.SS2.2.p1.5.m5.1.1.2.3" xref="A1.SS2.2.p1.5.m5.1.1.2.3.cmml">i</mi><mo id="A1.SS2.2.p1.5.m5.1.1.2.2.3" xref="A1.SS2.2.p1.5.m5.1.1.2.2.3.cmml">′</mo></msubsup><mo id="A1.SS2.2.p1.5.m5.1.1.1" xref="A1.SS2.2.p1.5.m5.1.1.1.cmml">=</mo><mrow id="A1.SS2.2.p1.5.m5.1.1.3" xref="A1.SS2.2.p1.5.m5.1.1.3.cmml"><msub id="A1.SS2.2.p1.5.m5.1.1.3.2" xref="A1.SS2.2.p1.5.m5.1.1.3.2.cmml"><mi id="A1.SS2.2.p1.5.m5.1.1.3.2.2" xref="A1.SS2.2.p1.5.m5.1.1.3.2.2.cmml">z</mi><mi id="A1.SS2.2.p1.5.m5.1.1.3.2.3" xref="A1.SS2.2.p1.5.m5.1.1.3.2.3.cmml">i</mi></msub><mo id="A1.SS2.2.p1.5.m5.1.1.3.1" xref="A1.SS2.2.p1.5.m5.1.1.3.1.cmml">+</mo><mrow id="A1.SS2.2.p1.5.m5.1.1.3.3" xref="A1.SS2.2.p1.5.m5.1.1.3.3.cmml"><mi id="A1.SS2.2.p1.5.m5.1.1.3.3.2" xref="A1.SS2.2.p1.5.m5.1.1.3.3.2.cmml">α</mi><mo id="A1.SS2.2.p1.5.m5.1.1.3.3.1" xref="A1.SS2.2.p1.5.m5.1.1.3.3.1.cmml">⁢</mo><msub id="A1.SS2.2.p1.5.m5.1.1.3.3.3" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3.cmml"><mi id="A1.SS2.2.p1.5.m5.1.1.3.3.3.2" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3.2.cmml">v</mi><mi id="A1.SS2.2.p1.5.m5.1.1.3.3.3.3" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.5.m5.1b"><apply id="A1.SS2.2.p1.5.m5.1.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1"><eq id="A1.SS2.2.p1.5.m5.1.1.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.1"></eq><apply id="A1.SS2.2.p1.5.m5.1.1.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.5.m5.1.1.2.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2">subscript</csymbol><apply id="A1.SS2.2.p1.5.m5.1.1.2.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.5.m5.1.1.2.2.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2">superscript</csymbol><ci id="A1.SS2.2.p1.5.m5.1.1.2.2.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2.2.2">𝑧</ci><ci id="A1.SS2.2.p1.5.m5.1.1.2.2.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2.2.3">′</ci></apply><ci id="A1.SS2.2.p1.5.m5.1.1.2.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="A1.SS2.2.p1.5.m5.1.1.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3"><plus id="A1.SS2.2.p1.5.m5.1.1.3.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.1"></plus><apply id="A1.SS2.2.p1.5.m5.1.1.3.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.5.m5.1.1.3.2.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="A1.SS2.2.p1.5.m5.1.1.3.2.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.2.2">𝑧</ci><ci id="A1.SS2.2.p1.5.m5.1.1.3.2.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.2.3">𝑖</ci></apply><apply id="A1.SS2.2.p1.5.m5.1.1.3.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3"><times id="A1.SS2.2.p1.5.m5.1.1.3.3.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.1"></times><ci id="A1.SS2.2.p1.5.m5.1.1.3.3.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.2">𝛼</ci><apply id="A1.SS2.2.p1.5.m5.1.1.3.3.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.5.m5.1.1.3.3.3.1.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3">subscript</csymbol><ci id="A1.SS2.2.p1.5.m5.1.1.3.3.3.2.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3.2">𝑣</ci><ci id="A1.SS2.2.p1.5.m5.1.1.3.3.3.3.cmml" xref="A1.SS2.2.p1.5.m5.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.5.m5.1c">z^{\prime}_{i}=z_{i}+\alpha v_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.5.m5.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_α italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="i\in[d]" class="ltx_Math" display="inline" id="A1.SS2.2.p1.6.m6.1"><semantics id="A1.SS2.2.p1.6.m6.1a"><mrow id="A1.SS2.2.p1.6.m6.1.2" xref="A1.SS2.2.p1.6.m6.1.2.cmml"><mi id="A1.SS2.2.p1.6.m6.1.2.2" xref="A1.SS2.2.p1.6.m6.1.2.2.cmml">i</mi><mo id="A1.SS2.2.p1.6.m6.1.2.1" xref="A1.SS2.2.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="A1.SS2.2.p1.6.m6.1.2.3.2" xref="A1.SS2.2.p1.6.m6.1.2.3.1.cmml"><mo id="A1.SS2.2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="A1.SS2.2.p1.6.m6.1.2.3.1.1.cmml">[</mo><mi id="A1.SS2.2.p1.6.m6.1.1" xref="A1.SS2.2.p1.6.m6.1.1.cmml">d</mi><mo id="A1.SS2.2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="A1.SS2.2.p1.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.6.m6.1b"><apply id="A1.SS2.2.p1.6.m6.1.2.cmml" xref="A1.SS2.2.p1.6.m6.1.2"><in id="A1.SS2.2.p1.6.m6.1.2.1.cmml" xref="A1.SS2.2.p1.6.m6.1.2.1"></in><ci id="A1.SS2.2.p1.6.m6.1.2.2.cmml" xref="A1.SS2.2.p1.6.m6.1.2.2">𝑖</ci><apply id="A1.SS2.2.p1.6.m6.1.2.3.1.cmml" xref="A1.SS2.2.p1.6.m6.1.2.3.2"><csymbol cd="latexml" id="A1.SS2.2.p1.6.m6.1.2.3.1.1.cmml" xref="A1.SS2.2.p1.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="A1.SS2.2.p1.6.m6.1.1.cmml" xref="A1.SS2.2.p1.6.m6.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.6.m6.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.6.m6.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math> and <math alttext="\alpha>0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.7.m7.1"><semantics id="A1.SS2.2.p1.7.m7.1a"><mrow id="A1.SS2.2.p1.7.m7.1.1" xref="A1.SS2.2.p1.7.m7.1.1.cmml"><mi id="A1.SS2.2.p1.7.m7.1.1.2" xref="A1.SS2.2.p1.7.m7.1.1.2.cmml">α</mi><mo id="A1.SS2.2.p1.7.m7.1.1.1" xref="A1.SS2.2.p1.7.m7.1.1.1.cmml">></mo><mn id="A1.SS2.2.p1.7.m7.1.1.3" xref="A1.SS2.2.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.7.m7.1b"><apply id="A1.SS2.2.p1.7.m7.1.1.cmml" xref="A1.SS2.2.p1.7.m7.1.1"><gt id="A1.SS2.2.p1.7.m7.1.1.1.cmml" xref="A1.SS2.2.p1.7.m7.1.1.1"></gt><ci id="A1.SS2.2.p1.7.m7.1.1.2.cmml" xref="A1.SS2.2.p1.7.m7.1.1.2">𝛼</ci><cn id="A1.SS2.2.p1.7.m7.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.7.m7.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.7.m7.1d">italic_α > 0</annotation></semantics></math>, and <math alttext="z^{\prime}_{j}=z_{j}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.8.m8.1"><semantics id="A1.SS2.2.p1.8.m8.1a"><mrow id="A1.SS2.2.p1.8.m8.1.1" xref="A1.SS2.2.p1.8.m8.1.1.cmml"><msubsup id="A1.SS2.2.p1.8.m8.1.1.2" xref="A1.SS2.2.p1.8.m8.1.1.2.cmml"><mi id="A1.SS2.2.p1.8.m8.1.1.2.2.2" xref="A1.SS2.2.p1.8.m8.1.1.2.2.2.cmml">z</mi><mi id="A1.SS2.2.p1.8.m8.1.1.2.3" xref="A1.SS2.2.p1.8.m8.1.1.2.3.cmml">j</mi><mo id="A1.SS2.2.p1.8.m8.1.1.2.2.3" xref="A1.SS2.2.p1.8.m8.1.1.2.2.3.cmml">′</mo></msubsup><mo id="A1.SS2.2.p1.8.m8.1.1.1" xref="A1.SS2.2.p1.8.m8.1.1.1.cmml">=</mo><msub id="A1.SS2.2.p1.8.m8.1.1.3" xref="A1.SS2.2.p1.8.m8.1.1.3.cmml"><mi id="A1.SS2.2.p1.8.m8.1.1.3.2" xref="A1.SS2.2.p1.8.m8.1.1.3.2.cmml">z</mi><mi id="A1.SS2.2.p1.8.m8.1.1.3.3" xref="A1.SS2.2.p1.8.m8.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.8.m8.1b"><apply id="A1.SS2.2.p1.8.m8.1.1.cmml" xref="A1.SS2.2.p1.8.m8.1.1"><eq id="A1.SS2.2.p1.8.m8.1.1.1.cmml" xref="A1.SS2.2.p1.8.m8.1.1.1"></eq><apply id="A1.SS2.2.p1.8.m8.1.1.2.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.8.m8.1.1.2.1.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2">subscript</csymbol><apply id="A1.SS2.2.p1.8.m8.1.1.2.2.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.8.m8.1.1.2.2.1.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2">superscript</csymbol><ci id="A1.SS2.2.p1.8.m8.1.1.2.2.2.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2.2.2">𝑧</ci><ci id="A1.SS2.2.p1.8.m8.1.1.2.2.3.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2.2.3">′</ci></apply><ci id="A1.SS2.2.p1.8.m8.1.1.2.3.cmml" xref="A1.SS2.2.p1.8.m8.1.1.2.3">𝑗</ci></apply><apply id="A1.SS2.2.p1.8.m8.1.1.3.cmml" xref="A1.SS2.2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.8.m8.1.1.3.1.cmml" xref="A1.SS2.2.p1.8.m8.1.1.3">subscript</csymbol><ci id="A1.SS2.2.p1.8.m8.1.1.3.2.cmml" xref="A1.SS2.2.p1.8.m8.1.1.3.2">𝑧</ci><ci id="A1.SS2.2.p1.8.m8.1.1.3.3.cmml" xref="A1.SS2.2.p1.8.m8.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.8.m8.1c">z^{\prime}_{j}=z_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.8.m8.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="j\in[d]\setminus\{i\}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.9.m9.2"><semantics id="A1.SS2.2.p1.9.m9.2a"><mrow id="A1.SS2.2.p1.9.m9.2.3" xref="A1.SS2.2.p1.9.m9.2.3.cmml"><mi id="A1.SS2.2.p1.9.m9.2.3.2" xref="A1.SS2.2.p1.9.m9.2.3.2.cmml">j</mi><mo id="A1.SS2.2.p1.9.m9.2.3.1" xref="A1.SS2.2.p1.9.m9.2.3.1.cmml">∈</mo><mrow id="A1.SS2.2.p1.9.m9.2.3.3" xref="A1.SS2.2.p1.9.m9.2.3.3.cmml"><mrow id="A1.SS2.2.p1.9.m9.2.3.3.2.2" xref="A1.SS2.2.p1.9.m9.2.3.3.2.1.cmml"><mo id="A1.SS2.2.p1.9.m9.2.3.3.2.2.1" stretchy="false" xref="A1.SS2.2.p1.9.m9.2.3.3.2.1.1.cmml">[</mo><mi id="A1.SS2.2.p1.9.m9.1.1" xref="A1.SS2.2.p1.9.m9.1.1.cmml">d</mi><mo id="A1.SS2.2.p1.9.m9.2.3.3.2.2.2" stretchy="false" xref="A1.SS2.2.p1.9.m9.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="A1.SS2.2.p1.9.m9.2.3.3.1" xref="A1.SS2.2.p1.9.m9.2.3.3.1.cmml">∖</mo><mrow id="A1.SS2.2.p1.9.m9.2.3.3.3.2" xref="A1.SS2.2.p1.9.m9.2.3.3.3.1.cmml"><mo id="A1.SS2.2.p1.9.m9.2.3.3.3.2.1" stretchy="false" xref="A1.SS2.2.p1.9.m9.2.3.3.3.1.cmml">{</mo><mi id="A1.SS2.2.p1.9.m9.2.2" xref="A1.SS2.2.p1.9.m9.2.2.cmml">i</mi><mo id="A1.SS2.2.p1.9.m9.2.3.3.3.2.2" stretchy="false" xref="A1.SS2.2.p1.9.m9.2.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.9.m9.2b"><apply id="A1.SS2.2.p1.9.m9.2.3.cmml" xref="A1.SS2.2.p1.9.m9.2.3"><in id="A1.SS2.2.p1.9.m9.2.3.1.cmml" xref="A1.SS2.2.p1.9.m9.2.3.1"></in><ci id="A1.SS2.2.p1.9.m9.2.3.2.cmml" xref="A1.SS2.2.p1.9.m9.2.3.2">𝑗</ci><apply id="A1.SS2.2.p1.9.m9.2.3.3.cmml" xref="A1.SS2.2.p1.9.m9.2.3.3"><setdiff id="A1.SS2.2.p1.9.m9.2.3.3.1.cmml" xref="A1.SS2.2.p1.9.m9.2.3.3.1"></setdiff><apply id="A1.SS2.2.p1.9.m9.2.3.3.2.1.cmml" xref="A1.SS2.2.p1.9.m9.2.3.3.2.2"><csymbol cd="latexml" id="A1.SS2.2.p1.9.m9.2.3.3.2.1.1.cmml" xref="A1.SS2.2.p1.9.m9.2.3.3.2.2.1">delimited-[]</csymbol><ci id="A1.SS2.2.p1.9.m9.1.1.cmml" xref="A1.SS2.2.p1.9.m9.1.1">𝑑</ci></apply><set id="A1.SS2.2.p1.9.m9.2.3.3.3.1.cmml" xref="A1.SS2.2.p1.9.m9.2.3.3.3.2"><ci id="A1.SS2.2.p1.9.m9.2.2.cmml" xref="A1.SS2.2.p1.9.m9.2.2">𝑖</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.9.m9.2c">j\in[d]\setminus\{i\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.9.m9.2d">italic_j ∈ [ italic_d ] ∖ { italic_i }</annotation></semantics></math>. By definition, we must have</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lVert x-z\rVert^{p}_{p}\leq\lVert x-\varepsilon v-z\rVert^{p}_{p}" class="ltx_Math" display="block" id="A1.Ex12.m1.2"><semantics id="A1.Ex12.m1.2a"><mrow id="A1.Ex12.m1.2.2" xref="A1.Ex12.m1.2.2.cmml"><msubsup id="A1.Ex12.m1.1.1.1" xref="A1.Ex12.m1.1.1.1.cmml"><mrow id="A1.Ex12.m1.1.1.1.1.1.1" xref="A1.Ex12.m1.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex12.m1.1.1.1.1.1.1.2" rspace="0em" xref="A1.Ex12.m1.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex12.m1.1.1.1.1.1.1.1" xref="A1.Ex12.m1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex12.m1.1.1.1.1.1.1.1.2" xref="A1.Ex12.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex12.m1.1.1.1.1.1.1.1.1" xref="A1.Ex12.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex12.m1.1.1.1.1.1.1.1.3" xref="A1.Ex12.m1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.Ex12.m1.1.1.1.1.1.1.3" lspace="0em" xref="A1.Ex12.m1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex12.m1.1.1.1.3" xref="A1.Ex12.m1.1.1.1.3.cmml">p</mi><mi id="A1.Ex12.m1.1.1.1.1.3" xref="A1.Ex12.m1.1.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.Ex12.m1.2.2.3" rspace="0.1389em" xref="A1.Ex12.m1.2.2.3.cmml">≤</mo><msubsup id="A1.Ex12.m1.2.2.2" xref="A1.Ex12.m1.2.2.2.cmml"><mrow id="A1.Ex12.m1.2.2.2.1.1.1" xref="A1.Ex12.m1.2.2.2.1.1.2.cmml"><mo fence="true" id="A1.Ex12.m1.2.2.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex12.m1.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex12.m1.2.2.2.1.1.1.1" xref="A1.Ex12.m1.2.2.2.1.1.1.1.cmml"><mi id="A1.Ex12.m1.2.2.2.1.1.1.1.2" xref="A1.Ex12.m1.2.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex12.m1.2.2.2.1.1.1.1.1" xref="A1.Ex12.m1.2.2.2.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex12.m1.2.2.2.1.1.1.1.3" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.cmml"><mi id="A1.Ex12.m1.2.2.2.1.1.1.1.3.2" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex12.m1.2.2.2.1.1.1.1.3.1" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex12.m1.2.2.2.1.1.1.1.3.3" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.Ex12.m1.2.2.2.1.1.1.1.1a" xref="A1.Ex12.m1.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex12.m1.2.2.2.1.1.1.1.4" xref="A1.Ex12.m1.2.2.2.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.Ex12.m1.2.2.2.1.1.1.3" lspace="0em" xref="A1.Ex12.m1.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex12.m1.2.2.2.3" xref="A1.Ex12.m1.2.2.2.3.cmml">p</mi><mi id="A1.Ex12.m1.2.2.2.1.3" xref="A1.Ex12.m1.2.2.2.1.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex12.m1.2b"><apply id="A1.Ex12.m1.2.2.cmml" xref="A1.Ex12.m1.2.2"><leq id="A1.Ex12.m1.2.2.3.cmml" xref="A1.Ex12.m1.2.2.3"></leq><apply id="A1.Ex12.m1.1.1.1.cmml" xref="A1.Ex12.m1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex12.m1.1.1.1.2.cmml" xref="A1.Ex12.m1.1.1.1">subscript</csymbol><apply id="A1.Ex12.m1.1.1.1.1.cmml" xref="A1.Ex12.m1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex12.m1.1.1.1.1.2.cmml" xref="A1.Ex12.m1.1.1.1">superscript</csymbol><apply id="A1.Ex12.m1.1.1.1.1.1.2.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.Ex12.m1.1.1.1.1.1.2.1.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex12.m1.1.1.1.1.1.1.1.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1.1"><minus id="A1.Ex12.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1.1.1"></minus><ci id="A1.Ex12.m1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.Ex12.m1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex12.m1.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.Ex12.m1.1.1.1.1.3.cmml" xref="A1.Ex12.m1.1.1.1.1.3">𝑝</ci></apply><ci id="A1.Ex12.m1.1.1.1.3.cmml" xref="A1.Ex12.m1.1.1.1.3">𝑝</ci></apply><apply id="A1.Ex12.m1.2.2.2.cmml" xref="A1.Ex12.m1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex12.m1.2.2.2.2.cmml" xref="A1.Ex12.m1.2.2.2">subscript</csymbol><apply id="A1.Ex12.m1.2.2.2.1.cmml" xref="A1.Ex12.m1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex12.m1.2.2.2.1.2.cmml" xref="A1.Ex12.m1.2.2.2">superscript</csymbol><apply id="A1.Ex12.m1.2.2.2.1.1.2.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1"><csymbol cd="latexml" id="A1.Ex12.m1.2.2.2.1.1.2.1.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex12.m1.2.2.2.1.1.1.1.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1"><minus id="A1.Ex12.m1.2.2.2.1.1.1.1.1.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.1"></minus><ci id="A1.Ex12.m1.2.2.2.1.1.1.1.2.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.2">𝑥</ci><apply id="A1.Ex12.m1.2.2.2.1.1.1.1.3.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3"><times id="A1.Ex12.m1.2.2.2.1.1.1.1.3.1.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.1"></times><ci id="A1.Ex12.m1.2.2.2.1.1.1.1.3.2.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.2">𝜀</ci><ci id="A1.Ex12.m1.2.2.2.1.1.1.1.3.3.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.Ex12.m1.2.2.2.1.1.1.1.4.cmml" xref="A1.Ex12.m1.2.2.2.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.Ex12.m1.2.2.2.1.3.cmml" xref="A1.Ex12.m1.2.2.2.1.3">𝑝</ci></apply><ci id="A1.Ex12.m1.2.2.2.3.cmml" xref="A1.Ex12.m1.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex12.m1.2c">\lVert x-z\rVert^{p}_{p}\leq\lVert x-\varepsilon v-z\rVert^{p}_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex12.m1.2d">∥ italic_x - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.2.p1.10">for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.10.m1.1"><semantics id="A1.SS2.2.p1.10.m1.1a"><mrow id="A1.SS2.2.p1.10.m1.1.1" xref="A1.SS2.2.p1.10.m1.1.1.cmml"><mi id="A1.SS2.2.p1.10.m1.1.1.2" xref="A1.SS2.2.p1.10.m1.1.1.2.cmml">ε</mi><mo id="A1.SS2.2.p1.10.m1.1.1.1" xref="A1.SS2.2.p1.10.m1.1.1.1.cmml">></mo><mn id="A1.SS2.2.p1.10.m1.1.1.3" xref="A1.SS2.2.p1.10.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.10.m1.1b"><apply id="A1.SS2.2.p1.10.m1.1.1.cmml" xref="A1.SS2.2.p1.10.m1.1.1"><gt id="A1.SS2.2.p1.10.m1.1.1.1.cmml" xref="A1.SS2.2.p1.10.m1.1.1.1"></gt><ci id="A1.SS2.2.p1.10.m1.1.1.2.cmml" xref="A1.SS2.2.p1.10.m1.1.1.2">𝜀</ci><cn id="A1.SS2.2.p1.10.m1.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.10.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.10.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.10.m1.1d">italic_ε > 0</annotation></semantics></math>. If we can prove that</p> <table class="ltx_equation ltx_eqn_table" id="A1.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|x_{i}-\varepsilon v_{i}-z_{i}|^{p}-|x_{i}-z_{i}|^{p}\leq|x_{i}-\varepsilon v_% {i}-z^{\prime}_{i}|^{p}-|x_{i}-z^{\prime}_{i}|^{p}" class="ltx_Math" display="block" id="A1.E1.m1.4"><semantics id="A1.E1.m1.4a"><mrow id="A1.E1.m1.4.4" xref="A1.E1.m1.4.4.cmml"><mrow id="A1.E1.m1.2.2.2" xref="A1.E1.m1.2.2.2.cmml"><msup id="A1.E1.m1.1.1.1.1" xref="A1.E1.m1.1.1.1.1.cmml"><mrow id="A1.E1.m1.1.1.1.1.1.1" xref="A1.E1.m1.1.1.1.1.1.2.cmml"><mo id="A1.E1.m1.1.1.1.1.1.1.2" stretchy="false" xref="A1.E1.m1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="A1.E1.m1.1.1.1.1.1.1.1" xref="A1.E1.m1.1.1.1.1.1.1.1.cmml"><msub id="A1.E1.m1.1.1.1.1.1.1.1.2" xref="A1.E1.m1.1.1.1.1.1.1.1.2.cmml"><mi id="A1.E1.m1.1.1.1.1.1.1.1.2.2" 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id="A1.E1.m1.3.3.3.1.3.cmml" xref="A1.E1.m1.3.3.3.1.3">𝑝</ci></apply><apply id="A1.E1.m1.4.4.4.2.cmml" xref="A1.E1.m1.4.4.4.2"><csymbol cd="ambiguous" id="A1.E1.m1.4.4.4.2.2.cmml" xref="A1.E1.m1.4.4.4.2">superscript</csymbol><apply id="A1.E1.m1.4.4.4.2.1.2.cmml" xref="A1.E1.m1.4.4.4.2.1.1"><abs id="A1.E1.m1.4.4.4.2.1.2.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.2"></abs><apply id="A1.E1.m1.4.4.4.2.1.1.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1"><minus id="A1.E1.m1.4.4.4.2.1.1.1.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.1"></minus><apply id="A1.E1.m1.4.4.4.2.1.1.1.2.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="A1.E1.m1.4.4.4.2.1.1.1.2.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.2">subscript</csymbol><ci id="A1.E1.m1.4.4.4.2.1.1.1.2.2.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.2.2">𝑥</ci><ci id="A1.E1.m1.4.4.4.2.1.1.1.2.3.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.2.3">𝑖</ci></apply><apply id="A1.E1.m1.4.4.4.2.1.1.1.3.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.E1.m1.4.4.4.2.1.1.1.3.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3">subscript</csymbol><apply id="A1.E1.m1.4.4.4.2.1.1.1.3.2.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.E1.m1.4.4.4.2.1.1.1.3.2.1.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3">superscript</csymbol><ci id="A1.E1.m1.4.4.4.2.1.1.1.3.2.2.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3.2.2">𝑧</ci><ci id="A1.E1.m1.4.4.4.2.1.1.1.3.2.3.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3.2.3">′</ci></apply><ci id="A1.E1.m1.4.4.4.2.1.1.1.3.3.cmml" xref="A1.E1.m1.4.4.4.2.1.1.1.3.3">𝑖</ci></apply></apply></apply><ci id="A1.E1.m1.4.4.4.2.3.cmml" xref="A1.E1.m1.4.4.4.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E1.m1.4c">|x_{i}-\varepsilon v_{i}-z_{i}|^{p}-|x_{i}-z_{i}|^{p}\leq|x_{i}-\varepsilon v_% {i}-z^{\prime}_{i}|^{p}-|x_{i}-z^{\prime}_{i}|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.E1.m1.4d">| italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ≤ | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.2.p1.11">for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.11.m1.1"><semantics id="A1.SS2.2.p1.11.m1.1a"><mrow id="A1.SS2.2.p1.11.m1.1.1" xref="A1.SS2.2.p1.11.m1.1.1.cmml"><mi id="A1.SS2.2.p1.11.m1.1.1.2" xref="A1.SS2.2.p1.11.m1.1.1.2.cmml">ε</mi><mo id="A1.SS2.2.p1.11.m1.1.1.1" xref="A1.SS2.2.p1.11.m1.1.1.1.cmml">></mo><mn id="A1.SS2.2.p1.11.m1.1.1.3" xref="A1.SS2.2.p1.11.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.11.m1.1b"><apply id="A1.SS2.2.p1.11.m1.1.1.cmml" xref="A1.SS2.2.p1.11.m1.1.1"><gt id="A1.SS2.2.p1.11.m1.1.1.1.cmml" xref="A1.SS2.2.p1.11.m1.1.1.1"></gt><ci id="A1.SS2.2.p1.11.m1.1.1.2.cmml" xref="A1.SS2.2.p1.11.m1.1.1.2">𝜀</ci><cn id="A1.SS2.2.p1.11.m1.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.11.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.11.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.11.m1.1d">italic_ε > 0</annotation></semantics></math>, we are done, because then</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="0\leq\lVert x-\varepsilon v-z\rVert^{p}_{p}-\lVert x-z\rVert^{p}_{p}\leq\lVert x% -\varepsilon v-z^{\prime}\rVert^{p}_{p}-\lVert x-z^{\prime}\rVert^{p}_{p}" class="ltx_Math" display="block" id="A1.Ex13.m1.4"><semantics id="A1.Ex13.m1.4a"><mrow id="A1.Ex13.m1.4.4" xref="A1.Ex13.m1.4.4.cmml"><mn id="A1.Ex13.m1.4.4.6" xref="A1.Ex13.m1.4.4.6.cmml">0</mn><mo id="A1.Ex13.m1.4.4.7" rspace="0.1389em" xref="A1.Ex13.m1.4.4.7.cmml">≤</mo><mrow id="A1.Ex13.m1.2.2.2" xref="A1.Ex13.m1.2.2.2.cmml"><msubsup id="A1.Ex13.m1.1.1.1.1" xref="A1.Ex13.m1.1.1.1.1.cmml"><mrow id="A1.Ex13.m1.1.1.1.1.1.1.1" xref="A1.Ex13.m1.1.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex13.m1.1.1.1.1.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex13.m1.1.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex13.m1.1.1.1.1.1.1.1.1" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex13.m1.1.1.1.1.1.1.1.1.2" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex13.m1.1.1.1.1.1.1.1.1.1" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex13.m1.1.1.1.1.1.1.1.1.3" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.1" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.3" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.Ex13.m1.1.1.1.1.1.1.1.1.1a" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex13.m1.1.1.1.1.1.1.1.1.4" xref="A1.Ex13.m1.1.1.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.Ex13.m1.1.1.1.1.1.1.1.3" lspace="0em" rspace="0em" xref="A1.Ex13.m1.1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex13.m1.1.1.1.1.3" xref="A1.Ex13.m1.1.1.1.1.3.cmml">p</mi><mi id="A1.Ex13.m1.1.1.1.1.1.3" xref="A1.Ex13.m1.1.1.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.Ex13.m1.2.2.2.3" xref="A1.Ex13.m1.2.2.2.3.cmml">−</mo><msubsup id="A1.Ex13.m1.2.2.2.2" xref="A1.Ex13.m1.2.2.2.2.cmml"><mrow id="A1.Ex13.m1.2.2.2.2.1.1.1" xref="A1.Ex13.m1.2.2.2.2.1.1.2.cmml"><mo fence="true" id="A1.Ex13.m1.2.2.2.2.1.1.1.2" lspace="0em" rspace="0em" xref="A1.Ex13.m1.2.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex13.m1.2.2.2.2.1.1.1.1" xref="A1.Ex13.m1.2.2.2.2.1.1.1.1.cmml"><mi id="A1.Ex13.m1.2.2.2.2.1.1.1.1.2" xref="A1.Ex13.m1.2.2.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex13.m1.2.2.2.2.1.1.1.1.1" xref="A1.Ex13.m1.2.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex13.m1.2.2.2.2.1.1.1.1.3" xref="A1.Ex13.m1.2.2.2.2.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.Ex13.m1.2.2.2.2.1.1.1.3" lspace="0em" rspace="0.1389em" xref="A1.Ex13.m1.2.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex13.m1.2.2.2.2.3" xref="A1.Ex13.m1.2.2.2.2.3.cmml">p</mi><mi id="A1.Ex13.m1.2.2.2.2.1.3" xref="A1.Ex13.m1.2.2.2.2.1.3.cmml">p</mi></msubsup></mrow><mo id="A1.Ex13.m1.4.4.8" lspace="0.1389em" rspace="0.1389em" xref="A1.Ex13.m1.4.4.8.cmml">≤</mo><mrow id="A1.Ex13.m1.4.4.4" xref="A1.Ex13.m1.4.4.4.cmml"><msubsup id="A1.Ex13.m1.3.3.3.1" xref="A1.Ex13.m1.3.3.3.1.cmml"><mrow id="A1.Ex13.m1.3.3.3.1.1.1.1" xref="A1.Ex13.m1.3.3.3.1.1.1.2.cmml"><mo fence="true" id="A1.Ex13.m1.3.3.3.1.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex13.m1.3.3.3.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex13.m1.3.3.3.1.1.1.1.1" xref="A1.Ex13.m1.3.3.3.1.1.1.1.1.cmml"><mi id="A1.Ex13.m1.3.3.3.1.1.1.1.1.2" xref="A1.Ex13.m1.3.3.3.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex13.m1.3.3.3.1.1.1.1.1.1" 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xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.1"></minus><ci id="A1.Ex13.m1.4.4.4.2.1.1.1.1.2.cmml" xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.2">𝑥</ci><apply id="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.cmml" xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.1.cmml" xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.3">superscript</csymbol><ci id="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.2.cmml" xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.2">𝑧</ci><ci id="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.3.cmml" xref="A1.Ex13.m1.4.4.4.2.1.1.1.1.3.3">′</ci></apply></apply></apply><ci id="A1.Ex13.m1.4.4.4.2.1.3.cmml" xref="A1.Ex13.m1.4.4.4.2.1.3">𝑝</ci></apply><ci id="A1.Ex13.m1.4.4.4.2.3.cmml" xref="A1.Ex13.m1.4.4.4.2.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex13.m1.4c">0\leq\lVert x-\varepsilon v-z\rVert^{p}_{p}-\lVert x-z\rVert^{p}_{p}\leq\lVert x% -\varepsilon v-z^{\prime}\rVert^{p}_{p}-\lVert x-z^{\prime}\rVert^{p}_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex13.m1.4d">0 ≤ ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_x - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ ∥ italic_x - italic_ε italic_v - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_x - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.2.p1.13">holds for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.12.m1.1"><semantics id="A1.SS2.2.p1.12.m1.1a"><mrow id="A1.SS2.2.p1.12.m1.1.1" xref="A1.SS2.2.p1.12.m1.1.1.cmml"><mi id="A1.SS2.2.p1.12.m1.1.1.2" xref="A1.SS2.2.p1.12.m1.1.1.2.cmml">ε</mi><mo id="A1.SS2.2.p1.12.m1.1.1.1" xref="A1.SS2.2.p1.12.m1.1.1.1.cmml">></mo><mn id="A1.SS2.2.p1.12.m1.1.1.3" xref="A1.SS2.2.p1.12.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.12.m1.1b"><apply id="A1.SS2.2.p1.12.m1.1.1.cmml" xref="A1.SS2.2.p1.12.m1.1.1"><gt id="A1.SS2.2.p1.12.m1.1.1.1.cmml" xref="A1.SS2.2.p1.12.m1.1.1.1"></gt><ci id="A1.SS2.2.p1.12.m1.1.1.2.cmml" xref="A1.SS2.2.p1.12.m1.1.1.2">𝜀</ci><cn id="A1.SS2.2.p1.12.m1.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.12.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.12.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.12.m1.1d">italic_ε > 0</annotation></semantics></math> and thus <math alttext="z^{\prime}\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.13.m2.2"><semantics id="A1.SS2.2.p1.13.m2.2a"><mrow id="A1.SS2.2.p1.13.m2.2.3" xref="A1.SS2.2.p1.13.m2.2.3.cmml"><msup id="A1.SS2.2.p1.13.m2.2.3.2" xref="A1.SS2.2.p1.13.m2.2.3.2.cmml"><mi id="A1.SS2.2.p1.13.m2.2.3.2.2" xref="A1.SS2.2.p1.13.m2.2.3.2.2.cmml">z</mi><mo id="A1.SS2.2.p1.13.m2.2.3.2.3" xref="A1.SS2.2.p1.13.m2.2.3.2.3.cmml">′</mo></msup><mo id="A1.SS2.2.p1.13.m2.2.3.1" xref="A1.SS2.2.p1.13.m2.2.3.1.cmml">∈</mo><msubsup id="A1.SS2.2.p1.13.m2.2.3.3" xref="A1.SS2.2.p1.13.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.2.p1.13.m2.2.3.3.2.2" xref="A1.SS2.2.p1.13.m2.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.2.p1.13.m2.2.2.2.4" xref="A1.SS2.2.p1.13.m2.2.2.2.3.cmml"><mi id="A1.SS2.2.p1.13.m2.1.1.1.1" xref="A1.SS2.2.p1.13.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS2.2.p1.13.m2.2.2.2.4.1" xref="A1.SS2.2.p1.13.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS2.2.p1.13.m2.2.2.2.2" xref="A1.SS2.2.p1.13.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.2.p1.13.m2.2.3.3.2.3" xref="A1.SS2.2.p1.13.m2.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.13.m2.2b"><apply id="A1.SS2.2.p1.13.m2.2.3.cmml" xref="A1.SS2.2.p1.13.m2.2.3"><in id="A1.SS2.2.p1.13.m2.2.3.1.cmml" xref="A1.SS2.2.p1.13.m2.2.3.1"></in><apply id="A1.SS2.2.p1.13.m2.2.3.2.cmml" xref="A1.SS2.2.p1.13.m2.2.3.2"><csymbol cd="ambiguous" id="A1.SS2.2.p1.13.m2.2.3.2.1.cmml" xref="A1.SS2.2.p1.13.m2.2.3.2">superscript</csymbol><ci id="A1.SS2.2.p1.13.m2.2.3.2.2.cmml" xref="A1.SS2.2.p1.13.m2.2.3.2.2">𝑧</ci><ci id="A1.SS2.2.p1.13.m2.2.3.2.3.cmml" xref="A1.SS2.2.p1.13.m2.2.3.2.3">′</ci></apply><apply id="A1.SS2.2.p1.13.m2.2.3.3.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.13.m2.2.3.3.1.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3">subscript</csymbol><apply id="A1.SS2.2.p1.13.m2.2.3.3.2.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.13.m2.2.3.3.2.1.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3">superscript</csymbol><ci id="A1.SS2.2.p1.13.m2.2.3.3.2.2.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.2.p1.13.m2.2.3.3.2.3.cmml" xref="A1.SS2.2.p1.13.m2.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.2.p1.13.m2.2.2.2.3.cmml" xref="A1.SS2.2.p1.13.m2.2.2.2.4"><ci id="A1.SS2.2.p1.13.m2.1.1.1.1.cmml" xref="A1.SS2.2.p1.13.m2.1.1.1.1">𝑥</ci><ci id="A1.SS2.2.p1.13.m2.2.2.2.2.cmml" xref="A1.SS2.2.p1.13.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.13.m2.2c">z^{\prime}\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.13.m2.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.E1" title="In Proof. ‣ A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">A.1</span></a> follows from the observation that</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|x_{i}-\varepsilon v_{i}-z^{\prime}_{i}|^{p}-|x_{i}-z^{\prime}_{i}|^{p}=|x_{i}% -(\alpha+\varepsilon)v_{i}-z_{i}|^{p}-|x_{i}-\alpha v_{i}-z_{i}|^{p}" class="ltx_Math" display="block" id="A1.Ex14.m1.4"><semantics id="A1.Ex14.m1.4a"><mrow id="A1.Ex14.m1.4.4" xref="A1.Ex14.m1.4.4.cmml"><mrow id="A1.Ex14.m1.2.2.2" xref="A1.Ex14.m1.2.2.2.cmml"><msup id="A1.Ex14.m1.1.1.1.1" xref="A1.Ex14.m1.1.1.1.1.cmml"><mrow id="A1.Ex14.m1.1.1.1.1.1.1" xref="A1.Ex14.m1.1.1.1.1.1.2.cmml"><mo id="A1.Ex14.m1.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex14.m1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="A1.Ex14.m1.1.1.1.1.1.1.1" xref="A1.Ex14.m1.1.1.1.1.1.1.1.cmml"><msub id="A1.Ex14.m1.1.1.1.1.1.1.1.2" xref="A1.Ex14.m1.1.1.1.1.1.1.1.2.cmml"><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.2.2" xref="A1.Ex14.m1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.2.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="A1.Ex14.m1.1.1.1.1.1.1.1.1" xref="A1.Ex14.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex14.m1.1.1.1.1.1.1.1.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.3.2" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex14.m1.1.1.1.1.1.1.1.3.1" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A1.Ex14.m1.1.1.1.1.1.1.1.3.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.3.3.2" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.3.2.cmml">v</mi><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.3.3.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.3.3.3.cmml">i</mi></msub></mrow><mo id="A1.Ex14.m1.1.1.1.1.1.1.1.1a" xref="A1.Ex14.m1.1.1.1.1.1.1.1.1.cmml">−</mo><msubsup id="A1.Ex14.m1.1.1.1.1.1.1.1.4" xref="A1.Ex14.m1.1.1.1.1.1.1.1.4.cmml"><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.4.2.2" xref="A1.Ex14.m1.1.1.1.1.1.1.1.4.2.2.cmml">z</mi><mi id="A1.Ex14.m1.1.1.1.1.1.1.1.4.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.4.3.cmml">i</mi><mo id="A1.Ex14.m1.1.1.1.1.1.1.1.4.2.3" xref="A1.Ex14.m1.1.1.1.1.1.1.1.4.2.3.cmml">′</mo></msubsup></mrow><mo id="A1.Ex14.m1.1.1.1.1.1.1.3" stretchy="false" xref="A1.Ex14.m1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mi id="A1.Ex14.m1.1.1.1.1.3" xref="A1.Ex14.m1.1.1.1.1.3.cmml">p</mi></msup><mo id="A1.Ex14.m1.2.2.2.3" xref="A1.Ex14.m1.2.2.2.3.cmml">−</mo><msup id="A1.Ex14.m1.2.2.2.2" xref="A1.Ex14.m1.2.2.2.2.cmml"><mrow id="A1.Ex14.m1.2.2.2.2.1.1" xref="A1.Ex14.m1.2.2.2.2.1.2.cmml"><mo id="A1.Ex14.m1.2.2.2.2.1.1.2" stretchy="false" xref="A1.Ex14.m1.2.2.2.2.1.2.1.cmml">|</mo><mrow id="A1.Ex14.m1.2.2.2.2.1.1.1" xref="A1.Ex14.m1.2.2.2.2.1.1.1.cmml"><msub 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xref="A1.Ex14.m1.4.4.4.2">superscript</csymbol><apply id="A1.Ex14.m1.4.4.4.2.1.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1"><abs id="A1.Ex14.m1.4.4.4.2.1.2.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.2"></abs><apply id="A1.Ex14.m1.4.4.4.2.1.1.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1"><minus id="A1.Ex14.m1.4.4.4.2.1.1.1.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.1"></minus><apply id="A1.Ex14.m1.4.4.4.2.1.1.1.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex14.m1.4.4.4.2.1.1.1.2.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.2">subscript</csymbol><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.2.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.2.2">𝑥</ci><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.2.3.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.2.3">𝑖</ci></apply><apply id="A1.Ex14.m1.4.4.4.2.1.1.1.3.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3"><times id="A1.Ex14.m1.4.4.4.2.1.1.1.3.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.1"></times><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.3.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.2">𝛼</ci><apply id="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.3">subscript</csymbol><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.2">𝑣</ci><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.3.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.3.3.3">𝑖</ci></apply></apply><apply id="A1.Ex14.m1.4.4.4.2.1.1.1.4.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex14.m1.4.4.4.2.1.1.1.4.1.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.4">subscript</csymbol><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.4.2.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.4.2">𝑧</ci><ci id="A1.Ex14.m1.4.4.4.2.1.1.1.4.3.cmml" xref="A1.Ex14.m1.4.4.4.2.1.1.1.4.3">𝑖</ci></apply></apply></apply><ci id="A1.Ex14.m1.4.4.4.2.3.cmml" xref="A1.Ex14.m1.4.4.4.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex14.m1.4c">|x_{i}-\varepsilon v_{i}-z^{\prime}_{i}|^{p}-|x_{i}-z^{\prime}_{i}|^{p}=|x_{i}% -(\alpha+\varepsilon)v_{i}-z_{i}|^{p}-|x_{i}-\alpha v_{i}-z_{i}|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex14.m1.4d">| italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT = | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ( italic_α + italic_ε ) italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_α italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.2.p1.21">and that the function <math alttext="f(\beta)\coloneqq|\beta+\gamma|^{p}-|\beta|^{p}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.14.m1.3"><semantics id="A1.SS2.2.p1.14.m1.3a"><mrow id="A1.SS2.2.p1.14.m1.3.3" xref="A1.SS2.2.p1.14.m1.3.3.cmml"><mrow id="A1.SS2.2.p1.14.m1.3.3.3" xref="A1.SS2.2.p1.14.m1.3.3.3.cmml"><mi id="A1.SS2.2.p1.14.m1.3.3.3.2" xref="A1.SS2.2.p1.14.m1.3.3.3.2.cmml">f</mi><mo id="A1.SS2.2.p1.14.m1.3.3.3.1" xref="A1.SS2.2.p1.14.m1.3.3.3.1.cmml">⁢</mo><mrow id="A1.SS2.2.p1.14.m1.3.3.3.3.2" xref="A1.SS2.2.p1.14.m1.3.3.3.cmml"><mo id="A1.SS2.2.p1.14.m1.3.3.3.3.2.1" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.3.cmml">(</mo><mi id="A1.SS2.2.p1.14.m1.1.1" xref="A1.SS2.2.p1.14.m1.1.1.cmml">β</mi><mo id="A1.SS2.2.p1.14.m1.3.3.3.3.2.2" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="A1.SS2.2.p1.14.m1.3.3.2" xref="A1.SS2.2.p1.14.m1.3.3.2.cmml">≔</mo><mrow id="A1.SS2.2.p1.14.m1.3.3.1" xref="A1.SS2.2.p1.14.m1.3.3.1.cmml"><msup id="A1.SS2.2.p1.14.m1.3.3.1.1" xref="A1.SS2.2.p1.14.m1.3.3.1.1.cmml"><mrow id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.2.cmml"><mo id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.2" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.2.1.cmml">|</mo><mrow id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.cmml"><mi id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.2" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.2.cmml">β</mi><mo id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.1" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.3" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.3.cmml">γ</mi></mrow><mo id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.3" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.2.1.cmml">|</mo></mrow><mi id="A1.SS2.2.p1.14.m1.3.3.1.1.3" xref="A1.SS2.2.p1.14.m1.3.3.1.1.3.cmml">p</mi></msup><mo id="A1.SS2.2.p1.14.m1.3.3.1.2" xref="A1.SS2.2.p1.14.m1.3.3.1.2.cmml">−</mo><msup id="A1.SS2.2.p1.14.m1.3.3.1.3" xref="A1.SS2.2.p1.14.m1.3.3.1.3.cmml"><mrow id="A1.SS2.2.p1.14.m1.3.3.1.3.2.2" xref="A1.SS2.2.p1.14.m1.3.3.1.3.2.1.cmml"><mo id="A1.SS2.2.p1.14.m1.3.3.1.3.2.2.1" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.1.3.2.1.1.cmml">|</mo><mi id="A1.SS2.2.p1.14.m1.2.2" xref="A1.SS2.2.p1.14.m1.2.2.cmml">β</mi><mo id="A1.SS2.2.p1.14.m1.3.3.1.3.2.2.2" stretchy="false" xref="A1.SS2.2.p1.14.m1.3.3.1.3.2.1.1.cmml">|</mo></mrow><mi id="A1.SS2.2.p1.14.m1.3.3.1.3.3" xref="A1.SS2.2.p1.14.m1.3.3.1.3.3.cmml">p</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.14.m1.3b"><apply id="A1.SS2.2.p1.14.m1.3.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3"><ci id="A1.SS2.2.p1.14.m1.3.3.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.2">≔</ci><apply id="A1.SS2.2.p1.14.m1.3.3.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3.3"><times id="A1.SS2.2.p1.14.m1.3.3.3.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.3.1"></times><ci id="A1.SS2.2.p1.14.m1.3.3.3.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.3.2">𝑓</ci><ci id="A1.SS2.2.p1.14.m1.1.1.cmml" xref="A1.SS2.2.p1.14.m1.1.1">𝛽</ci></apply><apply id="A1.SS2.2.p1.14.m1.3.3.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1"><minus id="A1.SS2.2.p1.14.m1.3.3.1.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.2"></minus><apply id="A1.SS2.2.p1.14.m1.3.3.1.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.14.m1.3.3.1.1.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1">superscript</csymbol><apply id="A1.SS2.2.p1.14.m1.3.3.1.1.1.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1"><abs id="A1.SS2.2.p1.14.m1.3.3.1.1.1.2.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.2"></abs><apply id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1"><plus id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.1"></plus><ci id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.2.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.2">𝛽</ci><ci id="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.1.1.1.3">𝛾</ci></apply></apply><ci id="A1.SS2.2.p1.14.m1.3.3.1.1.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.1.3">𝑝</ci></apply><apply id="A1.SS2.2.p1.14.m1.3.3.1.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.14.m1.3.3.1.3.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.3">superscript</csymbol><apply id="A1.SS2.2.p1.14.m1.3.3.1.3.2.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.3.2.2"><abs id="A1.SS2.2.p1.14.m1.3.3.1.3.2.1.1.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.3.2.2.1"></abs><ci id="A1.SS2.2.p1.14.m1.2.2.cmml" xref="A1.SS2.2.p1.14.m1.2.2">𝛽</ci></apply><ci id="A1.SS2.2.p1.14.m1.3.3.1.3.3.cmml" xref="A1.SS2.2.p1.14.m1.3.3.1.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.14.m1.3c">f(\beta)\coloneqq|\beta+\gamma|^{p}-|\beta|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.14.m1.3d">italic_f ( italic_β ) ≔ | italic_β + italic_γ | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_β | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is for all <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="A1.SS2.2.p1.15.m2.2"><semantics id="A1.SS2.2.p1.15.m2.2a"><mrow id="A1.SS2.2.p1.15.m2.2.3" xref="A1.SS2.2.p1.15.m2.2.3.cmml"><mi id="A1.SS2.2.p1.15.m2.2.3.2" xref="A1.SS2.2.p1.15.m2.2.3.2.cmml">p</mi><mo id="A1.SS2.2.p1.15.m2.2.3.1" xref="A1.SS2.2.p1.15.m2.2.3.1.cmml">∈</mo><mrow id="A1.SS2.2.p1.15.m2.2.3.3.2" xref="A1.SS2.2.p1.15.m2.2.3.3.1.cmml"><mo id="A1.SS2.2.p1.15.m2.2.3.3.2.1" stretchy="false" xref="A1.SS2.2.p1.15.m2.2.3.3.1.cmml">[</mo><mn id="A1.SS2.2.p1.15.m2.1.1" xref="A1.SS2.2.p1.15.m2.1.1.cmml">1</mn><mo id="A1.SS2.2.p1.15.m2.2.3.3.2.2" xref="A1.SS2.2.p1.15.m2.2.3.3.1.cmml">,</mo><mi id="A1.SS2.2.p1.15.m2.2.2" mathvariant="normal" xref="A1.SS2.2.p1.15.m2.2.2.cmml">∞</mi><mo id="A1.SS2.2.p1.15.m2.2.3.3.2.3" stretchy="false" xref="A1.SS2.2.p1.15.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.15.m2.2b"><apply id="A1.SS2.2.p1.15.m2.2.3.cmml" xref="A1.SS2.2.p1.15.m2.2.3"><in id="A1.SS2.2.p1.15.m2.2.3.1.cmml" xref="A1.SS2.2.p1.15.m2.2.3.1"></in><ci id="A1.SS2.2.p1.15.m2.2.3.2.cmml" xref="A1.SS2.2.p1.15.m2.2.3.2">𝑝</ci><interval closure="closed-open" id="A1.SS2.2.p1.15.m2.2.3.3.1.cmml" xref="A1.SS2.2.p1.15.m2.2.3.3.2"><cn id="A1.SS2.2.p1.15.m2.1.1.cmml" type="integer" xref="A1.SS2.2.p1.15.m2.1.1">1</cn><infinity id="A1.SS2.2.p1.15.m2.2.2.cmml" xref="A1.SS2.2.p1.15.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.15.m2.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.15.m2.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math> monotonically non-increasing for <math alttext="\gamma\leq 0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.16.m3.1"><semantics id="A1.SS2.2.p1.16.m3.1a"><mrow id="A1.SS2.2.p1.16.m3.1.1" xref="A1.SS2.2.p1.16.m3.1.1.cmml"><mi id="A1.SS2.2.p1.16.m3.1.1.2" xref="A1.SS2.2.p1.16.m3.1.1.2.cmml">γ</mi><mo id="A1.SS2.2.p1.16.m3.1.1.1" xref="A1.SS2.2.p1.16.m3.1.1.1.cmml">≤</mo><mn id="A1.SS2.2.p1.16.m3.1.1.3" xref="A1.SS2.2.p1.16.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.16.m3.1b"><apply id="A1.SS2.2.p1.16.m3.1.1.cmml" xref="A1.SS2.2.p1.16.m3.1.1"><leq id="A1.SS2.2.p1.16.m3.1.1.1.cmml" xref="A1.SS2.2.p1.16.m3.1.1.1"></leq><ci id="A1.SS2.2.p1.16.m3.1.1.2.cmml" xref="A1.SS2.2.p1.16.m3.1.1.2">𝛾</ci><cn id="A1.SS2.2.p1.16.m3.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.16.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.16.m3.1c">\gamma\leq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.16.m3.1d">italic_γ ≤ 0</annotation></semantics></math> and monotonically non-decreasing for <math alttext="\gamma\geq 0" class="ltx_Math" display="inline" id="A1.SS2.2.p1.17.m4.1"><semantics id="A1.SS2.2.p1.17.m4.1a"><mrow id="A1.SS2.2.p1.17.m4.1.1" xref="A1.SS2.2.p1.17.m4.1.1.cmml"><mi id="A1.SS2.2.p1.17.m4.1.1.2" xref="A1.SS2.2.p1.17.m4.1.1.2.cmml">γ</mi><mo id="A1.SS2.2.p1.17.m4.1.1.1" xref="A1.SS2.2.p1.17.m4.1.1.1.cmml">≥</mo><mn id="A1.SS2.2.p1.17.m4.1.1.3" xref="A1.SS2.2.p1.17.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.17.m4.1b"><apply id="A1.SS2.2.p1.17.m4.1.1.cmml" xref="A1.SS2.2.p1.17.m4.1.1"><geq id="A1.SS2.2.p1.17.m4.1.1.1.cmml" xref="A1.SS2.2.p1.17.m4.1.1.1"></geq><ci id="A1.SS2.2.p1.17.m4.1.1.2.cmml" xref="A1.SS2.2.p1.17.m4.1.1.2">𝛾</ci><cn id="A1.SS2.2.p1.17.m4.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.17.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.17.m4.1c">\gamma\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.17.m4.1d">italic_γ ≥ 0</annotation></semantics></math>. The proof for <math alttext="p=\infty" class="ltx_Math" display="inline" id="A1.SS2.2.p1.18.m5.1"><semantics id="A1.SS2.2.p1.18.m5.1a"><mrow id="A1.SS2.2.p1.18.m5.1.1" xref="A1.SS2.2.p1.18.m5.1.1.cmml"><mi id="A1.SS2.2.p1.18.m5.1.1.2" xref="A1.SS2.2.p1.18.m5.1.1.2.cmml">p</mi><mo id="A1.SS2.2.p1.18.m5.1.1.1" xref="A1.SS2.2.p1.18.m5.1.1.1.cmml">=</mo><mi id="A1.SS2.2.p1.18.m5.1.1.3" mathvariant="normal" xref="A1.SS2.2.p1.18.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.18.m5.1b"><apply id="A1.SS2.2.p1.18.m5.1.1.cmml" xref="A1.SS2.2.p1.18.m5.1.1"><eq id="A1.SS2.2.p1.18.m5.1.1.1.cmml" xref="A1.SS2.2.p1.18.m5.1.1.1"></eq><ci id="A1.SS2.2.p1.18.m5.1.1.2.cmml" xref="A1.SS2.2.p1.18.m5.1.1.2">𝑝</ci><infinity id="A1.SS2.2.p1.18.m5.1.1.3.cmml" xref="A1.SS2.2.p1.18.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.18.m5.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.18.m5.1d">italic_p = ∞</annotation></semantics></math> follows from <math alttext="p=1" class="ltx_Math" display="inline" id="A1.SS2.2.p1.19.m6.1"><semantics id="A1.SS2.2.p1.19.m6.1a"><mrow id="A1.SS2.2.p1.19.m6.1.1" xref="A1.SS2.2.p1.19.m6.1.1.cmml"><mi id="A1.SS2.2.p1.19.m6.1.1.2" xref="A1.SS2.2.p1.19.m6.1.1.2.cmml">p</mi><mo id="A1.SS2.2.p1.19.m6.1.1.1" xref="A1.SS2.2.p1.19.m6.1.1.1.cmml">=</mo><mn id="A1.SS2.2.p1.19.m6.1.1.3" xref="A1.SS2.2.p1.19.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.19.m6.1b"><apply id="A1.SS2.2.p1.19.m6.1.1.cmml" xref="A1.SS2.2.p1.19.m6.1.1"><eq id="A1.SS2.2.p1.19.m6.1.1.1.cmml" xref="A1.SS2.2.p1.19.m6.1.1.1"></eq><ci id="A1.SS2.2.p1.19.m6.1.1.2.cmml" xref="A1.SS2.2.p1.19.m6.1.1.2">𝑝</ci><cn id="A1.SS2.2.p1.19.m6.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.19.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.19.m6.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.19.m6.1d">italic_p = 1</annotation></semantics></math>, since the <math alttext="\ell_{\infty}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.20.m7.1"><semantics id="A1.SS2.2.p1.20.m7.1a"><msub id="A1.SS2.2.p1.20.m7.1.1" xref="A1.SS2.2.p1.20.m7.1.1.cmml"><mi id="A1.SS2.2.p1.20.m7.1.1.2" mathvariant="normal" xref="A1.SS2.2.p1.20.m7.1.1.2.cmml">ℓ</mi><mi id="A1.SS2.2.p1.20.m7.1.1.3" mathvariant="normal" xref="A1.SS2.2.p1.20.m7.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.20.m7.1b"><apply id="A1.SS2.2.p1.20.m7.1.1.cmml" xref="A1.SS2.2.p1.20.m7.