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<!DOCTYPE html> <html lang="en" prefix="dcterms: http://purl.org/dc/terms/"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²</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/2503.13001v1/"/></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/2503.13001v1#S1" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><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/2503.13001v1#S1.SS1" title="In 1 Introduction ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Related Work</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Definitions</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/2503.13001v1#S2.SS1" title="In 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Motivation for a Non-Standard Definition of Polygons</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS2" title="In 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Polygons</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS3" title="In 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Continuous Piecewise Affine Functions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS4" title="In 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Neural Networks</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Decomposition of CPA Functions</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/2503.13001v1#S3.SS1" title="In 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Proof of <span class="ltx_text ltx_ref_tag">3.2</span></span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS1" title="In 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1.1 </span>Only one cycle</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS2" title="In 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1.2 </span>Only arcs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS3" title="In 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1.3 </span>General case</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>max-Representation of CPA Functions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S5" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Neural Network Representation of CPA Functions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S6" title="In Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Discussion</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/2503.13001v1#S6.SS1" title="In 6 Discussion ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Comparison to the Result of <span class="ltx_ERROR undefined">\citet</span>Koutschan2023</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S6.SS2" title="In 6 Discussion ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Comparison with Convex Pieces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S6.SS3" title="In 6 Discussion ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.3 </span>Extension to Higher Dimensions</span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\addbibresource</span> <p class="ltx_p" id="p1.2">ref.bib </p> </div> <h1 class="ltx_title ltx_title_document">Linear-Size Neural Network Representation of Piecewise Affine Functions in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="id1.m1.1"><semantics id="id1.m1.1b"><msup id="id1.m1.1.1" xref="id1.m1.1.1.cmml"><mi id="id1.m1.1.1.2" xref="id1.m1.1.1.2.cmml">ℝ</mi><mn id="id1.m1.1.1.3" xref="id1.m1.1.1.3.cmml">2</mn></msup><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">superscript</csymbol><ci id="id1.m1.1.1.2.cmml" xref="id1.m1.1.1.2">ℝ</ci><cn id="id1.m1.1.1.3.cmml" type="integer" xref="id1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.m1.1d">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="id1.m1.1e">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> <a class="ltx_ref ltx_href" href="https://orcid.org/0009-0001-9695-3812" title=""><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="8" id="id2.1.1.g1" src="x1.png" width="8"/> Leo Zanotti</a> <br class="ltx_break"/>Institute of Optimization and Operations Research <br class="ltx_break"/>Ulm University, Germany <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id6.2.id1">leo.zanotti@uni-ulm.de</span> </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id5.3">It is shown that any continuous piecewise affine (CPA) function <math alttext="\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="id3.1.m1.1"><semantics id="id3.1.m1.1a"><mrow id="id3.1.m1.1.1" xref="id3.1.m1.1.1.cmml"><msup id="id3.1.m1.1.1.2" xref="id3.1.m1.1.1.2.cmml"><mi id="id3.1.m1.1.1.2.2" xref="id3.1.m1.1.1.2.2.cmml">ℝ</mi><mn id="id3.1.m1.1.1.2.3" xref="id3.1.m1.1.1.2.3.cmml">2</mn></msup><mo id="id3.1.m1.1.1.1" stretchy="false" xref="id3.1.m1.1.1.1.cmml">→</mo><mi id="id3.1.m1.1.1.3" xref="id3.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="id3.1.m1.1b"><apply id="id3.1.m1.1.1.cmml" xref="id3.1.m1.1.1"><ci id="id3.1.m1.1.1.1.cmml" xref="id3.1.m1.1.1.1">→</ci><apply id="id3.1.m1.1.1.2.cmml" xref="id3.1.m1.1.1.2"><csymbol cd="ambiguous" id="id3.1.m1.1.1.2.1.cmml" xref="id3.1.m1.1.1.2">superscript</csymbol><ci id="id3.1.m1.1.1.2.2.cmml" xref="id3.1.m1.1.1.2.2">ℝ</ci><cn id="id3.1.m1.1.1.2.3.cmml" type="integer" xref="id3.1.m1.1.1.2.3">2</cn></apply><ci id="id3.1.m1.1.1.3.cmml" xref="id3.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.1.m1.1c">\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="id3.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> with <math alttext="p" class="ltx_Math" display="inline" id="id4.2.m2.1"><semantics id="id4.2.m2.1a"><mi id="id4.2.m2.1.1" xref="id4.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="id4.2.m2.1b"><ci id="id4.2.m2.1.1.cmml" xref="id4.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="id4.2.m2.1d">italic_p</annotation></semantics></math> pieces can be represented by a ReLU neural network with two hidden layers and <math alttext="O(p)" class="ltx_Math" display="inline" id="id5.3.m3.1"><semantics id="id5.3.m3.1a"><mrow id="id5.3.m3.1.2" xref="id5.3.m3.1.2.cmml"><mi id="id5.3.m3.1.2.2" xref="id5.3.m3.1.2.2.cmml">O</mi><mo id="id5.3.m3.1.2.1" xref="id5.3.m3.1.2.1.cmml">⁢</mo><mrow id="id5.3.m3.1.2.3.2" xref="id5.3.m3.1.2.cmml"><mo id="id5.3.m3.1.2.3.2.1" stretchy="false" xref="id5.3.m3.1.2.cmml">(</mo><mi id="id5.3.m3.1.1" xref="id5.3.m3.1.1.cmml">p</mi><mo id="id5.3.m3.1.2.3.2.2" stretchy="false" xref="id5.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id5.3.m3.1b"><apply id="id5.3.m3.1.2.cmml" xref="id5.3.m3.1.2"><times id="id5.3.m3.1.2.1.cmml" xref="id5.3.m3.1.2.1"></times><ci id="id5.3.m3.1.2.2.cmml" xref="id5.3.m3.1.2.2">𝑂</ci><ci id="id5.3.m3.1.1.cmml" xref="id5.3.m3.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.3.m3.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="id5.3.m3.1d">italic_O ( italic_p )</annotation></semantics></math> neurons. Unlike prior work, which focused on convex pieces, this analysis considers CPA functions with connected but potentially non-convex pieces.</p> </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.p1.2">One way to assess a neural network’s ability to adapt to complex data is to examine the functions it can compute. This paper considers this question for neural networks with the Rectified Linear Unit (<math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">roman_ReLU</annotation></semantics></math>) activation function, focusing on networks that map from <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><msup id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mi id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">ℝ</mi><mn id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p1.2.m2.1.1.1.cmml" xref="S1.p1.2.m2.1.1">superscript</csymbol><ci id="S1.p1.2.m2.1.1.2.cmml" xref="S1.p1.2.m2.1.1.2">ℝ</ci><cn id="S1.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.2">As compositions of the piecewise linear <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">roman_ReLU</annotation></semantics></math> and affine functions, the input-output mappings of these networks are described by continuous piecewise affine (CPA) functions. That is, for every neural network, there exists a finite cover of the input space such that the restriction of the function to any set in the cover is affine. In this work, these sets are called pieces and must have connected interiors, though they may be non-convex and contain holes. The main contribution of this work is a linear bound on the size of a neural network architecture required to represent a CPA function in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><msup id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">ℝ</mi><mn id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1">superscript</csymbol><ci id="S1.p2.2.m2.1.1.2.cmml" xref="S1.p2.2.m2.1.1.2">ℝ</ci><cn id="S1.p2.2.m2.1.1.3.cmml" type="integer" xref="S1.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with a given number of pieces.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem1.1.1.1">Theorem 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.4.4">Any continuous piecewise affine function <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem1.p1.1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S1.Thmtheorem1.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><msup id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml"><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.2" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mn id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.3" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mo id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.1.m1.1b"><apply id="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1"><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3"><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.2">ℝ</ci><cn id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.2.3">2</cn></apply><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.1.m1.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> with <math alttext="p" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.2.m2.1a"><mi id="S1.Thmtheorem1.p1.2.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.2.m2.1b"><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.2.m2.1d">italic_p</annotation></semantics></math> pieces can be represented by a <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.3.3.m3.1"><semantics id="S1.Thmtheorem1.p1.3.3.m3.1a"><mi id="S1.Thmtheorem1.p1.3.3.m3.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.3.3.m3.1b"><ci id="S1.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.3.3.m3.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.3.3.m3.1d">roman_ReLU</annotation></semantics></math> neural network with two hidden layers and <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.4.4.m4.1"><semantics id="S1.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem1.p1.4.4.m4.1.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mi id="S1.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2.cmml">O</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml">(</mo><mi id="S1.Thmtheorem1.p1.4.4.m4.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml">p</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.4.4.m4.1b"><apply id="S1.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2"><times id="S1.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.1"></times><ci id="S1.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.2.2">𝑂</ci><ci id="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.4.4.m4.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.4.4.m4.1d">italic_O ( italic_p )</annotation></semantics></math> neurons.</span></p> </div> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.3">The constructed network is highly sparse, with the number of parameters also scaling as <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.2" xref="S1.p3.1.m1.1.2.cmml"><mi id="S1.p3.1.m1.1.2.2" xref="S1.p3.1.m1.1.2.2.cmml">O</mi><mo id="S1.p3.1.m1.1.2.1" xref="S1.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.1.2.3.2" xref="S1.p3.1.m1.1.2.cmml"><mo id="S1.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S1.p3.1.m1.1.2.cmml">(</mo><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">p</mi><mo id="S1.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S1.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.2.cmml" xref="S1.p3.1.m1.1.2"><times id="S1.p3.1.m1.1.2.1.cmml" xref="S1.p3.1.m1.1.2.1"></times><ci id="S1.p3.1.m1.1.2.2.cmml" xref="S1.p3.1.m1.1.2.2">𝑂</ci><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_O ( italic_p )</annotation></semantics></math>. It seems unlikely that fewer than <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.2" xref="S1.p3.2.m2.1.2.cmml"><mi id="S1.p3.2.m2.1.2.2" xref="S1.p3.2.m2.1.2.2.cmml">O</mi><mo id="S1.p3.2.m2.1.2.1" xref="S1.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.1.2.3.2" xref="S1.p3.2.m2.1.2.cmml"><mo id="S1.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S1.p3.2.m2.1.2.cmml">(</mo><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">p</mi><mo id="S1.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S1.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.2.cmml" xref="S1.p3.2.m2.1.2"><times id="S1.p3.2.m2.1.2.1.cmml" xref="S1.p3.2.m2.1.2.1"></times><ci id="S1.p3.2.m2.1.2.2.cmml" xref="S1.p3.2.m2.1.2.2">𝑂</ci><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_O ( italic_p )</annotation></semantics></math> parameters suffice to represent every CPA function with <math alttext="p" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_p</annotation></semantics></math> pieces. This suggests that, for this task in two dimensions, networks with more than two hidden layers are not more powerful than those with two hidden layers.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.2">To prove <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1 Introduction ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 1.1</span></a>, a representation of CPA functions in terms of maxima is introduced. A univariate CPA function <math alttext="f:\mathds{R}\to\mathds{R}" 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">f</mi><mo id="S1.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml"><mi id="S1.p4.1.m1.1.1.3.2" xref="S1.p4.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="S1.p4.1.m1.1.1.3.1" stretchy="false" xref="S1.p4.1.m1.1.1.3.1.cmml">→</mo><mi id="S1.p4.1.m1.1.1.3.3" xref="S1.p4.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></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"><ci id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1">:</ci><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝑓</ci><apply id="S1.p4.1.m1.1.1.3.cmml" xref="S1.p4.1.m1.1.1.3"><ci id="S1.p4.1.m1.1.1.3.1.cmml" xref="S1.p4.1.m1.1.1.3.1">→</ci><ci id="S1.p4.1.m1.1.1.3.2.cmml" xref="S1.p4.1.m1.1.1.3.2">ℝ</ci><ci id="S1.p4.1.m1.1.1.3.3.cmml" xref="S1.p4.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">f:\mathds{R}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_f : blackboard_R → blackboard_R</annotation></semantics></math> defined over <math alttext="p" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_p</annotation></semantics></math> intervals can be expressed as</p> <table class="ltx_equation ltx_eqn_table" 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id="S1.Ex1.m1.3.3.1.1.2.2.2.2.2.2.3.2.3.cmml" xref="S1.Ex1.m1.3.3.1.1.2.2.2.2.2.2.3.2.3">′</ci></apply><ci id="S1.Ex1.m1.3.3.1.1.2.2.2.2.2.2.3.3.cmml" xref="S1.Ex1.m1.3.3.1.1.2.2.2.2.2.2.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex1.m1.3c">f(x)=\sum_{i=1}^{p-1}\sigma_{i}\max(a_{i}x+b_{i},a_{i}^{\prime}x+b^{\prime}_{i% }),</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.3d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p - 1 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_max ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x + italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_x + italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 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="S1.p4.4">where <math alttext="\sigma_{i}\in\{-1,1\}" class="ltx_Math" display="inline" id="S1.p4.3.m1.2"><semantics id="S1.p4.3.m1.2a"><mrow id="S1.p4.3.m1.2.2" xref="S1.p4.3.m1.2.2.cmml"><msub id="S1.p4.3.m1.2.2.3" xref="S1.p4.3.m1.2.2.3.cmml"><mi id="S1.p4.3.m1.2.2.3.2" xref="S1.p4.3.m1.2.2.3.2.cmml">σ</mi><mi id="S1.p4.3.m1.2.2.3.3" xref="S1.p4.3.m1.2.2.3.3.cmml">i</mi></msub><mo id="S1.p4.3.m1.2.2.2" xref="S1.p4.3.m1.2.2.2.cmml">∈</mo><mrow id="S1.p4.3.m1.2.2.1.1" xref="S1.p4.3.m1.2.2.1.2.cmml"><mo id="S1.p4.3.m1.2.2.1.1.2" stretchy="false" xref="S1.p4.3.m1.2.2.1.2.cmml">{</mo><mrow id="S1.p4.3.m1.2.2.1.1.1" xref="S1.p4.3.m1.2.2.1.1.1.cmml"><mo id="S1.p4.3.m1.2.2.1.1.1a" xref="S1.p4.3.m1.2.2.1.1.1.cmml">−</mo><mn id="S1.p4.3.m1.2.2.1.1.1.2" xref="S1.p4.3.m1.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S1.p4.3.m1.2.2.1.1.3" xref="S1.p4.3.m1.2.2.1.2.cmml">,</mo><mn id="S1.p4.3.m1.1.1" xref="S1.p4.3.m1.1.1.cmml">1</mn><mo id="S1.p4.3.m1.2.2.1.1.4" stretchy="false" xref="S1.p4.3.m1.2.2.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.3.m1.2b"><apply id="S1.p4.3.m1.2.2.cmml" xref="S1.p4.3.m1.2.2"><in id="S1.p4.3.m1.2.2.2.cmml" xref="S1.p4.3.m1.2.2.2"></in><apply id="S1.p4.3.m1.2.2.3.cmml" xref="S1.p4.3.m1.2.2.3"><csymbol cd="ambiguous" id="S1.p4.3.m1.2.2.3.1.cmml" xref="S1.p4.3.m1.2.2.3">subscript</csymbol><ci id="S1.p4.3.m1.2.2.3.2.cmml" xref="S1.p4.3.m1.2.2.3.2">𝜎</ci><ci id="S1.p4.3.m1.2.2.3.3.cmml" xref="S1.p4.3.m1.2.2.3.3">𝑖</ci></apply><set id="S1.p4.3.m1.2.2.1.2.cmml" xref="S1.p4.3.m1.2.2.1.1"><apply id="S1.p4.3.m1.2.2.1.1.1.cmml" xref="S1.p4.3.m1.2.2.1.1.1"><minus id="S1.p4.3.m1.2.2.1.1.1.1.cmml" xref="S1.p4.3.m1.2.2.1.1.1"></minus><cn id="S1.p4.3.m1.2.2.1.1.1.2.cmml" type="integer" xref="S1.p4.3.m1.2.2.1.1.1.2">1</cn></apply><cn id="S1.p4.3.m1.1.1.cmml" type="integer" xref="S1.p4.3.m1.1.1">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m1.2c">\sigma_{i}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m1.2d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math> and <math alttext="a_{i},a_{i}^{\prime},b_{i},b_{i}^{\prime}\in\mathds{R}" class="ltx_Math" display="inline" id="S1.p4.4.m2.4"><semantics id="S1.p4.4.m2.4a"><mrow id="S1.p4.4.m2.4.4" xref="S1.p4.4.m2.4.4.cmml"><mrow id="S1.p4.4.m2.4.4.4.4" xref="S1.p4.4.m2.4.4.4.5.cmml"><msub id="S1.p4.4.m2.1.1.1.1.1" xref="S1.p4.4.m2.1.1.1.1.1.cmml"><mi id="S1.p4.4.m2.1.1.1.1.1.2" xref="S1.p4.4.m2.1.1.1.1.1.2.cmml">a</mi><mi id="S1.p4.4.m2.1.1.1.1.1.3" xref="S1.p4.4.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S1.p4.4.m2.4.4.4.4.5" xref="S1.p4.4.m2.4.4.4.5.cmml">,</mo><msubsup id="S1.p4.4.m2.2.2.2.2.2" xref="S1.p4.4.m2.2.2.2.2.2.cmml"><mi id="S1.p4.4.m2.2.2.2.2.2.2.2" xref="S1.p4.4.m2.2.2.2.2.2.2.2.cmml">a</mi><mi id="S1.p4.4.m2.2.2.2.2.2.2.3" xref="S1.p4.4.m2.2.2.2.2.2.2.3.cmml">i</mi><mo id="S1.p4.4.m2.2.2.2.2.2.3" xref="S1.p4.4.m2.2.2.2.2.2.3.cmml">′</mo></msubsup><mo id="S1.p4.4.m2.4.4.4.4.6" xref="S1.p4.4.m2.4.4.4.5.cmml">,</mo><msub id="S1.p4.4.m2.3.3.3.3.3" xref="S1.p4.4.m2.3.3.3.3.3.cmml"><mi id="S1.p4.4.m2.3.3.3.3.3.2" xref="S1.p4.4.m2.3.3.3.3.3.2.cmml">b</mi><mi id="S1.p4.4.m2.3.3.3.3.3.3" xref="S1.p4.4.m2.3.3.3.3.3.3.cmml">i</mi></msub><mo id="S1.p4.4.m2.4.4.4.4.7" xref="S1.p4.4.m2.4.4.4.5.cmml">,</mo><msubsup id="S1.p4.4.m2.4.4.4.4.4" xref="S1.p4.4.m2.4.4.4.4.4.cmml"><mi id="S1.p4.4.m2.4.4.4.4.4.2.2" xref="S1.p4.4.m2.4.4.4.4.4.2.2.cmml">b</mi><mi id="S1.p4.4.m2.4.4.4.4.4.2.3" xref="S1.p4.4.m2.4.4.4.4.4.2.3.cmml">i</mi><mo id="S1.p4.4.m2.4.4.4.4.4.3" xref="S1.p4.4.m2.4.4.4.4.4.3.cmml">′</mo></msubsup></mrow><mo id="S1.p4.4.m2.4.4.5" xref="S1.p4.4.m2.4.4.5.cmml">∈</mo><mi id="S1.p4.4.m2.4.4.6" xref="S1.p4.4.m2.4.4.6.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m2.4b"><apply id="S1.p4.4.m2.4.4.cmml" xref="S1.p4.4.m2.4.4"><in id="S1.p4.4.m2.4.4.5.cmml" xref="S1.p4.4.m2.4.4.5"></in><list id="S1.p4.4.m2.4.4.4.5.cmml" xref="S1.p4.4.m2.4.4.4.4"><apply id="S1.p4.4.m2.1.1.1.1.1.cmml" xref="S1.p4.4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p4.4.m2.1.1.1.1.1.1.cmml" xref="S1.p4.4.m2.1.1.1.1.1">subscript</csymbol><ci id="S1.p4.4.m2.1.1.1.1.1.2.cmml" xref="S1.p4.4.m2.1.1.1.1.1.2">𝑎</ci><ci id="S1.p4.4.m2.1.1.1.1.1.3.cmml" xref="S1.p4.4.m2.1.1.1.1.1.3">𝑖</ci></apply><apply id="S1.p4.4.m2.2.2.2.2.2.cmml" xref="S1.p4.4.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.p4.4.m2.2.2.2.2.2.1.cmml" xref="S1.p4.4.m2.2.2.2.2.2">superscript</csymbol><apply id="S1.p4.4.m2.2.2.2.2.2.2.cmml" xref="S1.p4.4.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.p4.4.m2.2.2.2.2.2.2.1.cmml" xref="S1.p4.4.m2.2.2.2.2.2">subscript</csymbol><ci id="S1.p4.4.m2.2.2.2.2.2.2.2.cmml" xref="S1.p4.4.m2.2.2.2.2.2.2.2">𝑎</ci><ci id="S1.p4.4.m2.2.2.2.2.2.2.3.cmml" xref="S1.p4.4.m2.2.2.2.2.2.2.3">𝑖</ci></apply><ci id="S1.p4.4.m2.2.2.2.2.2.3.cmml" xref="S1.p4.4.m2.2.2.2.2.2.3">′</ci></apply><apply id="S1.p4.4.m2.3.3.3.3.3.cmml" xref="S1.p4.4.m2.3.3.3.3.3"><csymbol cd="ambiguous" id="S1.p4.4.m2.3.3.3.3.3.1.cmml" xref="S1.p4.4.m2.3.3.3.3.3">subscript</csymbol><ci id="S1.p4.4.m2.3.3.3.3.3.2.cmml" xref="S1.p4.4.m2.3.3.3.3.3.2">𝑏</ci><ci id="S1.p4.4.m2.3.3.3.3.3.3.cmml" xref="S1.p4.4.m2.3.3.3.3.3.3">𝑖</ci></apply><apply id="S1.p4.4.m2.4.4.4.4.4.cmml" xref="S1.p4.4.m2.4.4.4.4.4"><csymbol cd="ambiguous" id="S1.p4.4.m2.4.4.4.4.4.1.cmml" xref="S1.p4.4.m2.4.4.4.4.4">superscript</csymbol><apply id="S1.p4.4.m2.4.4.4.4.4.2.cmml" xref="S1.p4.4.m2.4.4.4.4.4"><csymbol cd="ambiguous" id="S1.p4.4.m2.4.4.4.4.4.2.1.cmml" xref="S1.p4.4.m2.4.4.4.4.4">subscript</csymbol><ci id="S1.p4.4.m2.4.4.4.4.4.2.2.cmml" xref="S1.p4.4.m2.4.4.4.4.4.2.2">𝑏</ci><ci id="S1.p4.4.m2.4.4.4.4.4.2.3.cmml" xref="S1.p4.4.m2.4.4.4.4.4.2.3">𝑖</ci></apply><ci id="S1.p4.4.m2.4.4.4.4.4.3.cmml" xref="S1.p4.4.m2.4.4.4.4.4.3">′</ci></apply></list><ci id="S1.p4.4.m2.4.4.6.cmml" xref="S1.p4.4.m2.4.4.6">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m2.4c">a_{i},a_{i}^{\prime},b_{i},b_{i}^{\prime}\in\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m2.4d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R</annotation></semantics></math>. <span class="ltx_ERROR undefined" id="S1.p4.4.1">\citet</span>mukherjee2017 observed that in two dimensions, maxima of only two affine functions are insufficient as summands. However, the following theorem shows that using three affine functions per summand suffices while keeping the number of summands linear in the number of pieces.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem2.1.1.1">Theorem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.4.4">Let <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S1.Thmtheorem2.p1.1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S1.Thmtheorem2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><msup id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml"><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mn id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.3" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mo id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.1.m1.1b"><apply id="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3"><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.2">ℝ</ci><cn id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.2.3">2</cn></apply><ci id="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.1.m1.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> be a continuous piecewise affine function with <math alttext="p" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.2.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.2.m2.1a"><mi id="S1.Thmtheorem2.p1.2.2.m2.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.2.m2.1b"><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.1d">italic_p</annotation></semantics></math> pieces. Then, there exist affine functions <math alttext="f^{(k)}_{n}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><msubsup id="S1.Thmtheorem2.p1.3.3.m3.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.2.2.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.2.2.cmml">f</mi><mi id="S1.Thmtheorem2.p1.3.3.m3.1.2.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.3.cmml">n</mi><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml"><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml">(</mo><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.cmml">k</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3.2" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.3.3.m3.1.2.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2">subscript</csymbol><apply id="S1.Thmtheorem2.p1.3.3.m3.1.2.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.3.3.m3.1.2.2.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2">superscript</csymbol><ci id="S1.Thmtheorem2.p1.3.3.m3.1.2.2.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.2.2">𝑓</ci><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1">𝑘</ci></apply><ci id="S1.Thmtheorem2.p1.3.3.m3.1.2.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">f^{(k)}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">italic_f start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and signs <math alttext="\sigma_{n}^{(k)}\in\{-1,1\}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.4.4.m4.3"><semantics id="S1.Thmtheorem2.p1.4.4.m4.3a"><mrow id="S1.Thmtheorem2.p1.4.4.m4.3.3" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.cmml"><msubsup id="S1.Thmtheorem2.p1.4.4.m4.3.3.3" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.cmml"><mi id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.2" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.2.cmml">σ</mi><mi id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.3" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.3.cmml">n</mi><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.cmml"><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.cmml">(</mo><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.cmml">k</mi><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.3.2" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S1.Thmtheorem2.p1.4.4.m4.3.3.2" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.2.cmml">∈</mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml"><mo id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.2" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">{</mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1a" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">,</mo><mn id="S1.Thmtheorem2.p1.4.4.m4.2.2" xref="S1.Thmtheorem2.p1.4.4.m4.2.2.cmml">1</mn><mo id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.4" stretchy="false" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.4.4.m4.3b"><apply id="S1.Thmtheorem2.p1.4.4.m4.3.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3"><in id="S1.Thmtheorem2.p1.4.4.m4.3.3.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.2"></in><apply id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3">superscript</csymbol><apply id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3">subscript</csymbol><ci id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.2">𝜎</ci><ci id="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.3.2.3">𝑛</ci></apply><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.1">𝑘</ci></apply><set id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1"><apply id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1"><minus id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1"></minus><cn id="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2">1</cn></apply><cn id="S1.Thmtheorem2.p1.4.4.m4.2.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.4.4.m4.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.4.4.m4.3c">\sigma_{n}^{(k)}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.4.4.m4.3d">italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math>, such that</span></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="f(x)=\sum_{n=1}^{9p}\sigma_{n}^{(1)}\max(f_{n}^{(1)},\sigma_{n}^{(2)}\max(f_{n% }^{(2)},f_{n}^{(3)}))." class="ltx_Math" display="block" id="S1.Ex2.m1.9"><semantics id="S1.Ex2.m1.9a"><mrow id="S1.Ex2.m1.9.9.1" xref="S1.Ex2.m1.9.9.1.1.cmml"><mrow id="S1.Ex2.m1.9.9.1.1" xref="S1.Ex2.m1.9.9.1.1.cmml"><mrow id="S1.Ex2.m1.9.9.1.1.4" xref="S1.Ex2.m1.9.9.1.1.4.cmml"><mi id="S1.Ex2.m1.9.9.1.1.4.2" xref="S1.Ex2.m1.9.9.1.1.4.2.cmml">f</mi><mo id="S1.Ex2.m1.9.9.1.1.4.1" xref="S1.Ex2.m1.9.9.1.1.4.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.9.9.1.1.4.3.2" xref="S1.Ex2.m1.9.9.1.1.4.cmml"><mo id="S1.Ex2.m1.9.9.1.1.4.3.2.1" stretchy="false" xref="S1.Ex2.m1.9.9.1.1.4.cmml">(</mo><mi id="S1.Ex2.m1.6.6" xref="S1.Ex2.m1.6.6.cmml">x</mi><mo id="S1.Ex2.m1.9.9.1.1.4.3.2.2" stretchy="false" xref="S1.Ex2.m1.9.9.1.1.4.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.9.9.1.1.3" rspace="0.111em" xref="S1.Ex2.m1.9.9.1.1.3.cmml">=</mo><mrow id="S1.Ex2.m1.9.9.1.1.2" xref="S1.Ex2.m1.9.9.1.1.2.cmml"><munderover 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xref="S1.Ex2.m1.5.5.1.1">3</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.9c">f(x)=\sum_{n=1}^{9p}\sigma_{n}^{(1)}\max(f_{n}^{(1)},\sigma_{n}^{(2)}\max(f_{n% }^{(2)},f_{n}^{(3)})).</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.9d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 9 italic_p end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">To achieve this representation, a constructive proof for a conic decomposition of the pieces is provided (see <span class="ltx_ERROR undefined" id="S1.p5.1.1">\citet</span>Shephard1967AngleSums for conic decompositions of convex polytopes).</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>Related Work</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.1">Various works estimate the complexity that a given neural network architecture may achieve. For a <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><mi id="S1.SS1.p1.1.m1.1.1" xref="S1.SS1.p1.1.m1.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><ci id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">roman_ReLU</annotation></semantics></math> network, given that it computes a CPA function, the number of pieces of the represented function is a natural measure for its complexity. Here, the number of pieces is defined as the minimum required, as the set of pieces is not unique.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.1">The first significant lower and upper bounds on the complexity of neural networks were established by <span class="ltx_ERROR undefined" id="S1.SS1.p2.1.1">\citet</span>Montufar2014 and <span class="ltx_ERROR undefined" id="S1.SS1.p2.1.2">\citet</span>Montufar2017Notes, respectively, and were later slightly improved by <span class="ltx_ERROR undefined" id="S1.SS1.p2.1.3">\citet</span>Serra2018BoundingAndCounting. A unifying view on deriving such bounds was presented by <span class="ltx_ERROR undefined" id="S1.SS1.p2.1.4">\citet</span>Hinz2019. Both upper and lower bounds indicate that asymptotically deep networks with a fixed number of neurons can produce exponentially more pieces than shallow ones. Moreover, <span class="ltx_ERROR undefined" id="S1.SS1.p2.1.5">\citet</span>arora2018understanding show an exponential tradeoff between width and depth for a whole continuum of ’hard’ functions. These findings align with practitioners’ observations that deeper networks are more powerful.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.1">However, the lower bounds on the maximum achievable complexity of an architecture are based on very specific constructions. Therefore, there might be much more functions of the same complexity that can not be represented by architectures of comparable size.</p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.1">In contrast, this work focuses on architectures capable of representing <em class="ltx_emph ltx_font_italic" id="S1.SS1.p4.1.1">all</em> CPA functions of given complexity. <span class="ltx_ERROR undefined" id="S1.SS1.p4.1.2">\citet</span>arora2018understanding established that this is feasible, which lead to research into the architectural size required for such representations. Key contributions include results by <span class="ltx_ERROR undefined" id="S1.SS1.p4.1.3">\citet</span>brandenburg2024decompositionPolyhedra, Chen2022neurCompl, Hanin2019, He2020FEM, Hertrich2023TowardsLowerBounds, Koutschan2023.</p> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.8">More specifically, <span class="ltx_ERROR undefined" id="S1.SS1.p5.8.1">\citet</span>arora2018understanding demonstrated that any CPA function <math alttext="f:\mathds{R}^{d}\to\mathds{R}^{m}" class="ltx_Math" display="inline" id="S1.SS1.p5.1.m1.1"><semantics id="S1.SS1.p5.1.m1.1a"><mrow id="S1.SS1.p5.1.m1.1.1" xref="S1.SS1.p5.1.m1.1.1.cmml"><mi id="S1.SS1.p5.1.m1.1.1.2" xref="S1.SS1.p5.1.m1.1.1.2.cmml">f</mi><mo id="S1.SS1.p5.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.SS1.p5.1.m1.1.1.1.cmml">:</mo><mrow id="S1.SS1.p5.1.m1.1.1.3" xref="S1.SS1.p5.1.m1.1.1.3.cmml"><msup id="S1.SS1.p5.1.m1.1.1.3.2" xref="S1.SS1.p5.1.m1.1.1.3.2.cmml"><mi id="S1.SS1.p5.1.m1.1.1.3.2.2" 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xref="S1.SS1.p5.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.SS1.p5.1.m1.1.1.3.2.2.cmml" xref="S1.SS1.p5.1.m1.1.1.3.2.2">ℝ</ci><ci id="S1.SS1.p5.1.m1.1.1.3.2.3.cmml" xref="S1.SS1.p5.1.m1.1.1.3.2.3">𝑑</ci></apply><apply id="S1.SS1.p5.1.m1.1.1.3.3.cmml" xref="S1.SS1.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p5.1.m1.1.1.3.3.1.cmml" xref="S1.SS1.p5.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.SS1.p5.1.m1.1.1.3.3.2.cmml" xref="S1.SS1.p5.1.m1.1.1.3.3.2">ℝ</ci><ci id="S1.SS1.p5.1.m1.1.1.3.3.3.cmml" xref="S1.SS1.p5.1.m1.1.1.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.1.m1.1c">f:\mathds{R}^{d}\to\mathds{R}^{m}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> can be represented using <math 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xref="S1.SS1.p5.2.m2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p5.2.m2.2.3.2.2" stretchy="false" xref="S1.SS1.p5.2.m2.2.3.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.2.m2.2b"><apply id="S1.SS1.p5.2.m2.2.3.1.cmml" xref="S1.SS1.p5.2.m2.2.3.2"><ceiling id="S1.SS1.p5.2.m2.2.3.1.1.cmml" xref="S1.SS1.p5.2.m2.2.3.2.1"></ceiling><apply id="S1.SS1.p5.2.m2.2.2.3.cmml" xref="S1.SS1.p5.2.m2.2.2.4"><log id="S1.SS1.p5.2.m2.2.2.3.1.cmml" xref="S1.SS1.p5.2.m2.2.2.2.2"></log><apply id="S1.SS1.p5.2.m2.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1.1.1.1"><plus id="S1.SS1.p5.2.m2.1.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1.1.1.1.1"></plus><ci id="S1.SS1.p5.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS1.p5.2.m2.1.1.1.1.1.2">𝑑</ci><cn id="S1.SS1.p5.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p5.2.m2.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.2.m2.2c">\lceil\log(d+1)\rceil</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.2.m2.2d">⌈ roman_log ( start_ARG italic_d + 1 end_ARG ) ⌉</annotation></semantics></math> hidden layers. <span class="ltx_ERROR undefined" id="S1.SS1.p5.8.2">\citet</span>Hertrich2023TowardsLowerBounds conjectured that this bound is sharp, which means that for <math alttext="l\in\mathds{N}" class="ltx_Math" display="inline" id="S1.SS1.p5.3.m3.1"><semantics id="S1.SS1.p5.3.m3.1a"><mrow id="S1.SS1.p5.3.m3.1.1" xref="S1.SS1.p5.3.m3.1.1.cmml"><mi id="S1.SS1.p5.3.m3.1.1.2" xref="S1.SS1.p5.3.m3.1.1.2.cmml">l</mi><mo id="S1.SS1.p5.3.m3.1.1.1" xref="S1.SS1.p5.3.m3.1.1.1.cmml">∈</mo><mi id="S1.SS1.p5.3.m3.1.1.3" xref="S1.SS1.p5.3.m3.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.3.m3.1b"><apply id="S1.SS1.p5.3.m3.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1"><in id="S1.SS1.p5.3.m3.1.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1.1"></in><ci id="S1.SS1.p5.3.m3.1.1.2.cmml" xref="S1.SS1.p5.3.m3.1.1.2">𝑙</ci><ci id="S1.SS1.p5.3.m3.1.1.3.cmml" xref="S1.SS1.p5.3.m3.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.3.m3.1c">l\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.3.m3.1d">italic_l ∈ blackboard_N</annotation></semantics></math> and <math alttext="d\geq 2^{l}" class="ltx_Math" display="inline" id="S1.SS1.p5.4.m4.1"><semantics id="S1.SS1.p5.4.m4.1a"><mrow id="S1.SS1.p5.4.m4.1.1" xref="S1.SS1.p5.4.m4.1.1.cmml"><mi id="S1.SS1.p5.4.m4.1.1.2" xref="S1.SS1.p5.4.m4.1.1.2.cmml">d</mi><mo id="S1.SS1.p5.4.m4.1.1.1" xref="S1.SS1.p5.4.m4.1.1.1.cmml">≥</mo><msup id="S1.SS1.p5.4.m4.1.1.3" xref="S1.SS1.p5.4.m4.1.1.3.cmml"><mn id="S1.SS1.p5.4.m4.1.1.3.2" xref="S1.SS1.p5.4.m4.1.1.3.2.cmml">2</mn><mi id="S1.SS1.p5.4.m4.1.1.3.3" xref="S1.SS1.p5.4.m4.1.1.3.3.cmml">l</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.4.m4.1b"><apply id="S1.SS1.p5.4.m4.1.1.cmml" xref="S1.SS1.p5.4.m4.1.1"><geq id="S1.SS1.p5.4.m4.1.1.1.cmml" xref="S1.SS1.p5.4.m4.1.1.1"></geq><ci id="S1.SS1.p5.4.m4.1.1.2.cmml" xref="S1.SS1.p5.4.m4.1.1.2">𝑑</ci><apply id="S1.SS1.p5.4.m4.1.1.3.cmml" xref="S1.SS1.p5.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.p5.4.m4.1.1.3.1.cmml" xref="S1.SS1.p5.4.m4.1.1.3">superscript</csymbol><cn id="S1.SS1.p5.4.m4.1.1.3.2.cmml" type="integer" xref="S1.SS1.p5.4.m4.1.1.3.2">2</cn><ci id="S1.SS1.p5.4.m4.1.1.3.3.cmml" xref="S1.SS1.p5.4.m4.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.4.m4.1c">d\geq 2^{l}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.4.m4.1d">italic_d ≥ 2 start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT</annotation></semantics></math>, there exist CPA functions mapping from <math alttext="\mathds{R}^{d}" class="ltx_Math" display="inline" id="S1.SS1.p5.5.m5.1"><semantics id="S1.SS1.p5.5.m5.1a"><msup id="S1.SS1.p5.5.m5.1.1" xref="S1.SS1.p5.5.m5.1.1.cmml"><mi id="S1.SS1.p5.5.m5.1.1.2" xref="S1.SS1.p5.5.m5.1.1.2.cmml">ℝ</mi><mi id="S1.SS1.p5.5.m5.1.1.3" xref="S1.SS1.p5.5.m5.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.5.m5.1b"><apply id="S1.SS1.p5.5.m5.1.1.cmml" xref="S1.SS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.5.m5.1.1.1.cmml" xref="S1.SS1.p5.5.m5.1.1">superscript</csymbol><ci id="S1.SS1.p5.5.m5.1.1.2.cmml" xref="S1.SS1.p5.5.m5.1.1.2">ℝ</ci><ci id="S1.SS1.p5.5.m5.1.1.3.cmml" xref="S1.SS1.p5.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.5.m5.1c">\mathds{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> that cannot be represented using <math alttext="l" class="ltx_Math" display="inline" id="S1.SS1.p5.6.m6.1"><semantics id="S1.SS1.p5.6.m6.1a"><mi id="S1.SS1.p5.6.m6.1.1" xref="S1.SS1.p5.6.m6.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.6.m6.1b"><ci id="S1.SS1.p5.6.m6.1.1.cmml" xref="S1.SS1.p5.6.m6.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.6.m6.1c">l</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.6.m6.1d">italic_l</annotation></semantics></math> hidden layers. While this is straightforward for <math alttext="l=1" class="ltx_Math" display="inline" id="S1.SS1.p5.7.m7.1"><semantics id="S1.SS1.p5.7.m7.1a"><mrow id="S1.SS1.p5.7.m7.1.1" xref="S1.SS1.p5.7.m7.1.1.cmml"><mi id="S1.SS1.p5.7.m7.1.1.2" xref="S1.SS1.p5.7.m7.1.1.2.cmml">l</mi><mo id="S1.SS1.p5.7.m7.1.1.1" xref="S1.SS1.p5.7.m7.1.1.1.cmml">=</mo><mn id="S1.SS1.p5.7.m7.1.1.3" xref="S1.SS1.p5.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.7.m7.1b"><apply id="S1.SS1.p5.7.m7.1.1.cmml" xref="S1.SS1.p5.7.m7.1.1"><eq id="S1.SS1.p5.7.m7.1.1.1.cmml" xref="S1.SS1.p5.7.m7.1.1.1"></eq><ci id="S1.SS1.p5.7.m7.1.1.2.cmml" xref="S1.SS1.p5.7.m7.1.1.2">𝑙</ci><cn id="S1.SS1.p5.7.m7.1.1.3.cmml" type="integer" xref="S1.SS1.p5.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.7.m7.1c">l=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.7.m7.1d">italic_l = 1</annotation></semantics></math> <span class="ltx_ERROR undefined" id="S1.SS1.p5.8.3">\citep</span>mukherjee2017, <span class="ltx_ERROR undefined" id="S1.SS1.p5.8.4">\citet</span>Hertrich2023TowardsLowerBounds proved it for <math alttext="l=2" class="ltx_Math" display="inline" id="S1.SS1.p5.8.m8.1"><semantics id="S1.SS1.p5.8.m8.1a"><mrow id="S1.SS1.p5.8.m8.1.1" xref="S1.SS1.p5.8.m8.1.1.cmml"><mi id="S1.SS1.p5.8.m8.1.1.2" xref="S1.SS1.p5.8.m8.1.1.2.cmml">l</mi><mo id="S1.SS1.p5.8.m8.1.1.1" xref="S1.SS1.p5.8.m8.1.1.1.cmml">=</mo><mn id="S1.SS1.p5.8.m8.1.1.3" xref="S1.SS1.p5.8.m8.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.8.m8.1b"><apply id="S1.SS1.p5.8.m8.1.1.cmml" xref="S1.SS1.p5.8.m8.1.1"><eq id="S1.SS1.p5.8.m8.1.1.1.cmml" xref="S1.SS1.p5.8.m8.1.1.1"></eq><ci id="S1.SS1.p5.8.m8.1.1.2.cmml" xref="S1.SS1.p5.8.m8.1.1.2">𝑙</ci><cn id="S1.SS1.p5.8.m8.1.1.3.cmml" type="integer" xref="S1.SS1.p5.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.8.m8.1c">l=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.8.m8.1d">italic_l = 2</annotation></semantics></math> under assumptions on the functions represented by intermediate neurons, and <span class="ltx_ERROR undefined" id="S1.SS1.p5.8.5">\citet</span>haase2023lower proved it for higher dimensions under an integer-weight assumption.</p> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.1"><a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1 Introduction ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 1.1</span></a> falls within the ’shallow’ regime described by the conjecture, which means that the depth is minimal and, in particular, independent of the complexity of the function. The most general results of that type for neural network representations of CPA functions are given by <span class="ltx_ERROR undefined" id="S1.SS1.p6.1.1">\citet</span>Hertrich2023TowardsLowerBounds, Koutschan2023. In contrast, <span class="ltx_ERROR undefined" id="S1.SS1.p6.1.2">\citet</span>Hanin2019 introduced a narrow network architecture whose width depends only on <math alttext="d" class="ltx_Math" display="inline" id="S1.SS1.p6.1.m1.1"><semantics id="S1.SS1.p6.1.m1.1a"><mi id="S1.SS1.p6.1.m1.1.1" xref="S1.SS1.p6.1.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.1.m1.1b"><ci id="S1.SS1.p6.1.m1.1.1.cmml" xref="S1.SS1.p6.1.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.1.m1.1c">d</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.1.m1.1d">italic_d</annotation></semantics></math>, but at the cost of significant depth. <span class="ltx_ERROR undefined" id="S1.SS1.p6.1.3">\citet</span>Chen2022neurCompl aimed to minimise the total number of neurons, making both the depth and width dependent on the complexity. If the set of pieces satisfies a regularity condition, interpolation between a shallow and narrow configuration is possible <span class="ltx_ERROR undefined" id="S1.SS1.p6.1.4">\citep</span>brandenburg2024decompositionPolyhedra.</p> </div> <div class="ltx_para" id="S1.SS1.p7"> <p class="ltx_p" id="S1.SS1.p7.5">The bounds on the size of the representing architectures are often expressed in terms of the number of pieces <math alttext="p" class="ltx_Math" display="inline" id="S1.SS1.p7.1.m1.1"><semantics id="S1.SS1.p7.1.m1.1a"><mi id="S1.SS1.p7.1.m1.1.1" xref="S1.SS1.p7.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.1.m1.1b"><ci id="S1.SS1.p7.1.m1.1.1.cmml" xref="S1.SS1.p7.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.1.m1.1d">italic_p</annotation></semantics></math>, with the additional requirement that these pieces be convex. The reason is that this allows the direct application of the <math alttext="\max" class="ltx_Math" display="inline" id="S1.SS1.p7.2.m2.1"><semantics id="S1.SS1.p7.2.m2.1a"><mi id="S1.SS1.p7.2.m2.1.1" xref="S1.SS1.p7.2.m2.1.1.cmml">max</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.2.m2.1b"><max id="S1.SS1.p7.2.m2.1.1.cmml" xref="S1.SS1.p7.2.m2.1.1"></max></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.2.m2.1c">\max</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.2.m2.1d">roman_max</annotation></semantics></math>-<math alttext="\min" class="ltx_Math" display="inline" id="S1.SS1.p7.3.m3.1"><semantics id="S1.SS1.p7.3.m3.1a"><mi id="S1.SS1.p7.3.m3.1.1" xref="S1.SS1.p7.3.m3.1.1.cmml">min</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.3.m3.1b"><min id="S1.SS1.p7.3.m3.1.1.cmml" xref="S1.SS1.p7.3.m3.1.1"></min></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.3.m3.1c">\min</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.3.m3.1d">roman_min</annotation></semantics></math> representation by <span class="ltx_ERROR undefined" id="S1.SS1.p7.5.1">\citet</span>Tarela1990MaxMin. If the pieces were allowed to be arbitrary sets, the number of pieces would match the number <math alttext="n" class="ltx_Math" display="inline" id="S1.SS1.p7.4.m4.1"><semantics id="S1.SS1.p7.4.m4.1a"><mi id="S1.SS1.p7.4.m4.1.1" xref="S1.SS1.p7.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.4.m4.1b"><ci id="S1.SS1.p7.4.m4.1.1.cmml" xref="S1.SS1.p7.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.4.m4.1d">italic_n</annotation></semantics></math> of unique affine functions needed to describe the CPA function. <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S1.F1" title="Figure 1 ‣ 1.1 Related Work ‣ 1 Introduction ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 1</span></a> illustrates that <math alttext="n" class="ltx_Math" display="inline" id="S1.SS1.p7.5.m5.1"><semantics id="S1.SS1.p7.5.m5.1a"><mi id="S1.SS1.p7.5.m5.1.1" xref="S1.SS1.p7.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.5.m5.1b"><ci id="S1.SS1.p7.5.m5.1.1.cmml" xref="S1.SS1.p7.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.5.m5.1d">italic_n</annotation></semantics></math> and the number of convex pieces may not always serve as ideal complexity measures. This work is the first to assume that the pieces are connected, without requiring them to be convex.</p> </div> <div class="ltx_para" id="S1.SS1.p8"> <p class="ltx_p" id="S1.SS1.p8.8">While previous results apply to arbitrary dimensions, in the specific case of <math alttext="d=2" class="ltx_Math" display="inline" id="S1.SS1.p8.1.m1.1"><semantics id="S1.SS1.p8.1.m1.1a"><mrow id="S1.SS1.p8.1.m1.1.1" xref="S1.SS1.p8.1.m1.1.1.cmml"><mi id="S1.SS1.p8.1.m1.1.1.2" xref="S1.SS1.p8.1.m1.1.1.2.cmml">d</mi><mo id="S1.SS1.p8.1.m1.1.1.1" xref="S1.SS1.p8.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS1.p8.1.m1.1.1.3" xref="S1.SS1.p8.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.1.m1.1b"><apply id="S1.SS1.p8.1.m1.1.1.cmml" xref="S1.SS1.p8.1.m1.1.1"><eq id="S1.SS1.p8.1.m1.1.1.1.cmml" xref="S1.SS1.p8.1.m1.1.1.1"></eq><ci id="S1.SS1.p8.1.m1.1.1.2.cmml" xref="S1.SS1.p8.1.m1.1.1.2">𝑑</ci><cn id="S1.SS1.p8.1.m1.1.1.3.cmml" type="integer" xref="S1.SS1.p8.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.1.m1.1c">d=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.1.m1.1d">italic_d = 2</annotation></semantics></math>, none achieve a neuron count scaling linearly with the number of pieces. The closest result to <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1 Introduction ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 1.1</span></a>, by <span class="ltx_ERROR undefined" id="S1.SS1.p8.8.1">\citet</span>Koutschan2023, shows that a shallow network with width <math alttext="O(n^{d+1})" class="ltx_Math" display="inline" id="S1.SS1.p8.2.m2.1"><semantics id="S1.SS1.p8.2.m2.1a"><mrow id="S1.SS1.p8.2.m2.1.1" xref="S1.SS1.p8.2.m2.1.1.cmml"><mi id="S1.SS1.p8.2.m2.1.1.3" xref="S1.SS1.p8.2.m2.1.1.3.cmml">O</mi><mo id="S1.SS1.p8.2.m2.1.1.2" xref="S1.SS1.p8.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p8.2.m2.1.1.1.1" xref="S1.SS1.p8.2.m2.1.1.1.1.1.cmml"><mo id="S1.SS1.p8.2.m2.1.1.1.1.2" stretchy="false" xref="S1.SS1.p8.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S1.SS1.p8.2.m2.1.1.1.1.1" xref="S1.SS1.p8.2.m2.1.1.1.1.1.cmml"><mi 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id="S1.SS1.p8.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS1.p8.2.m2.1.1.1.1.1.2">𝑛</ci><apply id="S1.SS1.p8.2.m2.1.1.1.1.1.3.cmml" xref="S1.SS1.p8.2.m2.1.1.1.1.1.3"><plus id="S1.SS1.p8.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.SS1.p8.2.m2.1.1.1.1.1.3.1"></plus><ci id="S1.SS1.p8.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.SS1.p8.2.m2.1.1.1.1.1.3.2">𝑑</ci><cn id="S1.SS1.p8.2.m2.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.SS1.p8.2.m2.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.2.m2.1c">O(n^{d+1})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.2.m2.1d">italic_O ( italic_n start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> suffices to represent any CPA function in <math alttext="\mathds{R}^{d}" class="ltx_Math" display="inline" id="S1.SS1.p8.3.m3.1"><semantics id="S1.SS1.p8.3.m3.1a"><msup id="S1.SS1.p8.3.m3.1.1" xref="S1.SS1.p8.3.m3.1.1.cmml"><mi id="S1.SS1.p8.3.m3.1.1.2" xref="S1.SS1.p8.3.m3.1.1.2.cmml">ℝ</mi><mi id="S1.SS1.p8.3.m3.1.1.3" xref="S1.SS1.p8.3.m3.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.3.m3.1b"><apply id="S1.SS1.p8.3.m3.1.1.cmml" xref="S1.SS1.p8.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p8.3.m3.1.1.1.cmml" xref="S1.SS1.p8.3.m3.1.1">superscript</csymbol><ci id="S1.SS1.p8.3.m3.1.1.2.cmml" xref="S1.SS1.p8.3.m3.1.1.2">ℝ</ci><ci id="S1.SS1.p8.3.m3.1.1.3.cmml" xref="S1.SS1.p8.3.m3.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.3.m3.1c">\mathds{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, implying an <math alttext="O(p^{d+1})" class="ltx_Math" display="inline" id="S1.SS1.p8.4.m4.1"><semantics id="S1.SS1.p8.4.m4.1a"><mrow id="S1.SS1.p8.4.m4.1.1" xref="S1.SS1.p8.4.m4.1.1.cmml"><mi id="S1.SS1.p8.4.m4.1.1.3" xref="S1.SS1.p8.4.m4.1.1.3.cmml">O</mi><mo id="S1.SS1.p8.4.m4.1.1.2" xref="S1.SS1.p8.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p8.4.m4.1.1.1.1" xref="S1.SS1.p8.4.m4.1.1.1.1.1.cmml"><mo id="S1.SS1.p8.4.m4.1.1.1.1.2" stretchy="false" xref="S1.SS1.p8.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S1.SS1.p8.4.m4.1.1.1.1.1" xref="S1.SS1.p8.4.m4.1.1.1.1.1.cmml"><mi id="S1.SS1.p8.4.m4.1.1.1.1.1.2" xref="S1.SS1.p8.4.m4.1.1.1.1.1.2.cmml">p</mi><mrow id="S1.SS1.p8.4.m4.1.1.1.1.1.3" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.cmml"><mi id="S1.SS1.p8.4.m4.1.1.1.1.1.3.2" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.2.cmml">d</mi><mo id="S1.SS1.p8.4.m4.1.1.1.1.1.3.1" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.1.cmml">+</mo><mn id="S1.SS1.p8.4.m4.1.1.1.1.1.3.3" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S1.SS1.p8.4.m4.1.1.1.1.3" stretchy="false" xref="S1.SS1.p8.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.4.m4.1b"><apply id="S1.SS1.p8.4.m4.1.1.cmml" xref="S1.SS1.p8.4.m4.1.1"><times id="S1.SS1.p8.4.m4.1.1.2.cmml" xref="S1.SS1.p8.4.m4.1.1.2"></times><ci id="S1.SS1.p8.4.m4.1.1.3.cmml" xref="S1.SS1.p8.4.m4.1.1.3">𝑂</ci><apply id="S1.SS1.p8.4.m4.1.1.1.1.1.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p8.4.m4.1.1.1.1.1.1.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1">superscript</csymbol><ci id="S1.SS1.p8.4.m4.1.1.1.1.1.2.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1.1.2">𝑝</ci><apply id="S1.SS1.p8.4.m4.1.1.1.1.1.3.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3"><plus id="S1.SS1.p8.4.m4.1.1.1.1.1.3.1.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.1"></plus><ci id="S1.SS1.p8.4.m4.1.1.1.1.1.3.2.cmml" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.2">𝑑</ci><cn id="S1.SS1.p8.4.m4.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.SS1.p8.4.m4.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.4.m4.1c">O(p^{d+1})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.4.m4.1d">italic_O ( italic_p start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> bound also in terms of <math alttext="p" class="ltx_Math" display="inline" id="S1.SS1.p8.5.m5.1"><semantics id="S1.SS1.p8.5.m5.1a"><mi id="S1.SS1.p8.5.m5.1.1" xref="S1.SS1.p8.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.5.m5.1b"><ci id="S1.SS1.p8.5.m5.1.1.cmml" xref="S1.SS1.p8.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.5.m5.1d">italic_p</annotation></semantics></math>. This paper improves this bound for <math alttext="d=2" class="ltx_Math" display="inline" id="S1.SS1.p8.6.m6.1"><semantics id="S1.SS1.p8.6.m6.1a"><mrow id="S1.SS1.p8.6.m6.1.1" xref="S1.SS1.p8.6.m6.1.1.cmml"><mi id="S1.SS1.p8.6.m6.1.1.2" xref="S1.SS1.p8.6.m6.1.1.2.cmml">d</mi><mo id="S1.SS1.p8.6.m6.1.1.1" xref="S1.SS1.p8.6.m6.1.1.1.cmml">=</mo><mn id="S1.SS1.p8.6.m6.1.1.3" xref="S1.SS1.p8.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.6.m6.1b"><apply id="S1.SS1.p8.6.m6.1.1.cmml" xref="S1.SS1.p8.6.m6.1.1"><eq id="S1.SS1.p8.6.m6.1.1.1.cmml" xref="S1.SS1.p8.6.m6.1.1.1"></eq><ci id="S1.SS1.p8.6.m6.1.1.2.cmml" xref="S1.SS1.p8.6.m6.1.1.2">𝑑</ci><cn id="S1.SS1.p8.6.m6.1.1.3.cmml" type="integer" xref="S1.SS1.p8.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.6.m6.1c">d=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.6.m6.1d">italic_d = 2</annotation></semantics></math>, reducing it from <math alttext="O(p^{3})" class="ltx_Math" display="inline" id="S1.SS1.p8.7.m7.1"><semantics id="S1.SS1.p8.7.m7.1a"><mrow id="S1.SS1.p8.7.m7.1.1" xref="S1.SS1.p8.7.m7.1.1.cmml"><mi id="S1.SS1.p8.7.m7.1.1.3" xref="S1.SS1.p8.7.m7.1.1.3.cmml">O</mi><mo id="S1.SS1.p8.7.m7.1.1.2" xref="S1.SS1.p8.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p8.7.m7.1.1.1.1" xref="S1.SS1.p8.7.m7.1.1.1.1.1.cmml"><mo id="S1.SS1.p8.7.m7.1.1.1.1.2" stretchy="false" xref="S1.SS1.p8.7.m7.1.1.1.1.1.cmml">(</mo><msup id="S1.SS1.p8.7.m7.1.1.1.1.1" xref="S1.SS1.p8.7.m7.1.1.1.1.1.cmml"><mi id="S1.SS1.p8.7.m7.1.1.1.1.1.2" xref="S1.SS1.p8.7.m7.1.1.1.1.1.2.cmml">p</mi><mn id="S1.SS1.p8.7.m7.1.1.1.1.1.3" xref="S1.SS1.p8.7.m7.1.1.1.1.1.3.cmml">3</mn></msup><mo id="S1.SS1.p8.7.m7.1.1.1.1.3" stretchy="false" xref="S1.SS1.p8.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.7.m7.1b"><apply id="S1.SS1.p8.7.m7.1.1.cmml" xref="S1.SS1.p8.7.m7.1.1"><times id="S1.SS1.p8.7.m7.1.1.2.cmml" xref="S1.SS1.p8.7.m7.1.1.2"></times><ci id="S1.SS1.p8.7.m7.1.1.3.cmml" xref="S1.SS1.p8.7.m7.1.1.3">𝑂</ci><apply id="S1.SS1.p8.7.m7.1.1.1.1.1.cmml" xref="S1.SS1.p8.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p8.7.m7.1.1.1.1.1.1.cmml" xref="S1.SS1.p8.7.m7.1.1.1.1">superscript</csymbol><ci id="S1.SS1.p8.7.m7.1.1.1.1.1.2.cmml" xref="S1.SS1.p8.7.m7.1.1.1.1.1.2">𝑝</ci><cn id="S1.SS1.p8.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p8.7.m7.1.1.1.1.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.7.m7.1c">O(p^{3})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.7.m7.1d">italic_O ( italic_p start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math> to <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.SS1.p8.8.m8.1"><semantics id="S1.SS1.p8.8.m8.1a"><mrow id="S1.SS1.p8.8.m8.1.2" xref="S1.SS1.p8.8.m8.1.2.cmml"><mi id="S1.SS1.p8.8.m8.1.2.2" xref="S1.SS1.p8.8.m8.1.2.2.cmml">O</mi><mo id="S1.SS1.p8.8.m8.1.2.1" xref="S1.SS1.p8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p8.8.m8.1.2.3.2" xref="S1.SS1.p8.8.m8.1.2.cmml"><mo id="S1.SS1.p8.8.m8.1.2.3.2.1" stretchy="false" xref="S1.SS1.p8.8.m8.1.2.cmml">(</mo><mi id="S1.SS1.p8.8.m8.1.1" xref="S1.SS1.p8.8.m8.1.1.cmml">p</mi><mo id="S1.SS1.p8.8.m8.1.2.3.2.2" stretchy="false" xref="S1.SS1.p8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p8.8.m8.1b"><apply id="S1.SS1.p8.8.m8.1.2.cmml" xref="S1.SS1.p8.8.m8.1.2"><times id="S1.SS1.p8.8.m8.1.2.1.cmml" xref="S1.SS1.p8.8.m8.1.2.1"></times><ci id="S1.SS1.p8.8.m8.1.2.2.cmml" xref="S1.SS1.p8.8.m8.1.2.2">𝑂</ci><ci id="S1.SS1.p8.8.m8.1.1.cmml" xref="S1.SS1.p8.8.m8.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p8.8.m8.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p8.8.m8.1d">italic_O ( italic_p )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p9"> <p class="ltx_p" id="S1.SS1.p9.3">Representation properties of neural networks are crucial for quantitative function approximation results. A common approach is to use known approximation results for standard function classes and to emulate or approximate these classes efficiently with neural networks, thereby transferring their approximation rates (see, e.g., <span class="ltx_ERROR undefined" id="S1.SS1.p9.3.1">\citet</span>DeVore2021). In this context, <span class="ltx_ERROR undefined" id="S1.SS1.p9.3.2">\citet</span>He2020FEM demonstrated that, for any fixed dimension, linear finite element spaces with convex-support nodal basis functions can be represented using <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.SS1.p9.1.m1.1"><semantics id="S1.SS1.p9.1.m1.1a"><mrow id="S1.SS1.p9.1.m1.1.2" xref="S1.SS1.p9.1.m1.1.2.cmml"><mi id="S1.SS1.p9.1.m1.1.2.2" xref="S1.SS1.p9.1.m1.1.2.2.cmml">O</mi><mo id="S1.SS1.p9.1.m1.1.2.1" xref="S1.SS1.p9.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p9.1.m1.1.2.3.2" xref="S1.SS1.p9.1.m1.1.2.cmml"><mo id="S1.SS1.p9.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.p9.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.p9.1.m1.1.1" xref="S1.SS1.p9.1.m1.1.1.cmml">p</mi><mo id="S1.SS1.p9.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.p9.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p9.1.m1.1b"><apply id="S1.SS1.p9.1.m1.1.2.cmml" xref="S1.SS1.p9.1.m1.1.2"><times id="S1.SS1.p9.1.m1.1.2.1.cmml" xref="S1.SS1.p9.1.m1.1.2.1"></times><ci id="S1.SS1.p9.1.m1.1.2.2.cmml" xref="S1.SS1.p9.1.m1.1.2.2">𝑂</ci><ci id="S1.SS1.p9.1.m1.1.1.cmml" xref="S1.SS1.p9.1.m1.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p9.1.m1.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p9.1.m1.1d">italic_O ( italic_p )</annotation></semantics></math> neurons. The constant in this representation depends on the shape-regularity of the triangulation. Similarly, <span class="ltx_ERROR undefined" id="S1.SS1.p9.3.3">\citet</span>HE2022Hierarchical established an <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.SS1.p9.2.m2.1"><semantics id="S1.SS1.p9.2.m2.1a"><mrow id="S1.SS1.p9.2.m2.1.2" xref="S1.SS1.p9.2.m2.1.2.cmml"><mi id="S1.SS1.p9.2.m2.1.2.2" xref="S1.SS1.p9.2.m2.1.2.2.cmml">O</mi><mo id="S1.SS1.p9.2.m2.1.2.1" xref="S1.SS1.p9.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p9.2.m2.1.2.3.2" xref="S1.SS1.p9.2.m2.1.2.cmml"><mo id="S1.SS1.p9.2.m2.1.2.3.2.1" stretchy="false" xref="S1.SS1.p9.2.m2.1.2.cmml">(</mo><mi id="S1.SS1.p9.2.m2.1.1" xref="S1.SS1.p9.2.m2.1.1.cmml">p</mi><mo id="S1.SS1.p9.2.m2.1.2.3.2.2" stretchy="false" xref="S1.SS1.p9.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p9.2.m2.1b"><apply id="S1.SS1.p9.2.m2.1.2.cmml" xref="S1.SS1.p9.2.m2.1.2"><times id="S1.SS1.p9.2.m2.1.2.1.cmml" xref="S1.SS1.p9.2.m2.1.2.1"></times><ci id="S1.SS1.p9.2.m2.1.2.2.cmml" xref="S1.SS1.p9.2.m2.1.2.2">𝑂</ci><ci id="S1.SS1.p9.2.m2.1.1.cmml" xref="S1.SS1.p9.2.m2.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p9.2.m2.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p9.2.m2.1d">italic_O ( italic_p )</annotation></semantics></math> bound for linear finite elements on uniform triangulations in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S1.SS1.p9.3.m3.1"><semantics id="S1.SS1.p9.3.m3.1a"><msup id="S1.SS1.p9.3.m3.1.1" xref="S1.SS1.p9.3.m3.1.1.cmml"><mi id="S1.SS1.p9.3.m3.1.1.2" xref="S1.SS1.p9.3.m3.1.1.2.cmml">ℝ</mi><mn id="S1.SS1.p9.3.m3.1.1.3" xref="S1.SS1.p9.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S1.SS1.p9.3.m3.1b"><apply id="S1.SS1.p9.3.m3.1.1.cmml" xref="S1.SS1.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p9.3.m3.1.1.1.cmml" xref="S1.SS1.p9.3.m3.1.1">superscript</csymbol><ci id="S1.SS1.p9.3.m3.1.1.2.cmml" xref="S1.SS1.p9.3.m3.1.1.2">ℝ</ci><cn id="S1.SS1.p9.3.m3.1.1.3.cmml" type="integer" xref="S1.SS1.p9.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p9.3.m3.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p9.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. For univariate inputs, neural networks effectively serve as free-knot spline approximators <span class="ltx_ERROR undefined" id="S1.SS1.p9.3.4">\citep</span>Opschoor2020. Thus, neural networks function as adaptive linear finite element approximations that are able to recover the mentioned classical counterparts with the same parameter count.</p> </div> <div class="ltx_para" id="S1.SS1.p10"> <p class="ltx_p" id="S1.SS1.p10.2">This work extends these findings by proving that a neural network with two inputs and <math alttext="O(p)" class="ltx_Math" display="inline" id="S1.SS1.p10.1.m1.1"><semantics id="S1.SS1.p10.1.m1.1a"><mrow id="S1.SS1.p10.1.m1.1.2" xref="S1.SS1.p10.1.m1.1.2.cmml"><mi id="S1.SS1.p10.1.m1.1.2.2" xref="S1.SS1.p10.1.m1.1.2.2.cmml">O</mi><mo id="S1.SS1.p10.1.m1.1.2.1" xref="S1.SS1.p10.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p10.1.m1.1.2.3.2" xref="S1.SS1.p10.1.m1.1.2.cmml"><mo id="S1.SS1.p10.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.p10.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.p10.1.m1.1.1" xref="S1.SS1.p10.1.m1.1.1.cmml">p</mi><mo id="S1.SS1.p10.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.p10.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.1.m1.1b"><apply id="S1.SS1.p10.1.m1.1.2.cmml" xref="S1.SS1.p10.1.m1.1.2"><times id="S1.SS1.p10.1.m1.1.2.1.cmml" xref="S1.SS1.p10.1.m1.1.2.1"></times><ci id="S1.SS1.p10.1.m1.1.2.2.cmml" xref="S1.SS1.p10.1.m1.1.2.2">𝑂</ci><ci id="S1.SS1.p10.1.m1.1.1.cmml" xref="S1.SS1.p10.1.m1.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.1.m1.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.1.m1.1d">italic_O ( italic_p )</annotation></semantics></math> parameters can represent any CPA function with <math alttext="p" class="ltx_Math" display="inline" id="S1.SS1.p10.2.m2.1"><semantics id="S1.SS1.p10.2.m2.1a"><mi id="S1.SS1.p10.2.m2.1.1" xref="S1.SS1.p10.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p10.2.m2.1b"><ci id="S1.SS1.p10.2.m2.1.1.cmml" xref="S1.SS1.p10.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p10.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p10.2.m2.1d">italic_p</annotation></semantics></math> pieces, unrestricted by specific triangulations or meshes.</p> </div> <figure class="ltx_figure" id="S1.F1"> <p class="ltx_p ltx_align_center" id="S1.F1.1"><span class="ltx_text" id="S1.F1.1.1"><foreignobject height="117.1pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="226.9pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="226" id="S1.F1.1.1.1.g1" src="x2.png" width="435"/></foreignobject></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>A CPA function <math alttext="f" class="ltx_Math" display="inline" id="S1.F1.10.m1.1"><semantics id="S1.F1.10.m1.1b"><mi id="S1.F1.10.m1.1.1" xref="S1.F1.10.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.F1.10.m1.1c"><ci id="S1.F1.10.m1.1.1.cmml" xref="S1.F1.10.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.10.m1.1d">f</annotation><annotation encoding="application/x-llamapun" id="S1.F1.10.m1.1e">italic_f</annotation></semantics></math> described locally by affine functions. The labelled coordinates emphasise relevant positions. At least <math alttext="p=8" class="ltx_Math" display="inline" id="S1.F1.11.m2.1"><semantics id="S1.F1.11.m2.1b"><mrow id="S1.F1.11.m2.1.1" xref="S1.F1.11.m2.1.1.cmml"><mi id="S1.F1.11.m2.1.1.2" xref="S1.F1.11.m2.1.1.2.cmml">p</mi><mo id="S1.F1.11.m2.1.1.1" xref="S1.F1.11.m2.1.1.1.cmml">=</mo><mn id="S1.F1.11.m2.1.1.3" xref="S1.F1.11.m2.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.11.m2.1c"><apply id="S1.F1.11.m2.1.1.cmml" xref="S1.F1.11.m2.1.1"><eq id="S1.F1.11.m2.1.1.1.cmml" xref="S1.F1.11.m2.1.1.1"></eq><ci id="S1.F1.11.m2.1.1.2.cmml" xref="S1.F1.11.m2.1.1.2">𝑝</ci><cn id="S1.F1.11.m2.1.1.3.cmml" type="integer" xref="S1.F1.11.m2.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.11.m2.1d">p=8</annotation><annotation encoding="application/x-llamapun" id="S1.F1.11.m2.1e">italic_p = 8</annotation></semantics></math> connected pieces are required to define <math alttext="f" class="ltx_Math" display="inline" id="S1.F1.12.m3.1"><semantics id="S1.F1.12.m3.1b"><mi id="S1.F1.12.m3.1.1" xref="S1.F1.12.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.F1.12.m3.1c"><ci id="S1.F1.12.m3.1.1.cmml" xref="S1.F1.12.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.12.m3.1d">f</annotation><annotation encoding="application/x-llamapun" id="S1.F1.12.m3.1e">italic_f</annotation></semantics></math>. However, if restricted to convex pieces, at least <math alttext="10" class="ltx_Math" display="inline" id="S1.F1.13.m4.1"><semantics id="S1.F1.13.m4.1b"><mn id="S1.F1.13.m4.1.1" xref="S1.F1.13.m4.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S1.F1.13.m4.1c"><cn id="S1.F1.13.m4.1.1.cmml" type="integer" xref="S1.F1.13.m4.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.13.m4.1d">10</annotation><annotation encoding="application/x-llamapun" id="S1.F1.13.m4.1e">10</annotation></semantics></math> are needed. In contrast, <math alttext="n=7" class="ltx_Math" display="inline" id="S1.F1.14.m5.1"><semantics id="S1.F1.14.m5.1b"><mrow id="S1.F1.14.m5.1.1" xref="S1.F1.14.m5.1.1.cmml"><mi id="S1.F1.14.m5.1.1.2" xref="S1.F1.14.m5.1.1.2.cmml">n</mi><mo id="S1.F1.14.m5.1.1.1" xref="S1.F1.14.m5.1.1.1.cmml">=</mo><mn id="S1.F1.14.m5.1.1.3" xref="S1.F1.14.m5.1.1.3.cmml">7</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.F1.14.m5.1c"><apply id="S1.F1.14.m5.1.1.cmml" xref="S1.F1.14.m5.1.1"><eq id="S1.F1.14.m5.1.1.1.cmml" xref="S1.F1.14.m5.1.1.1"></eq><ci id="S1.F1.14.m5.1.1.2.cmml" xref="S1.F1.14.m5.1.1.2">𝑛</ci><cn id="S1.F1.14.m5.1.1.3.cmml" type="integer" xref="S1.F1.14.m5.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.14.m5.1d">n=7</annotation><annotation encoding="application/x-llamapun" id="S1.F1.14.m5.1e">italic_n = 7</annotation></semantics></math> suffice if the connectivity assumption is dropped. Arguably, this function is less complex than a modified version with three convex outer regions (dashed lines) corresponding to different affine functions, but more complex than a simplified version where the inner triangle is removed by expanding adjacent regions. The difference between the different counts can be made arbitrarily large by adding shifted copies of <math alttext="f" class="ltx_Math" display="inline" id="S1.F1.15.m6.1"><semantics id="S1.F1.15.m6.1b"><mi id="S1.F1.15.m6.1.1" xref="S1.F1.15.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.F1.15.m6.1c"><ci id="S1.F1.15.m6.1.1.cmml" xref="S1.F1.15.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.15.m6.1d">f</annotation><annotation encoding="application/x-llamapun" id="S1.F1.15.m6.1e">italic_f</annotation></semantics></math>. Asymptotic bounds on <math alttext="p" class="ltx_Math" display="inline" id="S1.F1.16.m7.1"><semantics id="S1.F1.16.m7.1b"><mi id="S1.F1.16.m7.1.1" xref="S1.F1.16.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.F1.16.m7.1c"><ci id="S1.F1.16.m7.1.1.cmml" xref="S1.F1.16.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.16.m7.1d">p</annotation><annotation encoding="application/x-llamapun" id="S1.F1.16.m7.1e">italic_p</annotation></semantics></math> in terms of <math alttext="n" class="ltx_Math" display="inline" id="S1.F1.17.m8.1"><semantics id="S1.F1.17.m8.1b"><mi id="S1.F1.17.m8.1.1" xref="S1.F1.17.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.F1.17.m8.1c"><ci id="S1.F1.17.m8.1.1.cmml" xref="S1.F1.17.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.17.m8.1d">n</annotation><annotation encoding="application/x-llamapun" id="S1.F1.17.m8.1e">italic_n</annotation></semantics></math> are given in <span class="ltx_ERROR undefined" id="S1.F1.19.1">\citet</span>zanotti2025pieces.</figcaption> </figure> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Definitions</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.8">This section introduces the definitions and notation used throughout this work, with a primary focus on continuous piecewise affine functions. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS1" title="2.1 Motivation for a Non-Standard Definition of Polygons ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2.1</span></a> provides the motivation for how I will define these functions in the subsequent sections <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS2" title="2.2 Polygons ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2.2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS3" title="2.3 Continuous Piecewise Affine Functions ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2.3</span></a>. Subsequently, the class of <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">roman_ReLU</annotation></semantics></math> neural networks is introduced in section <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.SS4" title="2.4 Neural Networks ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2.4</span></a>. For <math alttext="n\in\mathds{N}" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mrow id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mi id="S2.p1.2.m2.1.1.2" xref="S2.p1.2.m2.1.1.2.cmml">n</mi><mo id="S2.p1.2.m2.1.1.1" xref="S2.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S2.p1.2.m2.1.1.3" xref="S2.p1.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><apply id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1"><in id="S2.p1.2.m2.1.1.1.cmml" xref="S2.p1.2.m2.1.1.1"></in><ci id="S2.p1.2.m2.1.1.2.cmml" xref="S2.p1.2.m2.1.1.2">𝑛</ci><ci id="S2.p1.2.m2.1.1.3.cmml" xref="S2.p1.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">n\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, I use the notation <math alttext="[n]:=\{1,\dots,n\}" class="ltx_Math" display="inline" id="S2.p1.3.m3.4"><semantics id="S2.p1.3.m3.4a"><mrow id="S2.p1.3.m3.4.5" xref="S2.p1.3.m3.4.5.cmml"><mrow id="S2.p1.3.m3.4.5.2.2" xref="S2.p1.3.m3.4.5.2.1.cmml"><mo id="S2.p1.3.m3.4.5.2.2.1" stretchy="false" xref="S2.p1.3.m3.4.5.2.1.1.cmml">[</mo><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">n</mi><mo id="S2.p1.3.m3.4.5.2.2.2" rspace="0.278em" stretchy="false" xref="S2.p1.3.m3.4.5.2.1.1.cmml">]</mo></mrow><mo id="S2.p1.3.m3.4.5.1" rspace="0.278em" xref="S2.p1.3.m3.4.5.1.cmml">:=</mo><mrow id="S2.p1.3.m3.4.5.3.2" xref="S2.p1.3.m3.4.5.3.1.cmml"><mo id="S2.p1.3.m3.4.5.3.2.1" stretchy="false" xref="S2.p1.3.m3.4.5.3.1.cmml">{</mo><mn id="S2.p1.3.m3.2.2" xref="S2.p1.3.m3.2.2.cmml">1</mn><mo id="S2.p1.3.m3.4.5.3.2.2" xref="S2.p1.3.m3.4.5.3.1.cmml">,</mo><mi id="S2.p1.3.m3.3.3" mathvariant="normal" xref="S2.p1.3.m3.3.3.cmml">…</mi><mo id="S2.p1.3.m3.4.5.3.2.3" xref="S2.p1.3.m3.4.5.3.1.cmml">,</mo><mi id="S2.p1.3.m3.4.4" xref="S2.p1.3.m3.4.4.cmml">n</mi><mo id="S2.p1.3.m3.4.5.3.2.4" stretchy="false" xref="S2.p1.3.m3.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.4b"><apply id="S2.p1.3.m3.4.5.cmml" xref="S2.p1.3.m3.4.5"><csymbol cd="latexml" id="S2.p1.3.m3.4.5.1.cmml" xref="S2.p1.3.m3.4.5.1">assign</csymbol><apply id="S2.p1.3.m3.4.5.2.1.cmml" xref="S2.p1.3.m3.4.5.2.2"><csymbol cd="latexml" id="S2.p1.3.m3.4.5.2.1.1.cmml" xref="S2.p1.3.m3.4.5.2.2.1">delimited-[]</csymbol><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝑛</ci></apply><set id="S2.p1.3.m3.4.5.3.1.cmml" xref="S2.p1.3.m3.4.5.3.2"><cn id="S2.p1.3.m3.2.2.cmml" type="integer" xref="S2.p1.3.m3.2.2">1</cn><ci id="S2.p1.3.m3.3.3.cmml" xref="S2.p1.3.m3.3.3">…</ci><ci id="S2.p1.3.m3.4.4.cmml" xref="S2.p1.3.m3.4.4">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.4c">[n]:=\{1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.4d">[ italic_n ] := { 1 , … , italic_n }</annotation></semantics></math>. For a set <math alttext="X\subset\mathds{R}^{d}" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><mrow id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml">X</mi><mo id="S2.p1.4.m4.1.1.1" xref="S2.p1.4.m4.1.1.1.cmml">⊂</mo><msup id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml"><mi id="S2.p1.4.m4.1.1.3.2" xref="S2.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="S2.p1.4.m4.1.1.3.3" xref="S2.p1.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><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"><subset id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1.1"></subset><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">𝑋</ci><apply id="S2.p1.4.m4.1.1.3.cmml" xref="S2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p1.4.m4.1.1.3.1.cmml" xref="S2.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.p1.4.m4.1.1.3.2.cmml" xref="S2.p1.4.m4.1.1.3.2">ℝ</ci><ci id="S2.p1.4.m4.1.1.3.3.cmml" xref="S2.p1.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">X\subset\mathds{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">italic_X ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname*{int}{X}" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><mrow id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml"><mo id="S2.p1.5.m5.1.1.1" rspace="0.167em" xref="S2.p1.5.m5.1.1.1.cmml">int</mo><mi id="S2.p1.5.m5.1.1.2" xref="S2.p1.5.m5.1.1.2.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><apply id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1"><ci id="S2.p1.5.m5.1.1.1.cmml" xref="S2.p1.5.m5.1.1.1">int</ci><ci id="S2.p1.5.m5.1.1.2.cmml" xref="S2.p1.5.m5.1.1.2">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">\operatorname*{int}{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">roman_int italic_X</annotation></semantics></math> denotes the interior of <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_X</annotation></semantics></math>, and <math alttext="\overline{X}" class="ltx_Math" display="inline" id="S2.p1.7.m7.1"><semantics id="S2.p1.7.m7.1a"><mover accent="true" id="S2.p1.7.m7.1.1" xref="S2.p1.7.m7.1.1.cmml"><mi id="S2.p1.7.m7.1.1.2" xref="S2.p1.7.m7.1.1.2.cmml">X</mi><mo id="S2.p1.7.m7.1.1.1" xref="S2.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.7.m7.1b"><apply id="S2.p1.7.m7.1.1.cmml" xref="S2.p1.7.m7.1.1"><ci id="S2.p1.7.m7.1.1.1.cmml" xref="S2.p1.7.m7.1.1.1">¯</ci><ci id="S2.p1.7.m7.1.1.2.cmml" xref="S2.p1.7.m7.1.1.2">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m7.1c">\overline{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m7.1d">over¯ start_ARG italic_X end_ARG</annotation></semantics></math> denotes the closure of <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mi id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><ci id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">italic_X</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Motivation for a Non-Standard Definition of Polygons</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.7">Let us begin with the most general notion of a continuous piecewise affine function. Let <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">caligraphic_P</annotation></semantics></math> be a partition of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><msup id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">ℝ</mi><mn id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">superscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">ℝ</ci><cn id="S2.SS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, and let <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mrow id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml">f</mi><mo id="S2.SS1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml"><msup id="S2.SS1.p1.3.m3.1.1.3.2" xref="S2.SS1.p1.3.m3.1.1.3.2.cmml"><mi id="S2.SS1.p1.3.m3.1.1.3.2.2" xref="S2.SS1.p1.3.m3.1.1.3.2.2.cmml">ℝ</mi><mn id="S2.SS1.p1.3.m3.1.1.3.2.3" xref="S2.SS1.p1.3.m3.1.1.3.2.3.cmml">2</mn></msup><mo id="S2.SS1.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.SS1.p1.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.SS1.p1.3.m3.1.1.3.3" xref="S2.SS1.p1.3.m3.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><ci id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1.1">:</ci><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝑓</ci><apply id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3"><ci id="S2.SS1.p1.3.m3.1.1.3.1.cmml" xref="S2.SS1.p1.3.m3.1.1.3.1">→</ci><apply id="S2.SS1.p1.3.m3.1.1.3.2.cmml" xref="S2.SS1.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.p1.3.m3.1.1.3.2.2">ℝ</ci><cn id="S2.SS1.p1.3.m3.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p1.3.m3.1.1.3.2.3">2</cn></apply><ci id="S2.SS1.p1.3.m3.1.1.3.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> be continuous, such that for each <math alttext="P\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">P</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><in id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></in><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑃</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">P\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_P ∈ caligraphic_P</annotation></semantics></math>, there exists an affine function <math alttext="f_{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><msub id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">f</mi><mi id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝑓</ci><ci id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="f|_{P}=f_{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.2"><semantics id="S2.SS1.p1.6.m6.2a"><mrow id="S2.SS1.p1.6.m6.2.3" xref="S2.SS1.p1.6.m6.2.3.cmml"><msub id="S2.SS1.p1.6.m6.2.3.2.2" xref="S2.SS1.p1.6.m6.2.3.2.1.cmml"><mrow id="S2.SS1.p1.6.m6.2.3.2.2.2" xref="S2.SS1.p1.6.m6.2.3.2.1.cmml"><mi id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">f</mi><mo id="S2.SS1.p1.6.m6.2.3.2.2.2.1" stretchy="false" xref="S2.SS1.p1.6.m6.2.3.2.1.1.cmml">|</mo></mrow><mi id="S2.SS1.p1.6.m6.2.2.1" xref="S2.SS1.p1.6.m6.2.2.1.cmml">P</mi></msub><mo id="S2.SS1.p1.6.m6.2.3.1" xref="S2.SS1.p1.6.m6.2.3.1.cmml">=</mo><msub id="S2.SS1.p1.6.m6.2.3.3" xref="S2.SS1.p1.6.m6.2.3.3.cmml"><mi id="S2.SS1.p1.6.m6.2.3.3.2" xref="S2.SS1.p1.6.m6.2.3.3.2.cmml">f</mi><mi id="S2.SS1.p1.6.m6.2.3.3.3" xref="S2.SS1.p1.6.m6.2.3.3.3.cmml">P</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.2b"><apply id="S2.SS1.p1.6.m6.2.3.cmml" xref="S2.SS1.p1.6.m6.2.3"><eq id="S2.SS1.p1.6.m6.2.3.1.cmml" xref="S2.SS1.p1.6.m6.2.3.1"></eq><apply id="S2.SS1.p1.6.m6.2.3.2.1.cmml" xref="S2.SS1.p1.6.m6.2.3.2.2"><csymbol cd="latexml" id="S2.SS1.p1.6.m6.2.3.2.1.1.cmml" xref="S2.SS1.p1.6.m6.2.3.2.2.2.1">evaluated-at</csymbol><ci id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1">𝑓</ci><ci id="S2.SS1.p1.6.m6.2.2.1.cmml" xref="S2.SS1.p1.6.m6.2.2.1">𝑃</ci></apply><apply id="S2.SS1.p1.6.m6.2.3.3.cmml" xref="S2.SS1.p1.6.m6.2.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.2.3.3.1.cmml" xref="S2.SS1.p1.6.m6.2.3.3">subscript</csymbol><ci id="S2.SS1.p1.6.m6.2.3.3.2.cmml" xref="S2.SS1.p1.6.m6.2.3.3.2">𝑓</ci><ci id="S2.SS1.p1.6.m6.2.3.3.3.cmml" xref="S2.SS1.p1.6.m6.2.3.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.2c">f|_{P}=f_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.2d">italic_f | start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>. For now, I call such a function continuous piecewise affine. However, this definition is too broad, as choosing <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><ci id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">caligraphic_P</annotation></semantics></math> as singletons would make every continuous function piecewise affine.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.2">To avoid this and to use the number of sets in <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">caligraphic_P</annotation></semantics></math> as a measure of complexity, impose the condition that <math alttext="|\mathcal{P}|" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.2.2" xref="S2.SS1.p2.2.m2.1.2.1.cmml"><mo id="S2.SS1.p2.2.m2.1.2.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">𝒫</mi><mo id="S2.SS1.p2.2.m2.1.2.2.2" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.2"><abs id="S2.SS1.p2.2.m2.1.2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.2.2.1"></abs><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">|\mathcal{P}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">| caligraphic_P |</annotation></semantics></math> is finite.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.9">Additionally, we can assume <math alttext="f_{P}\neq f_{Q}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><msub id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.cmml">f</mi><mi id="S2.SS1.p3.1.m1.1.1.2.3" xref="S2.SS1.p3.1.m1.1.1.2.3.cmml">P</mi></msub><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">≠</mo><msub id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">f</mi><mi id="S2.SS1.p3.1.m1.1.1.3.3" xref="S2.SS1.p3.1.m1.1.1.3.3.cmml">Q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><neq id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></neq><apply id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.2.1.cmml" xref="S2.SS1.p3.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.2.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2.2">𝑓</ci><ci id="S2.SS1.p3.1.m1.1.1.2.3.cmml" xref="S2.SS1.p3.1.m1.1.1.2.3">𝑃</ci></apply><apply id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2">𝑓</ci><ci id="S2.SS1.p3.1.m1.1.1.3.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">f_{P}\neq f_{Q}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ≠ italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT</annotation></semantics></math> for distinct <math alttext="P,Q\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.2"><semantics id="S2.SS1.p3.2.m2.2a"><mrow id="S2.SS1.p3.2.m2.2.3" xref="S2.SS1.p3.2.m2.2.3.cmml"><mrow id="S2.SS1.p3.2.m2.2.3.2.2" xref="S2.SS1.p3.2.m2.2.3.2.1.cmml"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">P</mi><mo id="S2.SS1.p3.2.m2.2.3.2.2.1" xref="S2.SS1.p3.2.m2.2.3.2.1.cmml">,</mo><mi id="S2.SS1.p3.2.m2.2.2" xref="S2.SS1.p3.2.m2.2.2.cmml">Q</mi></mrow><mo id="S2.SS1.p3.2.m2.2.3.1" xref="S2.SS1.p3.2.m2.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.2.m2.2.3.3" xref="S2.SS1.p3.2.m2.2.3.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.2b"><apply id="S2.SS1.p3.2.m2.2.3.cmml" xref="S2.SS1.p3.2.m2.2.3"><in id="S2.SS1.p3.2.m2.2.3.1.cmml" xref="S2.SS1.p3.2.m2.2.3.1"></in><list id="S2.SS1.p3.2.m2.2.3.2.1.cmml" xref="S2.SS1.p3.2.m2.2.3.2.2"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝑃</ci><ci id="S2.SS1.p3.2.m2.2.2.cmml" xref="S2.SS1.p3.2.m2.2.2">𝑄</ci></list><ci id="S2.SS1.p3.2.m2.2.3.3.cmml" xref="S2.SS1.p3.2.m2.2.3.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.2c">P,Q\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.2d">italic_P , italic_Q ∈ caligraphic_P</annotation></semantics></math> by merging sets where the same affine function applies. Then, the set of ambiguities <math alttext="L:=\bigcup_{P\neq Q}\{x:f_{P}(x)=f_{Q}(x)\}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.4"><semantics id="S2.SS1.p3.3.m3.4a"><mrow id="S2.SS1.p3.3.m3.4.4" xref="S2.SS1.p3.3.m3.4.4.cmml"><mi id="S2.SS1.p3.3.m3.4.4.3" xref="S2.SS1.p3.3.m3.4.4.3.cmml">L</mi><mo id="S2.SS1.p3.3.m3.4.4.2" lspace="0.278em" rspace="0.111em" xref="S2.SS1.p3.3.m3.4.4.2.cmml">:=</mo><mrow id="S2.SS1.p3.3.m3.4.4.1" xref="S2.SS1.p3.3.m3.4.4.1.cmml"><msub id="S2.SS1.p3.3.m3.4.4.1.2" xref="S2.SS1.p3.3.m3.4.4.1.2.cmml"><mo id="S2.SS1.p3.3.m3.4.4.1.2.2" rspace="0em" xref="S2.SS1.p3.3.m3.4.4.1.2.2.cmml">⋃</mo><mrow id="S2.SS1.p3.3.m3.4.4.1.2.3" xref="S2.SS1.p3.3.m3.4.4.1.2.3.cmml"><mi id="S2.SS1.p3.3.m3.4.4.1.2.3.2" xref="S2.SS1.p3.3.m3.4.4.1.2.3.2.cmml">P</mi><mo id="S2.SS1.p3.3.m3.4.4.1.2.3.1" xref="S2.SS1.p3.3.m3.4.4.1.2.3.1.cmml">≠</mo><mi id="S2.SS1.p3.3.m3.4.4.1.2.3.3" xref="S2.SS1.p3.3.m3.4.4.1.2.3.3.cmml">Q</mi></mrow></msub><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1" xref="S2.SS1.p3.3.m3.4.4.1.1.2.cmml"><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.2" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.2.1.cmml">{</mo><mi id="S2.SS1.p3.3.m3.3.3" xref="S2.SS1.p3.3.m3.3.3.cmml">x</mi><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p3.3.m3.4.4.1.1.2.1.cmml">:</mo><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1.1" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.cmml"><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.cmml"><msub id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.cmml"><mi id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.2.cmml">f</mi><mi id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.3" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.3.cmml">P</mi></msub><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.1" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.3.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.cmml"><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.cmml">(</mo><mi id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">x</mi><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.1" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.cmml"><msub id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.cmml"><mi id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.2.cmml">f</mi><mi id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.3" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.3.cmml">Q</mi></msub><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.1" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.3.2" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.cmml"><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.cmml">(</mo><mi id="S2.SS1.p3.3.m3.2.2" xref="S2.SS1.p3.3.m3.2.2.cmml">x</mi><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS1.p3.3.m3.4.4.1.1.1.4" stretchy="false" xref="S2.SS1.p3.3.m3.4.4.1.1.2.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.4b"><apply id="S2.SS1.p3.3.m3.4.4.cmml" xref="S2.SS1.p3.3.m3.4.4"><csymbol cd="latexml" id="S2.SS1.p3.3.m3.4.4.2.cmml" xref="S2.SS1.p3.3.m3.4.4.2">assign</csymbol><ci id="S2.SS1.p3.3.m3.4.4.3.cmml" xref="S2.SS1.p3.3.m3.4.4.3">𝐿</ci><apply id="S2.SS1.p3.3.m3.4.4.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1"><apply id="S2.SS1.p3.3.m3.4.4.1.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.4.4.1.2.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2">subscript</csymbol><union id="S2.SS1.p3.3.m3.4.4.1.2.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2.2"></union><apply id="S2.SS1.p3.3.m3.4.4.1.2.3.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2.3"><neq id="S2.SS1.p3.3.m3.4.4.1.2.3.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2.3.1"></neq><ci id="S2.SS1.p3.3.m3.4.4.1.2.3.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2.3.2">𝑃</ci><ci id="S2.SS1.p3.3.m3.4.4.1.2.3.3.cmml" xref="S2.SS1.p3.3.m3.4.4.1.2.3.3">𝑄</ci></apply></apply><apply id="S2.SS1.p3.3.m3.4.4.1.1.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1"><csymbol cd="latexml" id="S2.SS1.p3.3.m3.4.4.1.1.2.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.2">conditional-set</csymbol><ci id="S2.SS1.p3.3.m3.3.3.cmml" xref="S2.SS1.p3.3.m3.3.3">𝑥</ci><apply id="S2.SS1.p3.3.m3.4.4.1.1.1.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1"><eq id="S2.SS1.p3.3.m3.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.1"></eq><apply id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2"><times id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.1"></times><apply id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2">subscript</csymbol><ci id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.2">𝑓</ci><ci id="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.3.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.2.2.3">𝑃</ci></apply><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">𝑥</ci></apply><apply id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3"><times id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.1"></times><apply id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.1.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.2.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.2">𝑓</ci><ci id="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.3.cmml" xref="S2.SS1.p3.3.m3.4.4.1.1.1.1.3.2.3">𝑄</ci></apply><ci id="S2.SS1.p3.3.m3.2.2.cmml" xref="S2.SS1.p3.3.m3.2.2">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.4c">L:=\bigcup_{P\neq Q}\{x:f_{P}(x)=f_{Q}(x)\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.4d">italic_L := ⋃ start_POSTSUBSCRIPT italic_P ≠ italic_Q end_POSTSUBSCRIPT { italic_x : italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) = italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_x ) }</annotation></semantics></math> consists of finitely many lines. Hence, the complement <math alttext="U:=\mathds{R}^{2}\setminus L" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><mrow id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><mi id="S2.SS1.p3.4.m4.1.1.2" xref="S2.SS1.p3.4.m4.1.1.2.cmml">U</mi><mo id="S2.SS1.p3.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p3.4.m4.1.1.1.cmml">:=</mo><mrow id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml"><msup id="S2.SS1.p3.4.m4.1.1.3.2" xref="S2.SS1.p3.4.m4.1.1.3.2.cmml"><mi id="S2.SS1.p3.4.m4.1.1.3.2.2" xref="S2.SS1.p3.4.m4.1.1.3.2.2.cmml">ℝ</mi><mn id="S2.SS1.p3.4.m4.1.1.3.2.3" xref="S2.SS1.p3.4.m4.1.1.3.2.3.cmml">2</mn></msup><mo id="S2.SS1.p3.4.m4.1.1.3.1" xref="S2.SS1.p3.4.m4.1.1.3.1.cmml">∖</mo><mi id="S2.SS1.p3.4.m4.1.1.3.3" xref="S2.SS1.p3.4.m4.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><apply id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1"><csymbol cd="latexml" id="S2.SS1.p3.4.m4.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1">assign</csymbol><ci id="S2.SS1.p3.4.m4.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.2">𝑈</ci><apply id="S2.SS1.p3.4.m4.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3"><setdiff id="S2.SS1.p3.4.m4.1.1.3.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3.1"></setdiff><apply id="S2.SS1.p3.4.m4.1.1.3.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.3.2.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.3.2.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2.2">ℝ</ci><cn id="S2.SS1.p3.4.m4.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p3.4.m4.1.1.3.2.3">2</cn></apply><ci id="S2.SS1.p3.4.m4.1.1.3.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">U:=\mathds{R}^{2}\setminus L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">italic_U := blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_L</annotation></semantics></math> is a dense, open subset of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><msup id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml"><mi id="S2.SS1.p3.5.m5.1.1.2" xref="S2.SS1.p3.5.m5.1.1.2.cmml">ℝ</mi><mn id="S2.SS1.p3.5.m5.1.1.3" xref="S2.SS1.p3.5.m5.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><apply id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.1.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">superscript</csymbol><ci id="S2.SS1.p3.5.m5.1.1.2.cmml" xref="S2.SS1.p3.5.m5.1.1.2">ℝ</ci><cn id="S2.SS1.p3.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p3.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. By continuity of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.1"><semantics id="S2.SS1.p3.6.m6.1a"><mi id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><ci id="S2.SS1.p3.6.m6.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.1d">italic_f</annotation></semantics></math>, each <math alttext="P^{o}:=P\cap U" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m7.1"><semantics id="S2.SS1.p3.7.m7.1a"><mrow id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml"><msup id="S2.SS1.p3.7.m7.1.1.2" xref="S2.SS1.p3.7.m7.1.1.2.cmml"><mi id="S2.SS1.p3.7.m7.1.1.2.2" xref="S2.SS1.p3.7.m7.1.1.2.2.cmml">P</mi><mi id="S2.SS1.p3.7.m7.1.1.2.3" xref="S2.SS1.p3.7.m7.1.1.2.3.cmml">o</mi></msup><mo id="S2.SS1.p3.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p3.7.m7.1.1.1.cmml">:=</mo><mrow id="S2.SS1.p3.7.m7.1.1.3" xref="S2.SS1.p3.7.m7.1.1.3.cmml"><mi id="S2.SS1.p3.7.m7.1.1.3.2" xref="S2.SS1.p3.7.m7.1.1.3.2.cmml">P</mi><mo id="S2.SS1.p3.7.m7.1.1.3.1" xref="S2.SS1.p3.7.m7.1.1.3.1.cmml">∩</mo><mi id="S2.SS1.p3.7.m7.1.1.3.3" xref="S2.SS1.p3.7.m7.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.1b"><apply id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1"><csymbol cd="latexml" id="S2.SS1.p3.7.m7.1.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1.1">assign</csymbol><apply id="S2.SS1.p3.7.m7.1.1.2.cmml" xref="S2.SS1.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m7.1.1.2.1.cmml" xref="S2.SS1.p3.7.m7.1.1.2">superscript</csymbol><ci id="S2.SS1.p3.7.m7.1.1.2.2.cmml" xref="S2.SS1.p3.7.m7.1.1.2.2">𝑃</ci><ci id="S2.SS1.p3.7.m7.1.1.2.3.cmml" xref="S2.SS1.p3.7.m7.1.1.2.3">𝑜</ci></apply><apply id="S2.SS1.p3.7.m7.1.1.3.cmml" xref="S2.SS1.p3.7.m7.1.1.3"><intersect id="S2.SS1.p3.7.m7.1.1.3.1.cmml" xref="S2.SS1.p3.7.m7.1.1.3.1"></intersect><ci id="S2.SS1.p3.7.m7.1.1.3.2.cmml" xref="S2.SS1.p3.7.m7.1.1.3.2">𝑃</ci><ci id="S2.SS1.p3.7.m7.1.1.3.3.cmml" xref="S2.SS1.p3.7.m7.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.1c">P^{o}:=P\cap U</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m7.1d">italic_P start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT := italic_P ∩ italic_U</annotation></semantics></math> is a union of connected components of <math alttext="U" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m8.1"><semantics id="S2.SS1.p3.8.m8.1a"><mi id="S2.SS1.p3.8.m8.1.1" xref="S2.SS1.p3.8.m8.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.1b"><ci id="S2.SS1.p3.8.m8.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m8.1d">italic_U</annotation></semantics></math>, with <math alttext="U=\bigcup_{P\in\mathcal{P}}P^{o}" class="ltx_Math" display="inline" id="S2.SS1.p3.9.m9.1"><semantics id="S2.SS1.p3.9.m9.1a"><mrow id="S2.SS1.p3.9.m9.1.1" xref="S2.SS1.p3.9.m9.1.1.cmml"><mi id="S2.SS1.p3.9.m9.1.1.2" xref="S2.SS1.p3.9.m9.1.1.2.cmml">U</mi><mo id="S2.SS1.p3.9.m9.1.1.1" rspace="0.111em" xref="S2.SS1.p3.9.m9.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.9.m9.1.1.3" xref="S2.SS1.p3.9.m9.1.1.3.cmml"><msub id="S2.SS1.p3.9.m9.1.1.3.1" xref="S2.SS1.p3.9.m9.1.1.3.1.cmml"><mo id="S2.SS1.p3.9.m9.1.1.3.1.2" xref="S2.SS1.p3.9.m9.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.SS1.p3.9.m9.1.1.3.1.3" xref="S2.SS1.p3.9.m9.1.1.3.1.3.cmml"><mi id="S2.SS1.p3.9.m9.1.1.3.1.3.2" xref="S2.SS1.p3.9.m9.1.1.3.1.3.2.cmml">P</mi><mo id="S2.SS1.p3.9.m9.1.1.3.1.3.1" xref="S2.SS1.p3.9.m9.1.1.3.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.9.m9.1.1.3.1.3.3" xref="S2.SS1.p3.9.m9.1.1.3.1.3.3.cmml">𝒫</mi></mrow></msub><msup id="S2.SS1.p3.9.m9.1.1.3.2" xref="S2.SS1.p3.9.m9.1.1.3.2.cmml"><mi id="S2.SS1.p3.9.m9.1.1.3.2.2" xref="S2.SS1.p3.9.m9.1.1.3.2.2.cmml">P</mi><mi id="S2.SS1.p3.9.m9.1.1.3.2.3" xref="S2.SS1.p3.9.m9.1.1.3.2.3.cmml">o</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m9.1b"><apply id="S2.SS1.p3.9.m9.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1"><eq id="S2.SS1.p3.9.m9.1.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1.1"></eq><ci id="S2.SS1.p3.9.m9.1.1.2.cmml" xref="S2.SS1.p3.9.m9.1.1.2">𝑈</ci><apply id="S2.SS1.p3.9.m9.1.1.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3"><apply id="S2.SS1.p3.9.m9.1.1.3.1.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m9.1.1.3.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1">subscript</csymbol><union id="S2.SS1.p3.9.m9.1.1.3.1.2.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1.2"></union><apply id="S2.SS1.p3.9.m9.1.1.3.1.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1.3"><in id="S2.SS1.p3.9.m9.1.1.3.1.3.1.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1.3.1"></in><ci id="S2.SS1.p3.9.m9.1.1.3.1.3.2.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1.3.2">𝑃</ci><ci id="S2.SS1.p3.9.m9.1.1.3.1.3.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3.1.3.3">𝒫</ci></apply></apply><apply id="S2.SS1.p3.9.m9.1.1.3.2.cmml" xref="S2.SS1.p3.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m9.1.1.3.2.1.cmml" xref="S2.SS1.p3.9.m9.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p3.9.m9.1.1.3.2.2.cmml" xref="S2.SS1.p3.9.m9.1.1.3.2.2">𝑃</ci><ci id="S2.SS1.p3.9.m9.1.1.3.2.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3.2.3">𝑜</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m9.1c">U=\bigcup_{P\in\mathcal{P}}P^{o}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m9.1d">italic_U = ⋃ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus,</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\bigcup_{P\in\mathcal{P}}\overline{\operatorname*{int}{P}}\supset\bigcup_{P\in% \mathcal{P}}\overline{P^{o}}=\overline{\bigcup_{P\in\mathcal{P}}P^{o}}=% \overline{U}=\mathds{R}^{2}." class="ltx_Math" display="block" id="S2.Ex3.m1.1"><semantics id="S2.Ex3.m1.1a"><mrow id="S2.Ex3.m1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml"><mrow id="S2.Ex3.m1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.2.cmml"><munder id="S2.Ex3.m1.1.1.1.1.2.1" xref="S2.Ex3.m1.1.1.1.1.2.1.cmml"><mo id="S2.Ex3.m1.1.1.1.1.2.1.2" movablelimits="false" xref="S2.Ex3.m1.1.1.1.1.2.1.2.cmml">⋃</mo><mrow id="S2.Ex3.m1.1.1.1.1.2.1.3" xref="S2.Ex3.m1.1.1.1.1.2.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.1.3.2" xref="S2.Ex3.m1.1.1.1.1.2.1.3.2.cmml">P</mi><mo id="S2.Ex3.m1.1.1.1.1.2.1.3.1" xref="S2.Ex3.m1.1.1.1.1.2.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.1.1.1.1.2.1.3.3" xref="S2.Ex3.m1.1.1.1.1.2.1.3.3.cmml">𝒫</mi></mrow></munder><mover accent="true" id="S2.Ex3.m1.1.1.1.1.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.cmml"><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.1" lspace="0.167em" rspace="0.167em" xref="S2.Ex3.m1.1.1.1.1.2.2.2.1.cmml">int</mo><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.cmml">P</mi></mrow><mo id="S2.Ex3.m1.1.1.1.1.2.2.1" xref="S2.Ex3.m1.1.1.1.1.2.2.1.cmml">¯</mo></mover></mrow><mo id="S2.Ex3.m1.1.1.1.1.3" rspace="0.111em" xref="S2.Ex3.m1.1.1.1.1.3.cmml">⊃</mo><mrow id="S2.Ex3.m1.1.1.1.1.4" xref="S2.Ex3.m1.1.1.1.1.4.cmml"><munder id="S2.Ex3.m1.1.1.1.1.4.1" xref="S2.Ex3.m1.1.1.1.1.4.1.cmml"><mo id="S2.Ex3.m1.1.1.1.1.4.1.2" movablelimits="false" xref="S2.Ex3.m1.1.1.1.1.4.1.2.cmml">⋃</mo><mrow id="S2.Ex3.m1.1.1.1.1.4.1.3" xref="S2.Ex3.m1.1.1.1.1.4.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.4.1.3.2" xref="S2.Ex3.m1.1.1.1.1.4.1.3.2.cmml">P</mi><mo id="S2.Ex3.m1.1.1.1.1.4.1.3.1" xref="S2.Ex3.m1.1.1.1.1.4.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.1.1.1.1.4.1.3.3" xref="S2.Ex3.m1.1.1.1.1.4.1.3.3.cmml">𝒫</mi></mrow></munder><mover accent="true" id="S2.Ex3.m1.1.1.1.1.4.2" xref="S2.Ex3.m1.1.1.1.1.4.2.cmml"><msup id="S2.Ex3.m1.1.1.1.1.4.2.2" xref="S2.Ex3.m1.1.1.1.1.4.2.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.4.2.2.2" xref="S2.Ex3.m1.1.1.1.1.4.2.2.2.cmml">P</mi><mi id="S2.Ex3.m1.1.1.1.1.4.2.2.3" xref="S2.Ex3.m1.1.1.1.1.4.2.2.3.cmml">o</mi></msup><mo id="S2.Ex3.m1.1.1.1.1.4.2.1" xref="S2.Ex3.m1.1.1.1.1.4.2.1.cmml">¯</mo></mover></mrow><mo id="S2.Ex3.m1.1.1.1.1.5" rspace="0.111em" xref="S2.Ex3.m1.1.1.1.1.5.cmml">=</mo><mover accent="true" id="S2.Ex3.m1.1.1.1.1.6" xref="S2.Ex3.m1.1.1.1.1.6.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.6.2" xref="S2.Ex3.m1.1.1.1.1.6.2.cmml"><munder id="S2.Ex3.m1.1.1.1.1.6.2.1" xref="S2.Ex3.m1.1.1.1.1.6.2.1.cmml"><mo id="S2.Ex3.m1.1.1.1.1.6.2.1.2" movablelimits="false" xref="S2.Ex3.m1.1.1.1.1.6.2.1.2.cmml">⋃</mo><mrow id="S2.Ex3.m1.1.1.1.1.6.2.1.3" xref="S2.Ex3.m1.1.1.1.1.6.2.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.6.2.1.3.2" xref="S2.Ex3.m1.1.1.1.1.6.2.1.3.2.cmml">P</mi><mo id="S2.Ex3.m1.1.1.1.1.6.2.1.3.1" xref="S2.Ex3.m1.1.1.1.1.6.2.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.1.1.1.1.6.2.1.3.3" xref="S2.Ex3.m1.1.1.1.1.6.2.1.3.3.cmml">𝒫</mi></mrow></munder><msup id="S2.Ex3.m1.1.1.1.1.6.2.2" xref="S2.Ex3.m1.1.1.1.1.6.2.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.6.2.2.2" xref="S2.Ex3.m1.1.1.1.1.6.2.2.2.cmml">P</mi><mi id="S2.Ex3.m1.1.1.1.1.6.2.2.3" xref="S2.Ex3.m1.1.1.1.1.6.2.2.3.cmml">o</mi></msup></mrow><mo id="S2.Ex3.m1.1.1.1.1.6.1" xref="S2.Ex3.m1.1.1.1.1.6.1.cmml">¯</mo></mover><mo id="S2.Ex3.m1.1.1.1.1.7" xref="S2.Ex3.m1.1.1.1.1.7.cmml">=</mo><mover accent="true" id="S2.Ex3.m1.1.1.1.1.8" xref="S2.Ex3.m1.1.1.1.1.8.cmml"><mi id="S2.Ex3.m1.1.1.1.1.8.2" xref="S2.Ex3.m1.1.1.1.1.8.2.cmml">U</mi><mo id="S2.Ex3.m1.1.1.1.1.8.1" xref="S2.Ex3.m1.1.1.1.1.8.1.cmml">¯</mo></mover><mo id="S2.Ex3.m1.1.1.1.1.9" xref="S2.Ex3.m1.1.1.1.1.9.cmml">=</mo><msup id="S2.Ex3.m1.1.1.1.1.10" xref="S2.Ex3.m1.1.1.1.1.10.cmml"><mi id="S2.Ex3.m1.1.1.1.1.10.2" xref="S2.Ex3.m1.1.1.1.1.10.2.cmml">ℝ</mi><mn id="S2.Ex3.m1.1.1.1.1.10.3" xref="S2.Ex3.m1.1.1.1.1.10.3.cmml">2</mn></msup></mrow><mo id="S2.Ex3.m1.1.1.1.2" lspace="0em" xref="S2.Ex3.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.1b"><apply id="S2.Ex3.m1.1.1.1.1.cmml" xref="S2.Ex3.m1.1.1.1"><and id="S2.Ex3.m1.1.1.1.1a.cmml" xref="S2.Ex3.m1.1.1.1"></and><apply id="S2.Ex3.m1.1.1.1.1b.cmml" xref="S2.Ex3.m1.1.1.1"><csymbol cd="latexml" id="S2.Ex3.m1.1.1.1.1.3.cmml" xref="S2.Ex3.m1.1.1.1.1.3">superset-of</csymbol><apply id="S2.Ex3.m1.1.1.1.1.2.cmml" xref="S2.Ex3.m1.1.1.1.1.2"><apply id="S2.Ex3.m1.1.1.1.1.2.1.cmml" 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encoding="application/x-llamapun" id="S2.Ex3.m1.1d">⋃ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT over¯ start_ARG roman_int italic_P end_ARG ⊃ ⋃ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT over¯ start_ARG italic_P start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG ⋃ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_U end_ARG = blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.10">Continuity further ensures that <math alttext="f|_{\overline{\operatorname*{int}{P}}}=f_{P}|_{\overline{\operatorname*{int}{P% }}}" class="ltx_Math" display="inline" id="S2.SS1.p3.10.m1.4"><semantics id="S2.SS1.p3.10.m1.4a"><mrow id="S2.SS1.p3.10.m1.4.4" xref="S2.SS1.p3.10.m1.4.4.cmml"><msub 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xref="S2.SS1.p3.10.m1.4.4.3.2"><csymbol cd="latexml" id="S2.SS1.p3.10.m1.4.4.3.1.1.cmml" xref="S2.SS1.p3.10.m1.4.4.3.2.2.1">evaluated-at</csymbol><ci id="S2.SS1.p3.10.m1.1.1.cmml" xref="S2.SS1.p3.10.m1.1.1">𝑓</ci><apply id="S2.SS1.p3.10.m1.2.2.1.cmml" xref="S2.SS1.p3.10.m1.2.2.1"><ci id="S2.SS1.p3.10.m1.2.2.1.1.cmml" xref="S2.SS1.p3.10.m1.2.2.1.1">¯</ci><apply id="S2.SS1.p3.10.m1.2.2.1.2.cmml" xref="S2.SS1.p3.10.m1.2.2.1.2"><ci id="S2.SS1.p3.10.m1.2.2.1.2.1.cmml" xref="S2.SS1.p3.10.m1.2.2.1.2.1">int</ci><ci id="S2.SS1.p3.10.m1.2.2.1.2.2.cmml" xref="S2.SS1.p3.10.m1.2.2.1.2.2">𝑃</ci></apply></apply></apply><apply id="S2.SS1.p3.10.m1.4.4.1.2.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1"><csymbol cd="latexml" id="S2.SS1.p3.10.m1.4.4.1.2.1.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1.1.2">evaluated-at</csymbol><apply id="S2.SS1.p3.10.m1.4.4.1.1.1.1.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m1.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.10.m1.4.4.1.1.1.1.2.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1.1.1.2">𝑓</ci><ci id="S2.SS1.p3.10.m1.4.4.1.1.1.1.3.cmml" xref="S2.SS1.p3.10.m1.4.4.1.1.1.1.3">𝑃</ci></apply><apply id="S2.SS1.p3.10.m1.3.3.1.cmml" xref="S2.SS1.p3.10.m1.3.3.1"><ci id="S2.SS1.p3.10.m1.3.3.1.1.cmml" xref="S2.SS1.p3.10.m1.3.3.1.1">¯</ci><apply id="S2.SS1.p3.10.m1.3.3.1.2.cmml" xref="S2.SS1.p3.10.m1.3.3.1.2"><ci id="S2.SS1.p3.10.m1.3.3.1.2.1.cmml" xref="S2.SS1.p3.10.m1.3.3.1.2.1">int</ci><ci id="S2.SS1.p3.10.m1.3.3.1.2.2.cmml" xref="S2.SS1.p3.10.m1.3.3.1.2.2">𝑃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.10.m1.4c">f|_{\overline{\operatorname*{int}{P}}}=f_{P}|_{\overline{\operatorname*{int}{P% }}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.10.m1.4d">italic_f | start_POSTSUBSCRIPT over¯ start_ARG roman_int italic_P end_ARG end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT | start_POSTSUBSCRIPT over¯ start_ARG roman_int italic_P end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.3">Consequently, by considering the closures of connected components of <math alttext="\operatorname*{int}{P}" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mrow id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml"><mo id="S2.SS1.p4.1.m1.1.1.1" rspace="0.167em" xref="S2.SS1.p4.1.m1.1.1.1.cmml">int</mo><mi id="S2.SS1.p4.1.m1.1.1.2" xref="S2.SS1.p4.1.m1.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><apply id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1"><ci id="S2.SS1.p4.1.m1.1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1.1">int</ci><ci id="S2.SS1.p4.1.m1.1.1.2.cmml" xref="S2.SS1.p4.1.m1.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">\operatorname*{int}{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">roman_int italic_P</annotation></semantics></math> as distinct sets, we can replace the partition <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><ci id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">caligraphic_P</annotation></semantics></math> with a finite cover of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><msup id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml"><mi id="S2.SS1.p4.3.m3.1.1.2" xref="S2.SS1.p4.3.m3.1.1.2.cmml">ℝ</mi><mn id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><apply id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.3.m3.1.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">superscript</csymbol><ci id="S2.SS1.p4.3.m3.1.1.2.cmml" xref="S2.SS1.p4.3.m3.1.1.2">ℝ</ci><cn id="S2.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, where each element is a regular closed set with connected interior. Moreover, the intersection of any two such sets is contained within a line.</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.1">This leads to the refined definition presented in the following sections.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Polygons</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">Since I will directly work with the geometric structure of continuous piecewise affine functions, this structure will be defined very explicitly in a bottom-up manner.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.17">Given <math alttext="v,u\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.2"><semantics id="S2.SS2.p2.1.m1.2a"><mrow id="S2.SS2.p2.1.m1.2.3" xref="S2.SS2.p2.1.m1.2.3.cmml"><mrow id="S2.SS2.p2.1.m1.2.3.2.2" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">v</mi><mo id="S2.SS2.p2.1.m1.2.3.2.2.1" xref="S2.SS2.p2.1.m1.2.3.2.1.cmml">,</mo><mi id="S2.SS2.p2.1.m1.2.2" xref="S2.SS2.p2.1.m1.2.2.cmml">u</mi></mrow><mo id="S2.SS2.p2.1.m1.2.3.1" xref="S2.SS2.p2.1.m1.2.3.1.cmml">∈</mo><msup id="S2.SS2.p2.1.m1.2.3.3" xref="S2.SS2.p2.1.m1.2.3.3.cmml"><mi id="S2.SS2.p2.1.m1.2.3.3.2" xref="S2.SS2.p2.1.m1.2.3.3.2.cmml">ℝ</mi><mn id="S2.SS2.p2.1.m1.2.3.3.3" xref="S2.SS2.p2.1.m1.2.3.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.2b"><apply id="S2.SS2.p2.1.m1.2.3.cmml" xref="S2.SS2.p2.1.m1.2.3"><in id="S2.SS2.p2.1.m1.2.3.1.cmml" xref="S2.SS2.p2.1.m1.2.3.1"></in><list id="S2.SS2.p2.1.m1.2.3.2.1.cmml" xref="S2.SS2.p2.1.m1.2.3.2.2"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝑣</ci><ci id="S2.SS2.p2.1.m1.2.2.cmml" xref="S2.SS2.p2.1.m1.2.2">𝑢</ci></list><apply id="S2.SS2.p2.1.m1.2.3.3.cmml" xref="S2.SS2.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.2.3.3.1.cmml" xref="S2.SS2.p2.1.m1.2.3.3">superscript</csymbol><ci id="S2.SS2.p2.1.m1.2.3.3.2.cmml" xref="S2.SS2.p2.1.m1.2.3.3.2">ℝ</ci><cn id="S2.SS2.p2.1.m1.2.3.3.3.cmml" type="integer" xref="S2.SS2.p2.1.m1.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.2c">v,u\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.2d">italic_v , italic_u ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="v\neq u" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">≠</mo><mi id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><apply id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1"><neq id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1"></neq><ci id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2">𝑣</ci><ci id="S2.SS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">v\neq u</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_v ≠ italic_u</annotation></semantics></math>, I call <math alttext="e:=\{v+tu:\,t\in[0,1]\}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.4"><semantics id="S2.SS2.p2.3.m3.4a"><mrow id="S2.SS2.p2.3.m3.4.4" xref="S2.SS2.p2.3.m3.4.4.cmml"><mi id="S2.SS2.p2.3.m3.4.4.4" xref="S2.SS2.p2.3.m3.4.4.4.cmml">e</mi><mo id="S2.SS2.p2.3.m3.4.4.3" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p2.3.m3.4.4.3.cmml">:=</mo><mrow id="S2.SS2.p2.3.m3.4.4.2.2" xref="S2.SS2.p2.3.m3.4.4.2.3.cmml"><mo id="S2.SS2.p2.3.m3.4.4.2.2.3" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.2.3.1.cmml">{</mo><mrow id="S2.SS2.p2.3.m3.3.3.1.1.1" xref="S2.SS2.p2.3.m3.3.3.1.1.1.cmml"><mi id="S2.SS2.p2.3.m3.3.3.1.1.1.2" xref="S2.SS2.p2.3.m3.3.3.1.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.3.m3.3.3.1.1.1.1" xref="S2.SS2.p2.3.m3.3.3.1.1.1.1.cmml">+</mo><mrow id="S2.SS2.p2.3.m3.3.3.1.1.1.3" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.cmml"><mi id="S2.SS2.p2.3.m3.3.3.1.1.1.3.2" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p2.3.m3.3.3.1.1.1.3.1" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.3.m3.3.3.1.1.1.3.3" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.3.cmml">u</mi></mrow></mrow><mo id="S2.SS2.p2.3.m3.4.4.2.2.4" lspace="0.278em" rspace="0.448em" xref="S2.SS2.p2.3.m3.4.4.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.3.m3.4.4.2.2.2" xref="S2.SS2.p2.3.m3.4.4.2.2.2.cmml"><mi id="S2.SS2.p2.3.m3.4.4.2.2.2.2" xref="S2.SS2.p2.3.m3.4.4.2.2.2.2.cmml">t</mi><mo id="S2.SS2.p2.3.m3.4.4.2.2.2.1" xref="S2.SS2.p2.3.m3.4.4.2.2.2.1.cmml">∈</mo><mrow id="S2.SS2.p2.3.m3.4.4.2.2.2.3.2" xref="S2.SS2.p2.3.m3.4.4.2.2.2.3.1.cmml"><mo id="S2.SS2.p2.3.m3.4.4.2.2.2.3.2.1" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.2.2.2.3.1.cmml">[</mo><mn id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">0</mn><mo id="S2.SS2.p2.3.m3.4.4.2.2.2.3.2.2" xref="S2.SS2.p2.3.m3.4.4.2.2.2.3.1.cmml">,</mo><mn id="S2.SS2.p2.3.m3.2.2" xref="S2.SS2.p2.3.m3.2.2.cmml">1</mn><mo id="S2.SS2.p2.3.m3.4.4.2.2.2.3.2.3" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.2.2.2.3.1.cmml">]</mo></mrow></mrow><mo id="S2.SS2.p2.3.m3.4.4.2.2.5" stretchy="false" xref="S2.SS2.p2.3.m3.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.4b"><apply id="S2.SS2.p2.3.m3.4.4.cmml" xref="S2.SS2.p2.3.m3.4.4"><csymbol cd="latexml" id="S2.SS2.p2.3.m3.4.4.3.cmml" xref="S2.SS2.p2.3.m3.4.4.3">assign</csymbol><ci id="S2.SS2.p2.3.m3.4.4.4.cmml" xref="S2.SS2.p2.3.m3.4.4.4">𝑒</ci><apply id="S2.SS2.p2.3.m3.4.4.2.3.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2"><csymbol cd="latexml" id="S2.SS2.p2.3.m3.4.4.2.3.1.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2.3">conditional-set</csymbol><apply id="S2.SS2.p2.3.m3.3.3.1.1.1.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1"><plus id="S2.SS2.p2.3.m3.3.3.1.1.1.1.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.1"></plus><ci id="S2.SS2.p2.3.m3.3.3.1.1.1.2.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.2">𝑣</ci><apply id="S2.SS2.p2.3.m3.3.3.1.1.1.3.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3"><times id="S2.SS2.p2.3.m3.3.3.1.1.1.3.1.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.1"></times><ci id="S2.SS2.p2.3.m3.3.3.1.1.1.3.2.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.2">𝑡</ci><ci id="S2.SS2.p2.3.m3.3.3.1.1.1.3.3.cmml" xref="S2.SS2.p2.3.m3.3.3.1.1.1.3.3">𝑢</ci></apply></apply><apply id="S2.SS2.p2.3.m3.4.4.2.2.2.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2.2"><in id="S2.SS2.p2.3.m3.4.4.2.2.2.1.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2.2.1"></in><ci id="S2.SS2.p2.3.m3.4.4.2.2.2.2.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2.2.2">𝑡</ci><interval closure="closed" id="S2.SS2.p2.3.m3.4.4.2.2.2.3.1.cmml" xref="S2.SS2.p2.3.m3.4.4.2.2.2.3.2"><cn id="S2.SS2.p2.3.m3.1.1.cmml" type="integer" xref="S2.SS2.p2.3.m3.1.1">0</cn><cn id="S2.SS2.p2.3.m3.2.2.cmml" type="integer" xref="S2.SS2.p2.3.m3.2.2">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.4c">e:=\{v+tu:\,t\in[0,1]\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.4d">italic_e := { italic_v + italic_t italic_u : italic_t ∈ [ 0 , 1 ] }</annotation></semantics></math> a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.17.1">line segment</em> connecting its <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.17.2">vertices</em> <math alttext="v" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><mi id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><ci id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">italic_v</annotation></semantics></math> and <math alttext="v+u" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m5.1"><semantics id="S2.SS2.p2.5.m5.1a"><mrow id="S2.SS2.p2.5.m5.1.1" xref="S2.SS2.p2.5.m5.1.1.cmml"><mi id="S2.SS2.p2.5.m5.1.1.2" xref="S2.SS2.p2.5.m5.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.5.m5.1.1.1" xref="S2.SS2.p2.5.m5.1.1.1.cmml">+</mo><mi id="S2.SS2.p2.5.m5.1.1.3" xref="S2.SS2.p2.5.m5.1.1.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m5.1b"><apply id="S2.SS2.p2.5.m5.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1"><plus id="S2.SS2.p2.5.m5.1.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1.1"></plus><ci id="S2.SS2.p2.5.m5.1.1.2.cmml" xref="S2.SS2.p2.5.m5.1.1.2">𝑣</ci><ci id="S2.SS2.p2.5.m5.1.1.3.cmml" xref="S2.SS2.p2.5.m5.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m5.1c">v+u</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m5.1d">italic_v + italic_u</annotation></semantics></math>. A line segment with vertices <math alttext="a" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m6.1"><semantics id="S2.SS2.p2.6.m6.1a"><mi id="S2.SS2.p2.6.m6.1.1" xref="S2.SS2.p2.6.m6.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m6.1b"><ci id="S2.SS2.p2.6.m6.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m6.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m6.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m7.1"><semantics id="S2.SS2.p2.7.m7.1a"><mi id="S2.SS2.p2.7.m7.1.1" xref="S2.SS2.p2.7.m7.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m7.1b"><ci id="S2.SS2.p2.7.m7.1.1.cmml" xref="S2.SS2.p2.7.m7.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m7.1c">b</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m7.1d">italic_b</annotation></semantics></math> will also be denoted by <math alttext="\overline{ab}" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m8.1"><semantics id="S2.SS2.p2.8.m8.1a"><mover accent="true" id="S2.SS2.p2.8.m8.1.1" xref="S2.SS2.p2.8.m8.1.1.cmml"><mrow id="S2.SS2.p2.8.m8.1.1.2" xref="S2.SS2.p2.8.m8.1.1.2.cmml"><mi id="S2.SS2.p2.8.m8.1.1.2.2" xref="S2.SS2.p2.8.m8.1.1.2.2.cmml">a</mi><mo id="S2.SS2.p2.8.m8.1.1.2.1" xref="S2.SS2.p2.8.m8.1.1.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.8.m8.1.1.2.3" xref="S2.SS2.p2.8.m8.1.1.2.3.cmml">b</mi></mrow><mo id="S2.SS2.p2.8.m8.1.1.1" xref="S2.SS2.p2.8.m8.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m8.1b"><apply id="S2.SS2.p2.8.m8.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1"><ci id="S2.SS2.p2.8.m8.1.1.1.cmml" xref="S2.SS2.p2.8.m8.1.1.1">¯</ci><apply id="S2.SS2.p2.8.m8.1.1.2.cmml" xref="S2.SS2.p2.8.m8.1.1.2"><times id="S2.SS2.p2.8.m8.1.1.2.1.cmml" xref="S2.SS2.p2.8.m8.1.1.2.1"></times><ci id="S2.SS2.p2.8.m8.1.1.2.2.cmml" xref="S2.SS2.p2.8.m8.1.1.2.2">𝑎</ci><ci id="S2.SS2.p2.8.m8.1.1.2.3.cmml" xref="S2.SS2.p2.8.m8.1.1.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m8.1c">\overline{ab}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m8.1d">over¯ start_ARG italic_a italic_b end_ARG</annotation></semantics></math>. Similarly, I call <math alttext="r:=\{v+tu:\,t\geq 0\}" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m9.2"><semantics id="S2.SS2.p2.9.m9.2a"><mrow id="S2.SS2.p2.9.m9.2.2" xref="S2.SS2.p2.9.m9.2.2.cmml"><mi id="S2.SS2.p2.9.m9.2.2.4" xref="S2.SS2.p2.9.m9.2.2.4.cmml">r</mi><mo id="S2.SS2.p2.9.m9.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p2.9.m9.2.2.3.cmml">:=</mo><mrow id="S2.SS2.p2.9.m9.2.2.2.2" xref="S2.SS2.p2.9.m9.2.2.2.3.cmml"><mo id="S2.SS2.p2.9.m9.2.2.2.2.3" stretchy="false" xref="S2.SS2.p2.9.m9.2.2.2.3.1.cmml">{</mo><mrow id="S2.SS2.p2.9.m9.1.1.1.1.1" xref="S2.SS2.p2.9.m9.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.9.m9.1.1.1.1.1.2" xref="S2.SS2.p2.9.m9.1.1.1.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.9.m9.1.1.1.1.1.1" xref="S2.SS2.p2.9.m9.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS2.p2.9.m9.1.1.1.1.1.3" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p2.9.m9.1.1.1.1.1.3.2" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p2.9.m9.1.1.1.1.1.3.1" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.9.m9.1.1.1.1.1.3.3" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.3.cmml">u</mi></mrow></mrow><mo id="S2.SS2.p2.9.m9.2.2.2.2.4" lspace="0.278em" rspace="0.448em" xref="S2.SS2.p2.9.m9.2.2.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.9.m9.2.2.2.2.2" xref="S2.SS2.p2.9.m9.2.2.2.2.2.cmml"><mi id="S2.SS2.p2.9.m9.2.2.2.2.2.2" xref="S2.SS2.p2.9.m9.2.2.2.2.2.2.cmml">t</mi><mo id="S2.SS2.p2.9.m9.2.2.2.2.2.1" xref="S2.SS2.p2.9.m9.2.2.2.2.2.1.cmml">≥</mo><mn id="S2.SS2.p2.9.m9.2.2.2.2.2.3" xref="S2.SS2.p2.9.m9.2.2.2.2.2.3.cmml">0</mn></mrow><mo id="S2.SS2.p2.9.m9.2.2.2.2.5" stretchy="false" xref="S2.SS2.p2.9.m9.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m9.2b"><apply id="S2.SS2.p2.9.m9.2.2.cmml" xref="S2.SS2.p2.9.m9.2.2"><csymbol cd="latexml" id="S2.SS2.p2.9.m9.2.2.3.cmml" xref="S2.SS2.p2.9.m9.2.2.3">assign</csymbol><ci id="S2.SS2.p2.9.m9.2.2.4.cmml" xref="S2.SS2.p2.9.m9.2.2.4">𝑟</ci><apply id="S2.SS2.p2.9.m9.2.2.2.3.cmml" xref="S2.SS2.p2.9.m9.2.2.2.2"><csymbol cd="latexml" id="S2.SS2.p2.9.m9.2.2.2.3.1.cmml" xref="S2.SS2.p2.9.m9.2.2.2.2.3">conditional-set</csymbol><apply id="S2.SS2.p2.9.m9.1.1.1.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1"><plus id="S2.SS2.p2.9.m9.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.1"></plus><ci id="S2.SS2.p2.9.m9.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.2">𝑣</ci><apply id="S2.SS2.p2.9.m9.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3"><times id="S2.SS2.p2.9.m9.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.1"></times><ci id="S2.SS2.p2.9.m9.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.2">𝑡</ci><ci id="S2.SS2.p2.9.m9.1.1.1.1.1.3.3.cmml" xref="S2.SS2.p2.9.m9.1.1.1.1.1.3.3">𝑢</ci></apply></apply><apply id="S2.SS2.p2.9.m9.2.2.2.2.2.cmml" xref="S2.SS2.p2.9.m9.2.2.2.2.2"><geq id="S2.SS2.p2.9.m9.2.2.2.2.2.1.cmml" xref="S2.SS2.p2.9.m9.2.2.2.2.2.1"></geq><ci id="S2.SS2.p2.9.m9.2.2.2.2.2.2.cmml" xref="S2.SS2.p2.9.m9.2.2.2.2.2.2">𝑡</ci><cn id="S2.SS2.p2.9.m9.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.p2.9.m9.2.2.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m9.2c">r:=\{v+tu:\,t\geq 0\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m9.2d">italic_r := { italic_v + italic_t italic_u : italic_t ≥ 0 }</annotation></semantics></math> a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.17.3">ray</em> with initial vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m10.1"><semantics id="S2.SS2.p2.10.m10.1a"><mi id="S2.SS2.p2.10.m10.1.1" xref="S2.SS2.p2.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m10.1b"><ci id="S2.SS2.p2.10.m10.1.1.cmml" xref="S2.SS2.p2.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m10.1d">italic_v</annotation></semantics></math>, and <math alttext="l:=\{v+tu:\,t\in\mathds{R}\}" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m11.2"><semantics id="S2.SS2.p2.11.m11.2a"><mrow id="S2.SS2.p2.11.m11.2.2" xref="S2.SS2.p2.11.m11.2.2.cmml"><mi id="S2.SS2.p2.11.m11.2.2.4" xref="S2.SS2.p2.11.m11.2.2.4.cmml">l</mi><mo id="S2.SS2.p2.11.m11.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p2.11.m11.2.2.3.cmml">:=</mo><mrow id="S2.SS2.p2.11.m11.2.2.2.2" xref="S2.SS2.p2.11.m11.2.2.2.3.cmml"><mo id="S2.SS2.p2.11.m11.2.2.2.2.3" stretchy="false" xref="S2.SS2.p2.11.m11.2.2.2.3.1.cmml">{</mo><mrow id="S2.SS2.p2.11.m11.1.1.1.1.1" xref="S2.SS2.p2.11.m11.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.11.m11.1.1.1.1.1.2" xref="S2.SS2.p2.11.m11.1.1.1.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.11.m11.1.1.1.1.1.1" xref="S2.SS2.p2.11.m11.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS2.p2.11.m11.1.1.1.1.1.3" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p2.11.m11.1.1.1.1.1.3.2" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p2.11.m11.1.1.1.1.1.3.1" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.11.m11.1.1.1.1.1.3.3" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.3.cmml">u</mi></mrow></mrow><mo id="S2.SS2.p2.11.m11.2.2.2.2.4" lspace="0.278em" rspace="0.448em" xref="S2.SS2.p2.11.m11.2.2.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.11.m11.2.2.2.2.2" xref="S2.SS2.p2.11.m11.2.2.2.2.2.cmml"><mi id="S2.SS2.p2.11.m11.2.2.2.2.2.2" xref="S2.SS2.p2.11.m11.2.2.2.2.2.2.cmml">t</mi><mo id="S2.SS2.p2.11.m11.2.2.2.2.2.1" xref="S2.SS2.p2.11.m11.2.2.2.2.2.1.cmml">∈</mo><mi id="S2.SS2.p2.11.m11.2.2.2.2.2.3" xref="S2.SS2.p2.11.m11.2.2.2.2.2.3.cmml">ℝ</mi></mrow><mo id="S2.SS2.p2.11.m11.2.2.2.2.5" stretchy="false" xref="S2.SS2.p2.11.m11.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.11.m11.2b"><apply id="S2.SS2.p2.11.m11.2.2.cmml" xref="S2.SS2.p2.11.m11.2.2"><csymbol cd="latexml" id="S2.SS2.p2.11.m11.2.2.3.cmml" xref="S2.SS2.p2.11.m11.2.2.3">assign</csymbol><ci id="S2.SS2.p2.11.m11.2.2.4.cmml" xref="S2.SS2.p2.11.m11.2.2.4">𝑙</ci><apply id="S2.SS2.p2.11.m11.2.2.2.3.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2"><csymbol cd="latexml" id="S2.SS2.p2.11.m11.2.2.2.3.1.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2.3">conditional-set</csymbol><apply id="S2.SS2.p2.11.m11.1.1.1.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1"><plus id="S2.SS2.p2.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.1"></plus><ci id="S2.SS2.p2.11.m11.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.2">𝑣</ci><apply id="S2.SS2.p2.11.m11.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3"><times id="S2.SS2.p2.11.m11.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.1"></times><ci id="S2.SS2.p2.11.m11.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.2">𝑡</ci><ci id="S2.SS2.p2.11.m11.1.1.1.1.1.3.3.cmml" xref="S2.SS2.p2.11.m11.1.1.1.1.1.3.3">𝑢</ci></apply></apply><apply id="S2.SS2.p2.11.m11.2.2.2.2.2.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2.2"><in id="S2.SS2.p2.11.m11.2.2.2.2.2.1.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2.2.1"></in><ci id="S2.SS2.p2.11.m11.2.2.2.2.2.2.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2.2.2">𝑡</ci><ci id="S2.SS2.p2.11.m11.2.2.2.2.2.3.cmml" xref="S2.SS2.p2.11.m11.2.2.2.2.2.3">ℝ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.11.m11.2c">l:=\{v+tu:\,t\in\mathds{R}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m11.2d">italic_l := { italic_v + italic_t italic_u : italic_t ∈ blackboard_R }</annotation></semantics></math> a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.17.4">line</em>. If <math alttext="e\subset\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.12.m12.1"><semantics id="S2.SS2.p2.12.m12.1a"><mrow id="S2.SS2.p2.12.m12.1.1" xref="S2.SS2.p2.12.m12.1.1.cmml"><mi id="S2.SS2.p2.12.m12.1.1.2" xref="S2.SS2.p2.12.m12.1.1.2.cmml">e</mi><mo id="S2.SS2.p2.12.m12.1.1.1" xref="S2.SS2.p2.12.m12.1.1.1.cmml">⊂</mo><msup id="S2.SS2.p2.12.m12.1.1.3" xref="S2.SS2.p2.12.m12.1.1.3.cmml"><mi id="S2.SS2.p2.12.m12.1.1.3.2" xref="S2.SS2.p2.12.m12.1.1.3.2.cmml">ℝ</mi><mn id="S2.SS2.p2.12.m12.1.1.3.3" xref="S2.SS2.p2.12.m12.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.12.m12.1b"><apply id="S2.SS2.p2.12.m12.1.1.cmml" xref="S2.SS2.p2.12.m12.1.1"><subset id="S2.SS2.p2.12.m12.1.1.1.cmml" xref="S2.SS2.p2.12.m12.1.1.1"></subset><ci id="S2.SS2.p2.12.m12.1.1.2.cmml" xref="S2.SS2.p2.12.m12.1.1.2">𝑒</ci><apply id="S2.SS2.p2.12.m12.1.1.3.cmml" xref="S2.SS2.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m12.1.1.3.1.cmml" xref="S2.SS2.p2.12.m12.1.1.3">superscript</csymbol><ci id="S2.SS2.p2.12.m12.1.1.3.2.cmml" xref="S2.SS2.p2.12.m12.1.1.3.2">ℝ</ci><cn id="S2.SS2.p2.12.m12.1.1.3.3.cmml" type="integer" xref="S2.SS2.p2.12.m12.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.12.m12.1c">e\subset\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.12.m12.1d">italic_e ⊂ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is of one of these three types, its <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.17.5">affine hull</em> is defined as the line containing <math alttext="e" class="ltx_Math" display="inline" id="S2.SS2.p2.13.m13.1"><semantics id="S2.SS2.p2.13.m13.1a"><mi id="S2.SS2.p2.13.m13.1.1" xref="S2.SS2.p2.13.m13.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.13.m13.1b"><ci id="S2.SS2.p2.13.m13.1.1.cmml" xref="S2.SS2.p2.13.m13.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.13.m13.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.13.m13.1d">italic_e</annotation></semantics></math>, i.e. if <math alttext="v" class="ltx_Math" display="inline" id="S2.SS2.p2.14.m14.1"><semantics id="S2.SS2.p2.14.m14.1a"><mi id="S2.SS2.p2.14.m14.1.1" xref="S2.SS2.p2.14.m14.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.14.m14.1b"><ci id="S2.SS2.p2.14.m14.1.1.cmml" xref="S2.SS2.p2.14.m14.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.14.m14.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.14.m14.1d">italic_v</annotation></semantics></math> and <math alttext="v+u" class="ltx_Math" display="inline" id="S2.SS2.p2.15.m15.1"><semantics id="S2.SS2.p2.15.m15.1a"><mrow id="S2.SS2.p2.15.m15.1.1" xref="S2.SS2.p2.15.m15.1.1.cmml"><mi id="S2.SS2.p2.15.m15.1.1.2" xref="S2.SS2.p2.15.m15.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.15.m15.1.1.1" xref="S2.SS2.p2.15.m15.1.1.1.cmml">+</mo><mi id="S2.SS2.p2.15.m15.1.1.3" xref="S2.SS2.p2.15.m15.1.1.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.15.m15.1b"><apply id="S2.SS2.p2.15.m15.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1"><plus id="S2.SS2.p2.15.m15.1.1.1.cmml" xref="S2.SS2.p2.15.m15.1.1.1"></plus><ci id="S2.SS2.p2.15.m15.1.1.2.cmml" xref="S2.SS2.p2.15.m15.1.1.2">𝑣</ci><ci id="S2.SS2.p2.15.m15.1.1.3.cmml" xref="S2.SS2.p2.15.m15.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.15.m15.1c">v+u</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.15.m15.1d">italic_v + italic_u</annotation></semantics></math> are two distinct points on <math alttext="e" class="ltx_Math" display="inline" id="S2.SS2.p2.16.m16.1"><semantics id="S2.SS2.p2.16.m16.1a"><mi id="S2.SS2.p2.16.m16.1.1" xref="S2.SS2.p2.16.m16.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.16.m16.1b"><ci id="S2.SS2.p2.16.m16.1.1.cmml" xref="S2.SS2.p2.16.m16.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.16.m16.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.16.m16.1d">italic_e</annotation></semantics></math>, then <math alttext="\operatorname*{aff}(e):=\{v+tu:\,t\in\mathds{R}\}" class="ltx_Math" display="inline" id="S2.SS2.p2.17.m17.4"><semantics id="S2.SS2.p2.17.m17.4a"><mrow id="S2.SS2.p2.17.m17.4.4" xref="S2.SS2.p2.17.m17.4.4.cmml"><mrow id="S2.SS2.p2.17.m17.4.4.4.2" xref="S2.SS2.p2.17.m17.4.4.4.1.cmml"><mo id="S2.SS2.p2.17.m17.1.1" rspace="0em" xref="S2.SS2.p2.17.m17.1.1.cmml">aff</mo><mrow id="S2.SS2.p2.17.m17.4.4.4.2.1" xref="S2.SS2.p2.17.m17.4.4.4.1.cmml"><mo id="S2.SS2.p2.17.m17.4.4.4.2.1.1" stretchy="false" xref="S2.SS2.p2.17.m17.4.4.4.1.cmml">(</mo><mi id="S2.SS2.p2.17.m17.2.2" xref="S2.SS2.p2.17.m17.2.2.cmml">e</mi><mo id="S2.SS2.p2.17.m17.4.4.4.2.1.2" rspace="0.278em" stretchy="false" xref="S2.SS2.p2.17.m17.4.4.4.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.17.m17.4.4.3" rspace="0.278em" xref="S2.SS2.p2.17.m17.4.4.3.cmml">:=</mo><mrow id="S2.SS2.p2.17.m17.4.4.2.2" xref="S2.SS2.p2.17.m17.4.4.2.3.cmml"><mo id="S2.SS2.p2.17.m17.4.4.2.2.3" stretchy="false" xref="S2.SS2.p2.17.m17.4.4.2.3.1.cmml">{</mo><mrow id="S2.SS2.p2.17.m17.3.3.1.1.1" xref="S2.SS2.p2.17.m17.3.3.1.1.1.cmml"><mi id="S2.SS2.p2.17.m17.3.3.1.1.1.2" xref="S2.SS2.p2.17.m17.3.3.1.1.1.2.cmml">v</mi><mo id="S2.SS2.p2.17.m17.3.3.1.1.1.1" xref="S2.SS2.p2.17.m17.3.3.1.1.1.1.cmml">+</mo><mrow id="S2.SS2.p2.17.m17.3.3.1.1.1.3" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.cmml"><mi id="S2.SS2.p2.17.m17.3.3.1.1.1.3.2" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.2.cmml">t</mi><mo id="S2.SS2.p2.17.m17.3.3.1.1.1.3.1" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.17.m17.3.3.1.1.1.3.3" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.3.cmml">u</mi></mrow></mrow><mo id="S2.SS2.p2.17.m17.4.4.2.2.4" lspace="0.278em" rspace="0.448em" xref="S2.SS2.p2.17.m17.4.4.2.3.1.cmml">:</mo><mrow id="S2.SS2.p2.17.m17.4.4.2.2.2" xref="S2.SS2.p2.17.m17.4.4.2.2.2.cmml"><mi id="S2.SS2.p2.17.m17.4.4.2.2.2.2" xref="S2.SS2.p2.17.m17.4.4.2.2.2.2.cmml">t</mi><mo id="S2.SS2.p2.17.m17.4.4.2.2.2.1" xref="S2.SS2.p2.17.m17.4.4.2.2.2.1.cmml">∈</mo><mi id="S2.SS2.p2.17.m17.4.4.2.2.2.3" xref="S2.SS2.p2.17.m17.4.4.2.2.2.3.cmml">ℝ</mi></mrow><mo id="S2.SS2.p2.17.m17.4.4.2.2.5" stretchy="false" xref="S2.SS2.p2.17.m17.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.17.m17.4b"><apply id="S2.SS2.p2.17.m17.4.4.cmml" xref="S2.SS2.p2.17.m17.4.4"><csymbol cd="latexml" id="S2.SS2.p2.17.m17.4.4.3.cmml" xref="S2.SS2.p2.17.m17.4.4.3">assign</csymbol><apply id="S2.SS2.p2.17.m17.4.4.4.1.cmml" xref="S2.SS2.p2.17.m17.4.4.4.2"><ci id="S2.SS2.p2.17.m17.1.1.cmml" xref="S2.SS2.p2.17.m17.1.1">aff</ci><ci id="S2.SS2.p2.17.m17.2.2.cmml" xref="S2.SS2.p2.17.m17.2.2">𝑒</ci></apply><apply id="S2.SS2.p2.17.m17.4.4.2.3.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2"><csymbol cd="latexml" id="S2.SS2.p2.17.m17.4.4.2.3.1.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2.3">conditional-set</csymbol><apply id="S2.SS2.p2.17.m17.3.3.1.1.1.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1"><plus id="S2.SS2.p2.17.m17.3.3.1.1.1.1.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.1"></plus><ci id="S2.SS2.p2.17.m17.3.3.1.1.1.2.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.2">𝑣</ci><apply id="S2.SS2.p2.17.m17.3.3.1.1.1.3.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3"><times id="S2.SS2.p2.17.m17.3.3.1.1.1.3.1.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.1"></times><ci id="S2.SS2.p2.17.m17.3.3.1.1.1.3.2.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.2">𝑡</ci><ci id="S2.SS2.p2.17.m17.3.3.1.1.1.3.3.cmml" xref="S2.SS2.p2.17.m17.3.3.1.1.1.3.3">𝑢</ci></apply></apply><apply id="S2.SS2.p2.17.m17.4.4.2.2.2.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2.2"><in id="S2.SS2.p2.17.m17.4.4.2.2.2.1.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2.2.1"></in><ci id="S2.SS2.p2.17.m17.4.4.2.2.2.2.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2.2.2">𝑡</ci><ci id="S2.SS2.p2.17.m17.4.4.2.2.2.3.cmml" xref="S2.SS2.p2.17.m17.4.4.2.2.2.3">ℝ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.17.m17.4c">\operatorname*{aff}(e):=\{v+tu:\,t\in\mathds{R}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.17.m17.4d">roman_aff ( italic_e ) := { italic_v + italic_t italic_u : italic_t ∈ blackboard_R }</annotation></semantics></math> is its affine hull.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.2">A <em class="ltx_emph ltx_font_italic" id="S2.SS2.p3.2.1">polygonal arc</em> is just one line, or a subset of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><msup id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">ℝ</mi><mn id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">superscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">ℝ</ci><cn id="S2.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> that is homeomorphic to a line, and which can be defined as the union of two rays and finitely many line segments, such that any two of the line segments and rays are disjoint or intersect at a common vertex. Similarly, I define a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p3.2.2">polygonal cycle</em> to be a subset of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><msup id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml"><mi id="S2.SS2.p3.2.m2.1.1.2" xref="S2.SS2.p3.2.m2.1.1.2.cmml">ℝ</mi><mn id="S2.SS2.p3.2.m2.1.1.3" xref="S2.SS2.p3.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><apply id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">superscript</csymbol><ci id="S2.SS2.p3.2.m2.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.2">ℝ</ci><cn id="S2.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> that is homeomorphic to the unit circle and which can be defined as the union of finitely many line segments, such that any two of the line segments are disjoint or intersect only at a vertex of both.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.7">The collection of line segments, rays, and lines, that were used in the definition of a polygonal arc or cycle <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mi id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><ci id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">italic_γ</annotation></semantics></math>, is called the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p4.7.1">edge</em> set of <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mi id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><ci id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">italic_γ</annotation></semantics></math> and is denoted by <math alttext="E(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mrow id="S2.SS2.p4.3.m3.1.2" xref="S2.SS2.p4.3.m3.1.2.cmml"><mi id="S2.SS2.p4.3.m3.1.2.2" xref="S2.SS2.p4.3.m3.1.2.2.cmml">E</mi><mo id="S2.SS2.p4.3.m3.1.2.1" xref="S2.SS2.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p4.3.m3.1.2.3.2" xref="S2.SS2.p4.3.m3.1.2.cmml"><mo id="S2.SS2.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.p4.3.m3.1.2.cmml">(</mo><mi id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml">γ</mi><mo id="S2.SS2.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.2.cmml" xref="S2.SS2.p4.3.m3.1.2"><times id="S2.SS2.p4.3.m3.1.2.1.cmml" xref="S2.SS2.p4.3.m3.1.2.1"></times><ci id="S2.SS2.p4.3.m3.1.2.2.cmml" xref="S2.SS2.p4.3.m3.1.2.2">𝐸</ci><ci id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">E(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">italic_E ( italic_γ )</annotation></semantics></math>. The set of <em class="ltx_emph ltx_font_italic" id="S2.SS2.p4.7.2">vertices</em> <math alttext="V(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mrow id="S2.SS2.p4.4.m4.1.2" xref="S2.SS2.p4.4.m4.1.2.cmml"><mi id="S2.SS2.p4.4.m4.1.2.2" xref="S2.SS2.p4.4.m4.1.2.2.cmml">V</mi><mo id="S2.SS2.p4.4.m4.1.2.1" xref="S2.SS2.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p4.4.m4.1.2.3.2" xref="S2.SS2.p4.4.m4.1.2.cmml"><mo id="S2.SS2.p4.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.p4.4.m4.1.2.cmml">(</mo><mi id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml">γ</mi><mo id="S2.SS2.p4.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.p4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><apply id="S2.SS2.p4.4.m4.1.2.cmml" xref="S2.SS2.p4.4.m4.1.2"><times id="S2.SS2.p4.4.m4.1.2.1.cmml" xref="S2.SS2.p4.4.m4.1.2.1"></times><ci id="S2.SS2.p4.4.m4.1.2.2.cmml" xref="S2.SS2.p4.4.m4.1.2.2">𝑉</ci><ci id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">V(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_V ( italic_γ )</annotation></semantics></math> of a polygonal arc or cycle <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.1"><semantics id="S2.SS2.p4.5.m5.1a"><mi id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.1b"><ci id="S2.SS2.p4.5.m5.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.1d">italic_γ</annotation></semantics></math> is defined as the union of the vertices of its edges. These sets are not unique, since there are multiple ways to define the same curve. Note that, due to the homeomorphisms, removing a point from <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.1"><semantics id="S2.SS2.p4.6.m6.1a"><mi id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.1b"><ci id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.1d">italic_γ</annotation></semantics></math> will locally disconnect the curve into exactly two parts. Therefore, for every vertex <math alttext="v\in V(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.1"><semantics id="S2.SS2.p4.7.m7.1a"><mrow id="S2.SS2.p4.7.m7.1.2" xref="S2.SS2.p4.7.m7.1.2.cmml"><mi id="S2.SS2.p4.7.m7.1.2.2" xref="S2.SS2.p4.7.m7.1.2.2.cmml">v</mi><mo id="S2.SS2.p4.7.m7.1.2.1" xref="S2.SS2.p4.7.m7.1.2.1.cmml">∈</mo><mrow id="S2.SS2.p4.7.m7.1.2.3" xref="S2.SS2.p4.7.m7.1.2.3.cmml"><mi id="S2.SS2.p4.7.m7.1.2.3.2" xref="S2.SS2.p4.7.m7.1.2.3.2.cmml">V</mi><mo id="S2.SS2.p4.7.m7.1.2.3.1" xref="S2.SS2.p4.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p4.7.m7.1.2.3.3.2" xref="S2.SS2.p4.7.m7.1.2.3.cmml"><mo id="S2.SS2.p4.7.m7.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p4.7.m7.1.2.3.cmml">(</mo><mi id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml">γ</mi><mo id="S2.SS2.p4.7.m7.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p4.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.1b"><apply id="S2.SS2.p4.7.m7.1.2.cmml" xref="S2.SS2.p4.7.m7.1.2"><in id="S2.SS2.p4.7.m7.1.2.1.cmml" xref="S2.SS2.p4.7.m7.1.2.1"></in><ci id="S2.SS2.p4.7.m7.1.2.2.cmml" xref="S2.SS2.p4.7.m7.1.2.2">𝑣</ci><apply id="S2.SS2.p4.7.m7.1.2.3.cmml" xref="S2.SS2.p4.7.m7.1.2.3"><times id="S2.SS2.p4.7.m7.1.2.3.1.cmml" xref="S2.SS2.p4.7.m7.1.2.3.1"></times><ci id="S2.SS2.p4.7.m7.1.2.3.2.cmml" xref="S2.SS2.p4.7.m7.1.2.3.2">𝑉</ci><ci id="S2.SS2.p4.7.m7.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.1c">v\in V(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.1d">italic_v ∈ italic_V ( italic_γ )</annotation></semantics></math>, there are exactly two incident edges.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.17">In this work, I adopt a more generalised notion of polygons than the standard definition. Specifically, a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p5.17.2">polygon</em> is defined as a closed subset <math alttext="P\subseteq\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><mrow id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><mi id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml">P</mi><mo id="S2.SS2.p5.1.m1.1.1.1" xref="S2.SS2.p5.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.3.2" xref="S2.SS2.p5.1.m1.1.1.3.2.cmml">ℝ</mi><mn id="S2.SS2.p5.1.m1.1.1.3.3" xref="S2.SS2.p5.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><subset id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1.1"></subset><ci id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2">𝑃</ci><apply id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.3.2">ℝ</ci><cn id="S2.SS2.p5.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p5.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">P\subseteq\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">italic_P ⊆ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with connected interior, and whose boundary <math alttext="\partial P" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><mrow id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml"><mo id="S2.SS2.p5.2.m2.1.1.1" rspace="0em" xref="S2.SS2.p5.2.m2.1.1.1.cmml">∂</mo><mi id="S2.SS2.p5.2.m2.1.1.2" xref="S2.SS2.p5.2.m2.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><apply id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1"><partialdiff id="S2.SS2.p5.2.m2.1.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1.1"></partialdiff><ci id="S2.SS2.p5.2.m2.1.1.2.cmml" xref="S2.SS2.p5.2.m2.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\partial P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">∂ italic_P</annotation></semantics></math> is the union of finitely many polygonal arcs and cycles, such that the intersection of any two of the arcs and cycles is either empty or a vertex of both curves. The corresponding arcs and cycles are called <em class="ltx_emph ltx_font_italic" id="S2.SS2.p5.17.3">boundary components</em> of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><mi id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><ci id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_P</annotation></semantics></math>. Examples of polygons are shown in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S2.F2" title="Figure 2 ‣ 2.2 Polygons ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 2</span></a>. The set of line segments of a polygon <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><mi id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><ci id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">italic_P</annotation></semantics></math> is defined as the collection of the edges of all boundary components of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><mi id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><ci id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">italic_P</annotation></semantics></math> that are line segments. This set will be denoted by <math alttext="E_{b}(P)" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><mrow id="S2.SS2.p5.6.m6.1.2" xref="S2.SS2.p5.6.m6.1.2.cmml"><msub id="S2.SS2.p5.6.m6.1.2.2" xref="S2.SS2.p5.6.m6.1.2.2.cmml"><mi id="S2.SS2.p5.6.m6.1.2.2.2" xref="S2.SS2.p5.6.m6.1.2.2.2.cmml">E</mi><mi id="S2.SS2.p5.6.m6.1.2.2.3" xref="S2.SS2.p5.6.m6.1.2.2.3.cmml">b</mi></msub><mo id="S2.SS2.p5.6.m6.1.2.1" xref="S2.SS2.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.6.m6.1.2.3.2" xref="S2.SS2.p5.6.m6.1.2.cmml"><mo id="S2.SS2.p5.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.6.m6.1.2.cmml">(</mo><mi id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml">P</mi><mo id="S2.SS2.p5.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><apply id="S2.SS2.p5.6.m6.1.2.cmml" xref="S2.SS2.p5.6.m6.1.2"><times id="S2.SS2.p5.6.m6.1.2.1.cmml" xref="S2.SS2.p5.6.m6.1.2.1"></times><apply id="S2.SS2.p5.6.m6.1.2.2.cmml" xref="S2.SS2.p5.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.1.2.2.1.cmml" xref="S2.SS2.p5.6.m6.1.2.2">subscript</csymbol><ci id="S2.SS2.p5.6.m6.1.2.2.2.cmml" xref="S2.SS2.p5.6.m6.1.2.2.2">𝐸</ci><ci id="S2.SS2.p5.6.m6.1.2.2.3.cmml" xref="S2.SS2.p5.6.m6.1.2.2.3">𝑏</ci></apply><ci id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">E_{b}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>, where the subscript indicates that these are all the bounded edges of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m7.1"><semantics id="S2.SS2.p5.7.m7.1a"><mi id="S2.SS2.p5.7.m7.1.1" xref="S2.SS2.p5.7.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m7.1b"><ci id="S2.SS2.p5.7.m7.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m7.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m7.1d">italic_P</annotation></semantics></math>. Similarly, the sets <math alttext="E_{r}(P)" class="ltx_Math" display="inline" id="S2.SS2.p5.8.m8.1"><semantics id="S2.SS2.p5.8.m8.1a"><mrow id="S2.SS2.p5.8.m8.1.2" xref="S2.SS2.p5.8.m8.1.2.cmml"><msub id="S2.SS2.p5.8.m8.1.2.2" xref="S2.SS2.p5.8.m8.1.2.2.cmml"><mi id="S2.SS2.p5.8.m8.1.2.2.2" xref="S2.SS2.p5.8.m8.1.2.2.2.cmml">E</mi><mi id="S2.SS2.p5.8.m8.1.2.2.3" xref="S2.SS2.p5.8.m8.1.2.2.3.cmml">r</mi></msub><mo id="S2.SS2.p5.8.m8.1.2.1" xref="S2.SS2.p5.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.8.m8.1.2.3.2" xref="S2.SS2.p5.8.m8.1.2.cmml"><mo id="S2.SS2.p5.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.8.m8.1.2.cmml">(</mo><mi id="S2.SS2.p5.8.m8.1.1" xref="S2.SS2.p5.8.m8.1.1.cmml">P</mi><mo id="S2.SS2.p5.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.8.m8.1b"><apply id="S2.SS2.p5.8.m8.1.2.cmml" xref="S2.SS2.p5.8.m8.1.2"><times id="S2.SS2.p5.8.m8.1.2.1.cmml" xref="S2.SS2.p5.8.m8.1.2.1"></times><apply id="S2.SS2.p5.8.m8.1.2.2.cmml" xref="S2.SS2.p5.8.m8.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.8.m8.1.2.2.1.cmml" xref="S2.SS2.p5.8.m8.1.2.2">subscript</csymbol><ci id="S2.SS2.p5.8.m8.1.2.2.2.cmml" xref="S2.SS2.p5.8.m8.1.2.2.2">𝐸</ci><ci id="S2.SS2.p5.8.m8.1.2.2.3.cmml" xref="S2.SS2.p5.8.m8.1.2.2.3">𝑟</ci></apply><ci id="S2.SS2.p5.8.m8.1.1.cmml" xref="S2.SS2.p5.8.m8.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.8.m8.1c">E_{r}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.8.m8.1d">italic_E start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> and <math alttext="E_{l}(P)" class="ltx_Math" display="inline" id="S2.SS2.p5.9.m9.1"><semantics id="S2.SS2.p5.9.m9.1a"><mrow id="S2.SS2.p5.9.m9.1.2" xref="S2.SS2.p5.9.m9.1.2.cmml"><msub id="S2.SS2.p5.9.m9.1.2.2" xref="S2.SS2.p5.9.m9.1.2.2.cmml"><mi id="S2.SS2.p5.9.m9.1.2.2.2" xref="S2.SS2.p5.9.m9.1.2.2.2.cmml">E</mi><mi id="S2.SS2.p5.9.m9.1.2.2.3" xref="S2.SS2.p5.9.m9.1.2.2.3.cmml">l</mi></msub><mo id="S2.SS2.p5.9.m9.1.2.1" xref="S2.SS2.p5.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.9.m9.1.2.3.2" xref="S2.SS2.p5.9.m9.1.2.cmml"><mo id="S2.SS2.p5.9.m9.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.9.m9.1.2.cmml">(</mo><mi id="S2.SS2.p5.9.m9.1.1" xref="S2.SS2.p5.9.m9.1.1.cmml">P</mi><mo id="S2.SS2.p5.9.m9.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.9.m9.1b"><apply id="S2.SS2.p5.9.m9.1.2.cmml" xref="S2.SS2.p5.9.m9.1.2"><times id="S2.SS2.p5.9.m9.1.2.1.cmml" xref="S2.SS2.p5.9.m9.1.2.1"></times><apply id="S2.SS2.p5.9.m9.1.2.2.cmml" xref="S2.SS2.p5.9.m9.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.9.m9.1.2.2.1.cmml" xref="S2.SS2.p5.9.m9.1.2.2">subscript</csymbol><ci id="S2.SS2.p5.9.m9.1.2.2.2.cmml" xref="S2.SS2.p5.9.m9.1.2.2.2">𝐸</ci><ci id="S2.SS2.p5.9.m9.1.2.2.3.cmml" xref="S2.SS2.p5.9.m9.1.2.2.3">𝑙</ci></apply><ci id="S2.SS2.p5.9.m9.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.9.m9.1c">E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.9.m9.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> are defined as the sets of rays and lines of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.10.m10.1"><semantics id="S2.SS2.p5.10.m10.1a"><mi id="S2.SS2.p5.10.m10.1.1" xref="S2.SS2.p5.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.10.m10.1b"><ci id="S2.SS2.p5.10.m10.1.1.cmml" xref="S2.SS2.p5.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.10.m10.1d">italic_P</annotation></semantics></math>, i.e. the collections of rays respectively lines of the boundary components. Therefore, the set of <em class="ltx_emph ltx_font_italic" id="S2.SS2.p5.17.4">edges</em> <math alttext="E(P):=E_{b}(P)\dot{\cup}E_{r}(P)\dot{\cup}E_{l}(P)" class="ltx_Math" display="inline" id="S2.SS2.p5.11.m11.4"><semantics id="S2.SS2.p5.11.m11.4a"><mrow id="S2.SS2.p5.11.m11.4.5" xref="S2.SS2.p5.11.m11.4.5.cmml"><mrow id="S2.SS2.p5.11.m11.4.5.2" xref="S2.SS2.p5.11.m11.4.5.2.cmml"><mi id="S2.SS2.p5.11.m11.4.5.2.2" xref="S2.SS2.p5.11.m11.4.5.2.2.cmml">E</mi><mo id="S2.SS2.p5.11.m11.4.5.2.1" xref="S2.SS2.p5.11.m11.4.5.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.11.m11.4.5.2.3.2" xref="S2.SS2.p5.11.m11.4.5.2.cmml"><mo id="S2.SS2.p5.11.m11.4.5.2.3.2.1" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.2.cmml">(</mo><mi id="S2.SS2.p5.11.m11.1.1" xref="S2.SS2.p5.11.m11.1.1.cmml">P</mi><mo id="S2.SS2.p5.11.m11.4.5.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p5.11.m11.4.5.1" rspace="0.278em" xref="S2.SS2.p5.11.m11.4.5.1.cmml">:=</mo><mrow id="S2.SS2.p5.11.m11.4.5.3" xref="S2.SS2.p5.11.m11.4.5.3.cmml"><msub id="S2.SS2.p5.11.m11.4.5.3.2" xref="S2.SS2.p5.11.m11.4.5.3.2.cmml"><mi id="S2.SS2.p5.11.m11.4.5.3.2.2" xref="S2.SS2.p5.11.m11.4.5.3.2.2.cmml">E</mi><mi id="S2.SS2.p5.11.m11.4.5.3.2.3" xref="S2.SS2.p5.11.m11.4.5.3.2.3.cmml">b</mi></msub><mo id="S2.SS2.p5.11.m11.4.5.3.1" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.11.m11.4.5.3.3.2" xref="S2.SS2.p5.11.m11.4.5.3.cmml"><mo id="S2.SS2.p5.11.m11.4.5.3.3.2.1" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">(</mo><mi id="S2.SS2.p5.11.m11.2.2" xref="S2.SS2.p5.11.m11.2.2.cmml">P</mi><mo id="S2.SS2.p5.11.m11.4.5.3.3.2.2" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">)</mo></mrow><mo id="S2.SS2.p5.11.m11.4.5.3.1a" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><mover accent="true" id="S2.SS2.p5.11.m11.4.5.3.4" xref="S2.SS2.p5.11.m11.4.5.3.4.cmml"><mo id="S2.SS2.p5.11.m11.4.5.3.4.2" xref="S2.SS2.p5.11.m11.4.5.3.4.2.cmml">∪</mo><mo id="S2.SS2.p5.11.m11.4.5.3.4.1" xref="S2.SS2.p5.11.m11.4.5.3.4.1.cmml">˙</mo></mover><mo id="S2.SS2.p5.11.m11.4.5.3.1b" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><msub id="S2.SS2.p5.11.m11.4.5.3.5" xref="S2.SS2.p5.11.m11.4.5.3.5.cmml"><mi id="S2.SS2.p5.11.m11.4.5.3.5.2" xref="S2.SS2.p5.11.m11.4.5.3.5.2.cmml">E</mi><mi id="S2.SS2.p5.11.m11.4.5.3.5.3" xref="S2.SS2.p5.11.m11.4.5.3.5.3.cmml">r</mi></msub><mo id="S2.SS2.p5.11.m11.4.5.3.1c" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.11.m11.4.5.3.6.2" xref="S2.SS2.p5.11.m11.4.5.3.cmml"><mo id="S2.SS2.p5.11.m11.4.5.3.6.2.1" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">(</mo><mi id="S2.SS2.p5.11.m11.3.3" xref="S2.SS2.p5.11.m11.3.3.cmml">P</mi><mo id="S2.SS2.p5.11.m11.4.5.3.6.2.2" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">)</mo></mrow><mo id="S2.SS2.p5.11.m11.4.5.3.1d" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><mover accent="true" id="S2.SS2.p5.11.m11.4.5.3.7" xref="S2.SS2.p5.11.m11.4.5.3.7.cmml"><mo id="S2.SS2.p5.11.m11.4.5.3.7.2" xref="S2.SS2.p5.11.m11.4.5.3.7.2.cmml">∪</mo><mo id="S2.SS2.p5.11.m11.4.5.3.7.1" xref="S2.SS2.p5.11.m11.4.5.3.7.1.cmml">˙</mo></mover><mo id="S2.SS2.p5.11.m11.4.5.3.1e" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><msub id="S2.SS2.p5.11.m11.4.5.3.8" xref="S2.SS2.p5.11.m11.4.5.3.8.cmml"><mi id="S2.SS2.p5.11.m11.4.5.3.8.2" xref="S2.SS2.p5.11.m11.4.5.3.8.2.cmml">E</mi><mi id="S2.SS2.p5.11.m11.4.5.3.8.3" xref="S2.SS2.p5.11.m11.4.5.3.8.3.cmml">l</mi></msub><mo id="S2.SS2.p5.11.m11.4.5.3.1f" xref="S2.SS2.p5.11.m11.4.5.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.11.m11.4.5.3.9.2" xref="S2.SS2.p5.11.m11.4.5.3.cmml"><mo id="S2.SS2.p5.11.m11.4.5.3.9.2.1" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">(</mo><mi id="S2.SS2.p5.11.m11.4.4" xref="S2.SS2.p5.11.m11.4.4.cmml">P</mi><mo id="S2.SS2.p5.11.m11.4.5.3.9.2.2" stretchy="false" xref="S2.SS2.p5.11.m11.4.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.11.m11.4b"><apply id="S2.SS2.p5.11.m11.4.5.cmml" xref="S2.SS2.p5.11.m11.4.5"><csymbol cd="latexml" id="S2.SS2.p5.11.m11.4.5.1.cmml" xref="S2.SS2.p5.11.m11.4.5.1">assign</csymbol><apply id="S2.SS2.p5.11.m11.4.5.2.cmml" xref="S2.SS2.p5.11.m11.4.5.2"><times id="S2.SS2.p5.11.m11.4.5.2.1.cmml" xref="S2.SS2.p5.11.m11.4.5.2.1"></times><ci id="S2.SS2.p5.11.m11.4.5.2.2.cmml" xref="S2.SS2.p5.11.m11.4.5.2.2">𝐸</ci><ci id="S2.SS2.p5.11.m11.1.1.cmml" xref="S2.SS2.p5.11.m11.1.1">𝑃</ci></apply><apply id="S2.SS2.p5.11.m11.4.5.3.cmml" xref="S2.SS2.p5.11.m11.4.5.3"><times id="S2.SS2.p5.11.m11.4.5.3.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.1"></times><apply id="S2.SS2.p5.11.m11.4.5.3.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.2"><csymbol cd="ambiguous" id="S2.SS2.p5.11.m11.4.5.3.2.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.2">subscript</csymbol><ci id="S2.SS2.p5.11.m11.4.5.3.2.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.2.2">𝐸</ci><ci id="S2.SS2.p5.11.m11.4.5.3.2.3.cmml" xref="S2.SS2.p5.11.m11.4.5.3.2.3">𝑏</ci></apply><ci id="S2.SS2.p5.11.m11.2.2.cmml" xref="S2.SS2.p5.11.m11.2.2">𝑃</ci><apply id="S2.SS2.p5.11.m11.4.5.3.4.cmml" xref="S2.SS2.p5.11.m11.4.5.3.4"><ci id="S2.SS2.p5.11.m11.4.5.3.4.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.4.1">˙</ci><union id="S2.SS2.p5.11.m11.4.5.3.4.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.4.2"></union></apply><apply id="S2.SS2.p5.11.m11.4.5.3.5.cmml" xref="S2.SS2.p5.11.m11.4.5.3.5"><csymbol cd="ambiguous" id="S2.SS2.p5.11.m11.4.5.3.5.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.5">subscript</csymbol><ci id="S2.SS2.p5.11.m11.4.5.3.5.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.5.2">𝐸</ci><ci id="S2.SS2.p5.11.m11.4.5.3.5.3.cmml" xref="S2.SS2.p5.11.m11.4.5.3.5.3">𝑟</ci></apply><ci id="S2.SS2.p5.11.m11.3.3.cmml" xref="S2.SS2.p5.11.m11.3.3">𝑃</ci><apply id="S2.SS2.p5.11.m11.4.5.3.7.cmml" xref="S2.SS2.p5.11.m11.4.5.3.7"><ci id="S2.SS2.p5.11.m11.4.5.3.7.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.7.1">˙</ci><union id="S2.SS2.p5.11.m11.4.5.3.7.