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.20.m7.1.1.1.cmml" xref="A1.SS2.2.p1.20.m7.1.1">subscript</csymbol><ci id="A1.SS2.2.p1.20.m7.1.1.2.cmml" xref="A1.SS2.2.p1.20.m7.1.1.2">ℓ</ci><infinity id="A1.SS2.2.p1.20.m7.1.1.3.cmml" xref="A1.SS2.2.p1.20.m7.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.20.m7.1c">\ell_{\infty}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.20.m7.1d">roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math>-norm is just taking the maximum over the constituents of the <math alttext="\ell_{1}" class="ltx_Math" display="inline" id="A1.SS2.2.p1.21.m8.1"><semantics id="A1.SS2.2.p1.21.m8.1a"><msub id="A1.SS2.2.p1.21.m8.1.1" xref="A1.SS2.2.p1.21.m8.1.1.cmml"><mi id="A1.SS2.2.p1.21.m8.1.1.2" mathvariant="normal" xref="A1.SS2.2.p1.21.m8.1.1.2.cmml">ℓ</mi><mn id="A1.SS2.2.p1.21.m8.1.1.3" xref="A1.SS2.2.p1.21.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.21.m8.1b"><apply id="A1.SS2.2.p1.21.m8.1.1.cmml" xref="A1.SS2.2.p1.21.m8.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.21.m8.1.1.1.cmml" xref="A1.SS2.2.p1.21.m8.1.1">subscript</csymbol><ci id="A1.SS2.2.p1.21.m8.1.1.2.cmml" xref="A1.SS2.2.p1.21.m8.1.1.2">ℓ</ci><cn id="A1.SS2.2.p1.21.m8.1.1.3.cmml" type="integer" xref="A1.SS2.2.p1.21.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.21.m8.1c">\ell_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p1.21.m8.1d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-norm.</p> </div> <div class="ltx_para" id="A1.SS2.3.p2"> <p class="ltx_p" id="A1.SS2.3.p2.10">Consider now <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.1.m1.2"><semantics id="A1.SS2.3.p2.1.m1.2a"><mrow id="A1.SS2.3.p2.1.m1.2.3" xref="A1.SS2.3.p2.1.m1.2.3.cmml"><mi id="A1.SS2.3.p2.1.m1.2.3.2" xref="A1.SS2.3.p2.1.m1.2.3.2.cmml">z</mi><mo id="A1.SS2.3.p2.1.m1.2.3.1" xref="A1.SS2.3.p2.1.m1.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.3.p2.1.m1.2.3.3" xref="A1.SS2.3.p2.1.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.3.p2.1.m1.2.3.3.2.2" xref="A1.SS2.3.p2.1.m1.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.3.p2.1.m1.2.2.2.4" xref="A1.SS2.3.p2.1.m1.2.2.2.3.cmml"><mi id="A1.SS2.3.p2.1.m1.1.1.1.1" xref="A1.SS2.3.p2.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.3.p2.1.m1.2.2.2.4.1" xref="A1.SS2.3.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.3.p2.1.m1.2.2.2.2" xref="A1.SS2.3.p2.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.3.p2.1.m1.2.3.3.2.3" xref="A1.SS2.3.p2.1.m1.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.1.m1.2b"><apply id="A1.SS2.3.p2.1.m1.2.3.cmml" xref="A1.SS2.3.p2.1.m1.2.3"><notin id="A1.SS2.3.p2.1.m1.2.3.1.cmml" xref="A1.SS2.3.p2.1.m1.2.3.1"></notin><ci id="A1.SS2.3.p2.1.m1.2.3.2.cmml" xref="A1.SS2.3.p2.1.m1.2.3.2">𝑧</ci><apply id="A1.SS2.3.p2.1.m1.2.3.3.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.1.m1.2.3.3.1.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3">subscript</csymbol><apply id="A1.SS2.3.p2.1.m1.2.3.3.2.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.1.m1.2.3.3.2.1.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3">superscript</csymbol><ci id="A1.SS2.3.p2.1.m1.2.3.3.2.2.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.3.p2.1.m1.2.3.3.2.3.cmml" xref="A1.SS2.3.p2.1.m1.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.3.p2.1.m1.2.2.2.3.cmml" xref="A1.SS2.3.p2.1.m1.2.2.2.4"><ci id="A1.SS2.3.p2.1.m1.1.1.1.1.cmml" xref="A1.SS2.3.p2.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.3.p2.1.m1.2.2.2.2.cmml" xref="A1.SS2.3.p2.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.1.m1.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.1.m1.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and consider again first the case <math alttext="p\neq\infty" class="ltx_Math" display="inline" id="A1.SS2.3.p2.2.m2.1"><semantics id="A1.SS2.3.p2.2.m2.1a"><mrow id="A1.SS2.3.p2.2.m2.1.1" xref="A1.SS2.3.p2.2.m2.1.1.cmml"><mi id="A1.SS2.3.p2.2.m2.1.1.2" xref="A1.SS2.3.p2.2.m2.1.1.2.cmml">p</mi><mo id="A1.SS2.3.p2.2.m2.1.1.1" xref="A1.SS2.3.p2.2.m2.1.1.1.cmml">≠</mo><mi id="A1.SS2.3.p2.2.m2.1.1.3" mathvariant="normal" xref="A1.SS2.3.p2.2.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.2.m2.1b"><apply id="A1.SS2.3.p2.2.m2.1.1.cmml" xref="A1.SS2.3.p2.2.m2.1.1"><neq id="A1.SS2.3.p2.2.m2.1.1.1.cmml" xref="A1.SS2.3.p2.2.m2.1.1.1"></neq><ci id="A1.SS2.3.p2.2.m2.1.1.2.cmml" xref="A1.SS2.3.p2.2.m2.1.1.2">𝑝</ci><infinity id="A1.SS2.3.p2.2.m2.1.1.3.cmml" xref="A1.SS2.3.p2.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.2.m2.1c">p\neq\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.2.m2.1d">italic_p ≠ ∞</annotation></semantics></math>. As before, we can assume without loss of generality that <math alttext="z^{\prime}_{i}=z_{i}-\alpha v_{i}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.3.m3.1"><semantics id="A1.SS2.3.p2.3.m3.1a"><mrow id="A1.SS2.3.p2.3.m3.1.1" xref="A1.SS2.3.p2.3.m3.1.1.cmml"><msubsup id="A1.SS2.3.p2.3.m3.1.1.2" xref="A1.SS2.3.p2.3.m3.1.1.2.cmml"><mi id="A1.SS2.3.p2.3.m3.1.1.2.2.2" xref="A1.SS2.3.p2.3.m3.1.1.2.2.2.cmml">z</mi><mi id="A1.SS2.3.p2.3.m3.1.1.2.3" xref="A1.SS2.3.p2.3.m3.1.1.2.3.cmml">i</mi><mo id="A1.SS2.3.p2.3.m3.1.1.2.2.3" xref="A1.SS2.3.p2.3.m3.1.1.2.2.3.cmml">′</mo></msubsup><mo id="A1.SS2.3.p2.3.m3.1.1.1" xref="A1.SS2.3.p2.3.m3.1.1.1.cmml">=</mo><mrow id="A1.SS2.3.p2.3.m3.1.1.3" xref="A1.SS2.3.p2.3.m3.1.1.3.cmml"><msub id="A1.SS2.3.p2.3.m3.1.1.3.2" xref="A1.SS2.3.p2.3.m3.1.1.3.2.cmml"><mi id="A1.SS2.3.p2.3.m3.1.1.3.2.2" xref="A1.SS2.3.p2.3.m3.1.1.3.2.2.cmml">z</mi><mi id="A1.SS2.3.p2.3.m3.1.1.3.2.3" xref="A1.SS2.3.p2.3.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="A1.SS2.3.p2.3.m3.1.1.3.1" xref="A1.SS2.3.p2.3.m3.1.1.3.1.cmml">−</mo><mrow id="A1.SS2.3.p2.3.m3.1.1.3.3" xref="A1.SS2.3.p2.3.m3.1.1.3.3.cmml"><mi id="A1.SS2.3.p2.3.m3.1.1.3.3.2" xref="A1.SS2.3.p2.3.m3.1.1.3.3.2.cmml">α</mi><mo id="A1.SS2.3.p2.3.m3.1.1.3.3.1" xref="A1.SS2.3.p2.3.m3.1.1.3.3.1.cmml">⁢</mo><msub id="A1.SS2.3.p2.3.m3.1.1.3.3.3" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3.cmml"><mi id="A1.SS2.3.p2.3.m3.1.1.3.3.3.2" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3.2.cmml">v</mi><mi id="A1.SS2.3.p2.3.m3.1.1.3.3.3.3" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.3.m3.1b"><apply id="A1.SS2.3.p2.3.m3.1.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1"><eq id="A1.SS2.3.p2.3.m3.1.1.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.1"></eq><apply id="A1.SS2.3.p2.3.m3.1.1.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.3.m3.1.1.2.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2">subscript</csymbol><apply id="A1.SS2.3.p2.3.m3.1.1.2.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.3.m3.1.1.2.2.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2">superscript</csymbol><ci id="A1.SS2.3.p2.3.m3.1.1.2.2.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2.2.2">𝑧</ci><ci id="A1.SS2.3.p2.3.m3.1.1.2.2.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2.2.3">′</ci></apply><ci id="A1.SS2.3.p2.3.m3.1.1.2.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.2.3">𝑖</ci></apply><apply id="A1.SS2.3.p2.3.m3.1.1.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3"><minus id="A1.SS2.3.p2.3.m3.1.1.3.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.1"></minus><apply id="A1.SS2.3.p2.3.m3.1.1.3.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.3.m3.1.1.3.2.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.2">subscript</csymbol><ci id="A1.SS2.3.p2.3.m3.1.1.3.2.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.2.2">𝑧</ci><ci id="A1.SS2.3.p2.3.m3.1.1.3.2.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.2.3">𝑖</ci></apply><apply id="A1.SS2.3.p2.3.m3.1.1.3.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3"><times id="A1.SS2.3.p2.3.m3.1.1.3.3.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.1"></times><ci id="A1.SS2.3.p2.3.m3.1.1.3.3.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.2">𝛼</ci><apply id="A1.SS2.3.p2.3.m3.1.1.3.3.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.3.m3.1.1.3.3.3.1.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3">subscript</csymbol><ci id="A1.SS2.3.p2.3.m3.1.1.3.3.3.2.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3.2">𝑣</ci><ci id="A1.SS2.3.p2.3.m3.1.1.3.3.3.3.cmml" xref="A1.SS2.3.p2.3.m3.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.3.m3.1c">z^{\prime}_{i}=z_{i}-\alpha v_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.3.m3.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_α italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="i\in[d]" class="ltx_Math" display="inline" id="A1.SS2.3.p2.4.m4.1"><semantics id="A1.SS2.3.p2.4.m4.1a"><mrow id="A1.SS2.3.p2.4.m4.1.2" xref="A1.SS2.3.p2.4.m4.1.2.cmml"><mi id="A1.SS2.3.p2.4.m4.1.2.2" xref="A1.SS2.3.p2.4.m4.1.2.2.cmml">i</mi><mo id="A1.SS2.3.p2.4.m4.1.2.1" xref="A1.SS2.3.p2.4.m4.1.2.1.cmml">∈</mo><mrow id="A1.SS2.3.p2.4.m4.1.2.3.2" xref="A1.SS2.3.p2.4.m4.1.2.3.1.cmml"><mo id="A1.SS2.3.p2.4.m4.1.2.3.2.1" stretchy="false" xref="A1.SS2.3.p2.4.m4.1.2.3.1.1.cmml">[</mo><mi id="A1.SS2.3.p2.4.m4.1.1" xref="A1.SS2.3.p2.4.m4.1.1.cmml">d</mi><mo id="A1.SS2.3.p2.4.m4.1.2.3.2.2" stretchy="false" xref="A1.SS2.3.p2.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.4.m4.1b"><apply id="A1.SS2.3.p2.4.m4.1.2.cmml" xref="A1.SS2.3.p2.4.m4.1.2"><in id="A1.SS2.3.p2.4.m4.1.2.1.cmml" xref="A1.SS2.3.p2.4.m4.1.2.1"></in><ci id="A1.SS2.3.p2.4.m4.1.2.2.cmml" xref="A1.SS2.3.p2.4.m4.1.2.2">𝑖</ci><apply id="A1.SS2.3.p2.4.m4.1.2.3.1.cmml" xref="A1.SS2.3.p2.4.m4.1.2.3.2"><csymbol cd="latexml" id="A1.SS2.3.p2.4.m4.1.2.3.1.1.cmml" xref="A1.SS2.3.p2.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="A1.SS2.3.p2.4.m4.1.1.cmml" xref="A1.SS2.3.p2.4.m4.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.4.m4.1c">i\in[d]</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.4.m4.1d">italic_i ∈ [ italic_d ]</annotation></semantics></math> and <math alttext="\alpha>0" class="ltx_Math" display="inline" id="A1.SS2.3.p2.5.m5.1"><semantics id="A1.SS2.3.p2.5.m5.1a"><mrow id="A1.SS2.3.p2.5.m5.1.1" xref="A1.SS2.3.p2.5.m5.1.1.cmml"><mi id="A1.SS2.3.p2.5.m5.1.1.2" xref="A1.SS2.3.p2.5.m5.1.1.2.cmml">α</mi><mo id="A1.SS2.3.p2.5.m5.1.1.1" xref="A1.SS2.3.p2.5.m5.1.1.1.cmml">></mo><mn id="A1.SS2.3.p2.5.m5.1.1.3" xref="A1.SS2.3.p2.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.5.m5.1b"><apply id="A1.SS2.3.p2.5.m5.1.1.cmml" xref="A1.SS2.3.p2.5.m5.1.1"><gt id="A1.SS2.3.p2.5.m5.1.1.1.cmml" xref="A1.SS2.3.p2.5.m5.1.1.1"></gt><ci id="A1.SS2.3.p2.5.m5.1.1.2.cmml" xref="A1.SS2.3.p2.5.m5.1.1.2">𝛼</ci><cn id="A1.SS2.3.p2.5.m5.1.1.3.cmml" type="integer" xref="A1.SS2.3.p2.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.5.m5.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.5.m5.1d">italic_α > 0</annotation></semantics></math>, and <math alttext="z^{\prime}_{j}=z_{j}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.6.m6.1"><semantics id="A1.SS2.3.p2.6.m6.1a"><mrow id="A1.SS2.3.p2.6.m6.1.1" xref="A1.SS2.3.p2.6.m6.1.1.cmml"><msubsup id="A1.SS2.3.p2.6.m6.1.1.2" xref="A1.SS2.3.p2.6.m6.1.1.2.cmml"><mi id="A1.SS2.3.p2.6.m6.1.1.2.2.2" xref="A1.SS2.3.p2.6.m6.1.1.2.2.2.cmml">z</mi><mi id="A1.SS2.3.p2.6.m6.1.1.2.3" xref="A1.SS2.3.p2.6.m6.1.1.2.3.cmml">j</mi><mo id="A1.SS2.3.p2.6.m6.1.1.2.2.3" xref="A1.SS2.3.p2.6.m6.1.1.2.2.3.cmml">′</mo></msubsup><mo id="A1.SS2.3.p2.6.m6.1.1.1" xref="A1.SS2.3.p2.6.m6.1.1.1.cmml">=</mo><msub id="A1.SS2.3.p2.6.m6.1.1.3" xref="A1.SS2.3.p2.6.m6.1.1.3.cmml"><mi id="A1.SS2.3.p2.6.m6.1.1.3.2" xref="A1.SS2.3.p2.6.m6.1.1.3.2.cmml">z</mi><mi id="A1.SS2.3.p2.6.m6.1.1.3.3" xref="A1.SS2.3.p2.6.m6.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.6.m6.1b"><apply id="A1.SS2.3.p2.6.m6.1.1.cmml" xref="A1.SS2.3.p2.6.m6.1.1"><eq id="A1.SS2.3.p2.6.m6.1.1.1.cmml" xref="A1.SS2.3.p2.6.m6.1.1.1"></eq><apply id="A1.SS2.3.p2.6.m6.1.1.2.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.6.m6.1.1.2.1.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2">subscript</csymbol><apply id="A1.SS2.3.p2.6.m6.1.1.2.2.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.6.m6.1.1.2.2.1.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2">superscript</csymbol><ci id="A1.SS2.3.p2.6.m6.1.1.2.2.2.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2.2.2">𝑧</ci><ci id="A1.SS2.3.p2.6.m6.1.1.2.2.3.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2.2.3">′</ci></apply><ci id="A1.SS2.3.p2.6.m6.1.1.2.3.cmml" xref="A1.SS2.3.p2.6.m6.1.1.2.3">𝑗</ci></apply><apply id="A1.SS2.3.p2.6.m6.1.1.3.cmml" xref="A1.SS2.3.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.6.m6.1.1.3.1.cmml" xref="A1.SS2.3.p2.6.m6.1.1.3">subscript</csymbol><ci id="A1.SS2.3.p2.6.m6.1.1.3.2.cmml" xref="A1.SS2.3.p2.6.m6.1.1.3.2">𝑧</ci><ci id="A1.SS2.3.p2.6.m6.1.1.3.3.cmml" xref="A1.SS2.3.p2.6.m6.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.6.m6.1c">z^{\prime}_{j}=z_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.6.m6.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="j\in[d]\setminus\{i\}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.7.m7.2"><semantics id="A1.SS2.3.p2.7.m7.2a"><mrow id="A1.SS2.3.p2.7.m7.2.3" xref="A1.SS2.3.p2.7.m7.2.3.cmml"><mi id="A1.SS2.3.p2.7.m7.2.3.2" xref="A1.SS2.3.p2.7.m7.2.3.2.cmml">j</mi><mo id="A1.SS2.3.p2.7.m7.2.3.1" xref="A1.SS2.3.p2.7.m7.2.3.1.cmml">∈</mo><mrow id="A1.SS2.3.p2.7.m7.2.3.3" xref="A1.SS2.3.p2.7.m7.2.3.3.cmml"><mrow id="A1.SS2.3.p2.7.m7.2.3.3.2.2" xref="A1.SS2.3.p2.7.m7.2.3.3.2.1.cmml"><mo id="A1.SS2.3.p2.7.m7.2.3.3.2.2.1" stretchy="false" xref="A1.SS2.3.p2.7.m7.2.3.3.2.1.1.cmml">[</mo><mi id="A1.SS2.3.p2.7.m7.1.1" xref="A1.SS2.3.p2.7.m7.1.1.cmml">d</mi><mo id="A1.SS2.3.p2.7.m7.2.3.3.2.2.2" stretchy="false" xref="A1.SS2.3.p2.7.m7.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="A1.SS2.3.p2.7.m7.2.3.3.1" xref="A1.SS2.3.p2.7.m7.2.3.3.1.cmml">∖</mo><mrow id="A1.SS2.3.p2.7.m7.2.3.3.3.2" xref="A1.SS2.3.p2.7.m7.2.3.3.3.1.cmml"><mo id="A1.SS2.3.p2.7.m7.2.3.3.3.2.1" stretchy="false" xref="A1.SS2.3.p2.7.m7.2.3.3.3.1.cmml">{</mo><mi id="A1.SS2.3.p2.7.m7.2.2" xref="A1.SS2.3.p2.7.m7.2.2.cmml">i</mi><mo id="A1.SS2.3.p2.7.m7.2.3.3.3.2.2" stretchy="false" xref="A1.SS2.3.p2.7.m7.2.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.7.m7.2b"><apply id="A1.SS2.3.p2.7.m7.2.3.cmml" xref="A1.SS2.3.p2.7.m7.2.3"><in id="A1.SS2.3.p2.7.m7.2.3.1.cmml" xref="A1.SS2.3.p2.7.m7.2.3.1"></in><ci id="A1.SS2.3.p2.7.m7.2.3.2.cmml" xref="A1.SS2.3.p2.7.m7.2.3.2">𝑗</ci><apply id="A1.SS2.3.p2.7.m7.2.3.3.cmml" xref="A1.SS2.3.p2.7.m7.2.3.3"><setdiff id="A1.SS2.3.p2.7.m7.2.3.3.1.cmml" xref="A1.SS2.3.p2.7.m7.2.3.3.1"></setdiff><apply id="A1.SS2.3.p2.7.m7.2.3.3.2.1.cmml" xref="A1.SS2.3.p2.7.m7.2.3.3.2.2"><csymbol cd="latexml" id="A1.SS2.3.p2.7.m7.2.3.3.2.1.1.cmml" xref="A1.SS2.3.p2.7.m7.2.3.3.2.2.1">delimited-[]</csymbol><ci id="A1.SS2.3.p2.7.m7.1.1.cmml" xref="A1.SS2.3.p2.7.m7.1.1">𝑑</ci></apply><set id="A1.SS2.3.p2.7.m7.2.3.3.3.1.cmml" xref="A1.SS2.3.p2.7.m7.2.3.3.3.2"><ci id="A1.SS2.3.p2.7.m7.2.2.cmml" xref="A1.SS2.3.p2.7.m7.2.2">𝑖</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.7.m7.2c">j\in[d]\setminus\{i\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.7.m7.2d">italic_j ∈ [ italic_d ] ∖ { italic_i }</annotation></semantics></math>. We want to prove <math alttext="z^{\prime}\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.8.m8.2"><semantics id="A1.SS2.3.p2.8.m8.2a"><mrow id="A1.SS2.3.p2.8.m8.2.3" xref="A1.SS2.3.p2.8.m8.2.3.cmml"><msup id="A1.SS2.3.p2.8.m8.2.3.2" xref="A1.SS2.3.p2.8.m8.2.3.2.cmml"><mi id="A1.SS2.3.p2.8.m8.2.3.2.2" xref="A1.SS2.3.p2.8.m8.2.3.2.2.cmml">z</mi><mo id="A1.SS2.3.p2.8.m8.2.3.2.3" xref="A1.SS2.3.p2.8.m8.2.3.2.3.cmml">′</mo></msup><mo id="A1.SS2.3.p2.8.m8.2.3.1" xref="A1.SS2.3.p2.8.m8.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.3.p2.8.m8.2.3.3" xref="A1.SS2.3.p2.8.m8.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.3.p2.8.m8.2.3.3.2.2" xref="A1.SS2.3.p2.8.m8.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.3.p2.8.m8.2.2.2.4" xref="A1.SS2.3.p2.8.m8.2.2.2.3.cmml"><mi id="A1.SS2.3.p2.8.m8.1.1.1.1" xref="A1.SS2.3.p2.8.m8.1.1.1.1.cmml">x</mi><mo id="A1.SS2.3.p2.8.m8.2.2.2.4.1" xref="A1.SS2.3.p2.8.m8.2.2.2.3.cmml">,</mo><mi id="A1.SS2.3.p2.8.m8.2.2.2.2" xref="A1.SS2.3.p2.8.m8.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.3.p2.8.m8.2.3.3.2.3" xref="A1.SS2.3.p2.8.m8.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.8.m8.2b"><apply id="A1.SS2.3.p2.8.m8.2.3.cmml" xref="A1.SS2.3.p2.8.m8.2.3"><notin id="A1.SS2.3.p2.8.m8.2.3.1.cmml" xref="A1.SS2.3.p2.8.m8.2.3.1"></notin><apply id="A1.SS2.3.p2.8.m8.2.3.2.cmml" xref="A1.SS2.3.p2.8.m8.2.3.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.8.m8.2.3.2.1.cmml" xref="A1.SS2.3.p2.8.m8.2.3.2">superscript</csymbol><ci id="A1.SS2.3.p2.8.m8.2.3.2.2.cmml" xref="A1.SS2.3.p2.8.m8.2.3.2.2">𝑧</ci><ci id="A1.SS2.3.p2.8.m8.2.3.2.3.cmml" xref="A1.SS2.3.p2.8.m8.2.3.2.3">′</ci></apply><apply id="A1.SS2.3.p2.8.m8.2.3.3.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.8.m8.2.3.3.1.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3">subscript</csymbol><apply id="A1.SS2.3.p2.8.m8.2.3.3.2.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.8.m8.2.3.3.2.1.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3">superscript</csymbol><ci id="A1.SS2.3.p2.8.m8.2.3.3.2.2.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.3.p2.8.m8.2.3.3.2.3.cmml" xref="A1.SS2.3.p2.8.m8.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.3.p2.8.m8.2.2.2.3.cmml" xref="A1.SS2.3.p2.8.m8.2.2.2.4"><ci id="A1.SS2.3.p2.8.m8.1.1.1.1.cmml" xref="A1.SS2.3.p2.8.m8.1.1.1.1">𝑥</ci><ci id="A1.SS2.3.p2.8.m8.2.2.2.2.cmml" xref="A1.SS2.3.p2.8.m8.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.8.m8.2c">z^{\prime}\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.8.m8.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. By <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.9.m9.2"><semantics id="A1.SS2.3.p2.9.m9.2a"><mrow id="A1.SS2.3.p2.9.m9.2.3" xref="A1.SS2.3.p2.9.m9.2.3.cmml"><mi id="A1.SS2.3.p2.9.m9.2.3.2" xref="A1.SS2.3.p2.9.m9.2.3.2.cmml">z</mi><mo id="A1.SS2.3.p2.9.m9.2.3.1" xref="A1.SS2.3.p2.9.m9.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.3.p2.9.m9.2.3.3" xref="A1.SS2.3.p2.9.m9.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.3.p2.9.m9.2.3.3.2.2" xref="A1.SS2.3.p2.9.m9.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.3.p2.9.m9.2.2.2.4" xref="A1.SS2.3.p2.9.m9.2.2.2.3.cmml"><mi id="A1.SS2.3.p2.9.m9.1.1.1.1" xref="A1.SS2.3.p2.9.m9.1.1.1.1.cmml">x</mi><mo id="A1.SS2.3.p2.9.m9.2.2.2.4.1" xref="A1.SS2.3.p2.9.m9.2.2.2.3.cmml">,</mo><mi id="A1.SS2.3.p2.9.m9.2.2.2.2" xref="A1.SS2.3.p2.9.m9.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.3.p2.9.m9.2.3.3.2.3" xref="A1.SS2.3.p2.9.m9.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.9.m9.2b"><apply id="A1.SS2.3.p2.9.m9.2.3.cmml" xref="A1.SS2.3.p2.9.m9.2.3"><notin id="A1.SS2.3.p2.9.m9.2.3.1.cmml" xref="A1.SS2.3.p2.9.m9.2.3.1"></notin><ci id="A1.SS2.3.p2.9.m9.2.3.2.cmml" xref="A1.SS2.3.p2.9.m9.2.3.2">𝑧</ci><apply id="A1.SS2.3.p2.9.m9.2.3.3.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.9.m9.2.3.3.1.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3">subscript</csymbol><apply id="A1.SS2.3.p2.9.m9.2.3.3.2.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.9.m9.2.3.3.2.1.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3">superscript</csymbol><ci id="A1.SS2.3.p2.9.m9.2.3.3.2.2.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.3.p2.9.m9.2.3.3.2.3.cmml" xref="A1.SS2.3.p2.9.m9.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.3.p2.9.m9.2.2.2.3.cmml" xref="A1.SS2.3.p2.9.m9.2.2.2.4"><ci id="A1.SS2.3.p2.9.m9.1.1.1.1.cmml" xref="A1.SS2.3.p2.9.m9.1.1.1.1">𝑥</ci><ci id="A1.SS2.3.p2.9.m9.2.2.2.2.cmml" xref="A1.SS2.3.p2.9.m9.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.9.m9.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.9.m9.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>, there must exist <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS2.3.p2.10.m10.1"><semantics id="A1.SS2.3.p2.10.m10.1a"><mrow id="A1.SS2.3.p2.10.m10.1.1" xref="A1.SS2.3.p2.10.m10.1.1.cmml"><mi id="A1.SS2.3.p2.10.m10.1.1.2" xref="A1.SS2.3.p2.10.m10.1.1.2.cmml">ε</mi><mo id="A1.SS2.3.p2.10.m10.1.1.1" xref="A1.SS2.3.p2.10.m10.1.1.1.cmml">></mo><mn id="A1.SS2.3.p2.10.m10.1.1.3" xref="A1.SS2.3.p2.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.10.m10.1b"><apply id="A1.SS2.3.p2.10.m10.1.1.cmml" xref="A1.SS2.3.p2.10.m10.1.1"><gt id="A1.SS2.3.p2.10.m10.1.1.1.cmml" xref="A1.SS2.3.p2.10.m10.1.1.1"></gt><ci id="A1.SS2.3.p2.10.m10.1.1.2.cmml" xref="A1.SS2.3.p2.10.m10.1.1.2">𝜀</ci><cn id="A1.SS2.3.p2.10.m10.1.1.3.cmml" type="integer" xref="A1.SS2.3.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.10.m10.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.10.m10.1d">italic_ε > 0</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lVert x-z\rVert^{p}_{p}>\lVert x-\varepsilon v-z\rVert^{p}_{p}." class="ltx_Math" display="block" id="A1.Ex15.m1.1"><semantics id="A1.Ex15.m1.1a"><mrow id="A1.Ex15.m1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.cmml"><mrow id="A1.Ex15.m1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.cmml"><msubsup id="A1.Ex15.m1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.cmml"><mrow id="A1.Ex15.m1.1.1.1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2" rspace="0em" xref="A1.Ex15.m1.1.1.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.2" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3" lspace="0em" xref="A1.Ex15.m1.1.1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex15.m1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.3.cmml">p</mi><mi id="A1.Ex15.m1.1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.Ex15.m1.1.1.1.1.3" rspace="0.1389em" xref="A1.Ex15.m1.1.1.1.1.3.cmml">></mo><msubsup id="A1.Ex15.m1.1.1.1.1.2" xref="A1.Ex15.m1.1.1.1.1.2.cmml"><mrow id="A1.Ex15.m1.1.1.1.1.2.1.1.1" xref="A1.Ex15.m1.1.1.1.1.2.1.1.2.cmml"><mo fence="true" id="A1.Ex15.m1.1.1.1.1.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex15.m1.1.1.1.1.2.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.cmml"><mi id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.2" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.cmml"><mi id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.2" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.1" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.3" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.1a" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.4" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.Ex15.m1.1.1.1.1.2.1.1.1.3" lspace="0em" rspace="0em" xref="A1.Ex15.m1.1.1.1.1.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex15.m1.1.1.1.1.2.3" xref="A1.Ex15.m1.1.1.1.1.2.3.cmml">p</mi><mi id="A1.Ex15.m1.1.1.1.1.2.1.3" xref="A1.Ex15.m1.1.1.1.1.2.1.3.cmml">p</mi></msubsup></mrow><mo id="A1.Ex15.m1.1.1.1.2" lspace="0em" xref="A1.Ex15.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex15.m1.1b"><apply id="A1.Ex15.m1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1"><gt id="A1.Ex15.m1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.3"></gt><apply id="A1.Ex15.m1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1">subscript</csymbol><apply id="A1.Ex15.m1.1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex15.m1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.Ex15.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1"><minus id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.1"></minus><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.Ex15.m1.1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.3">𝑝</ci></apply><ci id="A1.Ex15.m1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3">𝑝</ci></apply><apply id="A1.Ex15.m1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.2.2.cmml" xref="A1.Ex15.m1.1.1.1.1.2">subscript</csymbol><apply 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xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.4.cmml" xref="A1.Ex15.m1.1.1.1.1.2.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.Ex15.m1.1.1.1.1.2.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.2.1.3">𝑝</ci></apply><ci id="A1.Ex15.m1.1.1.1.1.2.3.cmml" xref="A1.Ex15.m1.1.1.1.1.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex15.m1.1c">\lVert x-z\rVert^{p}_{p}>\lVert x-\varepsilon v-z\rVert^{p}_{p}.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex15.m1.1d">∥ italic_x - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.3.p2.18">If we can prove</p> <table 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xref="A1.E2.m1.1.1.1.1.4.2.1.1.1.4.3">𝑖</ci></apply></apply></apply><ci id="A1.E2.m1.1.1.1.1.4.2.3.cmml" xref="A1.E2.m1.1.1.1.1.4.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E2.m1.1c">|x_{i}-z_{i}|^{p}-|x_{i}-\varepsilon v_{i}-z_{i}|^{p}\leq|x_{i}-z^{\prime}_{i}% |^{p}-|x_{i}-\varepsilon v_{i}-z^{\prime}_{i}|^{p},</annotation><annotation encoding="application/x-llamapun" id="A1.E2.m1.1d">| italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ≤ | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.3.p2.19">we get</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="0<\lVert x-z\rVert^{p}_{p}-\lVert x-\varepsilon v-z\rVert^{p}_{p}\leq\lVert x-% 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xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.Ex16.m1.1.1.1.1.1.1.1.3" lspace="0em" rspace="0em" xref="A1.Ex16.m1.1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex16.m1.1.1.1.1.3" xref="A1.Ex16.m1.1.1.1.1.3.cmml">p</mi><mi id="A1.Ex16.m1.1.1.1.1.1.3" xref="A1.Ex16.m1.1.1.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.Ex16.m1.2.2.2.3" xref="A1.Ex16.m1.2.2.2.3.cmml">−</mo><msubsup id="A1.Ex16.m1.2.2.2.2" xref="A1.Ex16.m1.2.2.2.2.cmml"><mrow id="A1.Ex16.m1.2.2.2.2.1.1.1" xref="A1.Ex16.m1.2.2.2.2.1.1.2.cmml"><mo fence="true" id="A1.Ex16.m1.2.2.2.2.1.1.1.2" lspace="0em" rspace="0em" xref="A1.Ex16.m1.2.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex16.m1.2.2.2.2.1.1.1.1" xref="A1.Ex16.m1.2.2.2.2.1.1.1.1.cmml"><mi id="A1.Ex16.m1.2.2.2.2.1.1.1.1.2" xref="A1.Ex16.m1.2.2.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex16.m1.2.2.2.2.1.1.1.1.1" xref="A1.Ex16.m1.2.2.2.2.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex16.m1.2.2.2.2.1.1.1.1.3" xref="A1.Ex16.m1.2.2.2.2.1.1.1.1.3.cmml"><mi 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xref="A1.Ex16.m1.4.4.4.2.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex16.m1.4c">0<\lVert x-z\rVert^{p}_{p}-\lVert x-\varepsilon v-z\rVert^{p}_{p}\leq\lVert x-% z^{\prime}\rVert^{p}_{p}-\lVert x-\varepsilon v-z^{\prime}\rVert^{p}_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex16.m1.4d">0 < ∥ italic_x - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ ∥ italic_x - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_x - italic_ε italic_v - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.3.p2.11">and thus <math alttext="z^{\prime}\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.3.p2.11.m1.2"><semantics id="A1.SS2.3.p2.11.m1.2a"><mrow id="A1.SS2.3.p2.11.m1.2.3" xref="A1.SS2.3.p2.11.m1.2.3.cmml"><msup id="A1.SS2.3.p2.11.m1.2.3.2" xref="A1.SS2.3.p2.11.m1.2.3.2.cmml"><mi id="A1.SS2.3.p2.11.m1.2.3.2.2" xref="A1.SS2.3.p2.11.m1.2.3.2.2.cmml">z</mi><mo id="A1.SS2.3.p2.11.m1.2.3.2.3" xref="A1.SS2.3.p2.11.m1.2.3.2.3.cmml">′</mo></msup><mo id="A1.SS2.3.p2.11.m1.2.3.1" xref="A1.SS2.3.p2.11.m1.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.3.p2.11.m1.2.3.3" xref="A1.SS2.3.p2.11.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.3.p2.11.m1.2.3.3.2.2" xref="A1.SS2.3.p2.11.m1.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.3.p2.11.m1.2.2.2.4" xref="A1.SS2.3.p2.11.m1.2.2.2.3.cmml"><mi id="A1.SS2.3.p2.11.m1.1.1.1.1" xref="A1.SS2.3.p2.11.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.3.p2.11.m1.2.2.2.4.1" xref="A1.SS2.3.p2.11.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.3.p2.11.m1.2.2.2.2" xref="A1.SS2.3.p2.11.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.3.p2.11.m1.2.3.3.2.3" xref="A1.SS2.3.p2.11.m1.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.11.m1.2b"><apply id="A1.SS2.3.p2.11.m1.2.3.cmml" xref="A1.SS2.3.p2.11.m1.2.3"><notin id="A1.SS2.3.p2.11.m1.2.3.1.cmml" xref="A1.SS2.3.p2.11.m1.2.3.1"></notin><apply id="A1.SS2.3.p2.11.m1.2.3.2.cmml" xref="A1.SS2.3.p2.11.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS2.3.p2.11.m1.2.3.2.1.cmml" xref="A1.SS2.3.p2.11.m1.2.3.2">superscript</csymbol><ci id="A1.SS2.3.p2.11.m1.2.3.2.2.cmml" xref="A1.SS2.3.p2.11.m1.2.3.2.2">𝑧</ci><ci id="A1.SS2.3.p2.11.m1.2.3.2.3.cmml" xref="A1.SS2.3.p2.11.m1.2.3.2.3">′</ci></apply><apply id="A1.SS2.3.p2.11.m1.2.3.3.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.11.m1.2.3.3.1.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3">subscript</csymbol><apply id="A1.SS2.3.p2.11.m1.2.3.3.2.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.11.m1.2.3.3.2.1.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3">superscript</csymbol><ci id="A1.SS2.3.p2.11.m1.2.3.3.2.2.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.3.p2.11.m1.2.3.3.2.3.cmml" xref="A1.SS2.3.p2.11.m1.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.3.p2.11.m1.2.2.2.3.cmml" xref="A1.SS2.3.p2.11.m1.2.2.2.4"><ci id="A1.SS2.3.p2.11.m1.1.1.1.1.cmml" xref="A1.SS2.3.p2.11.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.3.p2.11.m1.2.2.2.2.cmml" xref="A1.SS2.3.p2.11.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.11.m1.2c">z^{\prime}\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.11.m1.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. The inequality <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.E2" title="In Proof. ‣ A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">A.2</span></a> follows from the observation that</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|x_{i}-z^{\prime}_{i}|^{p}-|x_{i}-\varepsilon v_{i}-z^{\prime}_{i}|^{p}=|x_{i}% +\alpha v_{i}-z_{i}|^{p}-|x_{i}+(\alpha-\varepsilon)v_{i}-z_{i}|^{p}" class="ltx_Math" display="block" id="A1.Ex17.m1.4"><semantics id="A1.Ex17.m1.4a"><mrow id="A1.Ex17.m1.4.4" xref="A1.Ex17.m1.4.4.cmml"><mrow id="A1.Ex17.m1.2.2.2" xref="A1.Ex17.m1.2.2.2.cmml"><msup id="A1.Ex17.m1.1.1.1.1" 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v_{i}-z_{i}|^{p}-|x_{i}+(\alpha-\varepsilon)v_{i}-z_{i}|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex17.m1.4d">| italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_ε italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT = | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_α italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ( italic_α - italic_ε ) italic_v start_POSTSUBSCRIPT italic_i 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id="A1.SS2.3.p2.12.m1.3.3.3.cmml" xref="A1.SS2.3.p2.12.m1.3.3.3"><times id="A1.SS2.3.p2.12.m1.3.3.3.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.3.1"></times><ci id="A1.SS2.3.p2.12.m1.3.3.3.2.cmml" xref="A1.SS2.3.p2.12.m1.3.3.3.2">𝑓</ci><ci id="A1.SS2.3.p2.12.m1.1.1.cmml" xref="A1.SS2.3.p2.12.m1.1.1">𝛽</ci></apply><apply id="A1.SS2.3.p2.12.m1.3.3.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1"><minus id="A1.SS2.3.p2.12.m1.3.3.1.2.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.2"></minus><apply id="A1.SS2.3.p2.12.m1.3.3.1.3.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.3"><csymbol cd="ambiguous" id="A1.SS2.3.p2.12.m1.3.3.1.3.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.3">superscript</csymbol><apply id="A1.SS2.3.p2.12.m1.3.3.1.3.2.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.3.2.2"><abs id="A1.SS2.3.p2.12.m1.3.3.1.3.2.1.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.3.2.2.1"></abs><ci id="A1.SS2.3.p2.12.m1.2.2.cmml" xref="A1.SS2.3.p2.12.m1.2.2">𝛽</ci></apply><ci id="A1.SS2.3.p2.12.m1.3.3.1.3.3.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.3.3">𝑝</ci></apply><apply id="A1.SS2.3.p2.12.m1.3.3.1.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.3.p2.12.m1.3.3.1.1.2.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1">superscript</csymbol><apply id="A1.SS2.3.p2.12.m1.3.3.1.1.1.2.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1"><abs id="A1.SS2.3.p2.12.m1.3.3.1.1.1.2.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.2"></abs><apply id="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1"><minus id="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.1.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.1"></minus><ci id="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.2.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.2">𝛽</ci><ci id="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.3.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.1.1.1.3">𝛾</ci></apply></apply><ci id="A1.SS2.3.p2.12.m1.3.3.1.1.3.cmml" xref="A1.SS2.3.p2.12.m1.3.3.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.12.m1.3c">f(\beta)\coloneqq|\beta|^{p}-|\beta-\gamma|^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.12.m1.3d">italic_f ( italic_β ) ≔ | italic_β | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - | italic_β - italic_γ | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is for all <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="A1.SS2.3.p2.13.m2.2"><semantics id="A1.SS2.3.p2.13.m2.2a"><mrow id="A1.SS2.3.p2.13.m2.2.3" xref="A1.SS2.3.p2.13.m2.2.3.cmml"><mi id="A1.SS2.3.p2.13.m2.2.3.2" xref="A1.SS2.3.p2.13.m2.2.3.2.cmml">p</mi><mo id="A1.SS2.3.p2.13.m2.2.3.1" xref="A1.SS2.3.p2.13.m2.2.3.1.cmml">∈</mo><mrow id="A1.SS2.3.p2.13.m2.2.3.3.2" xref="A1.SS2.3.p2.13.m2.2.3.3.1.cmml"><mo id="A1.SS2.3.p2.13.m2.2.3.3.2.1" stretchy="false" xref="A1.SS2.3.p2.13.m2.2.3.3.1.cmml">[</mo><mn id="A1.SS2.3.p2.13.m2.1.1" xref="A1.SS2.3.p2.13.m2.1.1.cmml">1</mn><mo id="A1.SS2.3.p2.13.m2.2.3.3.2.2" xref="A1.SS2.3.p2.13.m2.2.3.3.1.cmml">,</mo><mi id="A1.SS2.3.p2.13.m2.2.2" mathvariant="normal" xref="A1.SS2.3.p2.13.m2.2.2.cmml">∞</mi><mo id="A1.SS2.3.p2.13.m2.2.3.3.2.3" stretchy="false" xref="A1.SS2.3.p2.13.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.13.m2.2b"><apply id="A1.SS2.3.p2.13.m2.2.3.cmml" xref="A1.SS2.3.p2.13.m2.2.3"><in id="A1.SS2.3.p2.13.m2.2.3.1.cmml" xref="A1.SS2.3.p2.13.m2.2.3.1"></in><ci id="A1.SS2.3.p2.13.m2.2.3.2.cmml" xref="A1.SS2.3.p2.13.m2.2.3.2">𝑝</ci><interval closure="closed-open" id="A1.SS2.3.p2.13.m2.2.3.3.1.cmml" xref="A1.SS2.3.p2.13.m2.2.3.3.2"><cn id="A1.SS2.3.p2.13.m2.1.1.cmml" type="integer" xref="A1.SS2.3.p2.13.m2.1.1">1</cn><infinity id="A1.SS2.3.p2.13.m2.2.2.cmml" xref="A1.SS2.3.p2.13.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.13.m2.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.13.m2.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math> monotonically non-increasing for <math alttext="\gamma\leq 0" class="ltx_Math" display="inline" id="A1.SS2.3.p2.14.m3.1"><semantics id="A1.SS2.3.p2.14.m3.1a"><mrow id="A1.SS2.3.p2.14.m3.1.1" xref="A1.SS2.3.p2.14.m3.1.1.cmml"><mi id="A1.SS2.3.p2.14.m3.1.1.2" xref="A1.SS2.3.p2.14.m3.1.1.2.cmml">γ</mi><mo id="A1.SS2.3.p2.14.m3.1.1.1" xref="A1.SS2.3.p2.14.m3.1.1.1.cmml">≤</mo><mn id="A1.SS2.3.p2.14.m3.1.1.3" xref="A1.SS2.3.p2.14.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.14.m3.1b"><apply id="A1.SS2.3.p2.14.m3.1.1.cmml" xref="A1.SS2.3.p2.14.m3.1.1"><leq id="A1.SS2.3.p2.14.m3.1.1.1.cmml" xref="A1.SS2.3.p2.14.m3.1.1.1"></leq><ci id="A1.SS2.3.p2.14.m3.1.1.2.cmml" xref="A1.SS2.3.p2.14.m3.1.1.2">𝛾</ci><cn id="A1.SS2.3.p2.14.m3.1.1.3.cmml" type="integer" xref="A1.SS2.3.p2.14.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.14.m3.1c">\gamma\leq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.14.m3.1d">italic_γ ≤ 0</annotation></semantics></math> and monotonically non-decreasing for <math alttext="\gamma\geq 0" class="ltx_Math" display="inline" id="A1.SS2.3.p2.15.m4.1"><semantics id="A1.SS2.3.p2.15.m4.1a"><mrow id="A1.SS2.3.p2.15.m4.1.1" xref="A1.SS2.3.p2.15.m4.1.1.cmml"><mi id="A1.SS2.3.p2.15.m4.1.1.2" xref="A1.SS2.3.p2.15.m4.1.1.2.cmml">γ</mi><mo id="A1.SS2.3.p2.15.m4.1.1.1" xref="A1.SS2.3.p2.15.m4.1.1.1.cmml">≥</mo><mn id="A1.SS2.3.p2.15.m4.1.1.3" xref="A1.SS2.3.p2.15.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.15.m4.1b"><apply id="A1.SS2.3.p2.15.m4.1.1.cmml" xref="A1.SS2.3.p2.15.m4.1.1"><geq id="A1.SS2.3.p2.15.m4.1.1.1.cmml" xref="A1.SS2.3.p2.15.m4.1.1.1"></geq><ci id="A1.SS2.3.p2.15.m4.1.1.2.cmml" xref="A1.SS2.3.p2.15.m4.1.1.2">𝛾</ci><cn id="A1.SS2.3.p2.15.m4.1.1.3.cmml" type="integer" xref="A1.SS2.3.p2.15.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.15.m4.1c">\gamma\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.15.m4.1d">italic_γ ≥ 0</annotation></semantics></math>. The proof for <math alttext="p=\infty" class="ltx_Math" display="inline" id="A1.SS2.3.p2.16.m5.1"><semantics id="A1.SS2.3.p2.16.m5.1a"><mrow id="A1.SS2.3.p2.16.m5.1.1" xref="A1.SS2.3.p2.16.m5.1.1.cmml"><mi id="A1.SS2.3.p2.16.m5.1.1.2" xref="A1.SS2.3.p2.16.m5.1.1.2.cmml">p</mi><mo id="A1.SS2.3.p2.16.m5.1.1.1" xref="A1.SS2.3.p2.16.m5.1.1.1.cmml">=</mo><mi id="A1.SS2.3.p2.16.m5.1.1.3" mathvariant="normal" xref="A1.SS2.3.p2.16.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.16.m5.1b"><apply id="A1.SS2.3.p2.16.m5.1.1.cmml" xref="A1.SS2.3.p2.16.m5.1.1"><eq id="A1.SS2.3.p2.16.m5.1.1.1.cmml" xref="A1.SS2.3.p2.16.m5.1.1.1"></eq><ci id="A1.SS2.3.p2.16.m5.1.1.2.cmml" xref="A1.SS2.3.p2.16.m5.1.1.2">𝑝</ci><infinity id="A1.SS2.3.p2.16.m5.1.1.3.cmml" xref="A1.SS2.3.p2.16.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.16.m5.1c">p=\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.16.m5.1d">italic_p = ∞</annotation></semantics></math> again follows from <math alttext="p=1" class="ltx_Math" display="inline" id="A1.SS2.3.p2.17.m6.1"><semantics id="A1.SS2.3.p2.17.m6.1a"><mrow id="A1.SS2.3.p2.17.m6.1.1" xref="A1.SS2.3.p2.17.m6.1.1.cmml"><mi id="A1.SS2.3.p2.17.m6.1.1.2" xref="A1.SS2.3.p2.17.m6.1.1.2.cmml">p</mi><mo id="A1.SS2.3.p2.17.m6.1.1.1" xref="A1.SS2.3.p2.17.m6.1.1.1.cmml">=</mo><mn id="A1.SS2.3.p2.17.m6.1.1.3" xref="A1.SS2.3.p2.17.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p2.17.m6.1b"><apply id="A1.SS2.3.p2.17.m6.1.1.cmml" xref="A1.SS2.3.p2.17.m6.1.1"><eq id="A1.SS2.3.p2.17.m6.1.1.1.cmml" xref="A1.SS2.3.p2.17.m6.1.1.1"></eq><ci id="A1.SS2.3.p2.17.m6.1.1.2.cmml" xref="A1.SS2.3.p2.17.m6.1.1.2">𝑝</ci><cn id="A1.SS2.3.p2.17.m6.1.1.3.cmml" type="integer" xref="A1.SS2.3.p2.17.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p2.17.m6.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.3.p2.17.m6.1d">italic_p = 1</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS2.p3"> <p class="ltx_p" id="A1.SS2.p3.1">This now lets us conclude that the boundary of the halfspace has measure <math alttext="0" class="ltx_Math" display="inline" id="A1.SS2.p3.1.m1.1"><semantics id="A1.SS2.p3.1.m1.1a"><mn id="A1.SS2.p3.1.m1.1.1" xref="A1.SS2.p3.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS2.p3.1.m1.1b"><cn id="A1.SS2.p3.1.m1.1.1.cmml" type="integer" xref="A1.SS2.p3.1.m1.1.1">0</cn></annotation-xml></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="A1.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem5.2.1.1">Corollary A.5</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem5.3.2"> </span>(Boundary has Measure <math alttext="0" class="ltx_Math" display="inline" id="A1.Thmtheorem5.1.m1.1"><semantics id="A1.Thmtheorem5.1.m1.1b"><mn id="A1.Thmtheorem5.1.m1.1.1" xref="A1.Thmtheorem5.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.1.m1.1c"><cn id="A1.Thmtheorem5.1.m1.1.1.cmml" type="integer" xref="A1.Thmtheorem5.1.m1.1.1">0</cn></annotation-xml></semantics></math>)<span class="ltx_text ltx_font_bold" id="A1.Thmtheorem5.4.3">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem5.p1"> <p class="ltx_p" id="A1.Thmtheorem5.p1.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem5.p1.7.7">Let <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.1.1.m1.3"><semantics id="A1.Thmtheorem5.p1.1.1.m1.3a"><mrow id="A1.Thmtheorem5.p1.1.1.m1.3.4" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.cmml"><mi id="A1.Thmtheorem5.p1.1.1.m1.3.4.2" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.2.cmml">p</mi><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.1" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="A1.Thmtheorem5.p1.1.1.m1.3.4.3" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.cmml"><mrow id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.2" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml"><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.1" stretchy="false" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">[</mo><mn id="A1.Thmtheorem5.p1.1.1.m1.1.1" xref="A1.Thmtheorem5.p1.1.1.m1.1.1.cmml">1</mn><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.2" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">,</mo><mi id="A1.Thmtheorem5.p1.1.1.m1.2.2" mathvariant="normal" xref="A1.Thmtheorem5.p1.1.1.m1.2.2.cmml">∞</mi><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.2.3" stretchy="false" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.1" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml">∪</mo><mrow id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.2" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml"><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.1" stretchy="false" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">{</mo><mi id="A1.Thmtheorem5.p1.1.1.m1.3.3" mathvariant="normal" xref="A1.Thmtheorem5.p1.1.1.m1.3.3.cmml">∞</mi><mo id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.2.2" stretchy="false" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.1.1.m1.3b"><apply id="A1.Thmtheorem5.p1.1.1.m1.3.4.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4"><in id="A1.Thmtheorem5.p1.1.1.m1.3.4.1.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.1"></in><ci id="A1.Thmtheorem5.p1.1.1.m1.3.4.2.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.2">𝑝</ci><apply id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3"><union id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.1.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.1"></union><interval closure="closed-open" id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.1.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.2.2"><cn id="A1.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="A1.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="A1.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval><set id="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.1.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.4.3.3.2"><infinity id="A1.Thmtheorem5.p1.1.1.m1.3.3.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.1.1.m1.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.1.1.m1.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. The boundary <math alttext="\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.2.2.m2.2"><semantics id="A1.Thmtheorem5.p1.2.2.m2.2a"><mrow id="A1.Thmtheorem5.p1.2.2.m2.2.3" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.cmml"><mo id="A1.Thmtheorem5.p1.2.2.m2.2.3.1" rspace="0em" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.1.cmml">∂</mo><msubsup id="A1.Thmtheorem5.p1.2.2.m2.2.3.2" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.2" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem5.p1.2.2.m2.2.2.2.4" xref="A1.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml"><mi id="A1.Thmtheorem5.p1.2.2.m2.1.1.1.1" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem5.p1.2.2.m2.2.2.2.4.1" xref="A1.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem5.p1.2.2.m2.2.2.2.2" xref="A1.Thmtheorem5.p1.2.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.3" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.2.2.m2.2b"><apply id="A1.Thmtheorem5.p1.2.2.m2.2.3.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3"><partialdiff id="A1.Thmtheorem5.p1.2.2.m2.2.3.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.1"></partialdiff><apply id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2">subscript</csymbol><apply id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2">superscript</csymbol><ci id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.2">ℋ</ci><ci id="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.3.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.3.2.2.3">𝑝</ci></apply><list id="A1.Thmtheorem5.p1.2.2.m2.2.2.2.3.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.2.2.4"><ci id="A1.Thmtheorem5.p1.2.2.m2.1.1.1.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem5.p1.2.2.m2.2.2.2.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.2.2.m2.2c">\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.2.2.m2.2d">∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> of the <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.3.3.m3.1"><semantics id="A1.Thmtheorem5.p1.3.3.m3.1a"><msub id="A1.Thmtheorem5.p1.3.3.m3.1.1" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="A1.Thmtheorem5.p1.3.3.m3.1.1.2" mathvariant="normal" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">ℓ</mi><mi id="A1.Thmtheorem5.p1.3.3.m3.1.1.3" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.3.3.m3.1b"><apply id="A1.