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.7.2"></union></apply><apply id="S2.SS2.p5.11.m11.4.5.3.8.cmml" xref="S2.SS2.p5.11.m11.4.5.3.8"><csymbol cd="ambiguous" id="S2.SS2.p5.11.m11.4.5.3.8.1.cmml" xref="S2.SS2.p5.11.m11.4.5.3.8">subscript</csymbol><ci id="S2.SS2.p5.11.m11.4.5.3.8.2.cmml" xref="S2.SS2.p5.11.m11.4.5.3.8.2">𝐸</ci><ci id="S2.SS2.p5.11.m11.4.5.3.8.3.cmml" xref="S2.SS2.p5.11.m11.4.5.3.8.3">𝑙</ci></apply><ci id="S2.SS2.p5.11.m11.4.4.cmml" xref="S2.SS2.p5.11.m11.4.4">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.11.m11.4c">E(P):=E_{b}(P)\dot{\cup}E_{r}(P)\dot{\cup}E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.11.m11.4d">italic_E ( italic_P ) := italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) over˙ start_ARG ∪ end_ARG italic_E start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_P ) over˙ start_ARG ∪ end_ARG italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> completely characterises the boundary of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.12.m12.1"><semantics id="S2.SS2.p5.12.m12.1a"><mi id="S2.SS2.p5.12.m12.1.1" xref="S2.SS2.p5.12.m12.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.12.m12.1b"><ci id="S2.SS2.p5.12.m12.1.1.cmml" xref="S2.SS2.p5.12.m12.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.12.m12.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.12.m12.1d">italic_P</annotation></semantics></math> via <math alttext="\partial P=\bigcup_{e\in E(P)}e" class="ltx_Math" display="inline" id="S2.SS2.p5.13.m13.1"><semantics id="S2.SS2.p5.13.m13.1a"><mrow id="S2.SS2.p5.13.m13.1.2" xref="S2.SS2.p5.13.m13.1.2.cmml"><mrow id="S2.SS2.p5.13.m13.1.2.2" xref="S2.SS2.p5.13.m13.1.2.2.cmml"><mo id="S2.SS2.p5.13.m13.1.2.2.1" rspace="0em" xref="S2.SS2.p5.13.m13.1.2.2.1.cmml">∂</mo><mi id="S2.SS2.p5.13.m13.1.2.2.2" xref="S2.SS2.p5.13.m13.1.2.2.2.cmml">P</mi></mrow><mo id="S2.SS2.p5.13.m13.1.2.1" rspace="0.111em" xref="S2.SS2.p5.13.m13.1.2.1.cmml">=</mo><mrow id="S2.SS2.p5.13.m13.1.2.3" xref="S2.SS2.p5.13.m13.1.2.3.cmml"><msub id="S2.SS2.p5.13.m13.1.2.3.1" xref="S2.SS2.p5.13.m13.1.2.3.1.cmml"><mo id="S2.SS2.p5.13.m13.1.2.3.1.2" xref="S2.SS2.p5.13.m13.1.2.3.1.2.cmml">⋃</mo><mrow id="S2.SS2.p5.13.m13.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.cmml"><mi id="S2.SS2.p5.13.m13.1.1.1.3" xref="S2.SS2.p5.13.m13.1.1.1.3.cmml">e</mi><mo id="S2.SS2.p5.13.m13.1.1.1.2" xref="S2.SS2.p5.13.m13.1.1.1.2.cmml">∈</mo><mrow id="S2.SS2.p5.13.m13.1.1.1.4" xref="S2.SS2.p5.13.m13.1.1.1.4.cmml"><mi id="S2.SS2.p5.13.m13.1.1.1.4.2" xref="S2.SS2.p5.13.m13.1.1.1.4.2.cmml">E</mi><mo id="S2.SS2.p5.13.m13.1.1.1.4.1" xref="S2.SS2.p5.13.m13.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.SS2.p5.13.m13.1.1.1.4.3.2" xref="S2.SS2.p5.13.m13.1.1.1.4.cmml"><mo id="S2.SS2.p5.13.m13.1.1.1.4.3.2.1" stretchy="false" xref="S2.SS2.p5.13.m13.1.1.1.4.cmml">(</mo><mi id="S2.SS2.p5.13.m13.1.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.1.cmml">P</mi><mo id="S2.SS2.p5.13.m13.1.1.1.4.3.2.2" stretchy="false" xref="S2.SS2.p5.13.m13.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mi id="S2.SS2.p5.13.m13.1.2.3.2" xref="S2.SS2.p5.13.m13.1.2.3.2.cmml">e</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.13.m13.1b"><apply id="S2.SS2.p5.13.m13.1.2.cmml" xref="S2.SS2.p5.13.m13.1.2"><eq id="S2.SS2.p5.13.m13.1.2.1.cmml" xref="S2.SS2.p5.13.m13.1.2.1"></eq><apply id="S2.SS2.p5.13.m13.1.2.2.cmml" xref="S2.SS2.p5.13.m13.1.2.2"><partialdiff id="S2.SS2.p5.13.m13.1.2.2.1.cmml" xref="S2.SS2.p5.13.m13.1.2.2.1"></partialdiff><ci id="S2.SS2.p5.13.m13.1.2.2.2.cmml" xref="S2.SS2.p5.13.m13.1.2.2.2">𝑃</ci></apply><apply id="S2.SS2.p5.13.m13.1.2.3.cmml" xref="S2.SS2.p5.13.m13.1.2.3"><apply id="S2.SS2.p5.13.m13.1.2.3.1.cmml" xref="S2.SS2.p5.13.m13.1.2.3.1"><csymbol cd="ambiguous" id="S2.SS2.p5.13.m13.1.2.3.1.1.cmml" xref="S2.SS2.p5.13.m13.1.2.3.1">subscript</csymbol><union id="S2.SS2.p5.13.m13.1.2.3.1.2.cmml" xref="S2.SS2.p5.13.m13.1.2.3.1.2"></union><apply id="S2.SS2.p5.13.m13.1.1.1.cmml" xref="S2.SS2.p5.13.m13.1.1.1"><in id="S2.SS2.p5.13.m13.1.1.1.2.cmml" xref="S2.SS2.p5.13.m13.1.1.1.2"></in><ci id="S2.SS2.p5.13.m13.1.1.1.3.cmml" xref="S2.SS2.p5.13.m13.1.1.1.3">𝑒</ci><apply id="S2.SS2.p5.13.m13.1.1.1.4.cmml" xref="S2.SS2.p5.13.m13.1.1.1.4"><times id="S2.SS2.p5.13.m13.1.1.1.4.1.cmml" xref="S2.SS2.p5.13.m13.1.1.1.4.1"></times><ci id="S2.SS2.p5.13.m13.1.1.1.4.2.cmml" xref="S2.SS2.p5.13.m13.1.1.1.4.2">𝐸</ci><ci id="S2.SS2.p5.13.m13.1.1.1.1.cmml" xref="S2.SS2.p5.13.m13.1.1.1.1">𝑃</ci></apply></apply></apply><ci id="S2.SS2.p5.13.m13.1.2.3.2.cmml" xref="S2.SS2.p5.13.m13.1.2.3.2">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.13.m13.1c">\partial P=\bigcup_{e\in E(P)}e</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.13.m13.1d">∂ italic_P = ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_P ) end_POSTSUBSCRIPT italic_e</annotation></semantics></math>. Similarly, the set of vertices of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.14.m14.1"><semantics id="S2.SS2.p5.14.m14.1a"><mi id="S2.SS2.p5.14.m14.1.1" xref="S2.SS2.p5.14.m14.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.14.m14.1b"><ci id="S2.SS2.p5.14.m14.1.1.cmml" xref="S2.SS2.p5.14.m14.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.14.m14.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.14.m14.1d">italic_P</annotation></semantics></math>, denoted by <math alttext="V(P)" class="ltx_Math" display="inline" id="S2.SS2.p5.15.m15.1"><semantics id="S2.SS2.p5.15.m15.1a"><mrow id="S2.SS2.p5.15.m15.1.2" xref="S2.SS2.p5.15.m15.1.2.cmml"><mi id="S2.SS2.p5.15.m15.1.2.2" xref="S2.SS2.p5.15.m15.1.2.2.cmml">V</mi><mo id="S2.SS2.p5.15.m15.1.2.1" xref="S2.SS2.p5.15.m15.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.15.m15.1.2.3.2" xref="S2.SS2.p5.15.m15.1.2.cmml"><mo id="S2.SS2.p5.15.m15.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.15.m15.1.2.cmml">(</mo><mi id="S2.SS2.p5.15.m15.1.1" xref="S2.SS2.p5.15.m15.1.1.cmml">P</mi><mo id="S2.SS2.p5.15.m15.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.15.m15.1b"><apply id="S2.SS2.p5.15.m15.1.2.cmml" xref="S2.SS2.p5.15.m15.1.2"><times id="S2.SS2.p5.15.m15.1.2.1.cmml" xref="S2.SS2.p5.15.m15.1.2.1"></times><ci id="S2.SS2.p5.15.m15.1.2.2.cmml" xref="S2.SS2.p5.15.m15.1.2.2">𝑉</ci><ci id="S2.SS2.p5.15.m15.1.1.cmml" xref="S2.SS2.p5.15.m15.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.15.m15.1c">V(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.15.m15.1d">italic_V ( italic_P )</annotation></semantics></math>, is the set of vertices of its boundary. A point <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.16.m16.1"><semantics id="S2.SS2.p5.16.m16.1a"><mrow id="S2.SS2.p5.16.m16.1.1" xref="S2.SS2.p5.16.m16.1.1.cmml"><mi id="S2.SS2.p5.16.m16.1.1.2" xref="S2.SS2.p5.16.m16.1.1.2.cmml">x</mi><mo id="S2.SS2.p5.16.m16.1.1.1" xref="S2.SS2.p5.16.m16.1.1.1.cmml">∈</mo><msup id="S2.SS2.p5.16.m16.1.1.3" xref="S2.SS2.p5.16.m16.1.1.3.cmml"><mi id="S2.SS2.p5.16.m16.1.1.3.2" xref="S2.SS2.p5.16.m16.1.1.3.2.cmml">ℝ</mi><mn id="S2.SS2.p5.16.m16.1.1.3.3" xref="S2.SS2.p5.16.m16.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.16.m16.1b"><apply id="S2.SS2.p5.16.m16.1.1.cmml" xref="S2.SS2.p5.16.m16.1.1"><in id="S2.SS2.p5.16.m16.1.1.1.cmml" xref="S2.SS2.p5.16.m16.1.1.1"></in><ci id="S2.SS2.p5.16.m16.1.1.2.cmml" xref="S2.SS2.p5.16.m16.1.1.2">𝑥</ci><apply id="S2.SS2.p5.16.m16.1.1.3.cmml" xref="S2.SS2.p5.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.16.m16.1.1.3.1.cmml" xref="S2.SS2.p5.16.m16.1.1.3">superscript</csymbol><ci id="S2.SS2.p5.16.m16.1.1.3.2.cmml" xref="S2.SS2.p5.16.m16.1.1.3.2">ℝ</ci><cn id="S2.SS2.p5.16.m16.1.1.3.3.cmml" type="integer" xref="S2.SS2.p5.16.m16.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.16.m16.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.16.m16.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is said to be in <em class="ltx_emph ltx_font_italic" id="S2.SS2.p5.17.1"><math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p5.17.1.m1.1"><semantics id="S2.SS2.p5.17.1.m1.1a"><mi id="S2.SS2.p5.17.1.m1.1.1" xref="S2.SS2.p5.17.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.17.1.m1.1b"><ci id="S2.SS2.p5.17.1.m1.1.1.cmml" xref="S2.SS2.p5.17.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.17.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.17.1.m1.1d">italic_P</annotation></semantics></math>-general position</em> if</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="x\not\in\bigcup_{e\in E(P)}\operatorname*{aff}(e)." class="ltx_Math" display="block" id="S2.Ex4.m1.4"><semantics id="S2.Ex4.m1.4a"><mrow id="S2.Ex4.m1.4.4.1" xref="S2.Ex4.m1.4.4.1.1.cmml"><mrow id="S2.Ex4.m1.4.4.1.1" xref="S2.Ex4.m1.4.4.1.1.cmml"><mi id="S2.Ex4.m1.4.4.1.1.2" xref="S2.Ex4.m1.4.4.1.1.2.cmml">x</mi><mo id="S2.Ex4.m1.4.4.1.1.1" rspace="0.111em" xref="S2.Ex4.m1.4.4.1.1.1.cmml">∉</mo><mrow id="S2.Ex4.m1.4.4.1.1.3" xref="S2.Ex4.m1.4.4.1.1.3.cmml"><munder id="S2.Ex4.m1.4.4.1.1.3.1" xref="S2.Ex4.m1.4.4.1.1.3.1.cmml"><mo id="S2.Ex4.m1.4.4.1.1.3.1.2" movablelimits="false" xref="S2.Ex4.m1.4.4.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.3.cmml">e</mi><mo id="S2.Ex4.m1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.Ex4.m1.1.1.1.4" xref="S2.Ex4.m1.1.1.1.4.cmml"><mi id="S2.Ex4.m1.1.1.1.4.2" xref="S2.Ex4.m1.1.1.1.4.2.cmml">E</mi><mo id="S2.Ex4.m1.1.1.1.4.1" xref="S2.Ex4.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.1.1.1.4.3.2" xref="S2.Ex4.m1.1.1.1.4.cmml"><mo id="S2.Ex4.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.Ex4.m1.1.1.1.4.cmml">(</mo><mi id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml">P</mi><mo id="S2.Ex4.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.Ex4.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.Ex4.m1.4.4.1.1.3.2.2" xref="S2.Ex4.m1.4.4.1.1.3.2.1.cmml"><mo id="S2.Ex4.m1.2.2" lspace="0em" rspace="0em" xref="S2.Ex4.m1.2.2.cmml">aff</mo><mrow id="S2.Ex4.m1.4.4.1.1.3.2.2.1" xref="S2.Ex4.m1.4.4.1.1.3.2.1.cmml"><mo id="S2.Ex4.m1.4.4.1.1.3.2.2.1.1" stretchy="false" xref="S2.Ex4.m1.4.4.1.1.3.2.1.cmml">(</mo><mi id="S2.Ex4.m1.3.3" xref="S2.Ex4.m1.3.3.cmml">e</mi><mo id="S2.Ex4.m1.4.4.1.1.3.2.2.1.2" stretchy="false" xref="S2.Ex4.m1.4.4.1.1.3.2.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex4.m1.4.4.1.2" lspace="0em" xref="S2.Ex4.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.4b"><apply id="S2.Ex4.m1.4.4.1.1.cmml" xref="S2.Ex4.m1.4.4.1"><notin id="S2.Ex4.m1.4.4.1.1.1.cmml" xref="S2.Ex4.m1.4.4.1.1.1"></notin><ci id="S2.Ex4.m1.4.4.1.1.2.cmml" xref="S2.Ex4.m1.4.4.1.1.2">𝑥</ci><apply id="S2.Ex4.m1.4.4.1.1.3.cmml" xref="S2.Ex4.m1.4.4.1.1.3"><apply id="S2.Ex4.m1.4.4.1.1.3.1.cmml" xref="S2.Ex4.m1.4.4.1.1.3.1"><csymbol cd="ambiguous" id="S2.Ex4.m1.4.4.1.1.3.1.1.cmml" xref="S2.Ex4.m1.4.4.1.1.3.1">subscript</csymbol><union id="S2.Ex4.m1.4.4.1.1.3.1.2.cmml" xref="S2.Ex4.m1.4.4.1.1.3.1.2"></union><apply id="S2.Ex4.m1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1"><in id="S2.Ex4.m1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.2"></in><ci id="S2.Ex4.m1.1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.1.3">𝑒</ci><apply id="S2.Ex4.m1.1.1.1.4.cmml" xref="S2.Ex4.m1.1.1.1.4"><times id="S2.Ex4.m1.1.1.1.4.1.cmml" xref="S2.Ex4.m1.1.1.1.4.1"></times><ci id="S2.Ex4.m1.1.1.1.4.2.cmml" xref="S2.Ex4.m1.1.1.1.4.2">𝐸</ci><ci id="S2.Ex4.m1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1">𝑃</ci></apply></apply></apply><apply id="S2.Ex4.m1.4.4.1.1.3.2.1.cmml" xref="S2.Ex4.m1.4.4.1.1.3.2.2"><ci id="S2.Ex4.m1.2.2.cmml" xref="S2.Ex4.m1.2.2">aff</ci><ci id="S2.Ex4.m1.3.3.cmml" xref="S2.Ex4.m1.3.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.4c">x\not\in\bigcup_{e\in E(P)}\operatorname*{aff}(e).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.4d">italic_x ∉ ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_P ) end_POSTSUBSCRIPT roman_aff ( italic_e ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.13">I note some basic facts about polygonal cycles. Since a polygonal cycle is homeomorphic to a circle, it is a simple closed curve. Due to the Jordan curve theorem, this means that any polygonal cycle <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.1"><semantics id="S2.SS2.p6.1.m1.1a"><mi id="S2.SS2.p6.1.m1.1.1" xref="S2.SS2.p6.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.1b"><ci id="S2.SS2.p6.1.m1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.1d">italic_γ</annotation></semantics></math> separates <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m2.1"><semantics id="S2.SS2.p6.2.m2.1a"><msup id="S2.SS2.p6.2.m2.1.1" xref="S2.SS2.p6.2.m2.1.1.cmml"><mi id="S2.SS2.p6.2.m2.1.1.2" xref="S2.SS2.p6.2.m2.1.1.2.cmml">ℝ</mi><mn id="S2.SS2.p6.2.m2.1.1.3" xref="S2.SS2.p6.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.2.m2.1b"><apply id="S2.SS2.p6.2.m2.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.2.m2.1.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1">superscript</csymbol><ci id="S2.SS2.p6.2.m2.1.1.2.cmml" xref="S2.SS2.p6.2.m2.1.1.2">ℝ</ci><cn id="S2.SS2.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p6.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> into two disjoint open connected components <math alttext="\operatorname*{int}(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m3.2"><semantics id="S2.SS2.p6.3.m3.2a"><mrow id="S2.SS2.p6.3.m3.2.3.2" xref="S2.SS2.p6.3.m3.2.3.1.cmml"><mo id="S2.SS2.p6.3.m3.1.1" rspace="0em" xref="S2.SS2.p6.3.m3.1.1.cmml">int</mo><mrow id="S2.SS2.p6.3.m3.2.3.2.1" xref="S2.SS2.p6.3.m3.2.3.1.cmml"><mo id="S2.SS2.p6.3.m3.2.3.2.1.1" stretchy="false" xref="S2.SS2.p6.3.m3.2.3.1.cmml">(</mo><mi id="S2.SS2.p6.3.m3.2.2" xref="S2.SS2.p6.3.m3.2.2.cmml">γ</mi><mo id="S2.SS2.p6.3.m3.2.3.2.1.2" stretchy="false" xref="S2.SS2.p6.3.m3.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.3.m3.2b"><apply id="S2.SS2.p6.3.m3.2.3.1.cmml" xref="S2.SS2.p6.3.m3.2.3.2"><ci id="S2.SS2.p6.3.m3.1.1.cmml" xref="S2.SS2.p6.3.m3.1.1">int</ci><ci id="S2.SS2.p6.3.m3.2.2.cmml" xref="S2.SS2.p6.3.m3.2.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.3.m3.2c">\operatorname*{int}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.3.m3.2d">roman_int ( italic_γ )</annotation></semantics></math> and <math alttext="\operatorname*{out}(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p6.4.m4.2"><semantics id="S2.SS2.p6.4.m4.2a"><mrow id="S2.SS2.p6.4.m4.2.3.2" xref="S2.SS2.p6.4.m4.2.3.1.cmml"><mo id="S2.SS2.p6.4.m4.1.1" rspace="0em" xref="S2.SS2.p6.4.m4.1.1.cmml">out</mo><mrow id="S2.SS2.p6.4.m4.2.3.2.1" xref="S2.SS2.p6.4.m4.2.3.1.cmml"><mo id="S2.SS2.p6.4.m4.2.3.2.1.1" stretchy="false" xref="S2.SS2.p6.4.m4.2.3.1.cmml">(</mo><mi id="S2.SS2.p6.4.m4.2.2" xref="S2.SS2.p6.4.m4.2.2.cmml">γ</mi><mo id="S2.SS2.p6.4.m4.2.3.2.1.2" stretchy="false" xref="S2.SS2.p6.4.m4.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.4.m4.2b"><apply id="S2.SS2.p6.4.m4.2.3.1.cmml" xref="S2.SS2.p6.4.m4.2.3.2"><ci id="S2.SS2.p6.4.m4.1.1.cmml" xref="S2.SS2.p6.4.m4.1.1">out</ci><ci id="S2.SS2.p6.4.m4.2.2.cmml" xref="S2.SS2.p6.4.m4.2.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.4.m4.2c">\operatorname*{out}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.m4.2d">roman_out ( italic_γ )</annotation></semantics></math>, whose boundary is <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p6.5.m5.1"><semantics id="S2.SS2.p6.5.m5.1a"><mi id="S2.SS2.p6.5.m5.1.1" xref="S2.SS2.p6.5.m5.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.5.m5.1b"><ci id="S2.SS2.p6.5.m5.1.1.cmml" xref="S2.SS2.p6.5.m5.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.5.m5.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.5.m5.1d">italic_γ</annotation></semantics></math>, such that <math alttext="\operatorname*{int}(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p6.6.m6.2"><semantics id="S2.SS2.p6.6.m6.2a"><mrow id="S2.SS2.p6.6.m6.2.3.2" xref="S2.SS2.p6.6.m6.2.3.1.cmml"><mo id="S2.SS2.p6.6.m6.1.1" rspace="0em" xref="S2.SS2.p6.6.m6.1.1.cmml">int</mo><mrow id="S2.SS2.p6.6.m6.2.3.2.1" xref="S2.SS2.p6.6.m6.2.3.1.cmml"><mo id="S2.SS2.p6.6.m6.2.3.2.1.1" stretchy="false" xref="S2.SS2.p6.6.m6.2.3.1.cmml">(</mo><mi id="S2.SS2.p6.6.m6.2.2" xref="S2.SS2.p6.6.m6.2.2.cmml">γ</mi><mo id="S2.SS2.p6.6.m6.2.3.2.1.2" stretchy="false" xref="S2.SS2.p6.6.m6.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.6.m6.2b"><apply id="S2.SS2.p6.6.m6.2.3.1.cmml" xref="S2.SS2.p6.6.m6.2.3.2"><ci id="S2.SS2.p6.6.m6.1.1.cmml" xref="S2.SS2.p6.6.m6.1.1">int</ci><ci id="S2.SS2.p6.6.m6.2.2.cmml" xref="S2.SS2.p6.6.m6.2.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.6.m6.2c">\operatorname*{int}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.6.m6.2d">roman_int ( italic_γ )</annotation></semantics></math> is bounded and <math alttext="\operatorname*{out}(\gamma)" class="ltx_Math" display="inline" id="S2.SS2.p6.7.m7.2"><semantics id="S2.SS2.p6.7.m7.2a"><mrow id="S2.SS2.p6.7.m7.2.3.2" xref="S2.SS2.p6.7.m7.2.3.1.cmml"><mo id="S2.SS2.p6.7.m7.1.1" rspace="0em" xref="S2.SS2.p6.7.m7.1.1.cmml">out</mo><mrow id="S2.SS2.p6.7.m7.2.3.2.1" xref="S2.SS2.p6.7.m7.2.3.1.cmml"><mo id="S2.SS2.p6.7.m7.2.3.2.1.1" stretchy="false" xref="S2.SS2.p6.7.m7.2.3.1.cmml">(</mo><mi id="S2.SS2.p6.7.m7.2.2" xref="S2.SS2.p6.7.m7.2.2.cmml">γ</mi><mo id="S2.SS2.p6.7.m7.2.3.2.1.2" stretchy="false" xref="S2.SS2.p6.7.m7.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.7.m7.2b"><apply id="S2.SS2.p6.7.m7.2.3.1.cmml" xref="S2.SS2.p6.7.m7.2.3.2"><ci id="S2.SS2.p6.7.m7.1.1.cmml" xref="S2.SS2.p6.7.m7.1.1">out</ci><ci id="S2.SS2.p6.7.m7.2.2.cmml" xref="S2.SS2.p6.7.m7.2.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.7.m7.2c">\operatorname*{out}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.7.m7.2d">roman_out ( italic_γ )</annotation></semantics></math> unbounded. Therefore, if a cycle <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p6.8.m8.1"><semantics id="S2.SS2.p6.8.m8.1a"><mi id="S2.SS2.p6.8.m8.1.1" xref="S2.SS2.p6.8.m8.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.8.m8.1b"><ci id="S2.SS2.p6.8.m8.1.1.cmml" xref="S2.SS2.p6.8.m8.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.8.m8.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.8.m8.1d">italic_γ</annotation></semantics></math> is a boundary component of some polygon <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p6.9.m9.1"><semantics id="S2.SS2.p6.9.m9.1a"><mi id="S2.SS2.p6.9.m9.1.1" xref="S2.SS2.p6.9.m9.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.9.m9.1b"><ci id="S2.SS2.p6.9.m9.1.1.cmml" xref="S2.SS2.p6.9.m9.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.9.m9.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.9.m9.1d">italic_P</annotation></semantics></math>, then the interior of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p6.10.m10.1"><semantics id="S2.SS2.p6.10.m10.1a"><mi id="S2.SS2.p6.10.m10.1.1" xref="S2.SS2.p6.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.10.m10.1b"><ci id="S2.SS2.p6.10.m10.1.1.cmml" xref="S2.SS2.p6.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.10.m10.1d">italic_P</annotation></semantics></math>, which is connected, is completely contained in exactly one of these regions. If <math alttext="P\subseteq\overline{\operatorname*{out}(\gamma)}" class="ltx_Math" display="inline" id="S2.SS2.p6.11.m11.2"><semantics id="S2.SS2.p6.11.m11.2a"><mrow id="S2.SS2.p6.11.m11.2.3" xref="S2.SS2.p6.11.m11.2.3.cmml"><mi id="S2.SS2.p6.11.m11.2.3.2" xref="S2.SS2.p6.11.m11.2.3.2.cmml">P</mi><mo id="S2.SS2.p6.11.m11.2.3.1" rspace="0.1389em" xref="S2.SS2.p6.11.m11.2.3.1.cmml">⊆</mo><mover accent="true" id="S2.SS2.p6.11.m11.2.2" xref="S2.SS2.p6.11.m11.2.2.cmml"><mrow id="S2.SS2.p6.11.m11.2.2.2.4" xref="S2.SS2.p6.11.m11.2.2.2.3.cmml"><mo id="S2.SS2.p6.11.m11.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S2.SS2.p6.11.m11.1.1.1.1.cmml">out</mo><mrow id="S2.SS2.p6.11.m11.2.2.2.4.1" xref="S2.SS2.p6.11.m11.2.2.2.3.cmml"><mo id="S2.SS2.p6.11.m11.2.2.2.4.1.1" stretchy="false" xref="S2.SS2.p6.11.m11.2.2.2.3.cmml">(</mo><mi id="S2.SS2.p6.11.m11.2.2.2.2" xref="S2.SS2.p6.11.m11.2.2.2.2.cmml">γ</mi><mo id="S2.SS2.p6.11.m11.2.2.2.4.1.2" stretchy="false" xref="S2.SS2.p6.11.m11.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p6.11.m11.2.2.3" xref="S2.SS2.p6.11.m11.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.11.m11.2b"><apply id="S2.SS2.p6.11.m11.2.3.cmml" xref="S2.SS2.p6.11.m11.2.3"><subset id="S2.SS2.p6.11.m11.2.3.1.cmml" xref="S2.SS2.p6.11.m11.2.3.1"></subset><ci id="S2.SS2.p6.11.m11.2.3.2.cmml" xref="S2.SS2.p6.11.m11.2.3.2">𝑃</ci><apply id="S2.SS2.p6.11.m11.2.2.cmml" xref="S2.SS2.p6.11.m11.2.2"><ci id="S2.SS2.p6.11.m11.2.2.3.cmml" xref="S2.SS2.p6.11.m11.2.2.3">¯</ci><apply id="S2.SS2.p6.11.m11.2.2.2.3.cmml" xref="S2.SS2.p6.11.m11.2.2.2.4"><ci id="S2.SS2.p6.11.m11.1.1.1.1.cmml" xref="S2.SS2.p6.11.m11.1.1.1.1">out</ci><ci id="S2.SS2.p6.11.m11.2.2.2.2.cmml" xref="S2.SS2.p6.11.m11.2.2.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.11.m11.2c">P\subseteq\overline{\operatorname*{out}(\gamma)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.11.m11.2d">italic_P ⊆ over¯ start_ARG roman_out ( italic_γ ) end_ARG</annotation></semantics></math>, I call <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS2.p6.12.m12.1"><semantics id="S2.SS2.p6.12.m12.1a"><mi id="S2.SS2.p6.12.m12.1.1" xref="S2.SS2.p6.12.m12.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.12.m12.1b"><ci id="S2.SS2.p6.12.m12.1.1.cmml" xref="S2.SS2.p6.12.m12.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.12.m12.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.12.m12.1d">italic_γ</annotation></semantics></math> a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p6.13.1">hole</em> of <math alttext="P" class="ltx_Math" display="inline" id="S2.SS2.p6.13.m13.1"><semantics id="S2.SS2.p6.13.m13.1a"><mi id="S2.SS2.p6.13.m13.1.1" xref="S2.SS2.p6.13.m13.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.13.m13.1b"><ci id="S2.SS2.p6.13.m13.1.1.cmml" xref="S2.SS2.p6.13.m13.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.13.m13.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.13.m13.1d">italic_P</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S2.F2"> <table class="ltx_tabular ltx_align_middle" id="S2.F2.3"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.F2.3.4.1"> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.3.4.1.1" style="width:113.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S2.F2.3.4.1.1.1"> <span class="ltx_p" id="S2.F2.3.4.1.1.1.1">bounded</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.3.4.1.2" style="width:113.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S2.F2.3.4.1.2.1"> <span class="ltx_p" id="S2.F2.3.4.1.2.1.1">unbounded</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.3.4.1.3" style="width:113.8pt;"> <table class="ltx_tabular ltx_align_top" id="S2.F2.3.4.1.3.1"> <tr class="ltx_tr" id="S2.F2.3.4.1.3.1.1"> <td class="ltx_td ltx_align_center" id="S2.F2.3.4.1.3.1.1.1">unbounded</td> </tr> <tr class="ltx_tr" id="S2.F2.3.4.1.3.1.2"> <td class="ltx_td ltx_align_center" id="S2.F2.3.4.1.3.1.2.1">with holes</td> </tr> </table> </td> </tr> <tr class="ltx_tr" id="S2.F2.3.3"> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.1.1.1" style="width:113.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S2.F2.1.1.1.1"> <span class="ltx_p" id="S2.F2.1.1.1.1.1"><foreignobject height="43.4pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="43.4pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="83" id="S2.F2.1.1.1.1.1.1.g1" src="x3.png" width="83"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.2.2.2" style="width:113.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S2.F2.2.2.2.1"> <span class="ltx_p" id="S2.F2.2.2.2.1.1"><foreignobject height="41.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="41.9pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="81" id="S2.F2.2.2.2.1.1.1.g1" src="x4.png" width="81"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S2.F2.3.3.3" style="width:113.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S2.F2.3.3.3.1"> <span class="ltx_p" id="S2.F2.3.3.3.1.1"><foreignobject height="41.2pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="52.0pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="80" id="S2.F2.3.3.3.1.1.1.g1" src="x5.png" width="101"/></foreignobject></span> </span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Examples of polygons according to my definition. Polygons are shown in grey, vertices by a dot, and the unbounded directions of lines and rays are indicated by dashed lines. The boundary of the left polygon consists of one polygonal cycle, the one in the middle of two polygonal arcs, and the right one consists of one polygonal arc and two polygonal cycles.</figcaption> </figure> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Continuous Piecewise Affine Functions</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.24">Let <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mrow id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml"><mi id="S2.SS3.p1.1.m1.1.1.2" xref="S2.SS3.p1.1.m1.1.1.2.cmml">f</mi><mo id="S2.SS3.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS3.p1.1.m1.1.1.3" xref="S2.SS3.p1.1.m1.1.1.3.cmml"><msup id="S2.SS3.p1.1.m1.1.1.3.2" xref="S2.SS3.p1.1.m1.1.1.3.2.cmml"><mi id="S2.SS3.p1.1.m1.1.1.3.2.2" xref="S2.SS3.p1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mn id="S2.SS3.p1.1.m1.1.1.3.2.3" xref="S2.SS3.p1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mo id="S2.SS3.p1.1.m1.1.1.3.1" stretchy="false" xref="S2.SS3.p1.1.m1.1.1.3.1.cmml">→</mo><mi id="S2.SS3.p1.1.m1.1.1.3.3" xref="S2.SS3.p1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><apply id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1"><ci id="S2.SS3.p1.1.m1.1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1">:</ci><ci id="S2.SS3.p1.1.m1.1.1.2.cmml" xref="S2.SS3.p1.1.m1.1.1.2">𝑓</ci><apply id="S2.SS3.p1.1.m1.1.1.3.cmml" xref="S2.SS3.p1.1.m1.1.1.3"><ci id="S2.SS3.p1.1.m1.1.1.3.1.cmml" xref="S2.SS3.p1.1.m1.1.1.3.1">→</ci><apply id="S2.SS3.p1.1.m1.1.1.3.2.cmml" xref="S2.SS3.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS3.p1.1.m1.1.1.3.2.1.cmml" xref="S2.SS3.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS3.p1.1.m1.1.1.3.2.2.cmml" xref="S2.SS3.p1.1.m1.1.1.3.2.2">ℝ</ci><cn id="S2.SS3.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS3.p1.1.m1.1.1.3.2.3">2</cn></apply><ci id="S2.SS3.p1.1.m1.1.1.3.3.cmml" xref="S2.SS3.p1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> be a continuous function. Let <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.1"><semantics id="S2.SS3.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.1b"><ci id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.1d">caligraphic_P</annotation></semantics></math> be a set of polygons that covers <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m3.1"><semantics id="S2.SS3.p1.3.m3.1a"><msup id="S2.SS3.p1.3.m3.1.1" xref="S2.SS3.p1.3.m3.1.1.cmml"><mi id="S2.SS3.p1.3.m3.1.1.2" xref="S2.SS3.p1.3.m3.1.1.2.cmml">ℝ</mi><mn id="S2.SS3.p1.3.m3.1.1.3" xref="S2.SS3.p1.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m3.1b"><apply id="S2.SS3.p1.3.m3.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.3.m3.1.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1">superscript</csymbol><ci id="S2.SS3.p1.3.m3.1.1.2.cmml" xref="S2.SS3.p1.3.m3.1.1.2">ℝ</ci><cn id="S2.SS3.p1.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m3.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, such that <math alttext="P\cap Q=\partial P\cap\partial Q" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m4.1"><semantics id="S2.SS3.p1.4.m4.1a"><mrow id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml"><mrow id="S2.SS3.p1.4.m4.1.1.2" xref="S2.SS3.p1.4.m4.1.1.2.cmml"><mi id="S2.SS3.p1.4.m4.1.1.2.2" xref="S2.SS3.p1.4.m4.1.1.2.2.cmml">P</mi><mo id="S2.SS3.p1.4.m4.1.1.2.1" xref="S2.SS3.p1.4.m4.1.1.2.1.cmml">∩</mo><mi id="S2.SS3.p1.4.m4.1.1.2.3" xref="S2.SS3.p1.4.m4.1.1.2.3.cmml">Q</mi></mrow><mo id="S2.SS3.p1.4.m4.1.1.1" rspace="0.1389em" xref="S2.SS3.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS3.p1.4.m4.1.1.3" xref="S2.SS3.p1.4.m4.1.1.3.cmml"><mrow id="S2.SS3.p1.4.m4.1.1.3.2" xref="S2.SS3.p1.4.m4.1.1.3.2.cmml"><mo id="S2.SS3.p1.4.m4.1.1.3.2.1" lspace="0.1389em" rspace="0em" xref="S2.SS3.p1.4.m4.1.1.3.2.1.cmml">∂</mo><mi id="S2.SS3.p1.4.m4.1.1.3.2.2" xref="S2.SS3.p1.4.m4.1.1.3.2.2.cmml">P</mi></mrow><mo id="S2.SS3.p1.4.m4.1.1.3.1" xref="S2.SS3.p1.4.m4.1.1.3.1.cmml">∩</mo><mrow id="S2.SS3.p1.4.m4.1.1.3.3" xref="S2.SS3.p1.4.m4.1.1.3.3.cmml"><mo id="S2.SS3.p1.4.m4.1.1.3.3.1" lspace="0em" rspace="0em" xref="S2.SS3.p1.4.m4.1.1.3.3.1.cmml">∂</mo><mi id="S2.SS3.p1.4.m4.1.1.3.3.2" xref="S2.SS3.p1.4.m4.1.1.3.3.2.cmml">Q</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m4.1b"><apply id="S2.SS3.p1.4.m4.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1"><eq id="S2.SS3.p1.4.m4.1.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1.1"></eq><apply id="S2.SS3.p1.4.m4.1.1.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2"><intersect id="S2.SS3.p1.4.m4.1.1.2.1.cmml" xref="S2.SS3.p1.4.m4.1.1.2.1"></intersect><ci id="S2.SS3.p1.4.m4.1.1.2.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2.2">𝑃</ci><ci id="S2.SS3.p1.4.m4.1.1.2.3.cmml" xref="S2.SS3.p1.4.m4.1.1.2.3">𝑄</ci></apply><apply id="S2.SS3.p1.4.m4.1.1.3.cmml" xref="S2.SS3.p1.4.m4.1.1.3"><intersect id="S2.SS3.p1.4.m4.1.1.3.1.cmml" xref="S2.SS3.p1.4.m4.1.1.3.1"></intersect><apply id="S2.SS3.p1.4.m4.1.1.3.2.cmml" xref="S2.SS3.p1.4.m4.1.1.3.2"><partialdiff id="S2.SS3.p1.4.m4.1.1.3.2.1.cmml" xref="S2.SS3.p1.4.m4.1.1.3.2.1"></partialdiff><ci id="S2.SS3.p1.4.m4.1.1.3.2.2.cmml" xref="S2.SS3.p1.4.m4.1.1.3.2.2">𝑃</ci></apply><apply id="S2.SS3.p1.4.m4.1.1.3.3.cmml" xref="S2.SS3.p1.4.m4.1.1.3.3"><partialdiff id="S2.SS3.p1.4.m4.1.1.3.3.1.cmml" xref="S2.SS3.p1.4.m4.1.1.3.3.1"></partialdiff><ci id="S2.SS3.p1.4.m4.1.1.3.3.2.cmml" xref="S2.SS3.p1.4.m4.1.1.3.3.2">𝑄</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m4.1c">P\cap Q=\partial P\cap\partial Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m4.1d">italic_P ∩ italic_Q = ∂ italic_P ∩ ∂ italic_Q</annotation></semantics></math> holds for any distinct <math alttext="P,Q\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m5.2"><semantics id="S2.SS3.p1.5.m5.2a"><mrow id="S2.SS3.p1.5.m5.2.3" xref="S2.SS3.p1.5.m5.2.3.cmml"><mrow id="S2.SS3.p1.5.m5.2.3.2.2" xref="S2.SS3.p1.5.m5.2.3.2.1.cmml"><mi id="S2.SS3.p1.5.m5.1.1" xref="S2.SS3.p1.5.m5.1.1.cmml">P</mi><mo id="S2.SS3.p1.5.m5.2.3.2.2.1" xref="S2.SS3.p1.5.m5.2.3.2.1.cmml">,</mo><mi id="S2.SS3.p1.5.m5.2.2" xref="S2.SS3.p1.5.m5.2.2.cmml">Q</mi></mrow><mo id="S2.SS3.p1.5.m5.2.3.1" xref="S2.SS3.p1.5.m5.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.5.m5.2.3.3" xref="S2.SS3.p1.5.m5.2.3.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m5.2b"><apply id="S2.SS3.p1.5.m5.2.3.cmml" xref="S2.SS3.p1.5.m5.2.3"><in id="S2.SS3.p1.5.m5.2.3.1.cmml" xref="S2.SS3.p1.5.m5.2.3.1"></in><list id="S2.SS3.p1.5.m5.2.3.2.1.cmml" xref="S2.SS3.p1.5.m5.2.3.2.2"><ci id="S2.SS3.p1.5.m5.1.1.cmml" xref="S2.SS3.p1.5.m5.1.1">𝑃</ci><ci id="S2.SS3.p1.5.m5.2.2.cmml" xref="S2.SS3.p1.5.m5.2.2">𝑄</ci></list><ci id="S2.SS3.p1.5.m5.2.3.3.cmml" xref="S2.SS3.p1.5.m5.2.3.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m5.2c">P,Q\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m5.2d">italic_P , italic_Q ∈ caligraphic_P</annotation></semantics></math>, and such that if <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m6.1"><semantics id="S2.SS3.p1.6.m6.1a"><mrow id="S2.SS3.p1.6.m6.1.2" xref="S2.SS3.p1.6.m6.1.2.cmml"><mi id="S2.SS3.p1.6.m6.1.2.2" xref="S2.SS3.p1.6.m6.1.2.2.cmml">v</mi><mo id="S2.SS3.p1.6.m6.1.2.1" xref="S2.SS3.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.SS3.p1.6.m6.1.2.3" xref="S2.SS3.p1.6.m6.1.2.3.cmml"><mi id="S2.SS3.p1.6.m6.1.2.3.2" xref="S2.SS3.p1.6.m6.1.2.3.2.cmml">V</mi><mo id="S2.SS3.p1.6.m6.1.2.3.1" xref="S2.SS3.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.p1.6.m6.1.2.3.3.2" xref="S2.SS3.p1.6.m6.1.2.3.cmml"><mo id="S2.SS3.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="S2.SS3.p1.6.m6.1.2.3.cmml">(</mo><mi id="S2.SS3.p1.6.m6.1.1" xref="S2.SS3.p1.6.m6.1.1.cmml">P</mi><mo id="S2.SS3.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="S2.SS3.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.6.m6.1b"><apply id="S2.SS3.p1.6.m6.1.2.cmml" xref="S2.SS3.p1.6.m6.1.2"><in id="S2.SS3.p1.6.m6.1.2.1.cmml" xref="S2.SS3.p1.6.m6.1.2.1"></in><ci id="S2.SS3.p1.6.m6.1.2.2.cmml" xref="S2.SS3.p1.6.m6.1.2.2">𝑣</ci><apply id="S2.SS3.p1.6.m6.1.2.3.cmml" xref="S2.SS3.p1.6.m6.1.2.3"><times id="S2.SS3.p1.6.m6.1.2.3.1.cmml" xref="S2.SS3.p1.6.m6.1.2.3.1"></times><ci id="S2.SS3.p1.6.m6.1.2.3.2.cmml" xref="S2.SS3.p1.6.m6.1.2.3.2">𝑉</ci><ci id="S2.SS3.p1.6.m6.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.6.m6.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.6.m6.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math> for some <math alttext="P\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.7.m7.1"><semantics id="S2.SS3.p1.7.m7.1a"><mrow id="S2.SS3.p1.7.m7.1.1" xref="S2.SS3.p1.7.m7.1.1.cmml"><mi id="S2.SS3.p1.7.m7.1.1.2" xref="S2.SS3.p1.7.m7.1.1.2.cmml">P</mi><mo id="S2.SS3.p1.7.m7.1.1.1" xref="S2.SS3.p1.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.7.m7.1.1.3" xref="S2.SS3.p1.7.m7.1.1.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m7.1b"><apply id="S2.SS3.p1.7.m7.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1"><in id="S2.SS3.p1.7.m7.1.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1.1"></in><ci id="S2.SS3.p1.7.m7.1.1.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2">𝑃</ci><ci id="S2.SS3.p1.7.m7.1.1.3.cmml" xref="S2.SS3.p1.7.m7.1.1.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m7.1c">P\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m7.1d">italic_P ∈ caligraphic_P</annotation></semantics></math>, then <math alttext="v\in V(Q)" class="ltx_Math" display="inline" id="S2.SS3.p1.8.m8.1"><semantics id="S2.SS3.p1.8.m8.1a"><mrow id="S2.SS3.p1.8.m8.1.2" xref="S2.SS3.p1.8.m8.1.2.cmml"><mi id="S2.SS3.p1.8.m8.1.2.2" xref="S2.SS3.p1.8.m8.1.2.2.cmml">v</mi><mo id="S2.SS3.p1.8.m8.1.2.1" xref="S2.SS3.p1.8.m8.1.2.1.cmml">∈</mo><mrow id="S2.SS3.p1.8.m8.1.2.3" xref="S2.SS3.p1.8.m8.1.2.3.cmml"><mi id="S2.SS3.p1.8.m8.1.2.3.2" xref="S2.SS3.p1.8.m8.1.2.3.2.cmml">V</mi><mo id="S2.SS3.p1.8.m8.1.2.3.1" xref="S2.SS3.p1.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.p1.8.m8.1.2.3.3.2" xref="S2.SS3.p1.8.m8.1.2.3.cmml"><mo id="S2.SS3.p1.8.m8.1.2.3.3.2.1" stretchy="false" xref="S2.SS3.p1.8.m8.1.2.3.cmml">(</mo><mi id="S2.SS3.p1.8.m8.1.1" xref="S2.SS3.p1.8.m8.1.1.cmml">Q</mi><mo id="S2.SS3.p1.8.m8.1.2.3.3.2.2" stretchy="false" xref="S2.SS3.p1.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.8.m8.1b"><apply id="S2.SS3.p1.8.m8.1.2.cmml" xref="S2.SS3.p1.8.m8.1.2"><in id="S2.SS3.p1.8.m8.1.2.1.cmml" xref="S2.SS3.p1.8.m8.1.2.1"></in><ci id="S2.SS3.p1.8.m8.1.2.2.cmml" xref="S2.SS3.p1.8.m8.1.2.2">𝑣</ci><apply id="S2.SS3.p1.8.m8.1.2.3.cmml" xref="S2.SS3.p1.8.m8.1.2.3"><times id="S2.SS3.p1.8.m8.1.2.3.1.cmml" xref="S2.SS3.p1.8.m8.1.2.3.1"></times><ci id="S2.SS3.p1.8.m8.1.2.3.2.cmml" xref="S2.SS3.p1.8.m8.1.2.3.2">𝑉</ci><ci id="S2.SS3.p1.8.m8.1.1.cmml" xref="S2.SS3.p1.8.m8.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.8.m8.1c">v\in V(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.8.m8.1d">italic_v ∈ italic_V ( italic_Q )</annotation></semantics></math> for all <math alttext="Q\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.9.m9.1"><semantics id="S2.SS3.p1.9.m9.1a"><mrow id="S2.SS3.p1.9.m9.1.1" xref="S2.SS3.p1.9.m9.1.1.cmml"><mi id="S2.SS3.p1.9.m9.1.1.2" xref="S2.SS3.p1.9.m9.1.1.2.cmml">Q</mi><mo id="S2.SS3.p1.9.m9.1.1.1" xref="S2.SS3.p1.9.m9.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.9.m9.1.1.3" xref="S2.SS3.p1.9.m9.1.1.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.9.m9.1b"><apply id="S2.SS3.p1.9.m9.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1"><in id="S2.SS3.p1.9.m9.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1"></in><ci id="S2.SS3.p1.9.m9.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.2">𝑄</ci><ci id="S2.SS3.p1.9.m9.1.1.3.cmml" xref="S2.SS3.p1.9.m9.1.1.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.9.m9.1c">Q\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.9.m9.1d">italic_Q ∈ caligraphic_P</annotation></semantics></math> containing <math alttext="v" class="ltx_Math" display="inline" id="S2.SS3.p1.10.m10.1"><semantics id="S2.SS3.p1.10.m10.1a"><mi id="S2.SS3.p1.10.m10.1.1" xref="S2.SS3.p1.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.10.m10.1b"><ci id="S2.SS3.p1.10.m10.1.1.cmml" xref="S2.SS3.p1.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.10.m10.1d">italic_v</annotation></semantics></math>. Assume that there are affine functions <math alttext="\{f_{P}\}_{P\in\mathcal{P}}" class="ltx_Math" display="inline" id="S2.SS3.p1.11.m11.1"><semantics id="S2.SS3.p1.11.m11.1a"><msub id="S2.SS3.p1.11.m11.1.1" xref="S2.SS3.p1.11.m11.1.1.cmml"><mrow id="S2.SS3.p1.11.m11.1.1.1.1" xref="S2.SS3.p1.11.m11.1.1.1.2.cmml"><mo id="S2.SS3.p1.11.m11.1.1.1.1.2" stretchy="false" xref="S2.SS3.p1.11.m11.1.1.1.2.cmml">{</mo><msub id="S2.SS3.p1.11.m11.1.1.1.1.1" xref="S2.SS3.p1.11.m11.1.1.1.1.1.cmml"><mi id="S2.SS3.p1.11.m11.1.1.1.1.1.2" xref="S2.SS3.p1.11.m11.1.1.1.1.1.2.cmml">f</mi><mi id="S2.SS3.p1.11.m11.1.1.1.1.1.3" xref="S2.SS3.p1.11.m11.1.1.1.1.1.3.cmml">P</mi></msub><mo id="S2.SS3.p1.11.m11.1.1.1.1.3" stretchy="false" xref="S2.SS3.p1.11.m11.1.1.1.2.cmml">}</mo></mrow><mrow id="S2.SS3.p1.11.m11.1.1.3" xref="S2.SS3.p1.11.m11.1.1.3.cmml"><mi id="S2.SS3.p1.11.m11.1.1.3.2" xref="S2.SS3.p1.11.m11.1.1.3.2.cmml">P</mi><mo id="S2.SS3.p1.11.m11.1.1.3.1" xref="S2.SS3.p1.11.m11.1.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.11.m11.1.1.3.3" xref="S2.SS3.p1.11.m11.1.1.3.3.cmml">𝒫</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.11.m11.1b"><apply id="S2.SS3.p1.11.m11.1.1.cmml" xref="S2.SS3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.11.m11.1.1.2.cmml" xref="S2.SS3.p1.11.m11.1.1">subscript</csymbol><set id="S2.SS3.p1.11.m11.1.1.1.2.cmml" xref="S2.SS3.p1.11.m11.1.1.1.1"><apply id="S2.SS3.p1.11.m11.1.1.1.1.1.cmml" xref="S2.SS3.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p1.11.m11.1.1.1.1.1.2.cmml" xref="S2.SS3.p1.11.m11.1.1.1.1.1.2">𝑓</ci><ci id="S2.SS3.p1.11.m11.1.1.1.1.1.3.cmml" xref="S2.SS3.p1.11.m11.1.1.1.1.1.3">𝑃</ci></apply></set><apply id="S2.SS3.p1.11.m11.1.1.3.cmml" xref="S2.SS3.p1.11.m11.1.1.3"><in id="S2.SS3.p1.11.m11.1.1.3.1.cmml" xref="S2.SS3.p1.11.m11.1.1.3.1"></in><ci id="S2.SS3.p1.11.m11.1.1.3.2.cmml" xref="S2.SS3.p1.11.m11.1.1.3.2">𝑃</ci><ci id="S2.SS3.p1.11.m11.1.1.3.3.cmml" xref="S2.SS3.p1.11.m11.1.1.3.3">𝒫</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.11.m11.1c">\{f_{P}\}_{P\in\mathcal{P}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.11.m11.1d">{ italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f|_{P}=f_{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.12.m12.2"><semantics id="S2.SS3.p1.12.m12.2a"><mrow id="S2.SS3.p1.12.m12.2.3" xref="S2.SS3.p1.12.m12.2.3.cmml"><msub id="S2.SS3.p1.12.m12.2.3.2.2" xref="S2.SS3.p1.12.m12.2.3.2.1.cmml"><mrow id="S2.SS3.p1.12.m12.2.3.2.2.2" xref="S2.SS3.p1.12.m12.2.3.2.1.cmml"><mi id="S2.SS3.p1.12.m12.1.1" xref="S2.SS3.p1.12.m12.1.1.cmml">f</mi><mo id="S2.SS3.p1.12.m12.2.3.2.2.2.1" stretchy="false" xref="S2.SS3.p1.12.m12.2.3.2.1.1.cmml">|</mo></mrow><mi id="S2.SS3.p1.12.m12.2.2.1" xref="S2.SS3.p1.12.m12.2.2.1.cmml">P</mi></msub><mo id="S2.SS3.p1.12.m12.2.3.1" xref="S2.SS3.p1.12.m12.2.3.1.cmml">=</mo><msub id="S2.SS3.p1.12.m12.2.3.3" xref="S2.SS3.p1.12.m12.2.3.3.cmml"><mi id="S2.SS3.p1.12.m12.2.3.3.2" xref="S2.SS3.p1.12.m12.2.3.3.2.cmml">f</mi><mi id="S2.SS3.p1.12.m12.2.3.3.3" xref="S2.SS3.p1.12.m12.2.3.3.3.cmml">P</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.12.m12.2b"><apply id="S2.SS3.p1.12.m12.2.3.cmml" xref="S2.SS3.p1.12.m12.2.3"><eq id="S2.SS3.p1.12.m12.2.3.1.cmml" xref="S2.SS3.p1.12.m12.2.3.1"></eq><apply id="S2.SS3.p1.12.m12.2.3.2.1.cmml" xref="S2.SS3.p1.12.m12.2.3.2.2"><csymbol cd="latexml" id="S2.SS3.p1.12.m12.2.3.2.1.1.cmml" xref="S2.SS3.p1.12.m12.2.3.2.2.2.1">evaluated-at</csymbol><ci id="S2.SS3.p1.12.m12.1.1.cmml" xref="S2.SS3.p1.12.m12.1.1">𝑓</ci><ci id="S2.SS3.p1.12.m12.2.2.1.cmml" xref="S2.SS3.p1.12.m12.2.2.1">𝑃</ci></apply><apply id="S2.SS3.p1.12.m12.2.3.3.cmml" xref="S2.SS3.p1.12.m12.2.3.3"><csymbol cd="ambiguous" id="S2.SS3.p1.12.m12.2.3.3.1.cmml" xref="S2.SS3.p1.12.m12.2.3.3">subscript</csymbol><ci id="S2.SS3.p1.12.m12.2.3.3.2.cmml" xref="S2.SS3.p1.12.m12.2.3.3.2">𝑓</ci><ci id="S2.SS3.p1.12.m12.2.3.3.3.cmml" xref="S2.SS3.p1.12.m12.2.3.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.12.m12.2c">f|_{P}=f_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.12.m12.2d">italic_f | start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="P\in\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.13.m13.1"><semantics id="S2.SS3.p1.13.m13.1a"><mrow id="S2.SS3.p1.13.m13.1.1" xref="S2.SS3.p1.13.m13.1.1.cmml"><mi id="S2.SS3.p1.13.m13.1.1.2" xref="S2.SS3.p1.13.m13.1.1.2.cmml">P</mi><mo id="S2.SS3.p1.13.m13.1.1.1" xref="S2.SS3.p1.13.m13.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.13.m13.1.1.3" xref="S2.SS3.p1.13.m13.1.1.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.13.m13.1b"><apply id="S2.SS3.p1.13.m13.1.1.cmml" xref="S2.SS3.p1.13.m13.1.1"><in id="S2.SS3.p1.13.m13.1.1.1.cmml" xref="S2.SS3.p1.13.m13.1.1.1"></in><ci id="S2.SS3.p1.13.m13.1.1.2.cmml" xref="S2.SS3.p1.13.m13.1.1.2">𝑃</ci><ci id="S2.SS3.p1.13.m13.1.1.3.cmml" xref="S2.SS3.p1.13.m13.1.1.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.13.m13.1c">P\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.13.m13.1d">italic_P ∈ caligraphic_P</annotation></semantics></math>. Then, <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p1.14.m14.1"><semantics id="S2.SS3.p1.14.m14.1a"><mi id="S2.SS3.p1.14.m14.1.1" xref="S2.SS3.p1.14.m14.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.14.m14.1b"><ci id="S2.SS3.p1.14.m14.1.1.cmml" xref="S2.SS3.p1.14.m14.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.14.m14.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.14.m14.1d">italic_f</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.24.1">continuous piecewise affine</em> (CPA), and <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.15.m15.1"><semantics id="S2.SS3.p1.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.15.m15.1.1" xref="S2.SS3.p1.15.m15.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.15.m15.1b"><ci id="S2.SS3.p1.15.m15.1.1.cmml" xref="S2.SS3.p1.15.m15.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.15.m15.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.15.m15.1d">caligraphic_P</annotation></semantics></math> is said to be an admissible set of <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.24.2">pieces</em> for <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p1.16.m16.1"><semantics id="S2.SS3.p1.16.m16.1a"><mi id="S2.SS3.p1.16.m16.1.1" xref="S2.SS3.p1.16.m16.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.16.m16.1b"><ci id="S2.SS3.p1.16.m16.1.1.cmml" xref="S2.SS3.p1.16.m16.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.16.m16.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.16.m16.1d">italic_f</annotation></semantics></math>. The affine function <math alttext="f_{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.17.m17.1"><semantics id="S2.SS3.p1.17.m17.1a"><msub id="S2.SS3.p1.17.m17.1.1" xref="S2.SS3.p1.17.m17.1.1.cmml"><mi id="S2.SS3.p1.17.m17.1.1.2" xref="S2.SS3.p1.17.m17.1.1.2.cmml">f</mi><mi id="S2.SS3.p1.17.m17.1.1.3" xref="S2.SS3.p1.17.m17.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.17.m17.1b"><apply id="S2.SS3.p1.17.m17.1.1.cmml" xref="S2.SS3.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.17.m17.1.1.1.cmml" xref="S2.SS3.p1.17.m17.1.1">subscript</csymbol><ci id="S2.SS3.p1.17.m17.1.1.2.cmml" xref="S2.SS3.p1.17.m17.1.1.2">𝑓</ci><ci id="S2.SS3.p1.17.m17.1.1.3.cmml" xref="S2.SS3.p1.17.m17.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.17.m17.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.17.m17.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> satisfying <math alttext="f|_{P}=f_{P}" class="ltx_Math" display="inline" id="S2.SS3.p1.18.m18.2"><semantics id="S2.SS3.p1.18.m18.2a"><mrow id="S2.SS3.p1.18.m18.2.3" xref="S2.SS3.p1.18.m18.2.3.cmml"><msub id="S2.SS3.p1.18.m18.2.3.2.2" xref="S2.SS3.p1.18.m18.2.3.2.1.cmml"><mrow id="S2.SS3.p1.18.m18.2.3.2.2.2" xref="S2.SS3.p1.18.m18.2.3.2.1.cmml"><mi id="S2.SS3.p1.18.m18.1.1" xref="S2.SS3.p1.18.m18.1.1.cmml">f</mi><mo id="S2.SS3.p1.18.m18.2.3.2.2.2.1" stretchy="false" xref="S2.SS3.p1.18.m18.2.3.2.1.1.cmml">|</mo></mrow><mi id="S2.SS3.p1.18.m18.2.2.1" xref="S2.SS3.p1.18.m18.2.2.1.cmml">P</mi></msub><mo id="S2.SS3.p1.18.m18.2.3.1" xref="S2.SS3.p1.18.m18.2.3.1.cmml">=</mo><msub id="S2.SS3.p1.18.m18.2.3.3" xref="S2.SS3.p1.18.m18.2.3.3.cmml"><mi id="S2.SS3.p1.18.m18.2.3.3.2" xref="S2.SS3.p1.18.m18.2.3.3.2.cmml">f</mi><mi id="S2.SS3.p1.18.m18.2.3.3.3" xref="S2.SS3.p1.18.m18.2.3.3.3.cmml">P</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.18.m18.2b"><apply id="S2.SS3.p1.18.m18.2.3.cmml" xref="S2.SS3.p1.18.m18.2.3"><eq id="S2.SS3.p1.18.m18.2.3.1.cmml" xref="S2.SS3.p1.18.m18.2.3.1"></eq><apply id="S2.SS3.p1.18.m18.2.3.2.1.cmml" xref="S2.SS3.p1.18.m18.2.3.2.2"><csymbol cd="latexml" id="S2.SS3.p1.18.m18.2.3.2.1.1.cmml" xref="S2.SS3.p1.18.m18.2.3.2.2.2.1">evaluated-at</csymbol><ci id="S2.SS3.p1.18.m18.1.1.cmml" xref="S2.SS3.p1.18.m18.1.1">𝑓</ci><ci id="S2.SS3.p1.18.m18.2.2.1.cmml" xref="S2.SS3.p1.18.m18.2.2.1">𝑃</ci></apply><apply id="S2.SS3.p1.18.m18.2.3.3.cmml" xref="S2.SS3.p1.18.m18.2.3.3"><csymbol cd="ambiguous" id="S2.SS3.p1.18.m18.2.3.3.1.cmml" xref="S2.SS3.p1.18.m18.2.3.3">subscript</csymbol><ci id="S2.SS3.p1.18.m18.2.3.3.2.cmml" xref="S2.SS3.p1.18.m18.2.3.3.2">𝑓</ci><ci id="S2.SS3.p1.18.m18.2.3.3.3.cmml" xref="S2.SS3.p1.18.m18.2.3.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.18.m18.2c">f|_{P}=f_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.18.m18.2d">italic_f | start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> for a piece <math alttext="P" class="ltx_Math" display="inline" id="S2.SS3.p1.19.m19.1"><semantics id="S2.SS3.p1.19.m19.1a"><mi id="S2.SS3.p1.19.m19.1.1" xref="S2.SS3.p1.19.m19.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.19.m19.1b"><ci id="S2.SS3.p1.19.m19.1.1.cmml" xref="S2.SS3.p1.19.m19.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.19.m19.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.19.m19.1d">italic_P</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.24.3">affine component</em> of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p1.20.m20.1"><semantics id="S2.SS3.p1.20.m20.1a"><mi id="S2.SS3.p1.20.m20.1.1" xref="S2.SS3.p1.20.m20.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.20.m20.1b"><ci id="S2.SS3.p1.20.m20.1.1.cmml" xref="S2.SS3.p1.20.m20.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.20.m20.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.20.m20.1d">italic_f</annotation></semantics></math> corresponding to <math alttext="P" class="ltx_Math" display="inline" id="S2.SS3.p1.21.m21.1"><semantics id="S2.SS3.p1.21.m21.1a"><mi id="S2.SS3.p1.21.m21.1.1" xref="S2.SS3.p1.21.m21.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.21.m21.1b"><ci id="S2.SS3.p1.21.m21.1.1.cmml" xref="S2.SS3.p1.21.m21.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.21.m21.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.21.m21.1d">italic_P</annotation></semantics></math>. The family of continuous piecewise affine functions in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS3.p1.22.m22.1"><semantics id="S2.SS3.p1.22.m22.1a"><msup id="S2.SS3.p1.22.m22.1.1" xref="S2.SS3.p1.22.m22.1.1.cmml"><mi id="S2.SS3.p1.22.m22.1.1.2" xref="S2.SS3.p1.22.m22.1.1.2.cmml">ℝ</mi><mn id="S2.SS3.p1.22.m22.1.1.3" xref="S2.SS3.p1.22.m22.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.22.m22.1b"><apply id="S2.SS3.p1.22.m22.1.1.cmml" xref="S2.SS3.p1.22.m22.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.22.m22.1.1.1.cmml" xref="S2.SS3.p1.22.m22.1.1">superscript</csymbol><ci id="S2.SS3.p1.22.m22.1.1.2.cmml" xref="S2.SS3.p1.22.m22.1.1.2">ℝ</ci><cn id="S2.SS3.p1.22.m22.1.1.3.cmml" type="integer" xref="S2.SS3.p1.22.m22.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.22.m22.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.22.m22.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> that can be defined by <math alttext="p" class="ltx_Math" display="inline" id="S2.SS3.p1.23.m23.1"><semantics id="S2.SS3.p1.23.m23.1a"><mi id="S2.SS3.p1.23.m23.1.1" xref="S2.SS3.p1.23.m23.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.23.m23.1b"><ci id="S2.SS3.p1.23.m23.1.1.cmml" xref="S2.SS3.p1.23.m23.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.23.m23.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.23.m23.1d">italic_p</annotation></semantics></math> pieces is denoted by <math alttext="\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S2.SS3.p1.24.m24.1"><semantics id="S2.SS3.p1.24.m24.1a"><msub id="S2.SS3.p1.24.m24.1.1" xref="S2.SS3.p1.24.m24.1.1.cmml"><mi id="S2.SS3.p1.24.m24.1.1.2" xref="S2.SS3.p1.24.m24.1.1.2.cmml">CPA</mi><mi id="S2.SS3.p1.24.m24.1.1.3" xref="S2.SS3.p1.24.m24.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.24.m24.1b"><apply id="S2.SS3.p1.24.m24.1.1.cmml" xref="S2.SS3.p1.24.m24.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.24.m24.1.1.1.cmml" xref="S2.SS3.p1.24.m24.1.1">subscript</csymbol><ci id="S2.SS3.p1.24.m24.1.1.2.cmml" xref="S2.SS3.p1.24.m24.1.1.2">CPA</ci><ci id="S2.SS3.p1.24.m24.1.1.3.cmml" xref="S2.SS3.p1.24.m24.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.24.m24.1c">\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.24.m24.1d">roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.3">Note that a piece can always be split into two which means that there is more than one set of admissible pieces and that <math alttext="\operatorname{CPA}_{p}\subseteq\operatorname{CPA}_{p+1}" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mrow id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml"><msub id="S2.SS3.p2.1.m1.1.1.2" xref="S2.SS3.p2.1.m1.1.1.2.cmml"><mi id="S2.SS3.p2.1.m1.1.1.2.2" xref="S2.SS3.p2.1.m1.1.1.2.2.cmml">CPA</mi><mi id="S2.SS3.p2.1.m1.1.1.2.3" xref="S2.SS3.p2.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="S2.SS3.p2.1.m1.1.1.1" xref="S2.SS3.p2.1.m1.1.1.1.cmml">⊆</mo><msub id="S2.SS3.p2.1.m1.1.1.3" xref="S2.SS3.p2.1.m1.1.1.3.cmml"><mi id="S2.SS3.p2.1.m1.1.1.3.2" xref="S2.SS3.p2.1.m1.1.1.3.2.cmml">CPA</mi><mrow id="S2.SS3.p2.1.m1.1.1.3.3" xref="S2.SS3.p2.1.m1.1.1.3.3.cmml"><mi id="S2.SS3.p2.1.m1.1.1.3.3.2" xref="S2.SS3.p2.1.m1.1.1.3.3.2.cmml">p</mi><mo id="S2.SS3.p2.1.m1.1.1.3.3.1" xref="S2.SS3.p2.1.m1.1.1.3.3.1.cmml">+</mo><mn id="S2.SS3.p2.1.m1.1.1.3.3.3" xref="S2.SS3.p2.1.m1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><apply id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1"><subset id="S2.SS3.p2.1.m1.1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1.1"></subset><apply id="S2.SS3.p2.1.m1.1.1.2.cmml" xref="S2.SS3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.1.m1.1.1.2.1.cmml" xref="S2.SS3.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.1.m1.1.1.2.2.cmml" xref="S2.SS3.p2.1.m1.1.1.2.2">CPA</ci><ci id="S2.SS3.p2.1.m1.1.1.2.3.cmml" xref="S2.SS3.p2.1.m1.1.1.2.3">𝑝</ci></apply><apply id="S2.SS3.p2.1.m1.1.1.3.cmml" xref="S2.SS3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p2.1.m1.1.1.3.1.cmml" xref="S2.SS3.p2.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS3.p2.1.m1.1.1.3.2.cmml" xref="S2.SS3.p2.1.m1.1.1.3.2">CPA</ci><apply id="S2.SS3.p2.1.m1.1.1.3.3.cmml" xref="S2.SS3.p2.1.m1.1.1.3.3"><plus id="S2.SS3.p2.1.m1.1.1.3.3.1.cmml" xref="S2.SS3.p2.1.m1.1.1.3.3.1"></plus><ci id="S2.SS3.p2.1.m1.1.1.3.3.2.cmml" xref="S2.SS3.p2.1.m1.1.1.3.3.2">𝑝</ci><cn id="S2.SS3.p2.1.m1.1.1.3.3.3.cmml" type="integer" xref="S2.SS3.p2.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\operatorname{CPA}_{p}\subseteq\operatorname{CPA}_{p+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⊆ roman_CPA start_POSTSUBSCRIPT italic_p + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, the restriction on the vertices of the pieces ensures only that the pieces are mutually compatible, but still does not make the choice of vertices unique. Therefore, when discussing a <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">roman_CPA</annotation></semantics></math> function, we must always fix a specific set of admissible pieces <math alttext="\mathcal{P}(f)" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mrow id="S2.SS3.p2.3.m3.1.2" xref="S2.SS3.p2.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.3.m3.1.2.2" xref="S2.SS3.p2.3.m3.1.2.2.cmml">𝒫</mi><mo id="S2.SS3.p2.3.m3.1.2.1" xref="S2.SS3.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.3.m3.1.2.3.2" xref="S2.SS3.p2.3.m3.1.2.cmml"><mo id="S2.SS3.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.3.m3.1.2.cmml">(</mo><mi id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml">f</mi><mo id="S2.SS3.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.2.cmml" xref="S2.SS3.p2.3.m3.1.2"><times id="S2.SS3.p2.3.m3.1.2.1.cmml" xref="S2.SS3.p2.3.m3.1.2.1"></times><ci id="S2.SS3.p2.3.m3.1.2.2.cmml" xref="S2.SS3.p2.3.m3.1.2.2">𝒫</ci><ci id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">caligraphic_P ( italic_f )</annotation></semantics></math> along with their vertex and edge sets.</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.9">For a function <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.1"><semantics id="S2.SS3.p3.1.m1.1a"><mrow id="S2.SS3.p3.1.m1.1.1" xref="S2.SS3.p3.1.m1.1.1.cmml"><mi id="S2.SS3.p3.1.m1.1.1.2" xref="S2.SS3.p3.1.m1.1.1.2.cmml">f</mi><mo id="S2.SS3.p3.1.m1.1.1.1" xref="S2.SS3.p3.1.m1.1.1.1.cmml">∈</mo><msub id="S2.SS3.p3.1.m1.1.1.3" xref="S2.SS3.p3.1.m1.1.1.3.cmml"><mi id="S2.SS3.p3.1.m1.1.1.3.2" xref="S2.SS3.p3.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S2.SS3.p3.1.m1.1.1.3.3" xref="S2.SS3.p3.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.1b"><apply id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1"><in id="S2.SS3.p3.1.m1.1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1.1"></in><ci id="S2.SS3.p3.1.m1.1.1.2.cmml" xref="S2.SS3.p3.1.m1.1.1.2">𝑓</ci><apply id="S2.SS3.p3.1.m1.1.1.3.cmml" xref="S2.SS3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p3.1.m1.1.1.3.1.cmml" xref="S2.SS3.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS3.p3.1.m1.1.1.3.2.cmml" xref="S2.SS3.p3.1.m1.1.1.3.2">CPA</ci><ci id="S2.SS3.p3.1.m1.1.1.3.3.cmml" xref="S2.SS3.p3.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> with pieces <math alttext="\mathcal{P}(f)" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><mrow id="S2.SS3.p3.2.m2.1.2" xref="S2.SS3.p3.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p3.2.m2.1.2.2" xref="S2.SS3.p3.2.m2.1.2.2.cmml">𝒫</mi><mo id="S2.SS3.p3.2.m2.1.2.1" xref="S2.SS3.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.2.m2.1.2.3.2" xref="S2.SS3.p3.2.m2.1.2.cmml"><mo id="S2.SS3.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml">f</mi><mo id="S2.SS3.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.1b"><apply id="S2.SS3.p3.2.m2.1.2.cmml" xref="S2.SS3.p3.2.m2.1.2"><times id="S2.SS3.p3.2.m2.1.2.1.cmml" xref="S2.SS3.p3.2.m2.1.2.1"></times><ci id="S2.SS3.p3.2.m2.1.2.2.cmml" xref="S2.SS3.p3.2.m2.1.2.2">𝒫</ci><ci id="S2.SS3.p3.2.m2.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.1c">\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.1d">caligraphic_P ( italic_f )</annotation></semantics></math>, I carry over the names and definitions of the sets <math alttext="V(P)" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.1"><semantics id="S2.SS3.p3.3.m3.1a"><mrow id="S2.SS3.p3.3.m3.1.2" xref="S2.SS3.p3.3.m3.1.2.cmml"><mi id="S2.SS3.p3.3.m3.1.2.2" xref="S2.SS3.p3.3.m3.1.2.2.cmml">V</mi><mo id="S2.SS3.p3.3.m3.1.2.1" xref="S2.SS3.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.3.m3.1.2.3.2" xref="S2.SS3.p3.3.m3.1.2.cmml"><mo id="S2.SS3.p3.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.3.m3.1.2.cmml">(</mo><mi id="S2.SS3.p3.3.m3.1.1" xref="S2.SS3.p3.3.m3.1.1.cmml">P</mi><mo id="S2.SS3.p3.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.3.m3.1b"><apply id="S2.SS3.p3.3.m3.1.2.cmml" xref="S2.SS3.p3.3.m3.1.2"><times id="S2.SS3.p3.3.m3.1.2.1.cmml" xref="S2.SS3.p3.3.m3.1.2.1"></times><ci id="S2.SS3.p3.3.m3.1.2.2.cmml" xref="S2.SS3.p3.3.m3.1.2.2">𝑉</ci><ci id="S2.SS3.p3.3.m3.1.1.cmml" xref="S2.SS3.p3.3.m3.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.3.m3.1c">V(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.3.m3.1d">italic_V ( italic_P )</annotation></semantics></math>, <math alttext="E_{b}(P)" class="ltx_Math" display="inline" id="S2.SS3.p3.4.m4.1"><semantics id="S2.SS3.p3.4.m4.1a"><mrow id="S2.SS3.p3.4.m4.1.2" xref="S2.SS3.p3.4.m4.1.2.cmml"><msub id="S2.SS3.p3.4.m4.1.2.2" xref="S2.SS3.p3.4.m4.1.2.2.cmml"><mi id="S2.SS3.p3.4.m4.1.2.2.2" xref="S2.SS3.p3.4.m4.1.2.2.2.cmml">E</mi><mi id="S2.SS3.p3.4.m4.1.2.2.3" xref="S2.SS3.p3.4.m4.1.2.2.3.cmml">b</mi></msub><mo id="S2.SS3.p3.4.m4.1.2.1" xref="S2.SS3.p3.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.4.m4.1.2.3.2" xref="S2.SS3.p3.4.m4.1.2.cmml"><mo id="S2.SS3.p3.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.4.m4.1.2.cmml">(</mo><mi id="S2.SS3.p3.4.m4.1.1" xref="S2.SS3.p3.4.m4.1.1.cmml">P</mi><mo id="S2.SS3.p3.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.4.m4.1b"><apply id="S2.SS3.p3.4.m4.1.2.cmml" xref="S2.SS3.p3.4.m4.1.2"><times id="S2.SS3.p3.4.m4.1.2.1.cmml" xref="S2.SS3.p3.4.m4.1.2.1"></times><apply id="S2.SS3.p3.4.m4.1.2.2.cmml" xref="S2.SS3.p3.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.4.m4.1.2.2.1.cmml" xref="S2.SS3.p3.4.m4.1.2.2">subscript</csymbol><ci id="S2.SS3.p3.4.m4.1.2.2.2.cmml" xref="S2.SS3.p3.4.m4.1.2.2.2">𝐸</ci><ci id="S2.SS3.p3.4.m4.1.2.2.3.cmml" xref="S2.SS3.p3.4.m4.1.2.2.3">𝑏</ci></apply><ci id="S2.SS3.p3.4.m4.1.1.cmml" xref="S2.SS3.p3.4.m4.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.4.m4.1c">E_{b}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>, <math alttext="E_{l}(P)" class="ltx_Math" display="inline" id="S2.SS3.p3.5.m5.1"><semantics id="S2.SS3.p3.5.m5.1a"><mrow id="S2.SS3.p3.5.m5.1.2" xref="S2.SS3.p3.5.m5.1.2.cmml"><msub id="S2.SS3.p3.5.m5.1.2.2" xref="S2.SS3.p3.5.m5.1.2.2.cmml"><mi id="S2.SS3.p3.5.m5.1.2.2.2" xref="S2.SS3.p3.5.m5.1.2.2.2.cmml">E</mi><mi id="S2.SS3.p3.5.m5.1.2.2.3" xref="S2.SS3.p3.5.m5.1.2.2.3.cmml">l</mi></msub><mo id="S2.SS3.p3.5.m5.1.2.1" xref="S2.SS3.p3.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.5.m5.1.2.3.2" xref="S2.SS3.p3.5.m5.1.2.cmml"><mo id="S2.SS3.p3.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS3.p3.5.m5.1.2.cmml">(</mo><mi id="S2.SS3.p3.5.m5.1.1" xref="S2.SS3.p3.5.m5.1.1.cmml">P</mi><mo id="S2.SS3.p3.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS3.p3.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.5.m5.1b"><apply id="S2.SS3.p3.5.m5.1.2.cmml" xref="S2.SS3.p3.5.m5.1.2"><times id="S2.SS3.p3.5.m5.1.2.1.cmml" xref="S2.SS3.p3.5.m5.1.2.1"></times><apply id="S2.SS3.p3.5.m5.1.2.2.cmml" xref="S2.SS3.p3.5.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.5.m5.1.2.2.1.cmml" xref="S2.SS3.p3.5.m5.1.2.2">subscript</csymbol><ci id="S2.SS3.p3.5.m5.1.2.2.2.cmml" xref="S2.SS3.p3.5.m5.1.2.2.2">𝐸</ci><ci id="S2.SS3.p3.5.m5.1.2.2.3.cmml" xref="S2.SS3.p3.5.m5.1.2.2.3">𝑙</ci></apply><ci id="S2.SS3.p3.5.m5.1.1.cmml" xref="S2.SS3.p3.5.m5.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.5.m5.1c">E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.5.m5.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> from polygons by taking the union of the respective sets over all the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p3.6.m6.1"><semantics id="S2.SS3.p3.6.m6.1a"><mi id="S2.SS3.p3.6.m6.1.1" xref="S2.SS3.p3.6.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.6.m6.1b"><ci id="S2.SS3.p3.6.m6.1.1.cmml" xref="S2.SS3.p3.6.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.6.m6.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.6.m6.1d">italic_f</annotation></semantics></math>. For example, the set of vertices of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p3.7.m7.1"><semantics id="S2.SS3.p3.7.m7.1a"><mi id="S2.SS3.p3.7.m7.1.1" xref="S2.SS3.p3.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.7.m7.1b"><ci id="S2.SS3.p3.7.m7.1.1.cmml" xref="S2.SS3.p3.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.7.m7.1d">italic_f</annotation></semantics></math> is defined by <math alttext="V(f):=\bigcup_{P\in\mathcal{P}(f)}V(P)" class="ltx_Math" display="inline" id="S2.SS3.p3.8.m8.3"><semantics id="S2.SS3.p3.8.m8.3a"><mrow id="S2.SS3.p3.8.m8.3.4" xref="S2.SS3.p3.8.m8.3.4.cmml"><mrow id="S2.SS3.p3.8.m8.3.4.2" xref="S2.SS3.p3.8.m8.3.4.2.cmml"><mi id="S2.SS3.p3.8.m8.3.4.2.2" xref="S2.SS3.p3.8.m8.3.4.2.2.cmml">V</mi><mo id="S2.SS3.p3.8.m8.3.4.2.1" xref="S2.SS3.p3.8.m8.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.8.m8.3.4.2.3.2" xref="S2.SS3.p3.8.m8.3.4.2.cmml"><mo id="S2.SS3.p3.8.m8.3.4.2.3.2.1" stretchy="false" xref="S2.SS3.p3.8.m8.3.4.2.cmml">(</mo><mi id="S2.SS3.p3.8.m8.2.2" xref="S2.SS3.p3.8.m8.2.2.cmml">f</mi><mo id="S2.SS3.p3.8.m8.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS3.p3.8.m8.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p3.8.m8.3.4.1" rspace="0.111em" xref="S2.SS3.p3.8.m8.3.4.1.cmml">:=</mo><mrow id="S2.SS3.p3.8.m8.3.4.3" xref="S2.SS3.p3.8.m8.3.4.3.cmml"><msub id="S2.SS3.p3.8.m8.3.4.3.1" xref="S2.SS3.p3.8.m8.3.4.3.1.cmml"><mo id="S2.SS3.p3.8.m8.3.4.3.1.2" xref="S2.SS3.p3.8.m8.3.4.3.1.2.cmml">⋃</mo><mrow id="S2.SS3.p3.8.m8.1.1.1" xref="S2.SS3.p3.8.m8.1.1.1.cmml"><mi id="S2.SS3.p3.8.m8.1.1.1.3" xref="S2.SS3.p3.8.m8.1.1.1.3.cmml">P</mi><mo id="S2.SS3.p3.8.m8.1.1.1.2" xref="S2.SS3.p3.8.m8.1.1.1.2.cmml">∈</mo><mrow id="S2.SS3.p3.8.m8.1.1.1.4" xref="S2.SS3.p3.8.m8.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p3.8.m8.1.1.1.4.2" xref="S2.SS3.p3.8.m8.1.1.1.4.2.cmml">𝒫</mi><mo id="S2.SS3.p3.8.m8.1.1.1.4.1" xref="S2.SS3.p3.8.m8.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.SS3.p3.8.m8.1.1.1.4.3.2" xref="S2.SS3.p3.8.m8.1.1.1.4.cmml"><mo id="S2.SS3.p3.8.m8.1.1.1.4.3.2.1" stretchy="false" xref="S2.SS3.p3.8.m8.1.1.1.4.cmml">(</mo><mi id="S2.SS3.p3.8.m8.1.1.1.1" xref="S2.SS3.p3.8.m8.1.1.1.1.cmml">f</mi><mo id="S2.SS3.p3.8.m8.1.1.1.4.3.2.2" stretchy="false" xref="S2.SS3.p3.8.m8.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S2.SS3.p3.8.m8.3.4.3.2" xref="S2.SS3.p3.8.m8.3.4.3.2.cmml"><mi id="S2.SS3.p3.8.m8.3.4.3.2.2" xref="S2.SS3.p3.8.m8.3.4.3.2.2.cmml">V</mi><mo id="S2.SS3.p3.8.m8.3.4.3.2.1" xref="S2.SS3.p3.8.m8.3.4.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.8.m8.3.4.3.2.3.2" xref="S2.SS3.p3.8.m8.3.4.3.2.cmml"><mo id="S2.SS3.p3.8.m8.3.4.3.2.3.2.1" stretchy="false" xref="S2.SS3.p3.8.m8.3.4.3.2.cmml">(</mo><mi id="S2.SS3.p3.8.m8.3.3" xref="S2.SS3.p3.8.m8.3.3.cmml">P</mi><mo id="S2.SS3.p3.8.m8.3.4.3.2.3.2.2" stretchy="false" xref="S2.SS3.p3.8.m8.3.4.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.8.m8.3b"><apply id="S2.SS3.p3.8.m8.3.4.cmml" xref="S2.SS3.p3.8.m8.3.4"><csymbol cd="latexml" id="S2.SS3.p3.8.m8.3.4.1.cmml" xref="S2.SS3.p3.8.m8.3.4.1">assign</csymbol><apply id="S2.SS3.p3.8.m8.3.4.2.cmml" xref="S2.SS3.p3.8.m8.3.4.2"><times id="S2.SS3.p3.8.m8.3.4.2.1.cmml" xref="S2.SS3.p3.8.m8.3.4.2.1"></times><ci id="S2.SS3.p3.8.m8.3.4.2.2.cmml" xref="S2.SS3.p3.8.m8.3.4.2.2">𝑉</ci><ci id="S2.SS3.p3.8.m8.2.2.cmml" xref="S2.SS3.p3.8.m8.2.2">𝑓</ci></apply><apply id="S2.SS3.p3.8.m8.3.4.3.cmml" xref="S2.SS3.p3.8.m8.3.4.3"><apply id="S2.SS3.p3.8.m8.3.4.3.1.cmml" xref="S2.SS3.p3.8.m8.3.4.3.1"><csymbol cd="ambiguous" id="S2.SS3.p3.8.m8.3.4.3.1.1.cmml" xref="S2.SS3.p3.8.m8.3.4.3.1">subscript</csymbol><union id="S2.SS3.p3.8.m8.3.4.3.1.2.cmml" xref="S2.SS3.p3.8.m8.3.4.3.1.2"></union><apply id="S2.SS3.p3.8.m8.1.1.1.cmml" xref="S2.SS3.p3.8.m8.1.1.1"><in id="S2.SS3.p3.8.m8.1.1.1.2.cmml" xref="S2.SS3.p3.8.m8.1.1.1.2"></in><ci id="S2.SS3.p3.8.m8.1.1.1.3.cmml" xref="S2.SS3.p3.8.m8.1.1.1.3">𝑃</ci><apply id="S2.SS3.p3.8.m8.1.1.1.4.cmml" xref="S2.SS3.p3.8.m8.1.1.1.4"><times id="S2.SS3.p3.8.m8.1.1.1.4.1.cmml" xref="S2.SS3.p3.8.m8.1.1.1.4.1"></times><ci id="S2.SS3.p3.8.m8.1.1.1.4.2.cmml" xref="S2.SS3.p3.8.m8.1.1.1.4.2">𝒫</ci><ci id="S2.SS3.p3.8.m8.1.1.1.1.cmml" xref="S2.SS3.p3.8.m8.1.1.1.1">𝑓</ci></apply></apply></apply><apply id="S2.SS3.p3.8.m8.3.4.3.2.cmml" xref="S2.SS3.p3.8.m8.3.4.3.2"><times id="S2.SS3.p3.8.m8.3.4.3.2.1.cmml" xref="S2.SS3.p3.8.m8.3.4.3.2.1"></times><ci id="S2.SS3.p3.8.m8.3.4.3.2.2.cmml" xref="S2.SS3.p3.8.m8.3.4.3.2.2">𝑉</ci><ci id="S2.SS3.p3.8.m8.3.3.cmml" xref="S2.SS3.p3.8.m8.3.3">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.8.m8.3c">V(f):=\bigcup_{P\in\mathcal{P}(f)}V(P)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.8.m8.3d">italic_V ( italic_f ) := ⋃ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT italic_V ( italic_P )</annotation></semantics></math>. Note that because of <math alttext="P\setminus\bigcup_{e\in E(f)}e=P\setminus\bigcup_{e\in E(P)}e=\operatorname*{% int}{P}" class="ltx_Math" display="inline" id="S2.SS3.p3.9.m9.2"><semantics id="S2.SS3.p3.9.m9.2a"><mrow id="S2.SS3.p3.9.m9.2.3" xref="S2.SS3.p3.9.m9.2.3.cmml"><mrow id="S2.SS3.p3.9.m9.2.3.2" xref="S2.SS3.p3.9.m9.2.3.2.cmml"><mi id="S2.SS3.p3.9.m9.2.3.2.2" xref="S2.SS3.p3.9.m9.2.3.2.2.cmml">P</mi><mo id="S2.SS3.p3.9.m9.2.3.2.1" rspace="0.055em" xref="S2.SS3.p3.9.m9.2.3.2.1.cmml">∖</mo><mrow id="S2.SS3.p3.9.m9.2.3.2.3" xref="S2.SS3.p3.9.m9.2.3.2.3.cmml"><msub id="S2.SS3.p3.9.m9.2.3.2.3.1" xref="S2.SS3.p3.9.m9.2.3.2.3.1.cmml"><mo id="S2.SS3.p3.9.m9.2.3.2.3.1.2" xref="S2.SS3.p3.9.m9.2.3.2.3.1.2.cmml">⋃</mo><mrow id="S2.SS3.p3.9.m9.1.1.1" xref="S2.SS3.p3.9.m9.1.1.1.cmml"><mi id="S2.SS3.p3.9.m9.1.1.1.3" xref="S2.SS3.p3.9.m9.1.1.1.3.cmml">e</mi><mo id="S2.SS3.p3.9.m9.1.1.1.2" xref="S2.SS3.p3.9.m9.1.1.1.2.cmml">∈</mo><mrow id="S2.SS3.p3.9.m9.1.1.1.4" xref="S2.SS3.p3.9.m9.1.1.1.4.cmml"><mi id="S2.SS3.p3.9.m9.1.1.1.4.2" xref="S2.SS3.p3.9.m9.1.1.1.4.2.cmml">E</mi><mo id="S2.SS3.p3.9.m9.1.1.1.4.1" xref="S2.SS3.p3.9.m9.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.SS3.p3.9.m9.1.1.1.4.3.2" xref="S2.SS3.p3.9.m9.1.1.1.4.cmml"><mo id="S2.SS3.p3.9.m9.1.1.1.4.3.2.1" stretchy="false" xref="S2.SS3.p3.9.m9.1.1.1.4.cmml">(</mo><mi id="S2.SS3.p3.9.m9.1.1.1.1" xref="S2.SS3.p3.9.m9.1.1.1.1.cmml">f</mi><mo id="S2.SS3.p3.9.m9.1.1.1.4.3.2.2" stretchy="false" xref="S2.SS3.p3.9.m9.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mi id="S2.SS3.p3.9.m9.2.3.2.3.2" xref="S2.SS3.p3.9.m9.2.3.2.3.2.cmml">e</mi></mrow></mrow><mo id="S2.SS3.p3.9.m9.2.3.3" xref="S2.SS3.p3.9.m9.2.3.3.cmml">=</mo><mrow id="S2.SS3.p3.9.m9.2.3.4" xref="S2.SS3.p3.9.m9.2.3.4.cmml"><mi id="S2.SS3.p3.9.m9.2.3.4.2" xref="S2.SS3.p3.9.m9.2.3.4.2.cmml">P</mi><mo id="S2.SS3.p3.9.m9.2.3.4.1" rspace="0.055em" xref="S2.SS3.p3.9.m9.2.3.4.1.cmml">∖</mo><mrow id="S2.SS3.p3.9.m9.2.3.4.3" xref="S2.SS3.p3.9.m9.2.3.4.3.cmml"><msub id="S2.SS3.p3.9.m9.2.3.4.3.1" xref="S2.SS3.p3.9.m9.2.3.4.3.1.cmml"><mo id="S2.SS3.p3.9.m9.2.3.4.3.1.2" xref="S2.SS3.p3.9.m9.2.3.4.3.1.2.cmml">⋃</mo><mrow id="S2.SS3.p3.9.m9.2.2.1" xref="S2.SS3.p3.9.m9.2.2.1.cmml"><mi id="S2.SS3.p3.9.m9.2.2.1.3" xref="S2.SS3.p3.9.m9.2.2.1.3.cmml">e</mi><mo id="S2.SS3.p3.9.m9.2.2.1.2" xref="S2.SS3.p3.9.m9.2.2.1.2.cmml">∈</mo><mrow id="S2.SS3.p3.9.m9.2.2.1.4" xref="S2.SS3.p3.9.m9.2.2.1.4.cmml"><mi id="S2.SS3.p3.9.m9.2.2.1.4.2" xref="S2.SS3.p3.9.m9.2.2.1.4.2.cmml">E</mi><mo id="S2.SS3.p3.9.m9.2.2.1.4.1" xref="S2.SS3.p3.9.m9.2.2.1.4.1.cmml">⁢</mo><mrow id="S2.SS3.p3.9.m9.2.2.1.4.3.2" xref="S2.SS3.p3.9.m9.2.2.1.4.cmml"><mo id="S2.SS3.p3.9.m9.2.2.1.4.3.2.1" stretchy="false" xref="S2.SS3.p3.9.m9.2.2.1.4.cmml">(</mo><mi id="S2.SS3.p3.9.m9.2.2.1.1" xref="S2.SS3.p3.9.m9.2.2.1.1.cmml">P</mi><mo id="S2.SS3.p3.9.m9.2.2.1.4.3.2.2" stretchy="false" xref="S2.SS3.p3.9.m9.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mi id="S2.SS3.p3.9.m9.2.3.4.3.2" xref="S2.SS3.p3.9.m9.2.3.4.3.2.cmml">e</mi></mrow></mrow><mo id="S2.SS3.p3.9.m9.2.3.5" rspace="0.1389em" xref="S2.SS3.p3.9.m9.2.3.5.cmml">=</mo><mrow id="S2.SS3.p3.9.m9.2.3.6" xref="S2.SS3.p3.9.m9.2.3.6.cmml"><mo id="S2.SS3.p3.9.m9.2.3.6.1" lspace="0.1389em" rspace="0.167em" xref="S2.SS3.p3.9.m9.2.3.6.1.cmml">int</mo><mi id="S2.SS3.p3.9.m9.2.3.6.2" xref="S2.SS3.p3.9.m9.2.3.6.2.cmml">P</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.9.m9.2b"><apply id="S2.SS3.p3.9.m9.2.3.cmml" xref="S2.SS3.p3.9.m9.2.3"><and id="S2.SS3.p3.9.m9.2.3a.cmml" xref="S2.SS3.p3.9.m9.2.3"></and><apply id="S2.SS3.p3.9.m9.2.3b.cmml" xref="S2.SS3.p3.9.m9.2.3"><eq id="S2.SS3.p3.9.m9.2.3.3.cmml" xref="S2.SS3.p3.9.m9.2.3.3"></eq><apply id="S2.SS3.p3.9.m9.2.3.2.cmml" xref="S2.SS3.p3.9.m9.2.3.2"><setdiff id="S2.SS3.p3.9.m9.2.3.2.1.cmml" xref="S2.SS3.p3.9.m9.2.3.2.1"></setdiff><ci id="S2.SS3.p3.9.m9.2.3.2.2.cmml" xref="S2.SS3.p3.9.m9.2.3.2.2">𝑃</ci><apply id="S2.SS3.p3.9.m9.2.3.2.3.cmml" xref="S2.SS3.p3.9.m9.2.3.2.3"><apply id="S2.SS3.p3.9.m9.2.3.2.3.1.cmml" xref="S2.SS3.p3.9.m9.2.3.2.3.1"><csymbol cd="ambiguous" id="S2.SS3.p3.9.m9.2.3.2.3.1.1.cmml" xref="S2.SS3.p3.9.m9.2.3.2.3.1">subscript</csymbol><union id="S2.SS3.p3.9.m9.2.3.2.3.1.2.cmml" xref="S2.SS3.p3.9.m9.2.3.2.3.1.2"></union><apply id="S2.SS3.p3.9.m9.1.1.1.cmml" xref="S2.SS3.p3.9.m9.1.1.1"><in id="S2.SS3.p3.9.m9.1.1.1.2.cmml" xref="S2.SS3.p3.9.m9.1.1.1.2"></in><ci id="S2.SS3.p3.9.m9.1.1.1.3.cmml" xref="S2.SS3.p3.9.m9.1.1.1.3">𝑒</ci><apply id="S2.SS3.p3.9.m9.1.1.1.4.cmml" xref="S2.SS3.p3.9.m9.1.1.1.4"><times id="S2.SS3.p3.9.m9.1.1.1.4.1.cmml" xref="S2.SS3.p3.9.m9.1.1.1.4.1"></times><ci id="S2.SS3.p3.9.m9.1.1.1.4.2.cmml" xref="S2.SS3.p3.9.m9.1.1.1.4.2">𝐸</ci><ci id="S2.SS3.p3.9.m9.1.1.1.1.cmml" xref="S2.SS3.p3.9.m9.1.1.1.1">𝑓</ci></apply></apply></apply><ci id="S2.SS3.p3.9.m9.2.3.2.3.2.cmml" xref="S2.SS3.p3.9.m9.2.3.2.3.2">𝑒</ci></apply></apply><apply id="S2.SS3.p3.9.m9.2.3.4.cmml" xref="S2.SS3.p3.9.m9.2.3.4"><setdiff id="S2.SS3.p3.9.m9.2.3.4.1.cmml" xref="S2.SS3.p3.9.m9.2.3.4.1"></setdiff><ci id="S2.SS3.p3.9.m9.2.3.4.2.cmml" xref="S2.SS3.p3.9.m9.2.3.4.2">𝑃</ci><apply id="S2.SS3.p3.9.m9.2.3.4.3.cmml" xref="S2.SS3.p3.9.m9.2.3.4.3"><apply id="S2.SS3.p3.9.m9.2.3.4.3.1.cmml" xref="S2.SS3.p3.9.m9.2.3.4.3.1"><csymbol cd="ambiguous" id="S2.SS3.p3.9.m9.2.3.4.3.1.1.cmml" xref="S2.SS3.p3.9.m9.2.3.4.3.1">subscript</csymbol><union id="S2.SS3.p3.9.m9.2.3.4.3.1.2.cmml" xref="S2.SS3.p3.9.m9.2.3.4.3.1.2"></union><apply id="S2.SS3.p3.9.m9.2.2.1.cmml" xref="S2.SS3.p3.9.m9.2.2.1"><in id="S2.SS3.p3.9.m9.2.2.1.2.cmml" xref="S2.SS3.p3.9.m9.2.2.1.2"></in><ci id="S2.SS3.p3.9.m9.2.2.1.3.cmml" xref="S2.SS3.p3.9.m9.2.2.1.3">𝑒</ci><apply id="S2.SS3.p3.9.m9.2.2.1.4.cmml" xref="S2.SS3.p3.9.m9.2.2.1.4"><times id="S2.SS3.p3.9.m9.2.2.1.4.1.cmml" xref="S2.SS3.p3.9.m9.2.2.1.4.1"></times><ci id="S2.SS3.p3.9.m9.2.2.1.4.2.cmml" xref="S2.SS3.p3.9.m9.2.2.1.4.2">𝐸</ci><ci id="S2.SS3.p3.9.m9.2.2.1.1.cmml" xref="S2.SS3.p3.9.m9.2.2.1.1">𝑃</ci></apply></apply></apply><ci id="S2.SS3.p3.9.m9.2.3.4.3.2.cmml" xref="S2.SS3.p3.9.m9.2.3.4.3.2">𝑒</ci></apply></apply></apply><apply id="S2.SS3.p3.9.m9.2.3c.cmml" xref="S2.SS3.p3.9.m9.2.3"><eq id="S2.SS3.p3.9.m9.2.3.5.cmml" xref="S2.SS3.p3.9.m9.2.3.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S2.SS3.p3.9.m9.2.3.4.cmml" id="S2.SS3.p3.9.m9.2.3d.cmml" xref="S2.SS3.p3.9.m9.2.3"></share><apply id="S2.SS3.p3.9.m9.2.3.6.cmml" xref="S2.SS3.p3.9.m9.2.3.6"><ci id="S2.SS3.p3.9.m9.2.3.6.1.cmml" xref="S2.SS3.p3.9.m9.2.3.6.1">int</ci><ci id="S2.SS3.p3.9.m9.2.3.6.2.cmml" xref="S2.SS3.p3.9.m9.2.3.6.2">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.9.m9.2c">P\setminus\bigcup_{e\in E(f)}e=P\setminus\bigcup_{e\in E(P)}e=\operatorname*{% int}{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.9.m9.2d">italic_P ∖ ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_f ) end_POSTSUBSCRIPT italic_e = italic_P ∖ ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_P ) end_POSTSUBSCRIPT italic_e = roman_int italic_P</annotation></semantics></math>, and because the interiors of distinct pieces are disjoint, we can write</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="f(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{P}(x)\cdot f_{P}(x)\qquad\forall x% \in\mathds{R}^{2}\setminus\bigcup_{e\in E(f)}e," class="ltx_Math" display="block" id="S2.E1.m1.6"><semantics id="S2.E1.m1.6a"><mrow id="S2.E1.m1.6.6.1"><mrow id="S2.E1.m1.6.6.1.1.2" xref="S2.E1.m1.6.6.1.1.3.cmml"><mrow id="S2.E1.m1.6.6.1.1.1.1" xref="S2.E1.m1.6.6.1.1.1.1.cmml"><mrow id="S2.E1.m1.6.6.1.1.1.1.2" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml"><mi id="S2.E1.m1.6.6.1.1.1.1.2.2" xref="S2.E1.m1.6.6.1.1.1.1.2.2.cmml">f</mi><mo id="S2.E1.m1.6.6.1.1.1.1.2.1" xref="S2.E1.m1.6.6.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.6.6.1.1.1.1.2.3.2" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml"><mo id="S2.E1.m1.6.6.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml">(</mo><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">x</mi><mo id="S2.E1.m1.6.6.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.6.6.1.1.1.1.1" rspace="0.111em" xref="S2.E1.m1.6.6.1.1.1.1.1.cmml">=</mo><mrow id="S2.E1.m1.6.6.1.1.1.1.3" xref="S2.E1.m1.6.6.1.1.1.1.3.cmml"><munder id="S2.E1.m1.6.6.1.1.1.1.3.1" xref="S2.E1.m1.6.6.1.1.1.1.3.1.cmml"><mo id="S2.E1.m1.6.6.1.1.1.1.3.1.2" movablelimits="false" xref="S2.E1.m1.6.6.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.3" xref="S2.E1.m1.1.1.1.3.cmml">P</mi><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.E1.m1.1.1.1.4" xref="S2.E1.m1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.1.1.4.2" xref="S2.E1.m1.1.1.1.4.2.cmml">𝒫</mi><mo id="S2.E1.m1.1.1.1.4.1" xref="S2.E1.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.4.3.2" xref="S2.E1.m1.1.1.1.4.cmml"><mo id="S2.E1.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.E1.m1.1.1.1.4.cmml">(</mo><mi id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml">f</mi><mo id="S2.E1.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.E1.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.E1.m1.6.6.1.1.1.1.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.cmml"><mrow id="S2.E1.m1.6.6.1.1.1.1.3.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.cmml"><mrow id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.cmml"><msub id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2.cmml"><mn id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2.2.cmml">𝟙</mn><mi id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2.3" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.2.3.cmml">P</mi></msub><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.1" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.cmml"><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.3.2.1" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.cmml">(</mo><mi id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml">x</mi><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.2.1" rspace="0.222em" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.1.cmml">⋅</mo><msub id="S2.E1.m1.6.6.1.1.1.1.3.2.2.3" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.3.cmml"><mi id="S2.E1.m1.6.6.1.1.1.1.3.2.2.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.3.2.cmml">f</mi><mi id="S2.E1.m1.6.6.1.1.1.1.3.2.2.3.3" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.3.3.cmml">P</mi></msub></mrow><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.1" xref="S2.E1.m1.6.6.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.6.6.1.1.1.1.3.2.