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem5.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem5.p1.3.3.m3.1.1">subscript</csymbol><ci id="A1.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.2">ℓ</ci><ci id="A1.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.3.3.m3.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.3.3.m3.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-halfspace <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.4.4.m4.2"><semantics id="A1.Thmtheorem5.p1.4.4.m4.2a"><msubsup id="A1.Thmtheorem5.p1.4.4.m4.2.3" xref="A1.Thmtheorem5.p1.4.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.2" xref="A1.Thmtheorem5.p1.4.4.m4.2.3.2.2.cmml">ℋ</mi><mrow id="A1.Thmtheorem5.p1.4.4.m4.2.2.2.4" xref="A1.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml"><mi id="A1.Thmtheorem5.p1.4.4.m4.1.1.1.1" xref="A1.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml">x</mi><mo id="A1.Thmtheorem5.p1.4.4.m4.2.2.2.4.1" xref="A1.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml">,</mo><mi id="A1.Thmtheorem5.p1.4.4.m4.2.2.2.2" xref="A1.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.3" xref="A1.Thmtheorem5.p1.4.4.m4.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.4.4.m4.2b"><apply id="A1.Thmtheorem5.p1.4.4.m4.2.3.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.4.4.m4.2.3.1.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3">subscript</csymbol><apply id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.1.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3">superscript</csymbol><ci id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.2.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3.2.2">ℋ</ci><ci id="A1.Thmtheorem5.p1.4.4.m4.2.3.2.3.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.3.2.3">𝑝</ci></apply><list id="A1.Thmtheorem5.p1.4.4.m4.2.2.2.3.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.2.2.4"><ci id="A1.Thmtheorem5.p1.4.4.m4.1.1.1.1.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.1.1.1.1">𝑥</ci><ci id="A1.Thmtheorem5.p1.4.4.m4.2.2.2.2.cmml" xref="A1.Thmtheorem5.p1.4.4.m4.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.4.4.m4.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.4.4.m4.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> has Lebesgue measure <math alttext="0" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.5.5.m5.1"><semantics id="A1.Thmtheorem5.p1.5.5.m5.1a"><mn id="A1.Thmtheorem5.p1.5.5.m5.1.1" xref="A1.Thmtheorem5.p1.5.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.5.5.m5.1b"><cn id="A1.Thmtheorem5.p1.5.5.m5.1.1.cmml" type="integer" xref="A1.Thmtheorem5.p1.5.5.m5.1.1">0</cn></annotation-xml></semantics></math> for all <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.6.6.m6.1"><semantics id="A1.Thmtheorem5.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem5.p1.6.6.m6.1.1" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.cmml"><mi id="A1.Thmtheorem5.p1.6.6.m6.1.1.2" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.2.cmml">x</mi><mo id="A1.Thmtheorem5.p1.6.6.m6.1.1.1" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem5.p1.6.6.m6.1.1.3" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3.cmml"><mi id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.2" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3.2.cmml">ℝ</mi><mi id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.3" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.6.6.m6.1b"><apply id="A1.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1"><in id="A1.Thmtheorem5.p1.6.6.m6.1.1.1.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.1"></in><ci id="A1.Thmtheorem5.p1.6.6.m6.1.1.2.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.2">𝑥</ci><apply id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.1.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.2.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3.2">ℝ</ci><ci id="A1.Thmtheorem5.p1.6.6.m6.1.1.3.3.cmml" xref="A1.Thmtheorem5.p1.6.6.m6.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.6.6.m6.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.6.6.m6.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.7.7.m7.1"><semantics id="A1.Thmtheorem5.p1.7.7.m7.1a"><mrow id="A1.Thmtheorem5.p1.7.7.m7.1.1" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.cmml"><mi id="A1.Thmtheorem5.p1.7.7.m7.1.1.2" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.2.cmml">v</mi><mo id="A1.Thmtheorem5.p1.7.7.m7.1.1.1" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.1.cmml">∈</mo><msup id="A1.Thmtheorem5.p1.7.7.m7.1.1.3" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.cmml"><mi id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.2" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.2.cmml">S</mi><mrow id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.cmml"><mi id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.2" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.2.cmml">d</mi><mo id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.1" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.1.cmml">−</mo><mn id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.3" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.7.7.m7.1b"><apply id="A1.Thmtheorem5.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1"><in id="A1.Thmtheorem5.p1.7.7.m7.1.1.1.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.1"></in><ci id="A1.Thmtheorem5.p1.7.7.m7.1.1.2.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.2">𝑣</ci><apply id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.1.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.2.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.2">𝑆</ci><apply id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3"><minus id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.1.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.1"></minus><ci id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.2.cmml" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.2">𝑑</ci><cn id="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.3.cmml" type="integer" xref="A1.Thmtheorem5.p1.7.7.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.7.7.m7.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.7.7.m7.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="A1.SS2.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS2.4.p1"> <p class="ltx_p" id="A1.SS2.4.p1.10">We prove that <math alttext="\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.1.m1.2"><semantics id="A1.SS2.4.p1.1.m1.2a"><mrow id="A1.SS2.4.p1.1.m1.2.3" xref="A1.SS2.4.p1.1.m1.2.3.cmml"><mo id="A1.SS2.4.p1.1.m1.2.3.1" rspace="0em" xref="A1.SS2.4.p1.1.m1.2.3.1.cmml">∂</mo><msubsup id="A1.SS2.4.p1.1.m1.2.3.2" xref="A1.SS2.4.p1.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.4.p1.1.m1.2.3.2.2.2" xref="A1.SS2.4.p1.1.m1.2.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS2.4.p1.1.m1.2.2.2.4" xref="A1.SS2.4.p1.1.m1.2.2.2.3.cmml"><mi id="A1.SS2.4.p1.1.m1.1.1.1.1" xref="A1.SS2.4.p1.1.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.4.p1.1.m1.2.2.2.4.1" xref="A1.SS2.4.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.4.p1.1.m1.2.2.2.2" xref="A1.SS2.4.p1.1.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.4.p1.1.m1.2.3.2.2.3" xref="A1.SS2.4.p1.1.m1.2.3.2.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.1.m1.2b"><apply id="A1.SS2.4.p1.1.m1.2.3.cmml" xref="A1.SS2.4.p1.1.m1.2.3"><partialdiff id="A1.SS2.4.p1.1.m1.2.3.1.cmml" xref="A1.SS2.4.p1.1.m1.2.3.1"></partialdiff><apply id="A1.SS2.4.p1.1.m1.2.3.2.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS2.4.p1.1.m1.2.3.2.1.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2">subscript</csymbol><apply id="A1.SS2.4.p1.1.m1.2.3.2.2.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="A1.SS2.4.p1.1.m1.2.3.2.2.1.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2">superscript</csymbol><ci id="A1.SS2.4.p1.1.m1.2.3.2.2.2.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2.2.2">ℋ</ci><ci id="A1.SS2.4.p1.1.m1.2.3.2.2.3.cmml" xref="A1.SS2.4.p1.1.m1.2.3.2.2.3">𝑝</ci></apply><list id="A1.SS2.4.p1.1.m1.2.2.2.3.cmml" xref="A1.SS2.4.p1.1.m1.2.2.2.4"><ci id="A1.SS2.4.p1.1.m1.1.1.1.1.cmml" xref="A1.SS2.4.p1.1.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.4.p1.1.m1.2.2.2.2.cmml" xref="A1.SS2.4.p1.1.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.1.m1.2c">\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.1.m1.2d">∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is porous, i.e., we prove that there exists <math alttext="0<\alpha<1" class="ltx_Math" display="inline" id="A1.SS2.4.p1.2.m2.1"><semantics id="A1.SS2.4.p1.2.m2.1a"><mrow id="A1.SS2.4.p1.2.m2.1.1" xref="A1.SS2.4.p1.2.m2.1.1.cmml"><mn id="A1.SS2.4.p1.2.m2.1.1.2" xref="A1.SS2.4.p1.2.m2.1.1.2.cmml">0</mn><mo id="A1.SS2.4.p1.2.m2.1.1.3" xref="A1.SS2.4.p1.2.m2.1.1.3.cmml"><</mo><mi id="A1.SS2.4.p1.2.m2.1.1.4" xref="A1.SS2.4.p1.2.m2.1.1.4.cmml">α</mi><mo id="A1.SS2.4.p1.2.m2.1.1.5" xref="A1.SS2.4.p1.2.m2.1.1.5.cmml"><</mo><mn id="A1.SS2.4.p1.2.m2.1.1.6" xref="A1.SS2.4.p1.2.m2.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.2.m2.1b"><apply id="A1.SS2.4.p1.2.m2.1.1.cmml" xref="A1.SS2.4.p1.2.m2.1.1"><and id="A1.SS2.4.p1.2.m2.1.1a.cmml" xref="A1.SS2.4.p1.2.m2.1.1"></and><apply id="A1.SS2.4.p1.2.m2.1.1b.cmml" xref="A1.SS2.4.p1.2.m2.1.1"><lt id="A1.SS2.4.p1.2.m2.1.1.3.cmml" xref="A1.SS2.4.p1.2.m2.1.1.3"></lt><cn id="A1.SS2.4.p1.2.m2.1.1.2.cmml" type="integer" xref="A1.SS2.4.p1.2.m2.1.1.2">0</cn><ci id="A1.SS2.4.p1.2.m2.1.1.4.cmml" xref="A1.SS2.4.p1.2.m2.1.1.4">𝛼</ci></apply><apply id="A1.SS2.4.p1.2.m2.1.1c.cmml" xref="A1.SS2.4.p1.2.m2.1.1"><lt id="A1.SS2.4.p1.2.m2.1.1.5.cmml" xref="A1.SS2.4.p1.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2503.16089v1#A1.SS2.4.p1.2.m2.1.1.4.cmml" id="A1.SS2.4.p1.2.m2.1.1d.cmml" xref="A1.SS2.4.p1.2.m2.1.1"></share><cn id="A1.SS2.4.p1.2.m2.1.1.6.cmml" type="integer" xref="A1.SS2.4.p1.2.m2.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.2.m2.1c">0<\alpha<1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.2.m2.1d">0 < italic_α < 1</annotation></semantics></math> such that for every sufficiently small <math alttext="r>0" class="ltx_Math" display="inline" id="A1.SS2.4.p1.3.m3.1"><semantics id="A1.SS2.4.p1.3.m3.1a"><mrow id="A1.SS2.4.p1.3.m3.1.1" xref="A1.SS2.4.p1.3.m3.1.1.cmml"><mi id="A1.SS2.4.p1.3.m3.1.1.2" xref="A1.SS2.4.p1.3.m3.1.1.2.cmml">r</mi><mo id="A1.SS2.4.p1.3.m3.1.1.1" xref="A1.SS2.4.p1.3.m3.1.1.1.cmml">></mo><mn id="A1.SS2.4.p1.3.m3.1.1.3" xref="A1.SS2.4.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.3.m3.1b"><apply id="A1.SS2.4.p1.3.m3.1.1.cmml" xref="A1.SS2.4.p1.3.m3.1.1"><gt id="A1.SS2.4.p1.3.m3.1.1.1.cmml" xref="A1.SS2.4.p1.3.m3.1.1.1"></gt><ci id="A1.SS2.4.p1.3.m3.1.1.2.cmml" xref="A1.SS2.4.p1.3.m3.1.1.2">𝑟</ci><cn id="A1.SS2.4.p1.3.m3.1.1.3.cmml" type="integer" xref="A1.SS2.4.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.3.m3.1c">r>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.3.m3.1d">italic_r > 0</annotation></semantics></math> and for every <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.4.m4.1"><semantics id="A1.SS2.4.p1.4.m4.1a"><mrow id="A1.SS2.4.p1.4.m4.1.1" xref="A1.SS2.4.p1.4.m4.1.1.cmml"><mi id="A1.SS2.4.p1.4.m4.1.1.2" xref="A1.SS2.4.p1.4.m4.1.1.2.cmml">z</mi><mo id="A1.SS2.4.p1.4.m4.1.1.1" xref="A1.SS2.4.p1.4.m4.1.1.1.cmml">∈</mo><msup id="A1.SS2.4.p1.4.m4.1.1.3" xref="A1.SS2.4.p1.4.m4.1.1.3.cmml"><mi id="A1.SS2.4.p1.4.m4.1.1.3.2" xref="A1.SS2.4.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.4.p1.4.m4.1.1.3.3" xref="A1.SS2.4.p1.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.4.m4.1b"><apply id="A1.SS2.4.p1.4.m4.1.1.cmml" xref="A1.SS2.4.p1.4.m4.1.1"><in id="A1.SS2.4.p1.4.m4.1.1.1.cmml" xref="A1.SS2.4.p1.4.m4.1.1.1"></in><ci id="A1.SS2.4.p1.4.m4.1.1.2.cmml" xref="A1.SS2.4.p1.4.m4.1.1.2">𝑧</ci><apply id="A1.SS2.4.p1.4.m4.1.1.3.cmml" xref="A1.SS2.4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.4.m4.1.1.3.1.cmml" xref="A1.SS2.4.p1.4.m4.1.1.3">superscript</csymbol><ci id="A1.SS2.4.p1.4.m4.1.1.3.2.cmml" xref="A1.SS2.4.p1.4.m4.1.1.3.2">ℝ</ci><ci id="A1.SS2.4.p1.4.m4.1.1.3.3.cmml" xref="A1.SS2.4.p1.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.4.m4.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.4.m4.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, there is <math alttext="y\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.5.m5.1"><semantics id="A1.SS2.4.p1.5.m5.1a"><mrow id="A1.SS2.4.p1.5.m5.1.1" xref="A1.SS2.4.p1.5.m5.1.1.cmml"><mi id="A1.SS2.4.p1.5.m5.1.1.2" xref="A1.SS2.4.p1.5.m5.1.1.2.cmml">y</mi><mo id="A1.SS2.4.p1.5.m5.1.1.1" xref="A1.SS2.4.p1.5.m5.1.1.1.cmml">∈</mo><msup id="A1.SS2.4.p1.5.m5.1.1.3" xref="A1.SS2.4.p1.5.m5.1.1.3.cmml"><mi id="A1.SS2.4.p1.5.m5.1.1.3.2" xref="A1.SS2.4.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.4.p1.5.m5.1.1.3.3" xref="A1.SS2.4.p1.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.5.m5.1b"><apply id="A1.SS2.4.p1.5.m5.1.1.cmml" xref="A1.SS2.4.p1.5.m5.1.1"><in id="A1.SS2.4.p1.5.m5.1.1.1.cmml" xref="A1.SS2.4.p1.5.m5.1.1.1"></in><ci id="A1.SS2.4.p1.5.m5.1.1.2.cmml" xref="A1.SS2.4.p1.5.m5.1.1.2">𝑦</ci><apply id="A1.SS2.4.p1.5.m5.1.1.3.cmml" xref="A1.SS2.4.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.5.m5.1.1.3.1.cmml" xref="A1.SS2.4.p1.5.m5.1.1.3">superscript</csymbol><ci id="A1.SS2.4.p1.5.m5.1.1.3.2.cmml" xref="A1.SS2.4.p1.5.m5.1.1.3.2">ℝ</ci><ci id="A1.SS2.4.p1.5.m5.1.1.3.3.cmml" xref="A1.SS2.4.p1.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.5.m5.1c">y\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.5.m5.1d">italic_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="B^{p}(y,\alpha r)\subseteq B^{p}(z,r)\setminus\partial\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.6.m6.6"><semantics id="A1.SS2.4.p1.6.m6.6a"><mrow id="A1.SS2.4.p1.6.m6.6.6" xref="A1.SS2.4.p1.6.m6.6.6.cmml"><mrow id="A1.SS2.4.p1.6.m6.6.6.1" xref="A1.SS2.4.p1.6.m6.6.6.1.cmml"><msup id="A1.SS2.4.p1.6.m6.6.6.1.3" xref="A1.SS2.4.p1.6.m6.6.6.1.3.cmml"><mi id="A1.SS2.4.p1.6.m6.6.6.1.3.2" xref="A1.SS2.4.p1.6.m6.6.6.1.3.2.cmml">B</mi><mi id="A1.SS2.4.p1.6.m6.6.6.1.3.3" xref="A1.SS2.4.p1.6.m6.6.6.1.3.3.cmml">p</mi></msup><mo id="A1.SS2.4.p1.6.m6.6.6.1.2" xref="A1.SS2.4.p1.6.m6.6.6.1.2.cmml">⁢</mo><mrow id="A1.SS2.4.p1.6.m6.6.6.1.1.1" xref="A1.SS2.4.p1.6.m6.6.6.1.1.2.cmml"><mo id="A1.SS2.4.p1.6.m6.6.6.1.1.1.2" stretchy="false" xref="A1.SS2.4.p1.6.m6.6.6.1.1.2.cmml">(</mo><mi id="A1.SS2.4.p1.6.m6.3.3" xref="A1.SS2.4.p1.6.m6.3.3.cmml">y</mi><mo id="A1.SS2.4.p1.6.m6.6.6.1.1.1.3" xref="A1.SS2.4.p1.6.m6.6.6.1.1.2.cmml">,</mo><mrow id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.cmml"><mi id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.2" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.2.cmml">α</mi><mo id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.1" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.1.cmml">⁢</mo><mi id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.3" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.3.cmml">r</mi></mrow><mo id="A1.SS2.4.p1.6.m6.6.6.1.1.1.4" stretchy="false" xref="A1.SS2.4.p1.6.m6.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.4.p1.6.m6.6.6.2" xref="A1.SS2.4.p1.6.m6.6.6.2.cmml">⊆</mo><mrow id="A1.SS2.4.p1.6.m6.6.6.3" xref="A1.SS2.4.p1.6.m6.6.6.3.cmml"><mrow id="A1.SS2.4.p1.6.m6.6.6.3.2" xref="A1.SS2.4.p1.6.m6.6.6.3.2.cmml"><msup id="A1.SS2.4.p1.6.m6.6.6.3.2.2" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2.cmml"><mi id="A1.SS2.4.p1.6.m6.6.6.3.2.2.2" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2.2.cmml">B</mi><mi id="A1.SS2.4.p1.6.m6.6.6.3.2.2.3" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2.3.cmml">p</mi></msup><mo id="A1.SS2.4.p1.6.m6.6.6.3.2.1" xref="A1.SS2.4.p1.6.m6.6.6.3.2.1.cmml">⁢</mo><mrow id="A1.SS2.4.p1.6.m6.6.6.3.2.3.2" xref="A1.SS2.4.p1.6.m6.6.6.3.2.3.1.cmml"><mo id="A1.SS2.4.p1.6.m6.6.6.3.2.3.2.1" stretchy="false" xref="A1.SS2.4.p1.6.m6.6.6.3.2.3.1.cmml">(</mo><mi id="A1.SS2.4.p1.6.m6.4.4" xref="A1.SS2.4.p1.6.m6.4.4.cmml">z</mi><mo id="A1.SS2.4.p1.6.m6.6.6.3.2.3.2.2" xref="A1.SS2.4.p1.6.m6.6.6.3.2.3.1.cmml">,</mo><mi id="A1.SS2.4.p1.6.m6.5.5" xref="A1.SS2.4.p1.6.m6.5.5.cmml">r</mi><mo id="A1.SS2.4.p1.6.m6.6.6.3.2.3.2.3" stretchy="false" xref="A1.SS2.4.p1.6.m6.6.6.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.4.p1.6.m6.6.6.3.1" xref="A1.SS2.4.p1.6.m6.6.6.3.1.cmml">∖</mo><mrow id="A1.SS2.4.p1.6.m6.6.6.3.3" xref="A1.SS2.4.p1.6.m6.6.6.3.3.cmml"><mo id="A1.SS2.4.p1.6.m6.6.6.3.3.1" lspace="0em" rspace="0em" xref="A1.SS2.4.p1.6.m6.6.6.3.3.1.cmml">∂</mo><msubsup id="A1.SS2.4.p1.6.m6.6.6.3.3.2" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.2" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.2.cmml">ℋ</mi><mrow id="A1.SS2.4.p1.6.m6.2.2.2.4" xref="A1.SS2.4.p1.6.m6.2.2.2.3.cmml"><mi id="A1.SS2.4.p1.6.m6.1.1.1.1" xref="A1.SS2.4.p1.6.m6.1.1.1.1.cmml">x</mi><mo id="A1.SS2.4.p1.6.m6.2.2.2.4.1" xref="A1.SS2.4.p1.6.m6.2.2.2.3.cmml">,</mo><mi id="A1.SS2.4.p1.6.m6.2.2.2.2" xref="A1.SS2.4.p1.6.m6.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.3" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.3.cmml">p</mi></msubsup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.6.m6.6b"><apply id="A1.SS2.4.p1.6.m6.6.6.cmml" xref="A1.SS2.4.p1.6.m6.6.6"><subset id="A1.SS2.4.p1.6.m6.6.6.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.2"></subset><apply id="A1.SS2.4.p1.6.m6.6.6.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1"><times id="A1.SS2.4.p1.6.m6.6.6.1.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.2"></times><apply id="A1.SS2.4.p1.6.m6.6.6.1.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.6.m6.6.6.1.3.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.3">superscript</csymbol><ci id="A1.SS2.4.p1.6.m6.6.6.1.3.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.3.2">𝐵</ci><ci id="A1.SS2.4.p1.6.m6.6.6.1.3.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.3.3">𝑝</ci></apply><interval closure="open" id="A1.SS2.4.p1.6.m6.6.6.1.1.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1"><ci id="A1.SS2.4.p1.6.m6.3.3.cmml" xref="A1.SS2.4.p1.6.m6.3.3">𝑦</ci><apply id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1"><times id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.1"></times><ci id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.2">𝛼</ci><ci id="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.1.1.1.1.3">𝑟</ci></apply></interval></apply><apply id="A1.SS2.4.p1.6.m6.6.6.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3"><setdiff id="A1.SS2.4.p1.6.m6.6.6.3.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.1"></setdiff><apply id="A1.SS2.4.p1.6.m6.6.6.3.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2"><times id="A1.SS2.4.p1.6.m6.6.6.3.2.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.1"></times><apply id="A1.SS2.4.p1.6.m6.6.6.3.2.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2"><csymbol cd="ambiguous" id="A1.SS2.4.p1.6.m6.6.6.3.2.2.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2">superscript</csymbol><ci id="A1.SS2.4.p1.6.m6.6.6.3.2.2.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2.2">𝐵</ci><ci id="A1.SS2.4.p1.6.m6.6.6.3.2.2.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.2.3">𝑝</ci></apply><interval closure="open" id="A1.SS2.4.p1.6.m6.6.6.3.2.3.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.2.3.2"><ci id="A1.SS2.4.p1.6.m6.4.4.cmml" xref="A1.SS2.4.p1.6.m6.4.4">𝑧</ci><ci id="A1.SS2.4.p1.6.m6.5.5.cmml" xref="A1.SS2.4.p1.6.m6.5.5">𝑟</ci></interval></apply><apply id="A1.SS2.4.p1.6.m6.6.6.3.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3"><partialdiff id="A1.SS2.4.p1.6.m6.6.6.3.3.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.1"></partialdiff><apply id="A1.SS2.4.p1.6.m6.6.6.3.3.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2"><csymbol cd="ambiguous" id="A1.SS2.4.p1.6.m6.6.6.3.3.2.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2">subscript</csymbol><apply id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2"><csymbol cd="ambiguous" id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.1.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2">superscript</csymbol><ci id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.2.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.2">ℋ</ci><ci id="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.3.cmml" xref="A1.SS2.4.p1.6.m6.6.6.3.3.2.2.3">𝑝</ci></apply><list id="A1.SS2.4.p1.6.m6.2.2.2.3.cmml" xref="A1.SS2.4.p1.6.m6.2.2.2.4"><ci id="A1.SS2.4.p1.6.m6.1.1.1.1.cmml" xref="A1.SS2.4.p1.6.m6.1.1.1.1">𝑥</ci><ci id="A1.SS2.4.p1.6.m6.2.2.2.2.cmml" xref="A1.SS2.4.p1.6.m6.2.2.2.2">𝑣</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.6.m6.6c">B^{p}(y,\alpha r)\subseteq B^{p}(z,r)\setminus\partial\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.6.m6.6d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_y , italic_α italic_r ) ⊆ italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_z , italic_r ) ∖ ∂ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. This is a sufficient condition for a subset of <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.7.m7.1"><semantics id="A1.SS2.4.p1.7.m7.1a"><msup id="A1.SS2.4.p1.7.m7.1.1" xref="A1.SS2.4.p1.7.m7.1.1.cmml"><mi id="A1.SS2.4.p1.7.m7.1.1.2" xref="A1.SS2.4.p1.7.m7.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.4.p1.7.m7.1.1.3" xref="A1.SS2.4.p1.7.m7.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.7.m7.1b"><apply id="A1.SS2.4.p1.7.m7.1.1.cmml" xref="A1.SS2.4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A1.SS2.4.p1.7.m7.1.1.1.cmml" xref="A1.SS2.4.p1.7.m7.1.1">superscript</csymbol><ci id="A1.SS2.4.p1.7.m7.1.1.2.cmml" xref="A1.SS2.4.p1.7.m7.1.1.2">ℝ</ci><ci id="A1.SS2.4.p1.7.m7.1.1.3.cmml" xref="A1.SS2.4.p1.7.m7.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.7.m7.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> to have measure <math alttext="0" class="ltx_Math" display="inline" id="A1.SS2.4.p1.8.m8.1"><semantics id="A1.SS2.4.p1.8.m8.1a"><mn id="A1.SS2.4.p1.8.m8.1.1" xref="A1.SS2.4.p1.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.8.m8.1b"><cn id="A1.SS2.4.p1.8.m8.1.1.cmml" type="integer" xref="A1.SS2.4.p1.8.m8.1.1">0</cn></annotation-xml></semantics></math>. Porosity follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem4" title="Lemma A.4 (Inside and Outside Orthant). ‣ A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">A.4</span></a> by distinguishing between the cases <math alttext="z\in\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.9.m9.2"><semantics id="A1.SS2.4.p1.9.m9.2a"><mrow id="A1.SS2.4.p1.9.m9.2.3" xref="A1.SS2.4.p1.9.m9.2.3.cmml"><mi id="A1.SS2.4.p1.9.m9.2.3.2" xref="A1.SS2.4.p1.9.m9.2.3.2.cmml">z</mi><mo id="A1.SS2.4.p1.9.m9.2.3.1" xref="A1.SS2.4.p1.9.m9.2.3.1.cmml">∈</mo><msubsup id="A1.SS2.4.p1.9.m9.2.3.3" xref="A1.SS2.4.p1.9.m9.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.4.p1.9.m9.2.3.3.2.2" xref="A1.SS2.4.p1.9.m9.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.4.p1.9.m9.2.2.2.4" xref="A1.SS2.4.p1.9.m9.2.2.2.3.cmml"><mi id="A1.SS2.4.p1.9.m9.1.1.1.1" xref="A1.SS2.4.p1.9.m9.1.1.1.1.cmml">x</mi><mo id="A1.SS2.4.p1.9.m9.2.2.2.4.1" xref="A1.SS2.4.p1.9.m9.2.2.2.3.cmml">,</mo><mi id="A1.SS2.4.p1.9.m9.2.2.2.2" xref="A1.SS2.4.p1.9.m9.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.4.p1.9.m9.2.3.3.2.3" xref="A1.SS2.4.p1.9.m9.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.9.m9.2b"><apply id="A1.SS2.4.p1.9.m9.2.3.cmml" xref="A1.SS2.4.p1.9.m9.2.3"><in id="A1.SS2.4.p1.9.m9.2.3.1.cmml" xref="A1.SS2.4.p1.9.m9.2.3.1"></in><ci id="A1.SS2.4.p1.9.m9.2.3.2.cmml" xref="A1.SS2.4.p1.9.m9.2.3.2">𝑧</ci><apply id="A1.SS2.4.p1.9.m9.2.3.3.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.9.m9.2.3.3.1.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3">subscript</csymbol><apply id="A1.SS2.4.p1.9.m9.2.3.3.2.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.9.m9.2.3.3.2.1.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3">superscript</csymbol><ci id="A1.SS2.4.p1.9.m9.2.3.3.2.2.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.4.p1.9.m9.2.3.3.2.3.cmml" xref="A1.SS2.4.p1.9.m9.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.4.p1.9.m9.2.2.2.3.cmml" xref="A1.SS2.4.p1.9.m9.2.2.2.4"><ci id="A1.SS2.4.p1.9.m9.1.1.1.1.cmml" xref="A1.SS2.4.p1.9.m9.1.1.1.1">𝑥</ci><ci id="A1.SS2.4.p1.9.m9.2.2.2.2.cmml" xref="A1.SS2.4.p1.9.m9.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.9.m9.2c">z\in\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.9.m9.2d">italic_z ∈ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.4.p1.10.m10.2"><semantics id="A1.SS2.4.p1.10.m10.2a"><mrow id="A1.SS2.4.p1.10.m10.2.3" xref="A1.SS2.4.p1.10.m10.2.3.cmml"><mi id="A1.SS2.4.p1.10.m10.2.3.2" xref="A1.SS2.4.p1.10.m10.2.3.2.cmml">z</mi><mo id="A1.SS2.4.p1.10.m10.2.3.1" xref="A1.SS2.4.p1.10.m10.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.4.p1.10.m10.2.3.3" xref="A1.SS2.4.p1.10.m10.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.4.p1.10.m10.2.3.3.2.2" xref="A1.SS2.4.p1.10.m10.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.4.p1.10.m10.2.2.2.4" xref="A1.SS2.4.p1.10.m10.2.2.2.3.cmml"><mi id="A1.SS2.4.p1.10.m10.1.1.1.1" xref="A1.SS2.4.p1.10.m10.1.1.1.1.cmml">x</mi><mo id="A1.SS2.4.p1.10.m10.2.2.2.4.1" xref="A1.SS2.4.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="A1.SS2.4.p1.10.m10.2.2.2.2" xref="A1.SS2.4.p1.10.m10.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.4.p1.10.m10.2.3.3.2.3" xref="A1.SS2.4.p1.10.m10.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.4.p1.10.m10.2b"><apply id="A1.SS2.4.p1.10.m10.2.3.cmml" xref="A1.SS2.4.p1.10.m10.2.3"><notin id="A1.SS2.4.p1.10.m10.2.3.1.cmml" xref="A1.SS2.4.p1.10.m10.2.3.1"></notin><ci id="A1.SS2.4.p1.10.m10.2.3.2.cmml" xref="A1.SS2.4.p1.10.m10.2.3.2">𝑧</ci><apply id="A1.SS2.4.p1.10.m10.2.3.3.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.10.m10.2.3.3.1.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3">subscript</csymbol><apply id="A1.SS2.4.p1.10.m10.2.3.3.2.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.4.p1.10.m10.2.3.3.2.1.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3">superscript</csymbol><ci id="A1.SS2.4.p1.10.m10.2.3.3.2.2.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.4.p1.10.m10.2.3.3.2.3.cmml" xref="A1.SS2.4.p1.10.m10.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.4.p1.10.m10.2.2.2.3.cmml" xref="A1.SS2.4.p1.10.m10.2.2.2.4"><ci id="A1.SS2.4.p1.10.m10.1.1.1.1.cmml" xref="A1.SS2.4.p1.10.m10.1.1.1.1">𝑥</ci><ci id="A1.SS2.4.p1.10.m10.2.2.2.2.cmml" xref="A1.SS2.4.p1.10.m10.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.4.p1.10.m10.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.4.p1.10.m10.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS2.p4"> <p class="ltx_p" id="A1.SS2.p4.1">We are now ready to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.10</span></a>. While this could be proven purely in terms of measure theory, we opted for the language of probability theory instead.</p> </div> <div class="ltx_para" id="A1.SS2.p5"> <p class="ltx_p" id="A1.SS2.p5.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.10</span></a></p> </div> <div class="ltx_proof" id="A1.SS2.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS2.5.p1"> <p class="ltx_p" id="A1.SS2.5.p1.16">Fix a mass distribution <math alttext="\mu" class="ltx_Math" display="inline" id="A1.SS2.5.p1.1.m1.1"><semantics id="A1.SS2.5.p1.1.m1.1a"><mi id="A1.SS2.5.p1.1.m1.1.1" xref="A1.SS2.5.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.1.m1.1b"><ci id="A1.SS2.5.p1.1.m1.1.1.cmml" xref="A1.SS2.5.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.1.m1.1d">italic_μ</annotation></semantics></math>, and fix <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.2.m2.1"><semantics id="A1.SS2.5.p1.2.m2.1a"><mrow id="A1.SS2.5.p1.2.m2.1.1" xref="A1.SS2.5.p1.2.m2.1.1.cmml"><mi id="A1.SS2.5.p1.2.m2.1.1.2" xref="A1.SS2.5.p1.2.m2.1.1.2.cmml">v</mi><mo id="A1.SS2.5.p1.2.m2.1.1.1" xref="A1.SS2.5.p1.2.m2.1.1.1.cmml">∈</mo><msup id="A1.SS2.5.p1.2.m2.1.1.3" xref="A1.SS2.5.p1.2.m2.1.1.3.cmml"><mi id="A1.SS2.5.p1.2.m2.1.1.3.2" xref="A1.SS2.5.p1.2.m2.1.1.3.2.cmml">S</mi><mrow id="A1.SS2.5.p1.2.m2.1.1.3.3" xref="A1.SS2.5.p1.2.m2.1.1.3.3.cmml"><mi id="A1.SS2.5.p1.2.m2.1.1.3.3.2" xref="A1.SS2.5.p1.2.m2.1.1.3.3.2.cmml">d</mi><mo id="A1.SS2.5.p1.2.m2.1.1.3.3.1" xref="A1.SS2.5.p1.2.m2.1.1.3.3.1.cmml">−</mo><mn id="A1.SS2.5.p1.2.m2.1.1.3.3.3" xref="A1.SS2.5.p1.2.m2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.2.m2.1b"><apply id="A1.SS2.5.p1.2.m2.1.1.cmml" xref="A1.SS2.5.p1.2.m2.1.1"><in id="A1.SS2.5.p1.2.m2.1.1.1.cmml" xref="A1.SS2.5.p1.2.m2.1.1.1"></in><ci id="A1.SS2.5.p1.2.m2.1.1.2.cmml" xref="A1.SS2.5.p1.2.m2.1.1.2">𝑣</ci><apply id="A1.SS2.5.p1.2.m2.1.1.3.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.2.m2.1.1.3.1.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3">superscript</csymbol><ci id="A1.SS2.5.p1.2.m2.1.1.3.2.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3.2">𝑆</ci><apply id="A1.SS2.5.p1.2.m2.1.1.3.3.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3.3"><minus id="A1.SS2.5.p1.2.m2.1.1.3.3.1.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3.3.1"></minus><ci id="A1.SS2.5.p1.2.m2.1.1.3.3.2.cmml" xref="A1.SS2.5.p1.2.m2.1.1.3.3.2">𝑑</ci><cn id="A1.SS2.5.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="A1.SS2.5.p1.2.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.2.m2.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.2.m2.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.3.m3.3"><semantics id="A1.SS2.5.p1.3.m3.3a"><mrow id="A1.SS2.5.p1.3.m3.3.4" xref="A1.SS2.5.p1.3.m3.3.4.cmml"><mi id="A1.SS2.5.p1.3.m3.3.4.2" xref="A1.SS2.5.p1.3.m3.3.4.2.cmml">p</mi><mo id="A1.SS2.5.p1.3.m3.3.4.1" xref="A1.SS2.5.p1.3.m3.3.4.1.cmml">∈</mo><mrow id="A1.SS2.5.p1.3.m3.3.4.3" xref="A1.SS2.5.p1.3.m3.3.4.3.cmml"><mrow id="A1.SS2.5.p1.3.m3.3.4.3.2.2" xref="A1.SS2.5.p1.3.m3.3.4.3.2.1.cmml"><mo id="A1.SS2.5.p1.3.m3.3.4.3.2.2.1" stretchy="false" xref="A1.SS2.5.p1.3.m3.3.4.3.2.1.cmml">[</mo><mn id="A1.SS2.5.p1.3.m3.1.1" xref="A1.SS2.5.p1.3.m3.1.1.cmml">1</mn><mo id="A1.SS2.5.p1.3.m3.3.4.3.2.2.2" xref="A1.SS2.5.p1.3.m3.3.4.3.2.1.cmml">,</mo><mi id="A1.SS2.5.p1.3.m3.2.2" mathvariant="normal" xref="A1.SS2.5.p1.3.m3.2.2.cmml">∞</mi><mo id="A1.SS2.5.p1.3.m3.3.4.3.2.2.3" stretchy="false" xref="A1.SS2.5.p1.3.m3.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.SS2.5.p1.3.m3.3.4.3.1" xref="A1.SS2.5.p1.3.m3.3.4.3.1.cmml">∪</mo><mrow id="A1.SS2.5.p1.3.m3.3.4.3.3.2" xref="A1.SS2.5.p1.3.m3.3.4.3.3.1.cmml"><mo id="A1.SS2.5.p1.3.m3.3.4.3.3.2.1" stretchy="false" xref="A1.SS2.5.p1.3.m3.3.4.3.3.1.cmml">{</mo><mi id="A1.SS2.5.p1.3.m3.3.3" mathvariant="normal" xref="A1.SS2.5.p1.3.m3.3.3.cmml">∞</mi><mo id="A1.SS2.5.p1.3.m3.3.4.3.3.2.2" stretchy="false" xref="A1.SS2.5.p1.3.m3.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.3.m3.3b"><apply id="A1.SS2.5.p1.3.m3.3.4.cmml" xref="A1.SS2.5.p1.3.m3.3.4"><in id="A1.SS2.5.p1.3.m3.3.4.1.cmml" xref="A1.SS2.5.p1.3.m3.3.4.1"></in><ci id="A1.SS2.5.p1.3.m3.3.4.2.cmml" xref="A1.SS2.5.p1.3.m3.3.4.2">𝑝</ci><apply id="A1.SS2.5.p1.3.m3.3.4.3.cmml" xref="A1.SS2.5.p1.3.m3.3.4.3"><union id="A1.SS2.5.p1.3.m3.3.4.3.1.cmml" xref="A1.SS2.5.p1.3.m3.3.4.3.1"></union><interval closure="closed-open" id="A1.SS2.5.p1.3.m3.3.4.3.2.1.cmml" xref="A1.SS2.5.p1.3.m3.3.4.3.2.2"><cn id="A1.SS2.5.p1.3.m3.1.1.cmml" type="integer" xref="A1.SS2.5.p1.3.m3.1.1">1</cn><infinity id="A1.SS2.5.p1.3.m3.2.2.cmml" xref="A1.SS2.5.p1.3.m3.2.2"></infinity></interval><set id="A1.SS2.5.p1.3.m3.3.4.3.3.1.cmml" xref="A1.SS2.5.p1.3.m3.3.4.3.3.2"><infinity id="A1.SS2.5.p1.3.m3.3.3.cmml" xref="A1.SS2.5.p1.3.m3.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.3.m3.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.3.m3.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math>. Without loss of generality, assume <math alttext="\mu(\mathbb{R}^{d})=1" class="ltx_Math" display="inline" id="A1.SS2.5.p1.4.m4.1"><semantics id="A1.SS2.5.p1.4.m4.1a"><mrow id="A1.SS2.5.p1.4.m4.1.1" xref="A1.SS2.5.p1.4.m4.1.1.cmml"><mrow id="A1.SS2.5.p1.4.m4.1.1.1" xref="A1.SS2.5.p1.4.m4.1.1.1.cmml"><mi id="A1.SS2.5.p1.4.m4.1.1.1.3" xref="A1.SS2.5.p1.4.m4.1.1.1.3.cmml">μ</mi><mo id="A1.SS2.5.p1.4.m4.1.1.1.2" xref="A1.SS2.5.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.5.p1.4.m4.1.1.1.1.1" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="A1.SS2.5.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.2" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.3" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="A1.SS2.5.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.5.p1.4.m4.1.1.2" xref="A1.SS2.5.p1.4.m4.1.1.2.cmml">=</mo><mn id="A1.SS2.5.p1.4.m4.1.1.3" xref="A1.SS2.5.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.4.m4.1b"><apply id="A1.SS2.5.p1.4.m4.1.1.cmml" xref="A1.SS2.5.p1.4.m4.1.1"><eq id="A1.SS2.5.p1.4.m4.1.1.2.cmml" xref="A1.SS2.5.p1.4.m4.1.1.2"></eq><apply id="A1.SS2.5.p1.4.m4.1.1.1.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1"><times id="A1.SS2.5.p1.4.m4.1.1.1.2.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.2"></times><ci id="A1.SS2.5.p1.4.m4.1.1.1.3.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.3">𝜇</ci><apply id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="A1.SS2.5.p1.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply><cn id="A1.SS2.5.p1.4.m4.1.1.3.cmml" type="integer" xref="A1.SS2.5.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.4.m4.1c">\mu(\mathbb{R}^{d})=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.4.m4.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>. Let <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.5.m5.1"><semantics id="A1.SS2.5.p1.5.m5.1a"><mrow id="A1.SS2.5.p1.5.m5.1.1" xref="A1.SS2.5.p1.5.m5.1.1.cmml"><mi id="A1.SS2.5.p1.5.m5.1.1.2" xref="A1.SS2.5.p1.5.m5.1.1.2.cmml">x</mi><mo id="A1.SS2.5.p1.5.m5.1.1.1" xref="A1.SS2.5.p1.5.m5.1.1.1.cmml">∈</mo><msup id="A1.SS2.5.p1.5.m5.1.1.3" xref="A1.SS2.5.p1.5.m5.1.1.3.cmml"><mi id="A1.SS2.5.p1.5.m5.1.1.3.2" xref="A1.SS2.5.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.5.p1.5.m5.1.1.3.3" xref="A1.SS2.5.p1.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.5.m5.1b"><apply id="A1.SS2.5.p1.5.m5.1.1.cmml" xref="A1.SS2.5.p1.5.m5.1.1"><in id="A1.SS2.5.p1.5.m5.1.1.1.cmml" xref="A1.SS2.5.p1.5.m5.1.1.1"></in><ci id="A1.SS2.5.p1.5.m5.1.1.2.cmml" xref="A1.SS2.5.p1.5.m5.1.1.2">𝑥</ci><apply id="A1.SS2.5.p1.5.m5.1.1.3.cmml" xref="A1.SS2.5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.5.m5.1.1.3.1.cmml" xref="A1.SS2.5.p1.5.m5.1.1.3">superscript</csymbol><ci id="A1.SS2.5.p1.5.m5.1.1.3.2.cmml" xref="A1.SS2.5.p1.5.m5.1.1.3.2">ℝ</ci><ci id="A1.SS2.5.p1.5.m5.1.1.3.3.cmml" xref="A1.SS2.5.p1.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.5.m5.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.5.m5.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be arbitrary, and consider an arbitrary sequence <math alttext="(x_{n})_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.6.m6.1"><semantics id="A1.SS2.5.p1.6.m6.1a"><msub id="A1.SS2.5.p1.6.m6.1.1" xref="A1.SS2.5.p1.6.m6.1.1.cmml"><mrow id="A1.SS2.5.p1.6.m6.1.1.1.1" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.cmml"><mo id="A1.SS2.5.p1.6.m6.1.1.1.1.2" stretchy="false" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.5.p1.6.m6.1.1.1.1.1" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.cmml"><mi id="A1.SS2.5.p1.6.m6.1.1.1.1.1.2" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.2.cmml">x</mi><mi id="A1.SS2.5.p1.6.m6.1.1.1.1.1.3" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.6.m6.1.1.1.1.3" stretchy="false" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS2.5.p1.6.m6.1.1.3" xref="A1.SS2.5.p1.6.m6.1.1.3.cmml"><mi id="A1.SS2.5.p1.6.m6.1.1.3.2" xref="A1.SS2.5.p1.6.m6.1.1.3.2.cmml">n</mi><mo id="A1.SS2.5.p1.6.m6.1.1.3.1" xref="A1.SS2.5.p1.6.m6.1.1.3.1.cmml">∈</mo><mi id="A1.SS2.5.p1.6.m6.1.1.3.3" xref="A1.SS2.5.p1.6.m6.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.6.m6.1b"><apply id="A1.SS2.5.p1.6.m6.1.1.cmml" xref="A1.SS2.5.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.6.m6.1.1.2.cmml" xref="A1.SS2.5.p1.6.m6.1.1">subscript</csymbol><apply id="A1.SS2.5.p1.6.m6.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.6.m6.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.6.m6.1.1.1.1">subscript</csymbol><ci id="A1.SS2.5.p1.6.m6.1.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.2">𝑥</ci><ci id="A1.SS2.5.p1.6.m6.1.1.1.1.1.3.cmml" xref="A1.SS2.5.p1.6.m6.1.1.1.1.1.3">𝑛</ci></apply><apply id="A1.SS2.5.p1.6.m6.1.1.3.cmml" xref="A1.SS2.5.p1.6.m6.1.1.3"><in id="A1.SS2.5.p1.6.m6.1.1.3.1.cmml" xref="A1.SS2.5.p1.6.m6.1.1.3.1"></in><ci id="A1.SS2.5.p1.6.m6.1.1.3.2.cmml" xref="A1.SS2.5.p1.6.m6.1.1.3.2">𝑛</ci><ci id="A1.SS2.5.p1.6.m6.1.1.3.3.cmml" xref="A1.SS2.5.p1.6.m6.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.6.m6.1c">(x_{n})_{n\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.6.m6.1d">( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converging to <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.5.p1.7.m7.1"><semantics id="A1.SS2.5.p1.7.m7.1a"><mi id="A1.SS2.5.p1.7.m7.1.1" xref="A1.SS2.5.p1.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.7.m7.1b"><ci id="A1.SS2.5.p1.7.m7.1.1.cmml" xref="A1.SS2.5.p1.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.7.m7.1d">italic_x</annotation></semantics></math>. For <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.8.m8.1"><semantics id="A1.SS2.5.p1.8.m8.1a"><mrow id="A1.SS2.5.p1.8.m8.1.1" xref="A1.SS2.5.p1.8.m8.1.1.cmml"><mi id="A1.SS2.5.p1.8.m8.1.1.2" xref="A1.SS2.5.p1.8.m8.1.1.2.cmml">n</mi><mo id="A1.SS2.5.p1.8.m8.1.1.1" xref="A1.SS2.5.p1.8.m8.1.1.1.cmml">∈</mo><mi id="A1.SS2.5.p1.8.m8.1.1.3" xref="A1.SS2.5.p1.8.m8.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.8.m8.1b"><apply id="A1.SS2.5.p1.8.m8.1.1.cmml" xref="A1.SS2.5.p1.8.m8.1.1"><in id="A1.SS2.5.p1.8.m8.1.1.1.cmml" xref="A1.SS2.5.p1.8.m8.1.1.1"></in><ci id="A1.SS2.5.p1.8.m8.1.1.2.cmml" xref="A1.SS2.5.p1.8.m8.1.1.2">𝑛</ci><ci id="A1.SS2.5.p1.8.m8.1.1.3.cmml" xref="A1.SS2.5.p1.8.m8.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.8.m8.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.8.m8.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, let <math alttext="X_{n}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.9.m9.1"><semantics id="A1.SS2.5.p1.9.m9.1a"><msub id="A1.SS2.5.p1.9.m9.1.1" xref="A1.SS2.5.p1.9.m9.1.1.cmml"><mi id="A1.SS2.5.p1.9.m9.1.1.2" xref="A1.SS2.5.p1.9.m9.1.1.2.cmml">X</mi><mi id="A1.SS2.5.p1.9.m9.1.1.3" xref="A1.SS2.5.p1.9.m9.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.9.m9.1b"><apply id="A1.SS2.5.p1.9.m9.1.1.cmml" xref="A1.SS2.5.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.9.m9.1.1.1.cmml" xref="A1.SS2.5.p1.9.m9.1.1">subscript</csymbol><ci id="A1.SS2.5.p1.9.m9.1.1.2.cmml" xref="A1.SS2.5.p1.9.m9.1.1.2">𝑋</ci><ci id="A1.SS2.5.p1.9.m9.1.1.3.cmml" xref="A1.SS2.5.p1.9.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.9.m9.1c">X_{n}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.9.m9.1d">italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> denote the indicator random variable for a point drawn at random (w.r.t. <math alttext="\mu" class="ltx_Math" display="inline" id="A1.SS2.5.p1.10.m10.1"><semantics id="A1.SS2.5.p1.10.m10.1a"><mi id="A1.SS2.5.p1.10.m10.1.1" xref="A1.SS2.5.p1.10.m10.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.10.m10.1b"><ci id="A1.SS2.5.p1.10.m10.1.1.cmml" xref="A1.SS2.5.p1.10.m10.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.10.m10.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.10.m10.1d">italic_μ</annotation></semantics></math>) to lie in the set <math alttext="\mathcal{H}^{p}_{x_{n},v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.11.m11.2"><semantics id="A1.SS2.5.p1.11.m11.2a"><msubsup id="A1.SS2.5.p1.11.m11.2.3" xref="A1.SS2.5.p1.11.m11.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.11.m11.2.3.2.2" xref="A1.SS2.5.p1.11.m11.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.11.m11.2.2.2.2" xref="A1.SS2.5.p1.11.m11.2.2.2.3.cmml"><msub id="A1.SS2.5.p1.11.m11.2.2.2.2.1" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1.cmml"><mi id="A1.SS2.5.p1.11.m11.2.2.2.2.1.2" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1.2.cmml">x</mi><mi id="A1.SS2.5.p1.11.m11.2.2.2.2.1.3" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.11.m11.2.2.2.2.2" xref="A1.SS2.5.p1.11.m11.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.11.m11.1.1.1.1" xref="A1.SS2.5.p1.11.m11.1.1.1.1.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.11.m11.2.3.2.3" xref="A1.SS2.5.p1.11.m11.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.11.m11.2b"><apply id="A1.SS2.5.p1.11.m11.2.3.cmml" xref="A1.SS2.5.p1.11.m11.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.11.m11.2.3.1.cmml" xref="A1.SS2.5.p1.11.m11.2.3">subscript</csymbol><apply id="A1.SS2.5.p1.11.m11.2.3.2.cmml" xref="A1.SS2.5.p1.11.m11.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.11.m11.2.3.2.1.cmml" xref="A1.SS2.5.p1.11.m11.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.11.m11.2.3.2.2.cmml" xref="A1.SS2.5.p1.11.m11.2.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.11.m11.2.3.2.3.cmml" xref="A1.SS2.5.p1.11.m11.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.11.m11.2.2.2.3.cmml" xref="A1.SS2.5.p1.11.m11.2.2.2.2"><apply id="A1.SS2.5.p1.11.m11.2.2.2.2.1.cmml" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.11.m11.2.2.2.2.1.1.cmml" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1">subscript</csymbol><ci id="A1.SS2.5.p1.11.m11.2.2.2.2.1.2.cmml" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1.2">𝑥</ci><ci id="A1.SS2.5.p1.11.m11.2.2.2.2.1.3.cmml" xref="A1.SS2.5.p1.11.m11.2.2.2.2.1.3">𝑛</ci></apply><ci id="A1.SS2.5.p1.11.m11.1.1.1.1.cmml" xref="A1.SS2.5.p1.11.m11.1.1.1.1">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.11.m11.2c">\mathcal{H}^{p}_{x_{n},v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.11.m11.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, let <math alttext="X" class="ltx_Math" display="inline" id="A1.SS2.5.p1.12.m12.1"><semantics id="A1.SS2.5.p1.12.m12.1a"><mi id="A1.SS2.5.p1.12.m12.1.1" xref="A1.SS2.5.p1.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.12.m12.1b"><ci id="A1.SS2.5.p1.12.m12.1.1.cmml" xref="A1.SS2.5.p1.12.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.12.m12.1d">italic_X</annotation></semantics></math> denote the indicator random variable for containment in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.13.m13.2"><semantics id="A1.SS2.5.p1.13.m13.2a"><msubsup id="A1.SS2.5.p1.13.m13.2.3" xref="A1.SS2.5.p1.13.m13.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.13.m13.2.3.2.2" xref="A1.SS2.5.p1.13.m13.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.13.m13.2.2.2.4" xref="A1.SS2.5.p1.13.m13.2.2.2.3.cmml"><mi id="A1.SS2.5.p1.13.m13.1.1.1.1" xref="A1.SS2.5.p1.13.m13.1.1.1.1.cmml">x</mi><mo id="A1.SS2.5.p1.13.m13.2.2.2.4.1" xref="A1.SS2.5.p1.13.m13.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.13.m13.2.2.2.2" xref="A1.SS2.5.p1.13.m13.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.13.m13.2.3.2.3" xref="A1.SS2.5.p1.13.m13.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.13.m13.2b"><apply id="A1.SS2.5.p1.13.m13.2.3.cmml" xref="A1.SS2.5.p1.13.m13.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.13.m13.2.3.1.cmml" xref="A1.SS2.5.p1.13.m13.2.3">subscript</csymbol><apply id="A1.SS2.5.p1.13.m13.2.3.2.cmml" xref="A1.SS2.5.p1.13.m13.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.13.m13.2.3.2.1.cmml" xref="A1.SS2.5.p1.13.m13.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.13.m13.2.3.2.2.cmml" xref="A1.SS2.5.p1.13.m13.2.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.13.m13.2.3.2.3.cmml" xref="A1.SS2.5.p1.13.m13.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.13.m13.2.2.2.3.cmml" xref="A1.SS2.5.p1.13.m13.2.2.2.4"><ci id="A1.SS2.5.p1.13.m13.1.1.1.1.cmml" xref="A1.SS2.5.p1.13.m13.1.1.1.1">𝑥</ci><ci id="A1.SS2.5.p1.13.m13.2.2.2.2.cmml" xref="A1.SS2.5.p1.13.m13.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.13.m13.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.13.m13.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. We claim that <math alttext="X_{n}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.14.m14.1"><semantics id="A1.SS2.5.p1.14.m14.1a"><msub id="A1.SS2.5.p1.14.m14.1.1" xref="A1.SS2.5.p1.14.m14.1.1.cmml"><mi id="A1.SS2.5.p1.14.m14.1.1.2" xref="A1.SS2.5.p1.14.m14.1.1.2.cmml">X</mi><mi id="A1.SS2.5.p1.14.m14.1.1.3" xref="A1.SS2.5.p1.14.m14.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.14.m14.1b"><apply id="A1.SS2.5.p1.14.m14.1.1.cmml" xref="A1.SS2.5.p1.14.m14.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.14.m14.1.1.1.cmml" xref="A1.SS2.5.p1.14.m14.1.1">subscript</csymbol><ci id="A1.SS2.5.p1.14.m14.1.1.2.cmml" xref="A1.SS2.5.p1.14.m14.1.1.2">𝑋</ci><ci id="A1.SS2.5.p1.14.m14.1.1.3.cmml" xref="A1.SS2.5.p1.14.m14.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.14.m14.1c">X_{n}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.14.m14.1d">italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> converges almost surely to <math alttext="X" class="ltx_Math" display="inline" id="A1.SS2.5.p1.15.m15.1"><semantics id="A1.SS2.5.p1.15.m15.1a"><mi id="A1.SS2.5.p1.15.m15.1.1" xref="A1.SS2.5.p1.15.m15.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.15.m15.1b"><ci id="A1.SS2.5.p1.15.m15.1.1.cmml" xref="A1.SS2.5.p1.15.m15.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.15.m15.1c">X</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.15.m15.1d">italic_X</annotation></semantics></math> as <math alttext="n" class="ltx_Math" display="inline" id="A1.SS2.5.p1.16.m16.1"><semantics id="A1.SS2.5.p1.16.m16.1a"><mi id="A1.SS2.5.p1.16.m16.1.1" xref="A1.SS2.5.p1.16.m16.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.16.m16.1b"><ci id="A1.SS2.5.p1.16.m16.1.1.cmml" xref="A1.SS2.5.p1.16.m16.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.16.m16.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.16.m16.1d">italic_n</annotation></semantics></math> goes to infinity. To see this, observe first that we naturally have</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{n\rightarrow\infty}\lVert z-x_{n}\rVert_{p}=\lVert z-x\rVert_{p}" class="ltx_Math" display="block" id="A1.Ex18.m1.2"><semantics id="A1.Ex18.m1.2a"><mrow id="A1.Ex18.m1.2.2" xref="A1.Ex18.m1.2.2.cmml"><mrow id="A1.Ex18.m1.1.1.1" xref="A1.Ex18.m1.1.1.1.cmml"><munder id="A1.Ex18.m1.1.1.1.2" xref="A1.Ex18.m1.1.1.1.2.cmml"><mo id="A1.Ex18.m1.1.1.1.2.2" movablelimits="false" xref="A1.Ex18.m1.1.1.1.2.2.cmml">lim</mo><mrow id="A1.Ex18.m1.1.1.1.2.3" xref="A1.Ex18.m1.1.1.1.2.3.cmml"><mi id="A1.Ex18.m1.1.1.1.2.3.2" xref="A1.Ex18.m1.1.1.1.2.3.2.cmml">n</mi><mo id="A1.Ex18.m1.1.1.1.2.3.1" stretchy="false" xref="A1.Ex18.m1.1.1.1.2.3.1.cmml">→</mo><mi id="A1.Ex18.m1.1.1.1.2.3.3" mathvariant="normal" xref="A1.Ex18.m1.1.1.1.2.3.3.cmml">∞</mi></mrow></munder><msub id="A1.Ex18.m1.1.1.1.1" xref="A1.Ex18.m1.1.1.1.1.cmml"><mrow id="A1.Ex18.m1.1.1.1.1.1.1" xref="A1.Ex18.m1.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex18.m1.1.1.1.1.1.1.2" lspace="0em" rspace="0em" xref="A1.Ex18.m1.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex18.m1.1.1.1.1.1.1.1" xref="A1.Ex18.m1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex18.m1.1.1.1.1.1.1.1.2" xref="A1.Ex18.m1.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="A1.Ex18.m1.1.1.1.1.1.1.1.1" xref="A1.Ex18.m1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="A1.Ex18.m1.1.1.1.1.1.1.1.3" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex18.m1.1.1.1.1.1.1.1.3.2" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3.2.cmml">x</mi><mi id="A1.Ex18.m1.1.1.1.1.1.1.1.3.3" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo fence="true" id="A1.Ex18.m1.1.1.1.1.1.1.3" lspace="0em" rspace="0.1389em" xref="A1.Ex18.m1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex18.m1.1.1.1.1.3" xref="A1.Ex18.m1.1.1.1.1.3.cmml">p</mi></msub></mrow><mo id="A1.Ex18.m1.2.2.3" lspace="0.1389em" rspace="0.1389em" xref="A1.Ex18.m1.2.2.3.cmml">=</mo><msub id="A1.Ex18.m1.2.2.2" xref="A1.Ex18.m1.2.2.2.cmml"><mrow id="A1.Ex18.m1.2.2.2.1.1" xref="A1.Ex18.m1.2.2.2.1.2.cmml"><mo fence="true" id="A1.Ex18.m1.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex18.m1.2.2.2.1.2.1.cmml">∥</mo><mrow id="A1.Ex18.m1.2.2.2.1.1.1" xref="A1.Ex18.m1.2.2.2.1.1.1.cmml"><mi id="A1.Ex18.m1.2.2.2.1.1.1.2" xref="A1.Ex18.m1.2.2.2.1.1.1.2.cmml">z</mi><mo id="A1.Ex18.m1.2.2.2.1.1.1.1" xref="A1.Ex18.m1.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.Ex18.m1.2.2.2.1.1.1.3" xref="A1.Ex18.m1.2.2.2.1.1.1.3.cmml">x</mi></mrow><mo fence="true" id="A1.Ex18.m1.2.2.2.1.1.3" lspace="0em" xref="A1.Ex18.m1.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex18.m1.2.2.2.3" xref="A1.Ex18.m1.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex18.m1.2b"><apply id="A1.Ex18.m1.2.2.cmml" xref="A1.Ex18.m1.2.2"><eq id="A1.Ex18.m1.2.2.3.cmml" xref="A1.Ex18.m1.2.2.3"></eq><apply id="A1.Ex18.m1.1.1.1.cmml" xref="A1.Ex18.m1.1.1.1"><apply id="A1.Ex18.m1.1.1.1.2.cmml" xref="A1.Ex18.m1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex18.m1.1.1.1.2.1.cmml" xref="A1.Ex18.m1.1.1.1.2">subscript</csymbol><limit id="A1.Ex18.m1.1.1.1.2.2.cmml" xref="A1.Ex18.m1.1.1.1.2.2"></limit><apply id="A1.Ex18.m1.1.1.1.2.3.cmml" xref="A1.Ex18.m1.1.1.1.2.3"><ci id="A1.Ex18.m1.1.1.1.2.3.1.cmml" xref="A1.Ex18.m1.1.1.1.2.3.1">→</ci><ci id="A1.Ex18.m1.1.1.1.2.3.2.cmml" xref="A1.Ex18.m1.1.1.1.2.3.2">𝑛</ci><infinity id="A1.Ex18.m1.1.1.1.2.3.3.cmml" xref="A1.Ex18.m1.1.1.1.2.3.3"></infinity></apply></apply><apply id="A1.Ex18.m1.1.1.1.1.cmml" xref="A1.Ex18.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex18.m1.1.1.1.1.2.cmml" xref="A1.Ex18.m1.1.1.1.1">subscript</csymbol><apply id="A1.Ex18.m1.1.1.1.1.1.2.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.Ex18.m1.1.1.1.1.1.2.1.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex18.m1.1.1.1.1.1.1.1.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1"><minus id="A1.Ex18.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.1"></minus><ci id="A1.Ex18.m1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.2">𝑧</ci><apply id="A1.Ex18.m1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex18.m1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex18.m1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3.2">𝑥</ci><ci id="A1.Ex18.m1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex18.m1.1.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply><ci id="A1.Ex18.m1.1.1.1.1.3.cmml" xref="A1.Ex18.m1.1.1.1.1.3">𝑝</ci></apply></apply><apply id="A1.Ex18.m1.2.2.2.cmml" xref="A1.Ex18.m1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex18.m1.2.2.2.2.cmml" xref="A1.Ex18.m1.2.2.2">subscript</csymbol><apply id="A1.Ex18.m1.2.2.2.1.2.cmml" xref="A1.Ex18.m1.2.2.2.1.1"><csymbol cd="latexml" id="A1.Ex18.m1.2.2.2.1.2.1.cmml" xref="A1.Ex18.m1.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex18.m1.2.2.2.1.1.1.cmml" xref="A1.Ex18.m1.2.2.2.1.1.1"><minus id="A1.Ex18.m1.2.2.2.1.1.1.1.cmml" xref="A1.Ex18.m1.2.2.2.1.1.1.1"></minus><ci id="A1.Ex18.m1.2.2.2.1.1.1.2.cmml" xref="A1.Ex18.m1.2.2.2.1.1.1.2">𝑧</ci><ci id="A1.Ex18.m1.2.2.2.1.1.1.3.cmml" xref="A1.Ex18.m1.2.2.2.1.1.1.3">𝑥</ci></apply></apply><ci id="A1.Ex18.m1.2.2.2.3.cmml" xref="A1.Ex18.m1.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex18.m1.2c">\lim_{n\rightarrow\infty}\lVert z-x_{n}\rVert_{p}=\lVert z-x\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex18.m1.2d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_z - italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = ∥ italic_z - italic_x ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.5.p1.29">for all <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.17.m1.1"><semantics id="A1.SS2.5.p1.17.m1.1a"><mrow id="A1.SS2.5.p1.17.m1.1.1" xref="A1.SS2.5.p1.17.m1.1.1.cmml"><mi id="A1.SS2.5.p1.17.m1.1.1.2" xref="A1.SS2.5.p1.17.m1.1.1.2.cmml">z</mi><mo id="A1.SS2.5.p1.17.m1.1.1.1" xref="A1.SS2.5.p1.17.m1.1.1.1.cmml">∈</mo><msup id="A1.SS2.5.p1.17.m1.1.1.3" xref="A1.SS2.5.p1.17.m1.1.1.3.cmml"><mi id="A1.SS2.5.p1.17.m1.1.1.3.2" xref="A1.SS2.5.p1.17.m1.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.5.p1.17.m1.1.1.3.3" xref="A1.SS2.5.p1.17.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.17.m1.1b"><apply id="A1.SS2.5.p1.17.m1.1.1.cmml" xref="A1.SS2.5.p1.17.m1.1.1"><in id="A1.SS2.5.p1.17.m1.1.1.1.cmml" xref="A1.SS2.5.p1.17.m1.1.1.1"></in><ci id="A1.SS2.5.p1.17.m1.1.1.2.cmml" xref="A1.SS2.5.p1.17.m1.1.1.2">𝑧</ci><apply id="A1.SS2.5.p1.17.m1.1.1.3.cmml" xref="A1.SS2.5.p1.17.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.17.m1.1.1.3.1.cmml" xref="A1.SS2.5.p1.17.m1.1.1.3">superscript</csymbol><ci id="A1.SS2.5.p1.17.m1.1.1.3.2.cmml" xref="A1.SS2.5.p1.17.m1.1.1.3.2">ℝ</ci><ci id="A1.SS2.5.p1.17.m1.1.1.3.3.cmml" xref="A1.SS2.5.p1.17.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.17.m1.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.17.m1.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. Now let <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.18.m2.2"><semantics id="A1.SS2.5.p1.18.m2.2a"><mrow id="A1.SS2.5.p1.18.m2.2.3" xref="A1.SS2.5.p1.18.m2.2.3.cmml"><mi id="A1.SS2.5.p1.18.m2.2.3.2" xref="A1.SS2.5.p1.18.m2.2.3.2.cmml">z</mi><mo id="A1.SS2.5.p1.18.m2.2.3.1" xref="A1.SS2.5.p1.18.m2.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.5.p1.18.m2.2.3.3" xref="A1.SS2.5.p1.18.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.18.m2.2.3.3.2.2" xref="A1.SS2.5.p1.18.m2.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.18.m2.2.2.2.4" xref="A1.SS2.5.p1.18.m2.2.2.2.3.cmml"><mi id="A1.SS2.5.p1.18.m2.1.1.1.1" xref="A1.SS2.5.p1.18.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS2.5.p1.18.m2.2.2.2.4.1" xref="A1.SS2.5.p1.18.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.18.m2.2.2.2.2" xref="A1.SS2.5.p1.18.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.18.m2.2.3.3.2.3" xref="A1.SS2.5.p1.18.m2.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.18.m2.2b"><apply id="A1.SS2.5.p1.18.m2.2.3.cmml" xref="A1.SS2.5.p1.18.m2.2.3"><notin id="A1.SS2.5.p1.18.m2.2.3.1.cmml" xref="A1.SS2.5.p1.18.m2.2.3.1"></notin><ci id="A1.SS2.5.p1.18.m2.2.3.2.cmml" xref="A1.SS2.5.p1.18.m2.2.3.2">𝑧</ci><apply id="A1.SS2.5.p1.18.m2.2.3.3.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.18.m2.2.3.3.1.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3">subscript</csymbol><apply id="A1.SS2.5.p1.18.m2.2.3.3.2.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.18.m2.2.3.3.2.1.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3">superscript</csymbol><ci id="A1.SS2.5.p1.18.m2.2.3.3.2.2.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.18.m2.2.3.3.2.3.cmml" xref="A1.SS2.5.p1.18.m2.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.18.m2.2.2.2.3.cmml" xref="A1.SS2.5.p1.18.m2.2.2.2.4"><ci id="A1.SS2.5.p1.18.m2.1.1.1.1.cmml" xref="A1.SS2.5.p1.18.m2.1.1.1.1">𝑥</ci><ci id="A1.SS2.5.p1.18.m2.2.2.2.2.cmml" xref="A1.SS2.5.p1.18.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.18.m2.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.18.m2.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be arbitrary (i.e., <math alttext="X(z)=0" class="ltx_Math" display="inline" id="A1.SS2.5.p1.19.m3.1"><semantics id="A1.SS2.5.p1.19.m3.1a"><mrow id="A1.SS2.5.p1.19.m3.1.2" xref="A1.SS2.5.p1.19.m3.1.2.cmml"><mrow id="A1.SS2.5.p1.19.m3.1.2.2" xref="A1.SS2.5.p1.19.m3.1.2.2.cmml"><mi id="A1.SS2.5.p1.19.m3.1.2.2.2" xref="A1.SS2.5.p1.19.m3.1.2.2.2.cmml">X</mi><mo id="A1.SS2.5.p1.19.m3.1.2.2.1" xref="A1.SS2.5.p1.19.m3.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.5.p1.19.m3.1.2.2.3.2" xref="A1.SS2.5.p1.19.m3.1.2.2.cmml"><mo id="A1.SS2.5.p1.19.m3.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.5.p1.19.m3.1.2.2.cmml">(</mo><mi id="A1.SS2.5.p1.19.m3.1.1" xref="A1.SS2.5.p1.19.m3.1.1.cmml">z</mi><mo id="A1.SS2.5.p1.19.m3.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.5.p1.19.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.5.p1.19.m3.1.2.1" xref="A1.SS2.5.p1.19.m3.1.2.1.cmml">=</mo><mn id="A1.SS2.5.p1.19.m3.1.2.3" xref="A1.SS2.5.p1.19.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.19.m3.1b"><apply id="A1.SS2.5.p1.19.m3.1.2.cmml" xref="A1.SS2.5.p1.19.m3.1.2"><eq id="A1.SS2.5.p1.19.m3.1.2.1.cmml" xref="A1.SS2.5.p1.19.m3.1.2.1"></eq><apply id="A1.SS2.5.p1.19.m3.1.2.2.cmml" xref="A1.SS2.5.p1.19.m3.1.2.2"><times id="A1.SS2.5.p1.19.m3.1.2.2.1.cmml" xref="A1.SS2.5.p1.19.m3.1.2.2.1"></times><ci id="A1.SS2.5.p1.19.m3.1.2.2.2.cmml" xref="A1.SS2.5.p1.19.m3.1.2.2.2">𝑋</ci><ci id="A1.SS2.5.p1.19.m3.1.1.cmml" xref="A1.SS2.5.p1.19.m3.1.1">𝑧</ci></apply><cn id="A1.SS2.5.p1.19.m3.1.2.3.cmml" type="integer" xref="A1.SS2.5.p1.19.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.19.m3.1c">X(z)=0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.19.m3.1d">italic_X ( italic_z ) = 0</annotation></semantics></math>). Then there exist <math alttext="\varepsilon,\delta>0" class="ltx_Math" display="inline" id="A1.SS2.5.p1.20.m4.2"><semantics id="A1.SS2.5.p1.20.m4.2a"><mrow id="A1.SS2.5.p1.20.m4.2.3" xref="A1.SS2.5.p1.20.m4.2.3.cmml"><mrow id="A1.SS2.5.p1.20.m4.2.3.2.2" xref="A1.SS2.5.p1.20.m4.2.3.2.1.cmml"><mi id="A1.SS2.5.p1.20.m4.1.1" xref="A1.SS2.5.p1.20.m4.1.1.cmml">ε</mi><mo id="A1.SS2.5.p1.20.m4.2.3.2.2.1" xref="A1.SS2.5.p1.20.m4.2.3.2.1.cmml">,</mo><mi id="A1.SS2.5.p1.20.m4.2.2" xref="A1.SS2.5.p1.20.m4.2.2.cmml">δ</mi></mrow><mo id="A1.SS2.5.p1.20.m4.2.3.1" xref="A1.SS2.5.p1.20.m4.2.3.1.cmml">></mo><mn id="A1.SS2.5.p1.20.m4.2.3.3" xref="A1.SS2.5.p1.20.m4.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.20.m4.2b"><apply id="A1.SS2.5.p1.20.m4.2.3.cmml" xref="A1.SS2.5.p1.20.m4.2.3"><gt id="A1.SS2.5.p1.20.m4.2.3.1.cmml" xref="A1.SS2.5.p1.20.m4.2.3.1"></gt><list id="A1.SS2.5.p1.20.m4.2.3.2.1.cmml" xref="A1.SS2.5.p1.20.m4.2.3.2.2"><ci id="A1.SS2.5.p1.20.m4.1.1.cmml" xref="A1.SS2.5.p1.20.m4.1.1">𝜀</ci><ci id="A1.SS2.5.p1.20.m4.2.2.cmml" xref="A1.SS2.5.p1.20.m4.2.2">𝛿</ci></list><cn id="A1.SS2.5.p1.20.m4.2.3.3.cmml" type="integer" xref="A1.SS2.5.p1.20.m4.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.20.m4.2c">\varepsilon,\delta>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.20.m4.2d">italic_ε , italic_δ > 0</annotation></semantics></math> such that <math alttext="\lVert x-\varepsilon v-z\rVert_{p}<\lVert x-z\rVert_{p}-\delta" class="ltx_Math" display="inline" id="A1.SS2.5.p1.21.m5.2"><semantics id="A1.SS2.5.p1.21.m5.2a"><mrow id="A1.SS2.5.p1.21.m5.2.2" xref="A1.SS2.5.p1.21.m5.2.2.cmml"><msub id="A1.SS2.5.p1.21.m5.1.1.1" xref="A1.SS2.5.p1.21.m5.1.1.1.cmml"><mrow id="A1.SS2.5.p1.21.m5.1.1.1.1.1" xref="A1.SS2.5.p1.21.m5.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS2.5.p1.21.m5.1.1.1.1.1.2" rspace="0em" xref="A1.SS2.5.p1.21.m5.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.cmml"><mi id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.2" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.cmml"><mi id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.2" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.1" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.3" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1a" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.4" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.5.p1.21.m5.1.1.1.1.1.3" lspace="0em" xref="A1.SS2.5.p1.21.m5.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.5.p1.21.m5.1.1.1.3" xref="A1.SS2.5.p1.21.m5.1.1.1.3.cmml">p</mi></msub><mo id="A1.SS2.5.p1.21.m5.2.2.3" rspace="0.1389em" xref="A1.SS2.5.p1.21.m5.2.2.3.cmml"><</mo><mrow id="A1.SS2.5.p1.21.m5.2.2.2" xref="A1.SS2.5.p1.21.m5.2.2.2.cmml"><msub id="A1.SS2.5.p1.21.m5.2.2.2.1" xref="A1.SS2.5.p1.21.m5.2.2.2.1.cmml"><mrow id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.2.cmml"><mo fence="true" id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.cmml"><mi id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.2" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.1" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.3" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.3" lspace="0em" rspace="0em" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.5.p1.21.m5.2.2.2.1.3" xref="A1.SS2.5.p1.21.m5.2.2.2.1.3.cmml">p</mi></msub><mo id="A1.SS2.5.p1.21.m5.2.2.2.2" xref="A1.SS2.5.p1.21.m5.2.2.2.2.cmml">−</mo><mi id="A1.SS2.5.p1.21.m5.2.2.2.3" xref="A1.SS2.5.p1.21.m5.2.2.2.3.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.21.m5.2b"><apply id="A1.SS2.5.p1.21.m5.2.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2"><lt id="A1.SS2.5.p1.21.m5.2.2.3.cmml" xref="A1.SS2.5.p1.21.m5.2.2.3"></lt><apply id="A1.SS2.5.p1.21.m5.1.1.1.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.21.m5.1.1.1.2.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1">subscript</csymbol><apply id="A1.SS2.5.p1.21.m5.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS2.5.p1.21.m5.1.1.1.1.2.1.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1"><minus id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.1"></minus><ci id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.2">𝑥</ci><apply id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3"><times id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.1.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.1"></times><ci id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.2.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.2">𝜀</ci><ci id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.3.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.4.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.SS2.5.p1.21.m5.1.1.1.3.cmml" xref="A1.SS2.5.p1.21.m5.1.1.1.3">𝑝</ci></apply><apply id="A1.SS2.5.p1.21.m5.2.2.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2"><minus id="A1.SS2.5.p1.21.m5.2.2.2.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.2"></minus><apply id="A1.SS2.5.p1.21.m5.2.2.2.1.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.21.m5.2.2.2.1.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1">subscript</csymbol><apply id="A1.SS2.5.p1.21.m5.2.2.2.1.1.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1"><csymbol cd="latexml" id="A1.SS2.5.p1.21.m5.2.2.2.1.1.2.1.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1"><minus id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.1"></minus><ci id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.2">𝑥</ci><ci id="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.3.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.SS2.5.p1.21.m5.2.2.2.1.3.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.1.3">𝑝</ci></apply><ci id="A1.SS2.5.p1.21.m5.2.2.2.3.cmml" xref="A1.SS2.5.p1.21.m5.2.2.2.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.21.m5.2c">\lVert x-\varepsilon v-z\rVert_{p}<\lVert x-z\rVert_{p}-\delta</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.21.m5.2d">∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT < ∥ italic_x - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_δ</annotation></semantics></math>. Thus, for <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.22.m6.1"><semantics id="A1.SS2.5.p1.22.m6.1a"><mrow id="A1.SS2.5.p1.22.m6.1.1" xref="A1.SS2.5.p1.22.m6.1.1.cmml"><mi id="A1.SS2.5.p1.22.m6.1.1.2" xref="A1.SS2.5.p1.22.m6.1.1.2.cmml">n</mi><mo id="A1.SS2.5.p1.22.m6.1.1.1" xref="A1.SS2.5.p1.22.m6.1.1.1.cmml">∈</mo><mi id="A1.SS2.5.p1.22.m6.1.1.3" xref="A1.SS2.5.p1.22.m6.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.22.m6.1b"><apply id="A1.SS2.5.p1.22.m6.1.1.cmml" xref="A1.SS2.5.p1.22.m6.1.1"><in id="A1.SS2.5.p1.22.m6.1.1.1.cmml" xref="A1.SS2.5.p1.22.m6.1.1.1"></in><ci id="A1.SS2.5.p1.22.m6.1.1.2.cmml" xref="A1.SS2.5.p1.22.m6.1.1.2">𝑛</ci><ci id="A1.SS2.5.p1.22.m6.1.1.3.cmml" xref="A1.SS2.5.p1.22.m6.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.22.m6.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.22.m6.1d">italic_n ∈ blackboard_N</annotation></semantics></math> large enough we also have <math alttext="X_{n}(z)=0" class="ltx_Math" display="inline" id="A1.SS2.5.p1.23.m7.1"><semantics id="A1.SS2.5.p1.23.m7.1a"><mrow id="A1.SS2.5.p1.23.m7.1.2" xref="A1.SS2.5.p1.23.m7.1.2.cmml"><mrow id="A1.SS2.5.p1.23.m7.1.2.2" xref="A1.SS2.5.p1.23.m7.1.2.2.cmml"><msub id="A1.SS2.5.p1.23.m7.1.2.2.2" xref="A1.SS2.5.p1.23.m7.1.2.2.2.cmml"><mi id="A1.SS2.5.p1.23.m7.1.2.2.2.2" xref="A1.SS2.5.p1.23.m7.1.2.2.2.2.cmml">X</mi><mi id="A1.SS2.5.p1.23.m7.1.2.2.2.3" xref="A1.SS2.5.p1.23.m7.1.2.2.2.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.23.m7.1.2.2.1" xref="A1.SS2.5.p1.23.m7.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.5.p1.23.m7.1.2.2.3.2" xref="A1.SS2.5.p1.23.m7.1.2.2.cmml"><mo id="A1.SS2.5.p1.23.m7.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.5.p1.23.m7.1.2.2.cmml">(</mo><mi id="A1.SS2.5.p1.23.m7.1.1" xref="A1.SS2.5.p1.23.m7.1.1.cmml">z</mi><mo id="A1.SS2.5.p1.23.m7.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.5.p1.23.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.5.p1.23.m7.1.2.1" xref="A1.SS2.5.p1.23.m7.1.2.1.cmml">=</mo><mn id="A1.SS2.5.p1.23.m7.1.2.3" xref="A1.SS2.5.p1.23.m7.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.23.m7.1b"><apply id="A1.SS2.5.p1.23.m7.1.2.cmml" xref="A1.SS2.5.p1.23.m7.1.2"><eq id="A1.SS2.5.p1.23.m7.1.2.1.cmml" xref="A1.SS2.5.p1.23.m7.1.2.1"></eq><apply id="A1.SS2.5.p1.23.m7.1.2.2.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2"><times id="A1.SS2.5.p1.23.m7.1.2.2.1.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2.1"></times><apply id="A1.SS2.5.p1.23.m7.1.2.2.2.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.5.p1.23.m7.1.2.2.2.1.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2.2">subscript</csymbol><ci id="A1.SS2.5.p1.23.m7.1.2.2.2.2.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2.2.2">𝑋</ci><ci id="A1.SS2.5.p1.23.m7.1.2.2.2.3.cmml" xref="A1.SS2.5.p1.23.m7.1.2.2.2.3">𝑛</ci></apply><ci id="A1.SS2.5.p1.23.m7.1.1.cmml" xref="A1.SS2.5.p1.23.m7.1.1">𝑧</ci></apply><cn id="A1.SS2.5.p1.23.m7.1.2.3.cmml" type="integer" xref="A1.SS2.5.p1.23.m7.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.23.m7.1c">X_{n}(z)=0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.23.m7.1d">italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_z ) = 0</annotation></semantics></math>. Consider now an arbitrary <math alttext="z" class="ltx_Math" display="inline" id="A1.SS2.5.p1.24.m8.1"><semantics id="A1.SS2.5.p1.24.m8.1a"><mi id="A1.SS2.5.p1.24.m8.1.1" xref="A1.SS2.5.p1.24.m8.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.24.m8.1b"><ci id="A1.SS2.5.p1.24.m8.1.1.cmml" xref="A1.SS2.5.p1.24.m8.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.24.m8.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.24.m8.1d">italic_z</annotation></semantics></math> in the interior of <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.25.m9.2"><semantics id="A1.SS2.5.p1.25.m9.2a"><msubsup id="A1.SS2.5.p1.25.m9.2.3" xref="A1.SS2.5.p1.25.m9.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.25.m9.2.3.2.2" xref="A1.SS2.5.p1.25.m9.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.25.m9.2.2.2.4" xref="A1.SS2.5.p1.25.m9.2.2.2.3.cmml"><mi id="A1.SS2.5.p1.25.m9.1.1.1.1" xref="A1.SS2.5.p1.25.m9.1.1.1.1.cmml">x</mi><mo id="A1.SS2.5.p1.25.m9.2.2.2.4.1" xref="A1.SS2.5.p1.25.m9.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.25.m9.2.2.2.2" xref="A1.SS2.5.p1.25.m9.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.25.m9.2.3.2.3" xref="A1.SS2.5.p1.25.m9.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.25.m9.2b"><apply id="A1.SS2.5.p1.25.m9.2.3.cmml" xref="A1.SS2.5.p1.25.m9.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.25.m9.2.3.1.cmml" xref="A1.SS2.5.p1.25.m9.2.3">subscript</csymbol><apply id="A1.SS2.5.p1.25.m9.2.3.2.cmml" xref="A1.SS2.5.p1.25.m9.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.25.m9.2.3.2.1.cmml" xref="A1.SS2.5.p1.25.m9.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.25.m9.2.3.2.2.cmml" xref="A1.SS2.5.p1.25.m9.2.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.25.m9.2.3.2.3.cmml" xref="A1.SS2.5.p1.25.m9.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.25.m9.2.2.2.3.cmml" xref="A1.SS2.5.p1.25.m9.2.2.2.4"><ci id="A1.SS2.5.p1.25.m9.1.1.1.1.cmml" xref="A1.SS2.5.p1.25.m9.1.1.1.1">𝑥</ci><ci id="A1.SS2.5.p1.25.m9.2.2.2.2.cmml" xref="A1.SS2.5.p1.25.m9.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.25.m9.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.25.m9.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. By definition, there exists <math alttext="\delta>0" class="ltx_Math" display="inline" id="A1.SS2.5.p1.26.m10.1"><semantics id="A1.SS2.5.p1.26.m10.1a"><mrow id="A1.SS2.5.p1.26.m10.1.1" xref="A1.SS2.5.p1.26.m10.1.1.cmml"><mi id="A1.SS2.5.p1.26.m10.1.1.2" xref="A1.SS2.5.p1.26.m10.1.1.2.cmml">δ</mi><mo id="A1.SS2.5.p1.26.m10.1.1.1" xref="A1.SS2.5.p1.26.m10.1.1.1.cmml">></mo><mn id="A1.SS2.5.p1.26.m10.1.1.3" xref="A1.SS2.5.p1.26.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.26.m10.1b"><apply id="A1.SS2.5.p1.26.m10.1.1.cmml" xref="A1.SS2.5.p1.26.m10.1.1"><gt id="A1.SS2.5.p1.26.m10.1.1.1.cmml" xref="A1.SS2.5.p1.26.m10.1.1.1"></gt><ci id="A1.SS2.5.p1.26.m10.1.1.2.cmml" xref="A1.SS2.5.p1.26.m10.1.1.2">𝛿</ci><cn id="A1.SS2.5.p1.26.m10.1.1.3.cmml" type="integer" xref="A1.SS2.5.p1.26.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.26.m10.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.26.m10.1d">italic_δ > 0</annotation></semantics></math> such that a <math alttext="\delta" class="ltx_Math" display="inline" id="A1.SS2.5.p1.27.m11.1"><semantics id="A1.SS2.5.p1.27.m11.1a"><mi id="A1.SS2.5.p1.27.m11.1.1" xref="A1.SS2.5.p1.27.m11.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.27.m11.1b"><ci id="A1.SS2.5.p1.27.m11.1.1.cmml" xref="A1.SS2.5.p1.27.m11.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.27.m11.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.27.m11.1d">italic_δ</annotation></semantics></math>-ball around <math alttext="z" class="ltx_Math" display="inline" id="A1.SS2.5.p1.28.m12.1"><semantics id="A1.SS2.5.p1.28.m12.1a"><mi id="A1.SS2.5.p1.28.m12.1.1" xref="A1.SS2.5.p1.28.m12.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.28.m12.1b"><ci id="A1.SS2.5.p1.28.m12.1.1.cmml" xref="A1.SS2.5.p1.28.m12.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.28.m12.1c">z</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.28.m12.1d">italic_z</annotation></semantics></math> is contained in <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.29.m13.2"><semantics id="A1.SS2.5.p1.29.m13.2a"><msubsup id="A1.SS2.5.p1.29.m13.2.3" xref="A1.SS2.5.p1.29.m13.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.29.m13.2.3.2.2" xref="A1.SS2.5.p1.29.m13.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.29.m13.2.2.2.4" xref="A1.SS2.5.p1.29.m13.2.2.2.3.cmml"><mi id="A1.SS2.5.p1.29.m13.1.1.1.1" xref="A1.SS2.5.p1.29.m13.1.1.1.1.cmml">x</mi><mo id="A1.SS2.5.p1.29.m13.2.2.2.4.1" xref="A1.SS2.5.p1.29.m13.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.29.m13.2.2.2.2" xref="A1.SS2.5.p1.29.m13.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.29.m13.2.3.2.3" xref="A1.SS2.5.p1.29.m13.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.29.m13.2b"><apply id="A1.SS2.5.p1.29.m13.2.3.cmml" xref="A1.SS2.5.p1.29.m13.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.29.m13.2.3.1.cmml" xref="A1.SS2.5.p1.29.m13.2.3">subscript</csymbol><apply id="A1.SS2.5.p1.29.m13.2.3.2.cmml" xref="A1.SS2.5.p1.29.m13.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.29.m13.2.3.2.1.cmml" xref="A1.SS2.5.p1.29.m13.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.29.m13.2.3.2.2.cmml" xref="A1.SS2.5.p1.29.m13.2.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.29.m13.2.3.2.3.cmml" xref="A1.SS2.5.p1.29.m13.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.29.m13.2.2.2.3.cmml" xref="A1.SS2.5.p1.29.m13.2.2.2.4"><ci id="A1.SS2.5.p1.29.m13.1.1.1.1.cmml" xref="A1.SS2.5.p1.29.m13.1.1.1.1">𝑥</ci><ci id="A1.SS2.5.p1.29.m13.2.2.2.2.cmml" xref="A1.SS2.5.p1.29.m13.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.29.m13.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.29.m13.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. In other words, we have</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lVert x-\varepsilon v-z^{\prime}\rVert_{p}\geq\lVert x-z^{\prime}\rVert_{p}" class="ltx_Math" display="block" id="A1.Ex19.m1.2"><semantics id="A1.Ex19.m1.2a"><mrow id="A1.Ex19.m1.2.2" xref="A1.Ex19.m1.2.2.cmml"><msub id="A1.Ex19.m1.1.1.1" xref="A1.Ex19.m1.1.1.1.cmml"><mrow id="A1.Ex19.m1.1.1.1.1.1" xref="A1.Ex19.m1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex19.m1.1.1.1.1.1.2" rspace="0em" xref="A1.Ex19.m1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex19.m1.1.1.1.1.1.1" xref="A1.Ex19.m1.1.1.1.1.1.1.cmml"><mi id="A1.Ex19.m1.1.1.1.1.1.1.2" xref="A1.Ex19.m1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.Ex19.m1.1.1.1.1.1.1.1" xref="A1.Ex19.m1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex19.m1.1.1.1.1.1.1.3" xref="A1.Ex19.m1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex19.m1.1.1.1.1.1.1.3.2" xref="A1.Ex19.m1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex19.m1.1.1.1.1.1.1.3.1" xref="A1.Ex19.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex19.m1.1.1.1.1.1.1.3.3" xref="A1.Ex19.m1.1.1.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.Ex19.m1.1.1.1.1.1.1.1a" xref="A1.Ex19.m1.1.1.1.1.1.1.1.cmml">−</mo><msup id="A1.Ex19.m1.1.1.1.1.1.1.4" xref="A1.Ex19.m1.1.1.1.1.1.1.4.cmml"><mi id="A1.Ex19.m1.1.1.1.1.1.1.4.2" xref="A1.Ex19.m1.1.1.1.1.1.1.4.2.cmml">z</mi><mo id="A1.Ex19.m1.1.1.1.1.1.1.4.3" xref="A1.Ex19.m1.1.1.1.1.1.1.4.3.cmml">′</mo></msup></mrow><mo fence="true" id="A1.Ex19.m1.1.1.1.1.1.3" lspace="0em" xref="A1.Ex19.m1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex19.m1.1.1.1.3" xref="A1.Ex19.m1.1.1.1.3.cmml">p</mi></msub><mo id="A1.Ex19.m1.2.2.3" rspace="0.1389em" xref="A1.Ex19.m1.2.2.3.cmml">≥</mo><msub id="A1.Ex19.m1.2.2.2" xref="A1.Ex19.m1.2.2.2.cmml"><mrow id="A1.Ex19.m1.2.2.2.1.1" xref="A1.Ex19.m1.2.2.2.1.2.cmml"><mo fence="true" id="A1.Ex19.m1.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex19.m1.2.2.2.1.2.1.cmml">∥</mo><mrow id="A1.Ex19.m1.2.2.2.1.1.1" xref="A1.Ex19.m1.2.2.2.1.1.1.cmml"><mi id="A1.Ex19.m1.2.2.2.1.1.1.2" xref="A1.Ex19.m1.2.2.2.1.1.1.2.cmml">x</mi><mo id="A1.Ex19.m1.2.2.2.1.1.1.1" xref="A1.Ex19.m1.2.2.2.1.1.1.1.cmml">−</mo><msup id="A1.Ex19.m1.2.2.2.1.1.1.3" xref="A1.Ex19.m1.2.2.2.1.1.1.3.cmml"><mi id="A1.Ex19.m1.2.2.2.1.1.1.3.2" xref="A1.Ex19.m1.2.2.2.1.1.1.3.2.cmml">z</mi><mo id="A1.Ex19.m1.2.2.2.1.1.1.3.3" xref="A1.Ex19.m1.2.2.2.1.1.1.3.3.cmml">′</mo></msup></mrow><mo fence="true" id="A1.Ex19.m1.2.2.2.1.1.3" lspace="0em" xref="A1.Ex19.m1.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex19.m1.2.2.2.3" xref="A1.Ex19.m1.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex19.m1.2b"><apply id="A1.Ex19.m1.2.2.cmml" xref="A1.Ex19.m1.2.2"><geq id="A1.Ex19.m1.2.2.3.cmml" xref="A1.Ex19.m1.2.2.3"></geq><apply id="A1.Ex19.m1.1.1.1.cmml" xref="A1.Ex19.m1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex19.m1.1.1.1.2.cmml" xref="A1.Ex19.m1.1.1.1">subscript</csymbol><apply id="A1.Ex19.m1.1.1.1.1.2.cmml" xref="A1.Ex19.m1.1.1.1.1.1"><csymbol cd="latexml" id="A1.Ex19.m1.1.1.1.1.2.1.cmml" xref="A1.Ex19.m1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex19.m1.1.1.1.1.1.1.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1"><minus id="A1.Ex19.m1.1.1.1.1.1.1.1.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.1"></minus><ci id="A1.Ex19.m1.1.1.1.1.1.1.2.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.2">𝑥</ci><apply id="A1.Ex19.m1.1.1.1.1.1.1.3.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.3"><times id="A1.Ex19.m1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.3.1"></times><ci id="A1.Ex19.m1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.3.2">𝜀</ci><ci id="A1.Ex19.m1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.3.3">𝑣</ci></apply><apply id="A1.Ex19.m1.1.1.1.1.1.1.4.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex19.m1.1.1.1.1.1.1.4.1.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.4">superscript</csymbol><ci id="A1.Ex19.m1.1.1.1.1.1.1.4.2.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.4.2">𝑧</ci><ci id="A1.Ex19.m1.1.1.1.1.1.1.4.3.cmml" xref="A1.Ex19.m1.1.1.1.1.1.1.4.3">′</ci></apply></apply></apply><ci id="A1.Ex19.m1.1.1.1.3.cmml" xref="A1.Ex19.m1.1.1.1.3">𝑝</ci></apply><apply id="A1.Ex19.m1.2.2.2.cmml" xref="A1.Ex19.m1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex19.m1.2.2.2.2.cmml" xref="A1.Ex19.m1.2.2.2">subscript</csymbol><apply id="A1.Ex19.m1.2.2.2.1.2.cmml" xref="A1.Ex19.m1.2.2.2.1.1"><csymbol cd="latexml" id="A1.Ex19.m1.2.2.2.1.2.1.cmml" xref="A1.Ex19.m1.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex19.m1.2.2.2.1.1.1.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1"><minus id="A1.Ex19.m1.2.2.2.1.1.1.1.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.1"></minus><ci id="A1.Ex19.m1.2.2.2.1.1.1.2.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.2">𝑥</ci><apply id="A1.Ex19.m1.2.2.2.1.1.1.3.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex19.m1.2.2.2.1.1.1.3.1.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.3">superscript</csymbol><ci id="A1.Ex19.m1.2.2.2.1.1.1.3.2.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.3.2">𝑧</ci><ci id="A1.Ex19.m1.2.2.2.1.1.1.3.3.cmml" xref="A1.Ex19.m1.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply><ci id="A1.Ex19.m1.2.2.2.3.cmml" xref="A1.Ex19.m1.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex19.m1.2c">\lVert x-\varepsilon v-z^{\prime}\rVert_{p}\geq\lVert x-z^{\prime}\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex19.m1.2d">∥ italic_x - italic_ε italic_v - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ ∥ italic_x - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.5.p1.35">for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.SS2.5.p1.30.m1.1"><semantics id="A1.SS2.5.p1.30.m1.1a"><mrow id="A1.SS2.5.p1.30.m1.1.1" xref="A1.SS2.5.p1.30.m1.1.1.cmml"><mi id="A1.SS2.5.p1.30.m1.1.1.2" xref="A1.SS2.5.p1.30.m1.1.1.2.cmml">ε</mi><mo id="A1.SS2.5.p1.30.m1.1.1.1" xref="A1.SS2.5.p1.30.m1.1.1.1.cmml">></mo><mn id="A1.SS2.5.p1.30.m1.1.1.3" xref="A1.SS2.5.p1.30.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.30.m1.1b"><apply id="A1.SS2.5.p1.30.m1.1.1.cmml" xref="A1.SS2.5.p1.30.m1.1.1"><gt id="A1.SS2.5.p1.30.m1.1.1.1.cmml" xref="A1.SS2.5.p1.30.m1.1.1.1"></gt><ci id="A1.SS2.5.p1.30.m1.1.1.2.cmml" xref="A1.SS2.5.p1.30.m1.1.1.2">𝜀</ci><cn id="A1.SS2.5.p1.30.m1.1.1.3.cmml" type="integer" xref="A1.SS2.5.p1.30.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.30.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.30.m1.1d">italic_ε > 0</annotation></semantics></math> and <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.31.m2.1"><semantics id="A1.SS2.5.p1.31.m2.1a"><msup id="A1.SS2.5.p1.31.m2.1.1" xref="A1.SS2.5.p1.31.m2.1.1.cmml"><mi id="A1.SS2.5.p1.31.m2.1.1.2" xref="A1.SS2.5.p1.31.m2.1.1.2.cmml">z</mi><mo id="A1.SS2.5.p1.31.m2.1.1.3" xref="A1.SS2.5.p1.31.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.31.m2.1b"><apply id="A1.SS2.5.p1.31.m2.1.1.cmml" xref="A1.SS2.5.p1.31.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.31.m2.1.1.1.cmml" xref="A1.SS2.5.p1.31.m2.1.1">superscript</csymbol><ci id="A1.SS2.5.p1.31.m2.1.1.2.cmml" xref="A1.SS2.5.p1.31.m2.1.1.2">𝑧</ci><ci id="A1.SS2.5.p1.31.m2.1.1.3.cmml" xref="A1.SS2.5.p1.31.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.31.m2.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.31.m2.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\lVert z^{\prime}-z\rVert_{p}\leq\delta" class="ltx_Math" display="inline" id="A1.SS2.5.p1.32.m3.1"><semantics id="A1.SS2.5.p1.32.m3.1a"><mrow id="A1.SS2.5.p1.32.m3.1.1" xref="A1.SS2.5.p1.32.m3.1.1.cmml"><msub id="A1.SS2.5.p1.32.m3.1.1.1" xref="A1.SS2.5.p1.32.m3.1.1.1.cmml"><mrow id="A1.SS2.5.p1.32.m3.1.1.1.1.1" xref="A1.SS2.5.p1.32.m3.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS2.5.p1.32.m3.1.1.1.1.1.2" rspace="0em" xref="A1.SS2.5.p1.32.m3.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.cmml"><msup id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.cmml"><mi id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.2" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.2.cmml">z</mi><mo id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.3" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.1" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.3" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.5.p1.32.m3.1.1.1.1.1.3" lspace="0em" xref="A1.SS2.5.p1.32.m3.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.5.p1.32.m3.1.1.1.3" xref="A1.SS2.5.p1.32.m3.1.1.1.3.cmml">p</mi></msub><mo id="A1.SS2.5.p1.32.m3.1.1.2" xref="A1.SS2.5.p1.32.m3.1.1.2.cmml">≤</mo><mi id="A1.SS2.5.p1.32.m3.1.1.3" xref="A1.SS2.5.p1.32.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.32.m3.1b"><apply id="A1.SS2.5.p1.32.m3.1.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1"><leq id="A1.SS2.5.p1.32.m3.1.1.2.cmml" xref="A1.SS2.5.p1.32.m3.1.1.2"></leq><apply id="A1.SS2.5.p1.32.m3.1.1.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.32.m3.1.1.1.2.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1">subscript</csymbol><apply id="A1.SS2.5.p1.32.m3.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS2.5.p1.32.m3.1.1.1.1.2.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1"><minus id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.1"></minus><apply id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.1.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2">superscript</csymbol><ci id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.2.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.2">𝑧</ci><ci id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.3.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.2.3">′</ci></apply><ci id="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.3.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.SS2.5.p1.32.m3.1.1.1.3.cmml" xref="A1.SS2.5.p1.32.m3.1.1.1.3">𝑝</ci></apply><ci id="A1.SS2.5.p1.32.m3.1.1.3.cmml" xref="A1.SS2.5.p1.32.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.32.m3.1c">\lVert z^{\prime}-z\rVert_{p}\leq\delta</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.32.m3.1d">∥ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_δ</annotation></semantics></math>. For large enough <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.33.m4.1"><semantics id="A1.SS2.5.p1.33.m4.1a"><mrow id="A1.SS2.5.p1.33.m4.1.1" xref="A1.SS2.5.p1.33.m4.1.1.cmml"><mi id="A1.SS2.5.p1.33.m4.1.1.2" xref="A1.SS2.5.p1.33.m4.1.1.2.cmml">n</mi><mo id="A1.SS2.5.p1.33.m4.1.1.1" xref="A1.SS2.5.p1.33.m4.1.1.1.cmml">∈</mo><mi id="A1.SS2.5.p1.33.m4.1.1.3" xref="A1.SS2.5.p1.33.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.33.m4.1b"><apply id="A1.SS2.5.p1.33.m4.1.1.cmml" xref="A1.SS2.5.p1.33.m4.1.1"><in id="A1.SS2.5.p1.33.m4.1.1.1.cmml" xref="A1.SS2.5.p1.33.m4.1.1.1"></in><ci id="A1.SS2.5.p1.33.m4.1.1.2.cmml" xref="A1.SS2.5.p1.33.m4.1.1.2">𝑛</ci><ci id="A1.SS2.5.p1.33.m4.1.1.3.cmml" xref="A1.SS2.5.p1.33.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.33.m4.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.33.m4.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, we can therefore choose <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.34.m5.1"><semantics id="A1.SS2.5.p1.34.m5.1a"><msup id="A1.SS2.5.p1.34.m5.1.1" xref="A1.SS2.5.p1.34.m5.1.1.cmml"><mi id="A1.SS2.5.p1.34.m5.1.1.2" xref="A1.SS2.5.p1.34.m5.1.1.2.cmml">z</mi><mo id="A1.SS2.5.p1.34.m5.1.1.3" xref="A1.SS2.5.p1.34.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.34.m5.1b"><apply id="A1.SS2.5.p1.34.m5.1.1.cmml" xref="A1.SS2.5.p1.34.m5.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.34.m5.1.1.1.cmml" xref="A1.SS2.5.p1.34.m5.1.1">superscript</csymbol><ci id="A1.SS2.5.p1.34.m5.1.1.2.cmml" xref="A1.SS2.5.p1.34.m5.1.1.2">𝑧</ci><ci id="A1.SS2.5.p1.34.m5.1.1.3.cmml" xref="A1.SS2.5.p1.34.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.34.m5.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.34.m5.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="x-z^{\prime}=x_{n}-z" class="ltx_Math" display="inline" id="A1.SS2.5.p1.35.m6.1"><semantics id="A1.SS2.5.p1.35.m6.1a"><mrow id="A1.SS2.5.p1.35.m6.1.1" xref="A1.SS2.5.p1.35.m6.1.1.cmml"><mrow id="A1.SS2.5.p1.35.m6.1.1.2" xref="A1.SS2.5.p1.35.m6.1.1.2.cmml"><mi id="A1.SS2.5.p1.35.m6.1.1.2.2" xref="A1.SS2.5.p1.35.m6.1.1.2.2.cmml">x</mi><mo id="A1.SS2.5.p1.35.m6.1.1.2.1" xref="A1.SS2.5.p1.35.m6.1.1.2.1.cmml">−</mo><msup id="A1.SS2.5.p1.35.m6.1.1.2.3" xref="A1.SS2.5.p1.35.m6.1.1.2.3.cmml"><mi id="A1.SS2.5.p1.35.m6.1.1.2.3.2" xref="A1.SS2.5.p1.35.m6.1.1.2.3.2.cmml">z</mi><mo id="A1.SS2.5.p1.35.m6.1.1.2.3.3" xref="A1.SS2.5.p1.35.m6.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="A1.SS2.5.p1.35.m6.1.1.1" xref="A1.SS2.5.p1.35.m6.1.1.1.cmml">=</mo><mrow id="A1.SS2.5.p1.35.m6.1.1.3" xref="A1.SS2.5.p1.35.m6.1.1.3.cmml"><msub id="A1.SS2.5.p1.35.m6.1.1.3.2" xref="A1.SS2.5.p1.35.m6.1.1.3.2.cmml"><mi id="A1.SS2.5.p1.35.m6.1.1.3.2.2" xref="A1.SS2.5.p1.35.m6.1.1.3.2.2.cmml">x</mi><mi id="A1.SS2.5.p1.35.m6.1.1.3.2.3" xref="A1.SS2.5.p1.35.m6.1.1.3.2.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.35.m6.1.1.3.1" xref="A1.SS2.5.p1.35.m6.1.1.3.1.cmml">−</mo><mi id="A1.SS2.5.p1.35.m6.1.1.3.3" xref="A1.SS2.5.p1.35.m6.1.1.3.3.cmml">z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.35.m6.1b"><apply id="A1.SS2.5.p1.35.m6.1.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1"><eq id="A1.SS2.5.p1.35.m6.1.1.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1.1"></eq><apply id="A1.SS2.5.p1.35.m6.1.1.2.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2"><minus id="A1.SS2.5.p1.35.m6.1.1.2.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.1"></minus><ci id="A1.SS2.5.p1.35.m6.1.1.2.2.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.2">𝑥</ci><apply id="A1.SS2.5.p1.35.m6.1.1.2.3.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.35.m6.1.1.2.3.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.35.m6.1.1.2.3.2.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.3.2">𝑧</ci><ci id="A1.SS2.5.p1.35.m6.1.1.2.3.3.cmml" xref="A1.SS2.5.p1.35.m6.1.1.2.3.3">′</ci></apply></apply><apply id="A1.SS2.5.p1.35.m6.1.1.3.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3"><minus id="A1.SS2.5.p1.35.m6.1.1.3.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.1"></minus><apply id="A1.SS2.5.p1.35.m6.1.1.3.2.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.5.p1.35.m6.1.1.3.2.1.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.2">subscript</csymbol><ci id="A1.SS2.5.p1.35.m6.1.1.3.2.2.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.2.2">𝑥</ci><ci id="A1.SS2.5.p1.35.m6.1.1.3.2.3.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.2.3">𝑛</ci></apply><ci id="A1.SS2.5.p1.35.m6.1.1.3.3.cmml" xref="A1.SS2.5.p1.35.m6.1.1.3.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.35.m6.1c">x-z^{\prime}=x_{n}-z</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.35.m6.1d">italic_x - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_z</annotation></semantics></math>, which yields</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lVert x_{n}-\varepsilon v-z\rVert_{p}\geq\lVert x_{n}-z\rVert_{p}" class="ltx_Math" display="block" id="A1.Ex20.m1.2"><semantics id="A1.Ex20.m1.2a"><mrow id="A1.Ex20.m1.2.2" xref="A1.Ex20.m1.2.2.cmml"><msub id="A1.Ex20.m1.1.1.1" xref="A1.Ex20.m1.1.1.1.cmml"><mrow id="A1.Ex20.m1.1.1.1.1.1" xref="A1.Ex20.m1.1.1.1.1.2.cmml"><mo fence="true" id="A1.Ex20.m1.1.1.1.1.1.2" rspace="0em" xref="A1.Ex20.m1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.Ex20.m1.1.1.1.1.1.1" xref="A1.Ex20.m1.1.1.1.1.1.1.cmml"><msub id="A1.Ex20.m1.1.1.1.1.1.1.2" xref="A1.Ex20.m1.1.1.1.1.1.1.2.cmml"><mi id="A1.Ex20.m1.1.1.1.1.1.1.2.2" xref="A1.Ex20.m1.1.1.1.1.1.1.2.2.cmml">x</mi><mi id="A1.Ex20.m1.1.1.1.1.1.1.2.3" xref="A1.Ex20.m1.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="A1.Ex20.m1.1.1.1.1.1.1.1" xref="A1.Ex20.m1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.Ex20.m1.1.1.1.1.1.1.3" xref="A1.Ex20.m1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex20.m1.1.1.1.1.1.1.3.2" xref="A1.Ex20.m1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.Ex20.m1.1.1.1.1.1.1.3.1" xref="A1.Ex20.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.Ex20.m1.1.1.1.1.1.1.3.3" xref="A1.Ex20.m1.1.1.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.Ex20.m1.1.1.1.1.1.1.1a" xref="A1.Ex20.m1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex20.m1.1.1.1.1.1.1.4" xref="A1.Ex20.m1.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.Ex20.m1.1.1.1.1.1.3" lspace="0em" xref="A1.Ex20.m1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex20.m1.1.1.1.3" xref="A1.Ex20.m1.1.1.1.3.cmml">p</mi></msub><mo id="A1.Ex20.m1.2.2.3" rspace="0.1389em" xref="A1.Ex20.m1.2.2.3.cmml">≥</mo><msub id="A1.Ex20.m1.2.2.2" xref="A1.Ex20.m1.2.2.2.cmml"><mrow id="A1.Ex20.m1.2.2.2.1.1" xref="A1.Ex20.m1.2.2.2.1.2.cmml"><mo fence="true" id="A1.Ex20.m1.