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.cmml"><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.2.cmml">(</mo><mi id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml">x</mi><mo id="S2.E1.m1.6.6.1.1.1.1.3.2.3.2.2" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mspace id="S2.E1.m1.6.6.1.1.2.3" width="2.167em" xref="S2.E1.m1.6.6.1.1.3a.cmml"></mspace><mrow id="S2.E1.m1.6.6.1.1.2.2" xref="S2.E1.m1.6.6.1.1.2.2.cmml"><mrow id="S2.E1.m1.6.6.1.1.2.2.2" xref="S2.E1.m1.6.6.1.1.2.2.2.cmml"><mo id="S2.E1.m1.6.6.1.1.2.2.2.1" rspace="0.167em" xref="S2.E1.m1.6.6.1.1.2.2.2.1.cmml">∀</mo><mi id="S2.E1.m1.6.6.1.1.2.2.2.2" xref="S2.E1.m1.6.6.1.1.2.2.2.2.cmml">x</mi></mrow><mo id="S2.E1.m1.6.6.1.1.2.2.1" xref="S2.E1.m1.6.6.1.1.2.2.1.cmml">∈</mo><mrow id="S2.E1.m1.6.6.1.1.2.2.3" xref="S2.E1.m1.6.6.1.1.2.2.3.cmml"><msup id="S2.E1.m1.6.6.1.1.2.2.3.2" xref="S2.E1.m1.6.6.1.1.2.2.3.2.cmml"><mi id="S2.E1.m1.6.6.1.1.2.2.3.2.2" xref="S2.E1.m1.6.6.1.1.2.2.3.2.2.cmml">ℝ</mi><mn id="S2.E1.m1.6.6.1.1.2.2.3.2.3" xref="S2.E1.m1.6.6.1.1.2.2.3.2.3.cmml">2</mn></msup><mo id="S2.E1.m1.6.6.1.1.2.2.3.1" rspace="0.055em" xref="S2.E1.m1.6.6.1.1.2.2.3.1.cmml">∖</mo><mrow id="S2.E1.m1.6.6.1.1.2.2.3.3" xref="S2.E1.m1.6.6.1.1.2.2.3.3.cmml"><munder id="S2.E1.m1.6.6.1.1.2.2.3.3.1" xref="S2.E1.m1.6.6.1.1.2.2.3.3.1.cmml"><mo id="S2.E1.m1.6.6.1.1.2.2.3.3.1.2" movablelimits="false" xref="S2.E1.m1.6.6.1.1.2.2.3.3.1.2.cmml">⋃</mo><mrow id="S2.E1.m1.2.2.1" xref="S2.E1.m1.2.2.1.cmml"><mi id="S2.E1.m1.2.2.1.3" xref="S2.E1.m1.2.2.1.3.cmml">e</mi><mo id="S2.E1.m1.2.2.1.2" xref="S2.E1.m1.2.2.1.2.cmml">∈</mo><mrow id="S2.E1.m1.2.2.1.4" xref="S2.E1.m1.2.2.1.4.cmml"><mi id="S2.E1.m1.2.2.1.4.2" xref="S2.E1.m1.2.2.1.4.2.cmml">E</mi><mo id="S2.E1.m1.2.2.1.4.1" xref="S2.E1.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.1.4.3.2" xref="S2.E1.m1.2.2.1.4.cmml"><mo id="S2.E1.m1.2.2.1.4.3.2.1" stretchy="false" xref="S2.E1.m1.2.2.1.4.cmml">(</mo><mi id="S2.E1.m1.2.2.1.1" xref="S2.E1.m1.2.2.1.1.cmml">f</mi><mo id="S2.E1.m1.2.2.1.4.3.2.2" stretchy="false" xref="S2.E1.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mi id="S2.E1.m1.6.6.1.1.2.2.3.3.2" xref="S2.E1.m1.6.6.1.1.2.2.3.3.2.cmml">e</mi></mrow></mrow></mrow></mrow><mo id="S2.E1.m1.6.6.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.6b"><apply id="S2.E1.m1.6.6.1.1.3.cmml" xref="S2.E1.m1.6.6.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.6.6.1.1.3a.cmml" xref="S2.E1.m1.6.6.1.1.2.3">formulae-sequence</csymbol><apply id="S2.E1.m1.6.6.1.1.1.1.cmml" xref="S2.E1.m1.6.6.1.1.1.1"><eq id="S2.E1.m1.6.6.1.1.1.1.1.cmml" xref="S2.E1.m1.6.6.1.1.1.1.1"></eq><apply id="S2.E1.m1.6.6.1.1.1.1.2.cmml" xref="S2.E1.m1.6.6.1.1.1.1.2"><times id="S2.E1.m1.6.6.1.1.1.1.2.1.cmml" xref="S2.E1.m1.6.6.1.1.1.1.2.1"></times><ci id="S2.E1.m1.6.6.1.1.1.1.2.2.cmml" xref="S2.E1.m1.6.6.1.1.1.1.2.2">𝑓</ci><ci id="S2.E1.m1.3.3.cmml" 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xref="S2.E1.m1.2.2.1.4.2">𝐸</ci><ci id="S2.E1.m1.2.2.1.1.cmml" xref="S2.E1.m1.2.2.1.1">𝑓</ci></apply></apply></apply><ci id="S2.E1.m1.6.6.1.1.2.2.3.3.2.cmml" xref="S2.E1.m1.6.6.1.1.2.2.3.3.2">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.6c">f(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{P}(x)\cdot f_{P}(x)\qquad\forall x% \in\mathds{R}^{2}\setminus\bigcup_{e\in E(f)}e,</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.6d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) ⋅ italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) ∀ italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_f ) end_POSTSUBSCRIPT italic_e ,</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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p3.14">where the indicator function <math alttext="\mathds{1}_{A}" class="ltx_Math" display="inline" id="S2.SS3.p3.10.m1.1"><semantics id="S2.SS3.p3.10.m1.1a"><msub id="S2.SS3.p3.10.m1.1.1" xref="S2.SS3.p3.10.m1.1.1.cmml"><mn id="S2.SS3.p3.10.m1.1.1.2" xref="S2.SS3.p3.10.m1.1.1.2.cmml">𝟙</mn><mi id="S2.SS3.p3.10.m1.1.1.3" xref="S2.SS3.p3.10.m1.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.10.m1.1b"><apply id="S2.SS3.p3.10.m1.1.1.cmml" xref="S2.SS3.p3.10.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p3.10.m1.1.1.1.cmml" xref="S2.SS3.p3.10.m1.1.1">subscript</csymbol><cn id="S2.SS3.p3.10.m1.1.1.2.cmml" type="integer" xref="S2.SS3.p3.10.m1.1.1.2">1</cn><ci id="S2.SS3.p3.10.m1.1.1.3.cmml" xref="S2.SS3.p3.10.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.10.m1.1c">\mathds{1}_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.10.m1.1d">blackboard_1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> for a set <math alttext="A" class="ltx_Math" display="inline" id="S2.SS3.p3.11.m2.1"><semantics id="S2.SS3.p3.11.m2.1a"><mi id="S2.SS3.p3.11.m2.1.1" xref="S2.SS3.p3.11.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.11.m2.1b"><ci id="S2.SS3.p3.11.m2.1.1.cmml" xref="S2.SS3.p3.11.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.11.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.11.m2.1d">italic_A</annotation></semantics></math> is defined by <math alttext="\mathds{1}_{A}(x)=1" class="ltx_Math" display="inline" id="S2.SS3.p3.12.m3.1"><semantics id="S2.SS3.p3.12.m3.1a"><mrow id="S2.SS3.p3.12.m3.1.2" xref="S2.SS3.p3.12.m3.1.2.cmml"><mrow id="S2.SS3.p3.12.m3.1.2.2" xref="S2.SS3.p3.12.m3.1.2.2.cmml"><msub id="S2.SS3.p3.12.m3.1.2.2.2" xref="S2.SS3.p3.12.m3.1.2.2.2.cmml"><mn id="S2.SS3.p3.12.m3.1.2.2.2.2" xref="S2.SS3.p3.12.m3.1.2.2.2.2.cmml">𝟙</mn><mi id="S2.SS3.p3.12.m3.1.2.2.2.3" xref="S2.SS3.p3.12.m3.1.2.2.2.3.cmml">A</mi></msub><mo id="S2.SS3.p3.12.m3.1.2.2.1" xref="S2.SS3.p3.12.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.12.m3.1.2.2.3.2" xref="S2.SS3.p3.12.m3.1.2.2.cmml"><mo id="S2.SS3.p3.12.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p3.12.m3.1.2.2.cmml">(</mo><mi id="S2.SS3.p3.12.m3.1.1" xref="S2.SS3.p3.12.m3.1.1.cmml">x</mi><mo id="S2.SS3.p3.12.m3.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p3.12.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p3.12.m3.1.2.1" xref="S2.SS3.p3.12.m3.1.2.1.cmml">=</mo><mn id="S2.SS3.p3.12.m3.1.2.3" xref="S2.SS3.p3.12.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.12.m3.1b"><apply 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encoding="application/x-llamapun" id="S2.SS3.p3.13.m4.1d">italic_x ∈ italic_A</annotation></semantics></math> and <math alttext="\mathds{1}_{A}(x)=0" class="ltx_Math" display="inline" id="S2.SS3.p3.14.m5.1"><semantics id="S2.SS3.p3.14.m5.1a"><mrow id="S2.SS3.p3.14.m5.1.2" xref="S2.SS3.p3.14.m5.1.2.cmml"><mrow id="S2.SS3.p3.14.m5.1.2.2" xref="S2.SS3.p3.14.m5.1.2.2.cmml"><msub id="S2.SS3.p3.14.m5.1.2.2.2" xref="S2.SS3.p3.14.m5.1.2.2.2.cmml"><mn id="S2.SS3.p3.14.m5.1.2.2.2.2" xref="S2.SS3.p3.14.m5.1.2.2.2.2.cmml">𝟙</mn><mi id="S2.SS3.p3.14.m5.1.2.2.2.3" xref="S2.SS3.p3.14.m5.1.2.2.2.3.cmml">A</mi></msub><mo id="S2.SS3.p3.14.m5.1.2.2.1" xref="S2.SS3.p3.14.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p3.14.m5.1.2.2.3.2" xref="S2.SS3.p3.14.m5.1.2.2.cmml"><mo id="S2.SS3.p3.14.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p3.14.m5.1.2.2.cmml">(</mo><mi id="S2.SS3.p3.14.m5.1.1" xref="S2.SS3.p3.14.m5.1.1.cmml">x</mi><mo id="S2.SS3.p3.14.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p3.14.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p3.14.m5.1.2.1" xref="S2.SS3.p3.14.m5.1.2.1.cmml">=</mo><mn id="S2.SS3.p3.14.m5.1.2.3" xref="S2.SS3.p3.14.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.14.m5.1b"><apply id="S2.SS3.p3.14.m5.1.2.cmml" xref="S2.SS3.p3.14.m5.1.2"><eq id="S2.SS3.p3.14.m5.1.2.1.cmml" xref="S2.SS3.p3.14.m5.1.2.1"></eq><apply id="S2.SS3.p3.14.m5.1.2.2.cmml" xref="S2.SS3.p3.14.m5.1.2.2"><times id="S2.SS3.p3.14.m5.1.2.2.1.cmml" xref="S2.SS3.p3.14.m5.1.2.2.1"></times><apply id="S2.SS3.p3.14.m5.1.2.2.2.cmml" xref="S2.SS3.p3.14.m5.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p3.14.m5.1.2.2.2.1.cmml" xref="S2.SS3.p3.14.m5.1.2.2.2">subscript</csymbol><cn id="S2.SS3.p3.14.m5.1.2.2.2.2.cmml" type="integer" xref="S2.SS3.p3.14.m5.1.2.2.2.2">1</cn><ci id="S2.SS3.p3.14.m5.1.2.2.2.3.cmml" xref="S2.SS3.p3.14.m5.1.2.2.2.3">𝐴</ci></apply><ci id="S2.SS3.p3.14.m5.1.1.cmml" xref="S2.SS3.p3.14.m5.1.1">𝑥</ci></apply><cn id="S2.SS3.p3.14.m5.1.2.3.cmml" type="integer" xref="S2.SS3.p3.14.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.14.m5.1c">\mathds{1}_{A}(x)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.14.m5.1d">blackboard_1 start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_x ) = 0</annotation></semantics></math> otherwise.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.9">If all affine components of a function <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.1"><semantics id="S2.SS3.p4.1.m1.1a"><mrow id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml"><mi id="S2.SS3.p4.1.m1.1.1.2" xref="S2.SS3.p4.1.m1.1.1.2.cmml">f</mi><mo id="S2.SS3.p4.1.m1.1.1.1" xref="S2.SS3.p4.1.m1.1.1.1.cmml">∈</mo><msub id="S2.SS3.p4.1.m1.1.1.3" xref="S2.SS3.p4.1.m1.1.1.3.cmml"><mi id="S2.SS3.p4.1.m1.1.1.3.2" xref="S2.SS3.p4.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S2.SS3.p4.1.m1.1.1.3.3" xref="S2.SS3.p4.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.1.m1.1b"><apply id="S2.SS3.p4.1.m1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1"><in id="S2.SS3.p4.1.m1.1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1.1"></in><ci id="S2.SS3.p4.1.m1.1.1.2.cmml" xref="S2.SS3.p4.1.m1.1.1.2">𝑓</ci><apply id="S2.SS3.p4.1.m1.1.1.3.cmml" xref="S2.SS3.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p4.1.m1.1.1.3.1.cmml" xref="S2.SS3.p4.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS3.p4.1.m1.1.1.3.2.cmml" xref="S2.SS3.p4.1.m1.1.1.3.2">CPA</ci><ci id="S2.SS3.p4.1.m1.1.1.3.3.cmml" xref="S2.SS3.p4.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> are linear, i.e. <math alttext="f_{P}(0)=0" class="ltx_Math" display="inline" id="S2.SS3.p4.2.m2.1"><semantics id="S2.SS3.p4.2.m2.1a"><mrow id="S2.SS3.p4.2.m2.1.2" xref="S2.SS3.p4.2.m2.1.2.cmml"><mrow id="S2.SS3.p4.2.m2.1.2.2" xref="S2.SS3.p4.2.m2.1.2.2.cmml"><msub id="S2.SS3.p4.2.m2.1.2.2.2" xref="S2.SS3.p4.2.m2.1.2.2.2.cmml"><mi id="S2.SS3.p4.2.m2.1.2.2.2.2" xref="S2.SS3.p4.2.m2.1.2.2.2.2.cmml">f</mi><mi id="S2.SS3.p4.2.m2.1.2.2.2.3" xref="S2.SS3.p4.2.m2.1.2.2.2.3.cmml">P</mi></msub><mo id="S2.SS3.p4.2.m2.1.2.2.1" xref="S2.SS3.p4.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p4.2.m2.1.2.2.3.2" xref="S2.SS3.p4.2.m2.1.2.2.cmml"><mo id="S2.SS3.p4.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p4.2.m2.1.2.2.cmml">(</mo><mn id="S2.SS3.p4.2.m2.1.1" xref="S2.SS3.p4.2.m2.1.1.cmml">0</mn><mo id="S2.SS3.p4.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p4.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p4.2.m2.1.2.1" xref="S2.SS3.p4.2.m2.1.2.1.cmml">=</mo><mn id="S2.SS3.p4.2.m2.1.2.3" xref="S2.SS3.p4.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.2.m2.1b"><apply id="S2.SS3.p4.2.m2.1.2.cmml" xref="S2.SS3.p4.2.m2.1.2"><eq id="S2.SS3.p4.2.m2.1.2.1.cmml" xref="S2.SS3.p4.2.m2.1.2.1"></eq><apply id="S2.SS3.p4.2.m2.1.2.2.cmml" xref="S2.SS3.p4.2.m2.1.2.2"><times id="S2.SS3.p4.2.m2.1.2.2.1.cmml" xref="S2.SS3.p4.2.m2.1.2.2.1"></times><apply id="S2.SS3.p4.2.m2.1.2.2.2.cmml" xref="S2.SS3.p4.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p4.2.m2.1.2.2.2.1.cmml" xref="S2.SS3.p4.2.m2.1.2.2.2">subscript</csymbol><ci id="S2.SS3.p4.2.m2.1.2.2.2.2.cmml" xref="S2.SS3.p4.2.m2.1.2.2.2.2">𝑓</ci><ci id="S2.SS3.p4.2.m2.1.2.2.2.3.cmml" xref="S2.SS3.p4.2.m2.1.2.2.2.3">𝑃</ci></apply><cn id="S2.SS3.p4.2.m2.1.1.cmml" type="integer" xref="S2.SS3.p4.2.m2.1.1">0</cn></apply><cn id="S2.SS3.p4.2.m2.1.2.3.cmml" type="integer" xref="S2.SS3.p4.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.2.m2.1c">f_{P}(0)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.2.m2.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( 0 ) = 0</annotation></semantics></math> for every piece <math alttext="P" class="ltx_Math" display="inline" id="S2.SS3.p4.3.m3.1"><semantics id="S2.SS3.p4.3.m3.1a"><mi id="S2.SS3.p4.3.m3.1.1" xref="S2.SS3.p4.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.3.m3.1b"><ci id="S2.SS3.p4.3.m3.1.1.cmml" xref="S2.SS3.p4.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.3.m3.1d">italic_P</annotation></semantics></math>, then I call the function <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.p4.4.m4.1"><semantics id="S2.SS3.p4.4.m4.1a"><mi id="S2.SS3.p4.4.m4.1.1" xref="S2.SS3.p4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.4.m4.1b"><ci id="S2.SS3.p4.4.m4.1.1.cmml" xref="S2.SS3.p4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.4.m4.1d">italic_f</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S2.SS3.p4.9.1">continuous piecewise linear</em>, and denote the respective set of such functions by <math alttext="\operatorname{CPL}_{p}\subsetneq\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S2.SS3.p4.5.m5.1"><semantics id="S2.SS3.p4.5.m5.1a"><mrow id="S2.SS3.p4.5.m5.1.1" xref="S2.SS3.p4.5.m5.1.1.cmml"><msub id="S2.SS3.p4.5.m5.1.1.2" xref="S2.SS3.p4.5.m5.1.1.2.cmml"><mi id="S2.SS3.p4.5.m5.1.1.2.2" xref="S2.SS3.p4.5.m5.1.1.2.2.cmml">CPL</mi><mi id="S2.SS3.p4.5.m5.1.1.2.3" xref="S2.SS3.p4.5.m5.1.1.2.3.cmml">p</mi></msub><mo id="S2.SS3.p4.5.m5.1.1.1" xref="S2.SS3.p4.5.m5.1.1.1.cmml">⊊</mo><msub id="S2.SS3.p4.5.m5.1.1.3" xref="S2.SS3.p4.5.m5.1.1.3.cmml"><mi id="S2.SS3.p4.5.m5.1.1.3.2" xref="S2.SS3.p4.5.m5.1.1.3.2.cmml">CPA</mi><mi id="S2.SS3.p4.5.m5.1.1.3.3" xref="S2.SS3.p4.5.m5.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.5.m5.1b"><apply id="S2.SS3.p4.5.m5.1.1.cmml" xref="S2.SS3.p4.5.m5.1.1"><prsubset id="S2.SS3.p4.5.m5.1.1.1.cmml" xref="S2.SS3.p4.5.m5.1.1.1"></prsubset><apply id="S2.SS3.p4.5.m5.1.1.2.cmml" xref="S2.SS3.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p4.5.m5.1.1.2.1.cmml" xref="S2.SS3.p4.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS3.p4.5.m5.1.1.2.2.cmml" xref="S2.SS3.p4.5.m5.1.1.2.2">CPL</ci><ci id="S2.SS3.p4.5.m5.1.1.2.3.cmml" xref="S2.SS3.p4.5.m5.1.1.2.3">𝑝</ci></apply><apply id="S2.SS3.p4.5.m5.1.1.3.cmml" xref="S2.SS3.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p4.5.m5.1.1.3.1.cmml" xref="S2.SS3.p4.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS3.p4.5.m5.1.1.3.2.cmml" xref="S2.SS3.p4.5.m5.1.1.3.2">CPA</ci><ci id="S2.SS3.p4.5.m5.1.1.3.3.cmml" xref="S2.SS3.p4.5.m5.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.5.m5.1c">\operatorname{CPL}_{p}\subsetneq\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.5.m5.1d">roman_CPL start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⊊ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. Since two linear functions in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS3.p4.6.m6.1"><semantics id="S2.SS3.p4.6.m6.1a"><msup id="S2.SS3.p4.6.m6.1.1" xref="S2.SS3.p4.6.m6.1.1.cmml"><mi id="S2.SS3.p4.6.m6.1.1.2" xref="S2.SS3.p4.6.m6.1.1.2.cmml">ℝ</mi><mn id="S2.SS3.p4.6.m6.1.1.3" xref="S2.SS3.p4.6.m6.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.6.m6.1b"><apply id="S2.SS3.p4.6.m6.1.1.cmml" xref="S2.SS3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.p4.6.m6.1.1.1.cmml" xref="S2.SS3.p4.6.m6.1.1">superscript</csymbol><ci id="S2.SS3.p4.6.m6.1.1.2.cmml" xref="S2.SS3.p4.6.m6.1.1.2">ℝ</ci><cn id="S2.SS3.p4.6.m6.1.1.3.cmml" type="integer" xref="S2.SS3.p4.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.6.m6.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> only agree on a line through zero (or on all of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S2.SS3.p4.7.m7.1"><semantics id="S2.SS3.p4.7.m7.1a"><msup id="S2.SS3.p4.7.m7.1.1" xref="S2.SS3.p4.7.m7.1.1.cmml"><mi id="S2.SS3.p4.7.m7.1.1.2" xref="S2.SS3.p4.7.m7.1.1.2.cmml">ℝ</mi><mn id="S2.SS3.p4.7.m7.1.1.3" xref="S2.SS3.p4.7.m7.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.7.m7.1b"><apply id="S2.SS3.p4.7.m7.1.1.cmml" xref="S2.SS3.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS3.p4.7.m7.1.1.1.cmml" xref="S2.SS3.p4.7.m7.1.1">superscript</csymbol><ci id="S2.SS3.p4.7.m7.1.1.2.cmml" xref="S2.SS3.p4.7.m7.1.1.2">ℝ</ci><cn id="S2.SS3.p4.7.m7.1.1.3.cmml" type="integer" xref="S2.SS3.p4.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.7.m7.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>), the pieces of <math alttext="f\in\operatorname{CPL}_{p}" class="ltx_Math" display="inline" id="S2.SS3.p4.8.m8.1"><semantics id="S2.SS3.p4.8.m8.1a"><mrow id="S2.SS3.p4.8.m8.1.1" xref="S2.SS3.p4.8.m8.1.1.cmml"><mi id="S2.SS3.p4.8.m8.1.1.2" xref="S2.SS3.p4.8.m8.1.1.2.cmml">f</mi><mo id="S2.SS3.p4.8.m8.1.1.1" xref="S2.SS3.p4.8.m8.1.1.1.cmml">∈</mo><msub id="S2.SS3.p4.8.m8.1.1.3" xref="S2.SS3.p4.8.m8.1.1.3.cmml"><mi id="S2.SS3.p4.8.m8.1.1.3.2" xref="S2.SS3.p4.8.m8.1.1.3.2.cmml">CPL</mi><mi id="S2.SS3.p4.8.m8.1.1.3.3" xref="S2.SS3.p4.8.m8.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.8.m8.1b"><apply id="S2.SS3.p4.8.m8.1.1.cmml" xref="S2.SS3.p4.8.m8.1.1"><in id="S2.SS3.p4.8.m8.1.1.1.cmml" xref="S2.SS3.p4.8.m8.1.1.1"></in><ci id="S2.SS3.p4.8.m8.1.1.2.cmml" xref="S2.SS3.p4.8.m8.1.1.2">𝑓</ci><apply id="S2.SS3.p4.8.m8.1.1.3.cmml" xref="S2.SS3.p4.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p4.8.m8.1.1.3.1.cmml" xref="S2.SS3.p4.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS3.p4.8.m8.1.1.3.2.cmml" xref="S2.SS3.p4.8.m8.1.1.3.2">CPL</ci><ci id="S2.SS3.p4.8.m8.1.1.3.3.cmml" xref="S2.SS3.p4.8.m8.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.8.m8.1c">f\in\operatorname{CPL}_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.8.m8.1d">italic_f ∈ roman_CPL start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> can always be chosen such that their boundary is the union of two rays with vertex <math alttext="0" class="ltx_Math" display="inline" id="S2.SS3.p4.9.m9.1"><semantics id="S2.SS3.p4.9.m9.1a"><mn id="S2.SS3.p4.9.m9.1.1" xref="S2.SS3.p4.9.m9.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.9.m9.1b"><cn id="S2.SS3.p4.9.m9.1.1.cmml" type="integer" xref="S2.SS3.p4.9.m9.1.1">0</cn></annotation-xml></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4 </span>Neural Networks</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.7">A second class of functions in this work are feed forward neural networks with the <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.1"><semantics id="S2.SS4.p1.1.m1.1a"><mi id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.1b"><ci id="S2.SS4.p1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.1d">roman_ReLU</annotation></semantics></math> activation function <math alttext="\rho(x):=\max(0,x)" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.4"><semantics id="S2.SS4.p1.2.m2.4a"><mrow id="S2.SS4.p1.2.m2.4.5" xref="S2.SS4.p1.2.m2.4.5.cmml"><mrow id="S2.SS4.p1.2.m2.4.5.2" xref="S2.SS4.p1.2.m2.4.5.2.cmml"><mi id="S2.SS4.p1.2.m2.4.5.2.2" xref="S2.SS4.p1.2.m2.4.5.2.2.cmml">ρ</mi><mo id="S2.SS4.p1.2.m2.4.5.2.1" xref="S2.SS4.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="S2.SS4.p1.2.m2.4.5.2.3.2" xref="S2.SS4.p1.2.m2.4.5.2.cmml"><mo id="S2.SS4.p1.2.m2.4.5.2.3.2.1" stretchy="false" xref="S2.SS4.p1.2.m2.4.5.2.cmml">(</mo><mi id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml">x</mi><mo id="S2.SS4.p1.2.m2.4.5.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS4.p1.2.m2.4.5.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p1.2.m2.4.5.1" rspace="0.278em" xref="S2.SS4.p1.2.m2.4.5.1.cmml">:=</mo><mrow id="S2.SS4.p1.2.m2.4.5.3.2" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml"><mi id="S2.SS4.p1.2.m2.2.2" xref="S2.SS4.p1.2.m2.2.2.cmml">max</mi><mo id="S2.SS4.p1.2.m2.4.5.3.2a" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml">⁡</mo><mrow id="S2.SS4.p1.2.m2.4.5.3.2.1" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml"><mo id="S2.SS4.p1.2.m2.4.5.3.2.1.1" stretchy="false" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml">(</mo><mn id="S2.SS4.p1.2.m2.3.3" xref="S2.SS4.p1.2.m2.3.3.cmml">0</mn><mo id="S2.SS4.p1.2.m2.4.5.3.2.1.2" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml">,</mo><mi id="S2.SS4.p1.2.m2.4.4" xref="S2.SS4.p1.2.m2.4.4.cmml">x</mi><mo id="S2.SS4.p1.2.m2.4.5.3.2.1.3" stretchy="false" xref="S2.SS4.p1.2.m2.4.5.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.4b"><apply id="S2.SS4.p1.2.m2.4.5.cmml" xref="S2.SS4.p1.2.m2.4.5"><csymbol cd="latexml" id="S2.SS4.p1.2.m2.4.5.1.cmml" xref="S2.SS4.p1.2.m2.4.5.1">assign</csymbol><apply id="S2.SS4.p1.2.m2.4.5.2.cmml" xref="S2.SS4.p1.2.m2.4.5.2"><times id="S2.SS4.p1.2.m2.4.5.2.1.cmml" xref="S2.SS4.p1.2.m2.4.5.2.1"></times><ci id="S2.SS4.p1.2.m2.4.5.2.2.cmml" xref="S2.SS4.p1.2.m2.4.5.2.2">𝜌</ci><ci id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1">𝑥</ci></apply><apply id="S2.SS4.p1.2.m2.4.5.3.1.cmml" xref="S2.SS4.p1.2.m2.4.5.3.2"><max id="S2.SS4.p1.2.m2.2.2.cmml" xref="S2.SS4.p1.2.m2.2.2"></max><cn id="S2.SS4.p1.2.m2.3.3.cmml" type="integer" xref="S2.SS4.p1.2.m2.3.3">0</cn><ci id="S2.SS4.p1.2.m2.4.4.cmml" xref="S2.SS4.p1.2.m2.4.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.4c">\rho(x):=\max(0,x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.4d">italic_ρ ( italic_x ) := roman_max ( 0 , italic_x )</annotation></semantics></math>, which is to be understood element-wise for inputs <math alttext="x\in\mathds{R}^{n}" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.1"><semantics id="S2.SS4.p1.3.m3.1a"><mrow id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml"><mi id="S2.SS4.p1.3.m3.1.1.2" xref="S2.SS4.p1.3.m3.1.1.2.cmml">x</mi><mo id="S2.SS4.p1.3.m3.1.1.1" xref="S2.SS4.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S2.SS4.p1.3.m3.1.1.3" xref="S2.SS4.p1.3.m3.1.1.3.cmml"><mi id="S2.SS4.p1.3.m3.1.1.3.2" xref="S2.SS4.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS4.p1.3.m3.1.1.3.3" xref="S2.SS4.p1.3.m3.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.1b"><apply id="S2.SS4.p1.3.m3.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1"><in id="S2.SS4.p1.3.m3.1.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1.1"></in><ci id="S2.SS4.p1.3.m3.1.1.2.cmml" xref="S2.SS4.p1.3.m3.1.1.2">𝑥</ci><apply id="S2.SS4.p1.3.m3.1.1.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.SS4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2">ℝ</ci><ci id="S2.SS4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.1c">x\in\mathds{R}^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Formally, I define a <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.7.1">neural network</em> of <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.7.2">depth</em> <math alttext="L\in\mathds{N}" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.1"><semantics id="S2.SS4.p1.4.m4.1a"><mrow id="S2.SS4.p1.4.m4.1.1" xref="S2.SS4.p1.4.m4.1.1.cmml"><mi id="S2.SS4.p1.4.m4.1.1.2" xref="S2.SS4.p1.4.m4.1.1.2.cmml">L</mi><mo id="S2.SS4.p1.4.m4.1.1.1" xref="S2.SS4.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S2.SS4.p1.4.m4.1.1.3" xref="S2.SS4.p1.4.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m4.1b"><apply id="S2.SS4.p1.4.m4.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1"><in id="S2.SS4.p1.4.m4.1.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1.1"></in><ci id="S2.SS4.p1.4.m4.1.1.2.cmml" xref="S2.SS4.p1.4.m4.1.1.2">𝐿</ci><ci id="S2.SS4.p1.4.m4.1.1.3.cmml" xref="S2.SS4.p1.4.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m4.1c">L\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m4.1d">italic_L ∈ blackboard_N</annotation></semantics></math> as a sequence <math alttext="s_{0},\dots,s_{L}\in\mathds{N}" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m5.3"><semantics id="S2.SS4.p1.5.m5.3a"><mrow id="S2.SS4.p1.5.m5.3.3" xref="S2.SS4.p1.5.m5.3.3.cmml"><mrow id="S2.SS4.p1.5.m5.3.3.2.2" xref="S2.SS4.p1.5.m5.3.3.2.3.cmml"><msub id="S2.SS4.p1.5.m5.2.2.1.1.1" xref="S2.SS4.p1.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS4.p1.5.m5.2.2.1.1.1.2" xref="S2.SS4.p1.5.m5.2.2.1.1.1.2.cmml">s</mi><mn id="S2.SS4.p1.5.m5.2.2.1.1.1.3" xref="S2.SS4.p1.5.m5.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS4.p1.5.m5.3.3.2.2.3" xref="S2.SS4.p1.5.m5.3.3.2.3.cmml">,</mo><mi id="S2.SS4.p1.5.m5.1.1" mathvariant="normal" xref="S2.SS4.p1.5.m5.1.1.cmml">…</mi><mo id="S2.SS4.p1.5.m5.3.3.2.2.4" xref="S2.SS4.p1.5.m5.3.3.2.3.cmml">,</mo><msub id="S2.SS4.p1.5.m5.3.3.2.2.2" xref="S2.SS4.p1.5.m5.3.3.2.2.2.cmml"><mi id="S2.SS4.p1.5.m5.3.3.2.2.2.2" xref="S2.SS4.p1.5.m5.3.3.2.2.2.2.cmml">s</mi><mi id="S2.SS4.p1.5.m5.3.3.2.2.2.3" xref="S2.SS4.p1.5.m5.3.3.2.2.2.3.cmml">L</mi></msub></mrow><mo id="S2.SS4.p1.5.m5.3.3.3" xref="S2.SS4.p1.5.m5.3.3.3.cmml">∈</mo><mi id="S2.SS4.p1.5.m5.3.3.4" xref="S2.SS4.p1.5.m5.3.3.4.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m5.3b"><apply id="S2.SS4.p1.5.m5.3.3.cmml" xref="S2.SS4.p1.5.m5.3.3"><in id="S2.SS4.p1.5.m5.3.3.3.cmml" xref="S2.SS4.p1.5.m5.3.3.3"></in><list id="S2.SS4.p1.5.m5.3.3.2.3.cmml" xref="S2.SS4.p1.5.m5.3.3.2.2"><apply id="S2.SS4.p1.5.m5.2.2.1.1.1.cmml" xref="S2.SS4.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS4.p1.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS4.p1.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS4.p1.5.m5.2.2.1.1.1.2">𝑠</ci><cn id="S2.SS4.p1.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.p1.5.m5.2.2.1.1.1.3">0</cn></apply><ci id="S2.SS4.p1.5.m5.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1">…</ci><apply id="S2.SS4.p1.5.m5.3.3.2.2.2.cmml" xref="S2.SS4.p1.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.p1.5.m5.3.3.2.2.2.1.cmml" xref="S2.SS4.p1.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S2.SS4.p1.5.m5.3.3.2.2.2.2.cmml" xref="S2.SS4.p1.5.m5.3.3.2.2.2.2">𝑠</ci><ci id="S2.SS4.p1.5.m5.3.3.2.2.2.3.cmml" xref="S2.SS4.p1.5.m5.3.3.2.2.2.3">𝐿</ci></apply></list><ci id="S2.SS4.p1.5.m5.3.3.4.cmml" xref="S2.SS4.p1.5.m5.3.3.4">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m5.3c">s_{0},\dots,s_{L}\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m5.3d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∈ blackboard_N</annotation></semantics></math> together with a sequence of affine functions <math alttext="T^{(l)}:\,\mathds{R}^{s_{l-1}}\to\mathds{R}^{s_{l}}" class="ltx_Math" display="inline" id="S2.SS4.p1.6.m6.1"><semantics id="S2.SS4.p1.6.m6.1a"><mrow id="S2.SS4.p1.6.m6.1.2" xref="S2.SS4.p1.6.m6.1.2.cmml"><msup id="S2.SS4.p1.6.m6.1.2.2" xref="S2.SS4.p1.6.m6.1.2.2.cmml"><mi id="S2.SS4.p1.6.m6.1.2.2.2" xref="S2.SS4.p1.6.m6.1.2.2.2.cmml">T</mi><mrow id="S2.SS4.p1.6.m6.1.1.1.3" xref="S2.SS4.p1.6.m6.1.2.2.cmml"><mo id="S2.SS4.p1.6.m6.1.1.1.3.1" stretchy="false" xref="S2.SS4.p1.6.m6.1.2.2.cmml">(</mo><mi id="S2.SS4.p1.6.m6.1.1.1.1" xref="S2.SS4.p1.6.m6.1.1.1.1.cmml">l</mi><mo id="S2.SS4.p1.6.m6.1.1.1.3.2" stretchy="false" xref="S2.SS4.p1.6.m6.1.2.2.cmml">)</mo></mrow></msup><mo id="S2.SS4.p1.6.m6.1.2.1" lspace="0.278em" rspace="0.448em" xref="S2.SS4.p1.6.m6.1.2.1.cmml">:</mo><mrow id="S2.SS4.p1.6.m6.1.2.3" xref="S2.SS4.p1.6.m6.1.2.3.cmml"><msup id="S2.SS4.p1.6.m6.1.2.3.2" xref="S2.SS4.p1.6.m6.1.2.3.2.cmml"><mi id="S2.SS4.p1.6.m6.1.2.3.2.2" xref="S2.SS4.p1.6.m6.1.2.3.2.2.cmml">ℝ</mi><msub id="S2.SS4.p1.6.m6.1.2.3.2.3" xref="S2.SS4.p1.6.m6.1.2.3.2.3.cmml"><mi id="S2.SS4.p1.6.m6.1.2.3.2.3.2" xref="S2.SS4.p1.6.m6.1.2.3.2.3.2.cmml">s</mi><mrow id="S2.SS4.p1.6.m6.1.2.3.2.3.3" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.cmml"><mi id="S2.SS4.p1.6.m6.1.2.3.2.3.3.2" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.2.cmml">l</mi><mo id="S2.SS4.p1.6.m6.1.2.3.2.3.3.1" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.1.cmml">−</mo><mn id="S2.SS4.p1.6.m6.1.2.3.2.3.3.3" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.3.cmml">1</mn></mrow></msub></msup><mo id="S2.SS4.p1.6.m6.1.2.3.1" stretchy="false" xref="S2.SS4.p1.6.m6.1.2.3.1.cmml">→</mo><msup id="S2.SS4.p1.6.m6.1.2.3.3" xref="S2.SS4.p1.6.m6.1.2.3.3.cmml"><mi id="S2.SS4.p1.6.m6.1.2.3.3.2" xref="S2.SS4.p1.6.m6.1.2.3.3.2.cmml">ℝ</mi><msub id="S2.SS4.p1.6.m6.1.2.3.3.3" xref="S2.SS4.p1.6.m6.1.2.3.3.3.cmml"><mi id="S2.SS4.p1.6.m6.1.2.3.3.3.2" xref="S2.SS4.p1.6.m6.1.2.3.3.3.2.cmml">s</mi><mi id="S2.SS4.p1.6.m6.1.2.3.3.3.3" xref="S2.SS4.p1.6.m6.1.2.3.3.3.3.cmml">l</mi></msub></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.6.m6.1b"><apply id="S2.SS4.p1.6.m6.1.2.cmml" xref="S2.SS4.p1.6.m6.1.2"><ci id="S2.SS4.p1.6.m6.1.2.1.cmml" xref="S2.SS4.p1.6.m6.1.2.1">:</ci><apply id="S2.SS4.p1.6.m6.1.2.2.cmml" xref="S2.SS4.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.2.2.1.cmml" xref="S2.SS4.p1.6.m6.1.2.2">superscript</csymbol><ci id="S2.SS4.p1.6.m6.1.2.2.2.cmml" xref="S2.SS4.p1.6.m6.1.2.2.2">𝑇</ci><ci id="S2.SS4.p1.6.m6.1.1.1.1.cmml" xref="S2.SS4.p1.6.m6.1.1.1.1">𝑙</ci></apply><apply id="S2.SS4.p1.6.m6.1.2.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3"><ci id="S2.SS4.p1.6.m6.1.2.3.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.1">→</ci><apply id="S2.SS4.p1.6.m6.1.2.3.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.2.3.2.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2">superscript</csymbol><ci id="S2.SS4.p1.6.m6.1.2.3.2.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.2">ℝ</ci><apply id="S2.SS4.p1.6.m6.1.2.3.2.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.2.3.2.3.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3">subscript</csymbol><ci id="S2.SS4.p1.6.m6.1.2.3.2.3.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3.2">𝑠</ci><apply id="S2.SS4.p1.6.m6.1.2.3.2.3.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3"><minus id="S2.SS4.p1.6.m6.1.2.3.2.3.3.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.1"></minus><ci id="S2.SS4.p1.6.m6.1.2.3.2.3.3.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.2">𝑙</ci><cn id="S2.SS4.p1.6.m6.1.2.3.2.3.3.3.cmml" type="integer" xref="S2.SS4.p1.6.m6.1.2.3.2.3.3.3">1</cn></apply></apply></apply><apply id="S2.SS4.p1.6.m6.1.2.3.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.2.3.3.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3">superscript</csymbol><ci id="S2.SS4.p1.6.m6.1.2.3.3.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3.2">ℝ</ci><apply id="S2.SS4.p1.6.m6.1.2.3.3.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.1.2.3.3.3.1.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3.3">subscript</csymbol><ci id="S2.SS4.p1.6.m6.1.2.3.3.3.2.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3.3.2">𝑠</ci><ci id="S2.SS4.p1.6.m6.1.2.3.3.3.3.cmml" xref="S2.SS4.p1.6.m6.1.2.3.3.3.3">𝑙</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.6.m6.1c">T^{(l)}:\,\mathds{R}^{s_{l-1}}\to\mathds{R}^{s_{l}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.m6.1d">italic_T start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_l - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, for <math alttext="l\in[L]" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m7.1"><semantics id="S2.SS4.p1.7.m7.1a"><mrow id="S2.SS4.p1.7.m7.1.2" xref="S2.SS4.p1.7.m7.1.2.cmml"><mi id="S2.SS4.p1.7.m7.1.2.2" xref="S2.SS4.p1.7.m7.1.2.2.cmml">l</mi><mo id="S2.SS4.p1.7.m7.1.2.1" xref="S2.SS4.p1.7.m7.1.2.1.cmml">∈</mo><mrow id="S2.SS4.p1.7.m7.1.2.3.2" xref="S2.SS4.p1.7.m7.1.2.3.1.cmml"><mo id="S2.SS4.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS4.p1.7.m7.1.2.3.1.1.cmml">[</mo><mi id="S2.SS4.p1.7.m7.1.1" xref="S2.SS4.p1.7.m7.1.1.cmml">L</mi><mo id="S2.SS4.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS4.p1.7.m7.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.7.m7.1b"><apply id="S2.SS4.p1.7.m7.1.2.cmml" xref="S2.SS4.p1.7.m7.1.2"><in id="S2.SS4.p1.7.m7.1.2.1.cmml" xref="S2.SS4.p1.7.m7.1.2.1"></in><ci id="S2.SS4.p1.7.m7.1.2.2.cmml" xref="S2.SS4.p1.7.m7.1.2.2">𝑙</ci><apply id="S2.SS4.p1.7.m7.1.2.3.1.cmml" xref="S2.SS4.p1.7.m7.1.2.3.2"><csymbol cd="latexml" id="S2.SS4.p1.7.m7.1.2.3.1.1.cmml" xref="S2.SS4.p1.7.m7.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.SS4.p1.7.m7.1.1.cmml" xref="S2.SS4.p1.7.m7.1.1">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m7.1c">l\in[L]</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m7.1d">italic_l ∈ [ italic_L ]</annotation></semantics></math>. Such a neural network is said to <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.7.3">represent</em> the function</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="f:\,\mathds{R}^{s_{0}}\to\mathds{R}^{s_{L}},\quad f=T^{(L)}\circ\rho\circ T^{(% L-1)}\circ\dots\circ\rho\circ T^{(1)}." class="ltx_Math" display="block" id="S2.Ex5.m1.4"><semantics id="S2.Ex5.m1.4a"><mrow id="S2.Ex5.m1.4.4.1" xref="S2.Ex5.m1.4.4.1.1.cmml"><mrow id="S2.Ex5.m1.4.4.1.1" xref="S2.Ex5.m1.4.4.1.1.cmml"><mi id="S2.Ex5.m1.4.4.1.1.4" xref="S2.Ex5.m1.4.4.1.1.4.cmml">f</mi><mo id="S2.Ex5.m1.4.4.1.1.3" lspace="0.278em" rspace="0.448em" xref="S2.Ex5.m1.4.4.1.1.3.cmml">:</mo><mrow id="S2.Ex5.m1.4.4.1.1.2.2" xref="S2.Ex5.m1.4.4.1.1.2.3.cmml"><mrow id="S2.Ex5.m1.4.4.1.1.1.1.1" xref="S2.Ex5.m1.4.4.1.1.1.1.1.cmml"><msup id="S2.Ex5.m1.4.4.1.1.1.1.1.2" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.cmml"><mi id="S2.Ex5.m1.4.4.1.1.1.1.1.2.2" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.2.cmml">ℝ</mi><msub id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.cmml"><mi id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.2" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.2.cmml">s</mi><mn id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.3" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.3.cmml">0</mn></msub></msup><mo id="S2.Ex5.m1.4.4.1.1.1.1.1.1" stretchy="false" xref="S2.Ex5.m1.4.4.1.1.1.1.1.1.cmml">→</mo><msup id="S2.Ex5.m1.4.4.1.1.1.1.1.3" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.cmml"><mi id="S2.Ex5.m1.4.4.1.1.1.1.1.3.2" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.2.cmml">ℝ</mi><msub id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.cmml"><mi id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.2" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.2.cmml">s</mi><mi id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.3" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.3.cmml">L</mi></msub></msup></mrow><mo id="S2.Ex5.m1.4.4.1.1.2.2.3" rspace="1.167em" xref="S2.Ex5.m1.4.4.1.1.2.3a.cmml">,</mo><mrow id="S2.Ex5.m1.4.4.1.1.2.2.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.cmml"><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.2.cmml">f</mi><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.1" xref="S2.Ex5.m1.4.4.1.1.2.2.2.1.cmml">=</mo><mrow id="S2.Ex5.m1.4.4.1.1.2.2.2.3" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.cmml"><msup id="S2.Ex5.m1.4.4.1.1.2.2.2.3.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.cmml"><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.2.cmml">T</mi><mrow id="S2.Ex5.m1.1.1.1.3" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.cmml"><mo id="S2.Ex5.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.cmml">(</mo><mi id="S2.Ex5.m1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.cmml">L</mi><mo id="S2.Ex5.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.cmml">)</mo></mrow></msup><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml">∘</mo><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.3" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.3.cmml">ρ</mi><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1a" lspace="0.222em" rspace="0.222em" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml">∘</mo><msup id="S2.Ex5.m1.4.4.1.1.2.2.2.3.4" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.cmml"><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.2.cmml">T</mi><mrow id="S2.Ex5.m1.2.2.1.1" xref="S2.Ex5.m1.2.2.1.1.1.cmml"><mo id="S2.Ex5.m1.2.2.1.1.2" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m1.2.2.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.cmml"><mi id="S2.Ex5.m1.2.2.1.1.1.2" xref="S2.Ex5.m1.2.2.1.1.1.2.cmml">L</mi><mo id="S2.Ex5.m1.2.2.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.1.cmml">−</mo><mn id="S2.Ex5.m1.2.2.1.1.1.3" xref="S2.Ex5.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S2.Ex5.m1.2.2.1.1.3" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.1.cmml">)</mo></mrow></msup><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1b" lspace="0.222em" rspace="0.222em" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml">∘</mo><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.5" mathvariant="normal" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.5.cmml">⋯</mi><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1c" lspace="0.222em" rspace="0.222em" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml">∘</mo><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.6" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.6.cmml">ρ</mi><mo id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1d" lspace="0.222em" rspace="0.222em" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml">∘</mo><msup id="S2.Ex5.m1.4.4.1.1.2.2.2.3.7" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.cmml"><mi id="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.2" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.2.cmml">T</mi><mrow id="S2.Ex5.m1.3.3.1.3" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.cmml"><mo id="S2.Ex5.m1.3.3.1.3.1" stretchy="false" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.cmml">(</mo><mn id="S2.Ex5.m1.3.3.1.1" xref="S2.Ex5.m1.3.3.1.1.cmml">1</mn><mo id="S2.Ex5.m1.3.3.1.3.2" stretchy="false" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.cmml">)</mo></mrow></msup></mrow></mrow></mrow></mrow><mo id="S2.Ex5.m1.4.4.1.2" lspace="0em" xref="S2.Ex5.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.4b"><apply id="S2.Ex5.m1.4.4.1.1.cmml" xref="S2.Ex5.m1.4.4.1"><ci id="S2.Ex5.m1.4.4.1.1.3.cmml" xref="S2.Ex5.m1.4.4.1.1.3">:</ci><ci id="S2.Ex5.m1.4.4.1.1.4.cmml" xref="S2.Ex5.m1.4.4.1.1.4">𝑓</ci><apply id="S2.Ex5.m1.4.4.1.1.2.3.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.2.3a.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.3">formulae-sequence</csymbol><apply id="S2.Ex5.m1.4.4.1.1.1.1.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1"><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.1">→</ci><apply id="S2.Ex5.m1.4.4.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.2.2.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.2">ℝ</ci><apply id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3">subscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.2.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.2">𝑠</ci><cn id="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.Ex5.m1.4.4.1.1.1.1.1.2.3.3">0</cn></apply></apply><apply id="S2.Ex5.m1.4.4.1.1.1.1.1.3.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.1.1.1.3.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.3.2.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.2">ℝ</ci><apply id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.1.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.2.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.2">𝑠</ci><ci id="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.3.cmml" xref="S2.Ex5.m1.4.4.1.1.1.1.1.3.3.3">𝐿</ci></apply></apply></apply><apply id="S2.Ex5.m1.4.4.1.1.2.2.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2"><eq id="S2.Ex5.m1.4.4.1.1.2.2.2.1.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.1"></eq><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.2">𝑓</ci><apply id="S2.Ex5.m1.4.4.1.1.2.2.2.3.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3"><compose id="S2.Ex5.m1.4.4.1.1.2.2.2.3.1.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.1"></compose><apply id="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.1.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2">superscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.2.2">𝑇</ci><ci id="S2.Ex5.m1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1">𝐿</ci></apply><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.3.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.3">𝜌</ci><apply id="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.4"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.1.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.4">superscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.4.2">𝑇</ci><apply id="S2.Ex5.m1.2.2.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1"><minus id="S2.Ex5.m1.2.2.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.1.1"></minus><ci id="S2.Ex5.m1.2.2.1.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.1.2">𝐿</ci><cn id="S2.Ex5.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.1.3">1</cn></apply></apply><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.5.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.5">⋯</ci><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.6.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.6">𝜌</ci><apply id="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.1.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7">superscript</csymbol><ci id="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.2.cmml" xref="S2.Ex5.m1.4.4.1.1.2.2.2.3.7.2">𝑇</ci><cn id="S2.Ex5.m1.3.3.1.1.cmml" type="integer" xref="S2.Ex5.m1.3.3.1.1">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.4c">f:\,\mathds{R}^{s_{0}}\to\mathds{R}^{s_{L}},\quad f=T^{(L)}\circ\rho\circ T^{(% L-1)}\circ\dots\circ\rho\circ T^{(1)}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.4d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_f = italic_T start_POSTSUPERSCRIPT ( italic_L ) end_POSTSUPERSCRIPT ∘ italic_ρ ∘ italic_T start_POSTSUPERSCRIPT ( italic_L - 1 ) end_POSTSUPERSCRIPT ∘ ⋯ ∘ italic_ρ ∘ italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p1.11">For <math alttext="l\in[L-1]" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m1.1"><semantics id="S2.SS4.p1.8.m1.1a"><mrow id="S2.SS4.p1.8.m1.1.1" xref="S2.SS4.p1.8.m1.1.1.cmml"><mi id="S2.SS4.p1.8.m1.1.1.3" xref="S2.SS4.p1.8.m1.1.1.3.cmml">l</mi><mo id="S2.SS4.p1.8.m1.1.1.2" xref="S2.SS4.p1.8.m1.1.1.2.cmml">∈</mo><mrow id="S2.SS4.p1.8.m1.1.1.1.1" xref="S2.SS4.p1.8.m1.1.1.1.2.cmml"><mo id="S2.SS4.p1.8.m1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p1.8.m1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS4.p1.8.m1.1.1.1.1.1" xref="S2.SS4.p1.8.m1.1.1.1.1.1.cmml"><mi id="S2.SS4.p1.8.m1.1.1.1.1.1.2" xref="S2.SS4.p1.8.m1.1.1.1.1.1.2.cmml">L</mi><mo id="S2.SS4.p1.8.m1.1.1.1.1.1.1" xref="S2.SS4.p1.8.m1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS4.p1.8.m1.1.1.1.1.1.3" xref="S2.SS4.p1.8.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS4.p1.8.m1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p1.8.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m1.1b"><apply id="S2.SS4.p1.8.m1.1.1.cmml" xref="S2.SS4.p1.8.m1.1.1"><in id="S2.SS4.p1.8.m1.1.1.2.cmml" xref="S2.SS4.p1.8.m1.1.1.2"></in><ci id="S2.SS4.p1.8.m1.1.1.3.cmml" xref="S2.SS4.p1.8.m1.1.1.3">𝑙</ci><apply id="S2.SS4.p1.8.m1.1.1.1.2.cmml" xref="S2.SS4.p1.8.m1.1.1.1.1"><csymbol cd="latexml" id="S2.SS4.p1.8.m1.1.1.1.2.1.cmml" xref="S2.SS4.p1.8.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS4.p1.8.m1.1.1.1.1.1.cmml" xref="S2.SS4.p1.8.m1.1.1.1.1.1"><minus id="S2.SS4.p1.8.m1.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.8.m1.1.1.1.1.1.1"></minus><ci id="S2.SS4.p1.8.m1.1.1.1.1.1.2.cmml" xref="S2.SS4.p1.8.m1.1.1.1.1.1.2">𝐿</ci><cn id="S2.SS4.p1.8.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p1.8.m1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m1.1c">l\in[L-1]</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m1.1d">italic_l ∈ [ italic_L - 1 ]</annotation></semantics></math>, <math alttext="s_{l}" class="ltx_Math" display="inline" id="S2.SS4.p1.9.m2.1"><semantics id="S2.SS4.p1.9.m2.1a"><msub id="S2.SS4.p1.9.m2.1.1" xref="S2.SS4.p1.9.m2.1.1.cmml"><mi id="S2.SS4.p1.9.m2.1.1.2" xref="S2.SS4.p1.9.m2.1.1.2.cmml">s</mi><mi id="S2.SS4.p1.9.m2.1.1.3" xref="S2.SS4.p1.9.m2.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.9.m2.1b"><apply id="S2.SS4.p1.9.m2.1.1.cmml" xref="S2.SS4.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.9.m2.1.1.1.cmml" xref="S2.SS4.p1.9.m2.1.1">subscript</csymbol><ci id="S2.SS4.p1.9.m2.1.1.2.cmml" xref="S2.SS4.p1.9.m2.1.1.2">𝑠</ci><ci id="S2.SS4.p1.9.m2.1.1.3.cmml" xref="S2.SS4.p1.9.m2.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.9.m2.1c">s_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.9.m2.1d">italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.11.1">width</em> of the <math alttext="l" class="ltx_Math" display="inline" id="S2.SS4.p1.10.m3.1"><semantics id="S2.SS4.p1.10.m3.1a"><mi id="S2.SS4.p1.10.m3.1.1" xref="S2.SS4.p1.10.m3.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.10.m3.1b"><ci id="S2.SS4.p1.10.m3.1.1.cmml" xref="S2.SS4.p1.10.m3.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.10.m3.1c">l</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.10.m3.1d">italic_l</annotation></semantics></math>-th <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.11.2">hidden layer</em>. The <em class="ltx_emph ltx_font_italic" id="S2.SS4.p1.11.3">width vector</em> of the network is defined as the vector <math alttext="(s_{l})_{l\in[L-1]}" class="ltx_Math" display="inline" id="S2.SS4.p1.11.m4.2"><semantics id="S2.SS4.p1.11.m4.2a"><msub id="S2.SS4.p1.11.m4.2.2" xref="S2.SS4.p1.11.m4.2.2.cmml"><mrow id="S2.SS4.p1.11.m4.2.2.1.1" xref="S2.SS4.p1.11.m4.2.2.1.1.1.cmml"><mo id="S2.SS4.p1.11.m4.2.2.1.1.2" stretchy="false" xref="S2.SS4.p1.11.m4.2.2.1.1.1.cmml">(</mo><msub id="S2.SS4.p1.11.m4.2.2.1.1.1" xref="S2.SS4.p1.11.m4.2.2.1.1.1.cmml"><mi id="S2.SS4.p1.11.m4.2.2.1.1.1.2" xref="S2.SS4.p1.11.m4.2.2.1.1.1.2.cmml">s</mi><mi id="S2.SS4.p1.11.m4.2.2.1.1.1.3" xref="S2.SS4.p1.11.m4.2.2.1.1.1.3.cmml">l</mi></msub><mo id="S2.SS4.p1.11.m4.2.2.1.1.3" stretchy="false" xref="S2.SS4.p1.11.m4.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS4.p1.11.m4.1.1.1" xref="S2.SS4.p1.11.m4.1.1.1.cmml"><mi id="S2.SS4.p1.11.m4.1.1.1.3" xref="S2.SS4.p1.11.m4.1.1.1.3.cmml">l</mi><mo id="S2.SS4.p1.11.m4.1.1.1.2" xref="S2.SS4.p1.11.m4.1.1.1.2.cmml">∈</mo><mrow id="S2.SS4.p1.11.m4.1.1.1.1.1" xref="S2.SS4.p1.11.m4.1.1.1.1.2.cmml"><mo id="S2.SS4.p1.11.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p1.11.m4.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS4.p1.11.m4.1.1.1.1.1.1" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p1.11.m4.1.1.1.1.1.1.2" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.2.cmml">L</mi><mo id="S2.SS4.p1.11.m4.1.1.1.1.1.1.1" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS4.p1.11.m4.1.1.1.1.1.1.3" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS4.p1.11.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p1.11.m4.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.11.m4.2b"><apply id="S2.SS4.p1.11.m4.2.2.cmml" xref="S2.SS4.p1.11.m4.2.2"><csymbol cd="ambiguous" id="S2.SS4.p1.11.m4.2.2.2.cmml" xref="S2.SS4.p1.11.m4.2.2">subscript</csymbol><apply id="S2.SS4.p1.11.m4.2.2.1.1.1.cmml" xref="S2.SS4.p1.11.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.11.m4.2.2.1.1.1.1.cmml" xref="S2.SS4.p1.11.m4.2.2.1.1">subscript</csymbol><ci id="S2.SS4.p1.11.m4.2.2.1.1.1.2.cmml" xref="S2.SS4.p1.11.m4.2.2.1.1.1.2">𝑠</ci><ci id="S2.SS4.p1.11.m4.2.2.1.1.1.3.cmml" xref="S2.SS4.p1.11.m4.2.2.1.1.1.3">𝑙</ci></apply><apply id="S2.SS4.p1.11.m4.1.1.1.cmml" xref="S2.SS4.p1.11.m4.1.1.1"><in id="S2.SS4.p1.11.m4.1.1.1.2.cmml" xref="S2.SS4.p1.11.m4.1.1.1.2"></in><ci id="S2.SS4.p1.11.m4.1.1.1.3.cmml" xref="S2.SS4.p1.11.m4.1.1.1.3">𝑙</ci><apply id="S2.SS4.p1.11.m4.1.1.1.1.2.cmml" xref="S2.SS4.p1.11.m4.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS4.p1.11.m4.1.1.1.1.2.1.cmml" xref="S2.SS4.p1.11.m4.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS4.p1.11.m4.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1"><minus id="S2.SS4.p1.11.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.1"></minus><ci id="S2.SS4.p1.11.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.2">𝐿</ci><cn id="S2.SS4.p1.11.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.p1.11.m4.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.11.m4.2c">(s_{l})_{l\in[L-1]}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.11.m4.2d">( italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_l ∈ [ italic_L - 1 ] end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.1">I will only consider neural networks with <math alttext="s_{L}=1" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.1"><semantics id="S2.SS4.p2.1.m1.1a"><mrow id="S2.SS4.p2.1.m1.1.1" xref="S2.SS4.p2.1.m1.1.1.cmml"><msub id="S2.SS4.p2.1.m1.1.1.2" xref="S2.SS4.p2.1.m1.1.1.2.cmml"><mi id="S2.SS4.p2.1.m1.1.1.2.2" xref="S2.SS4.p2.1.m1.1.1.2.2.cmml">s</mi><mi id="S2.SS4.p2.1.m1.1.1.2.3" xref="S2.SS4.p2.1.m1.1.1.2.3.cmml">L</mi></msub><mo id="S2.SS4.p2.1.m1.1.1.1" xref="S2.SS4.p2.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS4.p2.1.m1.1.1.3" xref="S2.SS4.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.1b"><apply id="S2.SS4.p2.1.m1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1"><eq id="S2.SS4.p2.1.m1.1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1.1"></eq><apply id="S2.SS4.p2.1.m1.1.1.2.cmml" xref="S2.SS4.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.p2.1.m1.1.1.2.1.cmml" xref="S2.SS4.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS4.p2.1.m1.1.1.2.2.cmml" xref="S2.SS4.p2.1.m1.1.1.2.2">𝑠</ci><ci id="S2.SS4.p2.1.m1.1.1.2.3.cmml" xref="S2.SS4.p2.1.m1.1.1.2.3">𝐿</ci></apply><cn id="S2.SS4.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS4.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.1c">s_{L}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.1d">italic_s start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = 1</annotation></semantics></math>, since the results can be easily generalised to higher dimensional outputs.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Decomposition of CPA Functions</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The first step towards a standardised representation of <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">roman_CPA</annotation></semantics></math> functions is to decompose them into a sum of simpler functions, each of which is associated with a vertex or edge of the considered function.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.10">To achieve this, the pieces incident to a vertex or edge are extended to infinity using a definition similar to tangent cones for convex polyhedra. For a polygon <math alttext="P" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mi id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_P</annotation></semantics></math> and an arbitrary point <math alttext="v\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><mi id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml">v</mi><mo id="S3.p2.2.m2.1.1.1" xref="S3.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S3.p2.2.m2.1.1.3" xref="S3.p2.2.m2.1.1.3.cmml"><mi id="S3.p2.2.m2.1.1.3.2" xref="S3.p2.2.m2.1.1.3.2.cmml">ℝ</mi><mn id="S3.p2.2.m2.1.1.3.3" xref="S3.p2.2.m2.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><in id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"></in><ci id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2">𝑣</ci><apply id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.3.1.cmml" xref="S3.p2.2.m2.1.1.3">superscript</csymbol><ci id="S3.p2.2.m2.1.1.3.2.cmml" xref="S3.p2.2.m2.1.1.3.2">ℝ</ci><cn id="S3.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.p2.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">v\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, I define the <em class="ltx_emph ltx_font_italic" id="S3.p2.3.1"><math alttext="P" class="ltx_Math" display="inline" id="S3.p2.3.1.m1.1"><semantics id="S3.p2.3.1.m1.1a"><mi id="S3.p2.3.1.m1.1.1" xref="S3.p2.3.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p2.3.1.m1.1b"><ci id="S3.p2.3.1.m1.1.1.cmml" xref="S3.p2.3.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.1.m1.1d">italic_P</annotation></semantics></math>-side</em> <math alttext="Q^{v}_{P}" class="ltx_Math" display="inline" id="S3.p2.4.m3.1"><semantics id="S3.p2.4.m3.1a"><msubsup id="S3.p2.4.m3.1.1" xref="S3.p2.4.m3.1.1.cmml"><mi id="S3.p2.4.m3.1.1.2.2" xref="S3.p2.4.m3.1.1.2.2.cmml">Q</mi><mi id="S3.p2.4.m3.1.1.3" xref="S3.p2.4.m3.1.1.3.cmml">P</mi><mi id="S3.p2.4.m3.1.1.2.3" xref="S3.p2.4.m3.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.4.m3.1b"><apply id="S3.p2.4.m3.1.1.cmml" xref="S3.p2.4.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m3.1.1.1.cmml" xref="S3.p2.4.m3.1.1">subscript</csymbol><apply id="S3.p2.4.m3.1.1.2.cmml" xref="S3.p2.4.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m3.1.1.2.1.cmml" xref="S3.p2.4.m3.1.1">superscript</csymbol><ci id="S3.p2.4.m3.1.1.2.2.cmml" xref="S3.p2.4.m3.1.1.2.2">𝑄</ci><ci id="S3.p2.4.m3.1.1.2.3.cmml" xref="S3.p2.4.m3.1.1.2.3">𝑣</ci></apply><ci id="S3.p2.4.m3.1.1.3.cmml" xref="S3.p2.4.m3.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m3.1c">Q^{v}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m3.1d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="v" class="ltx_Math" display="inline" id="S3.p2.5.m4.1"><semantics id="S3.p2.5.m4.1a"><mi id="S3.p2.5.m4.1.1" xref="S3.p2.5.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p2.5.m4.1b"><ci id="S3.p2.5.m4.1.1.cmml" xref="S3.p2.5.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m4.1d">italic_v</annotation></semantics></math> as follows. Let <math alttext="D" class="ltx_Math" display="inline" id="S3.p2.6.m5.1"><semantics id="S3.p2.6.m5.1a"><mi id="S3.p2.6.m5.1.1" xref="S3.p2.6.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.p2.6.m5.1b"><ci id="S3.p2.6.m5.1.1.cmml" xref="S3.p2.6.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m5.1d">italic_D</annotation></semantics></math> be a disk centered at <math alttext="v" class="ltx_Math" display="inline" id="S3.p2.7.m6.1"><semantics id="S3.p2.7.m6.1a"><mi id="S3.p2.7.m6.1.1" xref="S3.p2.7.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p2.7.m6.1b"><ci id="S3.p2.7.m6.1.1.cmml" xref="S3.p2.7.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m6.1d">italic_v</annotation></semantics></math>, small enough such that <math alttext="D" class="ltx_Math" display="inline" id="S3.p2.8.m7.1"><semantics id="S3.p2.8.m7.1a"><mi id="S3.p2.8.m7.1.1" xref="S3.p2.8.m7.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.p2.8.m7.1b"><ci id="S3.p2.8.m7.1.1.cmml" xref="S3.p2.8.m7.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m7.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m7.1d">italic_D</annotation></semantics></math> intersects only those edges of <math alttext="P" class="ltx_Math" display="inline" id="S3.p2.9.m8.1"><semantics id="S3.p2.9.m8.1a"><mi id="S3.p2.9.m8.1.1" xref="S3.p2.9.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p2.9.m8.1b"><ci id="S3.p2.9.m8.1.1.cmml" xref="S3.p2.9.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m8.1d">italic_P</annotation></semantics></math> that have <math alttext="v" class="ltx_Math" display="inline" id="S3.p2.10.m9.1"><semantics id="S3.p2.10.m9.1a"><mi id="S3.p2.10.m9.1.1" xref="S3.p2.10.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p2.10.m9.1b"><ci id="S3.p2.10.m9.1.1.cmml" xref="S3.p2.10.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m9.1d">italic_v</annotation></semantics></math> as a vertex. Then, define</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="Q^{v}_{P}:=\{v+t(x-v):t\in[0,\infty),\,x\in P\cap D\}." class="ltx_Math" display="block" id="S3.Ex6.m1.3"><semantics id="S3.Ex6.m1.3a"><mrow id="S3.Ex6.m1.3.3.1" xref="S3.Ex6.m1.3.3.1.1.cmml"><mrow id="S3.Ex6.m1.3.3.1.1" xref="S3.Ex6.m1.3.3.1.1.cmml"><msubsup id="S3.Ex6.m1.3.3.1.1.4" xref="S3.Ex6.m1.3.3.1.1.4.cmml"><mi id="S3.Ex6.m1.3.3.1.1.4.2.2" xref="S3.Ex6.m1.3.3.1.1.4.2.2.cmml">Q</mi><mi id="S3.Ex6.m1.3.3.1.1.4.3" xref="S3.Ex6.m1.3.3.1.1.4.3.cmml">P</mi><mi id="S3.Ex6.m1.3.3.1.1.4.2.3" xref="S3.Ex6.m1.3.3.1.1.4.2.3.cmml">v</mi></msubsup><mo id="S3.Ex6.m1.3.3.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex6.m1.3.3.1.1.3.cmml">:=</mo><mrow id="S3.Ex6.m1.3.3.1.1.2.2" xref="S3.Ex6.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex6.m1.3.3.1.1.2.2.3" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.2.3.1.cmml">{</mo><mrow id="S3.Ex6.m1.3.3.1.1.1.1.1" xref="S3.Ex6.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.Ex6.m1.3.3.1.1.1.1.1.3" xref="S3.Ex6.m1.3.3.1.1.1.1.1.3.cmml">v</mi><mo id="S3.Ex6.m1.3.3.1.1.1.1.1.2" xref="S3.Ex6.m1.3.3.1.1.1.1.1.2.cmml">+</mo><mrow id="S3.Ex6.m1.3.3.1.1.1.1.1.1" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex6.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.3.cmml">t</mi><mo id="S3.Ex6.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex6.m1.3.3.1.1.2.2.4" rspace="0.278em" xref="S3.Ex6.m1.3.3.1.1.2.3.1.cmml">:</mo><mrow id="S3.Ex6.m1.3.3.1.1.2.2.2.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.3.cmml"><mrow id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.cmml"><mi id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.2.cmml">t</mi><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.1" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.1.cmml">∈</mo><mrow id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.1.cmml"><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.2.1" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.1.cmml">[</mo><mn id="S3.Ex6.m1.1.1" xref="S3.Ex6.m1.1.1.cmml">0</mn><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.2.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.1.cmml">,</mo><mi id="S3.Ex6.m1.2.2" mathvariant="normal" xref="S3.Ex6.m1.2.2.cmml">∞</mi><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.2.3" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.2.3" rspace="0.337em" xref="S3.Ex6.m1.3.3.1.1.2.2.2.3a.cmml">,</mo><mrow id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.cmml"><mi id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.2.cmml">x</mi><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.1" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.1.cmml">∈</mo><mrow id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.cmml"><mi id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.2" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.2.cmml">P</mi><mo id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.1" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.1.cmml">∩</mo><mi id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.3" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.3.cmml">D</mi></mrow></mrow></mrow><mo id="S3.Ex6.m1.3.3.1.1.2.2.5" stretchy="false" xref="S3.Ex6.m1.3.3.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S3.Ex6.m1.3.3.1.2" lspace="0em" xref="S3.Ex6.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex6.m1.3b"><apply id="S3.Ex6.m1.3.3.1.1.cmml" xref="S3.Ex6.m1.3.3.1"><csymbol cd="latexml" id="S3.Ex6.m1.3.3.1.1.3.cmml" xref="S3.Ex6.m1.3.3.1.1.3">assign</csymbol><apply id="S3.Ex6.m1.3.3.1.1.4.cmml" xref="S3.Ex6.m1.3.3.1.1.4"><csymbol cd="ambiguous" id="S3.Ex6.m1.3.3.1.1.4.1.cmml" xref="S3.Ex6.m1.3.3.1.1.4">subscript</csymbol><apply id="S3.Ex6.m1.3.3.1.1.4.2.cmml" xref="S3.Ex6.m1.3.3.1.1.4"><csymbol cd="ambiguous" id="S3.Ex6.m1.3.3.1.1.4.2.1.cmml" xref="S3.Ex6.m1.3.3.1.1.4">superscript</csymbol><ci id="S3.Ex6.m1.3.3.1.1.4.2.2.cmml" xref="S3.Ex6.m1.3.3.1.1.4.2.2">𝑄</ci><ci id="S3.Ex6.m1.3.3.1.1.4.2.3.cmml" xref="S3.Ex6.m1.3.3.1.1.4.2.3">𝑣</ci></apply><ci id="S3.Ex6.m1.3.3.1.1.4.3.cmml" xref="S3.Ex6.m1.3.3.1.1.4.3">𝑃</ci></apply><apply id="S3.Ex6.m1.3.3.1.1.2.3.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2"><csymbol cd="latexml" id="S3.Ex6.m1.3.3.1.1.2.3.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.3">conditional-set</csymbol><apply id="S3.Ex6.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1"><plus id="S3.Ex6.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.2"></plus><ci id="S3.Ex6.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.3">𝑣</ci><apply id="S3.Ex6.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1"><times id="S3.Ex6.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.2"></times><ci id="S3.Ex6.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.3">𝑡</ci><apply id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1"><minus id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply><apply id="S3.Ex6.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex6.m1.3.3.1.1.2.2.2.3a.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.3">formulae-sequence</csymbol><apply id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1"><in id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.1"></in><ci id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.2.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.2">𝑡</ci><interval closure="closed-open" id="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.1.1.3.2"><cn id="S3.Ex6.m1.1.1.cmml" type="integer" xref="S3.Ex6.m1.1.1">0</cn><infinity id="S3.Ex6.m1.2.2.cmml" xref="S3.Ex6.m1.2.2"></infinity></interval></apply><apply id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2"><in id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.1"></in><ci id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.2">𝑥</ci><apply id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3"><intersect id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.1.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.1"></intersect><ci id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.2.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.2">𝑃</ci><ci id="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.3.cmml" xref="S3.Ex6.m1.3.3.1.1.2.2.2.2.2.3.3">𝐷</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m1.3c">Q^{v}_{P}:=\{v+t(x-v):t\in[0,\infty),\,x\in P\cap D\}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m1.3d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT := { italic_v + italic_t ( italic_x - italic_v ) : italic_t ∈ [ 0 , ∞ ) , italic_x ∈ italic_P ∩ italic_D } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.25">This definition is independent of the choice of <math alttext="D" class="ltx_Math" display="inline" id="S3.p2.11.m1.1"><semantics id="S3.p2.11.m1.1a"><mi id="S3.p2.11.m1.1.1" xref="S3.p2.11.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.p2.11.m1.1b"><ci id="S3.p2.11.m1.1.1.cmml" xref="S3.p2.11.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.11.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.11.m1.1d">italic_D</annotation></semantics></math> and satisfies the property that <math alttext="Q^{v}_{P}" class="ltx_Math" display="inline" id="S3.p2.12.m2.1"><semantics id="S3.p2.12.m2.1a"><msubsup id="S3.p2.12.m2.1.1" xref="S3.p2.12.m2.1.1.cmml"><mi id="S3.p2.12.m2.1.1.2.2" xref="S3.p2.12.m2.1.1.2.2.cmml">Q</mi><mi id="S3.p2.12.m2.1.1.3" xref="S3.p2.12.m2.1.1.3.cmml">P</mi><mi id="S3.p2.12.m2.1.1.2.3" xref="S3.p2.12.m2.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.12.m2.1b"><apply id="S3.p2.12.m2.1.1.cmml" xref="S3.p2.12.m2.1.1"><csymbol cd="ambiguous" id="S3.p2.12.m2.1.1.1.cmml" xref="S3.p2.12.m2.1.1">subscript</csymbol><apply id="S3.p2.12.m2.1.1.2.cmml" xref="S3.p2.12.m2.1.1"><csymbol cd="ambiguous" id="S3.p2.12.m2.1.1.2.1.cmml" xref="S3.p2.12.m2.1.1">superscript</csymbol><ci id="S3.p2.12.m2.1.1.2.2.cmml" xref="S3.p2.12.m2.1.1.2.2">𝑄</ci><ci id="S3.p2.12.m2.1.1.2.3.cmml" xref="S3.p2.12.m2.1.1.2.3">𝑣</ci></apply><ci id="S3.p2.12.m2.1.1.3.cmml" xref="S3.p2.12.m2.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.12.m2.1c">Q^{v}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.12.m2.1d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> is locally identical to <math alttext="P" class="ltx_Math" display="inline" id="S3.p2.13.m3.1"><semantics id="S3.p2.13.m3.1a"><mi id="S3.p2.13.m3.1.1" xref="S3.p2.13.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p2.13.m3.1b"><ci id="S3.p2.13.m3.1.1.cmml" xref="S3.p2.13.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.13.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p2.13.m3.1d">italic_P</annotation></semantics></math>, in the sense that <math alttext="P\cap D=Q^{v}_{P}\cap D" class="ltx_Math" display="inline" id="S3.p2.14.m4.1"><semantics id="S3.p2.14.m4.1a"><mrow id="S3.p2.14.m4.1.1" xref="S3.p2.14.m4.1.1.cmml"><mrow id="S3.p2.14.m4.1.1.2" xref="S3.p2.14.m4.1.1.2.cmml"><mi id="S3.p2.14.m4.1.1.2.2" xref="S3.p2.14.m4.1.1.2.2.cmml">P</mi><mo id="S3.p2.14.m4.1.1.2.1" xref="S3.p2.14.m4.1.1.2.1.cmml">∩</mo><mi id="S3.p2.14.m4.1.1.2.3" xref="S3.p2.14.m4.1.1.2.3.cmml">D</mi></mrow><mo id="S3.p2.14.m4.1.1.1" xref="S3.p2.14.m4.1.1.1.cmml">=</mo><mrow id="S3.p2.14.m4.1.1.3" xref="S3.p2.14.m4.1.1.3.cmml"><msubsup id="S3.p2.14.m4.1.1.3.2" xref="S3.p2.14.m4.1.1.3.2.cmml"><mi id="S3.p2.14.m4.1.1.3.2.2.2" xref="S3.p2.14.m4.1.1.3.2.2.2.cmml">Q</mi><mi id="S3.p2.14.m4.1.1.3.2.3" xref="S3.p2.14.m4.1.1.3.2.3.cmml">P</mi><mi id="S3.p2.14.m4.1.1.3.2.2.3" xref="S3.p2.14.m4.1.1.3.2.2.3.cmml">v</mi></msubsup><mo id="S3.p2.14.m4.1.1.3.1" xref="S3.p2.14.m4.1.1.3.1.cmml">∩</mo><mi id="S3.p2.14.m4.1.1.3.3" xref="S3.p2.14.m4.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.14.m4.1b"><apply id="S3.p2.14.m4.1.1.cmml" xref="S3.p2.14.m4.1.1"><eq id="S3.p2.14.m4.1.1.1.cmml" xref="S3.p2.14.m4.1.1.1"></eq><apply id="S3.p2.14.m4.1.1.2.cmml" xref="S3.p2.14.m4.1.1.2"><intersect id="S3.p2.14.m4.1.1.2.1.cmml" xref="S3.p2.14.m4.1.1.2.1"></intersect><ci id="S3.p2.14.m4.1.1.2.2.cmml" xref="S3.p2.14.m4.1.1.2.2">𝑃</ci><ci id="S3.p2.14.m4.1.1.2.3.cmml" xref="S3.p2.14.m4.1.1.2.3">𝐷</ci></apply><apply id="S3.p2.14.m4.1.1.3.cmml" xref="S3.p2.14.m4.1.1.3"><intersect id="S3.p2.14.m4.1.1.3.1.cmml" xref="S3.p2.14.m4.1.1.3.1"></intersect><apply id="S3.p2.14.m4.1.1.3.2.cmml" xref="S3.p2.14.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.14.m4.1.1.3.2.1.cmml" xref="S3.p2.14.m4.1.1.3.2">subscript</csymbol><apply id="S3.p2.14.m4.1.1.3.2.2.cmml" xref="S3.p2.14.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.14.m4.1.1.3.2.2.1.cmml" xref="S3.p2.14.m4.1.1.3.2">superscript</csymbol><ci id="S3.p2.14.m4.1.1.3.2.2.2.cmml" xref="S3.p2.14.m4.1.1.3.2.2.2">𝑄</ci><ci id="S3.p2.14.m4.1.1.3.2.2.3.cmml" xref="S3.p2.14.m4.1.1.3.2.2.3">𝑣</ci></apply><ci id="S3.p2.14.m4.1.1.3.2.3.cmml" xref="S3.p2.14.m4.1.1.3.2.3">𝑃</ci></apply><ci id="S3.p2.14.m4.1.1.3.3.cmml" xref="S3.p2.14.m4.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.14.m4.1c">P\cap D=Q^{v}_{P}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.14.m4.1d">italic_P ∩ italic_D = italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ∩ italic_D</annotation></semantics></math>. Note that the interior of <math alttext="Q^{v}_{P}" class="ltx_Math" display="inline" id="S3.p2.15.m5.1"><semantics id="S3.p2.15.m5.1a"><msubsup id="S3.p2.15.m5.1.1" xref="S3.p2.15.m5.1.1.cmml"><mi id="S3.p2.15.m5.1.1.2.2" xref="S3.p2.15.m5.1.1.2.2.cmml">Q</mi><mi id="S3.p2.15.m5.1.1.3" xref="S3.p2.15.m5.1.1.3.cmml">P</mi><mi id="S3.p2.15.m5.1.1.2.3" xref="S3.p2.15.m5.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.15.m5.1b"><apply id="S3.p2.15.m5.1.1.cmml" xref="S3.p2.15.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.15.m5.1.1.1.cmml" xref="S3.p2.15.m5.1.1">subscript</csymbol><apply id="S3.p2.15.m5.1.1.2.cmml" xref="S3.p2.15.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.15.m5.1.1.2.1.cmml" xref="S3.p2.15.m5.1.1">superscript</csymbol><ci id="S3.p2.15.m5.1.1.2.2.cmml" xref="S3.p2.15.m5.1.1.2.2">𝑄</ci><ci id="S3.p2.15.m5.1.1.2.3.cmml" xref="S3.p2.15.m5.1.1.2.3">𝑣</ci></apply><ci id="S3.p2.15.m5.1.1.3.cmml" xref="S3.p2.15.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.15.m5.1c">Q^{v}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.15.m5.1d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> is not necessarily connected. For an edge <math alttext="e\in E(P)" class="ltx_Math" display="inline" id="S3.p2.16.m6.1"><semantics id="S3.p2.16.m6.1a"><mrow id="S3.p2.16.m6.1.2" xref="S3.p2.16.m6.1.2.cmml"><mi id="S3.p2.16.m6.1.2.2" xref="S3.p2.16.m6.1.2.2.cmml">e</mi><mo id="S3.p2.16.m6.1.2.1" xref="S3.p2.16.m6.1.2.1.cmml">∈</mo><mrow id="S3.p2.16.m6.1.2.3" xref="S3.p2.16.m6.1.2.3.cmml"><mi id="S3.p2.16.m6.1.2.3.2" xref="S3.p2.16.m6.1.2.3.2.cmml">E</mi><mo id="S3.p2.16.m6.1.2.3.1" xref="S3.p2.16.m6.1.2.3.1.cmml">⁢</mo><mrow id="S3.p2.16.m6.1.2.3.3.2" xref="S3.p2.16.m6.1.2.3.cmml"><mo id="S3.p2.16.m6.1.2.3.3.2.1" stretchy="false" xref="S3.p2.16.m6.1.2.3.cmml">(</mo><mi id="S3.p2.16.m6.1.1" xref="S3.p2.16.m6.1.1.cmml">P</mi><mo id="S3.p2.16.m6.1.2.3.3.2.2" stretchy="false" xref="S3.p2.16.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.16.m6.1b"><apply id="S3.p2.16.m6.1.2.cmml" xref="S3.p2.16.m6.1.2"><in id="S3.p2.16.m6.1.2.1.cmml" xref="S3.p2.16.m6.1.2.1"></in><ci id="S3.p2.16.m6.1.2.2.cmml" xref="S3.p2.16.m6.1.2.2">𝑒</ci><apply id="S3.p2.16.m6.1.2.3.cmml" xref="S3.p2.16.m6.1.2.3"><times id="S3.p2.16.m6.1.2.3.1.cmml" xref="S3.p2.16.m6.1.2.3.1"></times><ci id="S3.p2.16.m6.1.2.3.2.cmml" xref="S3.p2.16.m6.1.2.3.2">𝐸</ci><ci id="S3.p2.16.m6.1.1.cmml" xref="S3.p2.16.m6.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.16.m6.1c">e\in E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.16.m6.1d">italic_e ∈ italic_E ( italic_P )</annotation></semantics></math>, I define the <em class="ltx_emph ltx_font_italic" id="S3.p2.17.1"><math alttext="P" class="ltx_Math" display="inline" id="S3.p2.17.1.m1.1"><semantics id="S3.p2.17.1.m1.1a"><mi id="S3.p2.17.1.m1.1.1" xref="S3.p2.17.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p2.17.1.m1.1b"><ci id="S3.p2.17.1.m1.1.1.cmml" xref="S3.p2.17.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.17.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p2.17.1.m1.1d">italic_P</annotation></semantics></math>-side</em> <math alttext="H^{e}_{P}" class="ltx_Math" display="inline" id="S3.p2.18.m7.1"><semantics id="S3.p2.18.m7.1a"><msubsup id="S3.p2.18.m7.1.1" xref="S3.p2.18.m7.1.1.cmml"><mi id="S3.p2.18.m7.1.1.2.2" xref="S3.p2.18.m7.1.1.2.2.cmml">H</mi><mi id="S3.p2.18.m7.1.1.3" xref="S3.p2.18.m7.1.1.3.cmml">P</mi><mi id="S3.p2.18.m7.1.1.2.3" xref="S3.p2.18.m7.1.1.2.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.18.m7.1b"><apply id="S3.p2.18.m7.1.1.cmml" xref="S3.p2.18.m7.1.1"><csymbol cd="ambiguous" id="S3.p2.18.m7.1.1.1.cmml" xref="S3.p2.18.m7.1.1">subscript</csymbol><apply id="S3.p2.18.m7.1.1.2.cmml" xref="S3.p2.18.m7.1.1"><csymbol cd="ambiguous" id="S3.p2.18.m7.1.1.2.1.cmml" xref="S3.p2.18.m7.1.1">superscript</csymbol><ci id="S3.p2.18.m7.1.1.2.2.cmml" xref="S3.p2.18.m7.1.1.2.2">𝐻</ci><ci id="S3.p2.18.m7.1.1.2.3.cmml" xref="S3.p2.18.m7.1.1.2.3">𝑒</ci></apply><ci id="S3.p2.18.m7.1.1.3.cmml" xref="S3.p2.18.m7.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.18.m7.1c">H^{e}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.18.m7.1d">italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="e" class="ltx_Math" display="inline" id="S3.p2.19.m8.1"><semantics id="S3.p2.19.m8.1a"><mi id="S3.p2.19.m8.1.1" xref="S3.p2.19.m8.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.p2.19.m8.1b"><ci id="S3.p2.19.m8.1.1.cmml" xref="S3.p2.19.m8.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.19.m8.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.p2.19.m8.1d">italic_e</annotation></semantics></math> as a closed half-plane such that for any point <math alttext="x\in e" class="ltx_Math" display="inline" id="S3.p2.20.m9.1"><semantics id="S3.p2.20.m9.1a"><mrow id="S3.p2.20.m9.1.1" xref="S3.p2.20.m9.1.1.cmml"><mi id="S3.p2.20.m9.1.1.2" xref="S3.p2.20.m9.1.1.2.cmml">x</mi><mo id="S3.p2.20.m9.1.1.1" xref="S3.p2.20.m9.1.1.1.cmml">∈</mo><mi id="S3.p2.20.m9.1.1.3" xref="S3.p2.20.m9.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.20.m9.1b"><apply id="S3.p2.20.m9.1.1.cmml" xref="S3.p2.20.m9.1.1"><in id="S3.p2.20.m9.1.1.1.cmml" xref="S3.p2.20.m9.1.1.1"></in><ci id="S3.p2.20.m9.1.1.2.cmml" xref="S3.p2.20.m9.1.1.2">𝑥</ci><ci id="S3.p2.20.m9.1.1.3.cmml" xref="S3.p2.20.m9.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.20.m9.1c">x\in e</annotation><annotation encoding="application/x-llamapun" id="S3.p2.20.m9.1d">italic_x ∈ italic_e</annotation></semantics></math>, that is not a vertex of <math alttext="e" class="ltx_Math" display="inline" id="S3.p2.21.m10.1"><semantics id="S3.p2.21.m10.1a"><mi id="S3.p2.21.m10.1.1" xref="S3.p2.21.m10.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.p2.21.m10.1b"><ci id="S3.p2.21.m10.1.1.cmml" xref="S3.p2.21.m10.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.21.m10.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.p2.21.m10.1d">italic_e</annotation></semantics></math>, there is a small disk <math alttext="D" class="ltx_Math" display="inline" id="S3.p2.22.m11.1"><semantics id="S3.p2.22.m11.1a"><mi id="S3.p2.22.m11.1.1" xref="S3.p2.22.m11.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.p2.22.m11.1b"><ci id="S3.p2.22.m11.1.1.cmml" xref="S3.p2.22.m11.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.22.m11.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.22.m11.1d">italic_D</annotation></semantics></math>, centered at <math alttext="x" class="ltx_Math" display="inline" id="S3.p2.23.m12.1"><semantics id="S3.p2.23.m12.1a"><mi id="S3.p2.23.m12.1.1" xref="S3.p2.23.m12.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.p2.23.m12.1b"><ci id="S3.p2.23.m12.1.1.cmml" xref="S3.p2.23.m12.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.23.m12.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.p2.23.m12.1d">italic_x</annotation></semantics></math>, such that <math alttext="P\cap D=H^{e}_{P}\cap D" class="ltx_Math" display="inline" id="S3.p2.24.m13.1"><semantics id="S3.p2.24.m13.1a"><mrow id="S3.p2.24.m13.1.1" xref="S3.p2.24.m13.1.1.cmml"><mrow id="S3.p2.24.m13.1.1.2" xref="S3.p2.24.m13.1.1.2.cmml"><mi id="S3.p2.24.m13.1.1.2.2" xref="S3.p2.24.m13.1.1.2.2.cmml">P</mi><mo id="S3.p2.24.m13.1.1.2.1" xref="S3.p2.24.m13.1.1.2.1.cmml">∩</mo><mi id="S3.p2.24.m13.1.1.2.3" xref="S3.p2.24.m13.1.1.2.3.cmml">D</mi></mrow><mo id="S3.p2.24.m13.1.1.1" xref="S3.p2.24.m13.1.1.1.cmml">=</mo><mrow id="S3.p2.24.m13.1.1.3" xref="S3.p2.24.m13.1.1.3.cmml"><msubsup id="S3.p2.24.m13.1.1.3.2" xref="S3.p2.24.m13.1.1.3.2.cmml"><mi id="S3.p2.24.m13.1.1.3.2.2.2" xref="S3.p2.24.m13.1.1.3.2.2.2.cmml">H</mi><mi id="S3.p2.24.m13.1.1.3.2.3" xref="S3.p2.24.m13.1.1.3.2.3.cmml">P</mi><mi id="S3.p2.24.m13.1.1.3.2.2.3" xref="S3.p2.24.m13.1.1.3.2.2.3.cmml">e</mi></msubsup><mo id="S3.p2.24.m13.1.1.3.1" xref="S3.p2.24.m13.1.1.3.1.cmml">∩</mo><mi id="S3.p2.24.m13.1.1.3.3" xref="S3.p2.24.m13.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.24.m13.1b"><apply id="S3.p2.24.m13.1.1.cmml" xref="S3.p2.24.m13.1.1"><eq id="S3.p2.24.m13.1.1.1.cmml" xref="S3.p2.24.m13.1.1.1"></eq><apply id="S3.p2.24.m13.1.1.2.cmml" xref="S3.p2.24.m13.1.1.2"><intersect id="S3.p2.24.m13.1.1.2.1.cmml" xref="S3.p2.24.m13.1.1.2.1"></intersect><ci id="S3.p2.24.m13.1.1.2.2.cmml" xref="S3.p2.24.m13.1.1.2.2">𝑃</ci><ci id="S3.p2.24.m13.1.1.2.3.cmml" xref="S3.p2.24.m13.1.1.2.3">𝐷</ci></apply><apply id="S3.p2.24.m13.1.1.3.cmml" xref="S3.p2.24.m13.1.1.3"><intersect id="S3.p2.24.m13.1.1.3.1.cmml" xref="S3.p2.24.m13.1.1.3.1"></intersect><apply id="S3.p2.24.m13.1.1.3.2.cmml" xref="S3.p2.24.m13.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.24.m13.1.1.3.2.1.cmml" xref="S3.p2.24.m13.1.1.3.2">subscript</csymbol><apply id="S3.p2.24.m13.1.1.3.2.2.cmml" xref="S3.p2.24.m13.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.24.m13.1.1.3.2.2.1.cmml" xref="S3.p2.24.m13.1.1.3.2">superscript</csymbol><ci id="S3.p2.24.m13.1.1.3.2.2.2.cmml" xref="S3.p2.24.m13.1.1.3.2.2.2">𝐻</ci><ci id="S3.p2.24.m13.1.1.3.2.2.3.cmml" xref="S3.p2.24.m13.1.1.3.2.2.3">𝑒</ci></apply><ci id="S3.p2.24.m13.1.1.3.2.3.cmml" xref="S3.p2.24.m13.1.1.3.2.3">𝑃</ci></apply><ci id="S3.p2.24.m13.1.1.3.3.cmml" xref="S3.p2.24.m13.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.24.m13.1c">P\cap D=H^{e}_{P}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.p2.24.m13.1d">italic_P ∩ italic_D = italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ∩ italic_D</annotation></semantics></math>. This implies that <math alttext="e\subseteq\partial H^{e}_{P}" class="ltx_Math" display="inline" id="S3.p2.25.m14.1"><semantics id="S3.p2.25.m14.1a"><mrow id="S3.p2.25.m14.1.1" xref="S3.p2.25.m14.1.1.cmml"><mi id="S3.p2.25.m14.1.1.2" xref="S3.p2.25.m14.1.1.2.cmml">e</mi><mo id="S3.p2.25.m14.1.1.1" rspace="0.1389em" xref="S3.p2.25.m14.1.1.1.cmml">⊆</mo><mrow id="S3.p2.25.m14.1.1.3" xref="S3.p2.25.m14.1.1.3.cmml"><mo id="S3.p2.25.m14.1.1.3.1" lspace="0.1389em" rspace="0em" xref="S3.p2.25.m14.1.1.3.1.cmml">∂</mo><msubsup id="S3.p2.25.m14.1.1.3.2" xref="S3.p2.25.m14.1.1.3.2.cmml"><mi id="S3.p2.25.m14.1.1.3.2.2.2" xref="S3.p2.25.m14.1.1.3.2.2.2.cmml">H</mi><mi id="S3.p2.25.m14.1.1.3.2.3" xref="S3.p2.25.m14.1.1.3.2.3.cmml">P</mi><mi id="S3.p2.25.m14.1.1.3.2.2.3" xref="S3.p2.25.m14.1.1.3.2.2.3.cmml">e</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.25.m14.1b"><apply id="S3.p2.25.m14.1.1.cmml" xref="S3.p2.25.m14.1.1"><subset id="S3.p2.25.m14.1.1.1.cmml" xref="S3.p2.25.m14.1.1.1"></subset><ci id="S3.p2.25.m14.1.1.2.cmml" xref="S3.p2.25.m14.1.1.2">𝑒</ci><apply id="S3.p2.25.m14.1.1.3.cmml" xref="S3.p2.25.m14.1.1.3"><partialdiff id="S3.p2.25.m14.1.1.3.1.cmml" xref="S3.p2.25.m14.1.1.3.1"></partialdiff><apply id="S3.p2.25.m14.1.1.3.2.cmml" xref="S3.p2.25.m14.