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.Ex20.m1.2.2.2.1.2.1.cmml">∥</mo><mrow id="A1.Ex20.m1.2.2.2.1.1.1" xref="A1.Ex20.m1.2.2.2.1.1.1.cmml"><msub id="A1.Ex20.m1.2.2.2.1.1.1.2" xref="A1.Ex20.m1.2.2.2.1.1.1.2.cmml"><mi id="A1.Ex20.m1.2.2.2.1.1.1.2.2" xref="A1.Ex20.m1.2.2.2.1.1.1.2.2.cmml">x</mi><mi id="A1.Ex20.m1.2.2.2.1.1.1.2.3" xref="A1.Ex20.m1.2.2.2.1.1.1.2.3.cmml">n</mi></msub><mo id="A1.Ex20.m1.2.2.2.1.1.1.1" xref="A1.Ex20.m1.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.Ex20.m1.2.2.2.1.1.1.3" xref="A1.Ex20.m1.2.2.2.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.Ex20.m1.2.2.2.1.1.3" lspace="0em" xref="A1.Ex20.m1.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A1.Ex20.m1.2.2.2.3" xref="A1.Ex20.m1.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex20.m1.2b"><apply id="A1.Ex20.m1.2.2.cmml" xref="A1.Ex20.m1.2.2"><geq id="A1.Ex20.m1.2.2.3.cmml" xref="A1.Ex20.m1.2.2.3"></geq><apply id="A1.Ex20.m1.1.1.1.cmml" xref="A1.Ex20.m1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex20.m1.1.1.1.2.cmml" xref="A1.Ex20.m1.1.1.1">subscript</csymbol><apply id="A1.Ex20.m1.1.1.1.1.2.cmml" xref="A1.Ex20.m1.1.1.1.1.1"><csymbol cd="latexml" id="A1.Ex20.m1.1.1.1.1.2.1.cmml" xref="A1.Ex20.m1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex20.m1.1.1.1.1.1.1.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1"><minus id="A1.Ex20.m1.1.1.1.1.1.1.1.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.1"></minus><apply id="A1.Ex20.m1.1.1.1.1.1.1.2.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex20.m1.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="A1.Ex20.m1.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.2.2">𝑥</ci><ci id="A1.Ex20.m1.1.1.1.1.1.1.2.3.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="A1.Ex20.m1.1.1.1.1.1.1.3.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.3"><times id="A1.Ex20.m1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.3.1"></times><ci id="A1.Ex20.m1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.3.2">𝜀</ci><ci id="A1.Ex20.m1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.Ex20.m1.1.1.1.1.1.1.4.cmml" xref="A1.Ex20.m1.1.1.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.Ex20.m1.1.1.1.3.cmml" xref="A1.Ex20.m1.1.1.1.3">𝑝</ci></apply><apply id="A1.Ex20.m1.2.2.2.cmml" xref="A1.Ex20.m1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex20.m1.2.2.2.2.cmml" xref="A1.Ex20.m1.2.2.2">subscript</csymbol><apply id="A1.Ex20.m1.2.2.2.1.2.cmml" xref="A1.Ex20.m1.2.2.2.1.1"><csymbol cd="latexml" id="A1.Ex20.m1.2.2.2.1.2.1.cmml" xref="A1.Ex20.m1.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A1.Ex20.m1.2.2.2.1.1.1.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1"><minus id="A1.Ex20.m1.2.2.2.1.1.1.1.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.1"></minus><apply id="A1.Ex20.m1.2.2.2.1.1.1.2.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex20.m1.2.2.2.1.1.1.2.1.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.2">subscript</csymbol><ci id="A1.Ex20.m1.2.2.2.1.1.1.2.2.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.2.2">𝑥</ci><ci id="A1.Ex20.m1.2.2.2.1.1.1.2.3.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.2.3">𝑛</ci></apply><ci id="A1.Ex20.m1.2.2.2.1.1.1.3.cmml" xref="A1.Ex20.m1.2.2.2.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.Ex20.m1.2.2.2.3.cmml" xref="A1.Ex20.m1.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex20.m1.2c">\lVert x_{n}-\varepsilon v-z\rVert_{p}\geq\lVert x_{n}-z\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex20.m1.2d">∥ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ ∥ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.5.p1.39">and thus <math alttext="\lim_{n\rightarrow\infty}X_{n}(z)=1" class="ltx_Math" display="inline" id="A1.SS2.5.p1.36.m1.1"><semantics id="A1.SS2.5.p1.36.m1.1a"><mrow id="A1.SS2.5.p1.36.m1.1.2" xref="A1.SS2.5.p1.36.m1.1.2.cmml"><mrow id="A1.SS2.5.p1.36.m1.1.2.2" xref="A1.SS2.5.p1.36.m1.1.2.2.cmml"><msub id="A1.SS2.5.p1.36.m1.1.2.2.1" xref="A1.SS2.5.p1.36.m1.1.2.2.1.cmml"><mo id="A1.SS2.5.p1.36.m1.1.2.2.1.2" xref="A1.SS2.5.p1.36.m1.1.2.2.1.2.cmml">lim</mo><mrow id="A1.SS2.5.p1.36.m1.1.2.2.1.3" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.cmml"><mi id="A1.SS2.5.p1.36.m1.1.2.2.1.3.2" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.2.cmml">n</mi><mo id="A1.SS2.5.p1.36.m1.1.2.2.1.3.1" stretchy="false" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.1.cmml">→</mo><mi id="A1.SS2.5.p1.36.m1.1.2.2.1.3.3" mathvariant="normal" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="A1.SS2.5.p1.36.m1.1.2.2.2" xref="A1.SS2.5.p1.36.m1.1.2.2.2.cmml"><msub id="A1.SS2.5.p1.36.m1.1.2.2.2.2" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2.cmml"><mi id="A1.SS2.5.p1.36.m1.1.2.2.2.2.2" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2.2.cmml">X</mi><mi id="A1.SS2.5.p1.36.m1.1.2.2.2.2.3" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.36.m1.1.2.2.2.1" xref="A1.SS2.5.p1.36.m1.1.2.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.5.p1.36.m1.1.2.2.2.3.2" xref="A1.SS2.5.p1.36.m1.1.2.2.2.cmml"><mo id="A1.SS2.5.p1.36.m1.1.2.2.2.3.2.1" stretchy="false" xref="A1.SS2.5.p1.36.m1.1.2.2.2.cmml">(</mo><mi id="A1.SS2.5.p1.36.m1.1.1" xref="A1.SS2.5.p1.36.m1.1.1.cmml">z</mi><mo id="A1.SS2.5.p1.36.m1.1.2.2.2.3.2.2" stretchy="false" xref="A1.SS2.5.p1.36.m1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A1.SS2.5.p1.36.m1.1.2.1" xref="A1.SS2.5.p1.36.m1.1.2.1.cmml">=</mo><mn id="A1.SS2.5.p1.36.m1.1.2.3" xref="A1.SS2.5.p1.36.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.36.m1.1b"><apply id="A1.SS2.5.p1.36.m1.1.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2"><eq id="A1.SS2.5.p1.36.m1.1.2.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.1"></eq><apply id="A1.SS2.5.p1.36.m1.1.2.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2"><apply id="A1.SS2.5.p1.36.m1.1.2.2.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.36.m1.1.2.2.1.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1">subscript</csymbol><limit id="A1.SS2.5.p1.36.m1.1.2.2.1.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1.2"></limit><apply id="A1.SS2.5.p1.36.m1.1.2.2.1.3.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3"><ci id="A1.SS2.5.p1.36.m1.1.2.2.1.3.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.1">→</ci><ci id="A1.SS2.5.p1.36.m1.1.2.2.1.3.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.2">𝑛</ci><infinity id="A1.SS2.5.p1.36.m1.1.2.2.1.3.3.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.1.3.3"></infinity></apply></apply><apply id="A1.SS2.5.p1.36.m1.1.2.2.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2"><times id="A1.SS2.5.p1.36.m1.1.2.2.2.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2.1"></times><apply id="A1.SS2.5.p1.36.m1.1.2.2.2.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.5.p1.36.m1.1.2.2.2.2.1.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2">subscript</csymbol><ci id="A1.SS2.5.p1.36.m1.1.2.2.2.2.2.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2.2">𝑋</ci><ci id="A1.SS2.5.p1.36.m1.1.2.2.2.2.3.cmml" xref="A1.SS2.5.p1.36.m1.1.2.2.2.2.3">𝑛</ci></apply><ci id="A1.SS2.5.p1.36.m1.1.1.cmml" xref="A1.SS2.5.p1.36.m1.1.1">𝑧</ci></apply></apply><cn id="A1.SS2.5.p1.36.m1.1.2.3.cmml" type="integer" xref="A1.SS2.5.p1.36.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.36.m1.1c">\lim_{n\rightarrow\infty}X_{n}(z)=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.36.m1.1d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_z ) = 1</annotation></semantics></math>. We conclude that the set <math alttext="\{z\in\mathbb{R}^{d}\mid\lim_{n\rightarrow\infty}X_{n}(z)=X(z)\}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.37.m2.4"><semantics id="A1.SS2.5.p1.37.m2.4a"><mrow id="A1.SS2.5.p1.37.m2.4.4.2" xref="A1.SS2.5.p1.37.m2.4.4.3.cmml"><mo id="A1.SS2.5.p1.37.m2.4.4.2.3" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.3.1.cmml">{</mo><mrow id="A1.SS2.5.p1.37.m2.3.3.1.1" xref="A1.SS2.5.p1.37.m2.3.3.1.1.cmml"><mi id="A1.SS2.5.p1.37.m2.3.3.1.1.2" xref="A1.SS2.5.p1.37.m2.3.3.1.1.2.cmml">z</mi><mo id="A1.SS2.5.p1.37.m2.3.3.1.1.1" xref="A1.SS2.5.p1.37.m2.3.3.1.1.1.cmml">∈</mo><msup id="A1.SS2.5.p1.37.m2.3.3.1.1.3" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3.cmml"><mi id="A1.SS2.5.p1.37.m2.3.3.1.1.3.2" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.5.p1.37.m2.3.3.1.1.3.3" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3.3.cmml">d</mi></msup></mrow><mo fence="true" id="A1.SS2.5.p1.37.m2.4.4.2.4" lspace="0em" rspace="0.0835em" xref="A1.SS2.5.p1.37.m2.4.4.3.1.cmml">∣</mo><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.cmml"><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.cmml"><msub id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.cmml"><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.2" lspace="0.0835em" rspace="0.167em" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.2.cmml">lim</mo><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.cmml"><mi id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.2.cmml">n</mi><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.1" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.1.cmml">→</mo><mi id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.3" mathvariant="normal" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.cmml"><msub id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.cmml"><mi id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.2.cmml">X</mi><mi id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.3" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.3.cmml">n</mi></msub><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.1" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.3.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.cmml"><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.3.2.1" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.cmml">(</mo><mi id="A1.SS2.5.p1.37.m2.1.1" xref="A1.SS2.5.p1.37.m2.1.1.cmml">z</mi><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.3.2.2" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.1" xref="A1.SS2.5.p1.37.m2.4.4.2.2.1.cmml">=</mo><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.3" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.cmml"><mi id="A1.SS2.5.p1.37.m2.4.4.2.2.3.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.2.cmml">X</mi><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.3.1" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.1.cmml">⁢</mo><mrow id="A1.SS2.5.p1.37.m2.4.4.2.2.3.3.2" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.cmml"><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.3.3.2.1" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.cmml">(</mo><mi id="A1.SS2.5.p1.37.m2.2.2" xref="A1.SS2.5.p1.37.m2.2.2.cmml">z</mi><mo id="A1.SS2.5.p1.37.m2.4.4.2.2.3.3.2.2" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="A1.SS2.5.p1.37.m2.4.4.2.5" stretchy="false" xref="A1.SS2.5.p1.37.m2.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.37.m2.4b"><apply id="A1.SS2.5.p1.37.m2.4.4.3.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2"><csymbol cd="latexml" id="A1.SS2.5.p1.37.m2.4.4.3.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.3">conditional-set</csymbol><apply id="A1.SS2.5.p1.37.m2.3.3.1.1.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1"><in id="A1.SS2.5.p1.37.m2.3.3.1.1.1.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.1"></in><ci id="A1.SS2.5.p1.37.m2.3.3.1.1.2.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.2">𝑧</ci><apply id="A1.SS2.5.p1.37.m2.3.3.1.1.3.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.37.m2.3.3.1.1.3.1.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3">superscript</csymbol><ci id="A1.SS2.5.p1.37.m2.3.3.1.1.3.2.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3.2">ℝ</ci><ci id="A1.SS2.5.p1.37.m2.3.3.1.1.3.3.cmml" xref="A1.SS2.5.p1.37.m2.3.3.1.1.3.3">𝑑</ci></apply></apply><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2"><eq id="A1.SS2.5.p1.37.m2.4.4.2.2.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.1"></eq><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2"><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1">subscript</csymbol><limit id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.2"></limit><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3"><ci id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.1">→</ci><ci id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.2">𝑛</ci><infinity id="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.3.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.1.3.3"></infinity></apply></apply><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2"><times id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.1"></times><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2">subscript</csymbol><ci id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.2">𝑋</ci><ci id="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.3.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.2.2.2.3">𝑛</ci></apply><ci id="A1.SS2.5.p1.37.m2.1.1.cmml" xref="A1.SS2.5.p1.37.m2.1.1">𝑧</ci></apply></apply><apply id="A1.SS2.5.p1.37.m2.4.4.2.2.3.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3"><times id="A1.SS2.5.p1.37.m2.4.4.2.2.3.1.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.1"></times><ci id="A1.SS2.5.p1.37.m2.4.4.2.2.3.2.cmml" xref="A1.SS2.5.p1.37.m2.4.4.2.2.3.2">𝑋</ci><ci id="A1.SS2.5.p1.37.m2.2.2.cmml" xref="A1.SS2.5.p1.37.m2.2.2">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.37.m2.4c">\{z\in\mathbb{R}^{d}\mid\lim_{n\rightarrow\infty}X_{n}(z)=X(z)\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.37.m2.4d">{ italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∣ roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_z ) = italic_X ( italic_z ) }</annotation></semantics></math> includes all of <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.38.m3.1"><semantics id="A1.SS2.5.p1.38.m3.1a"><msup id="A1.SS2.5.p1.38.m3.1.1" xref="A1.SS2.5.p1.38.m3.1.1.cmml"><mi id="A1.SS2.5.p1.38.m3.1.1.2" xref="A1.SS2.5.p1.38.m3.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.5.p1.38.m3.1.1.3" xref="A1.SS2.5.p1.38.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.38.m3.1b"><apply id="A1.SS2.5.p1.38.m3.1.1.cmml" xref="A1.SS2.5.p1.38.m3.1.1"><csymbol cd="ambiguous" id="A1.SS2.5.p1.38.m3.1.1.1.cmml" xref="A1.SS2.5.p1.38.m3.1.1">superscript</csymbol><ci id="A1.SS2.5.p1.38.m3.1.1.2.cmml" xref="A1.SS2.5.p1.38.m3.1.1.2">ℝ</ci><ci id="A1.SS2.5.p1.38.m3.1.1.3.cmml" xref="A1.SS2.5.p1.38.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.38.m3.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.38.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> except for the boundary of <math alttext="\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.5.p1.39.m4.2"><semantics id="A1.SS2.5.p1.39.m4.2a"><msubsup id="A1.SS2.5.p1.39.m4.2.3" xref="A1.SS2.5.p1.39.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.5.p1.39.m4.2.3.2.2" xref="A1.SS2.5.p1.39.m4.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.5.p1.39.m4.2.2.2.4" xref="A1.SS2.5.p1.39.m4.2.2.2.3.cmml"><mi id="A1.SS2.5.p1.39.m4.1.1.1.1" xref="A1.SS2.5.p1.39.m4.1.1.1.1.cmml">x</mi><mo id="A1.SS2.5.p1.39.m4.2.2.2.4.1" xref="A1.SS2.5.p1.39.m4.2.2.2.3.cmml">,</mo><mi id="A1.SS2.5.p1.39.m4.2.2.2.2" xref="A1.SS2.5.p1.39.m4.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.5.p1.39.m4.2.3.2.3" xref="A1.SS2.5.p1.39.m4.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.5.p1.39.m4.2b"><apply id="A1.SS2.5.p1.39.m4.2.3.cmml" xref="A1.SS2.5.p1.39.m4.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.39.m4.2.3.1.cmml" xref="A1.SS2.5.p1.39.m4.2.3">subscript</csymbol><apply id="A1.SS2.5.p1.39.m4.2.3.2.cmml" xref="A1.SS2.5.p1.39.m4.2.3"><csymbol cd="ambiguous" id="A1.SS2.5.p1.39.m4.2.3.2.1.cmml" xref="A1.SS2.5.p1.39.m4.2.3">superscript</csymbol><ci id="A1.SS2.5.p1.39.m4.2.3.2.2.cmml" xref="A1.SS2.5.p1.39.m4.2.3.2.2">ℋ</ci><ci id="A1.SS2.5.p1.39.m4.2.3.2.3.cmml" xref="A1.SS2.5.p1.39.m4.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.5.p1.39.m4.2.2.2.3.cmml" xref="A1.SS2.5.p1.39.m4.2.2.2.4"><ci id="A1.SS2.5.p1.39.m4.1.1.1.1.cmml" xref="A1.SS2.5.p1.39.m4.1.1.1.1">𝑥</ci><ci id="A1.SS2.5.p1.39.m4.2.2.2.2.cmml" xref="A1.SS2.5.p1.39.m4.2.2.2.2">𝑣</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.5.p1.39.m4.2c">\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.5.p1.39.m4.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#A1.Thmtheorem5" title="Corollary A.5 (Boundary has Measure 0). ‣ A.2 ℓ_𝑝-Halfspaces and Mass Distributions ‣ Appendix A More on ℓ_𝑝-Halfspaces ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.5</span></a>, this boundary has measure zero and we get almost sure convergence. Using the dominated convergence theorem, this implies convergence in mean and thus</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{n\rightarrow\infty}\mu(\mathcal{H}^{p}_{x_{n},v})=\lim_{n\rightarrow% \infty}\mathbb{E}(X_{n})=\mathbb{E}(X)=\mu(\mathcal{H}^{p}_{x,v})," class="ltx_Math" display="block" id="A1.Ex21.m1.6"><semantics id="A1.Ex21.m1.6a"><mrow id="A1.Ex21.m1.6.6.1" xref="A1.Ex21.m1.6.6.1.1.cmml"><mrow id="A1.Ex21.m1.6.6.1.1" xref="A1.Ex21.m1.6.6.1.1.cmml"><mrow id="A1.Ex21.m1.6.6.1.1.1" xref="A1.Ex21.m1.6.6.1.1.1.cmml"><munder id="A1.Ex21.m1.6.6.1.1.1.2" xref="A1.Ex21.m1.6.6.1.1.1.2.cmml"><mo id="A1.Ex21.m1.6.6.1.1.1.2.2" movablelimits="false" xref="A1.Ex21.m1.6.6.1.1.1.2.2.cmml">lim</mo><mrow id="A1.Ex21.m1.6.6.1.1.1.2.3" xref="A1.Ex21.m1.6.6.1.1.1.2.3.cmml"><mi 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xref="A1.Ex21.m1.6.6.1.1.3.1.1">superscript</csymbol><ci id="A1.Ex21.m1.6.6.1.1.3.1.1.1.2.2.cmml" xref="A1.Ex21.m1.6.6.1.1.3.1.1.1.2.2">ℋ</ci><ci id="A1.Ex21.m1.6.6.1.1.3.1.1.1.2.3.cmml" xref="A1.Ex21.m1.6.6.1.1.3.1.1.1.2.3">𝑝</ci></apply><list id="A1.Ex21.m1.4.4.2.3.cmml" xref="A1.Ex21.m1.4.4.2.4"><ci id="A1.Ex21.m1.3.3.1.1.cmml" xref="A1.Ex21.m1.3.3.1.1">𝑥</ci><ci id="A1.Ex21.m1.4.4.2.2.cmml" xref="A1.Ex21.m1.4.4.2.2">𝑣</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex21.m1.6c">\lim_{n\rightarrow\infty}\mu(\mathcal{H}^{p}_{x_{n},v})=\lim_{n\rightarrow% \infty}\mathbb{E}(X_{n})=\mathbb{E}(X)=\mu(\mathcal{H}^{p}_{x,v}),</annotation><annotation encoding="application/x-llamapun" id="A1.Ex21.m1.6d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ) = roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT blackboard_E ( italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = blackboard_E ( italic_X ) = italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.5.p1.40">as desired. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS2.p6"> <p class="ltx_p" id="A1.SS2.p6.7">While we cannot prove for all <math alttext="p" class="ltx_Math" display="inline" id="A1.SS2.p6.1.m1.1"><semantics id="A1.SS2.p6.1.m1.1a"><mi id="A1.SS2.p6.1.m1.1.1" xref="A1.SS2.p6.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.1.m1.1b"><ci id="A1.SS2.p6.1.m1.1.1.cmml" xref="A1.SS2.p6.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.1.m1.1d">italic_p</annotation></semantics></math> that <math alttext="\mu(\mathcal{H}_{x,v}^{p})" class="ltx_Math" display="inline" id="A1.SS2.p6.2.m2.3"><semantics id="A1.SS2.p6.2.m2.3a"><mrow id="A1.SS2.p6.2.m2.3.3" xref="A1.SS2.p6.2.m2.3.3.cmml"><mi id="A1.SS2.p6.2.m2.3.3.3" xref="A1.SS2.p6.2.m2.3.3.3.cmml">μ</mi><mo id="A1.SS2.p6.2.m2.3.3.2" xref="A1.SS2.p6.2.m2.3.3.2.cmml">⁢</mo><mrow id="A1.SS2.p6.2.m2.3.3.1.1" xref="A1.SS2.p6.2.m2.3.3.1.1.1.cmml"><mo id="A1.SS2.p6.2.m2.3.3.1.1.2" stretchy="false" xref="A1.SS2.p6.2.m2.3.3.1.1.1.cmml">(</mo><msubsup id="A1.SS2.p6.2.m2.3.3.1.1.1" xref="A1.SS2.p6.2.m2.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.p6.2.m2.3.3.1.1.1.2.2" xref="A1.SS2.p6.2.m2.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.SS2.p6.2.m2.2.2.2.4" xref="A1.SS2.p6.2.m2.2.2.2.3.cmml"><mi id="A1.SS2.p6.2.m2.1.1.1.1" xref="A1.SS2.p6.2.m2.1.1.1.1.cmml">x</mi><mo id="A1.SS2.p6.2.m2.2.2.2.4.1" xref="A1.SS2.p6.2.m2.2.2.2.3.cmml">,</mo><mi id="A1.SS2.p6.2.m2.2.2.2.2" xref="A1.SS2.p6.2.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.p6.2.m2.3.3.1.1.1.3" xref="A1.SS2.p6.2.m2.3.3.1.1.1.3.cmml">p</mi></msubsup><mo id="A1.SS2.p6.2.m2.3.3.1.1.3" stretchy="false" xref="A1.SS2.p6.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.2.m2.3b"><apply id="A1.SS2.p6.2.m2.3.3.cmml" xref="A1.SS2.p6.2.m2.3.3"><times id="A1.SS2.p6.2.m2.3.3.2.cmml" xref="A1.SS2.p6.2.m2.3.3.2"></times><ci id="A1.SS2.p6.2.m2.3.3.3.cmml" xref="A1.SS2.p6.2.m2.3.3.3">𝜇</ci><apply id="A1.SS2.p6.2.m2.3.3.1.1.1.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p6.2.m2.3.3.1.1.1.1.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1">superscript</csymbol><apply id="A1.SS2.p6.2.m2.3.3.1.1.1.2.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p6.2.m2.3.3.1.1.1.2.1.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1">subscript</csymbol><ci id="A1.SS2.p6.2.m2.3.3.1.1.1.2.2.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1.1.2.2">ℋ</ci><list id="A1.SS2.p6.2.m2.2.2.2.3.cmml" xref="A1.SS2.p6.2.m2.2.2.2.4"><ci id="A1.SS2.p6.2.m2.1.1.1.1.cmml" xref="A1.SS2.p6.2.m2.1.1.1.1">𝑥</ci><ci id="A1.SS2.p6.2.m2.2.2.2.2.cmml" xref="A1.SS2.p6.2.m2.2.2.2.2">𝑣</ci></list></apply><ci id="A1.SS2.p6.2.m2.3.3.1.1.1.3.cmml" xref="A1.SS2.p6.2.m2.3.3.1.1.1.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.2.m2.3c">\mu(\mathcal{H}_{x,v}^{p})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.2.m2.3d">italic_μ ( caligraphic_H start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT )</annotation></semantics></math> is continuous in <math alttext="v" class="ltx_Math" display="inline" id="A1.SS2.p6.3.m3.1"><semantics id="A1.SS2.p6.3.m3.1a"><mi id="A1.SS2.p6.3.m3.1.1" xref="A1.SS2.p6.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.3.m3.1b"><ci id="A1.SS2.p6.3.m3.1.1.cmml" xref="A1.SS2.p6.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.3.m3.1d">italic_v</annotation></semantics></math>, we at least prove <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem11" title="Lemma 3.11. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.11</span></a>, which states that the set <math alttext="V" class="ltx_Math" display="inline" id="A1.SS2.p6.4.m4.1"><semantics id="A1.SS2.p6.4.m4.1a"><mi id="A1.SS2.p6.4.m4.1.1" xref="A1.SS2.p6.4.m4.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.4.m4.1b"><ci id="A1.SS2.p6.4.m4.1.1.cmml" xref="A1.SS2.p6.4.m4.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.4.m4.1c">V</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.4.m4.1d">italic_V</annotation></semantics></math> of directions <math alttext="v" class="ltx_Math" display="inline" id="A1.SS2.p6.5.m5.1"><semantics id="A1.SS2.p6.5.m5.1a"><mi id="A1.SS2.p6.5.m5.1.1" xref="A1.SS2.p6.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.5.m5.1b"><ci id="A1.SS2.p6.5.m5.1.1.cmml" xref="A1.SS2.p6.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.5.m5.1d">italic_v</annotation></semantics></math> for which <math alttext="\mu(\mathcal{H}^{p}_{x,-v})" class="ltx_Math" display="inline" id="A1.SS2.p6.6.m6.3"><semantics id="A1.SS2.p6.6.m6.3a"><mrow id="A1.SS2.p6.6.m6.3.3" xref="A1.SS2.p6.6.m6.3.3.cmml"><mi id="A1.SS2.p6.6.m6.3.3.3" xref="A1.SS2.p6.6.m6.3.3.3.cmml">μ</mi><mo id="A1.SS2.p6.6.m6.3.3.2" xref="A1.SS2.p6.6.m6.3.3.2.cmml">⁢</mo><mrow id="A1.SS2.p6.6.m6.3.3.1.1" xref="A1.SS2.p6.6.m6.3.3.1.1.1.cmml"><mo id="A1.SS2.p6.6.m6.3.3.1.1.2" stretchy="false" xref="A1.SS2.p6.6.m6.3.3.1.1.1.cmml">(</mo><msubsup id="A1.SS2.p6.6.m6.3.3.1.1.1" xref="A1.SS2.p6.6.m6.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.p6.6.m6.3.3.1.1.1.2.2" xref="A1.SS2.p6.6.m6.3.3.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.SS2.p6.6.m6.2.2.2.2" xref="A1.SS2.p6.6.m6.2.2.2.3.cmml"><mi id="A1.SS2.p6.6.m6.1.1.1.1" xref="A1.SS2.p6.6.m6.1.1.1.1.cmml">x</mi><mo id="A1.SS2.p6.6.m6.2.2.2.2.2" xref="A1.SS2.p6.6.m6.2.2.2.3.cmml">,</mo><mrow id="A1.SS2.p6.6.m6.2.2.2.2.1" xref="A1.SS2.p6.6.m6.2.2.2.2.1.cmml"><mo id="A1.SS2.p6.6.m6.2.2.2.2.1a" xref="A1.SS2.p6.6.m6.2.2.2.2.1.cmml">−</mo><mi id="A1.SS2.p6.6.m6.2.2.2.2.1.2" xref="A1.SS2.p6.6.m6.2.2.2.2.1.2.cmml">v</mi></mrow></mrow><mi id="A1.SS2.p6.6.m6.3.3.1.1.1.2.3" xref="A1.SS2.p6.6.m6.3.3.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A1.SS2.p6.6.m6.3.3.1.1.3" stretchy="false" xref="A1.SS2.p6.6.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.6.m6.3b"><apply id="A1.SS2.p6.6.m6.3.3.cmml" xref="A1.SS2.p6.6.m6.3.3"><times id="A1.SS2.p6.6.m6.3.3.2.cmml" xref="A1.SS2.p6.6.m6.3.3.2"></times><ci id="A1.SS2.p6.6.m6.3.3.3.cmml" xref="A1.SS2.p6.6.m6.3.3.3">𝜇</ci><apply id="A1.SS2.p6.6.m6.3.3.1.1.1.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p6.6.m6.3.3.1.1.1.1.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1">subscript</csymbol><apply id="A1.SS2.p6.6.m6.3.3.1.1.1.2.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1"><csymbol cd="ambiguous" id="A1.SS2.p6.6.m6.3.3.1.1.1.2.1.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1">superscript</csymbol><ci id="A1.SS2.p6.6.m6.3.3.1.1.1.2.2.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1.1.2.2">ℋ</ci><ci id="A1.SS2.p6.6.m6.3.3.1.1.1.2.3.cmml" xref="A1.SS2.p6.6.m6.3.3.1.1.1.2.3">𝑝</ci></apply><list id="A1.SS2.p6.6.m6.2.2.2.3.cmml" xref="A1.SS2.p6.6.m6.2.2.2.2"><ci id="A1.SS2.p6.6.m6.1.1.1.1.cmml" xref="A1.SS2.p6.6.m6.1.1.1.1">𝑥</ci><apply id="A1.SS2.p6.6.m6.2.2.2.2.1.cmml" xref="A1.SS2.p6.6.m6.2.2.2.2.1"><minus id="A1.SS2.p6.6.m6.2.2.2.2.1.1.cmml" xref="A1.SS2.p6.6.m6.2.2.2.2.1"></minus><ci id="A1.SS2.p6.6.m6.2.2.2.2.1.2.cmml" xref="A1.SS2.p6.6.m6.2.2.2.2.1.2">𝑣</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.6.m6.3c">\mu(\mathcal{H}^{p}_{x,-v})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.6.m6.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , - italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> is strictly smaller than some threshold <math alttext="t" class="ltx_Math" display="inline" id="A1.SS2.p6.7.m7.1"><semantics id="A1.SS2.p6.7.m7.1a"><mi id="A1.SS2.p6.7.m7.1.1" xref="A1.SS2.p6.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p6.7.m7.1b"><ci id="A1.SS2.p6.7.m7.1.1.cmml" xref="A1.SS2.p6.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p6.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p6.7.m7.1d">italic_t</annotation></semantics></math>, is open.</p> </div> <div class="ltx_para" id="A1.SS2.p7"> <p class="ltx_p" id="A1.SS2.p7.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem11" title="Lemma 3.11. ‣ 3.2 Properties of ℓ_𝑝-Halfspaces ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.11</span></a></p> </div> <div class="ltx_proof" id="A1.SS2.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS2.6.p1"> <p class="ltx_p" id="A1.SS2.6.p1.5">Fix <math alttext="\mu" class="ltx_Math" display="inline" id="A1.SS2.6.p1.1.m1.1"><semantics id="A1.SS2.6.p1.1.m1.1a"><mi id="A1.SS2.6.p1.1.m1.1.1" xref="A1.SS2.6.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.6.p1.1.m1.1b"><ci id="A1.SS2.6.p1.1.m1.1.1.cmml" xref="A1.SS2.6.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.6.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.6.p1.1.m1.1d">italic_μ</annotation></semantics></math> and <math alttext="p\in[1,\infty)\cup\{\infty\}" class="ltx_Math" display="inline" id="A1.SS2.6.p1.2.m2.3"><semantics id="A1.SS2.6.p1.2.m2.3a"><mrow id="A1.SS2.6.p1.2.m2.3.4" xref="A1.SS2.6.p1.2.m2.3.4.cmml"><mi id="A1.SS2.6.p1.2.m2.3.4.2" xref="A1.SS2.6.p1.2.m2.3.4.2.cmml">p</mi><mo id="A1.SS2.6.p1.2.m2.3.4.1" xref="A1.SS2.6.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="A1.SS2.6.p1.2.m2.3.4.3" xref="A1.SS2.6.p1.2.m2.3.4.3.cmml"><mrow id="A1.SS2.6.p1.2.m2.3.4.3.2.2" xref="A1.SS2.6.p1.2.m2.3.4.3.2.1.cmml"><mo id="A1.SS2.6.p1.2.m2.3.4.3.2.2.1" stretchy="false" xref="A1.SS2.6.p1.2.m2.3.4.3.2.1.cmml">[</mo><mn id="A1.SS2.6.p1.2.m2.1.1" xref="A1.SS2.6.p1.2.m2.1.1.cmml">1</mn><mo id="A1.SS2.6.p1.2.m2.3.4.3.2.2.2" xref="A1.SS2.6.p1.2.m2.3.4.3.2.1.cmml">,</mo><mi id="A1.SS2.6.p1.2.m2.2.2" mathvariant="normal" xref="A1.SS2.6.p1.2.m2.2.2.cmml">∞</mi><mo id="A1.SS2.6.p1.2.m2.3.4.3.2.2.3" stretchy="false" xref="A1.SS2.6.p1.2.m2.3.4.3.2.1.cmml">)</mo></mrow><mo id="A1.SS2.6.p1.2.m2.3.4.3.1" xref="A1.SS2.6.p1.2.m2.3.4.3.1.cmml">∪</mo><mrow id="A1.SS2.6.p1.2.m2.3.4.3.3.2" xref="A1.SS2.6.p1.2.m2.3.4.3.3.1.cmml"><mo id="A1.SS2.6.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="A1.SS2.6.p1.2.m2.3.4.3.3.1.cmml">{</mo><mi id="A1.SS2.6.p1.2.m2.3.3" mathvariant="normal" xref="A1.SS2.6.p1.2.m2.3.3.cmml">∞</mi><mo id="A1.SS2.6.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="A1.SS2.6.p1.2.m2.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.6.p1.2.m2.3b"><apply id="A1.SS2.6.p1.2.m2.3.4.cmml" xref="A1.SS2.6.p1.2.m2.3.4"><in id="A1.SS2.6.p1.2.m2.3.4.1.cmml" xref="A1.SS2.6.p1.2.m2.3.4.1"></in><ci id="A1.SS2.6.p1.2.m2.3.4.2.cmml" xref="A1.SS2.6.p1.2.m2.3.4.2">𝑝</ci><apply id="A1.SS2.6.p1.2.m2.3.4.3.cmml" xref="A1.SS2.6.p1.2.m2.3.4.3"><union id="A1.SS2.6.p1.2.m2.3.4.3.1.cmml" xref="A1.SS2.6.p1.2.m2.3.4.3.1"></union><interval closure="closed-open" id="A1.SS2.6.p1.2.m2.3.4.3.2.1.cmml" xref="A1.SS2.6.p1.2.m2.3.4.3.2.2"><cn id="A1.SS2.6.p1.2.m2.1.1.cmml" type="integer" xref="A1.SS2.6.p1.2.m2.1.1">1</cn><infinity id="A1.SS2.6.p1.2.m2.2.2.cmml" xref="A1.SS2.6.p1.2.m2.2.2"></infinity></interval><set id="A1.SS2.6.p1.2.m2.3.4.3.3.1.cmml" xref="A1.SS2.6.p1.2.m2.3.4.3.3.2"><infinity id="A1.SS2.6.p1.2.m2.3.3.cmml" xref="A1.SS2.6.p1.2.m2.3.3"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.6.p1.2.m2.3c">p\in[1,\infty)\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.6.p1.2.m2.3d">italic_p ∈ [ 1 , ∞ ) ∪ { ∞ }</annotation></semantics></math> and let <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.6.p1.3.m3.1"><semantics id="A1.SS2.6.p1.3.m3.1a"><mrow id="A1.SS2.6.p1.3.m3.1.1" xref="A1.SS2.6.p1.3.m3.1.1.cmml"><mi id="A1.SS2.6.p1.3.m3.1.1.2" xref="A1.SS2.6.p1.3.m3.1.1.2.cmml">x</mi><mo id="A1.SS2.6.p1.3.m3.1.1.1" xref="A1.SS2.6.p1.3.m3.1.1.1.cmml">∈</mo><msup id="A1.SS2.6.p1.3.m3.1.1.3" xref="A1.SS2.6.p1.3.m3.1.1.3.cmml"><mi id="A1.SS2.6.p1.3.m3.1.1.3.2" xref="A1.SS2.6.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.6.p1.3.m3.1.1.3.3" xref="A1.SS2.6.p1.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.6.p1.3.m3.1b"><apply id="A1.SS2.6.p1.3.m3.1.1.cmml" xref="A1.SS2.6.p1.3.m3.1.1"><in id="A1.SS2.6.p1.3.m3.1.1.1.cmml" xref="A1.SS2.6.p1.3.m3.1.1.1"></in><ci id="A1.SS2.6.p1.3.m3.1.1.2.cmml" xref="A1.SS2.6.p1.3.m3.1.1.2">𝑥</ci><apply id="A1.SS2.6.p1.3.m3.1.1.3.cmml" xref="A1.SS2.6.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.6.p1.3.m3.1.1.3.1.cmml" xref="A1.SS2.6.p1.3.m3.1.1.3">superscript</csymbol><ci id="A1.SS2.6.p1.3.m3.1.1.3.2.cmml" xref="A1.SS2.6.p1.3.m3.1.1.3.2">ℝ</ci><ci id="A1.SS2.6.p1.3.m3.1.1.3.3.cmml" xref="A1.SS2.6.p1.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.6.p1.3.m3.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.6.p1.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be arbitrary. Assume without loss of generality that <math alttext="\mu(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="A1.SS2.6.p1.4.m4.1"><semantics id="A1.SS2.6.p1.4.m4.1a"><mrow id="A1.SS2.6.p1.4.m4.1.1" xref="A1.SS2.6.p1.4.m4.1.1.cmml"><mi id="A1.SS2.6.p1.4.m4.1.1.3" xref="A1.SS2.6.p1.4.m4.1.1.3.cmml">μ</mi><mo id="A1.SS2.6.p1.4.m4.1.1.2" xref="A1.SS2.6.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.6.p1.4.m4.1.1.1.1" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.cmml"><mo id="A1.SS2.6.p1.4.m4.1.1.1.1.2" stretchy="false" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.cmml">(</mo><msup id="A1.SS2.6.p1.4.m4.1.1.1.1.1" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.cmml"><mi id="A1.SS2.6.p1.4.m4.1.1.1.1.1.2" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.6.p1.4.m4.1.1.1.1.1.3" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.3.cmml">d</mi></msup><mo id="A1.SS2.6.p1.4.m4.1.1.1.1.3" stretchy="false" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.6.p1.4.m4.1b"><apply id="A1.SS2.6.p1.4.m4.1.1.cmml" xref="A1.SS2.6.p1.4.m4.1.1"><times id="A1.SS2.6.p1.4.m4.1.1.2.cmml" xref="A1.SS2.6.p1.4.m4.1.1.2"></times><ci id="A1.SS2.6.p1.4.m4.1.1.3.cmml" xref="A1.SS2.6.p1.4.m4.1.1.3">𝜇</ci><apply id="A1.SS2.6.p1.4.m4.1.1.1.1.1.cmml" xref="A1.SS2.6.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.6.p1.4.m4.1.1.1.1.1.1.cmml" xref="A1.SS2.6.p1.4.m4.1.1.1.1">superscript</csymbol><ci id="A1.SS2.6.p1.4.m4.1.1.1.1.1.2.cmml" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="A1.SS2.6.p1.4.m4.1.1.1.1.1.3.cmml" xref="A1.SS2.6.p1.4.m4.1.1.1.1.1.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.6.p1.4.m4.1c">\mu(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.6.p1.4.m4.1d">italic_μ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>. Instead of directly proving that <math alttext="V" class="ltx_Math" display="inline" id="A1.SS2.6.p1.5.m5.1"><semantics id="A1.SS2.6.p1.5.m5.1a"><mi id="A1.SS2.6.p1.5.m5.1.1" xref="A1.SS2.6.p1.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.6.p1.5.m5.1b"><ci id="A1.SS2.6.p1.5.m5.1.1.cmml" xref="A1.SS2.6.p1.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.6.p1.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.6.p1.5.m5.1d">italic_V</annotation></semantics></math> is open, we will prove that its complement</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S^{d-1}\setminus V=\{v\in S^{d-1}\mid\mu(\mathcal{H}^{p}_{x,v})\geq t\}" class="ltx_Math" display="block" id="A1.Ex22.m1.4"><semantics id="A1.Ex22.m1.4a"><mrow id="A1.Ex22.m1.4.4" xref="A1.Ex22.m1.4.4.cmml"><mrow id="A1.Ex22.m1.4.4.4" xref="A1.Ex22.m1.4.4.4.cmml"><msup id="A1.Ex22.m1.4.4.4.2" xref="A1.Ex22.m1.4.4.4.2.cmml"><mi id="A1.Ex22.m1.4.4.4.2.2" xref="A1.Ex22.m1.4.4.4.2.2.cmml">S</mi><mrow id="A1.Ex22.m1.4.4.4.2.3" xref="A1.Ex22.m1.4.4.4.2.3.cmml"><mi id="A1.Ex22.m1.4.4.4.2.3.2" xref="A1.Ex22.m1.4.4.4.2.3.2.cmml">d</mi><mo id="A1.Ex22.m1.4.4.4.2.3.1" xref="A1.Ex22.m1.4.4.4.2.3.1.cmml">−</mo><mn id="A1.Ex22.m1.4.4.4.2.3.3" xref="A1.Ex22.m1.4.4.4.2.3.3.cmml">1</mn></mrow></msup><mo id="A1.Ex22.m1.4.4.4.1" xref="A1.Ex22.m1.4.4.4.1.cmml">∖</mo><mi id="A1.Ex22.m1.4.4.4.3" xref="A1.Ex22.m1.4.4.4.3.cmml">V</mi></mrow><mo id="A1.Ex22.m1.4.4.3" xref="A1.Ex22.m1.4.4.3.cmml">=</mo><mrow id="A1.Ex22.m1.4.4.2.2" xref="A1.Ex22.m1.4.4.2.3.cmml"><mo id="A1.Ex22.m1.4.4.2.2.3" stretchy="false" xref="A1.Ex22.m1.4.4.2.3.1.cmml">{</mo><mrow id="A1.Ex22.m1.3.3.1.1.1" xref="A1.Ex22.m1.3.3.1.1.1.cmml"><mi id="A1.Ex22.m1.3.3.1.1.1.2" xref="A1.Ex22.m1.3.3.1.1.1.2.cmml">v</mi><mo id="A1.Ex22.m1.3.3.1.1.1.1" xref="A1.Ex22.m1.3.3.1.1.1.1.cmml">∈</mo><msup id="A1.Ex22.m1.3.3.1.1.1.3" xref="A1.Ex22.m1.3.3.1.1.1.3.cmml"><mi id="A1.Ex22.m1.3.3.1.1.1.3.2" xref="A1.Ex22.m1.3.3.1.1.1.3.2.cmml">S</mi><mrow id="A1.Ex22.m1.3.3.1.1.1.3.3" xref="A1.Ex22.m1.3.3.1.1.1.3.3.cmml"><mi id="A1.Ex22.m1.3.3.1.1.1.3.3.2" xref="A1.Ex22.m1.3.3.1.1.1.3.3.2.cmml">d</mi><mo id="A1.Ex22.m1.3.3.1.1.1.3.3.1" xref="A1.Ex22.m1.3.3.1.1.1.3.3.1.cmml">−</mo><mn id="A1.Ex22.m1.3.3.1.1.1.3.3.3" xref="A1.Ex22.m1.3.3.1.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><mo fence="true" id="A1.Ex22.m1.4.4.2.2.4" lspace="0em" rspace="0em" xref="A1.Ex22.m1.4.4.2.3.1.cmml">∣</mo><mrow id="A1.Ex22.m1.4.4.2.2.2" xref="A1.Ex22.m1.4.4.2.2.2.cmml"><mrow id="A1.Ex22.m1.4.4.2.2.2.1" xref="A1.Ex22.m1.4.4.2.2.2.1.cmml"><mi id="A1.Ex22.m1.4.4.2.2.2.1.3" xref="A1.Ex22.m1.4.4.2.2.2.1.3.cmml">μ</mi><mo id="A1.Ex22.m1.4.4.2.2.2.1.2" xref="A1.Ex22.m1.4.4.2.2.2.1.2.cmml">⁢</mo><mrow id="A1.Ex22.m1.4.4.2.2.2.1.1.1" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.cmml"><mo id="A1.Ex22.m1.4.4.2.2.2.1.1.1.2" stretchy="false" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex22.m1.4.4.2.2.2.1.1.1.1" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.2" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.Ex22.m1.2.2.2.4" xref="A1.Ex22.m1.2.2.2.3.cmml"><mi id="A1.Ex22.m1.1.1.1.1" xref="A1.Ex22.m1.1.1.1.1.cmml">x</mi><mo id="A1.Ex22.m1.2.2.2.4.1" xref="A1.Ex22.m1.2.2.2.3.cmml">,</mo><mi id="A1.Ex22.m1.2.2.2.2" xref="A1.Ex22.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.3" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A1.Ex22.m1.4.4.2.2.2.1.1.1.3" stretchy="false" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex22.m1.4.4.2.2.2.2" xref="A1.Ex22.m1.4.4.2.2.2.2.cmml">≥</mo><mi id="A1.Ex22.m1.4.4.2.2.2.3" xref="A1.Ex22.m1.4.4.2.2.2.3.cmml">t</mi></mrow><mo id="A1.Ex22.m1.4.4.2.2.5" stretchy="false" xref="A1.Ex22.m1.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex22.m1.4b"><apply id="A1.Ex22.m1.4.4.cmml" xref="A1.Ex22.m1.4.4"><eq id="A1.Ex22.m1.4.4.3.cmml" xref="A1.Ex22.m1.4.4.3"></eq><apply id="A1.Ex22.m1.4.4.4.cmml" xref="A1.Ex22.m1.4.4.4"><setdiff id="A1.Ex22.m1.4.4.4.1.cmml" xref="A1.Ex22.m1.4.4.4.1"></setdiff><apply id="A1.Ex22.m1.4.4.4.2.cmml" xref="A1.Ex22.m1.4.4.4.2"><csymbol cd="ambiguous" id="A1.Ex22.m1.4.4.4.2.1.cmml" xref="A1.Ex22.m1.4.4.4.2">superscript</csymbol><ci id="A1.Ex22.m1.4.4.4.2.2.cmml" xref="A1.Ex22.m1.4.4.4.2.2">𝑆</ci><apply id="A1.Ex22.m1.4.4.4.2.3.cmml" xref="A1.Ex22.m1.4.4.4.2.3"><minus id="A1.Ex22.m1.4.4.4.2.3.1.cmml" xref="A1.Ex22.m1.4.4.4.2.3.1"></minus><ci id="A1.Ex22.m1.4.4.4.2.3.2.cmml" xref="A1.Ex22.m1.4.4.4.2.3.2">𝑑</ci><cn id="A1.Ex22.m1.4.4.4.2.3.3.cmml" type="integer" 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xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.2">ℋ</ci><ci id="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.3.cmml" xref="A1.Ex22.m1.4.4.2.2.2.1.1.1.1.2.3">𝑝</ci></apply><list id="A1.Ex22.m1.2.2.2.3.cmml" xref="A1.Ex22.m1.2.2.2.4"><ci id="A1.Ex22.m1.1.1.1.1.cmml" xref="A1.Ex22.m1.1.1.1.1">𝑥</ci><ci id="A1.Ex22.m1.2.2.2.2.cmml" xref="A1.Ex22.m1.2.2.2.2">𝑣</ci></list></apply></apply><ci id="A1.Ex22.m1.4.4.2.2.2.3.cmml" xref="A1.Ex22.m1.4.4.2.2.2.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex22.m1.4c">S^{d-1}\setminus V=\{v\in S^{d-1}\mid\mu(\mathcal{H}^{p}_{x,v})\geq t\}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex22.m1.4d">italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∖ italic_V = { italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∣ italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) ≥ italic_t }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A1.SS2.7.p2"> <p class="ltx_p" id="A1.SS2.7.p2.14">is closed. To this end, consider an arbitrary sequence <math alttext="(v_{n})_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.1.m1.1"><semantics id="A1.SS2.7.p2.1.m1.1a"><msub id="A1.SS2.7.p2.1.m1.1.1" xref="A1.SS2.7.p2.1.m1.1.1.cmml"><mrow id="A1.SS2.7.p2.1.m1.1.1.1.1" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.cmml"><mo id="A1.SS2.7.p2.1.m1.1.1.1.1.2" stretchy="false" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.7.p2.1.m1.1.1.1.1.1" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.1.m1.1.1.1.1.1.2" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.2.cmml">v</mi><mi id="A1.SS2.7.p2.1.m1.1.1.1.1.1.3" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A1.SS2.7.p2.1.m1.1.1.1.1.3" stretchy="false" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS2.7.p2.1.m1.1.1.3" xref="A1.SS2.7.p2.1.m1.1.1.3.cmml"><mi id="A1.SS2.7.p2.1.m1.1.1.3.2" xref="A1.SS2.7.p2.1.m1.1.1.3.2.cmml">n</mi><mo id="A1.SS2.7.p2.1.m1.1.1.3.1" xref="A1.SS2.7.p2.1.m1.1.1.3.1.cmml">∈</mo><mi id="A1.SS2.7.p2.1.m1.1.1.3.3" xref="A1.SS2.7.p2.1.m1.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.1.m1.1b"><apply id="A1.SS2.7.p2.1.m1.1.1.cmml" xref="A1.SS2.7.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.1.m1.1.1.2.cmml" xref="A1.SS2.7.p2.1.m1.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.1.m1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.1.m1.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="A1.SS2.7.p2.1.m1.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.2">𝑣</ci><ci id="A1.SS2.7.p2.1.m1.1.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.1.m1.1.1.1.1.1.3">𝑛</ci></apply><apply id="A1.SS2.7.p2.1.m1.1.1.3.cmml" xref="A1.SS2.7.p2.1.m1.1.1.3"><in id="A1.SS2.7.p2.1.m1.1.1.3.1.cmml" xref="A1.SS2.7.p2.1.m1.1.1.3.1"></in><ci id="A1.SS2.7.p2.1.m1.1.1.3.2.cmml" xref="A1.SS2.7.p2.1.m1.1.1.3.2">𝑛</ci><ci id="A1.SS2.7.p2.1.m1.1.1.3.3.cmml" xref="A1.SS2.7.p2.1.m1.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.1.m1.1c">(v_{n})_{n\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.1.m1.1d">( italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converging to <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.2.m2.1"><semantics id="A1.SS2.7.p2.2.m2.1a"><mrow id="A1.SS2.7.p2.2.m2.1.1" xref="A1.SS2.7.p2.2.m2.1.1.cmml"><mi id="A1.SS2.7.p2.2.m2.1.1.2" xref="A1.SS2.7.p2.2.m2.1.1.2.cmml">v</mi><mo id="A1.SS2.7.p2.2.m2.1.1.1" xref="A1.SS2.7.p2.2.m2.1.1.1.cmml">∈</mo><msup id="A1.SS2.7.p2.2.m2.1.1.3" xref="A1.SS2.7.p2.2.m2.1.1.3.cmml"><mi id="A1.SS2.7.p2.2.m2.1.1.3.2" xref="A1.SS2.7.p2.2.m2.1.1.3.2.cmml">S</mi><mrow id="A1.SS2.7.p2.2.m2.1.1.3.3" xref="A1.SS2.7.p2.2.m2.1.1.3.3.cmml"><mi id="A1.SS2.7.p2.2.m2.1.1.3.3.2" xref="A1.SS2.7.p2.2.m2.1.1.3.3.2.cmml">d</mi><mo id="A1.SS2.7.p2.2.m2.1.1.3.3.1" xref="A1.SS2.7.p2.2.m2.1.1.3.3.1.cmml">−</mo><mn id="A1.SS2.7.p2.2.m2.1.1.3.3.3" xref="A1.SS2.7.p2.2.m2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.2.m2.1b"><apply id="A1.SS2.7.p2.2.m2.1.1.cmml" xref="A1.SS2.7.p2.2.m2.1.1"><in id="A1.SS2.7.p2.2.m2.1.1.1.cmml" xref="A1.SS2.7.p2.2.m2.1.1.1"></in><ci id="A1.SS2.7.p2.2.m2.1.1.2.cmml" xref="A1.SS2.7.p2.2.m2.1.1.2">𝑣</ci><apply id="A1.SS2.7.p2.2.m2.1.1.3.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.2.m2.1.1.3.1.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3">superscript</csymbol><ci id="A1.SS2.7.p2.2.m2.1.1.3.2.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3.2">𝑆</ci><apply id="A1.SS2.7.p2.2.m2.1.1.3.3.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3.3"><minus id="A1.SS2.7.p2.2.m2.1.1.3.3.1.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3.3.1"></minus><ci id="A1.SS2.7.p2.2.m2.1.1.3.3.2.cmml" xref="A1.SS2.7.p2.2.m2.1.1.3.3.2">𝑑</ci><cn id="A1.SS2.7.p2.2.m2.1.1.3.3.3.cmml" type="integer" xref="A1.SS2.7.p2.2.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.2.m2.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.2.m2.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> satisfying <math alttext="v_{n}\in S^{d-1}\setminus V" class="ltx_Math" display="inline" id="A1.SS2.7.p2.3.m3.1"><semantics id="A1.SS2.7.p2.3.m3.1a"><mrow id="A1.SS2.7.p2.3.m3.1.1" xref="A1.SS2.7.p2.3.m3.1.1.cmml"><msub id="A1.SS2.7.p2.3.m3.1.1.2" xref="A1.SS2.7.p2.3.m3.1.1.2.cmml"><mi id="A1.SS2.7.p2.3.m3.1.1.2.2" xref="A1.SS2.7.p2.3.m3.1.1.2.2.cmml">v</mi><mi id="A1.SS2.7.p2.3.m3.1.1.2.3" xref="A1.SS2.7.p2.3.m3.1.1.2.3.cmml">n</mi></msub><mo id="A1.SS2.7.p2.3.m3.1.1.1" xref="A1.SS2.7.p2.3.m3.1.1.1.cmml">∈</mo><mrow id="A1.SS2.7.p2.3.m3.1.1.3" xref="A1.SS2.7.p2.3.m3.1.1.3.cmml"><msup id="A1.SS2.7.p2.3.m3.1.1.3.2" xref="A1.SS2.7.p2.3.m3.1.1.3.2.cmml"><mi id="A1.SS2.7.p2.3.m3.1.1.3.2.2" xref="A1.SS2.7.p2.3.m3.1.1.3.2.2.cmml">S</mi><mrow id="A1.SS2.7.p2.3.m3.1.1.3.2.3" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.cmml"><mi id="A1.SS2.7.p2.3.m3.1.1.3.2.3.2" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.2.cmml">d</mi><mo id="A1.SS2.7.p2.3.m3.1.1.3.2.3.1" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.1.cmml">−</mo><mn id="A1.SS2.7.p2.3.m3.1.1.3.2.3.3" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.3.cmml">1</mn></mrow></msup><mo id="A1.SS2.7.p2.3.m3.1.1.3.1" xref="A1.SS2.7.p2.3.m3.1.1.3.1.cmml">∖</mo><mi id="A1.SS2.7.p2.3.m3.1.1.3.3" xref="A1.SS2.7.p2.3.m3.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.3.m3.1b"><apply id="A1.SS2.7.p2.3.m3.1.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1"><in id="A1.SS2.7.p2.3.m3.1.1.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1.1"></in><apply id="A1.SS2.7.p2.3.m3.1.1.2.cmml" xref="A1.SS2.7.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.3.m3.1.1.2.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1.2">subscript</csymbol><ci id="A1.SS2.7.p2.3.m3.1.1.2.2.cmml" xref="A1.SS2.7.p2.3.m3.1.1.2.2">𝑣</ci><ci id="A1.SS2.7.p2.3.m3.1.1.2.3.cmml" xref="A1.SS2.7.p2.3.m3.1.1.2.3">𝑛</ci></apply><apply id="A1.SS2.7.p2.3.m3.1.1.3.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3"><setdiff id="A1.SS2.7.p2.3.m3.1.1.3.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.1"></setdiff><apply id="A1.SS2.7.p2.3.m3.1.1.3.2.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.3.m3.1.1.3.2.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2">superscript</csymbol><ci id="A1.SS2.7.p2.3.m3.1.1.3.2.2.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2.2">𝑆</ci><apply id="A1.SS2.7.p2.3.m3.1.1.3.2.3.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3"><minus id="A1.SS2.7.p2.3.m3.1.1.3.2.3.1.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.1"></minus><ci id="A1.SS2.7.p2.3.m3.1.1.3.2.3.2.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.2">𝑑</ci><cn id="A1.SS2.7.p2.3.m3.1.1.3.2.3.3.cmml" type="integer" xref="A1.SS2.7.p2.3.m3.1.1.3.2.3.3">1</cn></apply></apply><ci id="A1.SS2.7.p2.3.m3.1.1.3.3.cmml" xref="A1.SS2.7.p2.3.m3.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.3.m3.1c">v_{n}\in S^{d-1}\setminus V</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.3.m3.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∖ italic_V</annotation></semantics></math> for all <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.4.m4.1"><semantics id="A1.SS2.7.p2.4.m4.1a"><mrow id="A1.SS2.7.p2.4.m4.1.1" xref="A1.SS2.7.p2.4.m4.1.1.cmml"><mi id="A1.SS2.7.p2.4.m4.1.1.2" xref="A1.SS2.7.p2.4.m4.1.1.2.cmml">n</mi><mo id="A1.SS2.7.p2.4.m4.1.1.1" xref="A1.SS2.7.p2.4.m4.1.1.1.cmml">∈</mo><mi id="A1.SS2.7.p2.4.m4.1.1.3" xref="A1.SS2.7.p2.4.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.4.m4.1b"><apply id="A1.SS2.7.p2.4.m4.1.1.cmml" xref="A1.SS2.7.p2.4.m4.1.1"><in id="A1.SS2.7.p2.4.m4.1.1.1.cmml" xref="A1.SS2.7.p2.4.m4.1.1.1"></in><ci id="A1.SS2.7.p2.4.m4.1.1.2.cmml" xref="A1.SS2.7.p2.4.m4.1.1.2">𝑛</ci><ci id="A1.SS2.7.p2.4.m4.1.1.3.cmml" xref="A1.SS2.7.p2.4.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.4.m4.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.4.m4.1d">italic_n ∈ blackboard_N</annotation></semantics></math>. We will prove that then <math alttext="v\in S^{d-1}\setminus V" class="ltx_Math" display="inline" id="A1.SS2.7.p2.5.m5.1"><semantics id="A1.SS2.7.p2.5.m5.1a"><mrow id="A1.SS2.7.p2.5.m5.1.1" xref="A1.SS2.7.p2.5.m5.1.1.cmml"><mi id="A1.SS2.7.p2.5.m5.1.1.2" xref="A1.SS2.7.p2.5.m5.1.1.2.cmml">v</mi><mo id="A1.SS2.7.p2.5.m5.1.1.1" xref="A1.SS2.7.p2.5.m5.1.1.1.cmml">∈</mo><mrow id="A1.SS2.7.p2.5.m5.1.1.3" xref="A1.SS2.7.p2.5.m5.1.1.3.cmml"><msup id="A1.SS2.7.p2.5.m5.1.1.3.2" xref="A1.SS2.7.p2.5.m5.1.1.3.2.cmml"><mi id="A1.SS2.7.p2.5.m5.1.1.3.2.2" xref="A1.SS2.7.p2.5.m5.1.1.3.2.2.cmml">S</mi><mrow id="A1.SS2.7.p2.5.m5.1.1.3.2.3" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.cmml"><mi id="A1.SS2.7.p2.5.m5.1.1.3.2.3.2" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.2.cmml">d</mi><mo id="A1.SS2.7.p2.5.m5.1.1.3.2.3.1" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.1.cmml">−</mo><mn id="A1.SS2.7.p2.5.m5.1.1.3.2.3.3" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.3.cmml">1</mn></mrow></msup><mo id="A1.SS2.7.p2.5.m5.1.1.3.1" xref="A1.SS2.7.p2.5.m5.1.1.3.1.cmml">∖</mo><mi id="A1.SS2.7.p2.5.m5.1.1.3.3" xref="A1.SS2.7.p2.5.m5.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.5.m5.1b"><apply id="A1.SS2.7.p2.5.m5.1.1.cmml" xref="A1.SS2.7.p2.5.m5.1.1"><in id="A1.SS2.7.p2.5.m5.1.1.1.cmml" xref="A1.SS2.7.p2.5.m5.1.1.1"></in><ci id="A1.SS2.7.p2.5.m5.1.1.2.cmml" xref="A1.SS2.7.p2.5.m5.1.1.2">𝑣</ci><apply id="A1.SS2.7.p2.5.m5.1.1.3.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3"><setdiff id="A1.SS2.7.p2.5.m5.1.1.3.1.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.1"></setdiff><apply id="A1.SS2.7.p2.5.m5.1.1.3.2.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.5.m5.1.1.3.2.1.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2">superscript</csymbol><ci id="A1.SS2.7.p2.5.m5.1.1.3.2.2.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2.2">𝑆</ci><apply id="A1.SS2.7.p2.5.m5.1.1.3.2.3.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3"><minus id="A1.SS2.7.p2.5.m5.1.1.3.2.3.1.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.1"></minus><ci id="A1.SS2.7.p2.5.m5.1.1.3.2.3.2.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.2">𝑑</ci><cn id="A1.SS2.7.p2.5.m5.1.1.3.2.3.3.cmml" type="integer" xref="A1.SS2.7.p2.5.m5.1.1.3.2.3.3">1</cn></apply></apply><ci id="A1.SS2.7.p2.5.m5.1.1.3.3.cmml" xref="A1.SS2.7.p2.5.m5.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.5.m5.1c">v\in S^{d-1}\setminus V</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.5.m5.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∖ italic_V</annotation></semantics></math>. Let <math alttext="X_{n}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.6.m6.1"><semantics id="A1.SS2.7.p2.6.m6.1a"><msub id="A1.SS2.7.p2.6.m6.1.1" xref="A1.SS2.7.p2.6.m6.1.1.cmml"><mi id="A1.SS2.7.p2.6.m6.1.1.2" xref="A1.SS2.7.p2.6.m6.1.1.2.cmml">X</mi><mi id="A1.SS2.7.p2.6.m6.1.1.3" xref="A1.SS2.7.p2.6.m6.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.6.m6.1b"><apply id="A1.SS2.7.p2.6.m6.1.1.cmml" xref="A1.SS2.7.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.6.m6.1.1.1.cmml" xref="A1.SS2.7.p2.6.m6.1.1">subscript</csymbol><ci id="A1.SS2.7.p2.6.m6.1.1.2.cmml" xref="A1.SS2.7.p2.6.m6.1.1.2">𝑋</ci><ci id="A1.SS2.7.p2.6.m6.1.1.3.cmml" xref="A1.SS2.7.p2.6.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.6.m6.1c">X_{n}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.6.m6.1d">italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> be the indicator random variable for containment of a point drawn according to <math alttext="\mu" class="ltx_Math" display="inline" id="A1.SS2.7.p2.7.m7.1"><semantics id="A1.SS2.7.p2.7.m7.1a"><mi id="A1.SS2.7.p2.7.m7.1.1" xref="A1.SS2.7.p2.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.7.m7.1b"><ci id="A1.SS2.7.p2.7.m7.1.1.cmml" xref="A1.SS2.7.p2.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.7.m7.1d">italic_μ</annotation></semantics></math> in <math alttext="\mathcal{H}^{p}_{x,v_{n}}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.8.m8.2"><semantics id="A1.SS2.7.p2.8.m8.2a"><msubsup id="A1.SS2.7.p2.8.m8.2.3" xref="A1.SS2.7.p2.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.7.p2.8.m8.2.3.2.2" xref="A1.SS2.7.p2.8.m8.2.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.7.p2.8.m8.2.2.2.2" xref="A1.SS2.7.p2.8.m8.2.2.2.3.cmml"><mi id="A1.SS2.7.p2.8.m8.1.1.1.1" xref="A1.SS2.7.p2.8.m8.1.1.1.1.cmml">x</mi><mo id="A1.SS2.7.p2.8.m8.2.2.2.2.2" xref="A1.SS2.7.p2.8.m8.2.2.2.3.cmml">,</mo><msub id="A1.SS2.7.p2.8.m8.2.2.2.2.1" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1.cmml"><mi id="A1.SS2.7.p2.8.m8.2.2.2.2.1.2" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1.2.cmml">v</mi><mi id="A1.SS2.7.p2.8.m8.2.2.2.2.1.3" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1.3.cmml">n</mi></msub></mrow><mi id="A1.SS2.7.p2.8.m8.2.3.2.3" xref="A1.SS2.7.p2.8.m8.2.3.2.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.8.m8.2b"><apply id="A1.SS2.7.p2.8.m8.2.3.cmml" xref="A1.SS2.7.p2.8.m8.2.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.8.m8.2.3.1.cmml" xref="A1.SS2.7.p2.8.m8.2.3">subscript</csymbol><apply id="A1.SS2.7.p2.8.m8.2.3.2.cmml" xref="A1.SS2.7.p2.8.m8.2.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.8.m8.2.3.2.1.cmml" xref="A1.SS2.7.p2.8.m8.2.3">superscript</csymbol><ci id="A1.SS2.7.p2.8.m8.2.3.2.2.cmml" xref="A1.SS2.7.p2.8.m8.2.3.2.2">ℋ</ci><ci id="A1.SS2.7.p2.8.m8.2.3.2.3.cmml" xref="A1.SS2.7.p2.8.m8.2.3.2.3">𝑝</ci></apply><list id="A1.SS2.7.p2.8.m8.2.2.2.3.cmml" xref="A1.SS2.7.p2.8.m8.2.2.2.2"><ci id="A1.SS2.7.p2.8.m8.1.1.1.1.cmml" xref="A1.SS2.7.p2.8.m8.1.1.1.1">𝑥</ci><apply id="A1.SS2.7.p2.8.m8.2.2.2.2.1.cmml" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.8.m8.2.2.2.2.1.1.cmml" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1">subscript</csymbol><ci id="A1.SS2.7.p2.8.m8.2.2.2.2.1.2.cmml" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1.2">𝑣</ci><ci id="A1.SS2.7.p2.8.m8.2.2.2.2.1.3.cmml" xref="A1.SS2.7.p2.8.m8.2.2.2.2.1.3">𝑛</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.8.m8.2c">\mathcal{H}^{p}_{x,v_{n}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.8.m8.2d">caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Observe that by <math alttext="\lim_{n\rightarrow\infty}\lVert x-\varepsilon v_{n}-z\rVert_{p}=\lVert x-% \varepsilon v-z\rVert_{p}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.9.m9.2"><semantics id="A1.SS2.7.p2.9.m9.2a"><mrow id="A1.SS2.7.p2.9.m9.2.2" xref="A1.SS2.7.p2.9.m9.2.2.cmml"><mrow id="A1.SS2.7.p2.9.m9.1.1.1" xref="A1.SS2.7.p2.9.m9.1.1.1.cmml"><msub id="A1.SS2.7.p2.9.m9.1.1.1.2" xref="A1.SS2.7.p2.9.m9.1.1.1.2.cmml"><mo id="A1.SS2.7.p2.9.m9.1.1.1.2.2" xref="A1.SS2.7.p2.9.m9.1.1.1.2.2.cmml">lim</mo><mrow id="A1.SS2.7.p2.9.m9.1.1.1.2.3" xref="A1.SS2.7.p2.9.m9.1.1.1.2.3.cmml"><mi id="A1.SS2.7.p2.9.m9.1.1.1.2.3.2" xref="A1.SS2.7.p2.9.m9.1.1.1.2.3.2.cmml">n</mi><mo id="A1.SS2.7.p2.9.m9.1.1.1.2.3.1" stretchy="false" xref="A1.SS2.7.p2.9.m9.1.1.1.2.3.1.cmml">→</mo><mi id="A1.SS2.7.p2.9.m9.1.1.1.2.3.3" mathvariant="normal" xref="A1.SS2.7.p2.9.m9.1.1.1.2.3.3.cmml">∞</mi></mrow></msub><msub id="A1.SS2.7.p2.9.m9.1.1.1.1" xref="A1.SS2.7.p2.9.m9.1.1.1.1.cmml"><mrow id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.2" lspace="0em" rspace="0em" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.2" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.1" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.2" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.1" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3.cmml"><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3.2" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3.2.cmml">v</mi><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3.3" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.3.3.3.cmml">n</mi></msub></mrow><mo id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.1a" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.4" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.9.m9.1.1.1.1.1.1.3" lspace="0em" rspace="0.1389em" xref="A1.SS2.7.p2.9.m9.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.9.m9.1.1.1.1.3" xref="A1.SS2.7.p2.9.m9.1.1.1.1.3.cmml">p</mi></msub></mrow><mo id="A1.SS2.7.p2.9.m9.2.2.3" lspace="0.1389em" rspace="0.1389em" xref="A1.SS2.7.p2.9.m9.2.2.3.cmml">=</mo><msub id="A1.SS2.7.p2.9.m9.2.2.2" xref="A1.SS2.7.p2.9.m9.2.2.2.cmml"><mrow id="A1.SS2.7.p2.9.m9.2.2.2.1.1" xref="A1.SS2.7.p2.9.m9.2.2.2.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.9.m9.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS2.7.p2.9.m9.2.2.2.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.cmml"><mi id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.2" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.1" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.2" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.1" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.3" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.1a" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.4" xref="A1.SS2.7.p2.9.m9.2.2.2.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.9.m9.2.2.2.1.1.3" lspace="0em" xref="A1.SS2.7.p2.9.m9.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.9.m9.2.2.2.3" xref="A1.SS2.7.p2.9.m9.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" 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start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_x - italic_ε italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="z\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.10.m10.1"><semantics id="A1.SS2.7.p2.10.m10.1a"><mrow id="A1.SS2.7.p2.10.m10.1.1" xref="A1.SS2.7.p2.10.m10.1.1.cmml"><mi id="A1.SS2.7.p2.10.m10.1.1.2" xref="A1.SS2.7.p2.10.m10.1.1.2.cmml">z</mi><mo id="A1.SS2.7.p2.10.m10.1.1.1" xref="A1.SS2.7.p2.10.m10.1.1.1.cmml">∈</mo><msup id="A1.SS2.7.p2.10.m10.1.1.3" xref="A1.SS2.7.p2.10.m10.1.1.3.cmml"><mi id="A1.SS2.7.p2.10.m10.1.1.3.2" xref="A1.SS2.7.p2.10.m10.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.7.p2.10.m10.1.1.3.3" xref="A1.SS2.7.p2.10.m10.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.10.m10.1b"><apply id="A1.SS2.7.p2.10.m10.1.1.cmml" xref="A1.SS2.7.p2.10.m10.1.1"><in id="A1.SS2.7.p2.10.m10.1.1.1.cmml" xref="A1.SS2.7.p2.10.m10.1.1.1"></in><ci id="A1.SS2.7.p2.10.m10.1.1.2.cmml" xref="A1.SS2.7.p2.10.m10.1.1.2">𝑧</ci><apply id="A1.SS2.7.p2.10.m10.1.1.3.cmml" xref="A1.SS2.7.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.10.m10.1.1.3.1.cmml" xref="A1.SS2.7.p2.10.m10.1.1.3">superscript</csymbol><ci id="A1.SS2.7.p2.10.m10.1.1.3.2.cmml" xref="A1.SS2.7.p2.10.m10.1.1.3.2">ℝ</ci><ci id="A1.SS2.7.p2.10.m10.1.1.3.3.cmml" xref="A1.SS2.7.p2.10.m10.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.10.m10.1c">z\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.10.m10.1d">italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, the sequence of random variables <math alttext="(X_{n})_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.11.m11.1"><semantics id="A1.SS2.7.p2.11.m11.1a"><msub id="A1.SS2.7.p2.11.m11.1.1" xref="A1.SS2.7.p2.11.m11.1.1.cmml"><mrow id="A1.SS2.7.p2.11.m11.1.1.1.1" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.cmml"><mo id="A1.SS2.7.p2.11.m11.1.1.1.1.2" stretchy="false" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.7.p2.11.m11.1.1.1.1.1" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.11.m11.1.1.1.1.1.2" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.2.cmml">X</mi><mi id="A1.SS2.7.p2.11.m11.1.1.1.1.1.3" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A1.SS2.7.p2.11.m11.1.1.1.1.3" stretchy="false" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS2.7.p2.11.m11.1.1.3" xref="A1.SS2.7.p2.11.m11.1.1.3.cmml"><mi id="A1.SS2.7.p2.11.m11.1.1.3.2" xref="A1.SS2.7.p2.11.m11.1.1.3.2.cmml">n</mi><mo id="A1.SS2.7.p2.11.m11.1.1.3.1" xref="A1.SS2.7.p2.11.m11.1.1.3.1.cmml">∈</mo><mi id="A1.SS2.7.p2.11.m11.1.1.3.3" xref="A1.SS2.7.p2.11.m11.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.11.m11.1b"><apply id="A1.SS2.7.p2.11.m11.1.1.cmml" xref="A1.SS2.7.p2.11.m11.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.11.m11.1.1.2.cmml" xref="A1.SS2.7.p2.11.m11.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.11.m11.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.11.m11.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.11.m11.1.1.1.1">subscript</csymbol><ci id="A1.SS2.7.p2.11.m11.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.2">𝑋</ci><ci id="A1.SS2.7.p2.11.m11.1.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.11.m11.1.1.1.1.1.3">𝑛</ci></apply><apply id="A1.SS2.7.p2.11.m11.1.1.3.cmml" xref="A1.SS2.7.p2.11.m11.1.1.3"><in id="A1.SS2.7.p2.11.m11.1.1.3.1.cmml" xref="A1.SS2.7.p2.11.m11.1.1.3.1"></in><ci id="A1.SS2.7.p2.11.m11.1.1.3.2.cmml" xref="A1.SS2.7.p2.11.m11.1.1.3.2">𝑛</ci><ci id="A1.SS2.7.p2.11.m11.1.1.3.3.cmml" xref="A1.SS2.7.p2.11.m11.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.11.m11.1c">(X_{n})_{n\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.11.m11.1d">( italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converges point-wise to a random variable <math alttext="X" class="ltx_Math" display="inline" id="A1.SS2.7.p2.12.m12.1"><semantics id="A1.SS2.7.p2.12.m12.1a"><mi id="A1.SS2.7.p2.12.m12.1.1" xref="A1.SS2.7.p2.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.12.m12.1b"><ci id="A1.SS2.7.p2.12.m12.1.1.cmml" xref="A1.SS2.7.p2.12.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.12.m12.1d">italic_X</annotation></semantics></math>. In other words, <math alttext="(X_{n})_{n\in\mathbb{N}}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.13.m13.1"><semantics id="A1.SS2.7.p2.13.m13.1a"><msub id="A1.SS2.7.p2.13.m13.1.1" xref="A1.SS2.7.p2.13.m13.1.1.cmml"><mrow id="A1.SS2.7.p2.13.m13.1.1.1.1" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.cmml"><mo id="A1.SS2.7.p2.13.m13.1.1.1.1.2" stretchy="false" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.7.p2.13.m13.1.1.1.1.1" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.13.m13.1.1.1.1.1.2" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.2.cmml">X</mi><mi id="A1.SS2.7.p2.13.m13.1.1.1.1.1.3" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A1.SS2.7.p2.13.m13.1.1.1.1.3" stretchy="false" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS2.7.p2.13.m13.1.1.3" xref="A1.SS2.7.p2.13.m13.1.1.3.cmml"><mi id="A1.SS2.7.p2.13.m13.1.1.3.2" xref="A1.SS2.7.p2.13.m13.1.1.3.2.cmml">n</mi><mo id="A1.SS2.7.p2.13.m13.1.1.3.1" xref="A1.SS2.7.p2.13.m13.1.1.3.1.cmml">∈</mo><mi id="A1.SS2.7.p2.13.m13.1.1.3.3" xref="A1.SS2.7.p2.13.m13.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.13.m13.1b"><apply id="A1.SS2.7.p2.13.m13.1.1.cmml" xref="A1.SS2.7.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.13.m13.1.1.2.cmml" xref="A1.SS2.7.p2.13.m13.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.13.m13.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.13.m13.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="A1.SS2.7.p2.13.m13.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.2">𝑋</ci><ci id="A1.SS2.7.p2.13.m13.1.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.13.m13.1.1.1.1.1.3">𝑛</ci></apply><apply id="A1.SS2.7.p2.13.m13.1.1.3.cmml" xref="A1.SS2.7.p2.13.m13.1.1.3"><in id="A1.SS2.7.p2.13.m13.1.1.3.1.cmml" xref="A1.SS2.7.p2.13.m13.1.1.3.1"></in><ci id="A1.SS2.7.p2.13.m13.1.1.3.2.cmml" xref="A1.SS2.7.p2.13.m13.1.1.3.2">𝑛</ci><ci id="A1.SS2.7.p2.13.m13.1.1.3.3.cmml" xref="A1.SS2.7.p2.13.m13.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.13.m13.1c">(X_{n})_{n\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.13.m13.1d">( italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converges almost surely to <math alttext="X" class="ltx_Math" display="inline" id="A1.SS2.7.p2.14.m14.1"><semantics id="A1.SS2.7.p2.14.m14.1a"><mi id="A1.SS2.7.p2.14.m14.1.1" xref="A1.SS2.7.p2.14.m14.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.14.m14.1b"><ci id="A1.SS2.7.p2.14.m14.1.1.cmml" xref="A1.SS2.7.p2.14.m14.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.14.m14.1c">X</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.14.m14.1d">italic_X</annotation></semantics></math>, and by the dominated convergence theorem we get convergence in mean. Concretely, this means that we have</p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{n\rightarrow\infty}\mu(\mathcal{H}^{p}_{x,v_{n}})=\mathbb{E}X\geq t." class="ltx_Math" display="block" id="A1.Ex23.m1.3"><semantics id="A1.Ex23.m1.3a"><mrow id="A1.Ex23.m1.3.3.1" xref="A1.Ex23.m1.3.3.1.1.cmml"><mrow id="A1.Ex23.m1.3.3.1.1" xref="A1.Ex23.m1.3.3.1.1.cmml"><mrow id="A1.Ex23.m1.3.3.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.cmml"><munder id="A1.Ex23.m1.3.3.1.1.1.2" xref="A1.Ex23.m1.3.3.1.1.1.2.cmml"><mo id="A1.Ex23.m1.3.3.1.1.1.2.2" movablelimits="false" xref="A1.Ex23.m1.3.3.1.1.1.2.2.cmml">lim</mo><mrow id="A1.Ex23.m1.3.3.1.1.1.2.3" xref="A1.Ex23.m1.3.3.1.1.1.2.3.cmml"><mi id="A1.Ex23.m1.3.3.1.1.1.2.3.2" xref="A1.Ex23.m1.3.3.1.1.1.2.3.2.cmml">n</mi><mo id="A1.Ex23.m1.3.3.1.1.1.2.3.1" stretchy="false" xref="A1.Ex23.m1.3.3.1.1.1.2.3.1.cmml">→</mo><mi id="A1.Ex23.m1.3.3.1.1.1.2.3.3" mathvariant="normal" xref="A1.Ex23.m1.3.3.1.1.1.2.3.3.cmml">∞</mi></mrow></munder><mrow id="A1.Ex23.m1.3.3.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.cmml"><mi id="A1.Ex23.m1.3.3.1.1.1.1.3" xref="A1.Ex23.m1.3.3.1.1.1.1.3.cmml">μ</mi><mo id="A1.Ex23.m1.3.3.1.1.1.1.2" xref="A1.Ex23.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex23.m1.3.3.1.1.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex23.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex23.m1.3.3.1.1.1.1.1.1.1" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.2.2" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.Ex23.m1.2.2.2.2" xref="A1.Ex23.m1.2.2.2.3.cmml"><mi id="A1.Ex23.m1.1.1.1.1" xref="A1.Ex23.m1.1.1.1.1.cmml">x</mi><mo id="A1.Ex23.m1.2.2.2.2.2" xref="A1.Ex23.m1.2.2.2.3.cmml">,</mo><msub id="A1.Ex23.m1.2.2.2.2.1" xref="A1.Ex23.m1.2.2.2.2.1.cmml"><mi id="A1.Ex23.m1.2.2.2.2.1.2" xref="A1.Ex23.m1.2.2.2.2.1.2.cmml">v</mi><mi id="A1.Ex23.m1.2.2.2.2.1.3" xref="A1.Ex23.m1.2.2.2.2.1.3.cmml">n</mi></msub></mrow><mi id="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.2.3" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A1.Ex23.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="A1.Ex23.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A1.Ex23.m1.3.3.1.1.3" xref="A1.Ex23.m1.3.3.1.1.3.cmml">=</mo><mrow id="A1.Ex23.m1.3.3.1.1.4" xref="A1.Ex23.m1.3.3.1.1.4.cmml"><mi id="A1.Ex23.m1.3.3.1.1.4.2" xref="A1.Ex23.m1.3.3.1.1.4.2.cmml">𝔼</mi><mo id="A1.Ex23.m1.3.3.1.1.4.1" xref="A1.Ex23.m1.3.3.1.1.4.1.cmml">⁢</mo><mi id="A1.Ex23.m1.3.3.1.1.4.3" xref="A1.Ex23.m1.3.3.1.1.4.3.cmml">X</mi></mrow><mo id="A1.Ex23.m1.3.3.1.1.5" xref="A1.Ex23.m1.3.3.1.1.5.cmml">≥</mo><mi id="A1.Ex23.m1.3.3.1.1.6" xref="A1.Ex23.m1.3.3.1.1.6.cmml">t</mi></mrow><mo id="A1.Ex23.m1.3.3.1.2" lspace="0em" xref="A1.Ex23.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex23.m1.3b"><apply id="A1.Ex23.m1.3.3.1.1.cmml" xref="A1.Ex23.m1.3.3.1"><and id="A1.Ex23.m1.3.3.1.1a.cmml" xref="A1.Ex23.m1.3.3.1"></and><apply id="A1.Ex23.m1.3.3.1.1b.cmml" xref="A1.Ex23.m1.3.3.1"><eq id="A1.Ex23.m1.3.3.1.1.3.cmml" xref="A1.Ex23.m1.3.3.1.1.3"></eq><apply id="A1.Ex23.m1.3.3.1.1.1.cmml" xref="A1.Ex23.m1.3.3.1.1.1"><apply id="A1.Ex23.m1.3.3.1.1.1.2.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex23.m1.3.3.1.1.1.2.1.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2">subscript</csymbol><limit id="A1.Ex23.m1.3.3.1.1.1.2.2.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2.2"></limit><apply id="A1.Ex23.m1.3.3.1.1.1.2.3.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2.3"><ci id="A1.Ex23.m1.3.3.1.1.1.2.3.1.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2.3.1">→</ci><ci id="A1.Ex23.m1.3.3.1.1.1.2.3.2.cmml" xref="A1.Ex23.m1.3.3.1.1.1.2.3.2">𝑛</ci><infinity id="A1.Ex23.m1.3.3.1.1.1.2.3.3.cmml" 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xref="A1.Ex23.m1.1.1.1.1">𝑥</ci><apply id="A1.Ex23.m1.2.2.2.2.1.cmml" xref="A1.Ex23.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="A1.Ex23.m1.2.2.2.2.1.1.cmml" xref="A1.Ex23.m1.2.2.2.2.1">subscript</csymbol><ci id="A1.Ex23.m1.2.2.2.2.1.2.cmml" xref="A1.Ex23.m1.2.2.2.2.1.2">𝑣</ci><ci id="A1.Ex23.m1.2.2.2.2.1.3.cmml" xref="A1.Ex23.m1.2.2.2.2.1.3">𝑛</ci></apply></list></apply></apply></apply><apply id="A1.Ex23.m1.3.3.1.1.4.cmml" xref="A1.Ex23.m1.3.3.1.1.4"><times id="A1.Ex23.m1.3.3.1.1.4.1.cmml" xref="A1.Ex23.m1.3.3.1.1.4.1"></times><ci id="A1.Ex23.m1.3.3.1.1.4.2.cmml" xref="A1.Ex23.m1.3.3.1.1.4.2">𝔼</ci><ci id="A1.Ex23.m1.3.3.1.1.4.3.cmml" xref="A1.Ex23.m1.3.3.1.1.4.3">𝑋</ci></apply></apply><apply id="A1.Ex23.m1.3.3.1.1c.cmml" xref="A1.Ex23.m1.3.3.1"><geq id="A1.Ex23.m1.3.3.1.1.5.cmml" xref="A1.Ex23.m1.3.3.1.1.5"></geq><share href="https://arxiv.org/html/2503.16089v1#A1.Ex23.m1.3.3.1.1.4.cmml" id="A1.Ex23.m1.3.3.1.1d.cmml" xref="A1.Ex23.m1.3.3.1"></share><ci id="A1.Ex23.m1.3.3.1.1.6.cmml" xref="A1.Ex23.m1.3.3.1.1.6">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex23.m1.3c">\lim_{n\rightarrow\infty}\mu(\mathcal{H}^{p}_{x,v_{n}})=\mathbb{E}X\geq t.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex23.m1.3d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = blackboard_E italic_X ≥ italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS2.7.p2.23">Let now <math alttext="z\notin\mathcal{H}^{p}_{x,v}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.15.m1.2"><semantics id="A1.SS2.7.p2.15.m1.2a"><mrow id="A1.SS2.7.p2.15.m1.2.3" xref="A1.SS2.7.p2.15.m1.2.3.cmml"><mi id="A1.SS2.7.p2.15.m1.2.3.2" xref="A1.SS2.7.p2.15.m1.2.3.2.cmml">z</mi><mo id="A1.SS2.7.p2.15.m1.2.3.1" xref="A1.SS2.7.p2.15.m1.2.3.1.cmml">∉</mo><msubsup id="A1.SS2.7.p2.15.m1.2.3.3" xref="A1.SS2.7.p2.15.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.7.p2.15.m1.2.3.3.2.2" xref="A1.SS2.7.p2.15.m1.2.3.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.7.p2.15.m1.2.2.2.4" xref="A1.SS2.7.p2.15.m1.2.2.2.3.cmml"><mi id="A1.SS2.7.p2.15.m1.1.1.1.1" xref="A1.SS2.7.p2.15.m1.1.1.1.1.cmml">x</mi><mo id="A1.SS2.7.p2.15.m1.2.2.2.4.1" xref="A1.SS2.7.p2.15.m1.2.2.2.3.cmml">,</mo><mi id="A1.SS2.7.p2.15.m1.2.2.2.2" xref="A1.SS2.7.p2.15.m1.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.7.p2.15.m1.2.3.3.2.3" xref="A1.SS2.7.p2.15.m1.2.3.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.15.m1.2b"><apply id="A1.SS2.7.p2.15.m1.2.3.cmml" xref="A1.SS2.7.p2.15.m1.2.3"><notin id="A1.SS2.7.p2.15.m1.2.3.1.cmml" xref="A1.SS2.7.p2.15.m1.2.3.1"></notin><ci id="A1.SS2.7.p2.15.m1.2.3.2.cmml" xref="A1.SS2.7.p2.15.m1.2.3.2">𝑧</ci><apply id="A1.SS2.7.p2.15.m1.2.3.3.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.15.m1.2.3.3.1.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3">subscript</csymbol><apply id="A1.SS2.7.p2.15.m1.2.3.3.2.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.15.m1.2.3.3.2.1.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3">superscript</csymbol><ci id="A1.SS2.7.p2.15.m1.2.3.3.2.2.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3.2.2">ℋ</ci><ci id="A1.SS2.7.p2.15.m1.2.3.3.2.3.cmml" xref="A1.SS2.7.p2.15.m1.2.3.3.2.3">𝑝</ci></apply><list id="A1.SS2.7.p2.15.m1.2.2.2.3.cmml" xref="A1.SS2.7.p2.15.m1.2.2.2.4"><ci id="A1.SS2.7.p2.15.m1.1.1.1.1.cmml" xref="A1.SS2.7.p2.15.m1.1.1.1.1">𝑥</ci><ci id="A1.SS2.7.p2.15.m1.2.2.2.2.cmml" xref="A1.SS2.7.p2.15.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.15.m1.2c">z\notin\mathcal{H}^{p}_{x,v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.15.m1.2d">italic_z ∉ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be arbitrary. Then there exist <math alttext="\varepsilon,\delta>0" class="ltx_Math" display="inline" id="A1.SS2.7.p2.16.m2.2"><semantics id="A1.SS2.7.p2.16.m2.2a"><mrow id="A1.SS2.7.p2.16.m2.2.3" xref="A1.SS2.7.p2.16.m2.2.3.cmml"><mrow id="A1.SS2.7.p2.16.m2.2.3.2.2" xref="A1.SS2.7.p2.16.m2.2.3.2.1.cmml"><mi id="A1.SS2.7.p2.16.m2.1.1" xref="A1.SS2.7.p2.16.m2.1.1.cmml">ε</mi><mo id="A1.SS2.7.p2.16.m2.2.3.2.2.1" xref="A1.SS2.7.p2.16.m2.2.3.2.1.cmml">,</mo><mi id="A1.SS2.7.p2.16.m2.2.2" xref="A1.SS2.7.p2.16.m2.2.2.cmml">δ</mi></mrow><mo id="A1.SS2.7.p2.16.m2.2.3.1" xref="A1.SS2.7.p2.16.m2.2.3.1.cmml">></mo><mn id="A1.SS2.7.p2.16.m2.2.3.3" xref="A1.SS2.7.p2.16.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.16.m2.2b"><apply id="A1.SS2.7.p2.16.m2.2.3.cmml" xref="A1.SS2.7.p2.16.m2.2.3"><gt id="A1.SS2.7.p2.16.m2.2.3.1.cmml" xref="A1.SS2.7.p2.16.m2.2.3.1"></gt><list id="A1.SS2.7.p2.16.m2.2.3.2.1.cmml" xref="A1.SS2.7.p2.16.m2.2.3.2.2"><ci id="A1.SS2.7.p2.16.m2.1.1.cmml" xref="A1.SS2.7.p2.16.m2.1.1">𝜀</ci><ci id="A1.SS2.7.p2.16.m2.2.2.cmml" xref="A1.SS2.7.p2.16.m2.2.2">𝛿</ci></list><cn id="A1.SS2.7.p2.16.m2.2.3.3.cmml" type="integer" xref="A1.SS2.7.p2.16.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.16.m2.2c">\varepsilon,\delta>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.16.m2.2d">italic_ε , italic_δ > 0</annotation></semantics></math> such that <math alttext="\lVert x-\varepsilon v-z\rVert_{p}<\lVert x-z\rVert_{p}-\delta" class="ltx_Math" display="inline" id="A1.SS2.7.p2.17.m3.2"><semantics id="A1.SS2.7.p2.17.m3.2a"><mrow id="A1.SS2.7.p2.17.m3.2.2" xref="A1.SS2.7.p2.17.m3.2.2.cmml"><msub id="A1.SS2.7.p2.17.m3.1.1.1" xref="A1.SS2.7.p2.17.m3.1.1.1.cmml"><mrow id="A1.SS2.7.p2.17.m3.1.1.1.1.1" xref="A1.SS2.7.p2.17.m3.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.17.m3.1.1.1.1.1.2" rspace="0em" xref="A1.SS2.7.p2.17.m3.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.2" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.2" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.1" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.3" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1a" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.4" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.17.m3.1.1.1.1.1.3" lspace="0em" xref="A1.SS2.7.p2.17.m3.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.17.m3.1.1.1.3" xref="A1.SS2.7.p2.17.m3.1.1.1.3.cmml">p</mi></msub><mo id="A1.SS2.7.p2.17.m3.2.2.3" rspace="0.1389em" xref="A1.SS2.7.p2.17.m3.2.2.3.cmml"><</mo><mrow id="A1.SS2.7.p2.17.m3.2.2.2" xref="A1.SS2.7.p2.17.m3.2.2.2.cmml"><msub id="A1.SS2.7.p2.17.m3.2.2.2.1" xref="A1.SS2.7.p2.17.m3.2.2.2.1.cmml"><mrow id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.2" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.1" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.3" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.3" lspace="0em" rspace="0em" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.17.m3.2.2.2.1.3" xref="A1.SS2.7.p2.17.m3.2.2.2.1.3.cmml">p</mi></msub><mo id="A1.SS2.7.p2.17.m3.2.2.2.2" xref="A1.SS2.7.p2.17.m3.2.2.2.2.cmml">−</mo><mi id="A1.SS2.7.p2.17.m3.2.2.2.3" xref="A1.SS2.7.p2.17.m3.2.2.2.3.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.17.m3.2b"><apply id="A1.SS2.7.p2.17.m3.2.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2"><lt id="A1.SS2.7.p2.17.m3.2.2.3.cmml" xref="A1.SS2.7.p2.17.m3.2.2.3"></lt><apply id="A1.SS2.7.p2.17.m3.1.1.1.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.17.m3.1.1.1.2.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.17.m3.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS2.7.p2.17.m3.1.1.1.1.2.1.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1"><minus id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.1"></minus><ci id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.2">𝑥</ci><apply id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3"><times id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.1.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.1"></times><ci id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.2.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.2">𝜀</ci><ci id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.3.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.3.3">𝑣</ci></apply><ci id="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.4.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.SS2.7.p2.17.m3.1.1.1.3.cmml" xref="A1.SS2.7.p2.17.m3.1.1.1.3">𝑝</ci></apply><apply id="A1.SS2.7.p2.17.m3.2.2.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2"><minus id="A1.SS2.7.p2.17.m3.2.2.2.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.2"></minus><apply id="A1.SS2.7.p2.17.m3.2.2.2.1.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.17.m3.2.2.2.1.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1">subscript</csymbol><apply id="A1.SS2.7.p2.17.m3.2.2.2.1.1.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1"><csymbol cd="latexml" id="A1.SS2.7.p2.17.m3.2.2.2.1.1.2.1.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1"><minus id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.1"></minus><ci id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.2">𝑥</ci><ci id="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A1.SS2.7.p2.17.m3.2.2.2.1.3.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.1.3">𝑝</ci></apply><ci id="A1.SS2.7.p2.17.m3.2.2.2.3.cmml" xref="A1.SS2.7.p2.17.m3.2.2.2.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.17.m3.2c">\lVert x-\varepsilon v-z\rVert_{p}<\lVert x-z\rVert_{p}-\delta</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.17.m3.2d">∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT < ∥ italic_x - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_δ</annotation></semantics></math>. Using <math alttext="\lim_{n\rightarrow\infty}\lVert x-\varepsilon v_{n}-z\rVert_{p}=\lVert x-% \varepsilon v-z\rVert_{p}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.18.m4.2"><semantics id="A1.SS2.7.p2.18.m4.2a"><mrow id="A1.SS2.7.p2.18.m4.2.2" xref="A1.SS2.7.p2.18.m4.2.2.cmml"><mrow id="A1.SS2.7.p2.18.m4.1.1.1" xref="A1.SS2.7.p2.18.m4.1.1.1.cmml"><msub id="A1.SS2.7.p2.18.m4.1.1.1.2" xref="A1.SS2.7.p2.18.m4.1.1.1.2.cmml"><mo id="A1.SS2.7.p2.18.m4.1.1.1.2.2" xref="A1.SS2.7.p2.18.m4.1.1.1.2.2.cmml">lim</mo><mrow id="A1.SS2.7.p2.18.m4.1.1.1.2.3" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.cmml"><mi id="A1.SS2.7.p2.18.m4.1.1.1.2.3.2" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.2.cmml">n</mi><mo id="A1.SS2.7.p2.18.m4.1.1.1.2.3.1" stretchy="false" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.1.cmml">→</mo><mi id="A1.SS2.7.p2.18.m4.1.1.1.2.3.3" mathvariant="normal" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.3.cmml">∞</mi></mrow></msub><msub id="A1.SS2.7.p2.18.m4.1.1.1.1" xref="A1.SS2.7.p2.18.m4.1.1.1.1.cmml"><mrow id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.2" lspace="0em" rspace="0em" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.cmml"><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.2" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.2" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.1" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.cmml"><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.2" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.2.cmml">v</mi><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.3" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.3.cmml">n</mi></msub></mrow><mo id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1a" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.4" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.3" lspace="0em" rspace="0.1389em" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.18.m4.1.1.1.1.3" xref="A1.SS2.7.p2.18.m4.1.1.1.1.3.cmml">p</mi></msub></mrow><mo id="A1.SS2.7.p2.18.m4.2.2.3" lspace="0.1389em" rspace="0.1389em" xref="A1.SS2.7.p2.18.m4.2.2.3.cmml">=</mo><msub id="A1.SS2.7.p2.18.m4.2.2.2" xref="A1.SS2.7.p2.18.m4.2.2.2.cmml"><mrow id="A1.SS2.7.p2.18.m4.2.2.2.1.1" xref="A1.SS2.7.p2.18.m4.2.2.2.1.2.cmml"><mo fence="true" id="A1.SS2.7.p2.18.m4.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A1.SS2.7.p2.18.m4.2.2.2.1.2.1.cmml">∥</mo><mrow id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.cmml"><mi id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.2" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.2.cmml">x</mi><mo id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1.cmml">−</mo><mrow id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.2" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.2.cmml">ε</mi><mo id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.1" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.3" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.3.cmml">v</mi></mrow><mo id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1a" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.4" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A1.SS2.7.p2.18.m4.2.2.2.1.1.3" lspace="0em" xref="A1.SS2.7.p2.18.m4.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A1.SS2.7.p2.18.m4.2.2.2.3" xref="A1.SS2.7.p2.18.m4.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.18.m4.2b"><apply id="A1.SS2.7.p2.18.m4.2.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2"><eq id="A1.SS2.7.p2.18.m4.2.2.3.cmml" xref="A1.SS2.7.p2.18.m4.2.2.3"></eq><apply id="A1.SS2.7.p2.18.m4.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1"><apply id="A1.SS2.7.p2.18.m4.1.1.1.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.18.m4.1.1.1.2.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2">subscript</csymbol><limit id="A1.SS2.7.p2.18.m4.1.1.1.2.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2.2"></limit><apply id="A1.SS2.7.p2.18.m4.1.1.1.2.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3"><ci id="A1.SS2.7.p2.18.m4.1.1.1.2.3.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.1">→</ci><ci id="A1.SS2.7.p2.18.m4.1.1.1.2.3.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.2">𝑛</ci><infinity id="A1.SS2.7.p2.18.m4.1.1.1.2.3.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.2.3.3"></infinity></apply></apply><apply id="A1.SS2.7.p2.18.m4.1.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.18.m4.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.18.m4.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS2.7.p2.18.m4.1.1.1.1.1.2.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1"><minus id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.1"></minus><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.2">𝑥</ci><apply id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3"><times id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.1"></times><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.2">𝜀</ci><apply id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.1.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.2.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.2">𝑣</ci><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.3.3.3">𝑛</ci></apply></apply><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.4.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.SS2.7.p2.18.m4.1.1.1.1.3.cmml" xref="A1.SS2.7.p2.18.m4.1.1.1.1.3">𝑝</ci></apply></apply><apply id="A1.SS2.7.p2.18.m4.2.2.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.18.m4.2.2.2.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2">subscript</csymbol><apply id="A1.SS2.7.p2.18.m4.2.2.2.1.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1"><csymbol cd="latexml" id="A1.SS2.7.p2.18.m4.2.2.2.1.2.1.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1"><minus id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.1"></minus><ci id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.2">𝑥</ci><apply id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3"><times id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.1.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.1"></times><ci id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.2.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.2">𝜀</ci><ci id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.3.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.3.3">𝑣</ci></apply><ci id="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.4.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.1.1.1.4">𝑧</ci></apply></apply><ci id="A1.SS2.7.p2.18.m4.2.2.2.3.cmml" xref="A1.SS2.7.p2.18.m4.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.18.m4.2c">\lim_{n\rightarrow\infty}\lVert x-\varepsilon v_{n}-z\rVert_{p}=\lVert x-% \varepsilon v-z\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.18.m4.2d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_x - italic_ε italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = ∥ italic_x - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, we conclude that <math alttext="X(z)=0" class="ltx_Math" display="inline" id="A1.SS2.7.p2.19.m5.1"><semantics id="A1.SS2.7.p2.19.m5.1a"><mrow id="A1.SS2.7.p2.19.m5.1.2" xref="A1.SS2.7.p2.19.m5.1.2.cmml"><mrow id="A1.SS2.7.p2.19.m5.1.2.2" xref="A1.SS2.7.p2.19.m5.1.2.2.cmml"><mi id="A1.SS2.7.p2.19.m5.1.2.2.2" xref="A1.SS2.7.p2.19.m5.1.2.2.2.cmml">X</mi><mo id="A1.SS2.7.p2.19.m5.1.2.2.1" xref="A1.SS2.7.p2.19.m5.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.7.p2.19.m5.1.2.2.3.2" xref="A1.SS2.7.p2.19.m5.1.2.2.cmml"><mo id="A1.SS2.7.p2.19.m5.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.7.p2.19.m5.1.2.2.cmml">(</mo><mi id="A1.SS2.7.p2.19.m5.1.1" xref="A1.SS2.7.p2.19.m5.1.1.cmml">z</mi><mo id="A1.SS2.7.p2.19.m5.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.7.p2.19.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.7.p2.19.m5.1.2.1" xref="A1.SS2.7.p2.19.m5.1.2.1.cmml">=</mo><mn id="A1.SS2.7.p2.19.m5.1.2.3" xref="A1.SS2.7.p2.19.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.19.m5.1b"><apply id="A1.SS2.7.p2.19.m5.1.2.cmml" xref="A1.SS2.7.p2.19.m5.1.2"><eq id="A1.SS2.7.p2.19.m5.1.2.1.cmml" xref="A1.SS2.7.p2.19.m5.1.2.1"></eq><apply id="A1.SS2.7.p2.19.m5.1.2.2.cmml" xref="A1.SS2.7.p2.19.m5.1.2.2"><times id="A1.SS2.7.p2.19.m5.1.2.2.1.cmml" xref="A1.SS2.7.p2.19.m5.1.2.2.1"></times><ci id="A1.SS2.7.p2.19.m5.1.2.2.2.cmml" xref="A1.SS2.7.p2.19.m5.1.2.2.2">𝑋</ci><ci id="A1.SS2.7.p2.19.m5.1.1.cmml" xref="A1.SS2.7.p2.19.m5.1.1">𝑧</ci></apply><cn id="A1.SS2.7.p2.19.m5.1.2.3.cmml" type="integer" xref="A1.SS2.7.p2.19.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.19.m5.1c">X(z)=0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.19.m5.1d">italic_X ( italic_z ) = 0</annotation></semantics></math>. In other words, we just proved that <math alttext="\mathbb{R}^{d}\setminus\mathcal{H}^{p}_{x,v}\subseteq\{z\in\mathbb{R}^{d}\mid X% (z)=0\}" class="ltx_Math" display="inline" id="A1.SS2.7.p2.20.m6.5"><semantics id="A1.SS2.7.p2.20.m6.5a"><mrow id="A1.SS2.7.p2.20.m6.5.5" xref="A1.SS2.7.p2.20.m6.5.5.cmml"><mrow id="A1.SS2.7.p2.20.m6.5.5.4" xref="A1.SS2.7.p2.20.m6.5.5.4.cmml"><msup id="A1.SS2.7.p2.20.m6.5.5.4.2" xref="A1.SS2.7.p2.20.m6.5.5.4.2.cmml"><mi id="A1.SS2.7.p2.20.m6.5.5.4.2.2" xref="A1.SS2.7.p2.20.m6.5.5.4.2.2.cmml">ℝ</mi><mi id="A1.SS2.7.p2.20.m6.5.5.4.2.3" xref="A1.SS2.7.p2.20.m6.5.5.4.2.3.cmml">d</mi></msup><mo id="A1.SS2.7.p2.20.m6.5.5.4.1" xref="A1.SS2.7.p2.20.m6.5.5.4.1.cmml">∖</mo><msubsup id="A1.SS2.7.p2.20.m6.5.5.4.3" xref="A1.SS2.7.p2.20.m6.5.5.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.7.p2.20.m6.5.5.4.3.2.2" xref="A1.SS2.7.p2.20.m6.5.5.4.3.2.2.cmml">ℋ</mi><mrow id="A1.SS2.7.p2.20.m6.2.2.2.4" xref="A1.SS2.7.p2.20.m6.2.2.2.3.cmml"><mi id="A1.SS2.7.p2.20.m6.1.1.1.1" xref="A1.SS2.7.p2.20.m6.1.1.1.1.cmml">x</mi><mo id="A1.SS2.7.p2.20.m6.2.2.2.4.1" xref="A1.SS2.7.p2.20.m6.2.2.2.3.cmml">,</mo><mi id="A1.SS2.7.p2.20.m6.2.2.2.2" xref="A1.SS2.7.p2.20.m6.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.7.p2.20.m6.5.5.4.3.2.3" xref="A1.SS2.7.p2.20.m6.5.5.4.3.2.3.cmml">p</mi></msubsup></mrow><mo id="A1.SS2.7.p2.20.m6.5.5.3" xref="A1.SS2.7.p2.20.m6.5.5.3.cmml">⊆</mo><mrow id="A1.SS2.7.p2.20.m6.5.5.2.2" xref="A1.SS2.7.p2.20.m6.5.5.2.3.cmml"><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.3" stretchy="false" xref="A1.SS2.7.p2.20.m6.5.5.2.3.1.cmml">{</mo><mrow id="A1.SS2.7.p2.20.m6.4.4.1.1.1" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.cmml"><mi id="A1.SS2.7.p2.20.m6.4.4.1.1.1.2" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.2.cmml">z</mi><mo id="A1.SS2.7.p2.20.m6.4.4.1.1.1.1" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.1.cmml">∈</mo><msup id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.cmml"><mi id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.2" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.3" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.3.cmml">d</mi></msup></mrow><mo fence="true" id="A1.SS2.7.p2.20.m6.5.5.2.2.4" lspace="0em" rspace="0em" xref="A1.SS2.7.p2.20.m6.5.5.2.3.1.cmml">∣</mo><mrow id="A1.SS2.7.p2.20.m6.5.5.2.2.2" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.cmml"><mrow id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.cmml"><mi id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.2" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.2.cmml">X</mi><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.1" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.3.2" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.cmml"><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.3.2.1" stretchy="false" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.cmml">(</mo><mi id="A1.SS2.7.p2.20.m6.3.3" xref="A1.SS2.7.p2.20.m6.3.3.cmml">z</mi><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.3.2.2" stretchy="false" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.2.1" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.1.cmml">=</mo><mn id="A1.SS2.7.p2.20.m6.5.5.2.2.2.3" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.3.cmml">0</mn></mrow><mo id="A1.SS2.7.p2.20.m6.5.5.2.2.5" stretchy="false" xref="A1.SS2.7.p2.20.m6.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.20.m6.5b"><apply id="A1.SS2.7.p2.20.m6.5.5.cmml" xref="A1.SS2.7.p2.20.m6.5.5"><subset id="A1.SS2.7.p2.20.m6.5.5.3.cmml" xref="A1.SS2.7.p2.20.m6.5.5.3"></subset><apply id="A1.SS2.7.p2.20.m6.5.5.4.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4"><setdiff id="A1.SS2.7.p2.20.m6.5.5.4.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.1"></setdiff><apply id="A1.SS2.7.p2.20.m6.5.5.4.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.20.m6.5.5.4.2.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.2">superscript</csymbol><ci id="A1.SS2.7.p2.20.m6.5.5.4.2.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.2.2">ℝ</ci><ci id="A1.SS2.7.p2.20.m6.5.5.4.2.3.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.2.3">𝑑</ci></apply><apply id="A1.SS2.7.p2.20.m6.5.5.4.3.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.20.m6.5.5.4.3.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3">subscript</csymbol><apply id="A1.SS2.7.p2.20.m6.5.5.4.3.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.20.m6.5.5.4.3.2.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3">superscript</csymbol><ci id="A1.SS2.7.p2.20.m6.5.5.4.3.2.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3.2.2">ℋ</ci><ci id="A1.SS2.7.p2.20.m6.5.5.4.3.2.3.cmml" xref="A1.SS2.7.p2.20.m6.5.5.4.3.2.3">𝑝</ci></apply><list id="A1.SS2.7.p2.20.m6.2.2.2.3.cmml" xref="A1.SS2.7.p2.20.m6.2.2.2.4"><ci id="A1.SS2.7.p2.20.m6.1.1.1.1.cmml" xref="A1.SS2.7.p2.20.m6.1.1.1.1">𝑥</ci><ci id="A1.SS2.7.p2.20.m6.2.2.2.2.cmml" xref="A1.SS2.7.p2.20.m6.2.2.2.2">𝑣</ci></list></apply></apply><apply id="A1.SS2.7.p2.20.m6.5.5.2.3.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2"><csymbol cd="latexml" id="A1.SS2.7.p2.20.m6.5.5.2.3.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.3">conditional-set</csymbol><apply id="A1.SS2.7.p2.20.m6.4.4.1.1.1.