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.25.m14.1.1.3.2.1.cmml" xref="S3.p2.25.m14.1.1.3.2">subscript</csymbol><apply id="S3.p2.25.m14.1.1.3.2.2.cmml" xref="S3.p2.25.m14.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.25.m14.1.1.3.2.2.1.cmml" xref="S3.p2.25.m14.1.1.3.2">superscript</csymbol><ci id="S3.p2.25.m14.1.1.3.2.2.2.cmml" xref="S3.p2.25.m14.1.1.3.2.2.2">𝐻</ci><ci id="S3.p2.25.m14.1.1.3.2.2.3.cmml" xref="S3.p2.25.m14.1.1.3.2.2.3">𝑒</ci></apply><ci id="S3.p2.25.m14.1.1.3.2.3.cmml" xref="S3.p2.25.m14.1.1.3.2.3">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.25.m14.1c">e\subseteq\partial H^{e}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.25.m14.1d">italic_e ⊆ ∂ italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.5">For any subset <math alttext="e\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mrow id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">e</mi><mo id="S3.p3.1.m1.1.1.1" xref="S3.p3.1.m1.1.1.1.cmml">∈</mo><msup id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml"><mi id="S3.p3.1.m1.1.1.3.2" xref="S3.p3.1.m1.1.1.3.2.cmml">ℝ</mi><mn id="S3.p3.1.m1.1.1.3.3" xref="S3.p3.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><in id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1.1"></in><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝑒</ci><apply id="S3.p3.1.m1.1.1.3.cmml" xref="S3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.3.1.cmml" xref="S3.p3.1.m1.1.1.3">superscript</csymbol><ci id="S3.p3.1.m1.1.1.3.2.cmml" xref="S3.p3.1.m1.1.1.3.2">ℝ</ci><cn id="S3.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">e\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_e ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="e\cap P=\emptyset" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mrow id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mrow id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml"><mi id="S3.p3.2.m2.1.1.2.2" xref="S3.p3.2.m2.1.1.2.2.cmml">e</mi><mo id="S3.p3.2.m2.1.1.2.1" xref="S3.p3.2.m2.1.1.2.1.cmml">∩</mo><mi id="S3.p3.2.m2.1.1.2.3" xref="S3.p3.2.m2.1.1.2.3.cmml">P</mi></mrow><mo id="S3.p3.2.m2.1.1.1" xref="S3.p3.2.m2.1.1.1.cmml">=</mo><mi id="S3.p3.2.m2.1.1.3" mathvariant="normal" xref="S3.p3.2.m2.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><eq id="S3.p3.2.m2.1.1.1.cmml" xref="S3.p3.2.m2.1.1.1"></eq><apply id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2"><intersect id="S3.p3.2.m2.1.1.2.1.cmml" xref="S3.p3.2.m2.1.1.2.1"></intersect><ci id="S3.p3.2.m2.1.1.2.2.cmml" xref="S3.p3.2.m2.1.1.2.2">𝑒</ci><ci id="S3.p3.2.m2.1.1.2.3.cmml" xref="S3.p3.2.m2.1.1.2.3">𝑃</ci></apply><emptyset id="S3.p3.2.m2.1.1.3.cmml" xref="S3.p3.2.m2.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">e\cap P=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_e ∩ italic_P = ∅</annotation></semantics></math>, I define its <math alttext="P" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><mi id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><ci id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">italic_P</annotation></semantics></math>-sides <math alttext="Q^{e}_{P}" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><msubsup id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml"><mi id="S3.p3.4.m4.1.1.2.2" xref="S3.p3.4.m4.1.1.2.2.cmml">Q</mi><mi id="S3.p3.4.m4.1.1.3" xref="S3.p3.4.m4.1.1.3.cmml">P</mi><mi id="S3.p3.4.m4.1.1.2.3" xref="S3.p3.4.m4.1.1.2.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><apply id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p3.4.m4.1.1.1.cmml" xref="S3.p3.4.m4.1.1">subscript</csymbol><apply id="S3.p3.4.m4.1.1.2.cmml" xref="S3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p3.4.m4.1.1.2.1.cmml" xref="S3.p3.4.m4.1.1">superscript</csymbol><ci id="S3.p3.4.m4.1.1.2.2.cmml" xref="S3.p3.4.m4.1.1.2.2">𝑄</ci><ci id="S3.p3.4.m4.1.1.2.3.cmml" xref="S3.p3.4.m4.1.1.2.3">𝑒</ci></apply><ci id="S3.p3.4.m4.1.1.3.cmml" xref="S3.p3.4.m4.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">Q^{e}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">italic_Q start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H^{e}_{P}" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><msubsup id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml"><mi id="S3.p3.5.m5.1.1.2.2" xref="S3.p3.5.m5.1.1.2.2.cmml">H</mi><mi id="S3.p3.5.m5.1.1.3" xref="S3.p3.5.m5.1.1.3.cmml">P</mi><mi id="S3.p3.5.m5.1.1.2.3" xref="S3.p3.5.m5.1.1.2.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><apply id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p3.5.m5.1.1.1.cmml" xref="S3.p3.5.m5.1.1">subscript</csymbol><apply id="S3.p3.5.m5.1.1.2.cmml" xref="S3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p3.5.m5.1.1.2.1.cmml" xref="S3.p3.5.m5.1.1">superscript</csymbol><ci id="S3.p3.5.m5.1.1.2.2.cmml" xref="S3.p3.5.m5.1.1.2.2">𝐻</ci><ci id="S3.p3.5.m5.1.1.2.3.cmml" xref="S3.p3.5.m5.1.1.2.3">𝑒</ci></apply><ci id="S3.p3.5.m5.1.1.3.cmml" xref="S3.p3.5.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">H^{e}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> to be empty.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.11">Let <math alttext="f\in\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mrow id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><mi id="S3.p4.1.m1.1.1.2" xref="S3.p4.1.m1.1.1.2.cmml">f</mi><mo id="S3.p4.1.m1.1.1.1" xref="S3.p4.1.m1.1.1.1.cmml">∈</mo><mi id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml">CPA</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><in id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1"></in><ci id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2">𝑓</ci><ci id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3">CPA</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">f\in\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_f ∈ roman_CPA</annotation></semantics></math> be a continuous piecewise affine function, and let <math alttext="f_{P}" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><msub id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><mi id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">f</mi><mi id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.1.1.cmml" xref="S3.p4.2.m2.1.1">subscript</csymbol><ci id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2">𝑓</ci><ci id="S3.p4.2.m2.1.1.3.cmml" xref="S3.p4.2.m2.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> denote the affine component corresponding to the piece <math alttext="P\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mrow id="S3.p4.3.m3.1.2" xref="S3.p4.3.m3.1.2.cmml"><mi id="S3.p4.3.m3.1.2.2" xref="S3.p4.3.m3.1.2.2.cmml">P</mi><mo id="S3.p4.3.m3.1.2.1" xref="S3.p4.3.m3.1.2.1.cmml">∈</mo><mrow id="S3.p4.3.m3.1.2.3" xref="S3.p4.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.3.m3.1.2.3.2" xref="S3.p4.3.m3.1.2.3.2.cmml">𝒫</mi><mo id="S3.p4.3.m3.1.2.3.1" xref="S3.p4.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.p4.3.m3.1.2.3.3.2" xref="S3.p4.3.m3.1.2.3.cmml"><mo id="S3.p4.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.p4.3.m3.1.2.3.cmml">(</mo><mi id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml">f</mi><mo id="S3.p4.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.p4.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.2.cmml" xref="S3.p4.3.m3.1.2"><in id="S3.p4.3.m3.1.2.1.cmml" xref="S3.p4.3.m3.1.2.1"></in><ci id="S3.p4.3.m3.1.2.2.cmml" xref="S3.p4.3.m3.1.2.2">𝑃</ci><apply id="S3.p4.3.m3.1.2.3.cmml" xref="S3.p4.3.m3.1.2.3"><times id="S3.p4.3.m3.1.2.3.1.cmml" xref="S3.p4.3.m3.1.2.3.1"></times><ci id="S3.p4.3.m3.1.2.3.2.cmml" xref="S3.p4.3.m3.1.2.3.2">𝒫</ci><ci id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">P\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">italic_P ∈ caligraphic_P ( italic_f )</annotation></semantics></math>. For a vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.p4.4.m4.1"><semantics id="S3.p4.4.m4.1a"><mi id="S3.p4.4.m4.1.1" xref="S3.p4.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p4.4.m4.1b"><ci id="S3.p4.4.m4.1.1.cmml" xref="S3.p4.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m4.1d">italic_v</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S3.p4.5.m5.1"><semantics id="S3.p4.5.m5.1a"><mi id="S3.p4.5.m5.1.1" xref="S3.p4.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.p4.5.m5.1b"><ci id="S3.p4.5.m5.1.1.cmml" xref="S3.p4.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m5.1d">italic_f</annotation></semantics></math>, I define the function <math alttext="f^{v}" class="ltx_Math" display="inline" id="S3.p4.6.m6.1"><semantics id="S3.p4.6.m6.1a"><msup id="S3.p4.6.m6.1.1" xref="S3.p4.6.m6.1.1.cmml"><mi id="S3.p4.6.m6.1.1.2" xref="S3.p4.6.m6.1.1.2.cmml">f</mi><mi id="S3.p4.6.m6.1.1.3" xref="S3.p4.6.m6.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p4.6.m6.1b"><apply id="S3.p4.6.m6.1.1.cmml" xref="S3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.p4.6.m6.1.1.1.cmml" xref="S3.p4.6.m6.1.1">superscript</csymbol><ci id="S3.p4.6.m6.1.1.2.cmml" xref="S3.p4.6.m6.1.1.2">𝑓</ci><ci id="S3.p4.6.m6.1.1.3.cmml" xref="S3.p4.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m6.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m6.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> piecewise by setting <math alttext="f^{v}|_{Q_{P}^{v}}:=f_{P}" class="ltx_Math" display="inline" id="S3.p4.7.m7.2"><semantics id="S3.p4.7.m7.2a"><mrow id="S3.p4.7.m7.2.2" xref="S3.p4.7.m7.2.2.cmml"><msub id="S3.p4.7.m7.2.2.1.1" xref="S3.p4.7.m7.2.2.1.2.cmml"><mrow id="S3.p4.7.m7.2.2.1.1.1" xref="S3.p4.7.m7.2.2.1.2.cmml"><msup id="S3.p4.7.m7.2.2.1.1.1.1" xref="S3.p4.7.m7.2.2.1.1.1.1.cmml"><mi id="S3.p4.7.m7.2.2.1.1.1.1.2" xref="S3.p4.7.m7.2.2.1.1.1.1.2.cmml">f</mi><mi id="S3.p4.7.m7.2.2.1.1.1.1.3" xref="S3.p4.7.m7.2.2.1.1.1.1.3.cmml">v</mi></msup><mo id="S3.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.p4.7.m7.2.2.1.2.1.cmml">|</mo></mrow><msubsup id="S3.p4.7.m7.1.1.1" xref="S3.p4.7.m7.1.1.1.cmml"><mi id="S3.p4.7.m7.1.1.1.2.2" xref="S3.p4.7.m7.1.1.1.2.2.cmml">Q</mi><mi id="S3.p4.7.m7.1.1.1.2.3" xref="S3.p4.7.m7.1.1.1.2.3.cmml">P</mi><mi id="S3.p4.7.m7.1.1.1.3" xref="S3.p4.7.m7.1.1.1.3.cmml">v</mi></msubsup></msub><mo id="S3.p4.7.m7.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.p4.7.m7.2.2.2.cmml">:=</mo><msub id="S3.p4.7.m7.2.2.3" xref="S3.p4.7.m7.2.2.3.cmml"><mi id="S3.p4.7.m7.2.2.3.2" xref="S3.p4.7.m7.2.2.3.2.cmml">f</mi><mi id="S3.p4.7.m7.2.2.3.3" xref="S3.p4.7.m7.2.2.3.3.cmml">P</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.7.m7.2b"><apply id="S3.p4.7.m7.2.2.cmml" xref="S3.p4.7.m7.2.2"><csymbol cd="latexml" id="S3.p4.7.m7.2.2.2.cmml" xref="S3.p4.7.m7.2.2.2">assign</csymbol><apply id="S3.p4.7.m7.2.2.1.2.cmml" xref="S3.p4.7.m7.2.2.1.1"><csymbol cd="latexml" id="S3.p4.7.m7.2.2.1.2.1.cmml" xref="S3.p4.7.m7.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S3.p4.7.m7.2.2.1.1.1.1.cmml" xref="S3.p4.7.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.p4.7.m7.2.2.1.1.1.1">superscript</csymbol><ci id="S3.p4.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.p4.7.m7.2.2.1.1.1.1.2">𝑓</ci><ci id="S3.p4.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.p4.7.m7.2.2.1.1.1.1.3">𝑣</ci></apply><apply id="S3.p4.7.m7.1.1.1.cmml" xref="S3.p4.7.m7.1.1.1"><csymbol cd="ambiguous" id="S3.p4.7.m7.1.1.1.1.cmml" xref="S3.p4.7.m7.1.1.1">superscript</csymbol><apply id="S3.p4.7.m7.1.1.1.2.cmml" xref="S3.p4.7.m7.1.1.1"><csymbol cd="ambiguous" id="S3.p4.7.m7.1.1.1.2.1.cmml" xref="S3.p4.7.m7.1.1.1">subscript</csymbol><ci id="S3.p4.7.m7.1.1.1.2.2.cmml" xref="S3.p4.7.m7.1.1.1.2.2">𝑄</ci><ci id="S3.p4.7.m7.1.1.1.2.3.cmml" xref="S3.p4.7.m7.1.1.1.2.3">𝑃</ci></apply><ci id="S3.p4.7.m7.1.1.1.3.cmml" xref="S3.p4.7.m7.1.1.1.3">𝑣</ci></apply></apply><apply id="S3.p4.7.m7.2.2.3.cmml" xref="S3.p4.7.m7.2.2.3"><csymbol cd="ambiguous" id="S3.p4.7.m7.2.2.3.1.cmml" xref="S3.p4.7.m7.2.2.3">subscript</csymbol><ci id="S3.p4.7.m7.2.2.3.2.cmml" xref="S3.p4.7.m7.2.2.3.2">𝑓</ci><ci id="S3.p4.7.m7.2.2.3.3.cmml" xref="S3.p4.7.m7.2.2.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m7.2c">f^{v}|_{Q_{P}^{v}}:=f_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m7.2d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="P\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S3.p4.8.m8.1"><semantics id="S3.p4.8.m8.1a"><mrow id="S3.p4.8.m8.1.2" xref="S3.p4.8.m8.1.2.cmml"><mi id="S3.p4.8.m8.1.2.2" xref="S3.p4.8.m8.1.2.2.cmml">P</mi><mo id="S3.p4.8.m8.1.2.1" xref="S3.p4.8.m8.1.2.1.cmml">∈</mo><mrow id="S3.p4.8.m8.1.2.3" xref="S3.p4.8.m8.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.8.m8.1.2.3.2" xref="S3.p4.8.m8.1.2.3.2.cmml">𝒫</mi><mo id="S3.p4.8.m8.1.2.3.1" xref="S3.p4.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="S3.p4.8.m8.1.2.3.3.2" xref="S3.p4.8.m8.1.2.3.cmml"><mo id="S3.p4.8.m8.1.2.3.3.2.1" stretchy="false" xref="S3.p4.8.m8.1.2.3.cmml">(</mo><mi id="S3.p4.8.m8.1.1" xref="S3.p4.8.m8.1.1.cmml">f</mi><mo id="S3.p4.8.m8.1.2.3.3.2.2" stretchy="false" xref="S3.p4.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.8.m8.1b"><apply id="S3.p4.8.m8.1.2.cmml" xref="S3.p4.8.m8.1.2"><in id="S3.p4.8.m8.1.2.1.cmml" xref="S3.p4.8.m8.1.2.1"></in><ci id="S3.p4.8.m8.1.2.2.cmml" xref="S3.p4.8.m8.1.2.2">𝑃</ci><apply id="S3.p4.8.m8.1.2.3.cmml" xref="S3.p4.8.m8.1.2.3"><times id="S3.p4.8.m8.1.2.3.1.cmml" xref="S3.p4.8.m8.1.2.3.1"></times><ci id="S3.p4.8.m8.1.2.3.2.cmml" xref="S3.p4.8.m8.1.2.3.2">𝒫</ci><ci id="S3.p4.8.m8.1.1.cmml" xref="S3.p4.8.m8.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m8.1c">P\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m8.1d">italic_P ∈ caligraphic_P ( italic_f )</annotation></semantics></math>. For <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.p4.9.m9.1"><semantics id="S3.p4.9.m9.1a"><mrow id="S3.p4.9.m9.1.1" xref="S3.p4.9.m9.1.1.cmml"><mi id="S3.p4.9.m9.1.1.2" xref="S3.p4.9.m9.1.1.2.cmml">x</mi><mo id="S3.p4.9.m9.1.1.1" xref="S3.p4.9.m9.1.1.1.cmml">∈</mo><msup id="S3.p4.9.m9.1.1.3" xref="S3.p4.9.m9.1.1.3.cmml"><mi id="S3.p4.9.m9.1.1.3.2" xref="S3.p4.9.m9.1.1.3.2.cmml">ℝ</mi><mn id="S3.p4.9.m9.1.1.3.3" xref="S3.p4.9.m9.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.9.m9.1b"><apply id="S3.p4.9.m9.1.1.cmml" xref="S3.p4.9.m9.1.1"><in id="S3.p4.9.m9.1.1.1.cmml" xref="S3.p4.9.m9.1.1.1"></in><ci id="S3.p4.9.m9.1.1.2.cmml" xref="S3.p4.9.m9.1.1.2">𝑥</ci><apply id="S3.p4.9.m9.1.1.3.cmml" xref="S3.p4.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.p4.9.m9.1.1.3.1.cmml" xref="S3.p4.9.m9.1.1.3">superscript</csymbol><ci id="S3.p4.9.m9.1.1.3.2.cmml" xref="S3.p4.9.m9.1.1.3.2">ℝ</ci><cn id="S3.p4.9.m9.1.1.3.3.cmml" type="integer" xref="S3.p4.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.9.m9.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.9.m9.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> that does not lie on the boundary of any <math alttext="Q_{P}^{v}" class="ltx_Math" display="inline" id="S3.p4.10.m10.1"><semantics id="S3.p4.10.m10.1a"><msubsup id="S3.p4.10.m10.1.1" xref="S3.p4.10.m10.1.1.cmml"><mi id="S3.p4.10.m10.1.1.2.2" xref="S3.p4.10.m10.1.1.2.2.cmml">Q</mi><mi id="S3.p4.10.m10.1.1.2.3" xref="S3.p4.10.m10.1.1.2.3.cmml">P</mi><mi id="S3.p4.10.m10.1.1.3" xref="S3.p4.10.m10.1.1.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p4.10.m10.1b"><apply id="S3.p4.10.m10.1.1.cmml" xref="S3.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p4.10.m10.1.1.1.cmml" xref="S3.p4.10.m10.1.1">superscript</csymbol><apply id="S3.p4.10.m10.1.1.2.cmml" xref="S3.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p4.10.m10.1.1.2.1.cmml" xref="S3.p4.10.m10.1.1">subscript</csymbol><ci id="S3.p4.10.m10.1.1.2.2.cmml" xref="S3.p4.10.m10.1.1.2.2">𝑄</ci><ci id="S3.p4.10.m10.1.1.2.3.cmml" xref="S3.p4.10.m10.1.1.2.3">𝑃</ci></apply><ci id="S3.p4.10.m10.1.1.3.cmml" xref="S3.p4.10.m10.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.10.m10.1c">Q_{P}^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.10.m10.1d">italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math>, we can express <math alttext="f^{v}" class="ltx_Math" display="inline" id="S3.p4.11.m11.1"><semantics id="S3.p4.11.m11.1a"><msup id="S3.p4.11.m11.1.1" xref="S3.p4.11.m11.1.1.cmml"><mi id="S3.p4.11.m11.1.1.2" xref="S3.p4.11.m11.1.1.2.cmml">f</mi><mi id="S3.p4.11.m11.1.1.3" xref="S3.p4.11.m11.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p4.11.m11.1b"><apply id="S3.p4.11.m11.1.1.cmml" xref="S3.p4.11.m11.1.1"><csymbol cd="ambiguous" id="S3.p4.11.m11.1.1.1.cmml" xref="S3.p4.11.m11.1.1">superscript</csymbol><ci id="S3.p4.11.m11.1.1.2.cmml" xref="S3.p4.11.m11.1.1.2">𝑓</ci><ci id="S3.p4.11.m11.1.1.3.cmml" xref="S3.p4.11.m11.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.11.m11.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.11.m11.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <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^{v}(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{Q_{P}^{v}}(x)\cdot f_{P}(x)." class="ltx_Math" display="block" id="S3.E2.m1.5"><semantics id="S3.E2.m1.5a"><mrow id="S3.E2.m1.5.5.1" xref="S3.E2.m1.5.5.1.1.cmml"><mrow id="S3.E2.m1.5.5.1.1" xref="S3.E2.m1.5.5.1.1.cmml"><mrow id="S3.E2.m1.5.5.1.1.2" xref="S3.E2.m1.5.5.1.1.2.cmml"><msup id="S3.E2.m1.5.5.1.1.2.2" xref="S3.E2.m1.5.5.1.1.2.2.cmml"><mi id="S3.E2.m1.5.5.1.1.2.2.2" xref="S3.E2.m1.5.5.1.1.2.2.2.cmml">f</mi><mi id="S3.E2.m1.5.5.1.1.2.2.3" xref="S3.E2.m1.5.5.1.1.2.2.3.cmml">v</mi></msup><mo id="S3.E2.m1.5.5.1.1.2.1" xref="S3.E2.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S3.E2.m1.5.5.1.1.2.3.2" xref="S3.E2.m1.5.5.1.1.2.cmml"><mo id="S3.E2.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S3.E2.m1.5.5.1.1.2.cmml">(</mo><mi id="S3.E2.m1.2.2" xref="S3.E2.m1.2.2.cmml">x</mi><mo id="S3.E2.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S3.E2.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E2.m1.5.5.1.1.1" rspace="0.111em" xref="S3.E2.m1.5.5.1.1.1.cmml">=</mo><mrow id="S3.E2.m1.5.5.1.1.3" xref="S3.E2.m1.5.5.1.1.3.cmml"><munder id="S3.E2.m1.5.5.1.1.3.1" xref="S3.E2.m1.5.5.1.1.3.1.cmml"><mo id="S3.E2.m1.5.5.1.1.3.1.2" movablelimits="false" xref="S3.E2.m1.5.5.1.1.3.1.2.cmml">∑</mo><mrow id="S3.E2.m1.1.1.1" xref="S3.E2.m1.1.1.1.cmml"><mi id="S3.E2.m1.1.1.1.3" xref="S3.E2.m1.1.1.1.3.cmml">P</mi><mo id="S3.E2.m1.1.1.1.2" xref="S3.E2.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E2.m1.1.1.1.4" xref="S3.E2.m1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.1.1.1.4.2" xref="S3.E2.m1.1.1.1.4.2.cmml">𝒫</mi><mo id="S3.E2.m1.1.1.1.4.1" xref="S3.E2.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.E2.m1.1.1.1.4.3.2" xref="S3.E2.m1.1.1.1.4.cmml"><mo id="S3.E2.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.E2.m1.1.1.1.4.cmml">(</mo><mi id="S3.E2.m1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml">f</mi><mo id="S3.E2.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.E2.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E2.m1.5.5.1.1.3.2" xref="S3.E2.m1.5.5.1.1.3.2.cmml"><mrow id="S3.E2.m1.5.5.1.1.3.2.2" xref="S3.E2.m1.5.5.1.1.3.2.2.cmml"><mrow id="S3.E2.m1.5.5.1.1.3.2.2.2" xref="S3.E2.m1.5.5.1.1.3.2.2.2.cmml"><msub id="S3.E2.m1.5.5.1.1.3.2.2.2.2" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.cmml"><mn id="S3.E2.m1.5.5.1.1.3.2.2.2.2.2" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.E2.m1.5.5.1.1.3.2.2.2.2.3" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.cmml"><mi id="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.2.2" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.2.2.cmml">Q</mi><mi id="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.2.3" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.2.3.cmml">P</mi><mi id="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.3" xref="S3.E2.m1.5.5.1.1.3.2.2.2.2.3.3.cmml">v</mi></msubsup></msub><mo id="S3.E2.m1.5.5.1.1.3.2.2.2.1" xref="S3.E2.m1.5.5.1.1.3.2.2.2.1.cmml">⁢</mo><mrow id="S3.E2.m1.5.5.1.1.3.2.2.2.3.2" xref="S3.E2.m1.5.5.1.1.3.2.2.2.cmml"><mo id="S3.E2.m1.5.5.1.1.3.2.2.2.3.2.1" stretchy="false" xref="S3.E2.m1.5.5.1.1.3.2.2.2.cmml">(</mo><mi id="S3.E2.m1.3.3" xref="S3.E2.m1.3.3.cmml">x</mi><mo id="S3.E2.m1.5.5.1.1.3.2.2.2.3.2.2" 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xref="S3.E2.m1.5.5.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.E2.m1.5.5.1.1.3.2.2.3.1.cmml" xref="S3.E2.m1.5.5.1.1.3.2.2.3">subscript</csymbol><ci id="S3.E2.m1.5.5.1.1.3.2.2.3.2.cmml" xref="S3.E2.m1.5.5.1.1.3.2.2.3.2">𝑓</ci><ci id="S3.E2.m1.5.5.1.1.3.2.2.3.3.cmml" xref="S3.E2.m1.5.5.1.1.3.2.2.3.3">𝑃</ci></apply></apply><ci id="S3.E2.m1.4.4.cmml" xref="S3.E2.m1.4.4">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.5c">f^{v}(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{Q_{P}^{v}}(x)\cdot f_{P}(x).</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.5d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ⋅ italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) .</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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.16">Note that, since the interior of <math alttext="Q_{P}^{v}" class="ltx_Math" display="inline" id="S3.p4.12.m1.1"><semantics id="S3.p4.12.m1.1a"><msubsup id="S3.p4.12.m1.1.1" xref="S3.p4.12.m1.1.1.cmml"><mi id="S3.p4.12.m1.1.1.2.2" xref="S3.p4.12.m1.1.1.2.2.cmml">Q</mi><mi id="S3.p4.12.m1.1.1.2.3" xref="S3.p4.12.m1.1.1.2.3.cmml">P</mi><mi id="S3.p4.12.m1.1.1.3" xref="S3.p4.12.m1.1.1.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p4.12.m1.1b"><apply id="S3.p4.12.m1.1.1.cmml" xref="S3.p4.12.m1.1.1"><csymbol cd="ambiguous" id="S3.p4.12.m1.1.1.1.cmml" xref="S3.p4.12.m1.1.1">superscript</csymbol><apply id="S3.p4.12.m1.1.1.2.cmml" xref="S3.p4.12.m1.1.1"><csymbol cd="ambiguous" id="S3.p4.12.m1.1.1.2.1.cmml" xref="S3.p4.12.m1.1.1">subscript</csymbol><ci id="S3.p4.12.m1.1.1.2.2.cmml" xref="S3.p4.12.m1.1.1.2.2">𝑄</ci><ci id="S3.p4.12.m1.1.1.2.3.cmml" xref="S3.p4.12.m1.1.1.2.3">𝑃</ci></apply><ci id="S3.p4.12.m1.1.1.3.cmml" xref="S3.p4.12.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.12.m1.1c">Q_{P}^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.12.m1.1d">italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> is not necessarily connected, the <math alttext="P" class="ltx_Math" display="inline" id="S3.p4.13.m2.1"><semantics id="S3.p4.13.m2.1a"><mi id="S3.p4.13.m2.1.1" xref="S3.p4.13.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p4.13.m2.1b"><ci id="S3.p4.13.m2.1.1.cmml" xref="S3.p4.13.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.13.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p4.13.m2.1d">italic_P</annotation></semantics></math>-sides may not define valid pieces for a <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.p4.14.m3.1"><semantics id="S3.p4.14.m3.1a"><mi id="S3.p4.14.m3.1.1" xref="S3.p4.14.m3.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S3.p4.14.m3.1b"><ci id="S3.p4.14.m3.1.1.cmml" xref="S3.p4.14.m3.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.14.m3.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.14.m3.1d">roman_CPA</annotation></semantics></math> function. However, their connected components do, and thus <math alttext="f^{v}\in\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.p4.15.m4.1"><semantics id="S3.p4.15.m4.1a"><mrow id="S3.p4.15.m4.1.1" xref="S3.p4.15.m4.1.1.cmml"><msup id="S3.p4.15.m4.1.1.2" xref="S3.p4.15.m4.1.1.2.cmml"><mi id="S3.p4.15.m4.1.1.2.2" xref="S3.p4.15.m4.1.1.2.2.cmml">f</mi><mi id="S3.p4.15.m4.1.1.2.3" xref="S3.p4.15.m4.1.1.2.3.cmml">v</mi></msup><mo id="S3.p4.15.m4.1.1.1" xref="S3.p4.15.m4.1.1.1.cmml">∈</mo><mi id="S3.p4.15.m4.1.1.3" xref="S3.p4.15.m4.1.1.3.cmml">CPA</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.15.m4.1b"><apply id="S3.p4.15.m4.1.1.cmml" xref="S3.p4.15.m4.1.1"><in id="S3.p4.15.m4.1.1.1.cmml" xref="S3.p4.15.m4.1.1.1"></in><apply id="S3.p4.15.m4.1.1.2.cmml" xref="S3.p4.15.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p4.15.m4.1.1.2.1.cmml" xref="S3.p4.15.m4.1.1.2">superscript</csymbol><ci id="S3.p4.15.m4.1.1.2.2.cmml" xref="S3.p4.15.m4.1.1.2.2">𝑓</ci><ci id="S3.p4.15.m4.1.1.2.3.cmml" xref="S3.p4.15.m4.1.1.2.3">𝑣</ci></apply><ci id="S3.p4.15.m4.1.1.3.cmml" xref="S3.p4.15.m4.1.1.3">CPA</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.15.m4.1c">f^{v}\in\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.15.m4.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ∈ roman_CPA</annotation></semantics></math>. Moreover, the expression (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E2" title="Equation 2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2</span></a>), together with the local properties of the <math alttext="P" class="ltx_Math" display="inline" id="S3.p4.16.m5.1"><semantics id="S3.p4.16.m5.1a"><mi id="S3.p4.16.m5.1.1" xref="S3.p4.16.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p4.16.m5.1b"><ci id="S3.p4.16.m5.1.1.cmml" xref="S3.p4.16.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.16.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p4.16.m5.1d">italic_P</annotation></semantics></math>-sides, implies 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="f^{v}|_{D}=f|_{D}," class="ltx_Math" display="block" id="S3.Ex7.m1.4"><semantics id="S3.Ex7.m1.4a"><mrow id="S3.Ex7.m1.4.4.1" xref="S3.Ex7.m1.4.4.1.1.cmml"><mrow id="S3.Ex7.m1.4.4.1.1" xref="S3.Ex7.m1.4.4.1.1.cmml"><msub id="S3.Ex7.m1.4.4.1.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.2.cmml"><mrow id="S3.Ex7.m1.4.4.1.1.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.2.cmml"><msup id="S3.Ex7.m1.4.4.1.1.1.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.Ex7.m1.4.4.1.1.1.1.1.1.2" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.2.cmml">f</mi><mi id="S3.Ex7.m1.4.4.1.1.1.1.1.1.3" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.3.cmml">v</mi></msup><mo id="S3.Ex7.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.4.4.1.1.1.2.1.cmml">|</mo></mrow><mi id="S3.Ex7.m1.1.1.1" xref="S3.Ex7.m1.1.1.1.cmml">D</mi></msub><mo id="S3.Ex7.m1.4.4.1.1.2" xref="S3.Ex7.m1.4.4.1.1.2.cmml">=</mo><msub id="S3.Ex7.m1.4.4.1.1.3.2" xref="S3.Ex7.m1.4.4.1.1.3.1.cmml"><mrow id="S3.Ex7.m1.4.4.1.1.3.2.2" xref="S3.Ex7.m1.4.4.1.1.3.1.cmml"><mi id="S3.Ex7.m1.2.2" xref="S3.Ex7.m1.2.2.cmml">f</mi><mo id="S3.Ex7.m1.4.4.1.1.3.2.2.1" stretchy="false" xref="S3.Ex7.m1.4.4.1.1.3.1.1.cmml">|</mo></mrow><mi id="S3.Ex7.m1.3.3.1" xref="S3.Ex7.m1.3.3.1.cmml">D</mi></msub></mrow><mo id="S3.Ex7.m1.4.4.1.2" xref="S3.Ex7.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex7.m1.4b"><apply id="S3.Ex7.m1.4.4.1.1.cmml" xref="S3.Ex7.m1.4.4.1"><eq id="S3.Ex7.m1.4.4.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.2"></eq><apply id="S3.Ex7.m1.4.4.1.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1"><csymbol cd="latexml" id="S3.Ex7.m1.4.4.1.1.1.2.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.2">evaluated-at</csymbol><apply id="S3.Ex7.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex7.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Ex7.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.2">𝑓</ci><ci id="S3.Ex7.m1.4.4.1.1.1.1.1.1.3.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.3">𝑣</ci></apply><ci id="S3.Ex7.m1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1">𝐷</ci></apply><apply id="S3.Ex7.m1.4.4.1.1.3.1.cmml" xref="S3.Ex7.m1.4.4.1.1.3.2"><csymbol cd="latexml" id="S3.Ex7.m1.4.4.1.1.3.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.3.2.2.1">evaluated-at</csymbol><ci id="S3.Ex7.m1.2.2.cmml" xref="S3.Ex7.m1.2.2">𝑓</ci><ci id="S3.Ex7.m1.3.3.1.cmml" xref="S3.Ex7.m1.3.3.1">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.4c">f^{v}|_{D}=f|_{D},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.4d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = italic_f | start_POSTSUBSCRIPT italic_D 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.p4.18">for sufficiently small disks <math alttext="D" class="ltx_Math" display="inline" id="S3.p4.17.m1.1"><semantics id="S3.p4.17.m1.1a"><mi id="S3.p4.17.m1.1.1" xref="S3.p4.17.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.p4.17.m1.1b"><ci id="S3.p4.17.m1.1.1.cmml" xref="S3.p4.17.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.17.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.p4.17.m1.1d">italic_D</annotation></semantics></math> centered at <math alttext="v" class="ltx_Math" display="inline" id="S3.p4.18.m2.1"><semantics id="S3.p4.18.m2.1a"><mi id="S3.p4.18.m2.1.1" xref="S3.p4.18.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.p4.18.m2.1b"><ci id="S3.p4.18.m2.1.1.cmml" xref="S3.p4.18.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.18.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.p4.18.m2.1d">italic_v</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.12">Similarly, I want to describe the local shape of <math alttext="f" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mi id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_f</annotation></semantics></math> at an edge by a global function. Let <math alttext="Q,R\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S3.p5.2.m2.3"><semantics id="S3.p5.2.m2.3a"><mrow id="S3.p5.2.m2.3.4" xref="S3.p5.2.m2.3.4.cmml"><mrow id="S3.p5.2.m2.3.4.2.2" xref="S3.p5.2.m2.3.4.2.1.cmml"><mi id="S3.p5.2.m2.2.2" xref="S3.p5.2.m2.2.2.cmml">Q</mi><mo id="S3.p5.2.m2.3.4.2.2.1" xref="S3.p5.2.m2.3.4.2.1.cmml">,</mo><mi id="S3.p5.2.m2.3.3" xref="S3.p5.2.m2.3.3.cmml">R</mi></mrow><mo id="S3.p5.2.m2.3.4.1" xref="S3.p5.2.m2.3.4.1.cmml">∈</mo><mrow id="S3.p5.2.m2.3.4.3" xref="S3.p5.2.m2.3.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p5.2.m2.3.4.3.2" xref="S3.p5.2.m2.3.4.3.2.cmml">𝒫</mi><mo id="S3.p5.2.m2.3.4.3.1" xref="S3.p5.2.m2.3.4.3.1.cmml">⁢</mo><mrow id="S3.p5.2.m2.3.4.3.3.2" xref="S3.p5.2.m2.3.4.3.cmml"><mo id="S3.p5.2.m2.3.4.3.3.2.1" stretchy="false" xref="S3.p5.2.m2.3.4.3.cmml">(</mo><mi id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml">f</mi><mo id="S3.p5.2.m2.3.4.3.3.2.2" stretchy="false" xref="S3.p5.2.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.3b"><apply id="S3.p5.2.m2.3.4.cmml" xref="S3.p5.2.m2.3.4"><in id="S3.p5.2.m2.3.4.1.cmml" xref="S3.p5.2.m2.3.4.1"></in><list id="S3.p5.2.m2.3.4.2.1.cmml" xref="S3.p5.2.m2.3.4.2.2"><ci id="S3.p5.2.m2.2.2.cmml" xref="S3.p5.2.m2.2.2">𝑄</ci><ci id="S3.p5.2.m2.3.3.cmml" xref="S3.p5.2.m2.3.3">𝑅</ci></list><apply id="S3.p5.2.m2.3.4.3.cmml" xref="S3.p5.2.m2.3.4.3"><times id="S3.p5.2.m2.3.4.3.1.cmml" xref="S3.p5.2.m2.3.4.3.1"></times><ci id="S3.p5.2.m2.3.4.3.2.cmml" xref="S3.p5.2.m2.3.4.3.2">𝒫</ci><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.3c">Q,R\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.3d">italic_Q , italic_R ∈ caligraphic_P ( italic_f )</annotation></semantics></math> be two distinct pieces of <math alttext="f" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mi id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_f</annotation></semantics></math> that share an edge <math alttext="e\in E(f)" class="ltx_Math" display="inline" id="S3.p5.4.m4.1"><semantics id="S3.p5.4.m4.1a"><mrow id="S3.p5.4.m4.1.2" xref="S3.p5.4.m4.1.2.cmml"><mi id="S3.p5.4.m4.1.2.2" xref="S3.p5.4.m4.1.2.2.cmml">e</mi><mo id="S3.p5.4.m4.1.2.1" xref="S3.p5.4.m4.1.2.1.cmml">∈</mo><mrow id="S3.p5.4.m4.1.2.3" xref="S3.p5.4.m4.1.2.3.cmml"><mi id="S3.p5.4.m4.1.2.3.2" xref="S3.p5.4.m4.1.2.3.2.cmml">E</mi><mo id="S3.p5.4.m4.1.2.3.1" xref="S3.p5.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S3.p5.4.m4.1.2.3.3.2" xref="S3.p5.4.m4.1.2.3.cmml"><mo id="S3.p5.4.m4.1.2.3.3.2.1" stretchy="false" xref="S3.p5.4.m4.1.2.3.cmml">(</mo><mi id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml">f</mi><mo id="S3.p5.4.m4.1.2.3.3.2.2" stretchy="false" xref="S3.p5.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.1b"><apply id="S3.p5.4.m4.1.2.cmml" xref="S3.p5.4.m4.1.2"><in id="S3.p5.4.m4.1.2.1.cmml" xref="S3.p5.4.m4.1.2.1"></in><ci id="S3.p5.4.m4.1.2.2.cmml" xref="S3.p5.4.m4.1.2.2">𝑒</ci><apply id="S3.p5.4.m4.1.2.3.cmml" xref="S3.p5.4.m4.1.2.3"><times id="S3.p5.4.m4.1.2.3.1.cmml" xref="S3.p5.4.m4.1.2.3.1"></times><ci id="S3.p5.4.m4.1.2.3.2.cmml" xref="S3.p5.4.m4.1.2.3.2">𝐸</ci><ci id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.1c">e\in E(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.1d">italic_e ∈ italic_E ( italic_f )</annotation></semantics></math>, i.e. <math alttext="e\subseteq Q\cap R" class="ltx_Math" display="inline" id="S3.p5.5.m5.1"><semantics id="S3.p5.5.m5.1a"><mrow id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml"><mi id="S3.p5.5.m5.1.1.2" xref="S3.p5.5.m5.1.1.2.cmml">e</mi><mo id="S3.p5.5.m5.1.1.1" xref="S3.p5.5.m5.1.1.1.cmml">⊆</mo><mrow id="S3.p5.5.m5.1.1.3" xref="S3.p5.5.m5.1.1.3.cmml"><mi id="S3.p5.5.m5.1.1.3.2" xref="S3.p5.5.m5.1.1.3.2.cmml">Q</mi><mo id="S3.p5.5.m5.1.1.3.1" xref="S3.p5.5.m5.1.1.3.1.cmml">∩</mo><mi id="S3.p5.5.m5.1.1.3.3" xref="S3.p5.5.m5.1.1.3.3.cmml">R</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.1b"><apply id="S3.p5.5.m5.1.1.cmml" xref="S3.p5.5.m5.1.1"><subset id="S3.p5.5.m5.1.1.1.cmml" xref="S3.p5.5.m5.1.1.1"></subset><ci id="S3.p5.5.m5.1.1.2.cmml" xref="S3.p5.5.m5.1.1.2">𝑒</ci><apply id="S3.p5.5.m5.1.1.3.cmml" xref="S3.p5.5.m5.1.1.3"><intersect id="S3.p5.5.m5.1.1.3.1.cmml" xref="S3.p5.5.m5.1.1.3.1"></intersect><ci id="S3.p5.5.m5.1.1.3.2.cmml" xref="S3.p5.5.m5.1.1.3.2">𝑄</ci><ci id="S3.p5.5.m5.1.1.3.3.cmml" xref="S3.p5.5.m5.1.1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.1c">e\subseteq Q\cap R</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.1d">italic_e ⊆ italic_Q ∩ italic_R</annotation></semantics></math>. Then, define <math alttext="f^{e}\in\operatorname{CPA}_{2}" class="ltx_Math" display="inline" id="S3.p5.6.m6.1"><semantics id="S3.p5.6.m6.1a"><mrow id="S3.p5.6.m6.1.1" xref="S3.p5.6.m6.1.1.cmml"><msup id="S3.p5.6.m6.1.1.2" xref="S3.p5.6.m6.1.1.2.cmml"><mi id="S3.p5.6.m6.1.1.2.2" xref="S3.p5.6.m6.1.1.2.2.cmml">f</mi><mi id="S3.p5.6.m6.1.1.2.3" xref="S3.p5.6.m6.1.1.2.3.cmml">e</mi></msup><mo id="S3.p5.6.m6.1.1.1" xref="S3.p5.6.m6.1.1.1.cmml">∈</mo><msub id="S3.p5.6.m6.1.1.3" xref="S3.p5.6.m6.1.1.3.cmml"><mi id="S3.p5.6.m6.1.1.3.2" xref="S3.p5.6.m6.1.1.3.2.cmml">CPA</mi><mn id="S3.p5.6.m6.1.1.3.3" xref="S3.p5.6.m6.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m6.1b"><apply id="S3.p5.6.m6.1.1.cmml" xref="S3.p5.6.m6.1.1"><in id="S3.p5.6.m6.1.1.1.cmml" xref="S3.p5.6.m6.1.1.1"></in><apply id="S3.p5.6.m6.1.1.2.cmml" xref="S3.p5.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.p5.6.m6.1.1.2.1.cmml" xref="S3.p5.6.m6.1.1.2">superscript</csymbol><ci id="S3.p5.6.m6.1.1.2.2.cmml" xref="S3.p5.6.m6.1.1.2.2">𝑓</ci><ci id="S3.p5.6.m6.1.1.2.3.cmml" xref="S3.p5.6.m6.1.1.2.3">𝑒</ci></apply><apply id="S3.p5.6.m6.1.1.3.cmml" xref="S3.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.p5.6.m6.1.1.3.1.cmml" xref="S3.p5.6.m6.1.1.3">subscript</csymbol><ci id="S3.p5.6.m6.1.1.3.2.cmml" xref="S3.p5.6.m6.1.1.3.2">CPA</ci><cn id="S3.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S3.p5.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m6.1c">f^{e}\in\operatorname{CPA}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m6.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ∈ roman_CPA start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> on the two pieces <math alttext="H_{Q}^{e}" class="ltx_Math" display="inline" id="S3.p5.7.m7.1"><semantics id="S3.p5.7.m7.1a"><msubsup id="S3.p5.7.m7.1.1" xref="S3.p5.7.m7.1.1.cmml"><mi id="S3.p5.7.m7.1.1.2.2" xref="S3.p5.7.m7.1.1.2.2.cmml">H</mi><mi id="S3.p5.7.m7.1.1.2.3" xref="S3.p5.7.m7.1.1.2.3.cmml">Q</mi><mi id="S3.p5.7.m7.1.1.3" xref="S3.p5.7.m7.1.1.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.7.m7.1b"><apply id="S3.p5.7.m7.1.1.cmml" xref="S3.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p5.7.m7.1.1.1.cmml" xref="S3.p5.7.m7.1.1">superscript</csymbol><apply id="S3.p5.7.m7.1.1.2.cmml" xref="S3.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p5.7.m7.1.1.2.1.cmml" xref="S3.p5.7.m7.1.1">subscript</csymbol><ci id="S3.p5.7.m7.1.1.2.2.cmml" xref="S3.p5.7.m7.1.1.2.2">𝐻</ci><ci id="S3.p5.7.m7.1.1.2.3.cmml" xref="S3.p5.7.m7.1.1.2.3">𝑄</ci></apply><ci id="S3.p5.7.m7.1.1.3.cmml" xref="S3.p5.7.m7.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m7.1c">H_{Q}^{e}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m7.1d">italic_H start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="H_{R}^{e}" class="ltx_Math" display="inline" id="S3.p5.8.m8.1"><semantics id="S3.p5.8.m8.1a"><msubsup id="S3.p5.8.m8.1.1" xref="S3.p5.8.m8.1.1.cmml"><mi id="S3.p5.8.m8.1.1.2.2" xref="S3.p5.8.m8.1.1.2.2.cmml">H</mi><mi id="S3.p5.8.m8.1.1.2.3" xref="S3.p5.8.m8.1.1.2.3.cmml">R</mi><mi id="S3.p5.8.m8.1.1.3" xref="S3.p5.8.m8.1.1.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.8.m8.1b"><apply id="S3.p5.8.m8.1.1.cmml" xref="S3.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p5.8.m8.1.1.1.cmml" xref="S3.p5.8.m8.1.1">superscript</csymbol><apply id="S3.p5.8.m8.1.1.2.cmml" xref="S3.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p5.8.m8.1.1.2.1.cmml" xref="S3.p5.8.m8.1.1">subscript</csymbol><ci id="S3.p5.8.m8.1.1.2.2.cmml" xref="S3.p5.8.m8.1.1.2.2">𝐻</ci><ci id="S3.p5.8.m8.1.1.2.3.cmml" xref="S3.p5.8.m8.1.1.2.3">𝑅</ci></apply><ci id="S3.p5.8.m8.1.1.3.cmml" xref="S3.p5.8.m8.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m8.1c">H_{R}^{e}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m8.1d">italic_H start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="f^{e}|_{H_{Q}^{e}}:=f_{Q}" class="ltx_Math" display="inline" id="S3.p5.9.m9.2"><semantics id="S3.p5.9.m9.2a"><mrow id="S3.p5.9.m9.2.2" xref="S3.p5.9.m9.2.2.cmml"><msub id="S3.p5.9.m9.2.2.1.1" xref="S3.p5.9.m9.2.2.1.2.cmml"><mrow id="S3.p5.9.m9.2.2.1.1.1" xref="S3.p5.9.m9.2.2.1.2.cmml"><msup id="S3.p5.9.m9.2.2.1.1.1.1" xref="S3.p5.9.m9.2.2.1.1.1.1.cmml"><mi id="S3.p5.9.m9.2.2.1.1.1.1.2" xref="S3.p5.9.m9.2.2.1.1.1.1.2.cmml">f</mi><mi id="S3.p5.9.m9.2.2.1.1.1.1.3" xref="S3.p5.9.m9.2.2.1.1.1.1.3.cmml">e</mi></msup><mo id="S3.p5.9.m9.2.2.1.1.1.2" stretchy="false" xref="S3.p5.9.m9.2.2.1.2.1.cmml">|</mo></mrow><msubsup id="S3.p5.9.m9.1.1.1" xref="S3.p5.9.m9.1.1.1.cmml"><mi id="S3.p5.9.m9.1.1.1.2.2" xref="S3.p5.9.m9.1.1.1.2.2.cmml">H</mi><mi id="S3.p5.9.m9.1.1.1.2.3" xref="S3.p5.9.m9.1.1.1.2.3.cmml">Q</mi><mi id="S3.p5.9.m9.1.1.1.3" xref="S3.p5.9.m9.1.1.1.3.cmml">e</mi></msubsup></msub><mo id="S3.p5.9.m9.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.p5.9.m9.2.2.2.cmml">:=</mo><msub id="S3.p5.9.m9.2.2.3" xref="S3.p5.9.m9.2.2.3.cmml"><mi id="S3.p5.9.m9.2.2.3.2" xref="S3.p5.9.m9.2.2.3.2.cmml">f</mi><mi id="S3.p5.9.m9.2.2.3.3" xref="S3.p5.9.m9.2.2.3.3.cmml">Q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.9.m9.2b"><apply id="S3.p5.9.m9.2.2.cmml" xref="S3.p5.9.m9.2.2"><csymbol cd="latexml" id="S3.p5.9.m9.2.2.2.cmml" xref="S3.p5.9.m9.2.2.2">assign</csymbol><apply id="S3.p5.9.m9.2.2.1.2.cmml" xref="S3.p5.9.m9.2.2.1.1"><csymbol cd="latexml" id="S3.p5.9.m9.2.2.1.2.1.cmml" xref="S3.p5.9.m9.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S3.p5.9.m9.2.2.1.1.1.1.cmml" xref="S3.p5.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p5.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.p5.9.m9.2.2.1.1.1.1">superscript</csymbol><ci id="S3.p5.9.m9.2.2.1.1.1.1.2.cmml" xref="S3.p5.9.m9.2.2.1.1.1.1.2">𝑓</ci><ci id="S3.p5.9.m9.2.2.1.1.1.1.3.cmml" xref="S3.p5.9.m9.2.2.1.1.1.1.3">𝑒</ci></apply><apply id="S3.p5.9.m9.1.1.1.cmml" xref="S3.p5.9.m9.1.1.1"><csymbol cd="ambiguous" id="S3.p5.9.m9.1.1.1.1.cmml" xref="S3.p5.9.m9.1.1.1">superscript</csymbol><apply id="S3.p5.9.m9.1.1.1.2.cmml" xref="S3.p5.9.m9.1.1.1"><csymbol cd="ambiguous" id="S3.p5.9.m9.1.1.1.2.1.cmml" xref="S3.p5.9.m9.1.1.1">subscript</csymbol><ci id="S3.p5.9.m9.1.1.1.2.2.cmml" xref="S3.p5.9.m9.1.1.1.2.2">𝐻</ci><ci id="S3.p5.9.m9.1.1.1.2.3.cmml" xref="S3.p5.9.m9.1.1.1.2.3">𝑄</ci></apply><ci id="S3.p5.9.m9.1.1.1.3.cmml" xref="S3.p5.9.m9.1.1.1.3">𝑒</ci></apply></apply><apply id="S3.p5.9.m9.2.2.3.cmml" xref="S3.p5.9.m9.2.2.3"><csymbol cd="ambiguous" id="S3.p5.9.m9.2.2.3.1.cmml" xref="S3.p5.9.m9.2.2.3">subscript</csymbol><ci id="S3.p5.9.m9.2.2.3.2.cmml" xref="S3.p5.9.m9.2.2.3.2">𝑓</ci><ci id="S3.p5.9.m9.2.2.3.3.cmml" xref="S3.p5.9.m9.2.2.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m9.2c">f^{e}|_{H_{Q}^{e}}:=f_{Q}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m9.2d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f^{e}|_{H_{R}^{e}}:=f_{R}" class="ltx_Math" display="inline" id="S3.p5.10.m10.2"><semantics id="S3.p5.10.m10.2a"><mrow id="S3.p5.10.m10.2.2" xref="S3.p5.10.m10.2.2.cmml"><msub id="S3.p5.10.m10.2.2.1.1" xref="S3.p5.10.m10.2.2.1.2.cmml"><mrow id="S3.p5.10.m10.2.2.1.1.1" xref="S3.p5.10.m10.2.2.1.2.cmml"><msup id="S3.p5.10.m10.2.2.1.1.1.1" xref="S3.p5.10.m10.2.2.1.1.1.1.cmml"><mi id="S3.p5.10.m10.2.2.1.1.1.1.2" xref="S3.p5.10.m10.2.2.1.1.1.1.2.cmml">f</mi><mi id="S3.p5.10.m10.2.2.1.1.1.1.3" xref="S3.p5.10.m10.2.2.1.1.1.1.3.cmml">e</mi></msup><mo id="S3.p5.10.m10.2.2.1.1.1.2" stretchy="false" xref="S3.p5.10.m10.2.2.1.2.1.cmml">|</mo></mrow><msubsup id="S3.p5.10.m10.1.1.1" xref="S3.p5.10.m10.1.1.1.cmml"><mi id="S3.p5.10.m10.1.1.1.2.2" xref="S3.p5.10.m10.1.1.1.2.2.cmml">H</mi><mi id="S3.p5.10.m10.1.1.1.2.3" xref="S3.p5.10.m10.1.1.1.2.3.cmml">R</mi><mi id="S3.p5.10.m10.1.1.1.3" xref="S3.p5.10.m10.1.1.1.3.cmml">e</mi></msubsup></msub><mo id="S3.p5.10.m10.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.p5.10.m10.2.2.2.cmml">:=</mo><msub id="S3.p5.10.m10.2.2.3" xref="S3.p5.10.m10.2.2.3.cmml"><mi id="S3.p5.10.m10.2.2.3.2" xref="S3.p5.10.m10.2.2.3.2.cmml">f</mi><mi id="S3.p5.10.m10.2.2.3.3" xref="S3.p5.10.m10.2.2.3.3.cmml">R</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.10.m10.2b"><apply id="S3.p5.10.m10.2.2.cmml" xref="S3.p5.10.m10.2.2"><csymbol cd="latexml" id="S3.p5.10.m10.2.2.2.cmml" xref="S3.p5.10.m10.2.2.2">assign</csymbol><apply id="S3.p5.10.m10.2.2.1.2.cmml" xref="S3.p5.10.m10.2.2.1.1"><csymbol cd="latexml" id="S3.p5.10.m10.2.2.1.2.1.cmml" xref="S3.p5.10.m10.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S3.p5.10.m10.2.2.1.1.1.1.cmml" xref="S3.p5.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m10.2.2.1.1.1.1.1.cmml" xref="S3.p5.10.m10.2.2.1.1.1.1">superscript</csymbol><ci id="S3.p5.10.m10.2.2.1.1.1.1.2.cmml" xref="S3.p5.10.m10.2.2.1.1.1.1.2">𝑓</ci><ci id="S3.p5.10.m10.2.2.1.1.1.1.3.cmml" xref="S3.p5.10.m10.2.2.1.1.1.1.3">𝑒</ci></apply><apply id="S3.p5.10.m10.1.1.1.cmml" xref="S3.p5.10.m10.1.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m10.1.1.1.1.cmml" xref="S3.p5.10.m10.1.1.1">superscript</csymbol><apply id="S3.p5.10.m10.1.1.1.2.cmml" xref="S3.p5.10.m10.1.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m10.1.1.1.2.1.cmml" xref="S3.p5.10.m10.1.1.1">subscript</csymbol><ci id="S3.p5.10.m10.1.1.1.2.2.cmml" xref="S3.p5.10.m10.1.1.1.2.2">𝐻</ci><ci id="S3.p5.10.m10.1.1.1.2.3.cmml" xref="S3.p5.10.m10.1.1.1.2.3">𝑅</ci></apply><ci id="S3.p5.10.m10.1.1.1.3.cmml" xref="S3.p5.10.m10.1.1.1.3">𝑒</ci></apply></apply><apply id="S3.p5.10.m10.2.2.3.cmml" xref="S3.p5.10.m10.2.2.3"><csymbol cd="ambiguous" id="S3.p5.10.m10.2.2.3.1.cmml" xref="S3.p5.10.m10.2.2.3">subscript</csymbol><ci id="S3.p5.10.m10.2.2.3.2.cmml" xref="S3.p5.10.m10.2.2.3.2">𝑓</ci><ci id="S3.p5.10.m10.2.2.3.3.cmml" xref="S3.p5.10.m10.2.2.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m10.2c">f^{e}|_{H_{R}^{e}}:=f_{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m10.2d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := italic_f start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="H_{P}^{e}=\emptyset" class="ltx_Math" display="inline" id="S3.p5.11.m11.1"><semantics id="S3.p5.11.m11.1a"><mrow id="S3.p5.11.m11.1.1" xref="S3.p5.11.m11.1.1.cmml"><msubsup id="S3.p5.11.m11.1.1.2" xref="S3.p5.11.m11.1.1.2.cmml"><mi id="S3.p5.11.m11.1.1.2.2.2" xref="S3.p5.11.m11.1.1.2.2.2.cmml">H</mi><mi id="S3.p5.11.m11.1.1.2.2.3" xref="S3.p5.11.m11.1.1.2.2.3.cmml">P</mi><mi id="S3.p5.11.m11.1.1.2.3" xref="S3.p5.11.m11.1.1.2.3.cmml">e</mi></msubsup><mo id="S3.p5.11.m11.1.1.1" xref="S3.p5.11.m11.1.1.1.cmml">=</mo><mi id="S3.p5.11.m11.1.1.3" mathvariant="normal" xref="S3.p5.11.m11.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.11.m11.1b"><apply id="S3.p5.11.m11.1.1.cmml" xref="S3.p5.11.m11.1.1"><eq id="S3.p5.11.m11.1.1.1.cmml" xref="S3.p5.11.m11.1.1.1"></eq><apply id="S3.p5.11.m11.1.1.2.cmml" xref="S3.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.p5.11.m11.1.1.2.1.cmml" xref="S3.p5.11.m11.1.1.2">superscript</csymbol><apply id="S3.p5.11.m11.1.1.2.2.cmml" xref="S3.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.p5.11.m11.1.1.2.2.1.cmml" xref="S3.p5.11.m11.1.1.2">subscript</csymbol><ci id="S3.p5.11.m11.1.1.2.2.2.cmml" xref="S3.p5.11.m11.1.1.2.2.2">𝐻</ci><ci id="S3.p5.11.m11.1.1.2.2.3.cmml" xref="S3.p5.11.m11.1.1.2.2.3">𝑃</ci></apply><ci id="S3.p5.11.m11.1.1.2.3.cmml" xref="S3.p5.11.m11.1.1.2.3">𝑒</ci></apply><emptyset id="S3.p5.11.m11.1.1.3.cmml" xref="S3.p5.11.m11.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m11.1c">H_{P}^{e}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m11.1d">italic_H start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT = ∅</annotation></semantics></math> for all <math alttext="P\in\mathcal{P}(f)\setminus\{Q,R\}" class="ltx_Math" display="inline" id="S3.p5.12.m12.3"><semantics id="S3.p5.12.m12.3a"><mrow id="S3.p5.12.m12.3.4" xref="S3.p5.12.m12.3.4.cmml"><mi id="S3.p5.12.m12.3.4.2" xref="S3.p5.12.m12.3.4.2.cmml">P</mi><mo id="S3.p5.12.m12.3.4.1" xref="S3.p5.12.m12.3.4.1.cmml">∈</mo><mrow id="S3.p5.12.m12.3.4.3" xref="S3.p5.12.m12.3.4.3.cmml"><mrow id="S3.p5.12.m12.3.4.3.2" xref="S3.p5.12.m12.3.4.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p5.12.m12.3.4.3.2.2" xref="S3.p5.12.m12.3.4.3.2.2.cmml">𝒫</mi><mo id="S3.p5.12.m12.3.4.3.2.1" xref="S3.p5.12.m12.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.p5.12.m12.3.4.3.2.3.2" xref="S3.p5.12.m12.3.4.3.2.cmml"><mo id="S3.p5.12.m12.3.4.3.2.3.2.1" stretchy="false" xref="S3.p5.12.m12.3.4.3.2.cmml">(</mo><mi id="S3.p5.12.m12.1.1" xref="S3.p5.12.m12.1.1.cmml">f</mi><mo id="S3.p5.12.m12.3.4.3.2.3.2.2" stretchy="false" xref="S3.p5.12.m12.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.12.m12.3.4.3.1" xref="S3.p5.12.m12.3.4.3.1.cmml">∖</mo><mrow id="S3.p5.12.m12.3.4.3.3.2" xref="S3.p5.12.m12.3.4.3.3.1.cmml"><mo id="S3.p5.12.m12.3.4.3.3.2.1" stretchy="false" xref="S3.p5.12.m12.3.4.3.3.1.cmml">{</mo><mi id="S3.p5.12.m12.2.2" xref="S3.p5.12.m12.2.2.cmml">Q</mi><mo id="S3.p5.12.m12.3.4.3.3.2.2" xref="S3.p5.12.m12.3.4.3.3.1.cmml">,</mo><mi id="S3.p5.12.m12.3.3" xref="S3.p5.12.m12.3.3.cmml">R</mi><mo id="S3.p5.12.m12.3.4.3.3.2.3" stretchy="false" xref="S3.p5.12.m12.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.12.m12.3b"><apply id="S3.p5.12.m12.3.4.cmml" xref="S3.p5.12.m12.3.4"><in id="S3.p5.12.m12.3.4.1.cmml" xref="S3.p5.12.m12.3.4.1"></in><ci id="S3.p5.12.m12.3.4.2.cmml" xref="S3.p5.12.m12.3.4.2">𝑃</ci><apply id="S3.p5.12.m12.3.4.3.cmml" xref="S3.p5.12.m12.3.4.3"><setdiff id="S3.p5.12.m12.3.4.3.1.cmml" xref="S3.p5.12.m12.3.4.3.1"></setdiff><apply id="S3.p5.12.m12.3.4.3.2.cmml" xref="S3.p5.12.m12.3.4.3.2"><times id="S3.p5.12.m12.3.4.3.2.1.cmml" xref="S3.p5.12.m12.3.4.3.2.1"></times><ci id="S3.p5.12.m12.3.4.3.2.2.cmml" xref="S3.p5.12.m12.3.4.3.2.2">𝒫</ci><ci id="S3.p5.12.m12.1.1.cmml" xref="S3.p5.12.m12.1.1">𝑓</ci></apply><set id="S3.p5.12.m12.3.4.3.3.1.cmml" xref="S3.p5.12.m12.3.4.3.3.2"><ci id="S3.p5.12.m12.2.2.cmml" xref="S3.p5.12.m12.2.2">𝑄</ci><ci id="S3.p5.12.m12.3.3.cmml" xref="S3.p5.12.m12.3.3">𝑅</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.12.m12.3c">P\in\mathcal{P}(f)\setminus\{Q,R\}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.12.m12.3d">italic_P ∈ caligraphic_P ( italic_f ) ∖ { italic_Q , italic_R }</annotation></semantics></math>, it follows that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <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^{e}(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{H_{P}^{e}}(x)\cdot f_{P}(x)\quad% \forall x\not\in\operatorname*{aff}(e)." class="ltx_Math" display="block" id="S3.E3.m1.7"><semantics id="S3.E3.m1.7a"><mrow id="S3.E3.m1.7.7.1"><mrow id="S3.E3.m1.7.7.1.1.2" xref="S3.E3.m1.7.7.1.1.3.cmml"><mrow id="S3.E3.m1.7.7.1.1.1.1" xref="S3.E3.m1.7.7.1.1.1.1.cmml"><mrow id="S3.E3.m1.7.7.1.1.1.1.2" xref="S3.E3.m1.7.7.1.1.1.1.2.cmml"><msup id="S3.E3.m1.7.7.1.1.1.1.2.2" xref="S3.E3.m1.7.7.1.1.1.1.2.2.cmml"><mi id="S3.E3.m1.7.7.1.1.1.1.2.2.2" xref="S3.E3.m1.7.7.1.1.1.1.2.2.2.cmml">f</mi><mi id="S3.E3.m1.7.7.1.1.1.1.2.2.3" xref="S3.E3.m1.7.7.1.1.1.1.2.2.3.cmml">e</mi></msup><mo id="S3.E3.m1.7.7.1.1.1.1.2.1" xref="S3.E3.m1.7.7.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.7.7.1.1.1.1.2.3.2" xref="S3.E3.m1.7.7.1.1.1.1.2.cmml"><mo id="S3.E3.m1.7.7.1.1.1.1.2.3.2.1" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.2.cmml">(</mo><mi id="S3.E3.m1.2.2" xref="S3.E3.m1.2.2.cmml">x</mi><mo id="S3.E3.m1.7.7.1.1.1.1.2.3.2.2" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.7.7.1.1.1.1.1" rspace="0.111em" xref="S3.E3.m1.7.7.1.1.1.1.1.cmml">=</mo><mrow id="S3.E3.m1.7.7.1.1.1.1.3" xref="S3.E3.m1.7.7.1.1.1.1.3.cmml"><munder id="S3.E3.m1.7.7.1.1.1.1.3.1" xref="S3.E3.m1.7.7.1.1.1.1.3.1.cmml"><mo id="S3.E3.m1.7.7.1.1.1.1.3.1.2" movablelimits="false" xref="S3.E3.m1.7.7.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S3.E3.m1.1.1.1" xref="S3.E3.m1.1.1.1.cmml"><mi id="S3.E3.m1.1.1.1.3" xref="S3.E3.m1.1.1.1.3.cmml">P</mi><mo id="S3.E3.m1.1.1.1.2" xref="S3.E3.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E3.m1.1.1.1.4" xref="S3.E3.m1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.1.1.1.4.2" xref="S3.E3.m1.1.1.1.4.2.cmml">𝒫</mi><mo id="S3.E3.m1.1.1.1.4.1" xref="S3.E3.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.E3.m1.1.1.1.4.3.2" xref="S3.E3.m1.1.1.1.4.cmml"><mo id="S3.E3.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.E3.m1.1.1.1.4.cmml">(</mo><mi id="S3.E3.m1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.cmml">f</mi><mo id="S3.E3.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.E3.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E3.m1.7.7.1.1.1.1.3.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.cmml"><mrow id="S3.E3.m1.7.7.1.1.1.1.3.2.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.cmml"><mrow id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.cmml"><msub id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.cmml"><mn id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.cmml"><mi id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.2.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.2.2.cmml">H</mi><mi id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.2.3" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.2.3.cmml">P</mi><mi id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.3" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.2.3.3.cmml">e</mi></msubsup></msub><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.1" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.3.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.cmml"><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.3.2.1" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.cmml">(</mo><mi id="S3.E3.m1.3.3" xref="S3.E3.m1.3.3.cmml">x</mi><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.2.1" rspace="0.222em" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.1.cmml">⋅</mo><msub id="S3.E3.m1.7.7.1.1.1.1.3.2.2.3" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.3.cmml"><mi id="S3.E3.m1.7.7.1.1.1.1.3.2.2.3.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.3.2.cmml">f</mi><mi id="S3.E3.m1.7.7.1.1.1.1.3.2.2.3.3" xref="S3.E3.m1.7.7.1.1.1.1.3.2.2.3.3.cmml">P</mi></msub></mrow><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.1" xref="S3.E3.m1.7.7.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.7.7.1.1.1.1.3.2.3.2" xref="S3.E3.m1.7.7.1.1.1.1.3.2.cmml"><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.3.2.cmml">(</mo><mi id="S3.E3.m1.4.4" xref="S3.E3.m1.4.4.cmml">x</mi><mo id="S3.E3.m1.7.7.1.1.1.1.3.2.3.2.2" stretchy="false" xref="S3.E3.m1.7.7.1.1.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mspace id="S3.E3.m1.7.7.1.1.2.3" width="1.167em" xref="S3.E3.m1.7.7.1.1.3a.cmml"></mspace><mrow id="S3.E3.m1.7.7.1.1.2.2" xref="S3.E3.m1.7.7.1.1.2.2.cmml"><mrow id="S3.E3.m1.7.7.1.1.2.2.2" xref="S3.E3.m1.7.7.1.1.2.2.2.cmml"><mo id="S3.E3.m1.7.7.1.1.2.2.2.1" rspace="0.167em" xref="S3.E3.m1.7.7.1.1.2.2.2.1.cmml">∀</mo><mi id="S3.E3.m1.7.7.1.1.2.2.2.2" xref="S3.E3.m1.7.7.1.1.2.2.2.2.cmml">x</mi></mrow><mo id="S3.E3.m1.7.7.1.1.2.2.1" rspace="0.1389em" xref="S3.E3.m1.7.7.1.1.2.2.1.cmml">∉</mo><mrow id="S3.E3.m1.7.7.1.1.2.2.3.2" xref="S3.E3.m1.7.7.1.1.2.2.3.1.cmml"><mo id="S3.E3.m1.5.5" lspace="0.1389em" rspace="0em" xref="S3.E3.m1.5.5.cmml">aff</mo><mrow id="S3.E3.m1.7.7.1.1.2.2.3.2.1" xref="S3.E3.m1.7.7.1.1.2.2.3.1.cmml"><mo id="S3.E3.m1.7.7.1.1.2.2.3.2.1.1" stretchy="false" xref="S3.E3.m1.7.7.1.1.2.2.3.1.cmml">(</mo><mi id="S3.E3.m1.6.6" xref="S3.E3.m1.6.6.cmml">e</mi><mo id="S3.E3.m1.7.7.1.1.2.2.3.2.1.2" stretchy="false" xref="S3.E3.m1.7.7.1.1.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.E3.m1.7.7.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.7b"><apply id="S3.E3.m1.7.7.1.1.3.cmml" xref="S3.E3.m1.7.7.1.1.2"><csymbol cd="ambiguous" id="S3.E3.m1.7.7.1.1.3a.cmml" xref="S3.E3.m1.7.7.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E3.m1.7.7.1.1.1.1.cmml" xref="S3.E3.m1.7.7.1.1.1.1"><eq id="S3.E3.m1.7.7.1.1.1.1.1.cmml" xref="S3.E3.m1.7.7.1.1.1.1.1"></eq><apply id="S3.E3.m1.7.7.1.1.1.1.2.cmml" xref="S3.E3.m1.7.7.1.1.1.1.2"><times id="S3.E3.m1.7.7.1.1.1.1.2.1.cmml" xref="S3.E3.m1.7.7.1.1.1.1.2.1"></times><apply id="S3.E3.m1.7.7.1.1.1.1.2.2.cmml" xref="S3.E3.m1.7.7.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.E3.m1.7.7.1.1.1.1.2.2.1.cmml" xref="S3.E3.m1.7.7.1.1.1.1.2.2">superscript</csymbol><ci id="S3.E3.m1.7.7.1.1.1.1.2.2.2.cmml" xref="S3.E3.m1.7.7.1.1.1.1.2.2.2">𝑓</ci><ci id="S3.E3.m1.7.7.1.1.1.1.2.2.3.cmml" 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id="S3.E3.m1.4.4.cmml" xref="S3.E3.m1.4.4">𝑥</ci></apply></apply></apply><apply id="S3.E3.m1.7.7.1.1.2.2.cmml" xref="S3.E3.m1.7.7.1.1.2.2"><notin id="S3.E3.m1.7.7.1.1.2.2.1.cmml" xref="S3.E3.m1.7.7.1.1.2.2.1"></notin><apply id="S3.E3.m1.7.7.1.1.2.2.2.cmml" xref="S3.E3.m1.7.7.1.1.2.2.2"><csymbol cd="latexml" id="S3.E3.m1.7.7.1.1.2.2.2.1.cmml" xref="S3.E3.m1.7.7.1.1.2.2.2.1">for-all</csymbol><ci id="S3.E3.m1.7.7.1.1.2.2.2.2.cmml" xref="S3.E3.m1.7.7.1.1.2.2.2.2">𝑥</ci></apply><apply id="S3.E3.m1.7.7.1.1.2.2.3.1.cmml" xref="S3.E3.m1.7.7.1.1.2.2.3.2"><ci id="S3.E3.m1.5.5.cmml" xref="S3.E3.m1.5.5">aff</ci><ci id="S3.E3.m1.6.6.cmml" xref="S3.E3.m1.6.6">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.7c">f^{e}(x)=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{H_{P}^{e}}(x)\cdot f_{P}(x)\quad% \forall x\not\in\operatorname*{aff}(e).</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.7d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ⋅ italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) ∀ italic_x ∉ roman_aff ( italic_e ) .</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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.14">Examples of <math alttext="f^{v}" class="ltx_Math" display="inline" id="S3.p5.13.m1.1"><semantics id="S3.p5.13.m1.1a"><msup id="S3.p5.13.m1.1.1" xref="S3.p5.13.m1.1.1.cmml"><mi id="S3.p5.13.m1.1.1.2" xref="S3.p5.13.m1.1.1.2.cmml">f</mi><mi id="S3.p5.13.m1.1.1.3" xref="S3.p5.13.m1.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p5.13.m1.1b"><apply id="S3.p5.13.m1.1.1.cmml" xref="S3.p5.13.m1.1.1"><csymbol cd="ambiguous" id="S3.p5.13.m1.1.1.1.cmml" xref="S3.p5.13.m1.1.1">superscript</csymbol><ci id="S3.p5.13.m1.1.1.2.cmml" xref="S3.p5.13.m1.1.1.2">𝑓</ci><ci id="S3.p5.13.m1.1.1.3.cmml" xref="S3.p5.13.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.13.m1.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.13.m1.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f^{e}" class="ltx_Math" display="inline" id="S3.p5.14.m2.1"><semantics id="S3.p5.14.m2.1a"><msup id="S3.p5.14.m2.1.1" xref="S3.p5.14.m2.1.1.cmml"><mi id="S3.p5.14.m2.1.1.2" xref="S3.p5.14.m2.1.1.2.cmml">f</mi><mi id="S3.p5.14.m2.1.1.3" xref="S3.p5.14.m2.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p5.14.m2.1b"><apply id="S3.p5.14.m2.1.1.cmml" xref="S3.p5.14.m2.1.1"><csymbol cd="ambiguous" id="S3.p5.14.m2.1.1.1.cmml" xref="S3.p5.14.m2.1.1">superscript</csymbol><ci id="S3.p5.14.m2.1.1.2.cmml" xref="S3.p5.14.m2.1.1.2">𝑓</ci><ci id="S3.p5.14.m2.1.1.3.cmml" xref="S3.p5.14.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.14.m2.1c">f^{e}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.14.m2.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> are shown in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F3" title="Figure 3 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 3</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S3.F3.1"><span class="ltx_text" id="S3.F3.1.1"><foreignobject height="55.6pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="256.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="107" id="S3.F3.1.1.1.g1" src="x6.png" width="493"/></foreignobject></span> <br class="ltx_break"/></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S3.F3.2"><span class="ltx_text" id="S3.F3.2.1"><foreignobject height="41.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="143.8pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="81" id="S3.F3.2.1.1.g1" src="x7.png" width="276"/></foreignobject></span></p> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Two <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.F3.10.m1.1"><semantics id="S3.F3.10.m1.1b"><mi id="S3.F3.10.m1.1.1" xref="S3.F3.10.m1.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S3.F3.10.m1.1c"><ci id="S3.F3.10.m1.1.1.cmml" xref="S3.F3.10.m1.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.10.m1.1d">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.10.m1.1e">roman_CPA</annotation></semantics></math> functions and some of the building blocks that will be used in their decomposition. In the upper example, <math alttext="e" class="ltx_Math" display="inline" id="S3.F3.11.m2.1"><semantics id="S3.F3.11.m2.1b"><mi id="S3.F3.11.m2.1.1" xref="S3.F3.11.m2.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.F3.11.m2.1c"><ci id="S3.F3.11.m2.1.1.cmml" xref="S3.F3.11.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.11.m2.1d">e</annotation><annotation encoding="application/x-llamapun" id="S3.F3.11.m2.1e">italic_e</annotation></semantics></math> is the edge between the pieces <math alttext="P_{6}" class="ltx_Math" display="inline" id="S3.F3.12.m3.1"><semantics id="S3.F3.12.m3.1b"><msub id="S3.F3.12.m3.1.1" xref="S3.F3.12.m3.1.1.cmml"><mi id="S3.F3.12.m3.1.1.2" xref="S3.F3.12.m3.1.1.2.cmml">P</mi><mn id="S3.F3.12.m3.1.1.3" xref="S3.F3.12.m3.1.1.3.cmml">6</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F3.12.m3.1c"><apply id="S3.F3.12.m3.1.1.cmml" xref="S3.F3.12.m3.1.1"><csymbol cd="ambiguous" id="S3.F3.12.m3.1.1.1.cmml" xref="S3.F3.12.m3.1.1">subscript</csymbol><ci id="S3.F3.12.m3.1.1.2.cmml" xref="S3.F3.12.m3.1.1.2">𝑃</ci><cn id="S3.F3.12.m3.1.1.3.cmml" type="integer" xref="S3.F3.12.m3.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.12.m3.1d">P_{6}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.12.m3.1e">italic_P start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{7}" class="ltx_Math" display="inline" id="S3.F3.13.m4.1"><semantics id="S3.F3.13.m4.1b"><msub id="S3.F3.13.m4.1.1" xref="S3.F3.13.m4.1.1.cmml"><mi id="S3.F3.13.m4.1.1.2" xref="S3.F3.13.m4.1.1.2.cmml">P</mi><mn id="S3.F3.13.m4.1.1.3" xref="S3.F3.13.m4.1.1.3.cmml">7</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F3.13.m4.1c"><apply id="S3.F3.13.m4.1.1.cmml" xref="S3.F3.13.m4.1.1"><csymbol cd="ambiguous" id="S3.F3.13.m4.1.1.1.cmml" xref="S3.F3.13.m4.1.1">subscript</csymbol><ci id="S3.F3.13.m4.1.1.2.cmml" xref="S3.F3.13.m4.1.1.2">𝑃</ci><cn id="S3.F3.13.m4.1.1.3.cmml" type="integer" xref="S3.F3.13.m4.1.1.3">7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.13.m4.1d">P_{7}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.13.m4.1e">italic_P start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT</annotation></semantics></math>. The example below shows <math alttext="f^{v_{1}}" class="ltx_Math" display="inline" id="S3.F3.14.m5.1"><semantics id="S3.F3.14.m5.1b"><msup id="S3.F3.14.m5.1.1" xref="S3.F3.14.m5.1.1.cmml"><mi id="S3.F3.14.m5.1.1.2" xref="S3.F3.14.m5.1.1.2.cmml">f</mi><msub id="S3.F3.14.m5.1.1.3" xref="S3.F3.14.m5.1.1.3.cmml"><mi id="S3.F3.14.m5.1.1.3.2" xref="S3.F3.14.m5.1.1.3.2.cmml">v</mi><mn id="S3.F3.14.m5.1.1.3.3" xref="S3.F3.14.m5.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S3.F3.14.m5.1c"><apply id="S3.F3.14.m5.1.1.cmml" xref="S3.F3.14.m5.1.1"><csymbol cd="ambiguous" id="S3.F3.14.m5.1.1.1.cmml" xref="S3.F3.14.m5.1.1">superscript</csymbol><ci id="S3.F3.14.m5.1.1.2.cmml" xref="S3.F3.14.m5.1.1.2">𝑓</ci><apply id="S3.F3.14.m5.1.1.3.cmml" xref="S3.F3.14.m5.1.1.3"><csymbol cd="ambiguous" id="S3.F3.14.m5.1.1.3.1.cmml" xref="S3.F3.14.m5.1.1.3">subscript</csymbol><ci id="S3.F3.14.m5.1.1.3.2.cmml" xref="S3.F3.14.m5.1.1.3.2">𝑣</ci><cn id="S3.F3.14.m5.1.1.3.3.cmml" type="integer" xref="S3.F3.14.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.14.m5.1d">f^{v_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.14.m5.1e">italic_f start_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> for a vertex <math alttext="v_{1}" class="ltx_Math" display="inline" id="S3.F3.15.m6.1"><semantics id="S3.F3.15.m6.1b"><msub id="S3.F3.15.m6.1.1" xref="S3.F3.15.m6.1.1.cmml"><mi id="S3.F3.15.m6.1.1.2" xref="S3.F3.15.m6.1.1.2.cmml">v</mi><mn id="S3.F3.15.m6.1.1.3" xref="S3.F3.15.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F3.15.m6.1c"><apply id="S3.F3.15.m6.1.1.cmml" xref="S3.F3.15.m6.1.1"><csymbol cd="ambiguous" id="S3.F3.15.m6.1.1.1.cmml" xref="S3.F3.15.m6.1.1">subscript</csymbol><ci id="S3.F3.15.m6.1.1.2.cmml" xref="S3.F3.15.m6.1.1.2">𝑣</ci><cn id="S3.F3.15.m6.1.1.3.cmml" type="integer" xref="S3.F3.15.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.15.m6.1d">v_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.15.m6.1e">italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> whose <math alttext="P_{2}" class="ltx_Math" display="inline" id="S3.F3.16.m7.1"><semantics id="S3.F3.16.m7.1b"><msub id="S3.F3.16.m7.1.1" xref="S3.F3.16.m7.1.1.cmml"><mi id="S3.F3.16.m7.1.1.2" xref="S3.F3.16.m7.1.1.2.cmml">P</mi><mn id="S3.F3.16.m7.1.1.3" xref="S3.F3.16.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.F3.16.m7.1c"><apply id="S3.F3.16.m7.1.1.cmml" xref="S3.F3.16.m7.1.1"><csymbol cd="ambiguous" id="S3.F3.16.m7.1.1.1.cmml" xref="S3.F3.16.m7.1.1">subscript</csymbol><ci id="S3.F3.16.m7.1.1.2.cmml" xref="S3.F3.16.m7.1.1.2">𝑃</ci><cn id="S3.F3.16.m7.1.1.3.cmml" type="integer" xref="S3.F3.16.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.16.m7.1d">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.16.m7.1e">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-side has disconnected interior, resulting in two separate pieces.</figcaption> </figure> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.7">For a polygon <math alttext="P" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><mi id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><ci id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">italic_P</annotation></semantics></math>, let <math alttext="n_{h}(P)" class="ltx_Math" display="inline" id="S3.p6.2.m2.1"><semantics id="S3.p6.2.m2.1a"><mrow id="S3.p6.2.m2.1.2" xref="S3.p6.2.m2.1.2.cmml"><msub id="S3.p6.2.m2.1.2.2" xref="S3.p6.2.m2.1.2.2.cmml"><mi id="S3.p6.2.m2.1.2.2.2" xref="S3.p6.2.m2.1.2.2.2.cmml">n</mi><mi id="S3.p6.2.m2.1.2.2.3" xref="S3.p6.2.m2.1.2.2.3.cmml">h</mi></msub><mo id="S3.p6.2.m2.1.2.1" xref="S3.p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.p6.2.m2.1.2.3.2" xref="S3.p6.2.m2.1.2.cmml"><mo id="S3.p6.2.m2.1.2.3.2.1" stretchy="false" xref="S3.p6.2.m2.1.2.cmml">(</mo><mi id="S3.p6.2.m2.1.1" xref="S3.p6.2.m2.1.1.cmml">P</mi><mo id="S3.p6.2.m2.1.2.3.2.2" stretchy="false" xref="S3.p6.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.2.m2.1b"><apply id="S3.p6.2.m2.1.2.cmml" xref="S3.p6.2.m2.1.2"><times id="S3.p6.2.m2.1.2.1.cmml" xref="S3.p6.2.m2.1.2.1"></times><apply id="S3.p6.2.m2.1.2.2.cmml" xref="S3.p6.2.m2.1.2.2"><csymbol cd="ambiguous" id="S3.p6.2.m2.1.2.2.1.cmml" xref="S3.p6.2.m2.1.2.2">subscript</csymbol><ci id="S3.p6.2.m2.1.2.2.2.cmml" xref="S3.p6.2.m2.1.2.2.2">𝑛</ci><ci id="S3.p6.2.m2.1.2.2.3.cmml" xref="S3.p6.2.m2.1.2.2.3">ℎ</ci></apply><ci id="S3.p6.2.m2.1.1.cmml" xref="S3.p6.2.m2.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.1c">n_{h}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> and <math alttext="n_{a}(P)" class="ltx_Math" display="inline" id="S3.p6.3.m3.1"><semantics id="S3.p6.3.m3.1a"><mrow id="S3.p6.3.m3.1.2" xref="S3.p6.3.m3.1.2.cmml"><msub id="S3.p6.3.m3.1.2.2" xref="S3.p6.3.m3.1.2.2.cmml"><mi id="S3.p6.3.m3.1.2.2.2" xref="S3.p6.3.m3.1.2.2.2.cmml">n</mi><mi id="S3.p6.3.m3.1.2.2.3" xref="S3.p6.3.m3.1.2.2.3.cmml">a</mi></msub><mo id="S3.p6.3.m3.1.2.1" xref="S3.p6.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.p6.3.m3.1.2.3.2" xref="S3.p6.3.m3.1.2.cmml"><mo id="S3.p6.3.m3.1.2.3.2.1" stretchy="false" xref="S3.p6.3.m3.1.2.cmml">(</mo><mi id="S3.p6.3.m3.1.1" xref="S3.p6.3.m3.1.1.cmml">P</mi><mo id="S3.p6.3.m3.1.2.3.2.2" stretchy="false" xref="S3.p6.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.1b"><apply id="S3.p6.3.m3.1.2.cmml" xref="S3.p6.3.m3.1.2"><times id="S3.p6.3.m3.1.2.1.cmml" xref="S3.p6.3.m3.1.2.1"></times><apply id="S3.p6.3.m3.1.2.2.cmml" xref="S3.p6.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p6.3.m3.1.2.2.1.cmml" xref="S3.p6.3.m3.1.2.2">subscript</csymbol><ci id="S3.p6.3.m3.1.2.2.2.cmml" xref="S3.p6.3.m3.1.2.2.2">𝑛</ci><ci id="S3.p6.3.m3.1.2.2.3.cmml" xref="S3.p6.3.m3.1.2.2.3">𝑎</ci></apply><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.1c">n_{a}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> denote the number of holes respectively arcs that are boundary components of <math alttext="P" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><mi id="S3.p6.4.m4.1.1" xref="S3.p6.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><ci id="S3.p6.4.m4.1.1.cmml" xref="S3.p6.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">italic_P</annotation></semantics></math>. With <math alttext="{\deg_{P}(v):=\absolutevalue{\{e\in E(P):\,v\in e\}}}" class="ltx_Math" display="inline" id="S3.p6.5.m5.3"><semantics id="S3.p6.5.m5.3a"><mrow id="S3.p6.5.m5.3.3" xref="S3.p6.5.m5.3.3.cmml"><mrow id="S3.p6.5.m5.3.3.1.1" xref="S3.p6.5.m5.3.3.1.2.cmml"><msub id="S3.p6.5.m5.3.3.1.1.1" xref="S3.p6.5.m5.3.3.1.1.1.cmml"><mi id="S3.p6.5.m5.3.3.1.1.1.2" xref="S3.p6.5.m5.3.3.1.1.1.2.cmml">deg</mi><mi id="S3.p6.5.m5.3.3.1.1.1.3" xref="S3.p6.5.m5.3.3.1.1.1.3.cmml">P</mi></msub><mo id="S3.p6.5.m5.3.3.1.1a" xref="S3.p6.5.m5.3.3.1.2.cmml">⁡</mo><mrow id="S3.p6.5.m5.3.3.1.1.2" xref="S3.p6.5.m5.3.3.1.2.cmml"><mo id="S3.p6.5.m5.3.3.1.1.2.1" stretchy="false" xref="S3.p6.5.m5.3.3.1.2.cmml">(</mo><mi id="S3.p6.5.m5.2.2" xref="S3.p6.5.m5.2.2.cmml">v</mi><mo id="S3.p6.5.m5.3.3.1.1.2.2" rspace="0.278em" stretchy="false" xref="S3.p6.5.m5.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S3.p6.5.m5.3.3.2" rspace="0.278em" xref="S3.p6.5.m5.3.3.2.cmml">:=</mo><mrow id="S3.p6.5.m5.1.1.3" xref="S3.p6.5.m5.1.1.2.cmml"><mo id="S3.p6.5.m5.1.1.3.1" xref="S3.p6.5.m5.1.1.2.1.cmml">|</mo><mrow id="S3.p6.5.m5.1.1.1.1.1.3" xref="S3.p6.5.m5.1.1.1.1.1.4.cmml"><mo id="S3.p6.5.m5.1.1.1.1.1.3.3" stretchy="false" xref="S3.p6.5.m5.1.1.1.1.1.4.1.cmml">{</mo><mrow id="S3.p6.5.m5.1.1.1.1.1.2.1" xref="S3.p6.5.m5.1.1.1.1.1.2.1.cmml"><mi id="S3.p6.5.m5.1.1.1.1.1.2.1.2" xref="S3.p6.5.m5.1.1.1.1.1.2.1.2.cmml">e</mi><mo id="S3.p6.5.m5.1.1.1.1.1.2.1.1" xref="S3.p6.5.m5.1.1.1.1.1.2.1.1.cmml">∈</mo><mrow id="S3.p6.5.m5.1.1.1.1.1.2.1.3" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.cmml"><mi id="S3.p6.5.m5.1.1.1.1.1.2.1.3.2" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.2.cmml">E</mi><mo id="S3.p6.5.m5.1.1.1.1.1.2.1.3.1" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.1.cmml">⁢</mo><mrow id="S3.p6.5.m5.1.1.1.1.1.2.1.3.3.2" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.cmml"><mo id="S3.p6.5.m5.1.1.1.1.1.2.1.3.3.2.1" stretchy="false" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.cmml">(</mo><mi id="S3.p6.5.m5.1.1.1.1.1.1" xref="S3.p6.5.m5.1.1.1.1.1.1.cmml">P</mi><mo id="S3.p6.5.m5.1.1.1.1.1.2.1.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p6.5.m5.1.1.1.1.1.3.4" rspace="0.448em" xref="S3.p6.5.m5.1.1.1.1.1.4.1.cmml">:</mo><mrow id="S3.p6.5.m5.1.1.1.1.1.3.2" xref="S3.p6.5.m5.1.1.1.1.1.3.2.cmml"><mi id="S3.p6.5.m5.1.1.1.1.1.3.2.2" xref="S3.p6.5.m5.1.1.1.1.1.3.2.2.cmml">v</mi><mo id="S3.p6.5.m5.1.1.1.1.1.3.2.1" xref="S3.p6.5.m5.1.1.1.1.1.3.2.1.cmml">∈</mo><mi id="S3.p6.5.m5.1.1.1.1.1.3.2.3" xref="S3.p6.5.m5.1.1.1.1.1.3.2.3.cmml">e</mi></mrow><mo id="S3.p6.5.m5.1.1.1.1.1.3.5" stretchy="false" xref="S3.p6.5.m5.1.1.1.1.1.4.1.cmml">}</mo></mrow><mo id="S3.p6.5.m5.1.1.3.2" xref="S3.p6.5.m5.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.3b"><apply id="S3.p6.5.m5.3.3.cmml" xref="S3.p6.5.m5.3.3"><csymbol cd="latexml" id="S3.p6.5.m5.3.3.2.cmml" xref="S3.p6.5.m5.3.3.2">assign</csymbol><apply id="S3.p6.5.m5.3.3.1.2.cmml" xref="S3.p6.5.m5.3.3.1.1"><apply id="S3.p6.5.m5.3.3.1.1.1.cmml" xref="S3.p6.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.p6.5.m5.3.3.1.1.1.1.cmml" xref="S3.p6.5.m5.3.3.1.1.1">subscript</csymbol><csymbol cd="latexml" id="S3.p6.5.m5.3.3.1.1.1.2.cmml" xref="S3.p6.5.m5.3.3.1.1.1.2">degree</csymbol><ci id="S3.p6.5.m5.3.3.1.1.1.3.cmml" xref="S3.p6.5.m5.3.3.1.1.1.3">𝑃</ci></apply><ci id="S3.p6.5.m5.2.2.cmml" xref="S3.p6.5.m5.2.2">𝑣</ci></apply><apply id="S3.p6.5.m5.1.1.2.cmml" xref="S3.p6.5.m5.1.1.3"><abs id="S3.p6.5.m5.1.1.2.1.cmml" xref="S3.p6.5.m5.1.1.3.1"></abs><apply id="S3.p6.5.m5.1.1.1.1.1.4.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3"><csymbol cd="latexml" id="S3.p6.5.m5.1.1.1.1.1.4.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3.3">conditional-set</csymbol><apply id="S3.p6.5.m5.1.1.1.1.1.2.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1"><in id="S3.p6.5.m5.1.1.1.1.1.2.1.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1.1"></in><ci id="S3.p6.5.m5.1.1.1.1.1.2.1.2.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1.2">𝑒</ci><apply id="S3.p6.5.m5.1.1.1.1.1.2.1.3.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3"><times id="S3.p6.5.m5.1.1.1.1.1.2.1.3.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.1"></times><ci id="S3.p6.5.m5.1.1.1.1.1.2.1.3.2.cmml" xref="S3.p6.5.m5.1.1.1.1.1.2.1.3.2">𝐸</ci><ci id="S3.p6.5.m5.1.1.1.1.1.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.1">𝑃</ci></apply></apply><apply id="S3.p6.5.m5.1.1.1.1.1.3.2.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3.2"><in id="S3.p6.5.m5.1.1.1.1.1.3.2.1.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3.2.1"></in><ci id="S3.p6.5.m5.1.1.1.1.1.3.2.2.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3.2.2">𝑣</ci><ci id="S3.p6.5.m5.1.1.1.1.1.3.2.3.cmml" xref="S3.p6.5.m5.1.1.1.1.1.3.2.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.3c">{\deg_{P}(v):=\absolutevalue{\{e\in E(P):\,v\in e\}}}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.3d">roman_deg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_v ) := | start_ARG { italic_e ∈ italic_E ( italic_P ) : italic_v ∈ italic_e } end_ARG |</annotation></semantics></math>, the number <math alttext="d(P):=\sum_{v\in V(P)}(\deg_{P}(v)/2-1)" class="ltx_Math" display="inline" id="S3.p6.6.m6.4"><semantics id="S3.p6.6.m6.4a"><mrow id="S3.p6.6.m6.4.4" xref="S3.p6.6.m6.4.4.cmml"><mrow id="S3.p6.6.m6.4.4.3" xref="S3.p6.6.m6.4.4.3.cmml"><mi id="S3.p6.6.m6.4.4.3.2" xref="S3.p6.6.m6.4.4.3.2.cmml">d</mi><mo id="S3.p6.6.m6.4.4.3.1" xref="S3.p6.6.m6.4.4.3.1.cmml">⁢</mo><mrow id="S3.p6.6.m6.4.4.3.3.2" xref="S3.p6.6.m6.4.4.3.cmml"><mo id="S3.p6.6.m6.4.4.3.3.2.1" stretchy="false" xref="S3.p6.6.m6.4.4.3.cmml">(</mo><mi id="S3.p6.6.m6.2.2" xref="S3.p6.6.m6.2.2.cmml">P</mi><mo id="S3.p6.6.m6.4.4.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p6.6.m6.4.4.3.cmml">)</mo></mrow></mrow><mo id="S3.p6.6.m6.4.4.2" rspace="0.111em" xref="S3.p6.6.m6.4.4.2.cmml">:=</mo><mrow id="S3.p6.6.m6.4.4.1" xref="S3.p6.6.m6.4.4.1.cmml"><msub id="S3.p6.6.m6.4.4.1.2" xref="S3.p6.6.m6.4.4.1.2.cmml"><mo id="S3.p6.6.m6.4.4.1.2.2" rspace="0em" xref="S3.p6.6.m6.4.4.1.2.2.cmml">∑</mo><mrow id="S3.p6.6.m6.1.1.1" xref="S3.p6.6.m6.1.1.1.cmml"><mi id="S3.p6.6.m6.1.1.1.3" xref="S3.p6.6.m6.1.1.1.3.cmml">v</mi><mo id="S3.p6.6.m6.1.1.1.2" xref="S3.p6.6.m6.1.1.1.2.cmml">∈</mo><mrow id="S3.p6.6.m6.1.1.1.4" xref="S3.p6.6.m6.1.1.1.4.cmml"><mi id="S3.p6.6.m6.1.1.1.4.2" xref="S3.p6.6.m6.1.1.1.4.2.cmml">V</mi><mo id="S3.p6.6.m6.1.1.1.4.1" xref="S3.p6.6.m6.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.p6.6.m6.1.1.1.4.3.2" xref="S3.p6.6.m6.1.1.1.4.cmml"><mo id="S3.p6.6.m6.1.1.1.4.3.2.1" stretchy="false" xref="S3.p6.6.m6.1.1.1.4.cmml">(</mo><mi id="S3.p6.6.m6.1.1.1.1" xref="S3.p6.6.m6.1.1.1.1.cmml">P</mi><mo id="S3.p6.6.m6.1.1.1.4.3.2.2" stretchy="false" xref="S3.p6.6.m6.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S3.p6.6.m6.4.4.1.1.1" xref="S3.p6.6.m6.4.4.1.1.1.1.cmml"><mo id="S3.p6.6.m6.4.4.1.1.1.2" stretchy="false" xref="S3.p6.6.m6.4.4.1.1.1.1.cmml">(</mo><mrow id="S3.p6.6.m6.4.4.1.1.1.1" xref="S3.p6.6.m6.4.4.1.1.1.1.cmml"><mrow id="S3.p6.6.m6.4.4.1.1.1.1.1" xref="S3.p6.6.m6.4.4.1.1.1.1.1.cmml"><mrow id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml"><msub id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.cmml"><mi id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.2" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.2.cmml">deg</mi><mi id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.3" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.3.cmml">P</mi></msub><mo id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1a" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.2" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml"><mo id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.2.1" stretchy="false" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml">(</mo><mi id="S3.p6.6.m6.3.3" xref="S3.p6.6.m6.3.3.cmml">v</mi><mo id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.2.2" stretchy="false" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.p6.6.m6.4.4.1.1.1.1.1.2" xref="S3.p6.6.m6.4.4.1.1.1.1.1.2.cmml">/</mo><mn id="S3.p6.6.m6.4.4.1.1.1.1.1.3" xref="S3.p6.6.m6.4.4.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.p6.6.m6.4.4.1.1.1.1.2" xref="S3.p6.6.m6.4.4.1.1.1.1.2.cmml">−</mo><mn id="S3.p6.6.m6.4.4.1.1.1.1.3" xref="S3.p6.6.m6.4.4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p6.6.m6.4.4.1.1.1.3" stretchy="false" xref="S3.p6.6.m6.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.4b"><apply id="S3.p6.6.m6.4.4.cmml" xref="S3.p6.6.m6.4.4"><csymbol cd="latexml" id="S3.p6.6.m6.4.4.2.cmml" xref="S3.p6.6.m6.4.4.2">assign</csymbol><apply id="S3.p6.6.m6.4.4.3.cmml" xref="S3.p6.6.m6.4.4.3"><times id="S3.p6.6.m6.4.4.3.1.cmml" xref="S3.p6.6.m6.4.4.3.1"></times><ci id="S3.p6.6.m6.4.4.3.2.cmml" xref="S3.p6.6.m6.4.4.3.2">𝑑</ci><ci id="S3.p6.6.m6.2.2.cmml" xref="S3.p6.6.m6.2.2">𝑃</ci></apply><apply id="S3.p6.6.m6.4.4.1.cmml" xref="S3.p6.6.m6.4.4.1"><apply id="S3.p6.6.m6.4.4.1.2.cmml" xref="S3.p6.6.m6.4.4.1.2"><csymbol cd="ambiguous" id="S3.p6.6.m6.4.4.1.2.1.cmml" xref="S3.p6.6.m6.4.4.1.2">subscript</csymbol><sum id="S3.p6.6.m6.4.4.1.2.2.cmml" xref="S3.p6.6.m6.4.4.1.2.2"></sum><apply id="S3.p6.6.m6.1.1.1.cmml" xref="S3.p6.6.m6.1.1.1"><in id="S3.p6.6.m6.1.1.1.2.cmml" xref="S3.p6.6.m6.1.1.1.2"></in><ci id="S3.p6.6.m6.1.1.1.3.cmml" xref="S3.p6.6.m6.1.1.1.3">𝑣</ci><apply id="S3.p6.6.m6.1.1.1.4.cmml" xref="S3.p6.6.m6.1.1.1.4"><times id="S3.p6.6.m6.1.1.1.4.1.cmml" xref="S3.p6.6.m6.1.1.1.4.1"></times><ci id="S3.p6.6.m6.1.1.1.4.2.cmml" xref="S3.p6.6.m6.1.1.1.4.2">𝑉</ci><ci id="S3.p6.6.m6.1.1.1.1.cmml" xref="S3.p6.6.m6.1.1.1.1">𝑃</ci></apply></apply></apply><apply id="S3.p6.6.m6.4.4.1.1.1.1.cmml" xref="S3.p6.6.m6.4.4.1.1.1"><minus id="S3.p6.6.m6.4.4.1.1.1.1.2.