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1"><in id="A1.SS2.7.p2.20.m6.4.4.1.1.1.1.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.1"></in><ci id="A1.SS2.7.p2.20.m6.4.4.1.1.1.2.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.2">𝑧</ci><apply id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.1.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3">superscript</csymbol><ci id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.2.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.2">ℝ</ci><ci id="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.3.cmml" xref="A1.SS2.7.p2.20.m6.4.4.1.1.1.3.3">𝑑</ci></apply></apply><apply id="A1.SS2.7.p2.20.m6.5.5.2.2.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2"><eq id="A1.SS2.7.p2.20.m6.5.5.2.2.2.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.1"></eq><apply id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2"><times id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.1.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.1"></times><ci id="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.2.cmml" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.2.2">𝑋</ci><ci id="A1.SS2.7.p2.20.m6.3.3.cmml" xref="A1.SS2.7.p2.20.m6.3.3">𝑧</ci></apply><cn id="A1.SS2.7.p2.20.m6.5.5.2.2.2.3.cmml" type="integer" xref="A1.SS2.7.p2.20.m6.5.5.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.20.m6.5c">\mathbb{R}^{d}\setminus\mathcal{H}^{p}_{x,v}\subseteq\{z\in\mathbb{R}^{d}\mid X% (z)=0\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.20.m6.5d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ⊆ { italic_z ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∣ italic_X ( italic_z ) = 0 }</annotation></semantics></math> and thus <math alttext="\mu(\mathcal{H}^{p}_{x,v})\geq\mathbb{E}X" class="ltx_Math" display="inline" id="A1.SS2.7.p2.21.m7.3"><semantics id="A1.SS2.7.p2.21.m7.3a"><mrow id="A1.SS2.7.p2.21.m7.3.3" xref="A1.SS2.7.p2.21.m7.3.3.cmml"><mrow id="A1.SS2.7.p2.21.m7.3.3.1" xref="A1.SS2.7.p2.21.m7.3.3.1.cmml"><mi id="A1.SS2.7.p2.21.m7.3.3.1.3" xref="A1.SS2.7.p2.21.m7.3.3.1.3.cmml">μ</mi><mo id="A1.SS2.7.p2.21.m7.3.3.1.2" xref="A1.SS2.7.p2.21.m7.3.3.1.2.cmml">⁢</mo><mrow id="A1.SS2.7.p2.21.m7.3.3.1.1.1" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.cmml"><mo id="A1.SS2.7.p2.21.m7.3.3.1.1.1.2" stretchy="false" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.cmml">(</mo><msubsup id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.2" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.SS2.7.p2.21.m7.2.2.2.4" xref="A1.SS2.7.p2.21.m7.2.2.2.3.cmml"><mi id="A1.SS2.7.p2.21.m7.1.1.1.1" xref="A1.SS2.7.p2.21.m7.1.1.1.1.cmml">x</mi><mo id="A1.SS2.7.p2.21.m7.2.2.2.4.1" xref="A1.SS2.7.p2.21.m7.2.2.2.3.cmml">,</mo><mi id="A1.SS2.7.p2.21.m7.2.2.2.2" xref="A1.SS2.7.p2.21.m7.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.3" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A1.SS2.7.p2.21.m7.3.3.1.1.1.3" stretchy="false" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.7.p2.21.m7.3.3.2" xref="A1.SS2.7.p2.21.m7.3.3.2.cmml">≥</mo><mrow id="A1.SS2.7.p2.21.m7.3.3.3" xref="A1.SS2.7.p2.21.m7.3.3.3.cmml"><mi id="A1.SS2.7.p2.21.m7.3.3.3.2" xref="A1.SS2.7.p2.21.m7.3.3.3.2.cmml">𝔼</mi><mo id="A1.SS2.7.p2.21.m7.3.3.3.1" xref="A1.SS2.7.p2.21.m7.3.3.3.1.cmml">⁢</mo><mi id="A1.SS2.7.p2.21.m7.3.3.3.3" xref="A1.SS2.7.p2.21.m7.3.3.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.21.m7.3b"><apply id="A1.SS2.7.p2.21.m7.3.3.cmml" xref="A1.SS2.7.p2.21.m7.3.3"><geq id="A1.SS2.7.p2.21.m7.3.3.2.cmml" xref="A1.SS2.7.p2.21.m7.3.3.2"></geq><apply id="A1.SS2.7.p2.21.m7.3.3.1.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1"><times id="A1.SS2.7.p2.21.m7.3.3.1.2.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.2"></times><ci id="A1.SS2.7.p2.21.m7.3.3.1.3.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.3">𝜇</ci><apply id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.1.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1">superscript</csymbol><ci id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.2.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.2">ℋ</ci><ci id="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.3.cmml" xref="A1.SS2.7.p2.21.m7.3.3.1.1.1.1.2.3">𝑝</ci></apply><list id="A1.SS2.7.p2.21.m7.2.2.2.3.cmml" xref="A1.SS2.7.p2.21.m7.2.2.2.4"><ci id="A1.SS2.7.p2.21.m7.1.1.1.1.cmml" xref="A1.SS2.7.p2.21.m7.1.1.1.1">𝑥</ci><ci id="A1.SS2.7.p2.21.m7.2.2.2.2.cmml" xref="A1.SS2.7.p2.21.m7.2.2.2.2">𝑣</ci></list></apply></apply><apply id="A1.SS2.7.p2.21.m7.3.3.3.cmml" xref="A1.SS2.7.p2.21.m7.3.3.3"><times id="A1.SS2.7.p2.21.m7.3.3.3.1.cmml" xref="A1.SS2.7.p2.21.m7.3.3.3.1"></times><ci id="A1.SS2.7.p2.21.m7.3.3.3.2.cmml" xref="A1.SS2.7.p2.21.m7.3.3.3.2">𝔼</ci><ci id="A1.SS2.7.p2.21.m7.3.3.3.3.cmml" xref="A1.SS2.7.p2.21.m7.3.3.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.21.m7.3c">\mu(\mathcal{H}^{p}_{x,v})\geq\mathbb{E}X</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.21.m7.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) ≥ blackboard_E italic_X</annotation></semantics></math>. Plugging everything together, we obtain <math alttext="\mu(\mathcal{H}^{p}_{x,v})\geq t" class="ltx_Math" display="inline" id="A1.SS2.7.p2.22.m8.3"><semantics id="A1.SS2.7.p2.22.m8.3a"><mrow id="A1.SS2.7.p2.22.m8.3.3" xref="A1.SS2.7.p2.22.m8.3.3.cmml"><mrow id="A1.SS2.7.p2.22.m8.3.3.1" xref="A1.SS2.7.p2.22.m8.3.3.1.cmml"><mi id="A1.SS2.7.p2.22.m8.3.3.1.3" xref="A1.SS2.7.p2.22.m8.3.3.1.3.cmml">μ</mi><mo id="A1.SS2.7.p2.22.m8.3.3.1.2" xref="A1.SS2.7.p2.22.m8.3.3.1.2.cmml">⁢</mo><mrow id="A1.SS2.7.p2.22.m8.3.3.1.1.1" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.cmml"><mo id="A1.SS2.7.p2.22.m8.3.3.1.1.1.2" stretchy="false" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.cmml">(</mo><msubsup id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.2" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="A1.SS2.7.p2.22.m8.2.2.2.4" xref="A1.SS2.7.p2.22.m8.2.2.2.3.cmml"><mi id="A1.SS2.7.p2.22.m8.1.1.1.1" xref="A1.SS2.7.p2.22.m8.1.1.1.1.cmml">x</mi><mo id="A1.SS2.7.p2.22.m8.2.2.2.4.1" xref="A1.SS2.7.p2.22.m8.2.2.2.3.cmml">,</mo><mi id="A1.SS2.7.p2.22.m8.2.2.2.2" xref="A1.SS2.7.p2.22.m8.2.2.2.2.cmml">v</mi></mrow><mi id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.3" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A1.SS2.7.p2.22.m8.3.3.1.1.1.3" stretchy="false" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.7.p2.22.m8.3.3.2" xref="A1.SS2.7.p2.22.m8.3.3.2.cmml">≥</mo><mi id="A1.SS2.7.p2.22.m8.3.3.3" xref="A1.SS2.7.p2.22.m8.3.3.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.22.m8.3b"><apply id="A1.SS2.7.p2.22.m8.3.3.cmml" xref="A1.SS2.7.p2.22.m8.3.3"><geq id="A1.SS2.7.p2.22.m8.3.3.2.cmml" xref="A1.SS2.7.p2.22.m8.3.3.2"></geq><apply id="A1.SS2.7.p2.22.m8.3.3.1.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1"><times id="A1.SS2.7.p2.22.m8.3.3.1.2.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.2"></times><ci id="A1.SS2.7.p2.22.m8.3.3.1.3.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.3">𝜇</ci><apply id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.1.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1">subscript</csymbol><apply id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.1.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1">superscript</csymbol><ci id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.2.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.2">ℋ</ci><ci id="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.3.cmml" xref="A1.SS2.7.p2.22.m8.3.3.1.1.1.1.2.3">𝑝</ci></apply><list id="A1.SS2.7.p2.22.m8.2.2.2.3.cmml" xref="A1.SS2.7.p2.22.m8.2.2.2.4"><ci id="A1.SS2.7.p2.22.m8.1.1.1.1.cmml" xref="A1.SS2.7.p2.22.m8.1.1.1.1">𝑥</ci><ci id="A1.SS2.7.p2.22.m8.2.2.2.2.cmml" xref="A1.SS2.7.p2.22.m8.2.2.2.2">𝑣</ci></list></apply></apply><ci id="A1.SS2.7.p2.22.m8.3.3.3.cmml" xref="A1.SS2.7.p2.22.m8.3.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.22.m8.3c">\mu(\mathcal{H}^{p}_{x,v})\geq t</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.22.m8.3d">italic_μ ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x , italic_v end_POSTSUBSCRIPT ) ≥ italic_t</annotation></semantics></math> and hence <math alttext="v\in S^{d-1}\setminus V" class="ltx_Math" display="inline" id="A1.SS2.7.p2.23.m9.1"><semantics id="A1.SS2.7.p2.23.m9.1a"><mrow id="A1.SS2.7.p2.23.m9.1.1" xref="A1.SS2.7.p2.23.m9.1.1.cmml"><mi id="A1.SS2.7.p2.23.m9.1.1.2" xref="A1.SS2.7.p2.23.m9.1.1.2.cmml">v</mi><mo id="A1.SS2.7.p2.23.m9.1.1.1" xref="A1.SS2.7.p2.23.m9.1.1.1.cmml">∈</mo><mrow id="A1.SS2.7.p2.23.m9.1.1.3" xref="A1.SS2.7.p2.23.m9.1.1.3.cmml"><msup id="A1.SS2.7.p2.23.m9.1.1.3.2" xref="A1.SS2.7.p2.23.m9.1.1.3.2.cmml"><mi id="A1.SS2.7.p2.23.m9.1.1.3.2.2" xref="A1.SS2.7.p2.23.m9.1.1.3.2.2.cmml">S</mi><mrow id="A1.SS2.7.p2.23.m9.1.1.3.2.3" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.cmml"><mi id="A1.SS2.7.p2.23.m9.1.1.3.2.3.2" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.2.cmml">d</mi><mo id="A1.SS2.7.p2.23.m9.1.1.3.2.3.1" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.1.cmml">−</mo><mn id="A1.SS2.7.p2.23.m9.1.1.3.2.3.3" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.3.cmml">1</mn></mrow></msup><mo id="A1.SS2.7.p2.23.m9.1.1.3.1" xref="A1.SS2.7.p2.23.m9.1.1.3.1.cmml">∖</mo><mi id="A1.SS2.7.p2.23.m9.1.1.3.3" xref="A1.SS2.7.p2.23.m9.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.7.p2.23.m9.1b"><apply id="A1.SS2.7.p2.23.m9.1.1.cmml" xref="A1.SS2.7.p2.23.m9.1.1"><in id="A1.SS2.7.p2.23.m9.1.1.1.cmml" xref="A1.SS2.7.p2.23.m9.1.1.1"></in><ci id="A1.SS2.7.p2.23.m9.1.1.2.cmml" xref="A1.SS2.7.p2.23.m9.1.1.2">𝑣</ci><apply id="A1.SS2.7.p2.23.m9.1.1.3.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3"><setdiff id="A1.SS2.7.p2.23.m9.1.1.3.1.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.1"></setdiff><apply id="A1.SS2.7.p2.23.m9.1.1.3.2.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.7.p2.23.m9.1.1.3.2.1.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2">superscript</csymbol><ci id="A1.SS2.7.p2.23.m9.1.1.3.2.2.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2.2">𝑆</ci><apply id="A1.SS2.7.p2.23.m9.1.1.3.2.3.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3"><minus id="A1.SS2.7.p2.23.m9.1.1.3.2.3.1.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.1"></minus><ci id="A1.SS2.7.p2.23.m9.1.1.3.2.3.2.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.2">𝑑</ci><cn id="A1.SS2.7.p2.23.m9.1.1.3.2.3.3.cmml" type="integer" xref="A1.SS2.7.p2.23.m9.1.1.3.2.3.3">1</cn></apply></apply><ci id="A1.SS2.7.p2.23.m9.1.1.3.3.cmml" xref="A1.SS2.7.p2.23.m9.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.7.p2.23.m9.1c">v\in S^{d-1}\setminus V</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.7.p2.23.m9.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∖ italic_V</annotation></semantics></math>, as desired. ∎</p> </div> </div> </section> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.18</span></a> </h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1">See <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem18" title="Theorem 3.18 (ℓ_𝑝-Centerpoint Theorem for Finite Point Sets). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">3.18</span></a></p> </div> <div class="ltx_proof" id="A2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A2.1.p1"> <p class="ltx_p" id="A2.1.p1.21">Without loss of generality, assume that <math alttext="P\subseteq(0,1)^{d}" class="ltx_Math" display="inline" id="A2.1.p1.1.m1.2"><semantics id="A2.1.p1.1.m1.2a"><mrow id="A2.1.p1.1.m1.2.3" xref="A2.1.p1.1.m1.2.3.cmml"><mi id="A2.1.p1.1.m1.2.3.2" xref="A2.1.p1.1.m1.2.3.2.cmml">P</mi><mo id="A2.1.p1.1.m1.2.3.1" xref="A2.1.p1.1.m1.2.3.1.cmml">⊆</mo><msup id="A2.1.p1.1.m1.2.3.3" xref="A2.1.p1.1.m1.2.3.3.cmml"><mrow id="A2.1.p1.1.m1.2.3.3.2.2" xref="A2.1.p1.1.m1.2.3.3.2.1.cmml"><mo id="A2.1.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="A2.1.p1.1.m1.2.3.3.2.1.cmml">(</mo><mn id="A2.1.p1.1.m1.1.1" xref="A2.1.p1.1.m1.1.1.cmml">0</mn><mo id="A2.1.p1.1.m1.2.3.3.2.2.2" xref="A2.1.p1.1.m1.2.3.3.2.1.cmml">,</mo><mn id="A2.1.p1.1.m1.2.2" xref="A2.1.p1.1.m1.2.2.cmml">1</mn><mo id="A2.1.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="A2.1.p1.1.m1.2.3.3.2.1.cmml">)</mo></mrow><mi id="A2.1.p1.1.m1.2.3.3.3" xref="A2.1.p1.1.m1.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.1.m1.2b"><apply id="A2.1.p1.1.m1.2.3.cmml" xref="A2.1.p1.1.m1.2.3"><subset id="A2.1.p1.1.m1.2.3.1.cmml" xref="A2.1.p1.1.m1.2.3.1"></subset><ci id="A2.1.p1.1.m1.2.3.2.cmml" xref="A2.1.p1.1.m1.2.3.2">𝑃</ci><apply id="A2.1.p1.1.m1.2.3.3.cmml" xref="A2.1.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="A2.1.p1.1.m1.2.3.3.1.cmml" xref="A2.1.p1.1.m1.2.3.3">superscript</csymbol><interval closure="open" id="A2.1.p1.1.m1.2.3.3.2.1.cmml" xref="A2.1.p1.1.m1.2.3.3.2.2"><cn id="A2.1.p1.1.m1.1.1.cmml" type="integer" xref="A2.1.p1.1.m1.1.1">0</cn><cn id="A2.1.p1.1.m1.2.2.cmml" type="integer" xref="A2.1.p1.1.m1.2.2">1</cn></interval><ci id="A2.1.p1.1.m1.2.3.3.3.cmml" xref="A2.1.p1.1.m1.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.1.m1.2c">P\subseteq(0,1)^{d}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.1.m1.2d">italic_P ⊆ ( 0 , 1 ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="r>0" class="ltx_Math" display="inline" id="A2.1.p1.2.m2.1"><semantics id="A2.1.p1.2.m2.1a"><mrow id="A2.1.p1.2.m2.1.1" xref="A2.1.p1.2.m2.1.1.cmml"><mi id="A2.1.p1.2.m2.1.1.2" xref="A2.1.p1.2.m2.1.1.2.cmml">r</mi><mo id="A2.1.p1.2.m2.1.1.1" xref="A2.1.p1.2.m2.1.1.1.cmml">></mo><mn id="A2.1.p1.2.m2.1.1.3" xref="A2.1.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.2.m2.1b"><apply id="A2.1.p1.2.m2.1.1.cmml" xref="A2.1.p1.2.m2.1.1"><gt id="A2.1.p1.2.m2.1.1.1.cmml" xref="A2.1.p1.2.m2.1.1.1"></gt><ci id="A2.1.p1.2.m2.1.1.2.cmml" xref="A2.1.p1.2.m2.1.1.2">𝑟</ci><cn id="A2.1.p1.2.m2.1.1.3.cmml" type="integer" xref="A2.1.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.2.m2.1c">r>0</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.2.m2.1d">italic_r > 0</annotation></semantics></math> be arbitrary and consider the mass distribution <math alttext="\mu_{r}" class="ltx_Math" display="inline" id="A2.1.p1.3.m3.1"><semantics id="A2.1.p1.3.m3.1a"><msub id="A2.1.p1.3.m3.1.1" xref="A2.1.p1.3.m3.1.1.cmml"><mi id="A2.1.p1.3.m3.1.1.2" xref="A2.1.p1.3.m3.1.1.2.cmml">μ</mi><mi id="A2.1.p1.3.m3.1.1.3" xref="A2.1.p1.3.m3.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.p1.3.m3.1b"><apply id="A2.1.p1.3.m3.1.1.cmml" xref="A2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A2.1.p1.3.m3.1.1.1.cmml" xref="A2.1.p1.3.m3.1.1">subscript</csymbol><ci id="A2.1.p1.3.m3.1.1.2.cmml" xref="A2.1.p1.3.m3.1.1.2">𝜇</ci><ci id="A2.1.p1.3.m3.1.1.3.cmml" xref="A2.1.p1.3.m3.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.3.m3.1c">\mu_{r}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.3.m3.1d">italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="A2.1.p1.4.m4.1"><semantics id="A2.1.p1.4.m4.1a"><msup id="A2.1.p1.4.m4.1.1" xref="A2.1.p1.4.m4.1.1.cmml"><mi id="A2.1.p1.4.m4.1.1.2" xref="A2.1.p1.4.m4.1.1.2.cmml">ℝ</mi><mi id="A2.1.p1.4.m4.1.1.3" xref="A2.1.p1.4.m4.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.4.m4.1b"><apply id="A2.1.p1.4.m4.1.1.cmml" xref="A2.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A2.1.p1.4.m4.1.1.1.cmml" xref="A2.1.p1.4.m4.1.1">superscript</csymbol><ci id="A2.1.p1.4.m4.1.1.2.cmml" xref="A2.1.p1.4.m4.1.1.2">ℝ</ci><ci id="A2.1.p1.4.m4.1.1.3.cmml" xref="A2.1.p1.4.m4.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.4.m4.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> obtained by replacing each point <math alttext="z" class="ltx_Math" display="inline" id="A2.1.p1.5.m5.1"><semantics id="A2.1.p1.5.m5.1a"><mi id="A2.1.p1.5.m5.1.1" xref="A2.1.p1.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.5.m5.1b"><ci id="A2.1.p1.5.m5.1.1.cmml" xref="A2.1.p1.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.5.m5.1d">italic_z</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="A2.1.p1.6.m6.1"><semantics id="A2.1.p1.6.m6.1a"><mi id="A2.1.p1.6.m6.1.1" xref="A2.1.p1.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.6.m6.1b"><ci id="A2.1.p1.6.m6.1.1.cmml" xref="A2.1.p1.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.6.m6.1d">italic_P</annotation></semantics></math> with an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A2.1.p1.7.m7.1"><semantics id="A2.1.p1.7.m7.1a"><msub id="A2.1.p1.7.m7.1.1" xref="A2.1.p1.7.m7.1.1.cmml"><mi id="A2.1.p1.7.m7.1.1.2" mathvariant="normal" xref="A2.1.p1.7.m7.1.1.2.cmml">ℓ</mi><mi id="A2.1.p1.7.m7.1.1.3" xref="A2.1.p1.7.m7.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.p1.7.m7.1b"><apply id="A2.1.p1.7.m7.1.1.cmml" xref="A2.1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A2.1.p1.7.m7.1.1.1.cmml" xref="A2.1.p1.7.m7.1.1">subscript</csymbol><ci id="A2.1.p1.7.m7.1.1.2.cmml" xref="A2.1.p1.7.m7.1.1.2">ℓ</ci><ci id="A2.1.p1.7.m7.1.1.3.cmml" xref="A2.1.p1.7.m7.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.7.m7.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.7.m7.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-norm ball <math alttext="B^{p}(z,r)" class="ltx_Math" display="inline" id="A2.1.p1.8.m8.2"><semantics id="A2.1.p1.8.m8.2a"><mrow id="A2.1.p1.8.m8.2.3" xref="A2.1.p1.8.m8.2.3.cmml"><msup id="A2.1.p1.8.m8.2.3.2" xref="A2.1.p1.8.m8.2.3.2.cmml"><mi id="A2.1.p1.8.m8.2.3.2.2" xref="A2.1.p1.8.m8.2.3.2.2.cmml">B</mi><mi id="A2.1.p1.8.m8.2.3.2.3" xref="A2.1.p1.8.m8.2.3.2.3.cmml">p</mi></msup><mo id="A2.1.p1.8.m8.2.3.1" xref="A2.1.p1.8.m8.2.3.1.cmml">⁢</mo><mrow id="A2.1.p1.8.m8.2.3.3.2" xref="A2.1.p1.8.m8.2.3.3.1.cmml"><mo id="A2.1.p1.8.m8.2.3.3.2.1" stretchy="false" xref="A2.1.p1.8.m8.2.3.3.1.cmml">(</mo><mi id="A2.1.p1.8.m8.1.1" xref="A2.1.p1.8.m8.1.1.cmml">z</mi><mo id="A2.1.p1.8.m8.2.3.3.2.2" xref="A2.1.p1.8.m8.2.3.3.1.cmml">,</mo><mi id="A2.1.p1.8.m8.2.2" xref="A2.1.p1.8.m8.2.2.cmml">r</mi><mo id="A2.1.p1.8.m8.2.3.3.2.3" stretchy="false" xref="A2.1.p1.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.8.m8.2b"><apply id="A2.1.p1.8.m8.2.3.cmml" xref="A2.1.p1.8.m8.2.3"><times id="A2.1.p1.8.m8.2.3.1.cmml" xref="A2.1.p1.8.m8.2.3.1"></times><apply id="A2.1.p1.8.m8.2.3.2.cmml" xref="A2.1.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="A2.1.p1.8.m8.2.3.2.1.cmml" xref="A2.1.p1.8.m8.2.3.2">superscript</csymbol><ci id="A2.1.p1.8.m8.2.3.2.2.cmml" xref="A2.1.p1.8.m8.2.3.2.2">𝐵</ci><ci id="A2.1.p1.8.m8.2.3.2.3.cmml" xref="A2.1.p1.8.m8.2.3.2.3">𝑝</ci></apply><interval closure="open" id="A2.1.p1.8.m8.2.3.3.1.cmml" xref="A2.1.p1.8.m8.2.3.3.2"><ci id="A2.1.p1.8.m8.1.1.cmml" xref="A2.1.p1.8.m8.1.1">𝑧</ci><ci id="A2.1.p1.8.m8.2.2.cmml" xref="A2.1.p1.8.m8.2.2">𝑟</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.8.m8.2c">B^{p}(z,r)</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.8.m8.2d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_z , italic_r )</annotation></semantics></math> of radius <math alttext="r" class="ltx_Math" display="inline" id="A2.1.p1.9.m9.1"><semantics id="A2.1.p1.9.m9.1a"><mi id="A2.1.p1.9.m9.1.1" xref="A2.1.p1.9.m9.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.9.m9.1b"><ci id="A2.1.p1.9.m9.1.1.cmml" xref="A2.1.p1.9.m9.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.9.m9.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.9.m9.1d">italic_r</annotation></semantics></math> around <math alttext="z" class="ltx_Math" display="inline" id="A2.1.p1.10.m10.1"><semantics id="A2.1.p1.10.m10.1a"><mi id="A2.1.p1.10.m10.1.1" xref="A2.1.p1.10.m10.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.10.m10.1b"><ci id="A2.1.p1.10.m10.1.1.cmml" xref="A2.1.p1.10.m10.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.10.m10.1c">z</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.10.m10.1d">italic_z</annotation></semantics></math> (without loss of generality, assume <math alttext="r" class="ltx_Math" display="inline" id="A2.1.p1.11.m11.1"><semantics id="A2.1.p1.11.m11.1a"><mi id="A2.1.p1.11.m11.1.1" xref="A2.1.p1.11.m11.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.11.m11.1b"><ci id="A2.1.p1.11.m11.1.1.cmml" xref="A2.1.p1.11.m11.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.11.m11.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.11.m11.1d">italic_r</annotation></semantics></math> is small enough such that the balls do not intersect and such that the support of <math alttext="\mu_{r}" class="ltx_Math" display="inline" id="A2.1.p1.12.m12.1"><semantics id="A2.1.p1.12.m12.1a"><msub id="A2.1.p1.12.m12.1.1" xref="A2.1.p1.12.m12.1.1.cmml"><mi id="A2.1.p1.12.m12.1.1.2" xref="A2.1.p1.12.m12.1.1.2.cmml">μ</mi><mi id="A2.1.p1.12.m12.1.1.3" xref="A2.1.p1.12.m12.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.p1.12.m12.1b"><apply id="A2.1.p1.12.m12.1.1.cmml" xref="A2.1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="A2.1.p1.12.m12.1.1.1.cmml" xref="A2.1.p1.12.m12.1.1">subscript</csymbol><ci id="A2.1.p1.12.m12.1.1.2.cmml" xref="A2.1.p1.12.m12.1.1.2">𝜇</ci><ci id="A2.1.p1.12.m12.1.1.3.cmml" xref="A2.1.p1.12.m12.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.12.m12.1c">\mu_{r}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.12.m12.1d">italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> is contained in <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="A2.1.p1.13.m13.2"><semantics id="A2.1.p1.13.m13.2a"><msup id="A2.1.p1.13.m13.2.3" xref="A2.1.p1.13.m13.2.3.cmml"><mrow id="A2.1.p1.13.m13.2.3.2.2" xref="A2.1.p1.13.m13.2.3.2.1.cmml"><mo id="A2.1.p1.13.m13.2.3.2.2.1" stretchy="false" xref="A2.1.p1.13.m13.2.3.2.1.cmml">[</mo><mn id="A2.1.p1.13.m13.1.1" xref="A2.1.p1.13.m13.1.1.cmml">0</mn><mo id="A2.1.p1.13.m13.2.3.2.2.2" xref="A2.1.p1.13.m13.2.3.2.1.cmml">,</mo><mn id="A2.1.p1.13.m13.2.2" xref="A2.1.p1.13.m13.2.2.cmml">1</mn><mo id="A2.1.p1.13.m13.2.3.2.2.3" stretchy="false" xref="A2.1.p1.13.m13.2.3.2.1.cmml">]</mo></mrow><mi id="A2.1.p1.13.m13.2.3.3" xref="A2.1.p1.13.m13.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.13.m13.2b"><apply id="A2.1.p1.13.m13.2.3.cmml" xref="A2.1.p1.13.m13.2.3"><csymbol cd="ambiguous" id="A2.1.p1.13.m13.2.3.1.cmml" xref="A2.1.p1.13.m13.2.3">superscript</csymbol><interval closure="closed" id="A2.1.p1.13.m13.2.3.2.1.cmml" xref="A2.1.p1.13.m13.2.3.2.2"><cn id="A2.1.p1.13.m13.1.1.cmml" type="integer" xref="A2.1.p1.13.m13.1.1">0</cn><cn id="A2.1.p1.13.m13.2.2.cmml" type="integer" xref="A2.1.p1.13.m13.2.2">1</cn></interval><ci id="A2.1.p1.13.m13.2.3.3.cmml" xref="A2.1.p1.13.m13.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.13.m13.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.13.m13.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>). <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem13" title="Theorem 3.13 (ℓ_𝑝-Centerpoint Theorem for Mass Distributions). ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3.13</span></a> yields a centerpoint <math alttext="c^{(r)}" class="ltx_Math" display="inline" id="A2.1.p1.14.m14.1"><semantics id="A2.1.p1.14.m14.1a"><msup id="A2.1.p1.14.m14.1.2" xref="A2.1.p1.14.m14.1.2.cmml"><mi id="A2.1.p1.14.m14.1.2.2" xref="A2.1.p1.14.m14.1.2.2.cmml">c</mi><mrow id="A2.1.p1.14.m14.1.1.1.3" xref="A2.1.p1.14.m14.1.2.cmml"><mo id="A2.1.p1.14.m14.1.1.1.3.1" stretchy="false" xref="A2.1.p1.14.m14.1.2.cmml">(</mo><mi id="A2.1.p1.14.m14.1.1.1.1" xref="A2.1.p1.14.m14.1.1.1.1.cmml">r</mi><mo id="A2.1.p1.14.m14.1.1.1.3.2" stretchy="false" xref="A2.1.p1.14.m14.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.14.m14.1b"><apply id="A2.1.p1.14.m14.1.2.cmml" xref="A2.1.p1.14.m14.1.2"><csymbol cd="ambiguous" id="A2.1.p1.14.m14.1.2.1.cmml" xref="A2.1.p1.14.m14.1.2">superscript</csymbol><ci id="A2.1.p1.14.m14.1.2.2.cmml" xref="A2.1.p1.14.m14.1.2.2">𝑐</ci><ci id="A2.1.p1.14.m14.1.1.1.1.cmml" xref="A2.1.p1.14.m14.1.1.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.14.m14.1c">c^{(r)}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.14.m14.1d">italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="\mu_{r}" class="ltx_Math" display="inline" id="A2.1.p1.15.m15.1"><semantics id="A2.1.p1.15.m15.1a"><msub id="A2.1.p1.15.m15.1.1" xref="A2.1.p1.15.m15.1.1.cmml"><mi id="A2.1.p1.15.m15.1.1.2" xref="A2.1.p1.15.m15.1.1.2.cmml">μ</mi><mi id="A2.1.p1.15.m15.1.1.3" xref="A2.1.p1.15.m15.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="A2.1.p1.15.m15.1b"><apply id="A2.1.p1.15.m15.1.1.cmml" xref="A2.1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="A2.1.p1.15.m15.1.1.1.cmml" xref="A2.1.p1.15.m15.1.1">subscript</csymbol><ci id="A2.1.p1.15.m15.1.1.2.cmml" xref="A2.1.p1.15.m15.1.1.2">𝜇</ci><ci id="A2.1.p1.15.m15.1.1.3.cmml" xref="A2.1.p1.15.m15.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.15.m15.1c">\mu_{r}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.15.m15.1d">italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, by <a class="ltx_ref" href="https://arxiv.org/html/2503.16089v1#S3.Thmtheorem19" title="Lemma 3.19. ‣ 3.3 ℓ_𝑝-Centerpoints of Mass Distributions ‣ 3 ℓ_𝑝-Halfspaces and ℓ_𝑝-Centerpoints ‣ Query-Efficient Fixpoints of ℓ_𝑝-Contractions"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.19</span></a>, these centerpoints are contained in <math alttext="[0,1]^{d}" class="ltx_Math" display="inline" id="A2.1.p1.16.m16.2"><semantics id="A2.1.p1.16.m16.2a"><msup id="A2.1.p1.16.m16.2.3" xref="A2.1.p1.16.m16.2.3.cmml"><mrow id="A2.1.p1.16.m16.2.3.2.2" xref="A2.1.p1.16.m16.2.3.2.1.cmml"><mo id="A2.1.p1.16.m16.2.3.2.2.1" stretchy="false" xref="A2.1.p1.16.m16.2.3.2.1.cmml">[</mo><mn id="A2.1.p1.16.m16.1.1" xref="A2.1.p1.16.m16.1.1.cmml">0</mn><mo id="A2.1.p1.16.m16.2.3.2.2.2" xref="A2.1.p1.16.m16.2.3.2.1.cmml">,</mo><mn id="A2.1.p1.16.m16.2.2" xref="A2.1.p1.16.m16.2.2.cmml">1</mn><mo id="A2.1.p1.16.m16.2.3.2.2.3" stretchy="false" xref="A2.1.p1.16.m16.2.3.2.1.cmml">]</mo></mrow><mi id="A2.1.p1.16.m16.2.3.3" xref="A2.1.p1.16.m16.2.3.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.16.m16.2b"><apply id="A2.1.p1.16.m16.2.3.cmml" xref="A2.1.p1.16.m16.2.3"><csymbol cd="ambiguous" id="A2.1.p1.16.m16.2.3.1.cmml" xref="A2.1.p1.16.m16.2.3">superscript</csymbol><interval closure="closed" id="A2.1.p1.16.m16.2.3.2.1.cmml" xref="A2.1.p1.16.m16.2.3.2.2"><cn id="A2.1.p1.16.m16.1.1.cmml" type="integer" xref="A2.1.p1.16.m16.1.1">0</cn><cn id="A2.1.p1.16.m16.2.2.cmml" type="integer" xref="A2.1.p1.16.m16.2.2">1</cn></interval><ci id="A2.1.p1.16.m16.2.3.3.cmml" xref="A2.1.p1.16.m16.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.16.m16.2c">[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.16.m16.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> as well. Now consider the sequence of centerpoints <math alttext="c^{(r)}" class="ltx_Math" display="inline" id="A2.1.p1.17.m17.1"><semantics id="A2.1.p1.17.m17.1a"><msup id="A2.1.p1.17.m17.1.2" xref="A2.1.p1.17.m17.1.2.cmml"><mi id="A2.1.p1.17.m17.1.2.2" xref="A2.1.p1.17.m17.1.2.2.cmml">c</mi><mrow id="A2.1.p1.17.m17.1.1.1.3" xref="A2.1.p1.17.m17.1.2.cmml"><mo id="A2.1.p1.17.m17.1.1.1.3.1" stretchy="false" xref="A2.1.p1.17.m17.1.2.cmml">(</mo><mi id="A2.1.p1.17.m17.1.1.1.1" xref="A2.1.p1.17.m17.1.1.1.1.cmml">r</mi><mo id="A2.1.p1.17.m17.1.1.1.3.2" stretchy="false" xref="A2.1.p1.17.m17.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.17.m17.1b"><apply id="A2.1.p1.17.m17.1.2.cmml" xref="A2.1.p1.17.m17.1.2"><csymbol cd="ambiguous" id="A2.1.p1.17.m17.1.2.1.cmml" xref="A2.1.p1.17.m17.1.2">superscript</csymbol><ci id="A2.1.p1.17.m17.1.2.2.cmml" xref="A2.1.p1.17.m17.1.2.2">𝑐</ci><ci id="A2.1.p1.17.m17.1.1.1.1.cmml" xref="A2.1.p1.17.m17.1.1.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.17.m17.1c">c^{(r)}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.17.m17.1d">italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT</annotation></semantics></math> as <math alttext="r" class="ltx_Math" display="inline" id="A2.1.p1.18.m18.1"><semantics id="A2.1.p1.18.m18.1a"><mi id="A2.1.p1.18.m18.1.1" xref="A2.1.p1.18.m18.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.1.p1.18.m18.1b"><ci id="A2.1.p1.18.m18.1.1.cmml" xref="A2.1.p1.18.m18.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.18.m18.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.18.m18.1d">italic_r</annotation></semantics></math> goes to <math alttext="0" class="ltx_Math" display="inline" id="A2.1.p1.19.m19.1"><semantics id="A2.1.p1.19.m19.1a"><mn id="A2.1.p1.19.m19.1.1" xref="A2.1.p1.19.m19.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A2.1.p1.19.m19.1b"><cn id="A2.1.p1.19.m19.1.1.cmml" type="integer" xref="A2.1.p1.19.m19.1.1">0</cn></annotation-xml></semantics></math>. By the theorem of Bolzano-Weierstrass, this sequence has a convergent subsequence with limit <math alttext="c\in[0,1]^{d}" class="ltx_Math" display="inline" id="A2.1.p1.20.m20.2"><semantics id="A2.1.p1.20.m20.2a"><mrow id="A2.1.p1.20.m20.2.3" xref="A2.1.p1.20.m20.2.3.cmml"><mi id="A2.1.p1.20.m20.2.3.2" xref="A2.1.p1.20.m20.2.3.2.cmml">c</mi><mo id="A2.1.p1.20.m20.2.3.1" xref="A2.1.p1.20.m20.2.3.1.cmml">∈</mo><msup id="A2.1.p1.20.m20.2.3.3" xref="A2.1.p1.20.m20.2.3.3.cmml"><mrow id="A2.1.p1.20.m20.2.3.3.2.2" xref="A2.1.p1.20.m20.2.3.3.2.1.cmml"><mo id="A2.1.p1.20.m20.2.3.3.2.2.1" stretchy="false" xref="A2.1.p1.20.m20.2.3.3.2.1.cmml">[</mo><mn id="A2.1.p1.20.m20.1.1" xref="A2.1.p1.20.m20.1.1.cmml">0</mn><mo id="A2.1.p1.20.m20.2.3.3.2.2.2" xref="A2.1.p1.20.m20.2.3.3.2.1.cmml">,</mo><mn id="A2.1.p1.20.m20.2.2" xref="A2.1.p1.20.m20.2.2.cmml">1</mn><mo id="A2.1.p1.20.m20.2.3.3.2.2.3" stretchy="false" xref="A2.1.p1.20.m20.2.3.3.2.1.cmml">]</mo></mrow><mi id="A2.1.p1.20.m20.2.3.3.3" xref="A2.1.p1.20.m20.2.3.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.1.p1.20.m20.2b"><apply id="A2.1.p1.20.m20.2.3.cmml" xref="A2.1.p1.20.m20.2.3"><in id="A2.1.p1.20.m20.2.3.1.cmml" xref="A2.1.p1.20.m20.2.3.1"></in><ci id="A2.1.p1.20.m20.2.3.2.cmml" xref="A2.1.p1.20.m20.2.3.2">𝑐</ci><apply id="A2.1.p1.20.m20.2.3.3.cmml" xref="A2.1.p1.20.m20.2.3.3"><csymbol cd="ambiguous" id="A2.1.p1.20.m20.2.3.3.1.cmml" xref="A2.1.p1.20.m20.2.3.3">superscript</csymbol><interval closure="closed" id="A2.1.p1.20.m20.2.3.3.2.1.cmml" xref="A2.1.p1.20.m20.2.3.3.2.2"><cn id="A2.1.p1.20.m20.1.1.cmml" type="integer" xref="A2.1.p1.20.m20.1.1">0</cn><cn id="A2.1.p1.20.m20.2.2.cmml" type="integer" xref="A2.1.p1.20.m20.2.2">1</cn></interval><ci id="A2.1.p1.20.m20.2.3.3.3.cmml" xref="A2.1.p1.20.m20.2.3.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.20.m20.2c">c\in[0,1]^{d}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.20.m20.2d">italic_c ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. From now on, we will use <math alttext="c^{(r)}" class="ltx_Math" display="inline" id="A2.1.p1.21.m21.1"><semantics id="A2.1.p1.21.m21.1a"><msup id="A2.1.p1.21.m21.1.2" xref="A2.1.p1.21.m21.1.2.cmml"><mi id="A2.1.p1.21.m21.1.2.2" xref="A2.1.p1.21.m21.1.2.2.cmml">c</mi><mrow id="A2.1.p1.21.m21.1.1.1.3" xref="A2.1.p1.21.m21.1.2.cmml"><mo id="A2.1.p1.21.m21.1.1.1.3.1" stretchy="false" xref="A2.1.p1.21.m21.1.2.cmml">(</mo><mi id="A2.1.p1.21.m21.1.1.1.1" xref="A2.1.p1.21.m21.1.1.1.1.cmml">r</mi><mo id="A2.1.p1.21.m21.1.1.1.3.2" stretchy="false" xref="A2.1.p1.21.m21.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A2.1.p1.21.m21.1b"><apply id="A2.1.p1.21.m21.1.2.cmml" xref="A2.1.p1.21.m21.1.2"><csymbol cd="ambiguous" id="A2.1.p1.21.m21.1.2.1.cmml" xref="A2.1.p1.21.m21.1.2">superscript</csymbol><ci id="A2.1.p1.21.m21.1.2.2.cmml" xref="A2.1.p1.21.m21.1.2.2">𝑐</ci><ci id="A2.1.p1.21.m21.1.1.1.1.cmml" xref="A2.1.p1.21.m21.1.1.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.1.p1.21.m21.1c">c^{(r)}</annotation><annotation encoding="application/x-llamapun" id="A2.1.p1.21.m21.1d">italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT</annotation></semantics></math> to refer to elements of this convergent subsequence.</p> </div> <div class="ltx_para" id="A2.2.p2"> <p class="ltx_p" id="A2.2.p2.12">We claim that <math alttext="c" class="ltx_Math" display="inline" id="A2.2.p2.1.m1.1"><semantics id="A2.2.p2.1.m1.1a"><mi id="A2.2.p2.1.m1.1.1" xref="A2.2.p2.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.1.m1.1b"><ci id="A2.2.p2.1.m1.1.1.cmml" xref="A2.2.p2.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.1.m1.1d">italic_c</annotation></semantics></math> is an <math alttext="\ell_{p}" class="ltx_Math" display="inline" id="A2.2.p2.2.m2.1"><semantics id="A2.2.p2.2.m2.1a"><msub id="A2.2.p2.2.m2.1.1" xref="A2.2.p2.2.m2.1.1.cmml"><mi id="A2.2.p2.2.m2.1.1.2" mathvariant="normal" xref="A2.2.p2.2.m2.1.1.2.cmml">ℓ</mi><mi id="A2.2.p2.2.m2.1.1.3" xref="A2.2.p2.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A2.2.p2.2.m2.1b"><apply id="A2.2.p2.2.m2.1.1.cmml" xref="A2.2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A2.2.p2.2.m2.1.1.1.cmml" xref="A2.2.p2.2.m2.1.1">subscript</csymbol><ci id="A2.2.p2.2.m2.1.1.2.cmml" xref="A2.2.p2.2.m2.1.1.2">ℓ</ci><ci id="A2.2.p2.2.m2.1.1.3.cmml" xref="A2.2.p2.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.2.m2.1c">\ell_{p}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.2.m2.1d">roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>-centerpoint of <math alttext="P" class="ltx_Math" display="inline" id="A2.2.p2.3.m3.1"><semantics id="A2.2.p2.3.m3.1a"><mi id="A2.2.p2.3.m3.1.1" xref="A2.2.p2.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.3.m3.1b"><ci id="A2.2.p2.3.m3.1.1.cmml" xref="A2.2.p2.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.3.m3.1d">italic_P</annotation></semantics></math>. To prove this, fix <math alttext="v\in S^{d-1}" class="ltx_Math" display="inline" id="A2.2.p2.4.m4.1"><semantics id="A2.2.p2.4.m4.1a"><mrow id="A2.2.p2.4.m4.1.1" xref="A2.2.p2.4.m4.1.1.cmml"><mi id="A2.2.p2.4.m4.1.1.2" xref="A2.2.p2.4.m4.1.1.2.cmml">v</mi><mo id="A2.2.p2.4.m4.1.1.1" xref="A2.2.p2.4.m4.1.1.1.cmml">∈</mo><msup id="A2.2.p2.4.m4.1.1.3" xref="A2.2.p2.4.m4.1.1.3.cmml"><mi id="A2.2.p2.4.m4.1.1.3.2" xref="A2.2.p2.4.m4.1.1.3.2.cmml">S</mi><mrow id="A2.2.p2.4.m4.1.1.3.3" xref="A2.2.p2.4.m4.1.1.3.3.cmml"><mi id="A2.2.p2.4.m4.1.1.3.3.2" xref="A2.2.p2.4.m4.1.1.3.3.2.cmml">d</mi><mo id="A2.2.p2.4.m4.1.1.3.3.1" xref="A2.2.p2.4.m4.1.1.3.3.1.cmml">−</mo><mn id="A2.2.p2.4.m4.1.1.3.3.3" xref="A2.2.p2.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.4.m4.1b"><apply id="A2.2.p2.4.m4.1.1.cmml" xref="A2.2.p2.4.m4.1.1"><in id="A2.2.p2.4.m4.1.1.1.cmml" xref="A2.2.p2.4.m4.1.1.1"></in><ci id="A2.2.p2.4.m4.1.1.2.cmml" xref="A2.2.p2.4.m4.1.1.2">𝑣</ci><apply id="A2.2.p2.4.m4.1.1.3.cmml" xref="A2.2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="A2.2.p2.4.m4.1.1.3.1.cmml" xref="A2.2.p2.4.m4.1.1.3">superscript</csymbol><ci id="A2.2.p2.4.m4.1.1.3.2.cmml" xref="A2.2.p2.4.m4.1.1.3.2">𝑆</ci><apply id="A2.2.p2.4.m4.1.1.3.3.cmml" xref="A2.2.p2.4.m4.1.1.3.3"><minus id="A2.2.p2.4.m4.1.1.3.3.1.cmml" xref="A2.2.p2.4.m4.1.1.3.3.1"></minus><ci id="A2.2.p2.4.m4.1.1.3.3.2.cmml" xref="A2.2.p2.4.m4.1.1.3.3.2">𝑑</ci><cn id="A2.2.p2.4.m4.1.1.3.3.3.cmml" type="integer" xref="A2.2.p2.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.4.m4.1c">v\in S^{d-1}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.4.m4.1d">italic_v ∈ italic_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and let <math alttext="z\in P\setminus\mathcal{H}^{p}_{c,v}" class="ltx_Math" display="inline" id="A2.2.p2.5.m5.2"><semantics id="A2.2.p2.5.m5.2a"><mrow id="A2.2.p2.5.m5.2.3" xref="A2.2.p2.5.m5.2.3.cmml"><mi id="A2.2.p2.5.m5.2.3.2" xref="A2.2.p2.5.m5.2.3.2.cmml">z</mi><mo id="A2.2.p2.5.m5.2.3.1" xref="A2.2.p2.5.m5.2.3.1.cmml">∈</mo><mrow id="A2.2.p2.5.m5.2.3.3" xref="A2.2.p2.5.m5.2.3.3.cmml"><mi id="A2.2.p2.5.m5.2.3.3.2" xref="A2.2.p2.5.m5.2.3.3.2.cmml">P</mi><mo id="A2.2.p2.5.m5.2.3.3.1" xref="A2.2.p2.5.m5.2.3.3.1.cmml">∖</mo><msubsup id="A2.2.p2.5.m5.2.3.3.3" xref="A2.2.p2.5.m5.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.5.m5.2.3.3.3.2.2" xref="A2.2.p2.5.m5.2.3.3.3.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.5.m5.2.2.2.4" xref="A2.2.p2.5.m5.2.2.2.3.cmml"><mi id="A2.2.p2.5.m5.1.1.1.1" xref="A2.2.p2.5.m5.1.1.1.1.cmml">c</mi><mo id="A2.2.p2.5.m5.2.2.2.4.1" xref="A2.2.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="A2.2.p2.5.m5.2.2.2.2" xref="A2.2.p2.5.m5.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.5.m5.2.3.3.3.2.3" xref="A2.2.p2.5.m5.2.3.3.3.2.3.cmml">p</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.5.m5.2b"><apply id="A2.2.p2.5.m5.2.3.cmml" xref="A2.2.p2.5.m5.2.3"><in id="A2.2.p2.5.m5.2.3.1.cmml" xref="A2.2.p2.5.m5.2.3.1"></in><ci id="A2.2.p2.5.m5.2.3.2.cmml" xref="A2.2.p2.5.m5.2.3.2">𝑧</ci><apply id="A2.2.p2.5.m5.2.3.3.cmml" xref="A2.2.p2.5.m5.2.3.3"><setdiff id="A2.2.p2.5.m5.2.3.3.1.cmml" xref="A2.2.p2.5.m5.2.3.3.1"></setdiff><ci id="A2.2.p2.5.m5.2.3.3.2.cmml" xref="A2.2.p2.5.m5.2.3.3.2">𝑃</ci><apply id="A2.2.p2.5.m5.2.3.3.3.cmml" xref="A2.2.p2.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="A2.2.p2.5.m5.2.3.3.3.1.cmml" xref="A2.2.p2.5.m5.2.3.3.3">subscript</csymbol><apply id="A2.2.p2.5.m5.2.3.3.3.2.cmml" xref="A2.2.p2.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="A2.2.p2.5.m5.2.3.3.3.2.1.cmml" xref="A2.2.p2.5.m5.2.3.3.3">superscript</csymbol><ci id="A2.2.p2.5.m5.2.3.3.3.2.2.cmml" xref="A2.2.p2.5.m5.2.3.3.3.2.2">ℋ</ci><ci id="A2.2.p2.5.m5.2.3.3.3.2.3.cmml" xref="A2.2.p2.5.m5.2.3.3.3.2.3">𝑝</ci></apply><list id="A2.2.p2.5.m5.2.2.2.3.cmml" xref="A2.2.p2.5.m5.2.2.2.4"><ci id="A2.2.p2.5.m5.1.1.1.1.cmml" xref="A2.2.p2.5.m5.1.1.1.1">𝑐</ci><ci id="A2.2.p2.5.m5.2.2.2.2.cmml" xref="A2.2.p2.5.m5.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.5.m5.2c">z\in P\setminus\mathcal{H}^{p}_{c,v}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.5.m5.2d">italic_z ∈ italic_P ∖ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> be arbitrary. This means that there exists <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A2.2.p2.6.m6.1"><semantics id="A2.2.p2.6.m6.1a"><mrow id="A2.2.p2.6.m6.1.1" xref="A2.2.p2.6.m6.1.1.cmml"><mi id="A2.2.p2.6.m6.1.1.2" xref="A2.2.p2.6.m6.1.1.2.cmml">ε</mi><mo id="A2.2.p2.6.m6.1.1.1" xref="A2.2.p2.6.m6.1.1.1.cmml">></mo><mn id="A2.2.p2.6.m6.1.1.3" xref="A2.2.p2.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.6.m6.1b"><apply id="A2.2.p2.6.m6.1.1.cmml" xref="A2.2.p2.6.m6.1.1"><gt id="A2.2.p2.6.m6.1.1.1.cmml" xref="A2.2.p2.6.m6.1.1.1"></gt><ci id="A2.2.p2.6.m6.1.1.2.cmml" xref="A2.2.p2.6.m6.1.1.2">𝜀</ci><cn id="A2.2.p2.6.m6.1.1.3.cmml" type="integer" xref="A2.2.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.6.m6.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.6.m6.1d">italic_ε > 0</annotation></semantics></math> such that <math alttext="\lVert c-z\rVert_{p}>\lVert c-\varepsilon v-z\rVert_{p}" class="ltx_Math" display="inline" id="A2.2.p2.7.m7.2"><semantics id="A2.2.p2.7.m7.2a"><mrow id="A2.2.p2.7.m7.2.2" xref="A2.2.p2.7.m7.2.2.cmml"><msub id="A2.2.p2.7.m7.1.1.1" xref="A2.2.p2.7.m7.1.1.1.cmml"><mrow id="A2.2.p2.7.m7.1.1.1.1.1" xref="A2.2.p2.7.m7.1.1.1.1.2.cmml"><mo fence="true" id="A2.2.p2.7.m7.1.1.1.1.1.2" rspace="0em" xref="A2.2.p2.7.m7.1.1.1.1.2.1.cmml">∥</mo><mrow id="A2.2.p2.7.m7.1.1.1.1.1.1" xref="A2.2.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="A2.2.p2.7.m7.1.1.1.1.1.1.2" xref="A2.2.p2.7.m7.1.1.1.1.1.1.2.cmml">c</mi><mo id="A2.2.p2.7.m7.1.1.1.1.1.1.1" xref="A2.2.p2.7.m7.1.1.1.1.1.1.1.cmml">−</mo><mi id="A2.2.p2.7.m7.1.1.1.1.1.1.3" xref="A2.2.p2.7.m7.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A2.2.p2.7.m7.1.1.1.1.1.3" lspace="0em" xref="A2.2.p2.7.m7.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A2.2.p2.7.m7.1.1.1.3" xref="A2.2.p2.7.m7.1.1.1.3.cmml">p</mi></msub><mo id="A2.2.p2.7.m7.2.2.3" rspace="0.1389em" xref="A2.2.p2.7.m7.2.2.3.cmml">></mo><msub id="A2.2.p2.7.m7.2.2.2" xref="A2.2.p2.7.m7.2.2.2.cmml"><mrow id="A2.2.p2.7.m7.2.2.2.1.1" xref="A2.2.p2.7.m7.2.2.2.1.2.cmml"><mo fence="true" id="A2.2.p2.7.m7.2.2.2.1.1.2" lspace="0.1389em" rspace="0em" xref="A2.2.p2.7.m7.2.2.2.1.2.1.cmml">∥</mo><mrow id="A2.2.p2.7.m7.2.2.2.1.1.1" xref="A2.2.p2.7.m7.2.2.2.1.1.1.cmml"><mi id="A2.2.p2.7.m7.2.2.2.1.1.1.2" xref="A2.2.p2.7.m7.2.2.2.1.1.1.2.cmml">c</mi><mo id="A2.2.p2.7.m7.2.2.2.1.1.1.1" xref="A2.2.p2.7.m7.2.2.2.1.1.1.1.cmml">−</mo><mrow id="A2.2.p2.7.m7.2.2.2.1.1.1.3" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.cmml"><mi id="A2.2.p2.7.m7.2.2.2.1.1.1.3.2" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.2.cmml">ε</mi><mo id="A2.2.p2.7.m7.2.2.2.1.1.1.3.1" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="A2.2.p2.7.m7.2.2.2.1.1.1.3.3" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.3.cmml">v</mi></mrow><mo id="A2.2.p2.7.m7.2.2.2.1.1.1.1a" xref="A2.2.p2.7.m7.2.2.2.1.1.1.1.cmml">−</mo><mi id="A2.2.p2.7.m7.2.2.2.1.1.1.4" xref="A2.2.p2.7.m7.2.2.2.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A2.2.p2.7.m7.2.2.2.1.1.3" lspace="0em" xref="A2.2.p2.7.m7.2.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A2.2.p2.7.m7.2.2.2.3" xref="A2.2.p2.7.m7.2.2.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.7.m7.2b"><apply id="A2.2.p2.7.m7.2.2.cmml" xref="A2.2.p2.7.m7.2.2"><gt id="A2.2.p2.7.m7.2.2.3.cmml" xref="A2.2.p2.7.m7.2.2.3"></gt><apply id="A2.2.p2.7.m7.1.1.1.cmml" xref="A2.2.p2.7.m7.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.7.m7.1.1.1.2.cmml" xref="A2.2.p2.7.m7.1.1.1">subscript</csymbol><apply id="A2.2.p2.7.m7.1.1.1.1.2.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1"><csymbol cd="latexml" id="A2.2.p2.7.m7.1.1.1.1.2.1.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A2.2.p2.7.m7.1.1.1.1.1.1.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1.1"><minus id="A2.2.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1.1.1"></minus><ci id="A2.2.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1.1.2">𝑐</ci><ci id="A2.2.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="A2.2.p2.7.m7.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A2.2.p2.7.m7.1.1.1.3.cmml" xref="A2.2.p2.7.m7.1.1.1.3">𝑝</ci></apply><apply id="A2.2.p2.7.m7.2.2.2.cmml" xref="A2.2.p2.7.m7.2.2.2"><csymbol cd="ambiguous" id="A2.2.p2.7.m7.2.2.2.2.cmml" xref="A2.2.p2.7.m7.2.2.2">subscript</csymbol><apply id="A2.2.p2.7.m7.2.2.2.1.2.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1"><csymbol cd="latexml" id="A2.2.p2.7.m7.2.2.2.1.2.1.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A2.2.p2.7.m7.2.2.2.1.1.1.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1"><minus id="A2.2.p2.7.m7.2.2.2.1.1.1.1.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.1"></minus><ci id="A2.2.p2.7.m7.2.2.2.1.1.1.2.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.2">𝑐</ci><apply id="A2.2.p2.7.m7.2.2.2.1.1.1.3.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3"><times id="A2.2.p2.7.m7.2.2.2.1.1.1.3.1.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.1"></times><ci id="A2.2.p2.7.m7.2.2.2.1.1.1.3.2.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.2">𝜀</ci><ci id="A2.2.p2.7.m7.2.2.2.1.1.1.3.3.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.3.3">𝑣</ci></apply><ci id="A2.2.p2.7.m7.2.2.2.1.1.1.4.cmml" xref="A2.2.p2.7.m7.2.2.2.1.1.1.4">𝑧</ci></apply></apply><ci id="A2.2.p2.7.m7.2.2.2.3.cmml" xref="A2.2.p2.7.m7.2.2.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.7.m7.2c">\lVert c-z\rVert_{p}>\lVert c-\varepsilon v-z\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.7.m7.2d">∥ italic_c - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > ∥ italic_c - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and thus even <math alttext="\lVert c-z\rVert_{p}>\lVert c-\varepsilon v-z\rVert_{p}+\delta" class="ltx_Math" display="inline" id="A2.2.p2.8.m8.2"><semantics id="A2.2.p2.8.m8.2a"><mrow id="A2.2.p2.8.m8.2.2" xref="A2.2.p2.8.m8.2.2.cmml"><msub id="A2.2.p2.8.m8.1.1.1" xref="A2.2.p2.8.m8.1.1.1.cmml"><mrow id="A2.2.p2.8.m8.1.1.1.1.1" xref="A2.2.p2.8.m8.1.1.1.1.2.cmml"><mo fence="true" id="A2.2.p2.8.m8.1.1.1.1.1.2" rspace="0em" xref="A2.2.p2.8.m8.1.1.1.1.2.1.cmml">∥</mo><mrow id="A2.2.p2.8.m8.1.1.1.1.1.1" xref="A2.2.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="A2.2.p2.8.m8.1.1.1.1.1.1.2" xref="A2.2.p2.8.m8.1.1.1.1.1.1.2.cmml">c</mi><mo id="A2.2.p2.8.m8.1.1.1.1.1.1.1" xref="A2.2.p2.8.m8.1.1.1.1.1.1.1.cmml">−</mo><mi id="A2.2.p2.8.m8.1.1.1.1.1.1.3" xref="A2.2.p2.8.m8.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo fence="true" id="A2.2.p2.8.m8.1.1.1.1.1.3" lspace="0em" xref="A2.2.p2.8.m8.