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.2"></minus><apply id="S3.p6.6.m6.4.4.1.1.1.1.1.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1"><divide id="S3.p6.6.m6.4.4.1.1.1.1.1.2.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.2"></divide><apply id="S3.p6.6.m6.4.4.1.1.1.1.1.1.2.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1"><apply id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><csymbol cd="latexml" id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.2.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.2">degree</csymbol><ci id="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="S3.p6.6.m6.4.4.1.1.1.1.1.1.1.1.3">𝑃</ci></apply><ci id="S3.p6.6.m6.3.3.cmml" xref="S3.p6.6.m6.3.3">𝑣</ci></apply><cn id="S3.p6.6.m6.4.4.1.1.1.1.1.3.cmml" type="integer" xref="S3.p6.6.m6.4.4.1.1.1.1.1.3">2</cn></apply><cn id="S3.p6.6.m6.4.4.1.1.1.1.3.cmml" type="integer" xref="S3.p6.6.m6.4.4.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.4c">d(P):=\sum_{v\in V(P)}(\deg_{P}(v)/2-1)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.4d">italic_d ( italic_P ) := ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT ( roman_deg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_v ) / 2 - 1 )</annotation></semantics></math> is only non-zero if there is a vertex in which at least two boundary components intersect. Finally, define <math alttext="c(P):=1+d(P)-n_{h}(P)-n_{a}(P)" class="ltx_Math" display="inline" id="S3.p6.7.m7.4"><semantics id="S3.p6.7.m7.4a"><mrow id="S3.p6.7.m7.4.5" xref="S3.p6.7.m7.4.5.cmml"><mrow id="S3.p6.7.m7.4.5.2" xref="S3.p6.7.m7.4.5.2.cmml"><mi id="S3.p6.7.m7.4.5.2.2" xref="S3.p6.7.m7.4.5.2.2.cmml">c</mi><mo id="S3.p6.7.m7.4.5.2.1" xref="S3.p6.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="S3.p6.7.m7.4.5.2.3.2" xref="S3.p6.7.m7.4.5.2.cmml"><mo id="S3.p6.7.m7.4.5.2.3.2.1" stretchy="false" xref="S3.p6.7.m7.4.5.2.cmml">(</mo><mi id="S3.p6.7.m7.1.1" xref="S3.p6.7.m7.1.1.cmml">P</mi><mo id="S3.p6.7.m7.4.5.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p6.7.m7.4.5.2.cmml">)</mo></mrow></mrow><mo id="S3.p6.7.m7.4.5.1" rspace="0.278em" xref="S3.p6.7.m7.4.5.1.cmml">:=</mo><mrow id="S3.p6.7.m7.4.5.3" xref="S3.p6.7.m7.4.5.3.cmml"><mrow id="S3.p6.7.m7.4.5.3.2" xref="S3.p6.7.m7.4.5.3.2.cmml"><mn id="S3.p6.7.m7.4.5.3.2.2" xref="S3.p6.7.m7.4.5.3.2.2.cmml">1</mn><mo id="S3.p6.7.m7.4.5.3.2.1" xref="S3.p6.7.m7.4.5.3.2.1.cmml">+</mo><mrow id="S3.p6.7.m7.4.5.3.2.3" xref="S3.p6.7.m7.4.5.3.2.3.cmml"><mi id="S3.p6.7.m7.4.5.3.2.3.2" xref="S3.p6.7.m7.4.5.3.2.3.2.cmml">d</mi><mo id="S3.p6.7.m7.4.5.3.2.3.1" xref="S3.p6.7.m7.4.5.3.2.3.1.cmml">⁢</mo><mrow id="S3.p6.7.m7.4.5.3.2.3.3.2" xref="S3.p6.7.m7.4.5.3.2.3.cmml"><mo id="S3.p6.7.m7.4.5.3.2.3.3.2.1" stretchy="false" xref="S3.p6.7.m7.4.5.3.2.3.cmml">(</mo><mi id="S3.p6.7.m7.2.2" xref="S3.p6.7.m7.2.2.cmml">P</mi><mo id="S3.p6.7.m7.4.5.3.2.3.3.2.2" stretchy="false" xref="S3.p6.7.m7.4.5.3.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p6.7.m7.4.5.3.1" xref="S3.p6.7.m7.4.5.3.1.cmml">−</mo><mrow id="S3.p6.7.m7.4.5.3.3" xref="S3.p6.7.m7.4.5.3.3.cmml"><msub id="S3.p6.7.m7.4.5.3.3.2" xref="S3.p6.7.m7.4.5.3.3.2.cmml"><mi id="S3.p6.7.m7.4.5.3.3.2.2" xref="S3.p6.7.m7.4.5.3.3.2.2.cmml">n</mi><mi id="S3.p6.7.m7.4.5.3.3.2.3" xref="S3.p6.7.m7.4.5.3.3.2.3.cmml">h</mi></msub><mo id="S3.p6.7.m7.4.5.3.3.1" xref="S3.p6.7.m7.4.5.3.3.1.cmml">⁢</mo><mrow id="S3.p6.7.m7.4.5.3.3.3.2" xref="S3.p6.7.m7.4.5.3.3.cmml"><mo id="S3.p6.7.m7.4.5.3.3.3.2.1" stretchy="false" xref="S3.p6.7.m7.4.5.3.3.cmml">(</mo><mi id="S3.p6.7.m7.3.3" xref="S3.p6.7.m7.3.3.cmml">P</mi><mo id="S3.p6.7.m7.4.5.3.3.3.2.2" stretchy="false" xref="S3.p6.7.m7.4.5.3.3.cmml">)</mo></mrow></mrow><mo id="S3.p6.7.m7.4.5.3.1a" xref="S3.p6.7.m7.4.5.3.1.cmml">−</mo><mrow id="S3.p6.7.m7.4.5.3.4" xref="S3.p6.7.m7.4.5.3.4.cmml"><msub id="S3.p6.7.m7.4.5.3.4.2" xref="S3.p6.7.m7.4.5.3.4.2.cmml"><mi id="S3.p6.7.m7.4.5.3.4.2.2" xref="S3.p6.7.m7.4.5.3.4.2.2.cmml">n</mi><mi id="S3.p6.7.m7.4.5.3.4.2.3" xref="S3.p6.7.m7.4.5.3.4.2.3.cmml">a</mi></msub><mo id="S3.p6.7.m7.4.5.3.4.1" xref="S3.p6.7.m7.4.5.3.4.1.cmml">⁢</mo><mrow id="S3.p6.7.m7.4.5.3.4.3.2" xref="S3.p6.7.m7.4.5.3.4.cmml"><mo id="S3.p6.7.m7.4.5.3.4.3.2.1" stretchy="false" xref="S3.p6.7.m7.4.5.3.4.cmml">(</mo><mi id="S3.p6.7.m7.4.4" xref="S3.p6.7.m7.4.4.cmml">P</mi><mo id="S3.p6.7.m7.4.5.3.4.3.2.2" stretchy="false" xref="S3.p6.7.m7.4.5.3.4.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.7.m7.4b"><apply id="S3.p6.7.m7.4.5.cmml" xref="S3.p6.7.m7.4.5"><csymbol cd="latexml" id="S3.p6.7.m7.4.5.1.cmml" xref="S3.p6.7.m7.4.5.1">assign</csymbol><apply id="S3.p6.7.m7.4.5.2.cmml" xref="S3.p6.7.m7.4.5.2"><times id="S3.p6.7.m7.4.5.2.1.cmml" xref="S3.p6.7.m7.4.5.2.1"></times><ci id="S3.p6.7.m7.4.5.2.2.cmml" xref="S3.p6.7.m7.4.5.2.2">𝑐</ci><ci id="S3.p6.7.m7.1.1.cmml" xref="S3.p6.7.m7.1.1">𝑃</ci></apply><apply id="S3.p6.7.m7.4.5.3.cmml" xref="S3.p6.7.m7.4.5.3"><minus id="S3.p6.7.m7.4.5.3.1.cmml" xref="S3.p6.7.m7.4.5.3.1"></minus><apply id="S3.p6.7.m7.4.5.3.2.cmml" xref="S3.p6.7.m7.4.5.3.2"><plus id="S3.p6.7.m7.4.5.3.2.1.cmml" xref="S3.p6.7.m7.4.5.3.2.1"></plus><cn id="S3.p6.7.m7.4.5.3.2.2.cmml" type="integer" xref="S3.p6.7.m7.4.5.3.2.2">1</cn><apply id="S3.p6.7.m7.4.5.3.2.3.cmml" xref="S3.p6.7.m7.4.5.3.2.3"><times id="S3.p6.7.m7.4.5.3.2.3.1.cmml" 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xref="S3.p6.7.m7.4.5.3.4.2">subscript</csymbol><ci id="S3.p6.7.m7.4.5.3.4.2.2.cmml" xref="S3.p6.7.m7.4.5.3.4.2.2">𝑛</ci><ci id="S3.p6.7.m7.4.5.3.4.2.3.cmml" xref="S3.p6.7.m7.4.5.3.4.2.3">𝑎</ci></apply><ci id="S3.p6.7.m7.4.4.cmml" xref="S3.p6.7.m7.4.4">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m7.4c">c(P):=1+d(P)-n_{h}(P)-n_{a}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m7.4d">italic_c ( italic_P ) := 1 + italic_d ( italic_P ) - italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.3">The following lemma describes the function <math alttext="f" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><mi id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><ci id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_f</annotation></semantics></math> using only the building blocks <math alttext="f^{v}" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><msup id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml"><mi id="S3.p7.2.m2.1.1.2" xref="S3.p7.2.m2.1.1.2.cmml">f</mi><mi id="S3.p7.2.m2.1.1.3" xref="S3.p7.2.m2.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><apply id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1">superscript</csymbol><ci id="S3.p7.2.m2.1.1.2.cmml" xref="S3.p7.2.m2.1.1.2">𝑓</ci><ci id="S3.p7.2.m2.1.1.3.cmml" xref="S3.p7.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="f^{e}" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><msup id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml"><mi id="S3.p7.3.m3.1.1.2" xref="S3.p7.3.m3.1.1.2.cmml">f</mi><mi id="S3.p7.3.m3.1.1.3" xref="S3.p7.3.m3.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><apply id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p7.3.m3.1.1.1.cmml" xref="S3.p7.3.m3.1.1">superscript</csymbol><ci id="S3.p7.3.m3.1.1.2.cmml" xref="S3.p7.3.m3.1.1.2">𝑓</ci><ci id="S3.p7.3.m3.1.1.3.cmml" xref="S3.p7.3.m3.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">f^{e}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math>, and a linear combination of its affine components. Note that a similar result can be derived from <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Tran2024MinimalTRF</span>, Prop. 18]</cite>, but only up to an affine function.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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.Thmtheorem1.1.1.1">Lemma 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</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">Let <math alttext="f\in\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">CPA</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.1b"><apply id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1"><in id="S3.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.1"></in><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.2">𝑓</ci><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.3">CPA</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">f\in\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_f ∈ roman_CPA</annotation></semantics></math> be a continuous piecewise affine function in <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.2.m2.1a"><msup id="S3.Thmtheorem1.p1.2.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mn id="S3.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.2.m2.1b"><apply id="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2">ℝ</ci><cn id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Denote the affine component corresponding to <math alttext="P\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem1.p1.3.3.m3.1.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.cmml"><mi id="S3.Thmtheorem1.p1.3.3.m3.1.2.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.2.cmml">P</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.1.2.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem1.p1.3.3.m3.1.2.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.2.cmml">𝒫</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.cmml"><mo id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml">f</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.3.m3.1.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2"><in id="S3.Thmtheorem1.p1.3.3.m3.1.2.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.1"></in><ci id="S3.Thmtheorem1.p1.3.3.m3.1.2.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.2">𝑃</ci><apply id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3"><times id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.1"></times><ci id="S3.Thmtheorem1.p1.3.3.m3.1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.2.3.2">𝒫</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">P\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.1d">italic_P ∈ caligraphic_P ( italic_f )</annotation></semantics></math> by <math alttext="f_{P}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.4.m4.1"><semantics id="S3.Thmtheorem1.p1.4.4.m4.1a"><msub id="S3.Thmtheorem1.p1.4.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">f</mi><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.4.m4.1b"><apply id="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.2">𝑓</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>. Then, it holds that</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <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)=\sum_{v\in V(f)}f^{{v}}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{% subarray}}f^{{e}}(x)-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end{subarray}}f^{{% e}}(x)+\sum_{P\in\mathcal{P}(f)}c(P)f_{P}(x)." class="ltx_Math" display="block" id="S3.E4.m1.11"><semantics id="S3.E4.m1.11a"><mrow id="S3.E4.m1.11.11.1" xref="S3.E4.m1.11.11.1.1.cmml"><mrow id="S3.E4.m1.11.11.1.1" xref="S3.E4.m1.11.11.1.1.cmml"><mrow id="S3.E4.m1.11.11.1.1.2" xref="S3.E4.m1.11.11.1.1.2.cmml"><mi id="S3.E4.m1.11.11.1.1.2.2" xref="S3.E4.m1.11.11.1.1.2.2.cmml">f</mi><mo id="S3.E4.m1.11.11.1.1.2.1" xref="S3.E4.m1.11.11.1.1.2.1.cmml">⁢</mo><mrow id="S3.E4.m1.11.11.1.1.2.3.2" xref="S3.E4.m1.11.11.1.1.2.cmml"><mo id="S3.E4.m1.11.11.1.1.2.3.2.1" stretchy="false" xref="S3.E4.m1.11.11.1.1.2.cmml">(</mo><mi id="S3.E4.m1.5.5" xref="S3.E4.m1.5.5.cmml">x</mi><mo id="S3.E4.m1.11.11.1.1.2.3.2.2" stretchy="false" xref="S3.E4.m1.11.11.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.11.11.1.1.1" rspace="0.111em" xref="S3.E4.m1.11.11.1.1.1.cmml">=</mo><mrow id="S3.E4.m1.11.11.1.1.3" xref="S3.E4.m1.11.11.1.1.3.cmml"><mrow id="S3.E4.m1.11.11.1.1.3.2" xref="S3.E4.m1.11.11.1.1.3.2.cmml"><mrow id="S3.E4.m1.11.11.1.1.3.2.2" xref="S3.E4.m1.11.11.1.1.3.2.2.cmml"><mrow id="S3.E4.m1.11.11.1.1.3.2.2.2" xref="S3.E4.m1.11.11.1.1.3.2.2.2.cmml"><munder id="S3.E4.m1.11.11.1.1.3.2.2.2.1" xref="S3.E4.m1.11.11.1.1.3.2.2.2.1.cmml"><mo id="S3.E4.m1.11.11.1.1.3.2.2.2.1.2" movablelimits="false" xref="S3.E4.m1.11.11.1.1.3.2.2.2.1.2.cmml">∑</mo><mrow id="S3.E4.m1.3.3.1" xref="S3.E4.m1.3.3.1.cmml"><mi id="S3.E4.m1.3.3.1.3" xref="S3.E4.m1.3.3.1.3.cmml">v</mi><mo id="S3.E4.m1.3.3.1.2" xref="S3.E4.m1.3.3.1.2.cmml">∈</mo><mrow id="S3.E4.m1.3.3.1.4" xref="S3.E4.m1.3.3.1.4.cmml"><mi id="S3.E4.m1.3.3.1.4.2" xref="S3.E4.m1.3.3.1.4.2.cmml">V</mi><mo id="S3.E4.m1.3.3.1.4.1" xref="S3.E4.m1.3.3.1.4.1.cmml">⁢</mo><mrow id="S3.E4.m1.3.3.1.4.3.2" xref="S3.E4.m1.3.3.1.4.cmml"><mo id="S3.E4.m1.3.3.1.4.3.2.1" stretchy="false" xref="S3.E4.m1.3.3.1.4.cmml">(</mo><mi id="S3.E4.m1.3.3.1.1" xref="S3.E4.m1.3.3.1.1.cmml">f</mi><mo id="S3.E4.m1.3.3.1.4.3.2.2" stretchy="false" xref="S3.E4.m1.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E4.m1.11.11.1.1.3.2.2.2.2" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.cmml"><msup id="S3.E4.m1.11.11.1.1.3.2.2.2.2.2" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.2.cmml"><mi id="S3.E4.m1.11.11.1.1.3.2.2.2.2.2.2" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.2.2.cmml">f</mi><mi id="S3.E4.m1.11.11.1.1.3.2.2.2.2.2.3" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.2.3.cmml">v</mi></msup><mo id="S3.E4.m1.11.11.1.1.3.2.2.2.2.1" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.E4.m1.11.11.1.1.3.2.2.2.2.3.2" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.cmml"><mo id="S3.E4.m1.11.11.1.1.3.2.2.2.2.3.2.1" stretchy="false" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.cmml">(</mo><mi id="S3.E4.m1.6.6" xref="S3.E4.m1.6.6.cmml">x</mi><mo id="S3.E4.m1.11.11.1.1.3.2.2.2.2.3.2.2" stretchy="false" xref="S3.E4.m1.11.11.1.1.3.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E4.m1.11.11.1.1.3.2.2.1" rspace="0.055em" xref="S3.E4.m1.11.11.1.1.3.2.2.1.cmml">+</mo><mrow id="S3.E4.m1.11.11.1.1.3.2.2.3" xref="S3.E4.m1.11.11.1.1.3.2.2.3.cmml"><munder id="S3.E4.m1.11.11.1.1.3.2.2.3.1" xref="S3.E4.m1.11.11.1.1.3.2.2.3.1.cmml"><mo id="S3.E4.m1.11.11.1.1.3.2.2.3.1.2" movablelimits="false" xref="S3.E4.m1.11.11.1.1.3.2.2.3.1.2.cmml">∑</mo><mtable id="S3.E4.m1.1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.2.cmml"><mtr id="S3.E4.m1.1.1.1.1.1.1a" xref="S3.E4.m1.1.1.1.2.cmml"><mtd id="S3.E4.m1.1.1.1.1.1.1b" xref="S3.E4.m1.1.1.1.2.cmml"><mrow id="S3.E4.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E4.m1.1.1.1.1.1.1.1.1.1.1.3" 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( italic_x ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_f ) end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT italic_c ( italic_P ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) .</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)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.1">For the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a>, the following lemma, proved in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.SS1" title="3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">subsection 3.1</span></a>, is essential. This lemma extends the conic decomposition of convex polytopes (see, e.g., <span class="ltx_ERROR undefined" id="S3.p8.1.1">\citet</span>Shephard1967AngleSums) to my definition of polygons.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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.Thmtheorem2.1.1.1">Lemma 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.3.3">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.1.m1.1a"><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.1b"><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon and let <math alttext="x\in\mathds{R}^{2}" 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><mn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">2</mn></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><cn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> be an arbitrary point in <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.3.m3.1a"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.1b"><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.1d">italic_P</annotation></semantics></math>-general position. Then, it holds that</span></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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{e\in E_{l}(P)}\mathds{1}_{H^{e% }_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P}}(x)+c(P)=\mathds{1}_{P}(x)." class="ltx_Math" display="block" id="S3.Ex8.m1.9"><semantics id="S3.Ex8.m1.9a"><mrow id="S3.Ex8.m1.9.9.1" xref="S3.Ex8.m1.9.9.1.1.cmml"><mrow id="S3.Ex8.m1.9.9.1.1" xref="S3.Ex8.m1.9.9.1.1.cmml"><mrow id="S3.Ex8.m1.9.9.1.1.2" xref="S3.Ex8.m1.9.9.1.1.2.cmml"><mrow id="S3.Ex8.m1.9.9.1.1.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.cmml"><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.cmml"><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.cmml"><munder id="S3.Ex8.m1.9.9.1.1.2.2.2.2.1" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.1.cmml"><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.2.1.2" movablelimits="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.1.2.cmml">∑</mo><mrow id="S3.Ex8.m1.1.1.1" xref="S3.Ex8.m1.1.1.1.cmml"><mi id="S3.Ex8.m1.1.1.1.3" xref="S3.Ex8.m1.1.1.1.3.cmml">v</mi><mo id="S3.Ex8.m1.1.1.1.2" xref="S3.Ex8.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex8.m1.1.1.1.4" xref="S3.Ex8.m1.1.1.1.4.cmml"><mi id="S3.Ex8.m1.1.1.1.4.2" xref="S3.Ex8.m1.1.1.1.4.2.cmml">V</mi><mo id="S3.Ex8.m1.1.1.1.4.1" xref="S3.Ex8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.1.1.1.4.3.2" xref="S3.Ex8.m1.1.1.1.4.cmml"><mo id="S3.Ex8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex8.m1.1.1.1.4.cmml">(</mo><mi id="S3.Ex8.m1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.cmml">P</mi><mo id="S3.Ex8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex8.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.cmml"><msub id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.cmml"><mn id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.cmml"><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.2.2.cmml">Q</mi><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.3.cmml">P</mi><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.2.3.2.3.cmml">v</mi></msubsup></msub><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.1" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.3.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.cmml"><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.cmml">(</mo><mi id="S3.Ex8.m1.4.4" xref="S3.Ex8.m1.4.4.cmml">x</mi><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.1" rspace="0.055em" xref="S3.Ex8.m1.9.9.1.1.2.2.2.1.cmml">+</mo><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.cmml"><munder id="S3.Ex8.m1.9.9.1.1.2.2.2.3.1" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.1.cmml"><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.3.1.2" movablelimits="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.1.2.cmml">∑</mo><mrow id="S3.Ex8.m1.2.2.1" xref="S3.Ex8.m1.2.2.1.cmml"><mi id="S3.Ex8.m1.2.2.1.3" xref="S3.Ex8.m1.2.2.1.3.cmml">e</mi><mo id="S3.Ex8.m1.2.2.1.2" xref="S3.Ex8.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex8.m1.2.2.1.4" xref="S3.Ex8.m1.2.2.1.4.cmml"><msub id="S3.Ex8.m1.2.2.1.4.2" xref="S3.Ex8.m1.2.2.1.4.2.cmml"><mi id="S3.Ex8.m1.2.2.1.4.2.2" xref="S3.Ex8.m1.2.2.1.4.2.2.cmml">E</mi><mi id="S3.Ex8.m1.2.2.1.4.2.3" xref="S3.Ex8.m1.2.2.1.4.2.3.cmml">l</mi></msub><mo id="S3.Ex8.m1.2.2.1.4.1" xref="S3.Ex8.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.4.3.2" xref="S3.Ex8.m1.2.2.1.4.cmml"><mo id="S3.Ex8.m1.2.2.1.4.3.2.1" stretchy="false" xref="S3.Ex8.m1.2.2.1.4.cmml">(</mo><mi id="S3.Ex8.m1.2.2.1.1" xref="S3.Ex8.m1.2.2.1.1.cmml">P</mi><mo id="S3.Ex8.m1.2.2.1.4.3.2.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.cmml"><msub id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.cmml"><mn id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.cmml"><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.2.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.2.2.cmml">H</mi><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.3.cmml">P</mi><mi id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.2.3.2.3.cmml">e</mi></msubsup></msub><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.1" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.3.2" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.cmml"><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.cmml">(</mo><mi id="S3.Ex8.m1.5.5" xref="S3.Ex8.m1.5.5.cmml">x</mi><mo id="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.9.9.1.1.2.2.2.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.Ex8.m1.9.9.1.1.2.2.1" rspace="0.055em" xref="S3.Ex8.m1.9.9.1.1.2.2.1.cmml">−</mo><mrow id="S3.Ex8.m1.9.9.1.1.2.2.3" xref="S3.Ex8.m1.9.9.1.1.2.2.3.cmml"><munder id="S3.Ex8.m1.9.9.1.1.2.2.3.1" xref="S3.Ex8.m1.9.9.1.1.2.2.3.1.cmml"><mo id="S3.Ex8.m1.9.9.1.1.2.2.3.1.2" movablelimits="false" xref="S3.Ex8.m1.9.9.1.1.2.2.3.1.2.cmml">∑</mo><mrow id="S3.Ex8.m1.3.3.1" xref="S3.Ex8.m1.3.3.1.cmml"><mi id="S3.Ex8.m1.3.3.1.3" xref="S3.Ex8.m1.3.3.1.3.cmml">e</mi><mo id="S3.Ex8.m1.3.3.1.2" xref="S3.Ex8.m1.3.3.1.2.cmml">∈</mo><mrow id="S3.Ex8.m1.3.3.1.4" xref="S3.Ex8.m1.3.3.1.4.cmml"><msub id="S3.Ex8.m1.3.3.1.4.2" xref="S3.Ex8.m1.3.3.1.4.2.cmml"><mi 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id="S3.Ex8.m1.7.7.cmml" xref="S3.Ex8.m1.7.7">𝑃</ci></apply></apply><apply id="S3.Ex8.m1.9.9.1.1.3.cmml" xref="S3.Ex8.m1.9.9.1.1.3"><times id="S3.Ex8.m1.9.9.1.1.3.1.cmml" xref="S3.Ex8.m1.9.9.1.1.3.1"></times><apply id="S3.Ex8.m1.9.9.1.1.3.2.cmml" xref="S3.Ex8.m1.9.9.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex8.m1.9.9.1.1.3.2.1.cmml" xref="S3.Ex8.m1.9.9.1.1.3.2">subscript</csymbol><cn id="S3.Ex8.m1.9.9.1.1.3.2.2.cmml" type="integer" xref="S3.Ex8.m1.9.9.1.1.3.2.2">1</cn><ci id="S3.Ex8.m1.9.9.1.1.3.2.3.cmml" xref="S3.Ex8.m1.9.9.1.1.3.2.3">𝑃</ci></apply><ci id="S3.Ex8.m1.8.8.cmml" xref="S3.Ex8.m1.8.8">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.9c">\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{e\in E_{l}(P)}\mathds{1}_{H^{e% }_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P}}(x)+c(P)=\mathds{1}_{P}(x).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.9d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + italic_c ( italic_P ) = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a>.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.5">First, note that it is sufficient to show (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E4" title="Equation 4 ‣ Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4</span></a>) for <math alttext="x\in\mathds{R}^{2}\setminus\bigcup_{e\in E(f)}\operatorname*{aff}(e)" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.3"><semantics id="S3.1.p1.1.m1.3a"><mrow id="S3.1.p1.1.m1.3.4" 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id="S3.1.p1.1.m1.3c">x\in\mathds{R}^{2}\setminus\bigcup_{e\in E(f)}\operatorname*{aff}(e)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.3d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ ⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_f ) end_POSTSUBSCRIPT roman_aff ( italic_e )</annotation></semantics></math>, since both sides describe continuous functions. As <math alttext="\bigcup_{e\in E(f)}\operatorname*{aff}(e)" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.3"><semantics id="S3.1.p1.2.m2.3a"><mrow id="S3.1.p1.2.m2.3.4" xref="S3.1.p1.2.m2.3.4.cmml"><msub id="S3.1.p1.2.m2.3.4.1" xref="S3.1.p1.2.m2.3.4.1.cmml"><mo id="S3.1.p1.2.m2.3.4.1.2" xref="S3.1.p1.2.m2.3.4.1.2.cmml">⋃</mo><mrow id="S3.1.p1.2.m2.1.1.1" xref="S3.1.p1.2.m2.1.1.1.cmml"><mi id="S3.1.p1.2.m2.1.1.1.3" xref="S3.1.p1.2.m2.1.1.1.3.cmml">e</mi><mo id="S3.1.p1.2.m2.1.1.1.2" xref="S3.1.p1.2.m2.1.1.1.2.cmml">∈</mo><mrow id="S3.1.p1.2.m2.1.1.1.4" xref="S3.1.p1.2.m2.1.1.1.4.cmml"><mi id="S3.1.p1.2.m2.1.1.1.4.2" xref="S3.1.p1.2.m2.1.1.1.4.2.cmml">E</mi><mo id="S3.1.p1.2.m2.1.1.1.4.1" xref="S3.1.p1.2.m2.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.1.p1.2.m2.1.1.1.4.3.2" xref="S3.1.p1.2.m2.1.1.1.4.cmml"><mo id="S3.1.p1.2.m2.1.1.1.4.3.2.1" stretchy="false" xref="S3.1.p1.2.m2.1.1.1.4.cmml">(</mo><mi id="S3.1.p1.2.m2.1.1.1.1" xref="S3.1.p1.2.m2.1.1.1.1.cmml">f</mi><mo id="S3.1.p1.2.m2.1.1.1.4.3.2.2" stretchy="false" xref="S3.1.p1.2.m2.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S3.1.p1.2.m2.3.4.2.2" xref="S3.1.p1.2.m2.3.4.2.1.cmml"><mo id="S3.1.p1.2.m2.2.2" lspace="0.167em" rspace="0em" xref="S3.1.p1.2.m2.2.2.cmml">aff</mo><mrow id="S3.1.p1.2.m2.3.4.2.2.1" xref="S3.1.p1.2.m2.3.4.2.1.cmml"><mo id="S3.1.p1.2.m2.3.4.2.2.1.1" stretchy="false" xref="S3.1.p1.2.m2.3.4.2.1.cmml">(</mo><mi id="S3.1.p1.2.m2.3.3" xref="S3.1.p1.2.m2.3.3.cmml">e</mi><mo id="S3.1.p1.2.m2.3.4.2.2.1.2" stretchy="false" xref="S3.1.p1.2.m2.3.4.2.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.3b"><apply id="S3.1.p1.2.m2.3.4.cmml" xref="S3.1.p1.2.m2.3.4"><apply id="S3.1.p1.2.m2.3.4.1.cmml" xref="S3.1.p1.2.m2.3.4.1"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.3.4.1.1.cmml" xref="S3.1.p1.2.m2.3.4.1">subscript</csymbol><union id="S3.1.p1.2.m2.3.4.1.2.cmml" xref="S3.1.p1.2.m2.3.4.1.2"></union><apply id="S3.1.p1.2.m2.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1.1"><in id="S3.1.p1.2.m2.1.1.1.2.cmml" xref="S3.1.p1.2.m2.1.1.1.2"></in><ci id="S3.1.p1.2.m2.1.1.1.3.cmml" xref="S3.1.p1.2.m2.1.1.1.3">𝑒</ci><apply id="S3.1.p1.2.m2.1.1.1.4.cmml" xref="S3.1.p1.2.m2.1.1.1.4"><times id="S3.1.p1.2.m2.1.1.1.4.1.cmml" xref="S3.1.p1.2.m2.1.1.1.4.1"></times><ci id="S3.1.p1.2.m2.1.1.1.4.2.cmml" xref="S3.1.p1.2.m2.1.1.1.4.2">𝐸</ci><ci id="S3.1.p1.2.m2.1.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1.1.1">𝑓</ci></apply></apply></apply><apply id="S3.1.p1.2.m2.3.4.2.1.cmml" xref="S3.1.p1.2.m2.3.4.2.2"><ci id="S3.1.p1.2.m2.2.2.cmml" xref="S3.1.p1.2.m2.2.2">aff</ci><ci id="S3.1.p1.2.m2.3.3.cmml" xref="S3.1.p1.2.m2.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.3c">\bigcup_{e\in E(f)}\operatorname*{aff}(e)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.3d">⋃ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_f ) end_POSTSUBSCRIPT roman_aff ( italic_e )</annotation></semantics></math> contains all edges of the functions <math alttext="f" class="ltx_Math" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><mi id="S3.1.p1.3.m3.1.1" xref="S3.1.p1.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.3.m3.1b"><ci id="S3.1.p1.3.m3.1.1.cmml" xref="S3.1.p1.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">italic_f</annotation></semantics></math>, <math alttext="f^{v}" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><msup id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml"><mi id="S3.1.p1.4.m4.1.1.2" xref="S3.1.p1.4.m4.1.1.2.cmml">f</mi><mi id="S3.1.p1.4.m4.1.1.3" xref="S3.1.p1.4.m4.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><apply id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.1.p1.4.m4.1.1.1.cmml" xref="S3.1.p1.4.m4.1.1">superscript</csymbol><ci id="S3.1.p1.4.m4.1.1.2.cmml" xref="S3.1.p1.4.m4.1.1.2">𝑓</ci><ci id="S3.1.p1.4.m4.1.1.3.cmml" xref="S3.1.p1.4.m4.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="f^{e}" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.1"><semantics id="S3.1.p1.5.m5.1a"><msup id="S3.1.p1.5.m5.1.1" xref="S3.1.p1.5.m5.1.1.cmml"><mi id="S3.1.p1.5.m5.1.1.2" xref="S3.1.p1.5.m5.1.1.2.cmml">f</mi><mi id="S3.1.p1.5.m5.1.1.3" xref="S3.1.p1.5.m5.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.1b"><apply id="S3.1.p1.5.m5.1.1.cmml" xref="S3.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.1.p1.5.m5.1.1.1.cmml" xref="S3.1.p1.5.m5.1.1">superscript</csymbol><ci id="S3.1.p1.5.m5.1.1.2.cmml" xref="S3.1.p1.5.m5.1.1.2">𝑓</ci><ci id="S3.1.p1.5.m5.1.1.3.cmml" xref="S3.1.p1.5.m5.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.1c">f^{e}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math>, we can apply (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S2.E1" title="Equation 1 ‣ 2.3 Continuous Piecewise Affine Functions ‣ 2 Definitions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">1</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E2" title="Equation 2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">2</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E3" title="Equation 3 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3</span></a>) to express these functions in an analytic form. The left hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E4" title="Equation 4 ‣ Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx1"> <tbody id="S3.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 f(x)" class="ltx_Math" display="inline" id="S3.E5.m1.1"><semantics id="S3.E5.m1.1a"><mrow id="S3.E5.m1.1.2" xref="S3.E5.m1.1.2.cmml"><mi id="S3.E5.m1.1.2.2" xref="S3.E5.m1.1.2.2.cmml">f</mi><mo id="S3.E5.m1.1.2.1" xref="S3.E5.m1.1.2.1.cmml">⁢</mo><mrow id="S3.E5.m1.1.2.3.2" xref="S3.E5.m1.1.2.cmml"><mo id="S3.E5.m1.1.2.3.2.1" stretchy="false" xref="S3.E5.m1.1.2.cmml">(</mo><mi id="S3.E5.m1.1.1" xref="S3.E5.m1.1.1.cmml">x</mi><mo id="S3.E5.m1.1.2.3.2.2" stretchy="false" xref="S3.E5.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E5.m1.1b"><apply id="S3.E5.m1.1.2.cmml" xref="S3.E5.m1.1.2"><times id="S3.E5.m1.1.2.1.cmml" xref="S3.E5.m1.1.2.1"></times><ci id="S3.E5.m1.1.2.2.cmml" xref="S3.E5.m1.1.2.2">𝑓</ci><ci id="S3.E5.m1.1.1.cmml" xref="S3.E5.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.1c">\displaystyle f(x)</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.1d">italic_f ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{P\in\mathcal{P}(f)}\mathds{1}_{P}(x)\cdot f_{P}(x)," class="ltx_Math" display="inline" id="S3.E5.m2.4"><semantics id="S3.E5.m2.4a"><mrow id="S3.E5.m2.4.4.1" xref="S3.E5.m2.4.4.1.1.cmml"><mrow id="S3.E5.m2.4.4.1.1" xref="S3.E5.m2.4.4.1.1.cmml"><mi id="S3.E5.m2.4.4.1.1.2" 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</tr></tbody> </table> <p class="ltx_p" id="S3.1.p1.12">and the right hand side becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx2"> <tbody id="S3.Ex9"><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\sum_{v\in V(f)}" class="ltx_Math" display="inline" id="S3.Ex9.m1.1"><semantics id="S3.Ex9.m1.1a"><mstyle displaystyle="true" id="S3.Ex9.m1.1.2" xref="S3.Ex9.m1.1.2.cmml"><munder id="S3.Ex9.m1.1.2a" xref="S3.Ex9.m1.1.2.cmml"><mo id="S3.Ex9.m1.1.2.2" movablelimits="false" xref="S3.Ex9.m1.1.2.2.cmml">∑</mo><mrow id="S3.Ex9.m1.1.1.1" xref="S3.Ex9.m1.1.1.1.cmml"><mi id="S3.Ex9.m1.1.1.1.3" xref="S3.Ex9.m1.1.1.1.3.cmml">v</mi><mo id="S3.Ex9.m1.1.1.1.2" xref="S3.Ex9.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex9.m1.1.1.1.4" xref="S3.Ex9.m1.1.1.1.4.cmml"><mi id="S3.Ex9.m1.1.1.1.4.2" xref="S3.Ex9.m1.1.1.1.4.2.cmml">V</mi><mo 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^{e}(x)+\sum_{P\in\mathcal{P}(f)}c(P)f_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m2.8d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT italic_c ( italic_P ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.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=\sum_{v\in V(f)}\sum_{P\in\mathcal{P}(f)}\mathds{1}_{Q^{v}_{P}}(% x)f_{P}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{subarray}}\sum_{P\in% \mathcal{P}(f)}\mathds{1}_{H^{e}_{P}}(x)f_{P}(x)" class="ltx_Math" display="inline" id="S3.Ex10.m1.8"><semantics id="S3.Ex10.m1.8a"><mrow id="S3.Ex10.m1.8.9" xref="S3.Ex10.m1.8.9.cmml"><mi id="S3.Ex10.m1.8.9.2" xref="S3.Ex10.m1.8.9.2.cmml"></mi><mo id="S3.Ex10.m1.8.9.1" xref="S3.Ex10.m1.8.9.1.cmml">=</mo><mrow id="S3.Ex10.m1.8.9.3" xref="S3.Ex10.m1.8.9.3.cmml"><mrow id="S3.Ex10.m1.8.9.3.2" xref="S3.Ex10.m1.8.9.3.2.cmml"><mstyle displaystyle="true" id="S3.Ex10.m1.8.9.3.2.1" xref="S3.Ex10.m1.8.9.3.2.1.cmml"><munder id="S3.Ex10.m1.8.9.3.2.1a" xref="S3.Ex10.m1.8.9.3.2.1.cmml"><mo id="S3.Ex10.m1.8.9.3.2.1.2" movablelimits="false" xref="S3.Ex10.m1.8.9.3.2.1.2.cmml">∑</mo><mrow id="S3.Ex10.m1.2.2.1" xref="S3.Ex10.m1.2.2.1.cmml"><mi id="S3.Ex10.m1.2.2.1.3" xref="S3.Ex10.m1.2.2.1.3.cmml">v</mi><mo id="S3.Ex10.m1.2.2.1.2" xref="S3.Ex10.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex10.m1.2.2.1.4" xref="S3.Ex10.m1.2.2.1.4.cmml"><mi id="S3.Ex10.m1.2.2.1.4.2" xref="S3.Ex10.m1.2.2.1.4.2.cmml">V</mi><mo id="S3.Ex10.m1.2.2.1.4.1" xref="S3.Ex10.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.2.2.1.4.3.2" xref="S3.Ex10.m1.2.2.1.4.cmml"><mo id="S3.Ex10.m1.2.2.1.4.3.2.1" stretchy="false" xref="S3.Ex10.m1.2.2.1.4.cmml">(</mo><mi id="S3.Ex10.m1.2.2.1.1" xref="S3.Ex10.m1.2.2.1.1.cmml">f</mi><mo id="S3.Ex10.m1.2.2.1.4.3.2.2" stretchy="false" xref="S3.Ex10.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S3.Ex10.m1.8.9.3.2.2" xref="S3.Ex10.m1.8.9.3.2.2.cmml"><mstyle displaystyle="true" id="S3.Ex10.m1.8.9.3.2.2.1" xref="S3.Ex10.m1.8.9.3.2.2.1.cmml"><munder id="S3.Ex10.m1.8.9.3.2.2.1a" xref="S3.Ex10.m1.8.9.3.2.2.1.cmml"><mo id="S3.Ex10.m1.8.9.3.2.2.1.2" movablelimits="false" xref="S3.Ex10.m1.8.9.3.2.2.1.2.cmml">∑</mo><mrow id="S3.Ex10.m1.3.3.1" xref="S3.Ex10.m1.3.3.1.cmml"><mi id="S3.Ex10.m1.3.3.1.3" xref="S3.Ex10.m1.3.3.1.3.cmml">P</mi><mo id="S3.Ex10.m1.3.3.1.2" xref="S3.Ex10.m1.3.3.1.2.cmml">∈</mo><mrow id="S3.Ex10.m1.3.3.1.4" xref="S3.Ex10.m1.3.3.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex10.m1.3.3.1.4.2" xref="S3.Ex10.m1.3.3.1.4.2.cmml">𝒫</mi><mo id="S3.Ex10.m1.3.3.1.4.1" xref="S3.Ex10.m1.3.3.1.4.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.3.3.1.4.3.2" xref="S3.Ex10.m1.3.3.1.4.cmml"><mo id="S3.Ex10.m1.3.3.1.4.3.2.1" stretchy="false" xref="S3.Ex10.m1.3.3.1.4.cmml">(</mo><mi id="S3.Ex10.m1.3.3.1.1" xref="S3.Ex10.m1.3.3.1.1.cmml">f</mi><mo id="S3.Ex10.m1.3.3.1.4.3.2.2" stretchy="false" xref="S3.Ex10.m1.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S3.Ex10.m1.8.9.3.2.2.2" xref="S3.Ex10.m1.8.9.3.2.2.2.cmml"><msub 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xref="S3.Ex10.m1.8.9.3.3.1.2.cmml">∑</mo><mtable id="S3.Ex10.m1.1.1.1.1.1.1" xref="S3.Ex10.m1.1.1.1a.2.cmml"><mtr id="S3.Ex10.m1.1.1.1.1.1.1a" xref="S3.Ex10.m1.1.1.1a.2.cmml"><mtd id="S3.Ex10.m1.1.1.1.1.1.1b" xref="S3.Ex10.m1.1.1.1a.2.cmml"><mrow id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">e</mi><mo id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.cmml"><msub id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2.cmml"><mi id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2.2" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml">E</mi><mi id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2.3" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml">l</mi></msub><mo id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.1" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.3.2" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.cmml"><mo id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">(</mo><mi id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">f</mi><mo id="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex10.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></munder></mstyle><mrow id="S3.Ex10.m1.8.9.3.3.2" xref="S3.Ex10.m1.8.9.3.3.2.cmml"><mstyle displaystyle="true" id="S3.Ex10.m1.8.9.3.3.2.1" xref="S3.Ex10.m1.8.9.3.3.2.1.cmml"><munder id="S3.Ex10.m1.8.9.3.3.2.1a" xref="S3.Ex10.m1.8.9.3.3.2.1.cmml"><mo id="S3.Ex10.m1.8.9.3.3.2.1.2" movablelimits="false" xref="S3.Ex10.m1.8.9.3.3.2.1.2.cmml">∑</mo><mrow id="S3.Ex10.m1.4.4.1" xref="S3.Ex10.m1.4.4.1.cmml"><mi id="S3.Ex10.m1.4.4.1.3" xref="S3.Ex10.m1.4.4.1.3.cmml">P</mi><mo 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id="S3.Ex10.m1.8.8.cmml" xref="S3.Ex10.m1.8.8">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.8c">\displaystyle=\sum_{v\in V(f)}\sum_{P\in\mathcal{P}(f)}\mathds{1}_{Q^{v}_{P}}(% x)f_{P}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{subarray}}\sum_{P\in% \mathcal{P}(f)}\mathds{1}_{H^{e}_{P}}(x)f_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.8d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_f ) end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.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\qquad-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end{subarray}}\sum_% {P\in\mathcal{P}(f)}\mathds{1}_{H^{e}_{P}}(x)f_{P}(x)+\sum_{P\in\mathcal{P}(f)% }c(P)f_{P}(x)" class="ltx_Math" display="inline" id="S3.Ex11.m1.7"><semantics id="S3.Ex11.m1.7a"><mrow id="S3.Ex11.m1.7.8" xref="S3.Ex11.m1.7.8.cmml"><mrow 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xref="S3.Ex11.m1.7.7">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex11.m1.7c">\displaystyle\qquad-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end{subarray}}\sum_% {P\in\mathcal{P}(f)}\mathds{1}_{H^{e}_{P}}(x)f_{P}(x)+\sum_{P\in\mathcal{P}(f)% }c(P)f_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.7d">- ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT italic_c ( italic_P ) italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E6"><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=\sum_{P\in\mathcal{P}(f)}f_{P}(x)\underbrace{\bigg{(}\sum_{v\in V% (f)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{% subarray}}\mathds{1}_{H^{e}_{P}}(x)-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end% {subarray}}\mathds{1}_{H^{e}_{P}}(x)+c(P)\bigg{)}}_{=:\,\text{$I_{P}(x)$}}" class="ltx_Math" display="inline" id="S3.E6.m1.14"><semantics id="S3.E6.m1.14a"><mrow id="S3.E6.m1.14.15" xref="S3.E6.m1.14.15.cmml"><mi id="S3.E6.m1.14.15.2" xref="S3.E6.m1.14.15.2.cmml"></mi><mo id="S3.E6.m1.14.15.1" xref="S3.E6.m1.14.15.1.cmml">=</mo><mrow id="S3.E6.m1.14.15.3" 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id="S3.E6.m1.14c">\displaystyle=\sum_{P\in\mathcal{P}(f)}f_{P}(x)\underbrace{\bigg{(}\sum_{v\in V% (f)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{% subarray}}\mathds{1}_{H^{e}_{P}}(x)-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end% {subarray}}\mathds{1}_{H^{e}_{P}}(x)+c(P)\bigg{)}}_{=:\,\text{$I_{P}(x)$}}</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.14d">= ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P ( italic_f ) end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) under⏟ start_ARG ( ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_f ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + italic_c ( italic_P ) ) end_ARG start_POSTSUBSCRIPT = : italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.1.p1.7">Comparing (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E5" title="Equation 5 ‣ Proof of 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">5</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.Ex9" title="Proof of 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3</span></a>), we see that (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E4" title="Equation 4 ‣ Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4</span></a>) is true, if <math alttext="\ref{tag:PiecesOverlap}=\mathds{1}_{P}(x)" class="ltx_Math" display="inline" id="S3.1.p1.6.m1.1"><semantics id="S3.1.p1.6.m1.1a"><mrow id="S3.1.p1.6.m1.1.2" xref="S3.1.p1.6.m1.1.2.cmml"><mtext class="ltx_missing_label ltx_mathvariant_italic" id="S3.1.p1.6.m1.1.2.2" xref="S3.1.p1.6.m1.1.2.2b.cmml"><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:tag:PiecesOverlap</span></mtext><mo id="S3.1.p1.6.m1.1.2.1" xref="S3.1.p1.6.m1.1.2.1.cmml">=</mo><mrow id="S3.1.p1.6.m1.1.2.3" xref="S3.1.p1.6.m1.1.2.3.cmml"><msub id="S3.1.p1.6.m1.1.2.3.2" xref="S3.1.p1.6.m1.1.2.3.2.cmml"><mn id="S3.1.p1.6.m1.1.2.3.2.2" xref="S3.1.p1.6.m1.1.2.3.2.2.cmml">𝟙</mn><mi id="S3.1.p1.6.m1.1.2.3.2.3" xref="S3.1.p1.6.m1.1.2.3.2.3.cmml">P</mi></msub><mo id="S3.1.p1.6.m1.1.2.3.1" xref="S3.1.p1.6.m1.1.2.3.1.cmml">⁢</mo><mrow id="S3.1.p1.6.m1.1.2.3.3.2" xref="S3.1.p1.6.m1.1.2.3.cmml"><mo id="S3.1.p1.6.m1.1.2.3.3.2.1" stretchy="false" xref="S3.1.p1.6.m1.1.2.3.cmml">(</mo><mi id="S3.1.p1.6.m1.1.1" xref="S3.1.p1.6.m1.1.1.cmml">x</mi><mo id="S3.1.p1.6.m1.1.2.3.3.2.2" stretchy="false" xref="S3.1.p1.6.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.6.m1.1b"><apply id="S3.1.p1.6.m1.1.2.cmml" xref="S3.1.p1.6.m1.1.2"><eq id="S3.1.p1.6.m1.1.2.1.cmml" xref="S3.1.p1.6.m1.1.2.1"></eq><ci id="S3.1.p1.6.m1.1.2.2b.cmml" xref="S3.1.p1.6.m1.1.2.2"><mtext class="ltx_missing_label ltx_mathvariant_italic" id="S3.1.p1.6.m1.1.2.2.cmml" xref="S3.1.p1.6.m1.1.2.2"><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:tag:PiecesOverlap</span></mtext></ci><apply id="S3.1.p1.6.m1.1.2.3.cmml" xref="S3.1.p1.6.m1.1.2.3"><times id="S3.1.p1.6.m1.1.2.3.1.cmml" xref="S3.1.p1.6.m1.1.2.3.1"></times><apply id="S3.1.p1.6.m1.1.2.3.2.cmml" xref="S3.1.p1.6.m1.1.2.3.2"><csymbol cd="ambiguous" id="S3.1.p1.6.m1.1.2.3.2.1.cmml" xref="S3.1.p1.6.m1.1.2.3.2">subscript</csymbol><cn id="S3.1.p1.6.m1.1.2.3.2.2.cmml" type="integer" xref="S3.1.p1.6.m1.1.2.3.2.2">1</cn><ci id="S3.1.p1.6.m1.1.2.3.2.3.cmml" xref="S3.1.p1.6.m1.1.2.3.2.3">𝑃</ci></apply><ci id="S3.1.p1.6.m1.1.1.cmml" xref="S3.1.p1.6.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.6.m1.1c">\ref{tag:PiecesOverlap}=\mathds{1}_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.6.m1.1d">= blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> for all <math alttext="P\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S3.1.p1.7.m2.1"><semantics id="S3.1.p1.7.m2.1a"><mrow id="S3.1.p1.7.m2.1.2" xref="S3.1.p1.7.m2.1.2.cmml"><mi id="S3.1.p1.7.m2.1.2.2" xref="S3.1.p1.7.m2.1.2.2.cmml">P</mi><mo id="S3.1.p1.7.m2.1.2.1" xref="S3.1.p1.7.m2.1.2.1.cmml">∈</mo><mrow id="S3.1.p1.7.m2.1.2.3" xref="S3.1.p1.7.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.1.p1.7.m2.1.2.3.2" xref="S3.1.p1.7.m2.1.2.3.2.cmml">𝒫</mi><mo id="S3.1.p1.7.m2.1.2.3.1" xref="S3.1.p1.7.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.1.p1.7.m2.1.2.3.3.2" xref="S3.1.p1.7.m2.1.2.3.cmml"><mo id="S3.1.p1.7.m2.1.2.3.3.2.1" stretchy="false" xref="S3.1.p1.7.m2.1.2.3.cmml">(</mo><mi id="S3.1.p1.7.m2.1.1" xref="S3.1.p1.7.m2.1.1.cmml">f</mi><mo id="S3.1.p1.7.m2.1.2.3.3.2.2" stretchy="false" xref="S3.1.p1.7.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.7.m2.1b"><apply id="S3.1.p1.7.m2.1.2.cmml" xref="S3.1.p1.7.m2.1.2"><in id="S3.1.p1.7.m2.1.2.1.cmml" xref="S3.1.p1.7.m2.1.2.1"></in><ci id="S3.1.p1.7.m2.1.2.2.cmml" xref="S3.1.p1.7.m2.1.2.2">𝑃</ci><apply id="S3.1.p1.7.m2.1.2.3.cmml" xref="S3.1.p1.7.m2.1.2.3"><times id="S3.1.p1.7.m2.1.2.3.1.cmml" xref="S3.1.p1.7.m2.1.2.3.1"></times><ci id="S3.1.p1.7.m2.1.2.3.2.cmml" xref="S3.1.p1.7.m2.1.2.3.2">𝒫</ci><ci id="S3.1.p1.7.m2.1.1.cmml" xref="S3.1.p1.7.m2.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.7.m2.1c">P\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.7.m2.1d">italic_P ∈ caligraphic_P ( italic_f )</annotation></semantics></math>. Using that</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="Q_{P}^{v}\neq\emptyset\quad\Leftrightarrow\quad v\in V(P)\qquad\text{and}% \qquad H_{P}^{e}\neq\emptyset\quad\Leftrightarrow\quad e\in E(P)," class="ltx_Math" display="block" id="S3.Ex12.m1.8"><semantics id="S3.Ex12.m1.8a"><mrow id="S3.Ex12.m1.8.8.1"><mrow id="S3.Ex12.m1.8.8.1.1.2" xref="S3.Ex12.m1.8.8.1.1.3.cmml"><mrow id="S3.Ex12.m1.8.8.1.1.1.1" xref="S3.Ex12.m1.8.8.1.1.1.1.cmml"><msubsup id="S3.Ex12.m1.8.8.1.1.1.1.2" xref="S3.Ex12.m1.8.8.1.1.1.1.2.cmml"><mi id="S3.Ex12.m1.8.8.1.1.1.1.2.2.2" xref="S3.Ex12.m1.8.8.1.1.1.1.2.2.2.cmml">Q</mi><mi id="S3.Ex12.m1.8.8.1.1.1.1.2.2.3" xref="S3.Ex12.m1.8.8.1.1.1.1.2.2.3.cmml">P</mi><mi id="S3.Ex12.m1.8.8.1.1.1.1.2.3" xref="S3.Ex12.m1.8.8.1.1.1.1.2.3.cmml">v</mi></msubsup><mo 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xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2"><in id="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.1.cmml" xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.1"></in><ci id="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.2.cmml" xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.2">𝑒</ci><apply id="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3.cmml" xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3"><times id="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3.1.cmml" xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3.1"></times><ci id="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3.2.cmml" xref="S3.Ex12.m1.8.8.1.1.2.2.2.2.2.2.3.2">𝐸</ci><ci id="S3.Ex12.m1.2.2.cmml" xref="S3.Ex12.m1.2.2">𝑃</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex12.m1.8c">Q_{P}^{v}\neq\emptyset\quad\Leftrightarrow\quad v\in V(P)\qquad\text{and}% \qquad H_{P}^{e}\neq\emptyset\quad\Leftrightarrow\quad e\in E(P),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex12.m1.8d">italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ≠ ∅ ⇔ italic_v ∈ italic_V ( italic_P ) and italic_H start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ≠ ∅ ⇔ italic_e ∈ italic_E ( italic_P ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.1.p1.8"><math alttext="I_{P}(x)" class="ltx_Math" display="inline" id="S3.1.p1.8.m1.1"><semantics id="S3.1.p1.8.m1.1a"><mrow id="S3.1.p1.8.m1.1.2" xref="S3.1.p1.8.m1.1.2.cmml"><msub id="S3.1.p1.8.m1.1.2.2" xref="S3.1.p1.8.m1.1.2.2.cmml"><mi id="S3.1.p1.8.m1.1.2.2.2" xref="S3.1.p1.8.m1.1.2.2.2.cmml">I</mi><mi id="S3.1.p1.8.m1.1.2.2.3" xref="S3.1.p1.8.m1.1.2.2.3.cmml">P</mi></msub><mo id="S3.1.p1.8.m1.1.2.1" xref="S3.1.p1.8.m1.1.2.1.cmml">⁢</mo><mrow id="S3.1.p1.8.m1.1.2.3.2" xref="S3.1.p1.8.m1.1.2.cmml"><mo id="S3.1.p1.8.m1.1.2.3.2.1" stretchy="false" xref="S3.1.p1.8.m1.1.2.cmml">(</mo><mi id="S3.1.p1.8.m1.1.1" xref="S3.1.p1.8.m1.1.1.cmml">x</mi><mo id="S3.1.p1.8.m1.1.2.3.2.2" stretchy="false" xref="S3.1.p1.8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.8.m1.1b"><apply id="S3.1.p1.8.m1.1.2.cmml" xref="S3.1.p1.8.m1.1.2"><times id="S3.1.p1.8.m1.1.2.1.cmml" xref="S3.1.p1.8.m1.1.2.1"></times><apply id="S3.1.p1.8.m1.1.2.2.cmml" xref="S3.1.p1.8.m1.1.2.2"><csymbol cd="ambiguous" id="S3.1.p1.8.m1.1.2.2.1.cmml" xref="S3.1.p1.8.m1.1.2.2">subscript</csymbol><ci id="S3.1.p1.8.m1.1.2.2.2.cmml" xref="S3.1.p1.8.m1.1.2.2.2">𝐼</ci><ci id="S3.1.p1.8.m1.1.2.2.3.cmml" xref="S3.1.p1.8.m1.1.2.2.3">𝑃</ci></apply><ci id="S3.1.p1.8.m1.1.1.cmml" xref="S3.1.p1.8.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.8.m1.1c">I_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.8.m1.1d">italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> can be written as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx3"> <tbody id="S3.Ex13"><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 I_{P}(x)" class="ltx_Math" display="inline" id="S3.Ex13.m1.1"><semantics id="S3.Ex13.m1.1a"><mrow id="S3.Ex13.m1.1.2" xref="S3.Ex13.m1.1.2.cmml"><msub id="S3.Ex13.m1.1.2.2" xref="S3.Ex13.m1.1.2.2.cmml"><mi id="S3.Ex13.m1.1.2.2.2" xref="S3.Ex13.m1.1.2.2.2.cmml">I</mi><mi id="S3.Ex13.m1.1.2.2.3" xref="S3.Ex13.m1.1.2.2.3.cmml">P</mi></msub><mo id="S3.Ex13.m1.1.2.1" xref="S3.Ex13.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex13.m1.1.2.3.2" xref="S3.Ex13.m1.1.2.cmml"><mo id="S3.Ex13.m1.1.2.3.2.1" stretchy="false" xref="S3.Ex13.m1.1.2.cmml">(</mo><mi id="S3.Ex13.m1.1.1" xref="S3.Ex13.m1.1.1.cmml">x</mi><mo id="S3.Ex13.m1.1.2.3.2.2" stretchy="false" xref="S3.Ex13.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex13.m1.1b"><apply id="S3.Ex13.m1.1.2.cmml" xref="S3.Ex13.m1.1.2"><times id="S3.Ex13.m1.1.2.1.cmml" xref="S3.Ex13.m1.1.2.1"></times><apply id="S3.Ex13.m1.1.2.2.cmml" xref="S3.Ex13.m1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex13.m1.1.2.2.1.cmml" xref="S3.Ex13.m1.1.2.2">subscript</csymbol><ci id="S3.Ex13.m1.1.2.2.2.cmml" xref="S3.Ex13.m1.1.2.2.2">𝐼</ci><ci id="S3.Ex13.m1.1.2.2.3.cmml" xref="S3.Ex13.m1.1.2.2.3">𝑃</ci></apply><ci id="S3.Ex13.m1.1.1.cmml" xref="S3.Ex13.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex13.m1.1c">\displaystyle I_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m1.1d">italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{\begin{subarray}% {c}e\in E_{l}(P)\end{subarray}}\mathds{1}_{H^{e}_{P}}(x)-\sum_{\begin{subarray% }{c}e\in E_{b}(P)\end{subarray}}\mathds{1}_{H^{e}_{P}}(x)+c(P)" class="ltx_Math" display="inline" id="S3.Ex13.m2.7"><semantics id="S3.Ex13.m2.7a"><mrow id="S3.Ex13.m2.7.8" xref="S3.Ex13.m2.7.8.cmml"><mi id="S3.Ex13.m2.7.8.2" xref="S3.Ex13.m2.7.8.2.cmml"></mi><mo id="S3.Ex13.m2.7.8.1" xref="S3.Ex13.m2.7.8.1.cmml">=</mo><mrow id="S3.Ex13.m2.7.8.3" xref="S3.Ex13.m2.7.8.3.cmml"><mrow id="S3.Ex13.m2.7.8.3.2" xref="S3.Ex13.m2.7.8.3.2.cmml"><mrow id="S3.Ex13.m2.7.8.3.2.2" xref="S3.Ex13.m2.7.8.3.2.2.cmml"><mrow id="S3.Ex13.m2.7.8.3.2.2.2" xref="S3.Ex13.m2.7.8.3.2.2.2.cmml"><mstyle displaystyle="true" id="S3.Ex13.m2.7.8.3.2.2.2.1" xref="S3.Ex13.m2.7.8.3.2.2.2.1.cmml"><munder id="S3.Ex13.m2.7.8.3.2.2.2.1a" xref="S3.Ex13.m2.7.8.3.2.2.2.1.cmml"><mo id="S3.Ex13.m2.7.8.3.2.2.2.1.2" movablelimits="false" xref="S3.Ex13.m2.7.8.3.2.2.2.1.2.cmml">∑</mo><mrow id="S3.Ex13.m2.3.3.1" xref="S3.Ex13.m2.3.3.1.cmml"><mi id="S3.Ex13.m2.3.3.1.3" xref="S3.Ex13.m2.3.3.1.3.cmml">v</mi><mo id="S3.Ex13.m2.3.3.1.2" xref="S3.Ex13.m2.3.3.1.2.cmml">∈</mo><mrow id="S3.Ex13.m2.3.3.1.4" xref="S3.Ex13.m2.3.3.1.4.cmml"><mi id="S3.Ex13.m2.3.3.1.4.2" xref="S3.Ex13.m2.3.3.1.4.2.cmml">V</mi><mo id="S3.Ex13.m2.3.3.1.4.1" xref="S3.Ex13.m2.3.3.1.4.1.cmml">⁢</mo><mrow id="S3.Ex13.m2.3.3.1.4.3.2" xref="S3.Ex13.m2.3.3.1.4.cmml"><mo id="S3.Ex13.m2.3.3.1.4.3.2.1" stretchy="false" xref="S3.Ex13.m2.3.3.1.4.cmml">(</mo><mi id="S3.Ex13.m2.3.3.1.1" xref="S3.Ex13.m2.3.3.1.1.cmml">P</mi><mo id="S3.Ex13.m2.3.3.1.4.3.2.2" stretchy="false" xref="S3.Ex13.m2.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S3.Ex13.m2.7.8.3.2.2.2.2" xref="S3.Ex13.m2.7.8.3.2.2.2.2.cmml"><msub id="S3.Ex13.m2.7.8.3.2.2.2.2.2" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.cmml"><mn id="S3.Ex13.m2.7.8.3.2.2.2.2.2.2" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex13.m2.7.8.3.2.2.2.2.2.3" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.cmml"><mi id="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.2.2" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.2.2.cmml">Q</mi><mi id="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.3" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.3.cmml">P</mi><mi id="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.2.3" xref="S3.Ex13.m2.7.8.3.2.2.2.2.2.3.2.3.cmml">v</mi></msubsup></msub><mo id="S3.Ex13.m2.7.8.3.2.2.2.2.1" xref="S3.Ex13.m2.7.8.3.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.Ex13.m2.7.8.3.2.2.2.2.3.2" xref="S3.Ex13.m2.7.8.3.2.2.2.2.cmml"><mo id="S3.Ex13.m2.7.8.3.2.2.2.2.3.2.1" stretchy="false" xref="S3.Ex13.m2.7.8.3.2.2.2.2.cmml">(</mo><mi id="S3.Ex13.m2.4.4" xref="S3.Ex13.m2.4.4.cmml">x</mi><mo id="S3.Ex13.m2.7.8.3.2.2.2.2.3.2.2" stretchy="false" xref="S3.Ex13.m2.7.8.3.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex13.m2.7.8.3.2.2.1" xref="S3.Ex13.m2.7.8.3.2.2.1.cmml">+</mo><mrow id="S3.Ex13.m2.7.8.3.2.2.3" xref="S3.Ex13.m2.7.8.3.2.2.3.cmml"><mstyle displaystyle="true" id="S3.Ex13.m2.7.8.3.2.2.3.1" xref="S3.Ex13.m2.7.8.3.2.2.3.1.cmml"><munder id="S3.Ex13.m2.7.8.3.2.2.3.1a" xref="S3.Ex13.m2.7.8.3.2.2.3.1.cmml"><mo id="S3.Ex13.m2.7.8.3.2.2.3.1.2" movablelimits="false" xref="S3.Ex13.m2.7.8.3.2.2.3.1.2.cmml">∑</mo><mtable id="S3.Ex13.m2.1.1.1.1.1.1" xref="S3.Ex13.m2.1.1.1a.2.cmml"><mtr id="S3.Ex13.m2.1.1.1.1.1.1a" xref="S3.Ex13.m2.1.1.1a.2.cmml"><mtd id="S3.Ex13.m2.1.1.1.1.1.1b" xref="S3.Ex13.m2.1.1.1a.2.cmml"><mrow id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.3.cmml">e</mi><mo id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.cmml"><msub id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2.cmml"><mi id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2.2" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml">E</mi><mi id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2.3" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml">l</mi></msub><mo id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.1" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.3.2" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.cmml"><mo id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.cmml">(</mo><mi id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.1.cmml">P</mi><mo id="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex13.m2.1.1.1.1.1.1.1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></munder></mstyle><mrow id="S3.Ex13.m2.7.8.3.2.2.3.2" xref="S3.Ex13.m2.7.8.3.2.2.3.2.cmml"><msub id="S3.Ex13.m2.7.8.3.2.2.3.2.2" xref="S3.Ex13.m2.7.8.3.2.2.3.2.2.cmml"><mn id="S3.Ex13.m2.7.8.3.2.2.3.2.2.2" xref="S3.Ex13.m2.7.8.3.2.2.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex13.m2.7.8.3.2.2.3.2.2.3" xref="S3.Ex13.m2.7.8.3.2.2.3.2.2.3.cmml"><mi id="S3.Ex13.m2.7.8.3.2.2.3.2.2.3.2.2" 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encoding="application/x-tex" id="S3.Ex13.m2.7c">\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{\begin{subarray}% {c}e\in E_{l}(P)\end{subarray}}\mathds{1}_{H^{e}_{P}}(x)-\sum_{\begin{subarray% }{c}e\in E_{b}(P)\end{subarray}}\mathds{1}_{H^{e}_{P}}(x)+c(P)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m2.7d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + italic_c ( italic_P )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.1.p1.11">Since we have restricted ourselves to <math alttext="x" class="ltx_Math" display="inline" id="S3.1.p1.9.m1.1"><semantics id="S3.1.p1.9.m1.1a"><mi id="S3.1.p1.9.m1.1.1" xref="S3.1.p1.9.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.9.m1.1b"><ci id="S3.1.p1.9.m1.1.1.cmml" xref="S3.1.p1.9.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.9.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.9.m1.1d">italic_x</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="S3.1.p1.10.m2.1"><semantics id="S3.1.p1.10.m2.1a"><mi id="S3.1.p1.10.m2.1.1" xref="S3.1.p1.10.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.10.m2.1b"><ci id="S3.1.p1.10.m2.1.1.cmml" xref="S3.1.p1.10.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.10.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.10.m2.1d">italic_P</annotation></semantics></math>-general position, we can apply <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>, which directly yields <math alttext="\ref{tag:PiecesOverlap}=\mathds{1}_{P}(x)" class="ltx_Math" display="inline" id="S3.1.p1.11.m3.1"><semantics id="S3.1.p1.11.m3.1a"><mrow id="S3.1.p1.11.m3.1.2" xref="S3.1.p1.11.m3.1.2.cmml"><mtext class="ltx_missing_label ltx_mathvariant_italic" id="S3.1.p1.11.m3.1.2.2" xref="S3.1.p1.11.m3.1.2.2b.cmml"><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:tag:PiecesOverlap</span></mtext><mo id="S3.1.p1.11.m3.1.2.1" xref="S3.1.p1.11.m3.1.2.1.cmml">=</mo><mrow id="S3.1.p1.11.m3.1.2.3" xref="S3.1.p1.11.m3.1.2.3.cmml"><msub id="S3.1.p1.11.m3.1.2.3.2" xref="S3.1.p1.11.m3.1.2.3.2.cmml"><mn id="S3.1.p1.11.m3.1.2.3.2.2" xref="S3.1.p1.11.m3.1.2.3.2.2.cmml">𝟙</mn><mi id="S3.1.p1.11.m3.1.2.3.2.3" xref="S3.1.p1.11.m3.1.2.3.2.3.cmml">P</mi></msub><mo id="S3.1.p1.11.m3.1.2.3.1" xref="S3.1.p1.11.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.1.p1.11.m3.1.2.3.3.2" xref="S3.1.p1.11.m3.1.2.3.cmml"><mo id="S3.1.p1.11.m3.1.2.3.3.2.1" stretchy="false" xref="S3.1.p1.11.m3.1.2.3.cmml">(</mo><mi id="S3.1.p1.11.m3.1.1" xref="S3.1.p1.11.m3.1.1.cmml">x</mi><mo id="S3.1.p1.11.m3.1.2.3.3.2.2" stretchy="false" xref="S3.1.p1.11.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.11.m3.1b"><apply id="S3.1.p1.11.m3.1.2.cmml" xref="S3.1.p1.11.m3.1.2"><eq id="S3.1.p1.11.m3.1.2.1.cmml" xref="S3.1.p1.11.m3.1.2.1"></eq><ci id="S3.1.p1.11.m3.1.2.2b.cmml" xref="S3.1.p1.11.m3.1.2.2"><mtext class="ltx_missing_label ltx_mathvariant_italic" id="S3.1.p1.11.m3.1.2.2.cmml" xref="S3.1.p1.11.m3.1.2.2"><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:tag:PiecesOverlap</span></mtext></ci><apply id="S3.1.p1.11.m3.1.2.3.cmml" xref="S3.1.p1.11.m3.1.2.3"><times id="S3.1.p1.11.m3.1.2.3.1.cmml" xref="S3.1.p1.11.m3.1.2.3.1"></times><apply id="S3.1.p1.11.m3.1.2.3.2.cmml" xref="S3.1.p1.11.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.1.p1.11.m3.1.2.3.2.1.cmml" xref="S3.1.p1.11.m3.1.2.3.2">subscript</csymbol><cn id="S3.1.p1.11.m3.1.2.3.2.2.cmml" type="integer" xref="S3.1.p1.11.m3.1.2.3.2.2">1</cn><ci id="S3.1.p1.11.m3.1.2.3.2.3.cmml" xref="S3.1.p1.11.m3.1.2.3.2.3">𝑃</ci></apply><ci id="S3.1.p1.11.m3.1.1.cmml" xref="S3.1.p1.11.m3.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.11.m3.1c">\ref{tag:PiecesOverlap}=\mathds{1}_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.11.m3.1d">= blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S3.Thmtheoremx1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S3.Thmtheoremx1.1.1.1">Remark</span></span><span class="ltx_text ltx_font_italic" id="S3.Thmtheoremx1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheoremx1.p1"> <p class="ltx_p" id="S3.Thmtheoremx1.p1.4">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.1.m1.2"><semantics id="S3.Thmtheoremx1.p1.1.m1.2a"><mrow id="S3.Thmtheoremx1.p1.1.m1.2.3" xref="S3.Thmtheoremx1.p1.1.m1.2.3.cmml"><mi id="S3.Thmtheoremx1.p1.1.m1.2.3.2" xref="S3.Thmtheoremx1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S3.Thmtheoremx1.p1.1.m1.2.3.1" xref="S3.Thmtheoremx1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S3.Thmtheoremx1.p1.1.m1.2.3.3.2" xref="S3.Thmtheoremx1.p1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheoremx1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheoremx1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheoremx1.p1.1.m1.1.1" xref="S3.Thmtheoremx1.p1.1.m1.1.1.cmml">V</mi><mo id="S3.Thmtheoremx1.p1.1.m1.2.3.3.2.2" xref="S3.Thmtheoremx1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheoremx1.p1.1.m1.2.2" xref="S3.Thmtheoremx1.p1.1.m1.2.2.cmml">E</mi><mo id="S3.Thmtheoremx1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheoremx1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.1.m1.2b"><apply id="S3.Thmtheoremx1.p1.1.m1.2.3.cmml" xref="S3.Thmtheoremx1.p1.1.m1.2.3"><eq id="S3.Thmtheoremx1.p1.1.m1.2.3.1.cmml" xref="S3.Thmtheoremx1.p1.1.m1.2.3.1"></eq><ci id="S3.Thmtheoremx1.p1.1.m1.2.3.2.cmml" xref="S3.Thmtheoremx1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S3.Thmtheoremx1.p1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheoremx1.p1.1.m1.2.3.3.2"><ci id="S3.Thmtheoremx1.p1.1.m1.1.1.cmml" xref="S3.Thmtheoremx1.p1.1.m1.1.1">𝑉</ci><ci id="S3.Thmtheoremx1.p1.1.m1.2.2.cmml" xref="S3.Thmtheoremx1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be a connected plane graph such that its set of faces <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.2.m2.1"><semantics id="S3.Thmtheoremx1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheoremx1.p1.2.m2.1.1" xref="S3.Thmtheoremx1.p1.2.m2.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.2.m2.1b"><ci id="S3.Thmtheoremx1.p1.2.m2.1.1.cmml" xref="S3.Thmtheoremx1.p1.2.m2.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.2.m2.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.2.m2.1d">caligraphic_P</annotation></semantics></math> consists of polygons. For <math alttext="f(x):=1" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.3.m3.1"><semantics id="S3.Thmtheoremx1.p1.3.m3.1a"><mrow id="S3.Thmtheoremx1.p1.3.m3.1.2" xref="S3.Thmtheoremx1.p1.3.m3.1.2.cmml"><mrow id="S3.Thmtheoremx1.p1.3.m3.1.2.2" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.cmml"><mi id="S3.Thmtheoremx1.p1.3.m3.1.2.2.2" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.2.cmml">f</mi><mo id="S3.Thmtheoremx1.p1.3.m3.1.2.2.1" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheoremx1.p1.3.m3.1.2.2.3.2" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.cmml"><mo id="S3.Thmtheoremx1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.cmml">(</mo><mi id="S3.Thmtheoremx1.p1.3.m3.1.1" xref="S3.Thmtheoremx1.p1.3.m3.1.1.cmml">x</mi><mo id="S3.Thmtheoremx1.p1.3.m3.1.2.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheoremx1.p1.3.m3.1.2.1" rspace="0.278em" xref="S3.Thmtheoremx1.p1.3.m3.1.2.1.cmml">:=</mo><mn id="S3.Thmtheoremx1.p1.3.m3.1.2.3" xref="S3.Thmtheoremx1.p1.3.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.3.m3.1b"><apply id="S3.Thmtheoremx1.p1.3.m3.1.2.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.2"><csymbol cd="latexml" id="S3.Thmtheoremx1.p1.3.m3.1.2.1.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.2.1">assign</csymbol><apply id="S3.Thmtheoremx1.p1.3.m3.1.2.2.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2"><times id="S3.Thmtheoremx1.p1.3.m3.1.2.2.1.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.1"></times><ci id="S3.Thmtheoremx1.p1.3.m3.1.2.2.2.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.2.2.2">𝑓</ci><ci id="S3.Thmtheoremx1.p1.3.m3.1.1.cmml" xref="S3.Thmtheoremx1.p1.3.m3.1.1">𝑥</ci></apply><cn id="S3.Thmtheoremx1.p1.3.m3.1.2.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.3.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.3.m3.1c">f(x):=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.3.m3.1d">italic_f ( italic_x ) := 1</annotation></semantics></math>, which is <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.4.m4.1"><semantics id="S3.Thmtheoremx1.p1.4.m4.1a"><mi id="S3.Thmtheoremx1.p1.4.m4.1.1" xref="S3.Thmtheoremx1.p1.4.m4.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.4.m4.1b"><ci id="S3.Thmtheoremx1.p1.4.m4.1.1.cmml" xref="S3.Thmtheoremx1.p1.4.m4.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.4.m4.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.4.m4.1d">roman_CPA</annotation></semantics></math> on any admissible set of pieces, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a> implies that</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="1=\sum_{v\in V}1-\sum_{e\in E}1+\sum_{P\in\mathcal{P}}(1+d(P)-n_{h}(P))." class="ltx_Math" display="block" id="S3.Ex14.m1.3"><semantics id="S3.Ex14.m1.3a"><mrow id="S3.Ex14.m1.3.3.1" xref="S3.Ex14.m1.3.3.1.1.cmml"><mrow id="S3.Ex14.m1.3.3.1.1" xref="S3.Ex14.m1.3.3.1.1.cmml"><mn id="S3.Ex14.m1.3.3.1.1.3" xref="S3.Ex14.m1.3.3.1.1.3.cmml">1</mn><mo id="S3.Ex14.m1.3.3.1.1.2" rspace="0.111em" xref="S3.Ex14.m1.3.3.1.1.2.cmml">=</mo><mrow id="S3.Ex14.m1.3.3.1.1.1" xref="S3.Ex14.m1.3.3.1.1.1.cmml"><mrow id="S3.Ex14.m1.3.3.1.1.1.3" xref="S3.Ex14.m1.3.3.1.1.1.3.cmml"><mrow id="S3.Ex14.m1.3.3.1.1.1.3.2" 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xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3.2.2">𝑛</ci><ci id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3.2.3.cmml" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3.2.3">ℎ</ci></apply><ci id="S3.Ex14.m1.2.2.cmml" xref="S3.Ex14.m1.2.2">𝑃</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex14.m1.3c">1=\sum_{v\in V}1-\sum_{e\in E}1+\sum_{P\in\mathcal{P}}(1+d(P)-n_{h}(P)).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex14.m1.3d">1 = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT 1 - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E end_POSTSUBSCRIPT 1 + ∑ start_POSTSUBSCRIPT italic_P ∈ caligraphic_P end_POSTSUBSCRIPT ( 1 + italic_d ( italic_P ) - italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheoremx1.p1.13">Since <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.5.m1.1"><semantics id="S3.Thmtheoremx1.p1.5.m1.1a"><mi id="S3.Thmtheoremx1.p1.5.m1.1.1" xref="S3.Thmtheoremx1.p1.5.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.5.m1.1b"><ci id="S3.Thmtheoremx1.p1.5.m1.1.1.cmml" xref="S3.Thmtheoremx1.p1.5.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.5.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.5.m1.1d">italic_G</annotation></semantics></math> is connected, there is one outer face, say <math alttext="P_{0}\in\mathcal{P}" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.6.m2.1"><semantics id="S3.Thmtheoremx1.p1.6.m2.1a"><mrow id="S3.Thmtheoremx1.p1.6.m2.1.1" xref="S3.Thmtheoremx1.p1.6.m2.1.1.cmml"><msub id="S3.Thmtheoremx1.p1.6.m2.1.1.2" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2.cmml"><mi id="S3.Thmtheoremx1.p1.6.m2.1.1.2.2" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2.2.cmml">P</mi><mn id="S3.Thmtheoremx1.p1.6.m2.1.1.2.3" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.Thmtheoremx1.p1.6.m2.1.1.1" xref="S3.Thmtheoremx1.p1.6.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheoremx1.p1.6.m2.1.1.3" xref="S3.Thmtheoremx1.p1.6.m2.1.1.3.cmml">𝒫</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.6.m2.1b"><apply id="S3.Thmtheoremx1.p1.6.m2.1.1.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1"><in id="S3.Thmtheoremx1.p1.6.m2.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1.1"></in><apply id="S3.Thmtheoremx1.p1.6.m2.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.6.m2.1.1.2.1.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.6.m2.1.1.2.2.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2.2">𝑃</ci><cn id="S3.Thmtheoremx1.p1.6.m2.1.1.2.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.6.m2.1.1.2.3">0</cn></apply><ci id="S3.Thmtheoremx1.p1.6.m2.1.1.3.cmml" xref="S3.Thmtheoremx1.p1.6.m2.1.1.3">𝒫</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.6.m2.1c">P_{0}\in\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.6.m2.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ caligraphic_P</annotation></semantics></math>, whose boundary is a hole, i.e. <math alttext="d(P_{0})=0" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.7.m3.1"><semantics id="S3.Thmtheoremx1.p1.7.m3.1a"><mrow id="S3.Thmtheoremx1.p1.7.m3.1.1" xref="S3.Thmtheoremx1.p1.7.m3.1.1.cmml"><mrow id="S3.Thmtheoremx1.p1.7.m3.1.1.1" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.cmml"><mi id="S3.Thmtheoremx1.p1.7.m3.1.1.1.3" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.3.cmml">d</mi><mo id="S3.Thmtheoremx1.p1.7.m3.1.1.1.2" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.2" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.2.cmml">P</mi><mn id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.3" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheoremx1.p1.7.m3.1.1.2" xref="S3.Thmtheoremx1.p1.7.m3.1.1.2.cmml">=</mo><mn id="S3.Thmtheoremx1.p1.7.m3.1.1.3" xref="S3.Thmtheoremx1.p1.7.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.7.m3.1b"><apply id="S3.Thmtheoremx1.p1.7.m3.1.1.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1"><eq id="S3.Thmtheoremx1.p1.7.m3.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.2"></eq><apply id="S3.Thmtheoremx1.p1.7.m3.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1"><times id="S3.Thmtheoremx1.p1.7.m3.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.2"></times><ci id="S3.Thmtheoremx1.p1.7.m3.1.1.1.3.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.3">𝑑</ci><apply id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.2">𝑃</ci><cn id="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.7.m3.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="S3.Thmtheoremx1.p1.7.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.7.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.7.m3.1c">d(P_{0})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.7.m3.1d">italic_d ( italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> and <math alttext="n_{h}(P_{0})=1" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.8.m4.1"><semantics id="S3.Thmtheoremx1.p1.8.m4.1a"><mrow id="S3.Thmtheoremx1.p1.8.m4.1.1" xref="S3.Thmtheoremx1.p1.8.m4.1.1.cmml"><mrow id="S3.Thmtheoremx1.p1.8.m4.1.1.1" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.cmml"><msub id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.cmml"><mi id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.2" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.2.cmml">n</mi><mi id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.3" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.3.cmml">h</mi></msub><mo id="S3.Thmtheoremx1.p1.8.m4.1.1.1.2" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.2" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.2.cmml">P</mi><mn id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.3" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheoremx1.p1.8.m4.1.1.2" xref="S3.Thmtheoremx1.p1.8.m4.1.1.2.cmml">=</mo><mn id="S3.Thmtheoremx1.p1.8.m4.1.1.3" xref="S3.Thmtheoremx1.p1.8.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.8.m4.1b"><apply id="S3.Thmtheoremx1.p1.8.m4.1.1.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1"><eq id="S3.Thmtheoremx1.p1.8.m4.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.2"></eq><apply id="S3.Thmtheoremx1.p1.8.m4.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1"><times id="S3.Thmtheoremx1.p1.8.m4.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.2"></times><apply id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.1.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.2.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.2">𝑛</ci><ci id="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.3.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.3.3">ℎ</ci></apply><apply id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.2">𝑃</ci><cn id="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.8.m4.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="S3.Thmtheoremx1.p1.8.m4.1.1.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.8.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.8.m4.1c">n_{h}(P_{0})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.8.m4.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 1</annotation></semantics></math>. For all the other faces <math alttext="P\in\mathcal{P}\setminus\{P_{0}\}" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.9.m5.1"><semantics id="S3.Thmtheoremx1.p1.9.m5.1a"><mrow id="S3.Thmtheoremx1.p1.9.m5.1.1" xref="S3.Thmtheoremx1.p1.9.m5.1.1.cmml"><mi id="S3.Thmtheoremx1.p1.9.m5.1.1.3" xref="S3.Thmtheoremx1.p1.9.m5.1.1.3.cmml">P</mi><mo id="S3.Thmtheoremx1.p1.9.m5.1.1.2" xref="S3.Thmtheoremx1.p1.9.m5.1.1.2.cmml">∈</mo><mrow id="S3.Thmtheoremx1.p1.9.m5.1.1.1" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheoremx1.p1.9.m5.1.1.1.3" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.3.cmml">𝒫</mi><mo id="S3.Thmtheoremx1.p1.9.m5.1.1.1.2" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.2.cmml">∖</mo><mrow id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.2.cmml"><mo id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.2.cmml">{</mo><msub id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.2" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.2.cmml">P</mi><mn id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.3" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.9.m5.1b"><apply id="S3.Thmtheoremx1.p1.9.m5.1.1.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1"><in id="S3.Thmtheoremx1.p1.9.m5.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.2"></in><ci id="S3.Thmtheoremx1.p1.9.m5.1.1.3.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.3">𝑃</ci><apply id="S3.Thmtheoremx1.p1.9.m5.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1"><setdiff id="S3.Thmtheoremx1.p1.9.m5.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.2"></setdiff><ci id="S3.Thmtheoremx1.p1.9.m5.1.1.1.3.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.3">𝒫</ci><set id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1"><apply id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.2">𝑃</ci><cn id="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheoremx1.p1.9.m5.1.1.1.1.1.1.3">0</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.9.m5.1c">P\in\mathcal{P}\setminus\{P_{0}\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.9.m5.1d">italic_P ∈ caligraphic_P ∖ { italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }</annotation></semantics></math>, every hole must intersect the boundary <math alttext="\partial P" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.10.m6.1"><semantics id="S3.Thmtheoremx1.p1.10.m6.1a"><mrow id="S3.Thmtheoremx1.p1.10.m6.1.1" xref="S3.Thmtheoremx1.p1.10.m6.1.1.cmml"><mo id="S3.Thmtheoremx1.p1.10.m6.1.1.1" rspace="0em" xref="S3.Thmtheoremx1.p1.10.m6.1.1.1.cmml">∂</mo><mi id="S3.Thmtheoremx1.p1.10.m6.1.1.2" xref="S3.Thmtheoremx1.p1.10.m6.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.10.m6.1b"><apply id="S3.Thmtheoremx1.p1.10.m6.1.1.cmml" xref="S3.Thmtheoremx1.p1.10.m6.1.1"><partialdiff id="S3.Thmtheoremx1.p1.10.m6.1.1.1.cmml" xref="S3.Thmtheoremx1.p1.10.m6.1.1.1"></partialdiff><ci id="S3.Thmtheoremx1.p1.10.m6.1.1.2.cmml" xref="S3.Thmtheoremx1.p1.10.m6.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.10.m6.1c">\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.10.m6.1d">∂ italic_P</annotation></semantics></math>, and there is exactly one intersection, since otherwise the hole would make <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.11.m7.1"><semantics id="S3.Thmtheoremx1.p1.11.m7.1a"><mi id="S3.Thmtheoremx1.p1.11.m7.1.1" xref="S3.Thmtheoremx1.p1.11.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.11.m7.1b"><ci id="S3.Thmtheoremx1.p1.11.m7.1.1.cmml" xref="S3.Thmtheoremx1.p1.11.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.11.m7.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.11.m7.1d">italic_P</annotation></semantics></math> disconnected. Therefore, <math alttext="d(P)=n_{h}(P)" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.12.m8.2"><semantics id="S3.Thmtheoremx1.p1.12.m8.2a"><mrow id="S3.Thmtheoremx1.p1.12.m8.2.3" xref="S3.Thmtheoremx1.p1.12.m8.2.3.cmml"><mrow id="S3.Thmtheoremx1.p1.12.m8.2.3.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.cmml"><mi id="S3.Thmtheoremx1.p1.12.m8.2.3.2.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.2.cmml">d</mi><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.2.1" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheoremx1.p1.12.m8.2.3.2.3.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.cmml"><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.cmml">(</mo><mi id="S3.Thmtheoremx1.p1.12.m8.1.1" xref="S3.Thmtheoremx1.p1.12.m8.1.1.cmml">P</mi><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.1" xref="S3.Thmtheoremx1.p1.12.m8.2.3.1.cmml">=</mo><mrow id="S3.Thmtheoremx1.p1.12.m8.2.3.3" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.cmml"><msub id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.cmml"><mi id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.2.cmml">n</mi><mi id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.3" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.3.cmml">h</mi></msub><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.3.1" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheoremx1.p1.12.m8.2.3.3.3.2" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.cmml"><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.cmml">(</mo><mi id="S3.Thmtheoremx1.p1.12.m8.2.2" xref="S3.Thmtheoremx1.p1.12.m8.2.2.cmml">P</mi><mo id="S3.Thmtheoremx1.p1.12.m8.2.3.3.3.2.2" stretchy="false" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.12.m8.2b"><apply id="S3.Thmtheoremx1.p1.12.m8.2.3.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3"><eq id="S3.Thmtheoremx1.p1.12.m8.2.3.1.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.1"></eq><apply id="S3.Thmtheoremx1.p1.12.m8.2.3.2.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2"><times id="S3.Thmtheoremx1.p1.12.m8.2.3.2.1.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.1"></times><ci id="S3.Thmtheoremx1.p1.12.m8.2.3.2.2.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.2.2">𝑑</ci><ci id="S3.Thmtheoremx1.p1.12.m8.1.1.cmml" xref="S3.Thmtheoremx1.p1.12.m8.1.1">𝑃</ci></apply><apply id="S3.Thmtheoremx1.p1.12.m8.2.3.3.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3"><times id="S3.Thmtheoremx1.p1.12.m8.2.3.3.1.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.1"></times><apply id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2"><csymbol cd="ambiguous" id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.1.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2">subscript</csymbol><ci id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.2.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.2">𝑛</ci><ci id="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.3.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.3.3.2.3">ℎ</ci></apply><ci id="S3.Thmtheoremx1.p1.12.m8.2.2.cmml" xref="S3.Thmtheoremx1.p1.12.m8.2.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.12.m8.2c">d(P)=n_{h}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.12.m8.2d">italic_d ( italic_P ) = italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> for all such <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheoremx1.p1.13.m9.1"><semantics id="S3.Thmtheoremx1.p1.13.m9.1a"><mi id="S3.Thmtheoremx1.p1.13.m9.1.1" xref="S3.Thmtheoremx1.p1.13.m9.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheoremx1.p1.13.m9.1b"><ci id="S3.Thmtheoremx1.p1.13.m9.1.1.cmml" xref="S3.Thmtheoremx1.p1.13.m9.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheoremx1.p1.13.m9.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheoremx1.p1.13.m9.1d">italic_P</annotation></semantics></math>. In total,</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="2=|V|-|E|+|\mathcal{P}|," class="ltx_Math" display="block" id="S3.Ex15.m1.4"><semantics id="S3.Ex15.m1.4a"><mrow id="S3.Ex15.m1.4.4.1" xref="S3.Ex15.m1.4.4.1.1.cmml"><mrow id="S3.Ex15.m1.4.4.1.1" xref="S3.Ex15.m1.4.4.1.1.cmml"><mn id="S3.Ex15.m1.4.4.1.1.2" xref="S3.Ex15.m1.4.4.1.1.2.cmml">2</mn><mo id="S3.Ex15.m1.4.4.1.1.1" xref="S3.Ex15.m1.4.4.1.1.1.cmml">=</mo><mrow id="S3.Ex15.m1.4.4.1.1.3" xref="S3.Ex15.m1.4.4.1.1.3.cmml"><mrow id="S3.Ex15.m1.4.4.1.1.3.2" xref="S3.Ex15.m1.4.4.1.1.3.2.cmml"><mrow id="S3.Ex15.m1.4.4.1.1.3.2.2.2" xref="S3.Ex15.m1.4.4.1.1.3.2.2.1.cmml"><mo id="S3.Ex15.m1.4.4.1.1.3.2.2.2.1" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.2.2.1.1.cmml">|</mo><mi id="S3.Ex15.m1.1.1" xref="S3.Ex15.m1.1.1.cmml">V</mi><mo id="S3.Ex15.m1.4.4.1.1.3.2.2.2.2" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.2.2.1.1.cmml">|</mo></mrow><mo id="S3.Ex15.m1.4.4.1.1.3.2.1" xref="S3.Ex15.m1.4.4.1.1.3.2.1.cmml">−</mo><mrow id="S3.Ex15.m1.4.4.1.1.3.2.3.2" xref="S3.Ex15.m1.4.4.1.1.3.2.3.1.cmml"><mo id="S3.Ex15.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.2.3.1.1.cmml">|</mo><mi id="S3.Ex15.m1.2.2" xref="S3.Ex15.m1.2.2.cmml">E</mi><mo id="S3.Ex15.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S3.Ex15.m1.4.4.1.1.3.1" xref="S3.Ex15.m1.4.4.1.1.3.1.cmml">+</mo><mrow id="S3.Ex15.m1.4.4.1.1.3.3.2" xref="S3.Ex15.m1.4.4.1.1.3.3.1.cmml"><mo id="S3.Ex15.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex15.m1.3.3" xref="S3.Ex15.m1.3.3.cmml">𝒫</mi><mo id="S3.Ex15.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S3.Ex15.m1.4.4.1.1.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><mo id="S3.Ex15.m1.4.4.1.2" xref="S3.Ex15.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex15.m1.4b"><apply id="S3.Ex15.m1.4.4.1.1.cmml" xref="S3.Ex15.m1.4.4.1"><eq id="S3.Ex15.m1.4.4.1.1.1.cmml" xref="S3.Ex15.m1.4.4.1.1.1"></eq><cn id="S3.Ex15.m1.4.4.1.1.2.cmml" type="integer" xref="S3.Ex15.m1.4.4.1.1.2">2</cn><apply id="S3.Ex15.m1.4.4.1.1.3.cmml" xref="S3.Ex15.m1.4.4.1.1.3"><plus id="S3.Ex15.m1.4.4.1.1.3.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.1"></plus><apply id="S3.Ex15.m1.4.4.1.1.3.2.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2"><minus id="S3.Ex15.m1.4.4.1.1.3.2.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2.1"></minus><apply id="S3.Ex15.m1.4.4.1.1.3.2.2.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2.2.2"><abs id="S3.Ex15.m1.4.4.1.1.3.2.2.1.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2.2.2.1"></abs><ci id="S3.Ex15.m1.1.1.cmml" xref="S3.Ex15.m1.1.1">𝑉</ci></apply><apply id="S3.Ex15.m1.4.4.1.1.3.2.3.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2.3.2"><abs id="S3.Ex15.m1.4.4.1.1.3.2.3.1.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.2.3.2.1"></abs><ci id="S3.Ex15.m1.2.2.cmml" xref="S3.Ex15.m1.2.2">𝐸</ci></apply></apply><apply id="S3.Ex15.m1.4.4.1.1.3.3.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.3.2"><abs id="S3.Ex15.m1.4.4.1.1.3.3.1.1.cmml" xref="S3.Ex15.m1.4.4.1.1.3.3.2.1"></abs><ci id="S3.Ex15.m1.3.3.cmml" xref="S3.Ex15.m1.3.3">𝒫</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m1.4c">2=|V|-|E|+|\mathcal{P}|,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m1.4d">2 = | italic_V | - | italic_E | + | caligraphic_P | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheoremx1.p1.14">which is Euler’s formula.</p> </div> </div> <div class="ltx_para" id="S3.p9"> <p class="ltx_p" id="S3.p9.1">Next is the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</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>Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">Since my definition of polygons allows for very complicated shapes, I will conduct the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a> in multiple steps. I begin in section <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS1" title="3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1.1</span></a> by proving the lemma for polygons whose only boundary component is a single cycle. This result will then be utilised in section <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS2" title="3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1.2</span></a>, where I consider the case in which the boundary does not contain cycles. By viewing a general polygon as an outer polygon with some holes cut out, I will combine the two special cases to prove the complete statement in section <a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS3" title="3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1.3</span></a>.</p> </div> <section class="ltx_subsubsection" id="S3.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.1.1 </span>Only one cycle</h4> <div class="ltx_para" id="S3.SS1.SSS1.p1"> <p class="ltx_p" id="S3.SS1.SSS1.p1.1">I start with a special case of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.4.4">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.1.m1.1b"><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon whose boundary consists of a single polygonal cycle <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.2.m2.1a"><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.2.m2.1b"><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.2.m2.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.1d">italic_γ</annotation></semantics></math>, and let <math alttext="x\in\mathds{R}^{2}" 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">x</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><mn id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">2</mn></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">𝑥</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><cn id="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.3.m3.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> be an arbitrary point in <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.4.m4.1a"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.4.m4.1b"><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.4.m4.1d">italic_P</annotation></semantics></math>-general position. Then, it holds that</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.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="\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{h}(P)=\mathds{1}_{P}(x)-1." class="ltx_Math" display="block" id="S3.E7.m1.7"><semantics id="S3.E7.m1.7a"><mrow id="S3.E7.m1.7.7.1" xref="S3.E7.m1.7.7.1.1.cmml"><mrow id="S3.E7.m1.7.7.1.1" xref="S3.E7.m1.7.7.1.1.cmml"><mrow id="S3.E7.m1.7.7.1.1.2" xref="S3.E7.m1.7.7.1.1.2.cmml"><mrow id="S3.E7.m1.7.7.1.1.2.2" xref="S3.E7.m1.7.7.1.1.2.2.cmml"><munder id="S3.E7.m1.7.7.1.1.2.2.1" xref="S3.E7.m1.7.7.1.1.2.2.1.cmml"><mo id="S3.E7.m1.7.7.1.1.2.2.1.2" movablelimits="false" xref="S3.E7.m1.7.7.1.1.2.2.1.2.cmml">∑</mo><mrow id="S3.E7.m1.1.1.1" xref="S3.E7.m1.1.1.1.cmml"><mi id="S3.E7.m1.1.1.1.3" xref="S3.E7.m1.1.1.1.3.cmml">v</mi><mo 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id="S3.E7.m1.7.7.1.1.3.2.1.cmml" xref="S3.E7.m1.7.7.1.1.3.2.1"></times><apply id="S3.E7.m1.7.7.1.1.3.2.2.cmml" xref="S3.E7.m1.7.7.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E7.m1.7.7.1.1.3.2.2.1.cmml" xref="S3.E7.m1.7.7.1.1.3.2.2">subscript</csymbol><cn id="S3.E7.m1.7.7.1.1.3.2.2.2.cmml" type="integer" xref="S3.E7.m1.7.7.1.1.3.2.2.2">1</cn><ci id="S3.E7.m1.7.7.1.1.3.2.2.3.cmml" xref="S3.E7.m1.7.7.1.1.3.2.2.3">𝑃</ci></apply><ci id="S3.E7.m1.6.6.cmml" xref="S3.E7.m1.6.6">𝑥</ci></apply><cn id="S3.E7.m1.7.7.1.1.3.3.cmml" type="integer" xref="S3.E7.m1.7.7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.7c">\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{h}(P)=\mathds{1}_{P}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.7d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 .</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">(7)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.SSS1.p2"> <p class="ltx_p" id="S3.SS1.SSS1.p2.10">Note that in this case <math alttext="n_{h}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.1.m1.1"><semantics id="S3.SS1.SSS1.p2.1.m1.1a"><mrow id="S3.SS1.SSS1.p2.1.m1.1.2" xref="S3.SS1.SSS1.p2.1.m1.1.2.cmml"><msub id="S3.SS1.SSS1.p2.1.m1.1.2.2" xref="S3.SS1.SSS1.p2.1.m1.1.2.2.cmml"><mi id="S3.SS1.SSS1.p2.1.m1.1.2.2.2" xref="S3.SS1.SSS1.p2.1.m1.1.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS1.p2.1.m1.1.2.2.3" xref="S3.SS1.SSS1.p2.1.m1.1.2.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS1.p2.1.m1.1.2.1" xref="S3.SS1.SSS1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.1.m1.1.2.3.2" xref="S3.SS1.SSS1.p2.1.m1.1.2.cmml"><mo id="S3.SS1.SSS1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.1.m1.1.2.cmml">(</mo><mi id="S3.SS1.SSS1.p2.1.m1.1.1" xref="S3.SS1.SSS1.p2.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.1.m1.1b"><apply id="S3.SS1.SSS1.p2.1.m1.1.2.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2"><times id="S3.SS1.SSS1.p2.1.m1.1.2.1.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2.1"></times><apply id="S3.SS1.SSS1.p2.1.m1.1.2.2.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p2.1.m1.1.2.2.1.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.p2.1.m1.1.2.2.2.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2.2.2">𝑛</ci><ci id="S3.SS1.SSS1.p2.1.m1.1.2.2.3.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.2.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS1.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS1.p2.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.1.m1.1c">n_{h}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> is <math alttext="1" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.2.m2.1"><semantics id="S3.SS1.SSS1.p2.2.m2.1a"><mn id="S3.SS1.SSS1.p2.2.m2.1.1" xref="S3.SS1.SSS1.p2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.2.m2.1b"><cn id="S3.SS1.SSS1.p2.2.m2.1.1.cmml" type="integer" xref="S3.SS1.SSS1.p2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.2.m2.1d">1</annotation></semantics></math> if <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.3.m3.1"><semantics id="S3.SS1.SSS1.p2.3.m3.1a"><mi id="S3.SS1.SSS1.p2.3.m3.1.1" xref="S3.SS1.SSS1.p2.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.3.m3.1b"><ci id="S3.SS1.SSS1.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS1.p2.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.3.m3.1d">italic_P</annotation></semantics></math> is unbounded, i.e. if <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.4.m4.1"><semantics id="S3.SS1.SSS1.p2.4.m4.1a"><mi id="S3.SS1.SSS1.p2.4.m4.1.1" xref="S3.SS1.SSS1.p2.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.4.m4.1b"><ci id="S3.SS1.SSS1.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS1.p2.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.4.m4.1d">italic_γ</annotation></semantics></math> describes a hole, and <math alttext="0" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.5.m5.1"><semantics id="S3.SS1.SSS1.p2.5.m5.1a"><mn id="S3.SS1.SSS1.p2.5.m5.1.1" xref="S3.SS1.SSS1.p2.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.5.m5.1b"><cn id="S3.SS1.SSS1.p2.5.m5.1.1.cmml" type="integer" xref="S3.SS1.SSS1.p2.5.m5.1.1">0</cn></annotation-xml></semantics></math> otherwise. Moreover, we have <math alttext="\deg_{P}(v)=2" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.6.m6.2"><semantics id="S3.SS1.SSS1.p2.6.m6.2a"><mrow id="S3.SS1.SSS1.p2.6.m6.2.2" xref="S3.SS1.SSS1.p2.6.m6.2.2.cmml"><mrow id="S3.SS1.SSS1.p2.6.m6.2.2.1.1" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml"><msub id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.2" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.2.cmml">deg</mi><mi id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.3" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.3.cmml">P</mi></msub><mo id="S3.SS1.SSS1.p2.6.m6.2.2.1.1a" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml">⁡</mo><mrow id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.2" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml"><mo id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml">(</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1" xref="S3.SS1.SSS1.p2.6.m6.1.1.cmml">v</mi><mo id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.p2.6.m6.2.2.2" xref="S3.SS1.SSS1.p2.6.m6.2.2.2.cmml">=</mo><mn id="S3.SS1.SSS1.p2.6.m6.2.2.3" xref="S3.SS1.SSS1.p2.6.m6.2.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.6.m6.2b"><apply id="S3.SS1.SSS1.p2.6.m6.2.2.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2"><eq id="S3.SS1.SSS1.p2.6.m6.2.2.2.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.2"></eq><apply id="S3.SS1.SSS1.p2.6.m6.2.2.1.2.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1"><apply id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1">subscript</csymbol><csymbol cd="latexml" id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.2">degree</csymbol><ci id="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS1.p2.6.m6.2.2.1.1.1.3">𝑃</ci></apply><ci id="S3.SS1.SSS1.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS1.p2.6.m6.1.1">𝑣</ci></apply><cn id="S3.SS1.SSS1.p2.6.m6.2.2.3.cmml" type="integer" xref="S3.SS1.SSS1.p2.6.m6.2.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.6.m6.2c">\deg_{P}(v)=2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.6.m6.2d">roman_deg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_v ) = 2</annotation></semantics></math> for all <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.7.m7.1"><semantics id="S3.SS1.SSS1.p2.7.m7.1a"><mrow id="S3.SS1.SSS1.p2.7.m7.1.2" xref="S3.SS1.SSS1.p2.7.m7.1.2.cmml"><mi id="S3.SS1.SSS1.p2.7.m7.1.2.2" xref="S3.SS1.SSS1.p2.7.m7.1.2.2.cmml">v</mi><mo id="S3.SS1.SSS1.p2.7.m7.1.2.1" xref="S3.SS1.SSS1.p2.7.m7.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.p2.7.m7.1.2.3" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.cmml"><mi id="S3.SS1.SSS1.p2.7.m7.1.2.3.2" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS1.p2.7.m7.1.2.3.1" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.7.m7.1.2.3.3.2" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.cmml"><mo id="S3.SS1.SSS1.p2.7.m7.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.p2.7.m7.1.1" xref="S3.SS1.SSS1.p2.7.m7.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.p2.7.m7.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.7.m7.1b"><apply id="S3.SS1.SSS1.p2.7.m7.1.2.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2"><in id="S3.SS1.SSS1.p2.7.m7.1.2.1.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2.1"></in><ci id="S3.SS1.SSS1.p2.7.m7.1.2.2.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2.2">𝑣</ci><apply id="S3.SS1.SSS1.p2.7.m7.1.2.3.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2.3"><times id="S3.SS1.SSS1.p2.7.m7.1.2.3.1.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.1"></times><ci id="S3.SS1.SSS1.p2.7.m7.1.2.3.2.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.2.3.2">𝑉</ci><ci id="S3.SS1.SSS1.p2.7.m7.1.1.cmml" xref="S3.SS1.SSS1.p2.7.m7.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.7.m7.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.7.m7.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math>, and thus <math alttext="d(P)=0" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.8.m8.1"><semantics id="S3.SS1.SSS1.p2.8.m8.1a"><mrow id="S3.SS1.SSS1.p2.8.m8.1.2" xref="S3.SS1.SSS1.p2.8.m8.1.2.cmml"><mrow id="S3.SS1.SSS1.p2.8.m8.1.2.2" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.cmml"><mi id="S3.SS1.SSS1.p2.8.m8.1.2.2.2" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.2.cmml">d</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.2.2.1" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.8.m8.1.2.2.3.2" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.cmml"><mo id="S3.SS1.SSS1.p2.8.m8.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1" xref="S3.SS1.SSS1.p2.8.m8.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.p2.8.m8.1.2.1" xref="S3.SS1.SSS1.p2.8.m8.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS1.p2.8.m8.1.2.3" xref="S3.SS1.SSS1.p2.8.m8.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.8.m8.1b"><apply id="S3.SS1.SSS1.p2.8.m8.1.2.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.2"><eq id="S3.SS1.SSS1.p2.8.m8.1.2.1.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.2.1"></eq><apply id="S3.SS1.SSS1.p2.8.m8.1.2.2.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.2.2"><times id="S3.SS1.SSS1.p2.8.m8.1.2.2.1.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.1"></times><ci id="S3.SS1.SSS1.p2.8.m8.1.2.2.2.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.2.2.2">𝑑</ci><ci id="S3.SS1.SSS1.p2.8.m8.1.1.cmml" xref="S3.SS1.SSS1.p2.8.m8.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS1.p2.8.m8.1.2.3.cmml" type="integer" xref="S3.SS1.SSS1.p2.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.8.m8.1c">d(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.8.m8.1d">italic_d ( italic_P ) = 0</annotation></semantics></math>. As the boundary does not contain any arcs, this implies <math alttext="c(P)=1-n_{h}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.9.m9.2"><semantics id="S3.SS1.SSS1.p2.9.m9.2a"><mrow id="S3.SS1.SSS1.p2.9.m9.2.3" xref="S3.SS1.SSS1.p2.9.m9.2.3.cmml"><mrow id="S3.SS1.SSS1.p2.9.m9.2.3.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.cmml"><mi id="S3.SS1.SSS1.p2.9.m9.2.3.2.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.2.cmml">c</mi><mo id="S3.SS1.SSS1.p2.9.m9.2.3.2.1" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.9.m9.2.3.2.3.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.cmml"><mo id="S3.SS1.SSS1.p2.9.m9.2.3.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.cmml">(</mo><mi id="S3.SS1.SSS1.p2.9.m9.1.1" xref="S3.SS1.SSS1.p2.9.m9.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.p2.9.m9.2.3.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.p2.9.m9.2.3.1" xref="S3.SS1.SSS1.p2.9.m9.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS1.p2.9.m9.2.3.3" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.cmml"><mn id="S3.SS1.SSS1.p2.9.m9.2.3.3.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.p2.9.m9.2.3.3.1" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.1.cmml">−</mo><mrow id="S3.SS1.SSS1.p2.9.m9.2.3.3.3" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.cmml"><msub id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.cmml"><mi id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.2.cmml">n</mi><mi id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.3" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.1" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.3.2" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.cmml"><mo id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.cmml">(</mo><mi id="S3.SS1.SSS1.p2.9.m9.2.2" xref="S3.SS1.SSS1.p2.9.m9.2.2.cmml">P</mi><mo id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.9.m9.2b"><apply id="S3.SS1.SSS1.p2.9.m9.2.3.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3"><eq id="S3.SS1.SSS1.p2.9.m9.2.3.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.1"></eq><apply id="S3.SS1.SSS1.p2.9.m9.2.3.2.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.2"><times id="S3.SS1.SSS1.p2.9.m9.2.3.2.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.1"></times><ci id="S3.SS1.SSS1.p2.9.m9.2.3.2.2.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.2.2">𝑐</ci><ci id="S3.SS1.SSS1.p2.9.m9.1.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS1.p2.9.m9.2.3.3.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3"><minus id="S3.SS1.SSS1.p2.9.m9.2.3.3.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.1"></minus><cn id="S3.SS1.SSS1.p2.9.m9.2.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.2">1</cn><apply id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3"><times id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.1"></times><apply id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.1.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.2.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.2">𝑛</ci><ci id="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.3.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.3.3.3.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS1.p2.9.m9.2.2.cmml" xref="S3.SS1.SSS1.p2.9.m9.2.2">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.9.m9.2c">c(P)=1-n_{h}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.9.m9.2d">italic_c ( italic_P ) = 1 - italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>. Therefore, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem3" title="Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.3</span></a> is indeed a special case of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a> with <math alttext="E_{l}(P)=\emptyset" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p2.10.m10.1"><semantics id="S3.SS1.SSS1.p2.10.m10.1a"><mrow id="S3.SS1.SSS1.p2.10.m10.1.2" xref="S3.SS1.SSS1.p2.10.m10.1.2.cmml"><mrow id="S3.SS1.SSS1.p2.10.m10.1.2.2" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.cmml"><msub id="S3.SS1.SSS1.p2.10.m10.1.2.2.2" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2.cmml"><mi id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.2" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2.2.cmml">E</mi><mi id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.3" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2.3.cmml">l</mi></msub><mo id="S3.SS1.SSS1.p2.10.m10.1.2.2.1" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p2.10.m10.1.2.2.3.2" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.cmml"><mo id="S3.SS1.SSS1.p2.10.m10.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS1.p2.10.m10.1.1" xref="S3.SS1.SSS1.p2.10.m10.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.p2.10.m10.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.p2.10.m10.1.2.1" xref="S3.SS1.SSS1.p2.10.m10.1.2.1.cmml">=</mo><mi id="S3.SS1.SSS1.p2.10.m10.1.2.3" mathvariant="normal" xref="S3.SS1.SSS1.p2.10.m10.1.2.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p2.10.m10.1b"><apply id="S3.SS1.SSS1.p2.10.m10.1.2.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2"><eq id="S3.SS1.SSS1.p2.10.m10.1.2.1.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.1"></eq><apply id="S3.SS1.SSS1.p2.10.m10.1.2.2.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2"><times id="S3.SS1.SSS1.p2.10.m10.1.2.2.1.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.1"></times><apply id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.1.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.2.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2.2">𝐸</ci><ci id="S3.SS1.SSS1.p2.10.m10.1.2.2.2.3.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.2.2.3">𝑙</ci></apply><ci id="S3.SS1.SSS1.p2.10.m10.1.1.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.1">𝑃</ci></apply><emptyset id="S3.SS1.SSS1.p2.10.m10.1.2.3.cmml" xref="S3.SS1.SSS1.p2.10.m10.1.2.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p2.10.m10.1c">E_{l}(P)=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p2.10.m10.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) = ∅</annotation></semantics></math>.</p> </div> <div class="ltx_proof" id="S3.SS1.SSS1.9"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS1.1.p1"> <p class="ltx_p" id="S3.SS1.SSS1.1.p1.5">Let us first assume that <math alttext="P=\overline{\operatorname*{int}(\gamma)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.1.m1.2"><semantics id="S3.SS1.SSS1.1.p1.1.m1.2a"><mrow id="S3.SS1.SSS1.1.p1.1.m1.2.3" xref="S3.SS1.SSS1.1.p1.1.m1.2.3.cmml"><mi id="S3.SS1.SSS1.1.p1.1.m1.2.3.2" xref="S3.SS1.SSS1.1.p1.1.m1.2.3.2.cmml">P</mi><mo id="S3.SS1.SSS1.1.p1.1.m1.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS1.1.p1.1.m1.2.3.1.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.1.p1.1.m1.2.2" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.cmml"><mrow id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.4" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.1.p1.1.m1.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.1.1.cmml">int</mo><mrow id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.4.1" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.4.1.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.2" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.2.cmml">γ</mi><mo id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.4.1.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.1.p1.1.m1.2.2.3" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.1.m1.2b"><apply id="S3.SS1.SSS1.1.p1.1.m1.2.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.3"><eq id="S3.SS1.SSS1.1.p1.1.m1.2.3.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.3.1"></eq><ci id="S3.SS1.SSS1.1.p1.1.m1.2.3.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.3.2">𝑃</ci><apply id="S3.SS1.SSS1.1.p1.1.m1.2.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.2"><ci id="S3.SS1.SSS1.1.p1.1.m1.2.2.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.3">¯</ci><apply id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.4"><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.1.1">int</ci><ci id="S3.SS1.SSS1.1.p1.1.m1.2.2.2.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.2.2.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.1.m1.2c">P=\overline{\operatorname*{int}(\gamma)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.1.m1.2d">italic_P = over¯ start_ARG roman_int ( italic_γ ) end_ARG</annotation></semantics></math>, which is equivalent to <math alttext="n_{h}(P)=0" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.2.m2.1"><semantics id="S3.SS1.SSS1.1.p1.2.m2.1a"><mrow id="S3.SS1.SSS1.1.p1.2.m2.1.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.cmml"><mrow id="S3.SS1.SSS1.1.p1.2.m2.1.2.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.cmml"><msub id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.cmml"><mi id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.3" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.1" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.3.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.cmml"><mo id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.2.m2.1.1" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.1.p1.2.m2.1.2.1" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS1.1.p1.2.m2.1.2.3" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.2.m2.1b"><apply id="S3.SS1.SSS1.1.p1.2.m2.1.2.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2"><eq id="S3.SS1.SSS1.1.p1.2.m2.1.2.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.1"></eq><apply id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2"><times id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.1"></times><apply id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.2.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.3.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.2.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS1.1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS1.1.p1.2.m2.1.2.3.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.2.m2.1c">n_{h}(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = 0</annotation></semantics></math>. Define <math alttext="v_{P}(x):=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.3.m3.3"><semantics id="S3.SS1.SSS1.1.p1.3.m3.3a"><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.cmml"><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.cmml"><msub id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2.cmml"><mi id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2.2.cmml">v</mi><mi id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.2.3.cmml">P</mi></msub><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.1" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.3.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.3.m3.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.2.2.cmml">x</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.1" rspace="0.111em" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.1.cmml">:=</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.cmml"><msub id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.1" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.1.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.1.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.1.2.cmml">∑</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.3" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.3.cmml">v</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.cmml"><mi id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.2.cmml">V</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.3.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.3.2.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.3.2.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.cmml"><msub id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.cmml"><mn id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.cmml"><mi id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.3.cmml">P</mi><mi id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.2.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.2.3.2.3.cmml">v</mi></msubsup></msub><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.1" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.3.2" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.3.m3.3.3" xref="S3.SS1.SSS1.1.p1.3.m3.3.3.cmml">x</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.3.m3.3b"><apply id="S3.SS1.SSS1.1.p1.3.m3.3.4.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.3.4"><csymbol cd="latexml" id="S3.SS1.SSS1.1.p1.3.m3.3.4.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.3.4.1">assign</csymbol><apply id="S3.SS1.SSS1.1.p1.3.m3.3.4.2.cmml" 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xref="S3.SS1.SSS1.1.p1.4.m4.3.4.3.2.2.3.3">𝑃</ci></apply></apply><ci id="S3.SS1.SSS1.1.p1.4.m4.3.3.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.4.m4.3c">e_{P}(x):=\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P}}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.4.m4.3d">italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) := ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. Note that a sum over indicators corresponds to counting the number of sets containing <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.5.m5.1"><semantics id="S3.SS1.SSS1.1.p1.5.m5.1a"><mi id="S3.SS1.SSS1.1.p1.5.m5.1.1" xref="S3.SS1.SSS1.1.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.5.m5.1b"><ci id="S3.SS1.SSS1.1.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS1.1.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.5.m5.1d">italic_x</annotation></semantics></math>. Therefore, we can reformulate the left hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E7" title="Equation 7 ‣ Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">7</span></a>) as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx4"> <tbody id="S3.E8"><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 v_{P}(x)-e_{P}(x)" class="ltx_Math" display="inline" id="S3.E8.m1.2"><semantics id="S3.E8.m1.2a"><mrow id="S3.E8.m1.2.3" xref="S3.E8.m1.2.3.cmml"><mrow id="S3.E8.m1.2.3.2" xref="S3.E8.m1.2.3.2.cmml"><msub id="S3.E8.m1.2.3.2.2" xref="S3.E8.m1.2.3.2.2.cmml"><mi id="S3.E8.m1.2.3.2.2.2" xref="S3.E8.m1.2.3.2.2.2.cmml">v</mi><mi id="S3.E8.m1.2.3.2.2.3" xref="S3.E8.m1.2.3.2.2.3.cmml">P</mi></msub><mo id="S3.E8.m1.2.3.2.1" xref="S3.E8.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.E8.m1.2.3.2.3.2" xref="S3.E8.m1.2.3.2.cmml"><mo id="S3.E8.m1.2.3.2.3.2.1" stretchy="false" xref="S3.E8.m1.2.3.2.cmml">(</mo><mi id="S3.E8.m1.1.1" xref="S3.E8.m1.1.1.cmml">x</mi><mo id="S3.E8.m1.2.3.2.3.2.2" stretchy="false" xref="S3.E8.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.E8.m1.2.3.1" xref="S3.E8.m1.2.3.1.cmml">−</mo><mrow id="S3.E8.m1.2.3.3" xref="S3.E8.m1.2.3.3.cmml"><msub id="S3.E8.m1.2.3.3.2" xref="S3.E8.m1.2.3.3.2.cmml"><mi id="S3.E8.m1.2.3.3.2.2" xref="S3.E8.m1.2.3.3.2.2.cmml">e</mi><mi id="S3.E8.m1.2.3.3.2.3" xref="S3.E8.m1.2.3.3.2.3.cmml">P</mi></msub><mo id="S3.E8.m1.2.3.3.1" xref="S3.E8.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.E8.m1.2.3.3.3.2" xref="S3.E8.m1.2.3.3.cmml"><mo id="S3.E8.m1.2.3.3.3.2.1" stretchy="false" xref="S3.E8.m1.2.3.3.cmml">(</mo><mi id="S3.E8.m1.2.2" xref="S3.E8.m1.2.2.cmml">x</mi><mo id="S3.E8.m1.2.3.3.3.2.2" stretchy="false" xref="S3.E8.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m1.2b"><apply id="S3.E8.m1.2.3.cmml" xref="S3.E8.m1.2.3"><minus id="S3.E8.m1.2.3.1.cmml" xref="S3.E8.m1.2.3.1"></minus><apply id="S3.E8.m1.2.3.2.cmml" xref="S3.E8.m1.2.3.2"><times id="S3.E8.m1.2.3.2.1.cmml" xref="S3.E8.m1.2.3.2.1"></times><apply id="S3.E8.m1.2.3.2.2.cmml" xref="S3.E8.m1.2.3.2.2"><csymbol cd="ambiguous" id="S3.E8.m1.2.3.2.2.1.cmml" xref="S3.E8.m1.2.3.2.2">subscript</csymbol><ci id="S3.E8.m1.2.3.2.2.2.cmml" xref="S3.E8.m1.2.3.2.2.2">𝑣</ci><ci id="S3.E8.m1.2.3.2.2.3.cmml" xref="S3.E8.m1.2.3.2.2.3">𝑃</ci></apply><ci id="S3.E8.m1.1.1.cmml" xref="S3.E8.m1.1.1">𝑥</ci></apply><apply id="S3.E8.m1.2.3.3.cmml" xref="S3.E8.m1.2.3.3"><times id="S3.E8.m1.2.3.3.1.cmml" xref="S3.E8.m1.2.3.3.1"></times><apply id="S3.E8.m1.2.3.3.2.cmml" xref="S3.E8.m1.2.3.3.2"><csymbol cd="ambiguous" id="S3.E8.m1.2.3.3.2.1.cmml" xref="S3.E8.m1.2.3.3.2">subscript</csymbol><ci id="S3.E8.m1.2.3.3.2.2.cmml" xref="S3.E8.m1.2.3.3.2.2">𝑒</ci><ci id="S3.E8.m1.2.3.3.2.3.cmml" xref="S3.E8.m1.2.3.3.2.3">𝑃</ci></apply><ci id="S3.E8.m1.2.2.cmml" xref="S3.E8.m1.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.2c">\displaystyle v_{P}(x)-e_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.2d">italic_v start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\absolutevalue{\{v\in V(P):x\in Q^{v}_{P}\}}-\absolutevalue{\{e% \in E_{b}(P):x\in H^{e}_{P}\}}," class="ltx_Math" display="inline" id="S3.E8.m2.3"><semantics id="S3.E8.m2.3a"><mrow id="S3.E8.m2.3.3.1" xref="S3.E8.m2.3.3.1.1.cmml"><mrow id="S3.E8.m2.3.3.1.1" xref="S3.E8.m2.3.3.1.1.cmml"><mi id="S3.E8.m2.3.3.1.1.2" xref="S3.E8.m2.3.3.1.1.2.cmml"></mi><mo 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id="S3.E8.m2.2.2.1.1.1.3.2.3.3" xref="S3.E8.m2.2.2.1.1.1.3.2.3.3.cmml">P</mi><mi id="S3.E8.m2.2.2.1.1.1.3.2.3.2.3" xref="S3.E8.m2.2.2.1.1.1.3.2.3.2.3.cmml">e</mi></msubsup></mrow><mo id="S3.E8.m2.2.2.1.1.1.3.5" stretchy="false" xref="S3.E8.m2.2.2.1.1.1.4.1.cmml">}</mo></mrow><mo id="S3.E8.m2.2.2a.3.2" xref="S3.E8.m2.2.2a.2.1.cmml">|</mo></mrow></mrow></mrow><mo id="S3.E8.m2.3.3.1.2" xref="S3.E8.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m2.3b"><apply id="S3.E8.m2.3.3.1.1.cmml" xref="S3.E8.m2.3.3.1"><eq id="S3.E8.m2.3.3.1.1.1.cmml" xref="S3.E8.m2.3.3.1.1.1"></eq><csymbol cd="latexml" id="S3.E8.m2.3.3.1.1.2.cmml" xref="S3.E8.m2.3.3.1.1.2">absent</csymbol><apply id="S3.E8.m2.3.3.1.1.3.cmml" xref="S3.E8.m2.3.3.1.1.3"><minus id="S3.E8.m2.3.3.1.1.3.1.cmml" xref="S3.E8.m2.3.3.1.1.3.1"></minus><apply id="S3.E8.m2.1.1a.2.cmml" xref="S3.E8.m2.1.1a.3"><abs id="S3.E8.m2.1.1a.2.1.cmml" xref="S3.E8.m2.1.1a.3.1"></abs><apply id="S3.E8.m2.1.1.1.1.1.4.cmml" xref="S3.E8.m2.1.1.1.1.1.3"><csymbol cd="latexml" id="S3.E8.m2.1.1.1.1.1.4.1.cmml" xref="S3.E8.m2.1.1.1.1.1.3.3">conditional-set</csymbol><apply id="S3.E8.m2.1.1.1.1.1.2.1.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1"><in id="S3.E8.m2.1.1.1.1.1.2.1.1.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1.1"></in><ci id="S3.E8.m2.1.1.1.1.1.2.1.2.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1.2">𝑣</ci><apply id="S3.E8.m2.1.1.1.1.1.2.1.3.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1.3"><times id="S3.E8.m2.1.1.1.1.1.2.1.3.1.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1.3.1"></times><ci id="S3.E8.m2.1.1.1.1.1.2.1.3.2.cmml" xref="S3.E8.m2.1.1.1.1.1.2.1.3.2">𝑉</ci><ci id="S3.E8.m2.1.1.1.1.1.1.cmml" xref="S3.E8.m2.1.1.1.1.1.1">𝑃</ci></apply></apply><apply id="S3.E8.m2.1.1.1.1.1.3.2.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2"><in id="S3.E8.m2.1.1.1.1.1.3.2.1.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.1"></in><ci id="S3.E8.m2.1.1.1.1.1.3.2.2.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.2">𝑥</ci><apply id="S3.E8.m2.1.1.1.1.1.3.2.3.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E8.m2.1.1.1.1.1.3.2.3.1.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3">subscript</csymbol><apply id="S3.E8.m2.1.1.1.1.1.3.2.3.2.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E8.m2.1.1.1.1.1.3.2.3.2.1.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3">superscript</csymbol><ci id="S3.E8.m2.1.1.1.1.1.3.2.3.2.2.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3.2.2">𝑄</ci><ci id="S3.E8.m2.1.1.1.1.1.3.2.3.2.3.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3.2.3">𝑣</ci></apply><ci id="S3.E8.m2.1.1.1.1.1.3.2.3.3.cmml" xref="S3.E8.m2.1.1.1.1.1.3.2.3.3">𝑃</ci></apply></apply></apply></apply><apply id="S3.E8.m2.2.2a.2.cmml" xref="S3.E8.m2.2.2a.3"><abs id="S3.E8.m2.2.2a.2.1.cmml" xref="S3.E8.m2.2.2a.3.1"></abs><apply id="S3.E8.m2.2.2.1.1.1.4.cmml" xref="S3.E8.m2.2.2.1.1.1.3"><csymbol cd="latexml" id="S3.E8.m2.2.2.1.1.1.4.1.cmml" xref="S3.E8.m2.2.2.1.1.1.3.3">conditional-set</csymbol><apply id="S3.E8.m2.2.2.1.1.1.2.1.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1"><in id="S3.E8.m2.2.2.1.1.1.2.1.1.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.1"></in><ci id="S3.E8.m2.2.2.1.1.1.2.1.2.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.2">𝑒</ci><apply id="S3.E8.m2.2.2.1.1.1.2.1.3.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3"><times id="S3.E8.m2.2.2.1.1.1.2.1.3.1.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3.1"></times><apply id="S3.E8.m2.2.2.1.1.1.2.1.3.2.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3.2"><csymbol cd="ambiguous" id="S3.E8.m2.2.2.1.1.1.2.1.3.2.1.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3.2">subscript</csymbol><ci id="S3.E8.m2.2.2.1.1.1.2.1.3.2.2.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3.2.2">𝐸</ci><ci id="S3.E8.m2.2.2.1.1.1.2.1.3.2.3.cmml" xref="S3.E8.m2.2.2.1.1.1.2.1.3.2.3">𝑏</ci></apply><ci id="S3.E8.m2.2.2.1.1.1.1.cmml" xref="S3.E8.m2.2.2.1.1.1.1">𝑃</ci></apply></apply><apply id="S3.E8.m2.2.2.1.1.1.3.2.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2"><in id="S3.E8.m2.2.2.1.1.1.3.2.1.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.1"></in><ci id="S3.E8.m2.2.2.1.1.1.3.2.2.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.2">𝑥</ci><apply id="S3.E8.m2.2.2.1.1.1.3.2.3.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E8.m2.2.2.1.1.1.3.2.3.1.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3">subscript</csymbol><apply id="S3.E8.m2.2.2.1.1.1.3.2.3.2.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.E8.m2.2.2.1.1.1.3.2.3.2.1.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3">superscript</csymbol><ci id="S3.E8.m2.2.2.1.1.1.3.2.3.2.2.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3.2.2">𝐻</ci><ci id="S3.E8.m2.2.2.1.1.1.3.2.3.2.3.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3.2.3">𝑒</ci></apply><ci id="S3.E8.m2.2.2.1.1.1.3.2.3.3.cmml" xref="S3.E8.m2.2.2.1.1.1.3.2.3.3">𝑃</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m2.3c">\displaystyle=\absolutevalue{\{v\in V(P):x\in Q^{v}_{P}\}}-\absolutevalue{\{e% \in E_{b}(P):x\in H^{e}_{P}\}},</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m2.3d">= | start_ARG { italic_v ∈ italic_V ( italic_P ) : italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT } end_ARG | - | start_ARG { italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) : italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT } end_ARG | ,</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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.1.p1.7">which is the difference between the number of vertices and edges that have <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.6.m1.1"><semantics id="S3.SS1.SSS1.1.p1.6.m1.1a"><mi id="S3.SS1.SSS1.1.p1.6.m1.1.1" xref="S3.SS1.SSS1.1.p1.6.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.6.m1.1b"><ci id="S3.SS1.SSS1.1.p1.6.m1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.6.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.6.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.6.m1.1d">italic_x</annotation></semantics></math> on their respective <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.7.m2.1"><semantics id="S3.SS1.SSS1.1.p1.7.m2.1a"><mi id="S3.SS1.SSS1.1.p1.7.m2.1.1" xref="S3.SS1.SSS1.1.p1.7.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.7.m2.1b"><ci id="S3.SS1.SSS1.1.p1.7.m2.1.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.7.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.7.m2.1d">italic_P</annotation></semantics></math>-side.</p> </div> <div class="ltx_para" id="S3.SS1.SSS1.2.p2"> <p class="ltx_p" id="S3.SS1.SSS1.2.p2.18">Let <math alttext="n:=|V(P)|" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.1.m1.2"><semantics id="S3.SS1.SSS1.2.p2.1.m1.2a"><mrow id="S3.SS1.SSS1.2.p2.1.m1.2.2" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.cmml"><mi id="S3.SS1.SSS1.2.p2.1.m1.2.2.3" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.3.cmml">n</mi><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.2.cmml">:=</mo><mrow id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.2.cmml"><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.2" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.2.cmml">V</mi><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.1" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.3.2" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.cmml"><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.cmml">(</mo><mi id="S3.SS1.SSS1.2.p2.1.m1.1.1" xref="S3.SS1.SSS1.2.p2.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.1.m1.2b"><apply id="S3.SS1.SSS1.2.p2.1.m1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2"><csymbol cd="latexml" id="S3.SS1.SSS1.2.p2.1.m1.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.2">assign</csymbol><ci id="S3.SS1.SSS1.2.p2.1.m1.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.3">𝑛</ci><apply id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.2.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1"><abs id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.2"></abs><apply id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1"><times id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.1"></times><ci id="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.2.2.1.1.1.2">𝑉</ci><ci id="S3.SS1.SSS1.2.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.1.m1.1.1">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.1.m1.2c">n:=|V(P)|</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.1.m1.2d">italic_n := | italic_V ( italic_P ) |</annotation></semantics></math> be the number of vertices of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.2.m2.1"><semantics id="S3.SS1.SSS1.2.p2.2.m2.1a"><mi id="S3.SS1.SSS1.2.p2.2.m2.1.1" xref="S3.SS1.SSS1.2.p2.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.2.m2.1b"><ci id="S3.SS1.SSS1.2.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.2.m2.1d">italic_P</annotation></semantics></math>. In the following, indices are always to be understood modulo <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.3.m3.1"><semantics id="S3.SS1.SSS1.2.p2.3.m3.1a"><mi id="S3.SS1.SSS1.2.p2.3.m3.1.1" xref="S3.SS1.SSS1.2.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.3.m3.1b"><ci id="S3.SS1.SSS1.2.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS1.2.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.3.m3.1d">italic_n</annotation></semantics></math>. Since <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.4.m4.1"><semantics id="S3.SS1.SSS1.2.p2.4.m4.1a"><mi id="S3.SS1.SSS1.2.p2.4.m4.1.1" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.4.m4.1b"><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.4.m4.1d">italic_P</annotation></semantics></math> is homeomorphic to a closed disk, we can enumerate its vertices <math alttext="V(P)=\{v_{k}\}_{k=1}^{n}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.5.m5.2"><semantics id="S3.SS1.SSS1.2.p2.5.m5.2a"><mrow id="S3.SS1.SSS1.2.p2.5.m5.2.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.cmml"><mrow id="S3.SS1.SSS1.2.p2.5.m5.2.2.3" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.cmml"><mi id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.1" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.3.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.cmml"><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.2.p2.5.m5.1.1" xref="S3.SS1.SSS1.2.p2.5.m5.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.2.cmml">=</mo><msubsup id="S3.SS1.SSS1.2.p2.5.m5.2.2.1" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.cmml"><mrow id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.2.cmml"><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.2.cmml">{</mo><msub id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.3" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.2" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.1" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.1.cmml">=</mo><mn id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.3" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.3.cmml">1</mn></mrow><mi id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.3" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.5.m5.2b"><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2"><eq id="S3.SS1.SSS1.2.p2.5.m5.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.2"></eq><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3"><times id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.1"></times><ci id="S3.SS1.SSS1.2.p2.5.m5.2.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.3.2">𝑉</ci><ci id="S3.SS1.SSS1.2.p2.5.m5.1.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1">superscript</csymbol><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1">subscript</csymbol><set id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1"><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.1.1.1.3">𝑘</ci></apply></set><apply id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3"><eq id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.1"></eq><ci id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.2">𝑘</ci><cn id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.SSS1.2.p2.5.m5.2.2.1.3.cmml" xref="S3.SS1.SSS1.2.p2.5.m5.2.2.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.5.m5.2c">V(P)=\{v_{k}\}_{k=1}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.5.m5.2d">italic_V ( italic_P ) = { italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and edges <math alttext="E_{b}(P)=\{e_{k}\}_{k=1}^{n}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.6.m6.2"><semantics id="S3.SS1.SSS1.2.p2.6.m6.2a"><mrow id="S3.SS1.SSS1.2.p2.6.m6.2.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.cmml"><mrow id="S3.SS1.SSS1.2.p2.6.m6.2.2.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.cmml"><msub id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.cmml"><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.2.cmml">E</mi><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.1" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.3.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.cmml"><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.2.p2.6.m6.1.1" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.2.cmml">=</mo><msubsup id="S3.SS1.SSS1.2.p2.6.m6.2.2.1" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.cmml"><mrow id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.2.cmml"><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.2.cmml">{</mo><msub id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.2" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.1" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.1.cmml">=</mo><mn id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.3.cmml">1</mn></mrow><mi id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.3" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.6.m6.2b"><apply id="S3.SS1.SSS1.2.p2.6.m6.2.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2"><eq id="S3.SS1.SSS1.2.p2.6.m6.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.2"></eq><apply id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3"><times id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.1"></times><apply id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.2">𝐸</ci><ci id="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.3.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS1.2.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1">superscript</csymbol><apply id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1"><csymbol cd="ambiguous" 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xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.SSS1.2.p2.6.m6.2.2.1.3.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.2.2.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.6.m6.2c">E_{b}(P)=\{e_{k}\}_{k=1}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.6.m6.2d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) = { italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> in a cyclic order such that <math alttext="e_{k}=\overline{v_{k-1}v_{k}}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.7.m7.1"><semantics id="S3.SS1.SSS1.2.p2.7.m7.1a"><mrow id="S3.SS1.SSS1.2.p2.7.m7.1.1" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.cmml"><msub id="S3.SS1.SSS1.2.p2.7.m7.1.1.2" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.cmml"><mi 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xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.2.3.1"></minus><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.2.3.3">1</cn></apply></apply><apply id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.2">𝑣</ci><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.2.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.7.m7.1c">e_{k}=\overline{v_{k-1}v_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = over¯ start_ARG italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. Moreover, by the Jordan–Schönflies theorem, we can choose the order in a counterclockwise sense, such that <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.8.m8.1"><semantics id="S3.SS1.SSS1.2.p2.8.m8.1a"><mi id="S3.SS1.SSS1.2.p2.8.m8.1.1" xref="S3.SS1.SSS1.2.p2.8.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.8.m8.1b"><ci id="S3.SS1.SSS1.2.p2.8.m8.1.1.cmml" xref="S3.SS1.SSS1.2.p2.8.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.8.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.8.m8.1d">italic_P</annotation></semantics></math> is always on the left hand side when traversing <math alttext="e_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.9.m9.1"><semantics id="S3.SS1.SSS1.2.p2.9.m9.1a"><msub id="S3.SS1.SSS1.2.p2.9.m9.1.1" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.9.m9.1.1.2" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS1.2.p2.9.m9.1.1.3" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.9.m9.1b"><apply id="S3.SS1.SSS1.2.p2.9.m9.1.1.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.2">𝑒</ci><ci id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.9.m9.1c">e_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.9.m9.1d">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="v_{k-1}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.10.m10.1"><semantics id="S3.SS1.SSS1.2.p2.10.m10.1a"><msub id="S3.SS1.SSS1.2.p2.10.m10.1.1" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.10.m10.1.1.2" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.2.cmml">v</mi><mrow id="S3.SS1.SSS1.2.p2.10.m10.1.1.3" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.2" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.1" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.1.cmml">−</mo><mn id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.3" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.10.m10.1b"><apply id="S3.SS1.SSS1.2.p2.10.m10.1.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.10.m10.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.2">𝑣</ci><apply id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3"><minus id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.1"></minus><ci id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.2">𝑘</ci><cn id="S3.SS1.SSS1.2.p2.10.m10.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.10.m10.1c">v_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.10.m10.1d">italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.11.m11.1"><semantics id="S3.SS1.SSS1.2.p2.11.m11.1a"><msub id="S3.SS1.SSS1.2.p2.11.m11.1.1" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.11.m11.1.1.2" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.2.p2.11.m11.1.1.3" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.11.m11.1b"><apply id="S3.SS1.SSS1.2.p2.11.m11.1.1.cmml" xref="S3.SS1.SSS1.2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.11.m11.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.11.m11.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.11.m11.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.2.p2.11.m11.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.11.m11.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.11.m11.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. We define <math alttext="H^{k}_{+}:=H^{e_{k}}_{P}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.12.m12.1"><semantics id="S3.SS1.SSS1.2.p2.12.m12.1a"><mrow id="S3.SS1.SSS1.2.p2.12.m12.1.1" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.cmml"><msubsup id="S3.SS1.SSS1.2.p2.12.m12.1.1.2" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.2" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.3.cmml">+</mo><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.3.cmml">k</mi></msubsup><mo id="S3.SS1.SSS1.2.p2.12.m12.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.1.cmml">:=</mo><msubsup id="S3.SS1.SSS1.2.p2.12.m12.1.1.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.2" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.2.cmml">H</mi><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.3.cmml">P</mi><msub id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.cmml"><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.2" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.2.cmml">e</mi><mi id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.3" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.3.cmml">k</mi></msub></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.12.m12.1b"><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1"><csymbol cd="latexml" id="S3.SS1.SSS1.2.p2.12.m12.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.1">assign</csymbol><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2">subscript</csymbol><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.2">𝐻</ci><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.2.3">𝑘</ci></apply><plus id="S3.SS1.SSS1.2.p2.12.m12.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.2.3"></plus></apply><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3">subscript</csymbol><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.2">𝑒</ci><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.2.3.3">𝑘</ci></apply></apply><ci id="S3.SS1.SSS1.2.p2.12.m12.1.1.3.3.cmml" xref="S3.SS1.SSS1.2.p2.12.m12.1.1.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.12.m12.1c">H^{k}_{+}:=H^{e_{k}}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.12.m12.1d">italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT := italic_H start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> to be the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.13.m13.1"><semantics id="S3.SS1.SSS1.2.p2.13.m13.1a"><mi id="S3.SS1.SSS1.2.p2.13.m13.1.1" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.13.m13.1b"><ci id="S3.SS1.SSS1.2.p2.13.m13.1.1.cmml" xref="S3.SS1.SSS1.2.p2.13.m13.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.13.m13.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.13.m13.1d">italic_P</annotation></semantics></math>-side of <math alttext="e_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.14.m14.1"><semantics id="S3.SS1.SSS1.2.p2.14.m14.1a"><msub id="S3.SS1.SSS1.2.p2.14.m14.1.1" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.14.m14.1.1.2" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS1.2.p2.14.m14.1.1.3" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.14.m14.1b"><apply id="S3.SS1.SSS1.2.p2.14.m14.1.1.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.14.m14.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.14.m14.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.2">𝑒</ci><ci id="S3.SS1.SSS1.2.p2.14.m14.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.14.m14.1c">e_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.14.m14.1d">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="H^{k}_{-}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.15.m15.1"><semantics id="S3.SS1.SSS1.2.p2.15.m15.1a"><msubsup id="S3.SS1.SSS1.2.p2.15.m15.1.1" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.2" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.2.p2.15.m15.1.1.3" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.3.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.3" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.2.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.15.m15.1b"><apply id="S3.SS1.SSS1.2.p2.15.m15.1.1.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.15.m15.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1">subscript</csymbol><apply id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.2.2">𝐻</ci><ci id="S3.SS1.SSS1.2.p2.15.m15.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.2.3">𝑘</ci></apply><minus id="S3.SS1.SSS1.2.p2.15.m15.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.15.m15.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.15.m15.1c">H^{k}_{-}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.15.m15.1d">italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> as the opposite half-plane. Further, <math alttext="Q^{k}:=Q^{v_{k}}_{P}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.16.m16.1"><semantics id="S3.SS1.SSS1.2.p2.16.m16.1a"><mrow id="S3.SS1.SSS1.2.p2.16.m16.1.1" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.cmml"><msup id="S3.SS1.SSS1.2.p2.16.m16.1.1.2" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.2" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.3" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2.3.cmml">k</mi></msup><mo id="S3.SS1.SSS1.2.p2.16.m16.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.1.cmml">:=</mo><msubsup id="S3.SS1.SSS1.2.p2.16.m16.1.1.3" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.2" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.3" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.3.cmml">P</mi><msub id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.cmml"><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.2" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.2.cmml">v</mi><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.3" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.3.cmml">k</mi></msub></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.16.m16.1b"><apply id="S3.SS1.SSS1.2.p2.16.m16.1.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1"><csymbol cd="latexml" id="S3.SS1.SSS1.2.p2.16.m16.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.1">assign</csymbol><apply id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2.2">𝑄</ci><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3">subscript</csymbol><apply id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.2">𝑄</ci><apply id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.2">𝑣</ci><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.2.3.3">𝑘</ci></apply></apply><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.3.3.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.16.m16.1c">Q^{k}:=Q^{v_{k}}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.16.m16.1d">italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT := italic_Q start_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> is short for the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.17.m17.1"><semantics id="S3.SS1.SSS1.2.p2.17.m17.1a"><mi id="S3.SS1.SSS1.2.p2.17.m17.1.1" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.17.m17.1b"><ci id="S3.SS1.SSS1.2.p2.17.m17.1.1.cmml" xref="S3.SS1.SSS1.2.p2.17.m17.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.17.m17.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.17.m17.1d">italic_P</annotation></semantics></math>-side of <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.18.m18.1"><semantics id="S3.SS1.SSS1.2.p2.18.m18.1a"><msub id="S3.SS1.SSS1.2.p2.18.m18.1.1" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.18.m18.1.1.2" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.2.p2.18.m18.1.1.3" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.18.m18.1b"><apply id="S3.SS1.SSS1.2.p2.18.m18.1.1.cmml" xref="S3.SS1.SSS1.2.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.18.m18.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.18.m18.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.18.m18.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.2.p2.18.m18.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.18.m18.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.18.m18.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. With this, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E8" title="Equation 8 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">8</span></a>) reads</p> <table class="ltx_equation ltx_eqn_table" id="S3.E9"> <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_{P}(x)-e_{P}(x)=|\{k\in[n]:x\in Q^{k}\}|-|\{k\in[n]:x\in H^{k}_{+}\}|." class="ltx_Math" display="block" id="S3.E9.m1.5"><semantics id="S3.E9.m1.5a"><mrow id="S3.E9.m1.5.5.1" xref="S3.E9.m1.5.5.1.1.cmml"><mrow id="S3.E9.m1.5.5.1.1" xref="S3.E9.m1.5.5.1.1.cmml"><mrow id="S3.E9.m1.5.5.1.1.4" xref="S3.E9.m1.5.5.1.1.4.cmml"><mrow id="S3.E9.m1.5.5.1.1.4.2" xref="S3.E9.m1.5.5.1.1.4.2.cmml"><msub id="S3.E9.m1.5.5.1.1.4.2.2" xref="S3.E9.m1.5.5.1.1.4.2.2.cmml"><mi id="S3.E9.m1.5.5.1.1.4.2.2.2" xref="S3.E9.m1.5.5.1.1.4.2.2.2.cmml">v</mi><mi id="S3.E9.m1.5.5.1.1.4.2.2.3" xref="S3.E9.m1.5.5.1.1.4.2.2.3.cmml">P</mi></msub><mo id="S3.E9.m1.5.5.1.1.4.2.1" xref="S3.E9.m1.5.5.1.1.4.2.1.cmml">⁢</mo><mrow id="S3.E9.m1.5.5.1.1.4.2.3.2" xref="S3.E9.m1.5.5.1.1.4.2.cmml"><mo id="S3.E9.m1.5.5.1.1.4.2.3.2.1" stretchy="false" xref="S3.E9.m1.5.5.1.1.4.2.cmml">(</mo><mi id="S3.E9.m1.1.1" xref="S3.E9.m1.1.1.cmml">x</mi><mo id="S3.E9.m1.5.5.1.1.4.2.3.2.2" stretchy="false" xref="S3.E9.m1.5.5.1.1.4.2.cmml">)</mo></mrow></mrow><mo id="S3.E9.m1.5.5.1.1.4.1" xref="S3.E9.m1.5.5.1.1.4.1.cmml">−</mo><mrow id="S3.E9.m1.5.5.1.1.4.3" xref="S3.E9.m1.5.5.1.1.4.3.cmml"><msub id="S3.E9.m1.5.5.1.1.4.3.2" xref="S3.E9.m1.5.5.1.1.4.3.2.cmml"><mi id="S3.E9.m1.5.5.1.1.4.3.2.2" xref="S3.E9.m1.5.5.1.1.4.3.2.2.cmml">e</mi><mi id="S3.E9.m1.5.5.1.1.4.3.2.3" xref="S3.E9.m1.5.5.1.1.4.3.2.3.cmml">P</mi></msub><mo id="S3.E9.m1.5.5.1.1.4.3.1" xref="S3.E9.m1.5.5.1.1.4.3.1.cmml">⁢</mo><mrow id="S3.E9.m1.5.5.1.1.4.3.3.2" xref="S3.E9.m1.5.5.1.1.4.3.cmml"><mo id="S3.E9.m1.5.5.1.1.4.3.3.2.1" stretchy="false" xref="S3.E9.m1.5.5.1.1.4.3.cmml">(</mo><mi id="S3.E9.m1.2.2" xref="S3.E9.m1.2.2.cmml">x</mi><mo id="S3.E9.m1.5.5.1.1.4.3.3.2.2" stretchy="false" xref="S3.E9.m1.5.5.1.1.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E9.m1.5.5.1.1.3" xref="S3.E9.m1.5.5.1.1.3.cmml">=</mo><mrow id="S3.E9.m1.5.5.1.1.2" xref="S3.E9.m1.5.5.1.1.2.cmml"><mrow id="S3.E9.m1.5.5.1.1.1.1.1" xref="S3.E9.m1.5.5.1.1.1.1.2.cmml"><mo id="S3.E9.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S3.E9.m1.5.5.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.E9.m1.5.5.1.1.1.1.1.1.2" xref="S3.E9.m1.5.5.1.1.1.1.1.1.3.cmml"><mo id="S3.E9.m1.5.5.1.1.1.1.1.1.2.3" stretchy="false" xref="S3.E9.m1.5.5.1.1.1.1.1.1.3.1.cmml">{</mo><mrow id="S3.E9.m1.5.5.1.1.1.1.1.1.1.1" xref="S3.E9.m1.5.5.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E9.m1.5.5.1.1.1.1.1.1.1.1.2" 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end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) = | { italic_k ∈ [ italic_n ] : italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } | - | { italic_k ∈ [ italic_n ] : italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + 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">(9)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.SSS1.3.p3"> <p class="ltx_p" id="S3.SS1.SSS1.3.p3.14">From now on, we assume <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.1.m1.1"><semantics id="S3.SS1.SSS1.3.p3.1.m1.1a"><mrow id="S3.SS1.SSS1.3.p3.1.m1.1.1" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.1.m1.1.1.2" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.1.m1.1.1.1" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS1.3.p3.1.m1.1.1.3" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3.cmml"><mi id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.2" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.3" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.1.m1.1b"><apply id="S3.SS1.SSS1.3.p3.1.m1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1"><in id="S3.SS1.SSS1.3.p3.1.m1.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3.2">ℝ</ci><cn id="S3.SS1.SSS1.3.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.1.m1.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.1.m1.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.2.m2.1"><semantics id="S3.SS1.SSS1.3.p3.2.m2.1a"><mi id="S3.SS1.SSS1.3.p3.2.m2.1.1" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.2.m2.1b"><ci id="S3.SS1.SSS1.3.p3.2.m2.1.1.cmml" xref="S3.SS1.SSS1.3.p3.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.2.m2.1d">italic_P</annotation></semantics></math>-general position to be fixed. Whether <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.3.m3.1"><semantics id="S3.SS1.SSS1.3.p3.3.m3.1a"><mi id="S3.SS1.SSS1.3.p3.3.m3.1.1" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.3.m3.1b"><ci id="S3.SS1.SSS1.3.p3.3.m3.1.1.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.3.m3.1d">italic_x</annotation></semantics></math> lies on the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.4.m4.1"><semantics id="S3.SS1.SSS1.3.p3.4.m4.1a"><mi id="S3.SS1.SSS1.3.p3.4.m4.1.1" xref="S3.SS1.SSS1.3.p3.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.4.m4.1b"><ci id="S3.SS1.SSS1.3.p3.4.m4.1.1.cmml" xref="S3.SS1.SSS1.3.p3.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.4.m4.1d">italic_P</annotation></semantics></math>-side of a vertex is fully determined by the incident edges. From <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F4" title="Figure 4 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 4</span></a>, we see that if the corner formed by the edges incident with vertex <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.5.m5.1"><semantics id="S3.SS1.SSS1.3.p3.5.m5.1a"><msub id="S3.SS1.SSS1.3.p3.5.m5.1.1" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.5.m5.1.1.2" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.3.p3.5.m5.1.1.3" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.5.m5.1b"><apply id="S3.SS1.SSS1.3.p3.5.m5.1.1.cmml" xref="S3.SS1.SSS1.3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.5.m5.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.5.m5.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.3.p3.5.m5.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.5.m5.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.5.m5.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is concave, <math alttext="x\in Q^{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.6.m6.1"><semantics id="S3.SS1.SSS1.3.p3.6.m6.1a"><mrow id="S3.SS1.SSS1.3.p3.6.m6.1.1" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.6.m6.1.1.2" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.6.m6.1.1.1" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS1.3.p3.6.m6.1.1.3" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.cmml"><mi id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.2" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.2.cmml">Q</mi><mi id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.3" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.6.m6.1b"><apply id="S3.SS1.SSS1.3.p3.6.m6.1.1.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1"><in id="S3.SS1.SSS1.3.p3.6.m6.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.6.m6.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.2">𝑄</ci><ci id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.6.m6.1c">x\in Q^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.6.m6.1d">italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> is equivalent to <math alttext="x\in H^{k}_{+}\cup H^{k+1}_{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.7.m7.1"><semantics id="S3.SS1.SSS1.3.p3.7.m7.1a"><mrow id="S3.SS1.SSS1.3.p3.7.m7.1.1" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1.2" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.7.m7.1.1.1" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.3.p3.7.m7.1.1.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.cmml"><msubsup id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.2" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.3.cmml">+</mo><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.3.cmml">k</mi></msubsup><mo id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.1" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.1.cmml">∪</mo><msubsup id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.2" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.3.cmml">+</mo><mrow id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.2" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.1" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.3" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.7.m7.1b"><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1"><in id="S3.SS1.SSS1.3.p3.7.m7.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3"><union id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.1"></union><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.2">𝐻</ci><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.2.3">𝑘</ci></apply><plus id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.2.3"></plus></apply><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.2.3.3">1</cn></apply></apply><plus id="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.7.m7.1c">x\in H^{k}_{+}\cup H^{k+1}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.7.m7.1d">italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ∪ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>. If it is convex, then <math alttext="x\in Q^{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.8.m8.1"><semantics id="S3.SS1.SSS1.3.p3.8.m8.1a"><mrow id="S3.SS1.SSS1.3.p3.8.m8.1.1" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.8.m8.1.1.2" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.8.m8.1.1.1" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS1.3.p3.8.m8.1.1.3" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.cmml"><mi id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.2" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.2.cmml">Q</mi><mi id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.3" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.8.m8.1b"><apply id="S3.SS1.SSS1.3.p3.8.m8.1.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1"><in id="S3.SS1.SSS1.3.p3.8.m8.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.2">𝑄</ci><ci id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.8.m8.1c">x\in Q^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.8.m8.1d">italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> if and only if <math alttext="x\in H^{k}_{+}\cap H^{k+1}_{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.9.m9.1"><semantics id="S3.SS1.SSS1.3.p3.9.m9.1a"><mrow id="S3.SS1.SSS1.3.p3.9.m9.1.1" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.9.m9.1.1.1" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.3.p3.9.m9.1.1.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.cmml"><msubsup id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.3.cmml">+</mo><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.3.cmml">k</mi></msubsup><mo id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.1" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.1.cmml">∩</mo><msubsup id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.3.cmml">+</mo><mrow id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.1" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.9.m9.1b"><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1"><in id="S3.SS1.SSS1.3.p3.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3"><intersect id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.1"></intersect><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.2">𝐻</ci><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.2.3">𝑘</ci></apply><plus id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.2.3"></plus></apply><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.2.3.3">1</cn></apply></apply><plus id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.9.m9.1c">x\in H^{k}_{+}\cap H^{k+1}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.9.m9.1d">italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ∩ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>. This is summarised in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.T1" title="Table 1 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Table 1</span></a>. In the following, I want to refer to the eight different cases listed in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.T1" title="Table 1 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Table 1</span></a> in a short form. For example, I write <math alttext="k=\texttt{+-convex}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.10.m10.1"><semantics id="S3.SS1.SSS1.3.p3.10.m10.1a"><mrow id="S3.SS1.SSS1.3.p3.10.m10.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.10.m10.1.1.2" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.10.m10.1.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.1.cmml">=</mo><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.3.p3.10.m10.1.1.3" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.3a.cmml">+-convex</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.10.m10.1b"><apply id="S3.SS1.SSS1.3.p3.10.m10.1.1.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.1.1"><eq id="S3.SS1.SSS1.3.p3.10.m10.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.1"></eq><ci id="S3.SS1.SSS1.3.p3.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.2">𝑘</ci><ci id="S3.SS1.SSS1.3.p3.10.m10.1.1.3a.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.3.p3.