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A2.2.p2.8.m8.1.1.1.3" xref="A2.2.p2.8.m8.1.1.1.3.cmml">p</mi></msub><mo id="A2.2.p2.8.m8.2.2.3" rspace="0.1389em" xref="A2.2.p2.8.m8.2.2.3.cmml">></mo><mrow id="A2.2.p2.8.m8.2.2.2" xref="A2.2.p2.8.m8.2.2.2.cmml"><msub id="A2.2.p2.8.m8.2.2.2.1" xref="A2.2.p2.8.m8.2.2.2.1.cmml"><mrow id="A2.2.p2.8.m8.2.2.2.1.1.1" xref="A2.2.p2.8.m8.2.2.2.1.1.2.cmml"><mo fence="true" id="A2.2.p2.8.m8.2.2.2.1.1.1.2" lspace="0.1389em" rspace="0em" xref="A2.2.p2.8.m8.2.2.2.1.1.2.1.cmml">∥</mo><mrow id="A2.2.p2.8.m8.2.2.2.1.1.1.1" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.cmml"><mi id="A2.2.p2.8.m8.2.2.2.1.1.1.1.2" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.2.cmml">c</mi><mo id="A2.2.p2.8.m8.2.2.2.1.1.1.1.1" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.1.cmml">−</mo><mrow id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.cmml"><mi id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.2" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.2.cmml">ε</mi><mo id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.1" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.3" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.3.cmml">v</mi></mrow><mo id="A2.2.p2.8.m8.2.2.2.1.1.1.1.1a" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="A2.2.p2.8.m8.2.2.2.1.1.1.1.4" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.4.cmml">z</mi></mrow><mo fence="true" id="A2.2.p2.8.m8.2.2.2.1.1.1.3" lspace="0em" rspace="0em" xref="A2.2.p2.8.m8.2.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A2.2.p2.8.m8.2.2.2.1.3" xref="A2.2.p2.8.m8.2.2.2.1.3.cmml">p</mi></msub><mo id="A2.2.p2.8.m8.2.2.2.2" xref="A2.2.p2.8.m8.2.2.2.2.cmml">+</mo><mi id="A2.2.p2.8.m8.2.2.2.3" xref="A2.2.p2.8.m8.2.2.2.3.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.8.m8.2b"><apply id="A2.2.p2.8.m8.2.2.cmml" xref="A2.2.p2.8.m8.2.2"><gt id="A2.2.p2.8.m8.2.2.3.cmml" xref="A2.2.p2.8.m8.2.2.3"></gt><apply id="A2.2.p2.8.m8.1.1.1.cmml" xref="A2.2.p2.8.m8.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.8.m8.1.1.1.2.cmml" xref="A2.2.p2.8.m8.1.1.1">subscript</csymbol><apply id="A2.2.p2.8.m8.1.1.1.1.2.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1"><csymbol cd="latexml" id="A2.2.p2.8.m8.1.1.1.1.2.1.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1.2">delimited-∥∥</csymbol><apply id="A2.2.p2.8.m8.1.1.1.1.1.1.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1.1"><minus id="A2.2.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1.1.1"></minus><ci id="A2.2.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1.1.2">𝑐</ci><ci id="A2.2.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="A2.2.p2.8.m8.1.1.1.1.1.1.3">𝑧</ci></apply></apply><ci id="A2.2.p2.8.m8.1.1.1.3.cmml" xref="A2.2.p2.8.m8.1.1.1.3">𝑝</ci></apply><apply id="A2.2.p2.8.m8.2.2.2.cmml" xref="A2.2.p2.8.m8.2.2.2"><plus id="A2.2.p2.8.m8.2.2.2.2.cmml" xref="A2.2.p2.8.m8.2.2.2.2"></plus><apply id="A2.2.p2.8.m8.2.2.2.1.cmml" xref="A2.2.p2.8.m8.2.2.2.1"><csymbol cd="ambiguous" id="A2.2.p2.8.m8.2.2.2.1.2.cmml" xref="A2.2.p2.8.m8.2.2.2.1">subscript</csymbol><apply id="A2.2.p2.8.m8.2.2.2.1.1.2.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1"><csymbol cd="latexml" id="A2.2.p2.8.m8.2.2.2.1.1.2.1.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A2.2.p2.8.m8.2.2.2.1.1.1.1.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1"><minus id="A2.2.p2.8.m8.2.2.2.1.1.1.1.1.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.1"></minus><ci id="A2.2.p2.8.m8.2.2.2.1.1.1.1.2.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.2">𝑐</ci><apply id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3"><times id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.1.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.1"></times><ci id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.2.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.2">𝜀</ci><ci id="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.3.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.3.3">𝑣</ci></apply><ci id="A2.2.p2.8.m8.2.2.2.1.1.1.1.4.cmml" xref="A2.2.p2.8.m8.2.2.2.1.1.1.1.4">𝑧</ci></apply></apply><ci id="A2.2.p2.8.m8.2.2.2.1.3.cmml" xref="A2.2.p2.8.m8.2.2.2.1.3">𝑝</ci></apply><ci id="A2.2.p2.8.m8.2.2.2.3.cmml" xref="A2.2.p2.8.m8.2.2.2.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.8.m8.2c">\lVert c-z\rVert_{p}>\lVert c-\varepsilon v-z\rVert_{p}+\delta</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.8.m8.2d">∥ italic_c - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT > ∥ italic_c - italic_ε italic_v - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_δ</annotation></semantics></math> for some <math alttext="\delta>0" class="ltx_Math" display="inline" id="A2.2.p2.9.m9.1"><semantics id="A2.2.p2.9.m9.1a"><mrow id="A2.2.p2.9.m9.1.1" xref="A2.2.p2.9.m9.1.1.cmml"><mi id="A2.2.p2.9.m9.1.1.2" xref="A2.2.p2.9.m9.1.1.2.cmml">δ</mi><mo id="A2.2.p2.9.m9.1.1.1" xref="A2.2.p2.9.m9.1.1.1.cmml">></mo><mn id="A2.2.p2.9.m9.1.1.3" xref="A2.2.p2.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.9.m9.1b"><apply id="A2.2.p2.9.m9.1.1.cmml" xref="A2.2.p2.9.m9.1.1"><gt id="A2.2.p2.9.m9.1.1.1.cmml" xref="A2.2.p2.9.m9.1.1.1"></gt><ci id="A2.2.p2.9.m9.1.1.2.cmml" xref="A2.2.p2.9.m9.1.1.2">𝛿</ci><cn id="A2.2.p2.9.m9.1.1.3.cmml" type="integer" xref="A2.2.p2.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.9.m9.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.9.m9.1d">italic_δ > 0</annotation></semantics></math>. By choosing <math alttext="r" class="ltx_Math" display="inline" id="A2.2.p2.10.m10.1"><semantics id="A2.2.p2.10.m10.1a"><mi id="A2.2.p2.10.m10.1.1" xref="A2.2.p2.10.m10.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.10.m10.1b"><ci id="A2.2.p2.10.m10.1.1.cmml" xref="A2.2.p2.10.m10.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.10.m10.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.10.m10.1d">italic_r</annotation></semantics></math> small enough, we can ensure both <math alttext="r\leq\delta/4" class="ltx_Math" display="inline" id="A2.2.p2.11.m11.1"><semantics id="A2.2.p2.11.m11.1a"><mrow id="A2.2.p2.11.m11.1.1" xref="A2.2.p2.11.m11.1.1.cmml"><mi id="A2.2.p2.11.m11.1.1.2" xref="A2.2.p2.11.m11.1.1.2.cmml">r</mi><mo id="A2.2.p2.11.m11.1.1.1" xref="A2.2.p2.11.m11.1.1.1.cmml">≤</mo><mrow id="A2.2.p2.11.m11.1.1.3" xref="A2.2.p2.11.m11.1.1.3.cmml"><mi id="A2.2.p2.11.m11.1.1.3.2" xref="A2.2.p2.11.m11.1.1.3.2.cmml">δ</mi><mo id="A2.2.p2.11.m11.1.1.3.1" xref="A2.2.p2.11.m11.1.1.3.1.cmml">/</mo><mn id="A2.2.p2.11.m11.1.1.3.3" xref="A2.2.p2.11.m11.1.1.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.11.m11.1b"><apply id="A2.2.p2.11.m11.1.1.cmml" xref="A2.2.p2.11.m11.1.1"><leq id="A2.2.p2.11.m11.1.1.1.cmml" xref="A2.2.p2.11.m11.1.1.1"></leq><ci id="A2.2.p2.11.m11.1.1.2.cmml" xref="A2.2.p2.11.m11.1.1.2">𝑟</ci><apply id="A2.2.p2.11.m11.1.1.3.cmml" xref="A2.2.p2.11.m11.1.1.3"><divide id="A2.2.p2.11.m11.1.1.3.1.cmml" xref="A2.2.p2.11.m11.1.1.3.1"></divide><ci id="A2.2.p2.11.m11.1.1.3.2.cmml" xref="A2.2.p2.11.m11.1.1.3.2">𝛿</ci><cn id="A2.2.p2.11.m11.1.1.3.3.cmml" type="integer" xref="A2.2.p2.11.m11.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.11.m11.1c">r\leq\delta/4</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.11.m11.1d">italic_r ≤ italic_δ / 4</annotation></semantics></math> and <math alttext="\lVert c^{(r)}-c\rVert_{p}\leq\delta/4" class="ltx_Math" display="inline" id="A2.2.p2.12.m12.2"><semantics id="A2.2.p2.12.m12.2a"><mrow id="A2.2.p2.12.m12.2.2" xref="A2.2.p2.12.m12.2.2.cmml"><msub id="A2.2.p2.12.m12.2.2.1" xref="A2.2.p2.12.m12.2.2.1.cmml"><mrow id="A2.2.p2.12.m12.2.2.1.1.1" xref="A2.2.p2.12.m12.2.2.1.1.2.cmml"><mo fence="true" id="A2.2.p2.12.m12.2.2.1.1.1.2" rspace="0em" xref="A2.2.p2.12.m12.2.2.1.1.2.1.cmml">∥</mo><mrow id="A2.2.p2.12.m12.2.2.1.1.1.1" xref="A2.2.p2.12.m12.2.2.1.1.1.1.cmml"><msup id="A2.2.p2.12.m12.2.2.1.1.1.1.2" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.cmml"><mi id="A2.2.p2.12.m12.2.2.1.1.1.1.2.2" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.2.cmml">c</mi><mrow id="A2.2.p2.12.m12.1.1.1.3" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.cmml"><mo id="A2.2.p2.12.m12.1.1.1.3.1" stretchy="false" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.cmml">(</mo><mi id="A2.2.p2.12.m12.1.1.1.1" xref="A2.2.p2.12.m12.1.1.1.1.cmml">r</mi><mo id="A2.2.p2.12.m12.1.1.1.3.2" stretchy="false" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.cmml">)</mo></mrow></msup><mo id="A2.2.p2.12.m12.2.2.1.1.1.1.1" xref="A2.2.p2.12.m12.2.2.1.1.1.1.1.cmml">−</mo><mi id="A2.2.p2.12.m12.2.2.1.1.1.1.3" xref="A2.2.p2.12.m12.2.2.1.1.1.1.3.cmml">c</mi></mrow><mo fence="true" id="A2.2.p2.12.m12.2.2.1.1.1.3" lspace="0em" xref="A2.2.p2.12.m12.2.2.1.1.2.1.cmml">∥</mo></mrow><mi id="A2.2.p2.12.m12.2.2.1.3" xref="A2.2.p2.12.m12.2.2.1.3.cmml">p</mi></msub><mo id="A2.2.p2.12.m12.2.2.2" xref="A2.2.p2.12.m12.2.2.2.cmml">≤</mo><mrow id="A2.2.p2.12.m12.2.2.3" xref="A2.2.p2.12.m12.2.2.3.cmml"><mi id="A2.2.p2.12.m12.2.2.3.2" xref="A2.2.p2.12.m12.2.2.3.2.cmml">δ</mi><mo id="A2.2.p2.12.m12.2.2.3.1" xref="A2.2.p2.12.m12.2.2.3.1.cmml">/</mo><mn id="A2.2.p2.12.m12.2.2.3.3" xref="A2.2.p2.12.m12.2.2.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.12.m12.2b"><apply id="A2.2.p2.12.m12.2.2.cmml" xref="A2.2.p2.12.m12.2.2"><leq id="A2.2.p2.12.m12.2.2.2.cmml" xref="A2.2.p2.12.m12.2.2.2"></leq><apply id="A2.2.p2.12.m12.2.2.1.cmml" xref="A2.2.p2.12.m12.2.2.1"><csymbol cd="ambiguous" id="A2.2.p2.12.m12.2.2.1.2.cmml" xref="A2.2.p2.12.m12.2.2.1">subscript</csymbol><apply id="A2.2.p2.12.m12.2.2.1.1.2.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1"><csymbol cd="latexml" id="A2.2.p2.12.m12.2.2.1.1.2.1.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.2">delimited-∥∥</csymbol><apply id="A2.2.p2.12.m12.2.2.1.1.1.1.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1"><minus id="A2.2.p2.12.m12.2.2.1.1.1.1.1.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1.1"></minus><apply id="A2.2.p2.12.m12.2.2.1.1.1.1.2.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.2.p2.12.m12.2.2.1.1.1.1.2.1.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2">superscript</csymbol><ci id="A2.2.p2.12.m12.2.2.1.1.1.1.2.2.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1.2.2">𝑐</ci><ci id="A2.2.p2.12.m12.1.1.1.1.cmml" xref="A2.2.p2.12.m12.1.1.1.1">𝑟</ci></apply><ci id="A2.2.p2.12.m12.2.2.1.1.1.1.3.cmml" xref="A2.2.p2.12.m12.2.2.1.1.1.1.3">𝑐</ci></apply></apply><ci id="A2.2.p2.12.m12.2.2.1.3.cmml" xref="A2.2.p2.12.m12.2.2.1.3">𝑝</ci></apply><apply id="A2.2.p2.12.m12.2.2.3.cmml" xref="A2.2.p2.12.m12.2.2.3"><divide id="A2.2.p2.12.m12.2.2.3.1.cmml" xref="A2.2.p2.12.m12.2.2.3.1"></divide><ci id="A2.2.p2.12.m12.2.2.3.2.cmml" xref="A2.2.p2.12.m12.2.2.3.2">𝛿</ci><cn id="A2.2.p2.12.m12.2.2.3.3.cmml" type="integer" xref="A2.2.p2.12.m12.2.2.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.12.m12.2c">\lVert c^{(r)}-c\rVert_{p}\leq\delta/4</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.12.m12.2d">∥ italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT - italic_c ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_δ / 4</annotation></semantics></math>. Using the triangle inequality, we then get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A2.EGx6"> <tbody id="A2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\lVert c^{(r)}-z^{\prime}\rVert_{p}" class="ltx_Math" display="inline" id="A2.Ex1.m1.2"><semantics id="A2.Ex1.m1.2a"><msub id="A2.Ex1.m1.2.2" xref="A2.Ex1.m1.2.2.cmml"><mrow id="A2.Ex1.m1.2.2.1.1" xref="A2.Ex1.m1.2.2.1.2.cmml"><mo fence="true" id="A2.Ex1.m1.2.2.1.1.2" rspace="0em" xref="A2.Ex1.m1.2.2.1.2.1.cmml">∥</mo><mrow id="A2.Ex1.m1.2.2.1.1.1" xref="A2.Ex1.m1.2.2.1.1.1.cmml"><msup id="A2.Ex1.m1.2.2.1.1.1.2" xref="A2.Ex1.m1.2.2.1.1.1.2.cmml"><mi id="A2.Ex1.m1.2.2.1.1.1.2.2" xref="A2.Ex1.m1.2.2.1.1.1.2.2.cmml">c</mi><mrow id="A2.Ex1.m1.1.1.1.3" xref="A2.Ex1.m1.2.2.1.1.1.2.cmml"><mo id="A2.Ex1.m1.1.1.1.3.1" stretchy="false" xref="A2.Ex1.m1.2.2.1.1.1.2.cmml">(</mo><mi id="A2.Ex1.m1.1.1.1.1" xref="A2.Ex1.m1.1.1.1.1.cmml">r</mi><mo id="A2.Ex1.m1.1.1.1.3.2" stretchy="false" xref="A2.Ex1.m1.2.2.1.1.1.2.cmml">)</mo></mrow></msup><mo id="A2.Ex1.m1.2.2.1.1.1.1" xref="A2.Ex1.m1.2.2.1.1.1.1.cmml">−</mo><msup id="A2.Ex1.m1.2.2.1.1.1.3" xref="A2.Ex1.m1.2.2.1.1.1.3.cmml"><mi id="A2.Ex1.m1.2.2.1.1.1.3.2" xref="A2.Ex1.m1.2.2.1.1.1.3.2.cmml">z</mi><mo id="A2.Ex1.m1.2.2.1.1.1.3.3" xref="A2.Ex1.m1.2.2.1.1.1.3.3.cmml">′</mo></msup></mrow><mo fence="true" id="A2.Ex1.m1.2.2.1.1.3" lspace="0em" xref="A2.Ex1.m1.2.2.1.2.1.cmml">∥</mo></mrow><mi id="A2.Ex1.m1.2.2.3" xref="A2.Ex1.m1.2.2.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A2.Ex1.m1.2b"><apply id="A2.Ex1.m1.2.2.cmml" xref="A2.Ex1.m1.2.2"><csymbol cd="ambiguous" id="A2.Ex1.m1.2.2.2.cmml" xref="A2.Ex1.m1.2.2">subscript</csymbol><apply id="A2.Ex1.m1.2.2.1.2.cmml" xref="A2.Ex1.m1.2.2.1.1"><csymbol cd="latexml" id="A2.Ex1.m1.2.2.1.2.1.cmml" xref="A2.Ex1.m1.2.2.1.1.2">delimited-∥∥</csymbol><apply id="A2.Ex1.m1.2.2.1.1.1.cmml" xref="A2.Ex1.m1.2.2.1.1.1"><minus id="A2.Ex1.m1.2.2.1.1.1.1.cmml" xref="A2.Ex1.m1.2.2.1.1.1.1"></minus><apply id="A2.Ex1.m1.2.2.1.1.1.2.cmml" xref="A2.Ex1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.Ex1.m1.2.2.1.1.1.2.1.cmml" xref="A2.Ex1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="A2.Ex1.m1.2.2.1.1.1.2.2.cmml" xref="A2.Ex1.m1.2.2.1.1.1.2.2">𝑐</ci><ci id="A2.Ex1.m1.1.1.1.1.cmml" xref="A2.Ex1.m1.1.1.1.1">𝑟</ci></apply><apply id="A2.Ex1.m1.2.2.1.1.1.3.cmml" xref="A2.Ex1.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex1.m1.2.2.1.1.1.3.1.cmml" xref="A2.Ex1.m1.2.2.1.1.1.3">superscript</csymbol><ci id="A2.Ex1.m1.2.2.1.1.1.3.2.cmml" xref="A2.Ex1.m1.2.2.1.1.1.3.2">𝑧</ci><ci id="A2.Ex1.m1.2.2.1.1.1.3.3.cmml" xref="A2.Ex1.m1.2.2.1.1.1.3.3">′</ci></apply></apply></apply><ci id="A2.Ex1.m1.2.2.3.cmml" xref="A2.Ex1.m1.2.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m1.2c">\displaystyle\lVert c^{(r)}-z^{\prime}\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex1.m1.2d">∥ italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\lVert c-z\rVert_{p}-\lVert z^{\prime}-z\rVert_{p}-\lVert c-c% ^{(r)}\rVert_{p}" class="ltx_Math" display="inline" id="A2.Ex1.m2.4"><semantics id="A2.Ex1.m2.4a"><mrow id="A2.Ex1.m2.4.4" xref="A2.Ex1.m2.4.4.cmml"><mi id="A2.Ex1.m2.4.4.5" xref="A2.Ex1.m2.4.4.5.cmml"></mi><mo id="A2.Ex1.m2.4.4.4" rspace="0.1389em" xref="A2.Ex1.m2.4.4.4.cmml">≥</mo><mrow id="A2.Ex1.m2.4.4.3" xref="A2.Ex1.m2.4.4.3.cmml"><msub id="A2.Ex1.m2.2.2.1.1" xref="A2.Ex1.m2.2.2.1.1.cmml"><mrow id="A2.Ex1.m2.2.2.1.1.1.1" xref="A2.Ex1.m2.2.2.1.1.1.2.cmml"><mo fence="true" 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xref="A2.Ex1.m2.3.3.2.2.3">𝑝</ci></apply><apply id="A2.Ex1.m2.4.4.3.3.cmml" xref="A2.Ex1.m2.4.4.3.3"><csymbol cd="ambiguous" id="A2.Ex1.m2.4.4.3.3.2.cmml" xref="A2.Ex1.m2.4.4.3.3">subscript</csymbol><apply id="A2.Ex1.m2.4.4.3.3.1.2.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1"><csymbol cd="latexml" id="A2.Ex1.m2.4.4.3.3.1.2.1.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.2">delimited-∥∥</csymbol><apply id="A2.Ex1.m2.4.4.3.3.1.1.1.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1"><minus id="A2.Ex1.m2.4.4.3.3.1.1.1.1.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1.1"></minus><ci id="A2.Ex1.m2.4.4.3.3.1.1.1.2.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1.2">𝑐</ci><apply id="A2.Ex1.m2.4.4.3.3.1.1.1.3.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A2.Ex1.m2.4.4.3.3.1.1.1.3.1.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1.3">superscript</csymbol><ci id="A2.Ex1.m2.4.4.3.3.1.1.1.3.2.cmml" xref="A2.Ex1.m2.4.4.3.3.1.1.1.3.2">𝑐</ci><ci id="A2.Ex1.m2.1.1.1.1.cmml" xref="A2.Ex1.m2.1.1.1.1">𝑟</ci></apply></apply></apply><ci id="A2.Ex1.m2.4.4.3.3.3.cmml" xref="A2.Ex1.m2.4.4.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m2.4c">\displaystyle\geq\lVert c-z\rVert_{p}-\lVert z^{\prime}-z\rVert_{p}-\lVert c-c% ^{(r)}\rVert_{p}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex1.m2.4d">≥ ∥ italic_c - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_c - italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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rspace="0em" xref="A2.Ex3.m1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mi id="A2.Ex3.m1.1.1.1.1.3" xref="A2.Ex3.m1.1.1.1.1.3.cmml">p</mi></msub><mo id="A2.Ex3.m1.1.1.1.2" xref="A2.Ex3.m1.1.1.1.2.cmml">+</mo><mrow id="A2.Ex3.m1.1.1.1.3" xref="A2.Ex3.m1.1.1.1.3.cmml"><mi id="A2.Ex3.m1.1.1.1.3.2" xref="A2.Ex3.m1.1.1.1.3.2.cmml">δ</mi><mo id="A2.Ex3.m1.1.1.1.3.1" xref="A2.Ex3.m1.1.1.1.3.1.cmml">/</mo><mn id="A2.Ex3.m1.1.1.1.3.3" xref="A2.Ex3.m1.1.1.1.3.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex3.m1.1b"><apply id="A2.Ex3.m1.1.1.cmml" xref="A2.Ex3.m1.1.1"><gt id="A2.Ex3.m1.1.1.2.cmml" xref="A2.Ex3.m1.1.1.2"></gt><csymbol cd="latexml" id="A2.Ex3.m1.1.1.3.cmml" xref="A2.Ex3.m1.1.1.3">absent</csymbol><apply id="A2.Ex3.m1.1.1.1.cmml" xref="A2.Ex3.m1.1.1.1"><plus id="A2.Ex3.m1.1.1.1.2.cmml" xref="A2.Ex3.m1.1.1.1.2"></plus><apply id="A2.Ex3.m1.1.1.1.1.cmml" xref="A2.Ex3.m1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.2.cmml" 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id="A2.Ex4.m1.5d">≥ ∥ italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT - italic_ε italic_v - italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_z ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - ∥ italic_c - italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_δ / 2</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\lVert c^{(r)}-\varepsilon v-z^{\prime}\rVert_{p}" class="ltx_Math" display="inline" id="A2.Ex5.m1.2"><semantics id="A2.Ex5.m1.2a"><mrow id="A2.Ex5.m1.2.2" 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italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.2.p2.22">for all <math alttext="z^{\prime}\in B^{p}(z,r)" class="ltx_Math" display="inline" id="A2.2.p2.13.m1.2"><semantics id="A2.2.p2.13.m1.2a"><mrow id="A2.2.p2.13.m1.2.3" xref="A2.2.p2.13.m1.2.3.cmml"><msup id="A2.2.p2.13.m1.2.3.2" xref="A2.2.p2.13.m1.2.3.2.cmml"><mi id="A2.2.p2.13.m1.2.3.2.2" xref="A2.2.p2.13.m1.2.3.2.2.cmml">z</mi><mo id="A2.2.p2.13.m1.2.3.2.3" xref="A2.2.p2.13.m1.2.3.2.3.cmml">′</mo></msup><mo id="A2.2.p2.13.m1.2.3.1" xref="A2.2.p2.13.m1.2.3.1.cmml">∈</mo><mrow id="A2.2.p2.13.m1.2.3.3" xref="A2.2.p2.13.m1.2.3.3.cmml"><msup id="A2.2.p2.13.m1.2.3.3.2" xref="A2.2.p2.13.m1.2.3.3.2.cmml"><mi id="A2.2.p2.13.m1.2.3.3.2.2" xref="A2.2.p2.13.m1.2.3.3.2.2.cmml">B</mi><mi id="A2.2.p2.13.m1.2.3.3.2.3" xref="A2.2.p2.13.m1.2.3.3.2.3.cmml">p</mi></msup><mo id="A2.2.p2.13.m1.2.3.3.1" xref="A2.2.p2.13.m1.2.3.3.1.cmml">⁢</mo><mrow id="A2.2.p2.13.m1.2.3.3.3.2" xref="A2.2.p2.13.m1.2.3.3.3.1.cmml"><mo id="A2.2.p2.13.m1.2.3.3.3.2.1" stretchy="false" xref="A2.2.p2.13.m1.2.3.3.3.1.cmml">(</mo><mi id="A2.2.p2.13.m1.1.1" xref="A2.2.p2.13.m1.1.1.cmml">z</mi><mo id="A2.2.p2.13.m1.2.3.3.3.2.2" xref="A2.2.p2.13.m1.2.3.3.3.1.cmml">,</mo><mi id="A2.2.p2.13.m1.2.2" xref="A2.2.p2.13.m1.2.2.cmml">r</mi><mo id="A2.2.p2.13.m1.2.3.3.3.2.3" stretchy="false" xref="A2.2.p2.13.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.13.m1.2b"><apply id="A2.2.p2.13.m1.2.3.cmml" xref="A2.2.p2.13.m1.2.3"><in id="A2.2.p2.13.m1.2.3.1.cmml" xref="A2.2.p2.13.m1.2.3.1"></in><apply id="A2.2.p2.13.m1.2.3.2.cmml" xref="A2.2.p2.13.m1.2.3.2"><csymbol cd="ambiguous" id="A2.2.p2.13.m1.2.3.2.1.cmml" xref="A2.2.p2.13.m1.2.3.2">superscript</csymbol><ci id="A2.2.p2.13.m1.2.3.2.2.cmml" xref="A2.2.p2.13.m1.2.3.2.2">𝑧</ci><ci id="A2.2.p2.13.m1.2.3.2.3.cmml" xref="A2.2.p2.13.m1.2.3.2.3">′</ci></apply><apply id="A2.2.p2.13.m1.2.3.3.cmml" xref="A2.2.p2.13.m1.2.3.3"><times id="A2.2.p2.13.m1.2.3.3.1.cmml" xref="A2.2.p2.13.m1.2.3.3.1"></times><apply id="A2.2.p2.13.m1.2.3.3.2.cmml" xref="A2.2.p2.13.m1.2.3.3.2"><csymbol cd="ambiguous" id="A2.2.p2.13.m1.2.3.3.2.1.cmml" xref="A2.2.p2.13.m1.2.3.3.2">superscript</csymbol><ci id="A2.2.p2.13.m1.2.3.3.2.2.cmml" xref="A2.2.p2.13.m1.2.3.3.2.2">𝐵</ci><ci id="A2.2.p2.13.m1.2.3.3.2.3.cmml" xref="A2.2.p2.13.m1.2.3.3.2.3">𝑝</ci></apply><interval closure="open" id="A2.2.p2.13.m1.2.3.3.3.1.cmml" xref="A2.2.p2.13.m1.2.3.3.3.2"><ci id="A2.2.p2.13.m1.1.1.cmml" xref="A2.2.p2.13.m1.1.1">𝑧</ci><ci id="A2.2.p2.13.m1.2.2.cmml" xref="A2.2.p2.13.m1.2.2">𝑟</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.13.m1.2c">z^{\prime}\in B^{p}(z,r)</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.13.m1.2d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_z , italic_r )</annotation></semantics></math>. We conclude that <math alttext="B^{p}(z,r)\cap\mathcal{H}^{p}_{c^{(r)},v}" class="ltx_Math" display="inline" id="A2.2.p2.14.m2.5"><semantics id="A2.2.p2.14.m2.5a"><mrow id="A2.2.p2.14.m2.5.6" xref="A2.2.p2.14.m2.5.6.cmml"><mrow id="A2.2.p2.14.m2.5.6.2" xref="A2.2.p2.14.m2.5.6.2.cmml"><msup id="A2.2.p2.14.m2.5.6.2.2" xref="A2.2.p2.14.m2.5.6.2.2.cmml"><mi id="A2.2.p2.14.m2.5.6.2.2.2" xref="A2.2.p2.14.m2.5.6.2.2.2.cmml">B</mi><mi id="A2.2.p2.14.m2.5.6.2.2.3" xref="A2.2.p2.14.m2.5.6.2.2.3.cmml">p</mi></msup><mo id="A2.2.p2.14.m2.5.6.2.1" xref="A2.2.p2.14.m2.5.6.2.1.cmml">⁢</mo><mrow id="A2.2.p2.14.m2.5.6.2.3.2" xref="A2.2.p2.14.m2.5.6.2.3.1.cmml"><mo id="A2.2.p2.14.m2.5.6.2.3.2.1" stretchy="false" xref="A2.2.p2.14.m2.5.6.2.3.1.cmml">(</mo><mi id="A2.2.p2.14.m2.4.4" xref="A2.2.p2.14.m2.4.4.cmml">z</mi><mo id="A2.2.p2.14.m2.5.6.2.3.2.2" xref="A2.2.p2.14.m2.5.6.2.3.1.cmml">,</mo><mi id="A2.2.p2.14.m2.5.5" xref="A2.2.p2.14.m2.5.5.cmml">r</mi><mo id="A2.2.p2.14.m2.5.6.2.3.2.3" stretchy="false" xref="A2.2.p2.14.m2.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.2.p2.14.m2.5.6.1" xref="A2.2.p2.14.m2.5.6.1.cmml">∩</mo><msubsup id="A2.2.p2.14.m2.5.6.3" xref="A2.2.p2.14.m2.5.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.14.m2.5.6.3.2.2" xref="A2.2.p2.14.m2.5.6.3.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.14.m2.3.3.3.3" xref="A2.2.p2.14.m2.3.3.3.4.cmml"><msup id="A2.2.p2.14.m2.3.3.3.3.1" xref="A2.2.p2.14.m2.3.3.3.3.1.cmml"><mi id="A2.2.p2.14.m2.3.3.3.3.1.2" xref="A2.2.p2.14.m2.3.3.3.3.1.2.cmml">c</mi><mrow id="A2.2.p2.14.m2.1.1.1.1.1.3" xref="A2.2.p2.14.m2.3.3.3.3.1.cmml"><mo id="A2.2.p2.14.m2.1.1.1.1.1.3.1" stretchy="false" xref="A2.2.p2.14.m2.3.3.3.3.1.cmml">(</mo><mi id="A2.2.p2.14.m2.1.1.1.1.1.1" xref="A2.2.p2.14.m2.1.1.1.1.1.1.cmml">r</mi><mo id="A2.2.p2.14.m2.1.1.1.1.1.3.2" stretchy="false" xref="A2.2.p2.14.m2.3.3.3.3.1.cmml">)</mo></mrow></msup><mo id="A2.2.p2.14.m2.3.3.3.3.2" xref="A2.2.p2.14.m2.3.3.3.4.cmml">,</mo><mi id="A2.2.p2.14.m2.2.2.2.2" xref="A2.2.p2.14.m2.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.14.m2.5.6.3.2.3" xref="A2.2.p2.14.m2.5.6.3.2.3.cmml">p</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.14.m2.5b"><apply id="A2.2.p2.14.m2.5.6.cmml" xref="A2.2.p2.14.m2.5.6"><intersect id="A2.2.p2.14.m2.5.6.1.cmml" xref="A2.2.p2.14.m2.5.6.1"></intersect><apply id="A2.2.p2.14.m2.5.6.2.cmml" xref="A2.2.p2.14.m2.5.6.2"><times id="A2.2.p2.14.m2.5.6.2.1.cmml" xref="A2.2.p2.14.m2.5.6.2.1"></times><apply id="A2.2.p2.14.m2.5.6.2.2.cmml" xref="A2.2.p2.14.m2.5.6.2.2"><csymbol cd="ambiguous" id="A2.2.p2.14.m2.5.6.2.2.1.cmml" xref="A2.2.p2.14.m2.5.6.2.2">superscript</csymbol><ci id="A2.2.p2.14.m2.5.6.2.2.2.cmml" xref="A2.2.p2.14.m2.5.6.2.2.2">𝐵</ci><ci id="A2.2.p2.14.m2.5.6.2.2.3.cmml" xref="A2.2.p2.14.m2.5.6.2.2.3">𝑝</ci></apply><interval closure="open" id="A2.2.p2.14.m2.5.6.2.3.1.cmml" xref="A2.2.p2.14.m2.5.6.2.3.2"><ci id="A2.2.p2.14.m2.4.4.cmml" xref="A2.2.p2.14.m2.4.4">𝑧</ci><ci id="A2.2.p2.14.m2.5.5.cmml" xref="A2.2.p2.14.m2.5.5">𝑟</ci></interval></apply><apply id="A2.2.p2.14.m2.5.6.3.cmml" xref="A2.2.p2.14.m2.5.6.3"><csymbol cd="ambiguous" id="A2.2.p2.14.m2.5.6.3.1.cmml" xref="A2.2.p2.14.m2.5.6.3">subscript</csymbol><apply id="A2.2.p2.14.m2.5.6.3.2.cmml" xref="A2.2.p2.14.m2.5.6.3"><csymbol cd="ambiguous" id="A2.2.p2.14.m2.5.6.3.2.1.cmml" xref="A2.2.p2.14.m2.5.6.3">superscript</csymbol><ci id="A2.2.p2.14.m2.5.6.3.2.2.cmml" xref="A2.2.p2.14.m2.5.6.3.2.2">ℋ</ci><ci id="A2.2.p2.14.m2.5.6.3.2.3.cmml" xref="A2.2.p2.14.m2.5.6.3.2.3">𝑝</ci></apply><list id="A2.2.p2.14.m2.3.3.3.4.cmml" xref="A2.2.p2.14.m2.3.3.3.3"><apply id="A2.2.p2.14.m2.3.3.3.3.1.cmml" xref="A2.2.p2.14.m2.3.3.3.3.1"><csymbol cd="ambiguous" id="A2.2.p2.14.m2.3.3.3.3.1.1.cmml" xref="A2.2.p2.14.m2.3.3.3.3.1">superscript</csymbol><ci id="A2.2.p2.14.m2.3.3.3.3.1.2.cmml" xref="A2.2.p2.14.m2.3.3.3.3.1.2">𝑐</ci><ci id="A2.2.p2.14.m2.1.1.1.1.1.1.cmml" xref="A2.2.p2.14.m2.1.1.1.1.1.1">𝑟</ci></apply><ci id="A2.2.p2.14.m2.2.2.2.2.cmml" xref="A2.2.p2.14.m2.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.14.m2.5c">B^{p}(z,r)\cap\mathcal{H}^{p}_{c^{(r)},v}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.14.m2.5d">italic_B start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_z , italic_r ) ∩ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> is empty. In fact, this derivation holds true for all sufficiently small <math alttext="r" class="ltx_Math" display="inline" id="A2.2.p2.15.m3.1"><semantics id="A2.2.p2.15.m3.1a"><mi id="A2.2.p2.15.m3.1.1" xref="A2.2.p2.15.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.15.m3.1b"><ci id="A2.2.p2.15.m3.1.1.cmml" xref="A2.2.p2.15.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.15.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.15.m3.1d">italic_r</annotation></semantics></math>. This means that all balls around points <math alttext="z\in P\setminus\mathcal{H}^{p}_{c,v}" class="ltx_Math" display="inline" id="A2.2.p2.16.m4.2"><semantics id="A2.2.p2.16.m4.2a"><mrow id="A2.2.p2.16.m4.2.3" xref="A2.2.p2.16.m4.2.3.cmml"><mi id="A2.2.p2.16.m4.2.3.2" xref="A2.2.p2.16.m4.2.3.2.cmml">z</mi><mo id="A2.2.p2.16.m4.2.3.1" xref="A2.2.p2.16.m4.2.3.1.cmml">∈</mo><mrow id="A2.2.p2.16.m4.2.3.3" xref="A2.2.p2.16.m4.2.3.3.cmml"><mi id="A2.2.p2.16.m4.2.3.3.2" xref="A2.2.p2.16.m4.2.3.3.2.cmml">P</mi><mo id="A2.2.p2.16.m4.2.3.3.1" xref="A2.2.p2.16.m4.2.3.3.1.cmml">∖</mo><msubsup id="A2.2.p2.16.m4.2.3.3.3" xref="A2.2.p2.16.m4.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.16.m4.2.3.3.3.2.2" xref="A2.2.p2.16.m4.2.3.3.3.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.16.m4.2.2.2.4" xref="A2.2.p2.16.m4.2.2.2.3.cmml"><mi id="A2.2.p2.16.m4.1.1.1.1" xref="A2.2.p2.16.m4.1.1.1.1.cmml">c</mi><mo id="A2.2.p2.16.m4.2.2.2.4.1" xref="A2.2.p2.16.m4.2.2.2.3.cmml">,</mo><mi id="A2.2.p2.16.m4.2.2.2.2" xref="A2.2.p2.16.m4.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.16.m4.2.3.3.3.2.3" xref="A2.2.p2.16.m4.2.3.3.3.2.3.cmml">p</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.16.m4.2b"><apply id="A2.2.p2.16.m4.2.3.cmml" xref="A2.2.p2.16.m4.2.3"><in id="A2.2.p2.16.m4.2.3.1.cmml" xref="A2.2.p2.16.m4.2.3.1"></in><ci id="A2.2.p2.16.m4.2.3.2.cmml" xref="A2.2.p2.16.m4.2.3.2">𝑧</ci><apply id="A2.2.p2.16.m4.2.3.3.cmml" xref="A2.2.p2.16.m4.2.3.3"><setdiff id="A2.2.p2.16.m4.2.3.3.1.cmml" xref="A2.2.p2.16.m4.2.3.3.1"></setdiff><ci id="A2.2.p2.16.m4.2.3.3.2.cmml" xref="A2.2.p2.16.m4.2.3.3.2">𝑃</ci><apply id="A2.2.p2.16.m4.2.3.3.3.cmml" xref="A2.2.p2.16.m4.2.3.3.3"><csymbol cd="ambiguous" id="A2.2.p2.16.m4.2.3.3.3.1.cmml" xref="A2.2.p2.16.m4.2.3.3.3">subscript</csymbol><apply id="A2.2.p2.16.m4.2.3.3.3.2.cmml" xref="A2.2.p2.16.m4.2.3.3.3"><csymbol cd="ambiguous" id="A2.2.p2.16.m4.2.3.3.3.2.1.cmml" xref="A2.2.p2.16.m4.2.3.3.3">superscript</csymbol><ci id="A2.2.p2.16.m4.2.3.3.3.2.2.cmml" xref="A2.2.p2.16.m4.2.3.3.3.2.2">ℋ</ci><ci id="A2.2.p2.16.m4.2.3.3.3.2.3.cmml" xref="A2.2.p2.16.m4.2.3.3.3.2.3">𝑝</ci></apply><list id="A2.2.p2.16.m4.2.2.2.3.cmml" xref="A2.2.p2.16.m4.2.2.2.4"><ci id="A2.2.p2.16.m4.1.1.1.1.cmml" xref="A2.2.p2.16.m4.1.1.1.1">𝑐</ci><ci id="A2.2.p2.16.m4.2.2.2.2.cmml" xref="A2.2.p2.16.m4.2.2.2.2">𝑣</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.16.m4.2c">z\in P\setminus\mathcal{H}^{p}_{c,v}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.16.m4.2d">italic_z ∈ italic_P ∖ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT</annotation></semantics></math> will eventually (for small enough <math alttext="r" class="ltx_Math" display="inline" id="A2.2.p2.17.m5.1"><semantics id="A2.2.p2.17.m5.1a"><mi id="A2.2.p2.17.m5.1.1" xref="A2.2.p2.17.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.17.m5.1b"><ci id="A2.2.p2.17.m5.1.1.cmml" xref="A2.2.p2.17.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.17.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.17.m5.1d">italic_r</annotation></semantics></math>) stop contributing to <math alttext="\mu_{r}(\mathcal{H}^{p}_{c^{(r)},v})" class="ltx_Math" display="inline" id="A2.2.p2.18.m6.4"><semantics id="A2.2.p2.18.m6.4a"><mrow id="A2.2.p2.18.m6.4.4" xref="A2.2.p2.18.m6.4.4.cmml"><msub id="A2.2.p2.18.m6.4.4.3" xref="A2.2.p2.18.m6.4.4.3.cmml"><mi id="A2.2.p2.18.m6.4.4.3.2" xref="A2.2.p2.18.m6.4.4.3.2.cmml">μ</mi><mi id="A2.2.p2.18.m6.4.4.3.3" xref="A2.2.p2.18.m6.4.4.3.3.cmml">r</mi></msub><mo id="A2.2.p2.18.m6.4.4.2" xref="A2.2.p2.18.m6.4.4.2.cmml">⁢</mo><mrow id="A2.2.p2.18.m6.4.4.1.1" xref="A2.2.p2.18.m6.4.4.1.1.1.cmml"><mo id="A2.2.p2.18.m6.4.4.1.1.2" stretchy="false" xref="A2.2.p2.18.m6.4.4.1.1.1.cmml">(</mo><msubsup id="A2.2.p2.18.m6.4.4.1.1.1" xref="A2.2.p2.18.m6.4.4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.18.m6.4.4.1.1.1.2.2" xref="A2.2.p2.18.m6.4.4.1.1.1.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.18.m6.3.3.3.3" xref="A2.2.p2.18.m6.3.3.3.4.cmml"><msup id="A2.2.p2.18.m6.3.3.3.3.1" xref="A2.2.p2.18.m6.3.3.3.3.1.cmml"><mi id="A2.2.p2.18.m6.3.3.3.3.1.2" xref="A2.2.p2.18.m6.3.3.3.3.1.2.cmml">c</mi><mrow id="A2.2.p2.18.m6.1.1.1.1.1.3" xref="A2.2.p2.18.m6.3.3.3.3.1.cmml"><mo id="A2.2.p2.18.m6.1.1.1.1.1.3.1" stretchy="false" xref="A2.2.p2.18.m6.3.3.3.3.1.cmml">(</mo><mi id="A2.2.p2.18.m6.1.1.1.1.1.1" xref="A2.2.p2.18.m6.1.1.1.1.1.1.cmml">r</mi><mo id="A2.2.p2.18.m6.1.1.1.1.1.3.2" stretchy="false" xref="A2.2.p2.18.m6.3.3.3.3.1.cmml">)</mo></mrow></msup><mo id="A2.2.p2.18.m6.3.3.3.3.2" xref="A2.2.p2.18.m6.3.3.3.4.cmml">,</mo><mi id="A2.2.p2.18.m6.2.2.2.2" xref="A2.2.p2.18.m6.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.18.m6.4.4.1.1.1.2.3" xref="A2.2.p2.18.m6.4.4.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A2.2.p2.18.m6.4.4.1.1.3" stretchy="false" xref="A2.2.p2.18.m6.4.4.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.18.m6.4b"><apply id="A2.2.p2.18.m6.4.4.cmml" xref="A2.2.p2.18.m6.4.4"><times id="A2.2.p2.18.m6.4.4.2.cmml" xref="A2.2.p2.18.m6.4.4.2"></times><apply id="A2.2.p2.18.m6.4.4.3.cmml" xref="A2.2.p2.18.m6.4.4.3"><csymbol cd="ambiguous" id="A2.2.p2.18.m6.4.4.3.1.cmml" xref="A2.2.p2.18.m6.4.4.3">subscript</csymbol><ci id="A2.2.p2.18.m6.4.4.3.2.cmml" xref="A2.2.p2.18.m6.4.4.3.2">𝜇</ci><ci id="A2.2.p2.18.m6.4.4.3.3.cmml" xref="A2.2.p2.18.m6.4.4.3.3">𝑟</ci></apply><apply id="A2.2.p2.18.m6.4.4.1.1.1.cmml" xref="A2.2.p2.18.m6.4.4.1.1"><csymbol cd="ambiguous" id="A2.2.p2.18.m6.4.4.1.1.1.1.cmml" xref="A2.2.p2.18.m6.4.4.1.1">subscript</csymbol><apply id="A2.2.p2.18.m6.4.4.1.1.1.2.cmml" xref="A2.2.p2.18.m6.4.4.1.1"><csymbol cd="ambiguous" id="A2.2.p2.18.m6.4.4.1.1.1.2.1.cmml" xref="A2.2.p2.18.m6.4.4.1.1">superscript</csymbol><ci id="A2.2.p2.18.m6.4.4.1.1.1.2.2.cmml" xref="A2.2.p2.18.m6.4.4.1.1.1.2.2">ℋ</ci><ci id="A2.2.p2.18.m6.4.4.1.1.1.2.3.cmml" xref="A2.2.p2.18.m6.4.4.1.1.1.2.3">𝑝</ci></apply><list id="A2.2.p2.18.m6.3.3.3.4.cmml" xref="A2.2.p2.18.m6.3.3.3.3"><apply id="A2.2.p2.18.m6.3.3.3.3.1.cmml" xref="A2.2.p2.18.m6.3.3.3.3.1"><csymbol cd="ambiguous" id="A2.2.p2.18.m6.3.3.3.3.1.1.cmml" xref="A2.2.p2.18.m6.3.3.3.3.1">superscript</csymbol><ci id="A2.2.p2.18.m6.3.3.3.3.1.2.cmml" xref="A2.2.p2.18.m6.3.3.3.3.1.2">𝑐</ci><ci id="A2.2.p2.18.m6.1.1.1.1.1.1.cmml" xref="A2.2.p2.18.m6.1.1.1.1.1.1">𝑟</ci></apply><ci id="A2.2.p2.18.m6.2.2.2.2.cmml" xref="A2.2.p2.18.m6.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.18.m6.4c">\mu_{r}(\mathcal{H}^{p}_{c^{(r)},v})</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.18.m6.4d">italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT )</annotation></semantics></math>. By <math alttext="\mu_{r}(\mathcal{H}^{p}_{c^{(r)},v})\geq\frac{1}{d+1}\mu_{r}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="A2.2.p2.19.m7.5"><semantics id="A2.2.p2.19.m7.5a"><mrow id="A2.2.p2.19.m7.5.5" xref="A2.2.p2.19.m7.5.5.cmml"><mrow id="A2.2.p2.19.m7.4.4.1" xref="A2.2.p2.19.m7.4.4.1.cmml"><msub id="A2.2.p2.19.m7.4.4.1.3" xref="A2.2.p2.19.m7.4.4.1.3.cmml"><mi id="A2.2.p2.19.m7.4.4.1.3.2" xref="A2.2.p2.19.m7.4.4.1.3.2.cmml">μ</mi><mi id="A2.2.p2.19.m7.4.4.1.3.3" xref="A2.2.p2.19.m7.4.4.1.3.3.cmml">r</mi></msub><mo id="A2.2.p2.19.m7.4.4.1.2" xref="A2.2.p2.19.m7.4.4.1.2.cmml">⁢</mo><mrow id="A2.2.p2.19.m7.4.4.1.1.1" xref="A2.2.p2.19.m7.4.4.1.1.1.1.cmml"><mo id="A2.2.p2.19.m7.4.4.1.1.1.2" stretchy="false" xref="A2.2.p2.19.m7.4.4.1.1.1.1.cmml">(</mo><msubsup id="A2.2.p2.19.m7.4.4.1.1.1.1" xref="A2.2.p2.19.m7.4.4.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.19.m7.4.4.1.1.1.1.2.2" xref="A2.2.p2.19.m7.4.4.1.1.1.1.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.19.m7.3.3.3.3" xref="A2.2.p2.19.m7.3.3.3.4.cmml"><msup id="A2.2.p2.19.m7.3.3.3.3.1" xref="A2.2.p2.19.m7.3.3.3.3.1.cmml"><mi id="A2.2.p2.19.m7.3.3.3.3.1.2" xref="A2.2.p2.19.m7.3.3.3.3.1.2.cmml">c</mi><mrow id="A2.2.p2.19.m7.1.1.1.1.1.3" xref="A2.2.p2.19.m7.3.3.3.3.1.cmml"><mo id="A2.2.p2.19.m7.1.1.1.1.1.3.1" stretchy="false" xref="A2.2.p2.19.m7.3.3.3.3.1.cmml">(</mo><mi id="A2.2.p2.19.m7.1.1.1.1.1.1" xref="A2.2.p2.19.m7.1.1.1.1.1.1.cmml">r</mi><mo id="A2.2.p2.19.m7.1.1.1.1.1.3.2" stretchy="false" xref="A2.2.p2.19.m7.3.3.3.3.1.cmml">)</mo></mrow></msup><mo id="A2.2.p2.19.m7.3.3.3.3.2" xref="A2.2.p2.19.m7.3.3.3.4.cmml">,</mo><mi id="A2.2.p2.19.m7.2.2.2.2" xref="A2.2.p2.19.m7.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.19.m7.4.4.1.1.1.1.2.3" xref="A2.2.p2.19.m7.4.4.1.1.1.1.2.3.cmml">p</mi></msubsup><mo id="A2.2.p2.19.m7.4.4.1.1.1.3" stretchy="false" xref="A2.2.p2.19.m7.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.2.p2.19.m7.5.5.3" xref="A2.2.p2.19.m7.5.5.3.cmml">≥</mo><mrow id="A2.2.p2.19.m7.5.5.2" xref="A2.2.p2.19.m7.5.5.2.cmml"><mfrac id="A2.2.p2.19.m7.5.5.2.3" xref="A2.2.p2.19.m7.5.5.2.3.cmml"><mn id="A2.2.p2.19.m7.5.5.2.3.2" xref="A2.2.p2.19.m7.5.5.2.3.2.cmml">1</mn><mrow id="A2.2.p2.19.m7.5.5.2.3.3" xref="A2.2.p2.19.m7.5.5.2.3.3.cmml"><mi id="A2.2.p2.19.m7.5.5.2.3.3.2" xref="A2.2.p2.19.m7.5.5.2.3.3.2.cmml">d</mi><mo id="A2.2.p2.19.m7.5.5.2.3.3.1" xref="A2.2.p2.19.m7.5.5.2.3.3.1.cmml">+</mo><mn id="A2.2.p2.19.m7.5.5.2.3.3.3" xref="A2.2.p2.19.m7.5.5.2.3.3.3.cmml">1</mn></mrow></mfrac><mo id="A2.2.p2.19.m7.5.5.2.2" xref="A2.2.p2.19.m7.5.5.2.2.cmml">⁢</mo><msub id="A2.2.p2.19.m7.5.5.2.4" xref="A2.2.p2.19.m7.5.5.2.4.cmml"><mi id="A2.2.p2.19.m7.5.5.2.4.2" xref="A2.2.p2.19.m7.5.5.2.4.2.cmml">μ</mi><mi id="A2.2.p2.19.m7.5.5.2.4.3" xref="A2.2.p2.19.m7.5.5.2.4.3.cmml">r</mi></msub><mo id="A2.2.p2.19.m7.5.5.2.2a" xref="A2.2.p2.19.m7.5.5.2.2.cmml">⁢</mo><mrow id="A2.2.p2.19.m7.5.5.2.1.1" xref="A2.2.p2.19.m7.5.5.2.1.1.1.cmml"><mo id="A2.2.p2.19.m7.5.5.2.1.1.2" stretchy="false" xref="A2.2.p2.19.m7.5.5.2.1.1.1.cmml">(</mo><msup id="A2.2.p2.19.m7.5.5.2.1.1.1" xref="A2.2.p2.19.m7.5.5.2.1.1.1.cmml"><mi id="A2.2.p2.19.m7.5.5.2.1.1.1.2" xref="A2.2.p2.19.m7.5.5.2.1.1.1.2.cmml">ℝ</mi><mi id="A2.2.p2.19.m7.5.5.2.1.1.1.3" xref="A2.2.p2.19.m7.5.5.2.1.1.1.3.cmml">d</mi></msup><mo id="A2.2.p2.19.m7.5.5.2.1.1.3" stretchy="false" xref="A2.2.p2.19.m7.5.5.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.19.m7.5b"><apply id="A2.2.p2.19.m7.5.5.cmml" xref="A2.2.p2.19.m7.5.5"><geq id="A2.2.p2.19.m7.5.5.3.cmml" xref="A2.2.p2.19.m7.5.5.3"></geq><apply id="A2.2.p2.19.m7.4.4.1.cmml" xref="A2.2.p2.19.m7.4.4.1"><times id="A2.2.p2.19.m7.4.4.1.2.cmml" xref="A2.2.p2.19.m7.4.4.1.2"></times><apply id="A2.2.p2.19.m7.4.4.1.3.cmml" xref="A2.2.p2.19.m7.4.4.1.3"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.4.4.1.3.1.cmml" xref="A2.2.p2.19.m7.4.4.1.3">subscript</csymbol><ci id="A2.2.p2.19.m7.4.4.1.3.2.cmml" xref="A2.2.p2.19.m7.4.4.1.3.2">𝜇</ci><ci id="A2.2.p2.19.m7.4.4.1.3.3.cmml" xref="A2.2.p2.19.m7.4.4.1.3.3">𝑟</ci></apply><apply id="A2.2.p2.19.m7.4.4.1.1.1.1.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.4.4.1.1.1.1.1.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1">subscript</csymbol><apply id="A2.2.p2.19.m7.4.4.1.1.1.1.2.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.4.4.1.1.1.1.2.1.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1">superscript</csymbol><ci id="A2.2.p2.19.m7.4.4.1.1.1.1.2.2.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1.1.2.2">ℋ</ci><ci id="A2.2.p2.19.m7.4.4.1.1.1.1.2.3.cmml" xref="A2.2.p2.19.m7.4.4.1.1.1.1.2.3">𝑝</ci></apply><list id="A2.2.p2.19.m7.3.3.3.4.cmml" xref="A2.2.p2.19.m7.3.3.3.3"><apply id="A2.2.p2.19.m7.3.3.3.3.1.cmml" xref="A2.2.p2.19.m7.3.3.3.3.1"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.3.3.3.3.1.1.cmml" xref="A2.2.p2.19.m7.3.3.3.3.1">superscript</csymbol><ci id="A2.2.p2.19.m7.3.3.3.3.1.2.cmml" xref="A2.2.p2.19.m7.3.3.3.3.1.2">𝑐</ci><ci id="A2.2.p2.19.m7.1.1.1.1.1.1.cmml" xref="A2.2.p2.19.m7.1.1.1.1.1.1">𝑟</ci></apply><ci id="A2.2.p2.19.m7.2.2.2.2.cmml" xref="A2.2.p2.19.m7.2.2.2.2">𝑣</ci></list></apply></apply><apply id="A2.2.p2.19.m7.5.5.2.cmml" xref="A2.2.p2.19.m7.5.5.2"><times id="A2.2.p2.19.m7.5.5.2.2.cmml" xref="A2.2.p2.19.m7.5.5.2.2"></times><apply id="A2.2.p2.19.m7.5.5.2.3.cmml" xref="A2.2.p2.19.m7.5.5.2.3"><divide id="A2.2.p2.19.m7.5.5.2.3.1.cmml" xref="A2.2.p2.19.m7.5.5.2.3"></divide><cn id="A2.2.p2.19.m7.5.5.2.3.2.cmml" type="integer" xref="A2.2.p2.19.m7.5.5.2.3.2">1</cn><apply id="A2.2.p2.19.m7.5.5.2.3.3.cmml" xref="A2.2.p2.19.m7.5.5.2.3.3"><plus id="A2.2.p2.19.m7.5.5.2.3.3.1.cmml" xref="A2.2.p2.19.m7.5.5.2.3.3.1"></plus><ci id="A2.2.p2.19.m7.5.5.2.3.3.2.cmml" xref="A2.2.p2.19.m7.5.5.2.3.3.2">𝑑</ci><cn id="A2.2.p2.19.m7.5.5.2.3.3.3.cmml" type="integer" xref="A2.2.p2.19.m7.5.5.2.3.3.3">1</cn></apply></apply><apply id="A2.2.p2.19.m7.5.5.2.4.cmml" xref="A2.2.p2.19.m7.5.5.2.4"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.5.5.2.4.1.cmml" xref="A2.2.p2.19.m7.5.5.2.4">subscript</csymbol><ci id="A2.2.p2.19.m7.5.5.2.4.2.cmml" xref="A2.2.p2.19.m7.5.5.2.4.2">𝜇</ci><ci id="A2.2.p2.19.m7.5.5.2.4.3.cmml" xref="A2.2.p2.19.m7.5.5.2.4.3">𝑟</ci></apply><apply id="A2.2.p2.19.m7.5.5.2.1.1.1.cmml" xref="A2.2.p2.19.m7.5.5.2.1.1"><csymbol cd="ambiguous" id="A2.2.p2.19.m7.5.5.2.1.1.1.1.cmml" xref="A2.2.p2.19.m7.5.5.2.1.1">superscript</csymbol><ci id="A2.2.p2.19.m7.5.5.2.1.1.1.2.cmml" xref="A2.2.p2.19.m7.5.5.2.1.1.1.2">ℝ</ci><ci id="A2.2.p2.19.m7.5.5.2.1.1.1.3.cmml" xref="A2.2.p2.19.m7.5.5.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.19.m7.5c">\mu_{r}(\mathcal{H}^{p}_{c^{(r)},v})\geq\frac{1}{d+1}\mu_{r}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.19.m7.5d">italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT , italic_v end_POSTSUBSCRIPT ) ≥ divide start_ARG 1 end_ARG start_ARG italic_d + 1 end_ARG italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>, we conclude that there can be at most <math alttext="\frac{d}{d+1}|P|" class="ltx_Math" display="inline" id="A2.2.p2.20.m8.1"><semantics id="A2.2.p2.20.m8.1a"><mrow id="A2.2.p2.20.m8.1.2" xref="A2.2.p2.20.m8.1.2.cmml"><mfrac id="A2.2.p2.20.m8.1.2.2" xref="A2.2.p2.20.m8.1.2.2.cmml"><mi id="A2.2.p2.20.m8.1.2.2.2" xref="A2.2.p2.20.m8.1.2.2.2.cmml">d</mi><mrow id="A2.2.p2.20.m8.1.2.2.3" xref="A2.2.p2.20.m8.1.2.2.3.cmml"><mi id="A2.2.p2.20.m8.1.2.2.3.2" xref="A2.2.p2.20.m8.1.2.2.3.2.cmml">d</mi><mo id="A2.2.p2.20.m8.1.2.2.3.1" xref="A2.2.p2.20.m8.1.2.2.3.1.cmml">+</mo><mn id="A2.2.p2.20.m8.1.2.2.3.3" xref="A2.2.p2.20.m8.1.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="A2.2.p2.20.m8.1.2.1" xref="A2.2.p2.20.m8.1.2.1.cmml">⁢</mo><mrow id="A2.2.p2.20.m8.1.2.3.2" xref="A2.2.p2.20.m8.1.2.3.1.cmml"><mo id="A2.2.p2.20.m8.1.2.3.2.1" stretchy="false" xref="A2.2.p2.20.m8.1.2.3.1.1.cmml">|</mo><mi id="A2.2.p2.20.m8.1.1" xref="A2.2.p2.20.m8.1.1.cmml">P</mi><mo id="A2.2.p2.20.m8.1.2.3.2.2" stretchy="false" xref="A2.2.p2.20.m8.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.20.m8.1b"><apply id="A2.2.p2.20.m8.1.2.cmml" xref="A2.2.p2.20.m8.1.2"><times id="A2.2.p2.20.m8.1.2.1.cmml" xref="A2.2.p2.20.m8.1.2.1"></times><apply id="A2.2.p2.20.m8.1.2.2.cmml" xref="A2.2.p2.20.m8.1.2.2"><divide id="A2.2.p2.20.m8.1.2.2.1.cmml" xref="A2.2.p2.20.m8.1.2.2"></divide><ci id="A2.2.p2.20.m8.1.2.2.2.cmml" xref="A2.2.p2.20.m8.1.2.2.2">𝑑</ci><apply id="A2.2.p2.20.m8.1.2.2.3.cmml" xref="A2.2.p2.20.m8.1.2.2.3"><plus id="A2.2.p2.20.m8.1.2.2.3.1.cmml" xref="A2.2.p2.20.m8.1.2.2.3.1"></plus><ci id="A2.2.p2.20.m8.1.2.2.3.2.cmml" xref="A2.2.p2.20.m8.1.2.2.3.2">𝑑</ci><cn id="A2.2.p2.20.m8.1.2.2.3.3.cmml" type="integer" xref="A2.2.p2.20.m8.1.2.2.3.3">1</cn></apply></apply><apply id="A2.2.p2.20.m8.1.2.3.1.cmml" xref="A2.2.p2.20.m8.1.2.3.2"><abs id="A2.2.p2.20.m8.1.2.3.1.1.cmml" xref="A2.2.p2.20.m8.1.2.3.2.1"></abs><ci id="A2.2.p2.20.m8.1.1.cmml" xref="A2.2.p2.20.m8.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.20.m8.1c">\frac{d}{d+1}|P|</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.20.m8.1d">divide start_ARG italic_d end_ARG start_ARG italic_d + 1 end_ARG | italic_P |</annotation></semantics></math> such points <math alttext="z" class="ltx_Math" display="inline" id="A2.2.p2.21.m9.1"><semantics id="A2.2.p2.21.m9.1a"><mi id="A2.2.p2.21.m9.1.1" xref="A2.2.p2.21.m9.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="A2.2.p2.21.m9.1b"><ci id="A2.2.p2.21.m9.1.1.cmml" xref="A2.2.p2.21.m9.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.21.m9.1c">z</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.21.m9.1d">italic_z</annotation></semantics></math> and thus we get <math alttext="|P\cap\mathcal{H}^{p}_{c,v}|\geq\frac{|P|}{d+1}" class="ltx_Math" display="inline" id="A2.2.p2.22.m10.4"><semantics id="A2.2.p2.22.m10.4a"><mrow id="A2.2.p2.22.m10.4.4" xref="A2.2.p2.22.m10.4.4.cmml"><mrow id="A2.2.p2.22.m10.4.4.1.1" xref="A2.2.p2.22.m10.4.4.1.2.cmml"><mo id="A2.2.p2.22.m10.4.4.1.1.2" stretchy="false" xref="A2.2.p2.22.m10.4.4.1.2.1.cmml">|</mo><mrow id="A2.2.p2.22.m10.4.4.1.1.1" xref="A2.2.p2.22.m10.4.4.1.1.1.cmml"><mi id="A2.2.p2.22.m10.4.4.1.1.1.2" xref="A2.2.p2.22.m10.4.4.1.1.1.2.cmml">P</mi><mo id="A2.2.p2.22.m10.4.4.1.1.1.1" xref="A2.2.p2.22.m10.4.4.1.1.1.1.cmml">∩</mo><msubsup id="A2.2.p2.22.m10.4.4.1.1.1.3" xref="A2.2.p2.22.m10.4.4.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.2.p2.22.m10.4.4.1.1.1.3.2.2" xref="A2.2.p2.22.m10.4.4.1.1.1.3.2.2.cmml">ℋ</mi><mrow id="A2.2.p2.22.m10.2.2.2.4" xref="A2.2.p2.22.m10.2.2.2.3.cmml"><mi id="A2.2.p2.22.m10.1.1.1.1" xref="A2.2.p2.22.m10.1.1.1.1.cmml">c</mi><mo id="A2.2.p2.22.m10.2.2.2.4.1" xref="A2.2.p2.22.m10.2.2.2.3.cmml">,</mo><mi id="A2.2.p2.22.m10.2.2.2.2" xref="A2.2.p2.22.m10.2.2.2.2.cmml">v</mi></mrow><mi id="A2.2.p2.22.m10.4.4.1.1.1.3.2.3" xref="A2.2.p2.22.m10.4.4.1.1.1.3.2.3.cmml">p</mi></msubsup></mrow><mo id="A2.2.p2.22.m10.4.4.1.1.3" stretchy="false" xref="A2.2.p2.22.m10.4.4.1.2.1.cmml">|</mo></mrow><mo id="A2.2.p2.22.m10.4.4.2" xref="A2.2.p2.22.m10.4.4.2.cmml">≥</mo><mfrac id="A2.2.p2.22.m10.3.3" xref="A2.2.p2.22.m10.3.3.cmml"><mrow id="A2.2.p2.22.m10.3.3.1.3" xref="A2.2.p2.22.m10.3.3.1.2.cmml"><mo id="A2.2.p2.22.m10.3.3.1.3.1" stretchy="false" xref="A2.2.p2.22.m10.3.3.1.2.1.cmml">|</mo><mi id="A2.2.p2.22.m10.3.3.1.1" xref="A2.2.p2.22.m10.3.3.1.1.cmml">P</mi><mo id="A2.2.p2.22.m10.3.3.1.3.2" stretchy="false" xref="A2.2.p2.22.m10.3.3.1.2.1.cmml">|</mo></mrow><mrow id="A2.2.p2.22.m10.3.3.3" xref="A2.2.p2.22.m10.3.3.3.cmml"><mi id="A2.2.p2.22.m10.3.3.3.2" xref="A2.2.p2.22.m10.3.3.3.2.cmml">d</mi><mo id="A2.2.p2.22.m10.3.3.3.1" xref="A2.2.p2.22.m10.3.3.3.1.cmml">+</mo><mn id="A2.2.p2.22.m10.3.3.3.3" xref="A2.2.p2.22.m10.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A2.2.p2.22.m10.4b"><apply id="A2.2.p2.22.m10.4.4.cmml" xref="A2.2.p2.22.m10.4.4"><geq id="A2.2.p2.22.m10.4.4.2.cmml" xref="A2.2.p2.22.m10.4.4.2"></geq><apply id="A2.2.p2.22.m10.4.4.1.2.cmml" xref="A2.2.p2.22.m10.4.4.1.1"><abs id="A2.2.p2.22.m10.4.4.1.2.1.cmml" xref="A2.2.p2.22.m10.4.4.1.1.2"></abs><apply id="A2.2.p2.22.m10.4.4.1.1.1.cmml" xref="A2.2.p2.22.m10.4.4.1.1.1"><intersect id="A2.2.p2.22.m10.4.4.1.1.1.1.cmml" xref="A2.2.p2.22.m10.4.4.1.1.1.1"></intersect><ci id="A2.2.p2.22.m10.4.4.1.1.1.2.cmml" xref="A2.2.p2.22.m10.4.4.1.1.1.2">𝑃</ci><apply id="A2.2.p2.22.m10.4.4.1.1.1.3.cmml" xref="A2.2.p2.22.m10.4.4.1.1.1.3"><csymbol cd="ambiguous" 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xref="A2.2.p2.22.m10.3.3.1.3.1"></abs><ci id="A2.2.p2.22.m10.3.3.1.1.cmml" xref="A2.2.p2.22.m10.3.3.1.1">𝑃</ci></apply><apply id="A2.2.p2.22.m10.3.3.3.cmml" xref="A2.2.p2.22.m10.3.3.3"><plus id="A2.2.p2.22.m10.3.3.3.1.cmml" xref="A2.2.p2.22.m10.3.3.3.1"></plus><ci id="A2.2.p2.22.m10.3.3.3.2.cmml" xref="A2.2.p2.22.m10.3.3.3.2">𝑑</ci><cn id="A2.2.p2.22.m10.3.3.3.3.cmml" type="integer" xref="A2.2.p2.22.m10.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.2.p2.22.m10.4c">|P\cap\mathcal{H}^{p}_{c,v}|\geq\frac{|P|}{d+1}</annotation><annotation encoding="application/x-llamapun" id="A2.2.p2.22.m10.4d">| italic_P ∩ caligraphic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c , italic_v end_POSTSUBSCRIPT | ≥ divide start_ARG | italic_P | end_ARG start_ARG italic_d + 1 end_ARG</annotation></semantics></math>, as desired. ∎</p> </div> </div> </section> </article> </div> <footer class="ltx_page_footer"> <div 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