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.3">+-convex</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.10.m10.1c">k=\texttt{+-convex}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.10.m10.1d">italic_k = +-convex</annotation></semantics></math> if <math alttext="x\in H^{k}_{+}\cap H^{k+1}_{-}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.11.m11.1"><semantics id="S3.SS1.SSS1.3.p3.11.m11.1a"><mrow id="S3.SS1.SSS1.3.p3.11.m11.1.1" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.11.m11.1.1.2" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.11.m11.1.1.1" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.3.p3.11.m11.1.1.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.cmml"><msubsup id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.2" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.3.cmml">+</mo><mi id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.3.cmml">k</mi></msubsup><mo id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.1" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.1.cmml">∩</mo><msubsup id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.2" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.3.cmml">−</mo><mrow id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.2" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.1" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.3" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.11.m11.1b"><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1"><in id="S3.SS1.SSS1.3.p3.11.m11.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.11.m11.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3"><intersect id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.1"></intersect><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.2">𝐻</ci><ci id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.2.3">𝑘</ci></apply><plus id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.2.3"></plus></apply><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.2.3.3">1</cn></apply></apply><minus id="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.3.p3.11.m11.1.1.3.3.3"></minus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.11.m11.1c">x\in H^{k}_{+}\cap H^{k+1}_{-}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.11.m11.1d">italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ∩ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> and the corner at <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.12.m12.1"><semantics id="S3.SS1.SSS1.3.p3.12.m12.1a"><msub id="S3.SS1.SSS1.3.p3.12.m12.1.1" xref="S3.SS1.SSS1.3.p3.12.m12.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.12.m12.1.1.2" xref="S3.SS1.SSS1.3.p3.12.m12.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.3.p3.12.m12.1.1.3" xref="S3.SS1.SSS1.3.p3.12.m12.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.12.m12.1b"><apply id="S3.SS1.SSS1.3.p3.12.m12.1.1.cmml" xref="S3.SS1.SSS1.3.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.12.m12.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.12.m12.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.12.m12.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.12.m12.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.3.p3.12.m12.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.12.m12.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.12.m12.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.12.m12.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is convex, and <math alttext="k=\texttt{-+concave}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.13.m13.1"><semantics id="S3.SS1.SSS1.3.p3.13.m13.1a"><mrow id="S3.SS1.SSS1.3.p3.13.m13.1.1" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.13.m13.1.1.2" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.13.m13.1.1.1" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.1.cmml">=</mo><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.3.p3.13.m13.1.1.3" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.3a.cmml">-+concave</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.13.m13.1b"><apply id="S3.SS1.SSS1.3.p3.13.m13.1.1.cmml" xref="S3.SS1.SSS1.3.p3.13.m13.1.1"><eq id="S3.SS1.SSS1.3.p3.13.m13.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.1"></eq><ci id="S3.SS1.SSS1.3.p3.13.m13.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.2">𝑘</ci><ci id="S3.SS1.SSS1.3.p3.13.m13.1.1.3a.cmml" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.3.p3.13.m13.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.13.m13.1.1.3">-+concave</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.13.m13.1c">k=\texttt{-+concave}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.13.m13.1d">italic_k = -+concave</annotation></semantics></math> if <math alttext="x\in H^{k}_{-}\cap H^{k+1}_{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.14.m14.1"><semantics id="S3.SS1.SSS1.3.p3.14.m14.1a"><mrow id="S3.SS1.SSS1.3.p3.14.m14.1.1" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.14.m14.1.1.2" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.3.p3.14.m14.1.1.1" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.3.p3.14.m14.1.1.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.cmml"><msubsup id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.2" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.3.cmml">−</mo><mi id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.3.cmml">k</mi></msubsup><mo id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.1" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.1.cmml">∩</mo><msubsup id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.2" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.3.cmml">+</mo><mrow id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.2" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.1" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.3" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.14.m14.1b"><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1"><in id="S3.SS1.SSS1.3.p3.14.m14.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.14.m14.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3"><intersect id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.1"></intersect><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.2">𝐻</ci><ci id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.2.3">𝑘</ci></apply><minus id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.2.3"></minus></apply><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3">subscript</csymbol><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.2.3.3">1</cn></apply></apply><plus id="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.3.p3.14.m14.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.14.m14.1c">x\in H^{k}_{-}\cap H^{k+1}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.14.m14.1d">italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ∩ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> and the corner is concave.</p> </div> <figure class="ltx_figure" id="S3.F4"> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S3.F4.4"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.F4.2.2"> <td class="ltx_td ltx_align_center" id="S3.F4.1.1.1"> <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.F4.1.1.1.m1.1"><semantics id="S3.F4.1.1.1.m1.1a"><msub id="S3.F4.1.1.1.m1.1.1" xref="S3.F4.1.1.1.m1.1.1.cmml"><mi id="S3.F4.1.1.1.m1.1.1.2" xref="S3.F4.1.1.1.m1.1.1.2.cmml">v</mi><mi id="S3.F4.1.1.1.m1.1.1.3" xref="S3.F4.1.1.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F4.1.1.1.m1.1b"><apply id="S3.F4.1.1.1.m1.1.1.cmml" xref="S3.F4.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F4.1.1.1.m1.1.1.1.cmml" xref="S3.F4.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.F4.1.1.1.m1.1.1.2.cmml" xref="S3.F4.1.1.1.m1.1.1.2">𝑣</ci><ci id="S3.F4.1.1.1.m1.1.1.3.cmml" xref="S3.F4.1.1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.1.1.1.m1.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.1.1.1.m1.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> concave</td> <td class="ltx_td ltx_align_center" id="S3.F4.2.2.2"> <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.F4.2.2.2.m1.1"><semantics id="S3.F4.2.2.2.m1.1a"><msub id="S3.F4.2.2.2.m1.1.1" xref="S3.F4.2.2.2.m1.1.1.cmml"><mi id="S3.F4.2.2.2.m1.1.1.2" xref="S3.F4.2.2.2.m1.1.1.2.cmml">v</mi><mi id="S3.F4.2.2.2.m1.1.1.3" xref="S3.F4.2.2.2.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F4.2.2.2.m1.1b"><apply id="S3.F4.2.2.2.m1.1.1.cmml" xref="S3.F4.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.F4.2.2.2.m1.1.1.1.cmml" xref="S3.F4.2.2.2.m1.1.1">subscript</csymbol><ci id="S3.F4.2.2.2.m1.1.1.2.cmml" xref="S3.F4.2.2.2.m1.1.1.2">𝑣</ci><ci id="S3.F4.2.2.2.m1.1.1.3.cmml" xref="S3.F4.2.2.2.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.2.2.2.m1.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.2.2.2.m1.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> convex</td> </tr> <tr class="ltx_tr" id="S3.F4.4.4"> <td class="ltx_td ltx_align_center" id="S3.F4.3.3.1"> <span class="ltx_text" id="S3.F4.3.3.1.1"><foreignobject height="60.0pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="102.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="115" id="S3.F4.3.3.1.1.1.g1" src="x8.png" width="197"/></foreignobject></span> </td> <td class="ltx_td ltx_align_center" id="S3.F4.4.4.2"> <span class="ltx_text" id="S3.F4.4.4.2.1"><foreignobject height="60.0pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="102.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="116" id="S3.F4.4.4.2.1.1.g1" src="x9.png" width="197"/></foreignobject></span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>The <math alttext="P" class="ltx_Math" display="inline" id="S3.F4.8.m1.1"><semantics id="S3.F4.8.m1.1b"><mi id="S3.F4.8.m1.1.1" xref="S3.F4.8.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F4.8.m1.1c"><ci id="S3.F4.8.m1.1.1.cmml" xref="S3.F4.8.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.8.m1.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F4.8.m1.1e">italic_P</annotation></semantics></math>-side <math alttext="Q^{k}" class="ltx_Math" display="inline" id="S3.F4.9.m2.1"><semantics id="S3.F4.9.m2.1b"><msup id="S3.F4.9.m2.1.1" xref="S3.F4.9.m2.1.1.cmml"><mi id="S3.F4.9.m2.1.1.2" xref="S3.F4.9.m2.1.1.2.cmml">Q</mi><mi id="S3.F4.9.m2.1.1.3" xref="S3.F4.9.m2.1.1.3.cmml">k</mi></msup><annotation-xml encoding="MathML-Content" id="S3.F4.9.m2.1c"><apply id="S3.F4.9.m2.1.1.cmml" xref="S3.F4.9.m2.1.1"><csymbol cd="ambiguous" id="S3.F4.9.m2.1.1.1.cmml" xref="S3.F4.9.m2.1.1">superscript</csymbol><ci id="S3.F4.9.m2.1.1.2.cmml" xref="S3.F4.9.m2.1.1.2">𝑄</ci><ci id="S3.F4.9.m2.1.1.3.cmml" xref="S3.F4.9.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.9.m2.1d">Q^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.9.m2.1e">italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.F4.10.m3.1"><semantics id="S3.F4.10.m3.1b"><msub id="S3.F4.10.m3.1.1" xref="S3.F4.10.m3.1.1.cmml"><mi id="S3.F4.10.m3.1.1.2" xref="S3.F4.10.m3.1.1.2.cmml">v</mi><mi id="S3.F4.10.m3.1.1.3" xref="S3.F4.10.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F4.10.m3.1c"><apply id="S3.F4.10.m3.1.1.cmml" xref="S3.F4.10.m3.1.1"><csymbol cd="ambiguous" id="S3.F4.10.m3.1.1.1.cmml" xref="S3.F4.10.m3.1.1">subscript</csymbol><ci id="S3.F4.10.m3.1.1.2.cmml" xref="S3.F4.10.m3.1.1.2">𝑣</ci><ci id="S3.F4.10.m3.1.1.3.cmml" xref="S3.F4.10.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.10.m3.1d">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.10.m3.1e">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is indicated in grey.</figcaption> </figure> <figure class="ltx_table" id="S3.T1"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T1.24"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.1.1.1"><math alttext="H^{k}_{\bullet}" class="ltx_Math" display="inline" id="S3.T1.1.1.1.m1.1"><semantics id="S3.T1.1.1.1.m1.1a"><msubsup id="S3.T1.1.1.1.m1.1.1" xref="S3.T1.1.1.1.m1.1.1.cmml"><mi id="S3.T1.1.1.1.m1.1.1.2.2" xref="S3.T1.1.1.1.m1.1.1.2.2.cmml">H</mi><mo id="S3.T1.1.1.1.m1.1.1.3" xref="S3.T1.1.1.1.m1.1.1.3.cmml">∙</mo><mi id="S3.T1.1.1.1.m1.1.1.2.3" xref="S3.T1.1.1.1.m1.1.1.2.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.T1.1.1.1.m1.1b"><apply id="S3.T1.1.1.1.m1.1.1.cmml" xref="S3.T1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.1.1.1.m1.1.1.1.cmml" xref="S3.T1.1.1.1.m1.1.1">subscript</csymbol><apply id="S3.T1.1.1.1.m1.1.1.2.cmml" xref="S3.T1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.1.1.1.m1.1.1.2.1.cmml" xref="S3.T1.1.1.1.m1.1.1">superscript</csymbol><ci id="S3.T1.1.1.1.m1.1.1.2.2.cmml" xref="S3.T1.1.1.1.m1.1.1.2.2">𝐻</ci><ci id="S3.T1.1.1.1.m1.1.1.2.3.cmml" xref="S3.T1.1.1.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S3.T1.1.1.1.m1.1.1.3.cmml" xref="S3.T1.1.1.1.m1.1.1.3">∙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.1.1.1.m1.1c">H^{k}_{\bullet}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.1.1.1.m1.1d">italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.2.2.2"><math alttext="H^{k+1}_{\bullet}" class="ltx_Math" display="inline" id="S3.T1.2.2.2.m1.1"><semantics id="S3.T1.2.2.2.m1.1a"><msubsup id="S3.T1.2.2.2.m1.1.1" xref="S3.T1.2.2.2.m1.1.1.cmml"><mi id="S3.T1.2.2.2.m1.1.1.2.2" xref="S3.T1.2.2.2.m1.1.1.2.2.cmml">H</mi><mo id="S3.T1.2.2.2.m1.1.1.3" xref="S3.T1.2.2.2.m1.1.1.3.cmml">∙</mo><mrow id="S3.T1.2.2.2.m1.1.1.2.3" xref="S3.T1.2.2.2.m1.1.1.2.3.cmml"><mi id="S3.T1.2.2.2.m1.1.1.2.3.2" xref="S3.T1.2.2.2.m1.1.1.2.3.2.cmml">k</mi><mo id="S3.T1.2.2.2.m1.1.1.2.3.1" xref="S3.T1.2.2.2.m1.1.1.2.3.1.cmml">+</mo><mn id="S3.T1.2.2.2.m1.1.1.2.3.3" xref="S3.T1.2.2.2.m1.1.1.2.3.3.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.T1.2.2.2.m1.1b"><apply id="S3.T1.2.2.2.m1.1.1.cmml" xref="S3.T1.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.2.2.2.m1.1.1.1.cmml" xref="S3.T1.2.2.2.m1.1.1">subscript</csymbol><apply id="S3.T1.2.2.2.m1.1.1.2.cmml" xref="S3.T1.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.2.2.2.m1.1.1.2.1.cmml" xref="S3.T1.2.2.2.m1.1.1">superscript</csymbol><ci id="S3.T1.2.2.2.m1.1.1.2.2.cmml" xref="S3.T1.2.2.2.m1.1.1.2.2">𝐻</ci><apply id="S3.T1.2.2.2.m1.1.1.2.3.cmml" xref="S3.T1.2.2.2.m1.1.1.2.3"><plus id="S3.T1.2.2.2.m1.1.1.2.3.1.cmml" xref="S3.T1.2.2.2.m1.1.1.2.3.1"></plus><ci id="S3.T1.2.2.2.m1.1.1.2.3.2.cmml" xref="S3.T1.2.2.2.m1.1.1.2.3.2">𝑘</ci><cn id="S3.T1.2.2.2.m1.1.1.2.3.3.cmml" type="integer" xref="S3.T1.2.2.2.m1.1.1.2.3.3">1</cn></apply></apply><ci id="S3.T1.2.2.2.m1.1.1.3.cmml" xref="S3.T1.2.2.2.m1.1.1.3">∙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.2.2.2.m1.1c">H^{k+1}_{\bullet}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.2.2.2.m1.1d">italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r" id="S3.T1.3.3.3">Convexity at <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.T1.3.3.3.m1.1"><semantics id="S3.T1.3.3.3.m1.1a"><msub id="S3.T1.3.3.3.m1.1.1" xref="S3.T1.3.3.3.m1.1.1.cmml"><mi id="S3.T1.3.3.3.m1.1.1.2" xref="S3.T1.3.3.3.m1.1.1.2.cmml">v</mi><mi id="S3.T1.3.3.3.m1.1.1.3" xref="S3.T1.3.3.3.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.3.3.3.m1.1b"><apply id="S3.T1.3.3.3.m1.1.1.cmml" xref="S3.T1.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.3.3.3.m1.1.1.1.cmml" xref="S3.T1.3.3.3.m1.1.1">subscript</csymbol><ci id="S3.T1.3.3.3.m1.1.1.2.cmml" xref="S3.T1.3.3.3.m1.1.1.2">𝑣</ci><ci id="S3.T1.3.3.3.m1.1.1.3.cmml" xref="S3.T1.3.3.3.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.3.3.3.m1.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.3.3.3.m1.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S3.T1.4.4.4"> <table class="ltx_tabular ltx_align_middle" id="S3.T1.4.4.4.1"> <tr class="ltx_tr" id="S3.T1.4.4.4.1.2"> <td class="ltx_td ltx_align_center" id="S3.T1.4.4.4.1.2.1">vertex</td> </tr> <tr class="ltx_tr" id="S3.T1.4.4.4.1.1"> <td class="ltx_td ltx_align_center" id="S3.T1.4.4.4.1.1.1"><math alttext="x\in Q^{k}" class="ltx_Math" display="inline" id="S3.T1.4.4.4.1.1.1.m1.1"><semantics id="S3.T1.4.4.4.1.1.1.m1.1a"><mrow id="S3.T1.4.4.4.1.1.1.m1.1.1" xref="S3.T1.4.4.4.1.1.1.m1.1.1.cmml"><mi id="S3.T1.4.4.4.1.1.1.m1.1.1.2" xref="S3.T1.4.4.4.1.1.1.m1.1.1.2.cmml">x</mi><mo id="S3.T1.4.4.4.1.1.1.m1.1.1.1" xref="S3.T1.4.4.4.1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.T1.4.4.4.1.1.1.m1.1.1.3" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3.cmml"><mi id="S3.T1.4.4.4.1.1.1.m1.1.1.3.2" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3.2.cmml">Q</mi><mi id="S3.T1.4.4.4.1.1.1.m1.1.1.3.3" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3.3.cmml">k</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.4.4.4.1.1.1.m1.1b"><apply id="S3.T1.4.4.4.1.1.1.m1.1.1.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1"><in id="S3.T1.4.4.4.1.1.1.m1.1.1.1.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.1"></in><ci id="S3.T1.4.4.4.1.1.1.m1.1.1.2.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.2">𝑥</ci><apply id="S3.T1.4.4.4.1.1.1.m1.1.1.3.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.T1.4.4.4.1.1.1.m1.1.1.3.1.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3">superscript</csymbol><ci id="S3.T1.4.4.4.1.1.1.m1.1.1.3.2.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3.2">𝑄</ci><ci id="S3.T1.4.4.4.1.1.1.m1.1.1.3.3.cmml" xref="S3.T1.4.4.4.1.1.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.4.4.4.1.1.1.m1.1c">x\in Q^{k}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.4.4.4.1.1.1.m1.1d">italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math></td> </tr> </table> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.7.7"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.5.5.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.5.5.1.m1.1"><semantics id="S3.T1.5.5.1.m1.1a"><mo id="S3.T1.5.5.1.m1.1.1" xref="S3.T1.5.5.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.5.5.1.m1.1b"><plus id="S3.T1.5.5.1.m1.1.1.cmml" xref="S3.T1.5.5.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.5.5.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.5.5.1.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.6.6.2"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.6.6.2.m1.1"><semantics id="S3.T1.6.6.2.m1.1a"><mo id="S3.T1.6.6.2.m1.1.1" xref="S3.T1.6.6.2.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.6.6.2.m1.1b"><plus id="S3.T1.6.6.2.m1.1.1.cmml" xref="S3.T1.6.6.2.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.6.6.2.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.6.6.2.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.7.7.4">convex</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.7.7.3"><math alttext="\checkmark" class="ltx_Math" display="inline" id="S3.T1.7.7.3.m1.1"><semantics id="S3.T1.7.7.3.m1.1a"><mi id="S3.T1.7.7.3.m1.1.1" mathvariant="normal" xref="S3.T1.7.7.3.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.T1.7.7.3.m1.1b"><ci id="S3.T1.7.7.3.m1.1.1.cmml" xref="S3.T1.7.7.3.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.7.7.3.m1.1c">\checkmark</annotation><annotation encoding="application/x-llamapun" id="S3.T1.7.7.3.m1.1d">✓</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.10.10"> <td class="ltx_td ltx_align_center" id="S3.T1.8.8.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.8.8.1.m1.1"><semantics id="S3.T1.8.8.1.m1.1a"><mo id="S3.T1.8.8.1.m1.1.1" xref="S3.T1.8.8.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.8.8.1.m1.1b"><plus id="S3.T1.8.8.1.m1.1.1.cmml" xref="S3.T1.8.8.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.8.8.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.8.8.1.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.9.9.2"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.9.9.2.m1.1"><semantics id="S3.T1.9.9.2.m1.1a"><mo id="S3.T1.9.9.2.m1.1.1" xref="S3.T1.9.9.2.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.9.9.2.m1.1b"><plus id="S3.T1.9.9.2.m1.1.1.cmml" xref="S3.T1.9.9.2.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.9.9.2.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.9.9.2.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.10.10.4">concave</td> <td class="ltx_td ltx_align_center" id="S3.T1.10.10.3"><math alttext="\checkmark" class="ltx_Math" display="inline" id="S3.T1.10.10.3.m1.1"><semantics id="S3.T1.10.10.3.m1.1a"><mi id="S3.T1.10.10.3.m1.1.1" mathvariant="normal" xref="S3.T1.10.10.3.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.T1.10.10.3.m1.1b"><ci id="S3.T1.10.10.3.m1.1.1.cmml" xref="S3.T1.10.10.3.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.10.10.3.m1.1c">\checkmark</annotation><annotation encoding="application/x-llamapun" id="S3.T1.10.10.3.m1.1d">✓</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.12.12"> <td class="ltx_td ltx_align_center" id="S3.T1.11.11.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.11.11.1.m1.1"><semantics id="S3.T1.11.11.1.m1.1a"><mo id="S3.T1.11.11.1.m1.1.1" xref="S3.T1.11.11.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.11.11.1.m1.1b"><plus id="S3.T1.11.11.1.m1.1.1.cmml" xref="S3.T1.11.11.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.11.11.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.11.11.1.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.12.12.2"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.12.12.2.m1.1"><semantics id="S3.T1.12.12.2.m1.1a"><mo id="S3.T1.12.12.2.m1.1.1" xref="S3.T1.12.12.2.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.12.12.2.m1.1b"><minus id="S3.T1.12.12.2.m1.1.1.cmml" xref="S3.T1.12.12.2.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.12.12.2.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.12.12.2.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.12.12.3">convex</td> <td class="ltx_td ltx_align_center" id="S3.T1.12.12.4">-</td> </tr> <tr class="ltx_tr" id="S3.T1.15.15"> <td class="ltx_td ltx_align_center" id="S3.T1.13.13.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.13.13.1.m1.1"><semantics id="S3.T1.13.13.1.m1.1a"><mo id="S3.T1.13.13.1.m1.1.1" xref="S3.T1.13.13.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.13.13.1.m1.1b"><plus id="S3.T1.13.13.1.m1.1.1.cmml" xref="S3.T1.13.13.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.13.13.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.13.13.1.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.14.14.2"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.14.14.2.m1.1"><semantics id="S3.T1.14.14.2.m1.1a"><mo id="S3.T1.14.14.2.m1.1.1" xref="S3.T1.14.14.2.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.14.14.2.m1.1b"><minus id="S3.T1.14.14.2.m1.1.1.cmml" xref="S3.T1.14.14.2.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.14.14.2.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.14.14.2.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.15.15.4">concave</td> <td class="ltx_td ltx_align_center" id="S3.T1.15.15.3"><math alttext="\checkmark" class="ltx_Math" display="inline" id="S3.T1.15.15.3.m1.1"><semantics id="S3.T1.15.15.3.m1.1a"><mi id="S3.T1.15.15.3.m1.1.1" mathvariant="normal" xref="S3.T1.15.15.3.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.T1.15.15.3.m1.1b"><ci id="S3.T1.15.15.3.m1.1.1.cmml" xref="S3.T1.15.15.3.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.15.15.3.m1.1c">\checkmark</annotation><annotation encoding="application/x-llamapun" id="S3.T1.15.15.3.m1.1d">✓</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.17.17"> <td class="ltx_td ltx_align_center" id="S3.T1.16.16.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.16.16.1.m1.1"><semantics id="S3.T1.16.16.1.m1.1a"><mo id="S3.T1.16.16.1.m1.1.1" xref="S3.T1.16.16.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.16.16.1.m1.1b"><minus id="S3.T1.16.16.1.m1.1.1.cmml" xref="S3.T1.16.16.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.16.16.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.16.16.1.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.17.17.2"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.17.17.2.m1.1"><semantics id="S3.T1.17.17.2.m1.1a"><mo id="S3.T1.17.17.2.m1.1.1" xref="S3.T1.17.17.2.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.17.17.2.m1.1b"><plus id="S3.T1.17.17.2.m1.1.1.cmml" xref="S3.T1.17.17.2.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.17.17.2.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.17.17.2.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.17.17.3">convex</td> <td class="ltx_td ltx_align_center" id="S3.T1.17.17.4">-</td> </tr> <tr class="ltx_tr" id="S3.T1.20.20"> <td class="ltx_td ltx_align_center" id="S3.T1.18.18.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.18.18.1.m1.1"><semantics id="S3.T1.18.18.1.m1.1a"><mo id="S3.T1.18.18.1.m1.1.1" xref="S3.T1.18.18.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.18.18.1.m1.1b"><minus id="S3.T1.18.18.1.m1.1.1.cmml" xref="S3.T1.18.18.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.18.18.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.18.18.1.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.19.19.2"><math alttext="+" class="ltx_Math" display="inline" id="S3.T1.19.19.2.m1.1"><semantics id="S3.T1.19.19.2.m1.1a"><mo id="S3.T1.19.19.2.m1.1.1" xref="S3.T1.19.19.2.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.19.19.2.m1.1b"><plus id="S3.T1.19.19.2.m1.1.1.cmml" xref="S3.T1.19.19.2.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.19.19.2.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.19.19.2.m1.1d">+</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.20.20.4">concave</td> <td class="ltx_td ltx_align_center" id="S3.T1.20.20.3"><math alttext="\checkmark" class="ltx_Math" display="inline" id="S3.T1.20.20.3.m1.1"><semantics id="S3.T1.20.20.3.m1.1a"><mi id="S3.T1.20.20.3.m1.1.1" mathvariant="normal" xref="S3.T1.20.20.3.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.T1.20.20.3.m1.1b"><ci id="S3.T1.20.20.3.m1.1.1.cmml" xref="S3.T1.20.20.3.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.20.20.3.m1.1c">\checkmark</annotation><annotation encoding="application/x-llamapun" id="S3.T1.20.20.3.m1.1d">✓</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.22.22"> <td class="ltx_td ltx_align_center" id="S3.T1.21.21.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.21.21.1.m1.1"><semantics id="S3.T1.21.21.1.m1.1a"><mo id="S3.T1.21.21.1.m1.1.1" xref="S3.T1.21.21.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.21.21.1.m1.1b"><minus id="S3.T1.21.21.1.m1.1.1.cmml" xref="S3.T1.21.21.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.21.21.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.21.21.1.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.22.22.2"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.22.22.2.m1.1"><semantics id="S3.T1.22.22.2.m1.1a"><mo id="S3.T1.22.22.2.m1.1.1" xref="S3.T1.22.22.2.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.22.22.2.m1.1b"><minus id="S3.T1.22.22.2.m1.1.1.cmml" xref="S3.T1.22.22.2.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.22.22.2.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.22.22.2.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.22.22.3">convex</td> <td class="ltx_td ltx_align_center" id="S3.T1.22.22.4">-</td> </tr> <tr class="ltx_tr" id="S3.T1.24.24"> <td class="ltx_td ltx_align_center" id="S3.T1.23.23.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.23.23.1.m1.1"><semantics id="S3.T1.23.23.1.m1.1a"><mo id="S3.T1.23.23.1.m1.1.1" xref="S3.T1.23.23.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.23.23.1.m1.1b"><minus id="S3.T1.23.23.1.m1.1.1.cmml" xref="S3.T1.23.23.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.23.23.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.23.23.1.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S3.T1.24.24.2"><math alttext="-" class="ltx_Math" display="inline" id="S3.T1.24.24.2.m1.1"><semantics id="S3.T1.24.24.2.m1.1a"><mo id="S3.T1.24.24.2.m1.1.1" xref="S3.T1.24.24.2.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.24.24.2.m1.1b"><minus id="S3.T1.24.24.2.m1.1.1.cmml" xref="S3.T1.24.24.2.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.24.24.2.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.24.24.2.m1.1d">-</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.24.24.3">concave</td> <td class="ltx_td ltx_align_center" id="S3.T1.24.24.4">-</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 1: </span>Summary of whether <math alttext="x" class="ltx_Math" display="inline" id="S3.T1.33.m1.1"><semantics id="S3.T1.33.m1.1b"><mi id="S3.T1.33.m1.1.1" xref="S3.T1.33.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.T1.33.m1.1c"><ci id="S3.T1.33.m1.1.1.cmml" xref="S3.T1.33.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.33.m1.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.T1.33.m1.1e">italic_x</annotation></semantics></math> is on the <math alttext="P" class="ltx_Math" display="inline" id="S3.T1.34.m2.1"><semantics id="S3.T1.34.m2.1b"><mi id="S3.T1.34.m2.1.1" xref="S3.T1.34.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.T1.34.m2.1c"><ci id="S3.T1.34.m2.1.1.cmml" xref="S3.T1.34.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.34.m2.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.T1.34.m2.1e">italic_P</annotation></semantics></math>-side of vertex <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.T1.35.m3.1"><semantics id="S3.T1.35.m3.1b"><msub id="S3.T1.35.m3.1.1" xref="S3.T1.35.m3.1.1.cmml"><mi id="S3.T1.35.m3.1.1.2" xref="S3.T1.35.m3.1.1.2.cmml">v</mi><mi id="S3.T1.35.m3.1.1.3" xref="S3.T1.35.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.35.m3.1c"><apply id="S3.T1.35.m3.1.1.cmml" xref="S3.T1.35.m3.1.1"><csymbol cd="ambiguous" id="S3.T1.35.m3.1.1.1.cmml" xref="S3.T1.35.m3.1.1">subscript</csymbol><ci id="S3.T1.35.m3.1.1.2.cmml" xref="S3.T1.35.m3.1.1.2">𝑣</ci><ci id="S3.T1.35.m3.1.1.3.cmml" xref="S3.T1.35.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.35.m3.1d">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.35.m3.1e">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, depending on the position of <math alttext="x" class="ltx_Math" display="inline" id="S3.T1.36.m4.1"><semantics id="S3.T1.36.m4.1b"><mi id="S3.T1.36.m4.1.1" xref="S3.T1.36.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.T1.36.m4.1c"><ci id="S3.T1.36.m4.1.1.cmml" xref="S3.T1.36.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.36.m4.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.T1.36.m4.1e">italic_x</annotation></semantics></math> w.r.t. the incident edges and the convexity of the corresponding corner. The signs <math alttext="+" class="ltx_Math" display="inline" id="S3.T1.37.m5.1"><semantics id="S3.T1.37.m5.1b"><mo id="S3.T1.37.m5.1.1" xref="S3.T1.37.m5.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T1.37.m5.1c"><plus id="S3.T1.37.m5.1.1.cmml" xref="S3.T1.37.m5.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.37.m5.1d">+</annotation><annotation encoding="application/x-llamapun" id="S3.T1.37.m5.1e">+</annotation></semantics></math> and <math alttext="-" class="ltx_Math" display="inline" id="S3.T1.38.m6.1"><semantics id="S3.T1.38.m6.1b"><mo id="S3.T1.38.m6.1.1" xref="S3.T1.38.m6.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.T1.38.m6.1c"><minus id="S3.T1.38.m6.1.1.cmml" xref="S3.T1.38.m6.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.38.m6.1d">-</annotation><annotation encoding="application/x-llamapun" id="S3.T1.38.m6.1e">-</annotation></semantics></math> in the first two columns mean <math alttext="x\in H_{+}" class="ltx_Math" display="inline" id="S3.T1.39.m7.1"><semantics id="S3.T1.39.m7.1b"><mrow id="S3.T1.39.m7.1.1" xref="S3.T1.39.m7.1.1.cmml"><mi id="S3.T1.39.m7.1.1.2" xref="S3.T1.39.m7.1.1.2.cmml">x</mi><mo id="S3.T1.39.m7.1.1.1" xref="S3.T1.39.m7.1.1.1.cmml">∈</mo><msub id="S3.T1.39.m7.1.1.3" xref="S3.T1.39.m7.1.1.3.cmml"><mi id="S3.T1.39.m7.1.1.3.2" xref="S3.T1.39.m7.1.1.3.2.cmml">H</mi><mo id="S3.T1.39.m7.1.1.3.3" xref="S3.T1.39.m7.1.1.3.3.cmml">+</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.39.m7.1c"><apply id="S3.T1.39.m7.1.1.cmml" xref="S3.T1.39.m7.1.1"><in id="S3.T1.39.m7.1.1.1.cmml" xref="S3.T1.39.m7.1.1.1"></in><ci id="S3.T1.39.m7.1.1.2.cmml" xref="S3.T1.39.m7.1.1.2">𝑥</ci><apply id="S3.T1.39.m7.1.1.3.cmml" xref="S3.T1.39.m7.1.1.3"><csymbol cd="ambiguous" id="S3.T1.39.m7.1.1.3.1.cmml" xref="S3.T1.39.m7.1.1.3">subscript</csymbol><ci id="S3.T1.39.m7.1.1.3.2.cmml" xref="S3.T1.39.m7.1.1.3.2">𝐻</ci><plus id="S3.T1.39.m7.1.1.3.3.cmml" xref="S3.T1.39.m7.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.39.m7.1d">x\in H_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.39.m7.1e">italic_x ∈ italic_H start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x\in H_{-}" class="ltx_Math" display="inline" id="S3.T1.40.m8.1"><semantics id="S3.T1.40.m8.1b"><mrow id="S3.T1.40.m8.1.1" xref="S3.T1.40.m8.1.1.cmml"><mi id="S3.T1.40.m8.1.1.2" xref="S3.T1.40.m8.1.1.2.cmml">x</mi><mo id="S3.T1.40.m8.1.1.1" xref="S3.T1.40.m8.1.1.1.cmml">∈</mo><msub id="S3.T1.40.m8.1.1.3" xref="S3.T1.40.m8.1.1.3.cmml"><mi id="S3.T1.40.m8.1.1.3.2" xref="S3.T1.40.m8.1.1.3.2.cmml">H</mi><mo id="S3.T1.40.m8.1.1.3.3" xref="S3.T1.40.m8.1.1.3.3.cmml">−</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.40.m8.1c"><apply id="S3.T1.40.m8.1.1.cmml" xref="S3.T1.40.m8.1.1"><in id="S3.T1.40.m8.1.1.1.cmml" xref="S3.T1.40.m8.1.1.1"></in><ci id="S3.T1.40.m8.1.1.2.cmml" xref="S3.T1.40.m8.1.1.2">𝑥</ci><apply id="S3.T1.40.m8.1.1.3.cmml" xref="S3.T1.40.m8.1.1.3"><csymbol cd="ambiguous" id="S3.T1.40.m8.1.1.3.1.cmml" xref="S3.T1.40.m8.1.1.3">subscript</csymbol><ci id="S3.T1.40.m8.1.1.3.2.cmml" xref="S3.T1.40.m8.1.1.3.2">𝐻</ci><minus id="S3.T1.40.m8.1.1.3.3.cmml" xref="S3.T1.40.m8.1.1.3.3"></minus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.40.m8.1d">x\in H_{-}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.40.m8.1e">italic_x ∈ italic_H start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> respectively.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS1.4.p4"> <p class="ltx_p" id="S3.SS1.SSS1.4.p4.4">To verify (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E7" title="Equation 7 ‣ Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">7</span></a>), we consider each pair <math alttext="\{(v_{k},e_{k+1})\}_{k=1}^{n}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p4.1.m1.1"><semantics id="S3.SS1.SSS1.4.p4.1.m1.1a"><msubsup id="S3.SS1.SSS1.4.p4.1.m1.1.1" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.cmml"><mrow id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.2.cmml"><mo id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.2.cmml">{</mo><mrow id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.2" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.3.cmml"><mo id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.2.3" stretchy="false" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.3.cmml">(</mo><msub id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.1.1.1.1.1.1.2.cmml">v</mi><mi 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id="S3.SS1.SSS1.4.p4.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p4.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p4.1.m1.1c">\{(v_{k},e_{k+1})\}_{k=1}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p4.1.m1.1d">{ ( italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> of a vertex and the succeeding edge separately. As the contribution of <math alttext="(v_{k},e_{k+1})" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p4.2.m2.2"><semantics id="S3.SS1.SSS1.4.p4.2.m2.2a"><mrow id="S3.SS1.SSS1.4.p4.2.m2.2.2.2" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.3.cmml"><mo id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.3" stretchy="false" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.3.cmml">(</mo><msub id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.2" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.3" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.4" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.cmml"><mi id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.2" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.2.cmml">e</mi><mrow id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.cmml"><mi id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.2" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.1" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.3" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.5" stretchy="false" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p4.2.m2.2b"><interval closure="open" id="S3.SS1.SSS1.4.p4.2.m2.2.2.3.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2"><apply id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.1.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.2.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.2">𝑒</ci><apply id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3"><plus id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.1.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.1"></plus><ci id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.2.cmml" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.4.p4.2.m2.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p4.2.m2.2c">(v_{k},e_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p4.2.m2.2d">( italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math> to (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E9" title="Equation 9 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">9</span></a>) cancels if <math alttext="x\in Q^{k}\cap H^{k+1}_{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p4.3.m3.1"><semantics id="S3.SS1.SSS1.4.p4.3.m3.1a"><mrow id="S3.SS1.SSS1.4.p4.3.m3.1.1" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.cmml"><mi id="S3.SS1.SSS1.4.p4.3.m3.1.1.2" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.4.p4.3.m3.1.1.1" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS1.4.p4.3.m3.1.1.3" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.cmml"><msup id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.2" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.2.2" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.2.2.cmml">Q</mi><mi 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id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.2.3.3">1</cn></apply></apply><plus id="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.4.p4.3.m3.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p4.3.m3.1c">x\in Q^{k}\cap H^{k+1}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p4.3.m3.1d">italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∩ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="x\not\in Q^{k}\cup H^{k+1}_{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p4.4.m4.1"><semantics id="S3.SS1.SSS1.4.p4.4.m4.1a"><mrow id="S3.SS1.SSS1.4.p4.4.m4.1.1" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.4.p4.4.m4.1.1.2" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS1.4.p4.4.m4.1.1.1" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.1.cmml">∉</mo><mrow id="S3.SS1.SSS1.4.p4.4.m4.1.1.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.cmml"><msup id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.2" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.3.cmml">k</mi></msup><mo id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.1" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.1.cmml">∪</mo><msubsup id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.2" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.2.cmml">H</mi><mo id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.3.cmml">+</mo><mrow id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.2" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.1" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.3" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p4.4.m4.1b"><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1"><notin id="S3.SS1.SSS1.4.p4.4.m4.1.1.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.1"></notin><ci id="S3.SS1.SSS1.4.p4.4.m4.1.1.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.2">𝑥</ci><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3"><union id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.1"></union><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.2">𝑄</ci><ci id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3">subscript</csymbol><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.2">𝐻</ci><apply id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3"><plus id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.1"></plus><ci id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.2.3.3">1</cn></apply></apply><plus id="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.4.p4.4.m4.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p4.4.m4.1c">x\not\in Q^{k}\cup H^{k+1}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p4.4.m4.1d">italic_x ∉ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∪ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>, we get</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="v_{P}(x)-e_{P}(x)=|\{k\in[n]:x\in Q^{k}\setminus H^{k+1}_{+}\}|-|\{k\in[n]:x% \in H^{k+1}_{+}\setminus Q^{k}\}|." class="ltx_Math" display="block" id="S3.Ex16.m1.5"><semantics id="S3.Ex16.m1.5a"><mrow id="S3.Ex16.m1.5.5.1" xref="S3.Ex16.m1.5.5.1.1.cmml"><mrow id="S3.Ex16.m1.5.5.1.1" xref="S3.Ex16.m1.5.5.1.1.cmml"><mrow id="S3.Ex16.m1.5.5.1.1.4" xref="S3.Ex16.m1.5.5.1.1.4.cmml"><mrow id="S3.Ex16.m1.5.5.1.1.4.2" xref="S3.Ex16.m1.5.5.1.1.4.2.cmml"><msub id="S3.Ex16.m1.5.5.1.1.4.2.2" 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xref="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.2.3"></plus></apply><apply id="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.cmml" xref="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.1.cmml" xref="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3">superscript</csymbol><ci id="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.2.cmml" xref="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.2">𝑄</ci><ci id="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.3.cmml" xref="S3.Ex16.m1.5.5.1.1.2.2.1.1.2.2.3.3.3">𝑘</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex16.m1.5c">v_{P}(x)-e_{P}(x)=|\{k\in[n]:x\in Q^{k}\setminus H^{k+1}_{+}\}|-|\{k\in[n]:x% \in H^{k+1}_{+}\setminus Q^{k}\}|.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex16.m1.5d">italic_v start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) = | { italic_k ∈ [ italic_n ] : italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∖ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT } | - | { italic_k ∈ [ italic_n ] : italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ∖ italic_Q start_POSTSUPERSCRIPT italic_k 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.SS1.SSS1.4.p4.5">Considering <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.T1" title="Table 1 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Table 1</span></a>, this is precisely</p> <table class="ltx_equation ltx_eqn_table" id="S3.E10"> <tbody><tr class="ltx_equation ltx_eqn_row 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xref="S3.E10.m1.2.2.2.2.2.2.1.1.1.3.2.3">-+convex</mtext></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.18c">v_{P}(x)-e_{P}(x)\\ =\absolutevalue{\{k\in[n]:k=\texttt{+-concave}\}}-\absolutevalue{\{k\in[n]:k=% \texttt{-+convex}\}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.18d">start_ROW start_CELL italic_v start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW start_ROW start_CELL = | start_ARG { italic_k ∈ [ italic_n ] : italic_k = +-concave } end_ARG | - | start_ARG { italic_k ∈ [ italic_n ] : italic_k = -+convex } end_ARG | . end_CELL end_ROW</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">(10)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.SSS1.5.p5"> <p class="ltx_p" id="S3.SS1.SSS1.5.p5.10">To determine (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E10" title="Equation 10 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">10</span></a>), let us define certain angles, see <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F5" title="Figure 5 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 5</span></a>, that help to distinguish between the different cases:</p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.5"><math alttext="\alpha_{k}:=\angle v_{k-1}xv_{k}\in(-\pi,\pi)" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.2"><semantics id="S3.I1.i1.p1.1.m1.2a"><mrow id="S3.I1.i1.p1.1.m1.2.2" xref="S3.I1.i1.p1.1.m1.2.2.cmml"><msub id="S3.I1.i1.p1.1.m1.2.2.3" xref="S3.I1.i1.p1.1.m1.2.2.3.cmml"><mi id="S3.I1.i1.p1.1.m1.2.2.3.2" xref="S3.I1.i1.p1.1.m1.2.2.3.2.cmml">α</mi><mi id="S3.I1.i1.p1.1.m1.2.2.3.3" xref="S3.I1.i1.p1.1.m1.2.2.3.3.cmml">k</mi></msub><mo id="S3.I1.i1.p1.1.m1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.I1.i1.p1.1.m1.2.2.4.cmml">:=</mo><mrow id="S3.I1.i1.p1.1.m1.2.2.5" xref="S3.I1.i1.p1.1.m1.2.2.5.cmml"><mi id="S3.I1.i1.p1.1.m1.2.2.5.2" mathvariant="normal" xref="S3.I1.i1.p1.1.m1.2.2.5.2.cmml">∠</mi><mo id="S3.I1.i1.p1.1.m1.2.2.5.1" xref="S3.I1.i1.p1.1.m1.2.2.5.1.cmml">⁢</mo><msub id="S3.I1.i1.p1.1.m1.2.2.5.3" xref="S3.I1.i1.p1.1.m1.2.2.5.3.cmml"><mi 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xref="S3.I1.i1.p1.1.m1.2.2"></share><interval closure="open" id="S3.I1.i1.p1.1.m1.2.2.1.2.cmml" xref="S3.I1.i1.p1.1.m1.2.2.1.1"><apply id="S3.I1.i1.p1.1.m1.2.2.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.2.2.1.1.1"><minus id="S3.I1.i1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.2.2.1.1.1"></minus><ci id="S3.I1.i1.p1.1.m1.2.2.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.2.2.1.1.1.2">𝜋</ci></apply><ci id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">𝜋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.2c">\alpha_{k}:=\angle v_{k-1}xv_{k}\in(-\pi,\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.2d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := ∠ italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_x italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ ( - italic_π , italic_π )</annotation></semantics></math>, where <math alttext="\angle abc" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mrow id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.2" mathvariant="normal" xref="S3.I1.i1.p1.2.m2.1.1.2.cmml">∠</mi><mo id="S3.I1.i1.p1.2.m2.1.1.1" xref="S3.I1.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.I1.i1.p1.2.m2.1.1.3" xref="S3.I1.i1.p1.2.m2.1.1.3.cmml">a</mi><mo id="S3.I1.i1.p1.2.m2.1.1.1a" xref="S3.I1.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.I1.i1.p1.2.m2.1.1.4" xref="S3.I1.i1.p1.2.m2.1.1.4.cmml">b</mi><mo id="S3.I1.i1.p1.2.m2.1.1.1b" xref="S3.I1.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.I1.i1.p1.2.m2.1.1.5" xref="S3.I1.i1.p1.2.m2.1.1.5.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><apply id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><times id="S3.I1.i1.p1.2.m2.1.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1.1"></times><ci id="S3.I1.i1.p1.2.m2.1.1.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2">∠</ci><ci id="S3.I1.i1.p1.2.m2.1.1.3.cmml" xref="S3.I1.i1.p1.2.m2.1.1.3">𝑎</ci><ci id="S3.I1.i1.p1.2.m2.1.1.4.cmml" xref="S3.I1.i1.p1.2.m2.1.1.4">𝑏</ci><ci id="S3.I1.i1.p1.2.m2.1.1.5.cmml" xref="S3.I1.i1.p1.2.m2.1.1.5">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">\angle abc</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">∠ italic_a italic_b italic_c</annotation></semantics></math> is to be understood as the angle from <math alttext="a" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><mi id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><ci id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_a</annotation></semantics></math> to <math alttext="c" class="ltx_Math" display="inline" id="S3.I1.i1.p1.4.m4.1"><semantics id="S3.I1.i1.p1.4.m4.1a"><mi id="S3.I1.i1.p1.4.m4.1.1" xref="S3.I1.i1.p1.4.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.4.m4.1b"><ci id="S3.I1.i1.p1.4.m4.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.4.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.4.m4.1d">italic_c</annotation></semantics></math> about <math alttext="b" class="ltx_Math" display="inline" id="S3.I1.i1.p1.5.m5.1"><semantics id="S3.I1.i1.p1.5.m5.1a"><mi id="S3.I1.i1.p1.5.m5.1.1" xref="S3.I1.i1.p1.5.m5.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.5.m5.1b"><ci id="S3.I1.i1.p1.5.m5.1.1.cmml" xref="S3.I1.i1.p1.5.m5.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.5.m5.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.5.m5.1d">italic_b</annotation></semantics></math>. We have</p> <table class="ltx_equation ltx_eqn_table" id="S3.E11"> <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="\alpha_{k}\begin{cases}>0,\quad x\in H^{k}_{+}\\ <0,\quad x\in H^{k}_{-}\end{cases}" class="ltx_Math" display="block" id="S3.E11.m1.2"><semantics id="S3.E11.m1.2a"><mrow id="S3.E11.m1.2.3" xref="S3.E11.m1.2.3.cmml"><msub id="S3.E11.m1.2.3.2" xref="S3.E11.m1.2.3.2.cmml"><mi id="S3.E11.m1.2.3.2.2" xref="S3.E11.m1.2.3.2.2.cmml">α</mi><mi id="S3.E11.m1.2.3.2.3" xref="S3.E11.m1.2.3.2.3.cmml">k</mi></msub><mo id="S3.E11.m1.2.3.1" xref="S3.E11.m1.2.3.1.cmml">⁢</mo><mrow id="S3.E11.m1.2.2" xref="S3.E11.m1.2.3.3.1.cmml"><mo id="S3.E11.m1.2.2.3" xref="S3.E11.m1.2.3.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S3.E11.m1.2.2.2" rowspacing="0pt" xref="S3.E11.m1.2.3.3.1.cmml"><mtr id="S3.E11.m1.2.2.2a" xref="S3.E11.m1.2.3.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E11.m1.2.2.2b" xref="S3.E11.m1.2.3.3.1.cmml"><mrow id="S3.E11.m1.1.1.1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.1.1.3.cmml"><mrow id="S3.E11.m1.1.1.1.1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E11.m1.1.1.1.1.1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2.cmml"></mi><mo id="S3.E11.m1.1.1.1.1.1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.1.1.1.1.1.cmml">></mo><mn id="S3.E11.m1.1.1.1.1.1.1.1.1.3" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.E11.m1.1.1.1.1.1.1.2.3" rspace="1.167em" xref="S3.E11.m1.1.1.1.1.1.1.3a.cmml">,</mo><mrow id="S3.E11.m1.1.1.1.1.1.1.2.2" xref="S3.E11.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S3.E11.m1.1.1.1.1.1.1.2.2.2" xref="S3.E11.m1.1.1.1.1.1.1.2.2.2.cmml">x</mi><mo id="S3.E11.m1.1.1.1.1.1.1.2.2.1" xref="S3.E11.m1.1.1.1.1.1.1.2.2.1.cmml">∈</mo><msubsup id="S3.E11.m1.1.1.1.1.1.1.2.2.3" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.cmml"><mi id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.2" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.2.cmml">H</mi><mo id="S3.E11.m1.1.1.1.1.1.1.2.2.3.3" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.3.cmml">+</mo><mi id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.3" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.3.cmml">k</mi></msubsup></mrow></mrow></mtd><mtd id="S3.E11.m1.2.2.2c" xref="S3.E11.m1.2.3.3.1.1.cmml"></mtd></mtr><mtr id="S3.E11.m1.2.2.2d" xref="S3.E11.m1.2.3.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E11.m1.2.2.2e" xref="S3.E11.m1.2.3.3.1.cmml"><mrow id="S3.E11.m1.2.2.2.2.1.1.4" xref="S3.E11.m1.2.2.2.2.1.1.5.cmml"><mrow id="S3.E11.m1.2.2.2.2.1.1.3.1" xref="S3.E11.m1.2.2.2.2.1.1.3.1.cmml"><mi id="S3.E11.m1.2.2.2.2.1.1.3.1.2" xref="S3.E11.m1.2.2.2.2.1.1.3.1.2.cmml"></mi><mo id="S3.E11.m1.2.2.2.2.1.1.3.1.1" xref="S3.E11.m1.2.2.2.2.1.1.3.1.1.cmml"><</mo><mn id="S3.E11.m1.2.2.2.2.1.1.1" xref="S3.E11.m1.2.2.2.2.1.1.1.cmml">0</mn></mrow><mo id="S3.E11.m1.2.2.2.2.1.1.4.3" rspace="1.167em" xref="S3.E11.m1.2.2.2.2.1.1.5a.cmml">,</mo><mrow id="S3.E11.m1.2.2.2.2.1.1.4.2" xref="S3.E11.m1.2.2.2.2.1.1.4.2.cmml"><mi id="S3.E11.m1.2.2.2.2.1.1.2" xref="S3.E11.m1.2.2.2.2.1.1.2.cmml">x</mi><mo id="S3.E11.m1.2.2.2.2.1.1.4.2.1" xref="S3.E11.m1.2.2.2.2.1.1.4.2.1.cmml">∈</mo><msubsup id="S3.E11.m1.2.2.2.2.1.1.4.2.2" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.cmml"><mi id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.2" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.2.cmml">H</mi><mo id="S3.E11.m1.2.2.2.2.1.1.4.2.2.3" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.3.cmml">−</mo><mi id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.3" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.3.cmml">k</mi></msubsup></mrow></mrow></mtd><mtd id="S3.E11.m1.2.2.2f" xref="S3.E11.m1.2.3.3.1.1.cmml"></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E11.m1.2b"><apply id="S3.E11.m1.2.3.cmml" xref="S3.E11.m1.2.3"><times id="S3.E11.m1.2.3.1.cmml" xref="S3.E11.m1.2.3.1"></times><apply id="S3.E11.m1.2.3.2.cmml" xref="S3.E11.m1.2.3.2"><csymbol cd="ambiguous" id="S3.E11.m1.2.3.2.1.cmml" xref="S3.E11.m1.2.3.2">subscript</csymbol><ci id="S3.E11.m1.2.3.2.2.cmml" xref="S3.E11.m1.2.3.2.2">𝛼</ci><ci id="S3.E11.m1.2.3.2.3.cmml" xref="S3.E11.m1.2.3.2.3">𝑘</ci></apply><apply id="S3.E11.m1.2.3.3.1.cmml" xref="S3.E11.m1.2.2"><csymbol cd="latexml" id="S3.E11.m1.2.3.3.1.1.cmml" xref="S3.E11.m1.2.2.3">cases</csymbol><apply id="S3.E11.m1.1.1.1.1.1.1.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.1.3a.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E11.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1"><gt id="S3.E11.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.1"></gt><csymbol cd="latexml" id="S3.E11.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.2">absent</csymbol><cn id="S3.E11.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3">0</cn></apply><apply id="S3.E11.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2"><in id="S3.E11.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.1"></in><ci id="S3.E11.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.2">𝑥</ci><apply id="S3.E11.m1.1.1.1.1.1.1.2.2.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.1.2.2.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3">subscript</csymbol><apply id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3">superscript</csymbol><ci id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.2">𝐻</ci><ci id="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.2.3">𝑘</ci></apply><plus id="S3.E11.m1.1.1.1.1.1.1.2.2.3.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.2.2.3.3"></plus></apply></apply></apply><ci id="S3.E11.m1.2.3.3.1.3a.cmml" xref="S3.E11.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S3.E11.m1.2.3.3.1.3.cmml" xref="S3.E11.m1.2.2.3">otherwise</mtext></ci><apply id="S3.E11.m1.2.2.2.2.1.1.5.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.E11.m1.2.2.2.2.1.1.5a.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.3">formulae-sequence</csymbol><apply id="S3.E11.m1.2.2.2.2.1.1.3.1.cmml" xref="S3.E11.m1.2.2.2.2.1.1.3.1"><lt id="S3.E11.m1.2.2.2.2.1.1.3.1.1.cmml" xref="S3.E11.m1.2.2.2.2.1.1.3.1.1"></lt><csymbol cd="latexml" id="S3.E11.m1.2.2.2.2.1.1.3.1.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.3.1.2">absent</csymbol><cn id="S3.E11.m1.2.2.2.2.1.1.1.cmml" type="integer" xref="S3.E11.m1.2.2.2.2.1.1.1">0</cn></apply><apply id="S3.E11.m1.2.2.2.2.1.1.4.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2"><in id="S3.E11.m1.2.2.2.2.1.1.4.2.1.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.1"></in><ci id="S3.E11.m1.2.2.2.2.1.1.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.2">𝑥</ci><apply id="S3.E11.m1.2.2.2.2.1.1.4.2.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2"><csymbol cd="ambiguous" id="S3.E11.m1.2.2.2.2.1.1.4.2.2.1.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2">subscript</csymbol><apply id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2"><csymbol cd="ambiguous" id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.1.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2">superscript</csymbol><ci id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.2.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.2">𝐻</ci><ci id="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.3.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.2.3">𝑘</ci></apply><minus id="S3.E11.m1.2.2.2.2.1.1.4.2.2.3.cmml" xref="S3.E11.m1.2.2.2.2.1.1.4.2.2.3"></minus></apply></apply></apply><ci id="S3.E11.m1.2.3.3.1.5a.cmml" xref="S3.E11.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S3.E11.m1.2.3.3.1.5.cmml" xref="S3.E11.m1.2.2.3">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.2c">\alpha_{k}\begin{cases}>0,\quad x\in H^{k}_{+}\\ <0,\quad x\in H^{k}_{-}\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.2d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { start_ROW start_CELL > 0 , italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL < 0 , italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW</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">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.I1.i1.p1.14">because <math alttext="P" class="ltx_Math" display="inline" id="S3.I1.i1.p1.6.m1.1"><semantics id="S3.I1.i1.p1.6.m1.1a"><mi id="S3.I1.i1.p1.6.m1.1.1" xref="S3.I1.i1.p1.6.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.6.m1.1b"><ci id="S3.I1.i1.p1.6.m1.1.1.cmml" xref="S3.I1.i1.p1.6.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.6.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.6.m1.1d">italic_P</annotation></semantics></math> is always on the left when traversing <math alttext="e_{k}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.7.m2.1"><semantics id="S3.I1.i1.p1.7.m2.1a"><msub id="S3.I1.i1.p1.7.m2.1.1" xref="S3.I1.i1.p1.7.m2.1.1.cmml"><mi id="S3.I1.i1.p1.7.m2.1.1.2" xref="S3.I1.i1.p1.7.m2.1.1.2.cmml">e</mi><mi id="S3.I1.i1.p1.7.m2.1.1.3" xref="S3.I1.i1.p1.7.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.7.m2.1b"><apply id="S3.I1.i1.p1.7.m2.1.1.cmml" xref="S3.I1.i1.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.7.m2.1.1.1.cmml" xref="S3.I1.i1.p1.7.m2.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.7.m2.1.1.2.cmml" xref="S3.I1.i1.p1.7.m2.1.1.2">𝑒</ci><ci id="S3.I1.i1.p1.7.m2.1.1.3.cmml" xref="S3.I1.i1.p1.7.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.7.m2.1c">e_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.7.m2.1d">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="v_{k-1}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.8.m3.1"><semantics id="S3.I1.i1.p1.8.m3.1a"><msub id="S3.I1.i1.p1.8.m3.1.1" xref="S3.I1.i1.p1.8.m3.1.1.cmml"><mi id="S3.I1.i1.p1.8.m3.1.1.2" xref="S3.I1.i1.p1.8.m3.1.1.2.cmml">v</mi><mrow id="S3.I1.i1.p1.8.m3.1.1.3" xref="S3.I1.i1.p1.8.m3.1.1.3.cmml"><mi id="S3.I1.i1.p1.8.m3.1.1.3.2" xref="S3.I1.i1.p1.8.m3.1.1.3.2.cmml">k</mi><mo id="S3.I1.i1.p1.8.m3.1.1.3.1" xref="S3.I1.i1.p1.8.m3.1.1.3.1.cmml">−</mo><mn id="S3.I1.i1.p1.8.m3.1.1.3.3" xref="S3.I1.i1.p1.8.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.8.m3.1b"><apply id="S3.I1.i1.p1.8.m3.1.1.cmml" xref="S3.I1.i1.p1.8.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.8.m3.1.1.1.cmml" xref="S3.I1.i1.p1.8.m3.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.8.m3.1.1.2.cmml" xref="S3.I1.i1.p1.8.m3.1.1.2">𝑣</ci><apply id="S3.I1.i1.p1.8.m3.1.1.3.cmml" xref="S3.I1.i1.p1.8.m3.1.1.3"><minus id="S3.I1.i1.p1.8.m3.1.1.3.1.cmml" xref="S3.I1.i1.p1.8.m3.1.1.3.1"></minus><ci id="S3.I1.i1.p1.8.m3.1.1.3.2.cmml" xref="S3.I1.i1.p1.8.m3.1.1.3.2">𝑘</ci><cn id="S3.I1.i1.p1.8.m3.1.1.3.3.cmml" type="integer" xref="S3.I1.i1.p1.8.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.8.m3.1c">v_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.8.m3.1d">italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.9.m4.1"><semantics id="S3.I1.i1.p1.9.m4.1a"><msub id="S3.I1.i1.p1.9.m4.1.1" xref="S3.I1.i1.p1.9.m4.1.1.cmml"><mi id="S3.I1.i1.p1.9.m4.1.1.2" xref="S3.I1.i1.p1.9.m4.1.1.2.cmml">v</mi><mi id="S3.I1.i1.p1.9.m4.1.1.3" xref="S3.I1.i1.p1.9.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.9.m4.1b"><apply id="S3.I1.i1.p1.9.m4.1.1.cmml" xref="S3.I1.i1.p1.9.m4.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.9.m4.1.1.1.cmml" xref="S3.I1.i1.p1.9.m4.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.9.m4.1.1.2.cmml" xref="S3.I1.i1.p1.9.m4.1.1.2">𝑣</ci><ci id="S3.I1.i1.p1.9.m4.1.1.3.cmml" xref="S3.I1.i1.p1.9.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.9.m4.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.9.m4.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. The sum of these angles is the winding number of <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.I1.i1.p1.10.m5.1"><semantics id="S3.I1.i1.p1.10.m5.1a"><mi id="S3.I1.i1.p1.10.m5.1.1" xref="S3.I1.i1.p1.10.m5.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.10.m5.1b"><ci id="S3.I1.i1.p1.10.m5.1.1.cmml" xref="S3.I1.i1.p1.10.m5.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.10.m5.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.10.m5.1d">italic_γ</annotation></semantics></math> with respect to <math alttext="x" class="ltx_Math" display="inline" id="S3.I1.i1.p1.11.m6.1"><semantics id="S3.I1.i1.p1.11.m6.1a"><mi id="S3.I1.i1.p1.11.m6.1.1" xref="S3.I1.i1.p1.11.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.11.m6.1b"><ci id="S3.I1.i1.p1.11.m6.1.1.cmml" xref="S3.I1.i1.p1.11.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.11.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.11.m6.1d">italic_x</annotation></semantics></math>, see, e.g., <span class="ltx_ERROR undefined" id="S3.I1.i1.p1.14.1">\citet</span>Hormann2001incremental. Because <math alttext="P=\overline{\operatorname*{int}(\gamma)}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.12.m7.2"><semantics id="S3.I1.i1.p1.12.m7.2a"><mrow id="S3.I1.i1.p1.12.m7.2.3" xref="S3.I1.i1.p1.12.m7.2.3.cmml"><mi id="S3.I1.i1.p1.12.m7.2.3.2" xref="S3.I1.i1.p1.12.m7.2.3.2.cmml">P</mi><mo id="S3.I1.i1.p1.12.m7.2.3.1" rspace="0.1389em" xref="S3.I1.i1.p1.12.m7.2.3.1.cmml">=</mo><mover accent="true" id="S3.I1.i1.p1.12.m7.2.2" xref="S3.I1.i1.p1.12.m7.2.2.cmml"><mrow id="S3.I1.i1.p1.12.m7.2.2.2.4" xref="S3.I1.i1.p1.12.m7.2.2.2.3.cmml"><mo id="S3.I1.i1.p1.12.m7.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.I1.i1.p1.12.m7.1.1.1.1.cmml">int</mo><mrow id="S3.I1.i1.p1.12.m7.2.2.2.4.1" xref="S3.I1.i1.p1.12.m7.2.2.2.3.cmml"><mo id="S3.I1.i1.p1.12.m7.2.2.2.4.1.1" stretchy="false" xref="S3.I1.i1.p1.12.m7.2.2.2.3.cmml">(</mo><mi id="S3.I1.i1.p1.12.m7.2.2.2.2" xref="S3.I1.i1.p1.12.m7.2.2.2.2.cmml">γ</mi><mo id="S3.I1.i1.p1.12.m7.2.2.2.4.1.2" stretchy="false" xref="S3.I1.i1.p1.12.m7.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I1.i1.p1.12.m7.2.2.3" xref="S3.I1.i1.p1.12.m7.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.12.m7.2b"><apply id="S3.I1.i1.p1.12.m7.2.3.cmml" xref="S3.I1.i1.p1.12.m7.2.3"><eq id="S3.I1.i1.p1.12.m7.2.3.1.cmml" xref="S3.I1.i1.p1.12.m7.2.3.1"></eq><ci id="S3.I1.i1.p1.12.m7.2.3.2.cmml" xref="S3.I1.i1.p1.12.m7.2.3.2">𝑃</ci><apply id="S3.I1.i1.p1.12.m7.2.2.cmml" xref="S3.I1.i1.p1.12.m7.2.2"><ci id="S3.I1.i1.p1.12.m7.2.2.3.cmml" xref="S3.I1.i1.p1.12.m7.2.2.3">¯</ci><apply id="S3.I1.i1.p1.12.m7.2.2.2.3.cmml" xref="S3.I1.i1.p1.12.m7.2.2.2.4"><ci id="S3.I1.i1.p1.12.m7.1.1.1.1.cmml" xref="S3.I1.i1.p1.12.m7.1.1.1.1">int</ci><ci id="S3.I1.i1.p1.12.m7.2.2.2.2.cmml" xref="S3.I1.i1.p1.12.m7.2.2.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.12.m7.2c">P=\overline{\operatorname*{int}(\gamma)}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.12.m7.2d">italic_P = over¯ start_ARG roman_int ( italic_γ ) end_ARG</annotation></semantics></math> is the bounded component defined by the simple (i.e., non-self-intersecting) closed curve <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.I1.i1.p1.13.m8.1"><semantics id="S3.I1.i1.p1.13.m8.1a"><mi id="S3.I1.i1.p1.13.m8.1.1" xref="S3.I1.i1.p1.13.m8.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.13.m8.1b"><ci id="S3.I1.i1.p1.13.m8.1.1.cmml" xref="S3.I1.i1.p1.13.m8.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.13.m8.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.13.m8.1d">italic_γ</annotation></semantics></math>, and due to the counterclockwise enumeration, we can use the winding number to distinguish between the interior and exterior of <math alttext="P" class="ltx_Math" display="inline" id="S3.I1.i1.p1.14.m9.1"><semantics id="S3.I1.i1.p1.14.m9.1a"><mi id="S3.I1.i1.p1.14.m9.1.1" xref="S3.I1.i1.p1.14.m9.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.14.m9.1b"><ci id="S3.I1.i1.p1.14.m9.1.1.cmml" xref="S3.I1.i1.p1.14.m9.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.14.m9.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.14.m9.1d">italic_P</annotation></semantics></math> (see, e.g., <span class="ltx_ERROR undefined" id="S3.I1.i1.p1.14.2">\citet</span>[Section 5-7]Carmo1976Differentialgeometry):</p> <table class="ltx_equation ltx_eqn_table" id="S3.E12"> <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_{k=1}^{n}\alpha_{k}=2\pi\cdot\mathds{1}_{P}(x)" class="ltx_Math" display="block" id="S3.E12.m1.1"><semantics id="S3.E12.m1.1a"><mrow id="S3.E12.m1.1.2" xref="S3.E12.m1.1.2.cmml"><mrow id="S3.E12.m1.1.2.2" xref="S3.E12.m1.1.2.2.cmml"><munderover id="S3.E12.m1.1.2.2.1" xref="S3.E12.m1.1.2.2.1.cmml"><mo id="S3.E12.m1.1.2.2.1.2.2" movablelimits="false" xref="S3.E12.m1.1.2.2.1.2.2.cmml">∑</mo><mrow id="S3.E12.m1.1.2.2.1.2.3" xref="S3.E12.m1.1.2.2.1.2.3.cmml"><mi id="S3.E12.m1.1.2.2.1.2.3.2" xref="S3.E12.m1.1.2.2.1.2.3.2.cmml">k</mi><mo id="S3.E12.m1.1.2.2.1.2.3.1" xref="S3.E12.m1.1.2.2.1.2.3.1.cmml">=</mo><mn id="S3.E12.m1.1.2.2.1.2.3.3" xref="S3.E12.m1.1.2.2.1.2.3.3.cmml">1</mn></mrow><mi id="S3.E12.m1.1.2.2.1.3" xref="S3.E12.m1.1.2.2.1.3.cmml">n</mi></munderover><msub id="S3.E12.m1.1.2.2.2" xref="S3.E12.m1.1.2.2.2.cmml"><mi id="S3.E12.m1.1.2.2.2.2" xref="S3.E12.m1.1.2.2.2.2.cmml">α</mi><mi id="S3.E12.m1.1.2.2.2.3" xref="S3.E12.m1.1.2.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.E12.m1.1.2.1" xref="S3.E12.m1.1.2.1.cmml">=</mo><mrow id="S3.E12.m1.1.2.3" xref="S3.E12.m1.1.2.3.cmml"><mrow id="S3.E12.m1.1.2.3.2" xref="S3.E12.m1.1.2.3.2.cmml"><mrow id="S3.E12.m1.1.2.3.2.2" xref="S3.E12.m1.1.2.3.2.2.cmml"><mn id="S3.E12.m1.1.2.3.2.2.2" xref="S3.E12.m1.1.2.3.2.2.2.cmml">2</mn><mo id="S3.E12.m1.1.2.3.2.2.1" xref="S3.E12.m1.1.2.3.2.2.1.cmml">⁢</mo><mi id="S3.E12.m1.1.2.3.2.2.3" xref="S3.E12.m1.1.2.3.2.2.3.cmml">π</mi></mrow><mo id="S3.E12.m1.1.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.E12.m1.1.2.3.2.1.cmml">⋅</mo><msub id="S3.E12.m1.1.2.3.2.3" xref="S3.E12.m1.1.2.3.2.3.cmml"><mn id="S3.E12.m1.1.2.3.2.3.2" xref="S3.E12.m1.1.2.3.2.3.2.cmml">𝟙</mn><mi id="S3.E12.m1.1.2.3.2.3.3" xref="S3.E12.m1.1.2.3.2.3.3.cmml">P</mi></msub></mrow><mo id="S3.E12.m1.1.2.3.1" xref="S3.E12.m1.1.2.3.1.cmml">⁢</mo><mrow id="S3.E12.m1.1.2.3.3.2" xref="S3.E12.m1.1.2.3.cmml"><mo id="S3.E12.m1.1.2.3.3.2.1" stretchy="false" xref="S3.E12.m1.1.2.3.cmml">(</mo><mi id="S3.E12.m1.1.1" xref="S3.E12.m1.1.1.cmml">x</mi><mo id="S3.E12.m1.1.2.3.3.2.2" stretchy="false" xref="S3.E12.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E12.m1.1b"><apply 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id="S3.E12.m1.1.2.3.2.3.2.cmml" type="integer" xref="S3.E12.m1.1.2.3.2.3.2">1</cn><ci id="S3.E12.m1.1.2.3.2.3.3.cmml" xref="S3.E12.m1.1.2.3.2.3.3">𝑃</ci></apply></apply><ci id="S3.E12.m1.1.1.cmml" xref="S3.E12.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.1c">\sum_{k=1}^{n}\alpha_{k}=2\pi\cdot\mathds{1}_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.1d">∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 2 italic_π ⋅ blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</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">(12)</span></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" 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xref="S3.I1.i2.p1.1.m1.2.2.6"></in><share href="https://arxiv.org/html/2503.13001v1#S3.I1.i2.p1.1.m1.2.2.5.cmml" id="S3.I1.i2.p1.1.m1.2.2d.cmml" xref="S3.I1.i2.p1.1.m1.2.2"></share><interval closure="open" id="S3.I1.i2.p1.1.m1.2.2.1.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1"><cn id="S3.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1">0</cn><apply id="S3.I1.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1.1"><times id="S3.I1.i2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1.1.1"></times><cn id="S3.I1.i2.p1.1.m1.2.2.1.1.1.2.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.2.2.1.1.1.2">2</cn><ci id="S3.I1.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1.1.3">𝜋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.2c">\eta_{k}:=\angle v_{k+1}v_{k}v_{k-1}\in(0,2\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.2d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := ∠ italic_v start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∈ ( 0 , 2 italic_π )</annotation></semantics></math>, measured counterclockwise. The corner at <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><msub id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml">v</mi><mi id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">𝑣</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is convex if <math alttext="\eta_{k}\leq\pi" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><mrow id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml"><msub id="S3.I1.i2.p1.3.m3.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.2.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.2.2" xref="S3.I1.i2.p1.3.m3.1.1.2.2.cmml">η</mi><mi id="S3.I1.i2.p1.3.m3.1.1.2.3" xref="S3.I1.i2.p1.3.m3.1.1.2.3.cmml">k</mi></msub><mo id="S3.I1.i2.p1.3.m3.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.cmml">≤</mo><mi id="S3.I1.i2.p1.3.m3.1.1.3" xref="S3.I1.i2.p1.3.m3.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.3.m3.1b"><apply id="S3.I1.i2.p1.3.m3.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1"><leq id="S3.I1.i2.p1.3.m3.1.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1.1"></leq><apply id="S3.I1.i2.p1.3.m3.1.1.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.3.m3.1.1.2.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.3.m3.1.1.2.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2.2">𝜂</ci><ci id="S3.I1.i2.p1.3.m3.1.1.2.3.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2.3">𝑘</ci></apply><ci id="S3.I1.i2.p1.3.m3.1.1.3.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1c">\eta_{k}\leq\pi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ italic_π</annotation></semantics></math> and concave if <math alttext="\eta_{k}\geq\pi" class="ltx_Math" display="inline" id="S3.I1.i2.p1.4.m4.1"><semantics id="S3.I1.i2.p1.4.m4.1a"><mrow id="S3.I1.i2.p1.4.m4.1.1" xref="S3.I1.i2.p1.4.m4.1.1.cmml"><msub id="S3.I1.i2.p1.4.m4.1.1.2" xref="S3.I1.i2.p1.4.m4.1.1.2.cmml"><mi id="S3.I1.i2.p1.4.m4.1.1.2.2" xref="S3.I1.i2.p1.4.m4.1.1.2.2.cmml">η</mi><mi id="S3.I1.i2.p1.4.m4.1.1.2.3" xref="S3.I1.i2.p1.4.m4.1.1.2.3.cmml">k</mi></msub><mo id="S3.I1.i2.p1.4.m4.1.1.1" xref="S3.I1.i2.p1.4.m4.1.1.1.cmml">≥</mo><mi id="S3.I1.i2.p1.4.m4.1.1.3" xref="S3.I1.i2.p1.4.m4.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.4.m4.1b"><apply id="S3.I1.i2.p1.4.m4.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1"><geq id="S3.I1.i2.p1.4.m4.1.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.1"></geq><apply id="S3.I1.i2.p1.4.m4.1.1.2.cmml" xref="S3.I1.i2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.4.m4.1.1.2.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.4.m4.1.1.2.2.cmml" xref="S3.I1.i2.p1.4.m4.1.1.2.2">𝜂</ci><ci id="S3.I1.i2.p1.4.m4.1.1.2.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.2.3">𝑘</ci></apply><ci id="S3.I1.i2.p1.4.m4.1.1.3.cmml" xref="S3.I1.i2.p1.4.m4.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.4.m4.1c">\eta_{k}\geq\pi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.4.m4.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≥ italic_π</annotation></semantics></math>. Further, since <math alttext="P" class="ltx_Math" display="inline" id="S3.I1.i2.p1.5.m5.1"><semantics id="S3.I1.i2.p1.5.m5.1a"><mi id="S3.I1.i2.p1.5.m5.1.1" xref="S3.I1.i2.p1.5.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.5.m5.1b"><ci id="S3.I1.i2.p1.5.m5.1.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.5.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.5.m5.1d">italic_P</annotation></semantics></math> is a simple polygon in the classical sense, we have the relation</p> <table class="ltx_equation ltx_eqn_table" id="S3.E13"> <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_{k=1}^{n}\eta_{k}=(n-2)\pi" class="ltx_Math" display="block" id="S3.E13.m1.1"><semantics id="S3.E13.m1.1a"><mrow id="S3.E13.m1.1.1" xref="S3.E13.m1.1.1.cmml"><mrow id="S3.E13.m1.1.1.3" xref="S3.E13.m1.1.1.3.cmml"><munderover id="S3.E13.m1.1.1.3.1" xref="S3.E13.m1.1.1.3.1.cmml"><mo id="S3.E13.m1.1.1.3.1.2.2" movablelimits="false" xref="S3.E13.m1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S3.E13.m1.1.1.3.1.2.3" xref="S3.E13.m1.1.1.3.1.2.3.cmml"><mi id="S3.E13.m1.1.1.3.1.2.3.2" xref="S3.E13.m1.1.1.3.1.2.3.2.cmml">k</mi><mo id="S3.E13.m1.1.1.3.1.2.3.1" xref="S3.E13.m1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S3.E13.m1.1.1.3.1.2.3.3" xref="S3.E13.m1.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S3.E13.m1.1.1.3.1.3" xref="S3.E13.m1.1.1.3.1.3.cmml">n</mi></munderover><msub id="S3.E13.m1.1.1.3.2" xref="S3.E13.m1.1.1.3.2.cmml"><mi id="S3.E13.m1.1.1.3.2.2" xref="S3.E13.m1.1.1.3.2.2.cmml">η</mi><mi id="S3.E13.m1.1.1.3.2.3" xref="S3.E13.m1.1.1.3.2.3.cmml">k</mi></msub></mrow><mo id="S3.E13.m1.1.1.2" xref="S3.E13.m1.1.1.2.cmml">=</mo><mrow id="S3.E13.m1.1.1.1" xref="S3.E13.m1.1.1.1.cmml"><mrow id="S3.E13.m1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.cmml"><mo id="S3.E13.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E13.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.cmml"><mi id="S3.E13.m1.1.1.1.1.1.1.2" xref="S3.E13.m1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S3.E13.m1.1.1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.E13.m1.1.1.1.1.1.1.3" xref="S3.E13.m1.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.E13.m1.1.1.1.1.1.3" stretchy="false" xref="S3.E13.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E13.m1.1.1.1.2" xref="S3.E13.m1.1.1.1.2.cmml">⁢</mo><mi id="S3.E13.m1.1.1.1.3" xref="S3.E13.m1.1.1.1.3.cmml">π</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E13.m1.1b"><apply id="S3.E13.m1.1.1.cmml" xref="S3.E13.m1.1.1"><eq id="S3.E13.m1.1.1.2.cmml" xref="S3.E13.m1.1.1.2"></eq><apply id="S3.E13.m1.1.1.3.cmml" xref="S3.E13.m1.1.1.3"><apply id="S3.E13.m1.1.1.3.1.cmml" xref="S3.E13.m1.1.1.3.1"><csymbol cd="ambiguous" id="S3.E13.m1.1.1.3.1.1.cmml" xref="S3.E13.m1.1.1.3.1">superscript</csymbol><apply id="S3.E13.m1.1.1.3.1.2.cmml" xref="S3.E13.m1.1.1.3.1"><csymbol cd="ambiguous" id="S3.E13.m1.1.1.3.1.2.1.cmml" xref="S3.E13.m1.1.1.3.1">subscript</csymbol><sum id="S3.E13.m1.1.1.3.1.2.2.cmml" xref="S3.E13.m1.1.1.3.1.2.2"></sum><apply id="S3.E13.m1.1.1.3.1.2.3.cmml" xref="S3.E13.m1.1.1.3.1.2.3"><eq id="S3.E13.m1.1.1.3.1.2.3.1.cmml" xref="S3.E13.m1.1.1.3.1.2.3.1"></eq><ci id="S3.E13.m1.1.1.3.1.2.3.2.cmml" xref="S3.E13.m1.1.1.3.1.2.3.2">𝑘</ci><cn id="S3.E13.m1.1.1.3.1.2.3.3.cmml" type="integer" xref="S3.E13.m1.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S3.E13.m1.1.1.3.1.3.cmml" xref="S3.E13.m1.1.1.3.1.3">𝑛</ci></apply><apply id="S3.E13.m1.1.1.3.2.cmml" xref="S3.E13.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.E13.m1.1.1.3.2.1.cmml" xref="S3.E13.m1.1.1.3.2">subscript</csymbol><ci id="S3.E13.m1.1.1.3.2.2.cmml" xref="S3.E13.m1.1.1.3.2.2">𝜂</ci><ci id="S3.E13.m1.1.1.3.2.3.cmml" xref="S3.E13.m1.1.1.3.2.3">𝑘</ci></apply></apply><apply id="S3.E13.m1.1.1.1.cmml" xref="S3.E13.m1.1.1.1"><times id="S3.E13.m1.1.1.1.2.cmml" xref="S3.E13.m1.1.1.1.2"></times><apply id="S3.E13.m1.1.1.1.1.1.1.cmml" xref="S3.E13.m1.1.1.1.1.1"><minus id="S3.E13.m1.1.1.1.1.1.1.1.cmml" xref="S3.E13.m1.1.1.1.1.1.1.1"></minus><ci id="S3.E13.m1.1.1.1.1.1.1.2.cmml" xref="S3.E13.m1.1.1.1.1.1.1.2">𝑛</ci><cn id="S3.E13.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.E13.m1.1.1.1.1.1.1.3">2</cn></apply><ci id="S3.E13.m1.1.1.1.3.cmml" xref="S3.E13.m1.1.1.1.3">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.1c">\sum_{k=1}^{n}\eta_{k}=(n-2)\pi</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.1d">∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( italic_n - 2 ) italic_π</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">(13)</span></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.6"><math alttext="\beta_{k}:=\angle xv_{k}v_{k+1}\in(-\pi,\pi)" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.2"><semantics id="S3.I1.i3.p1.1.m1.2a"><mrow id="S3.I1.i3.p1.1.m1.2.2" xref="S3.I1.i3.p1.1.m1.2.2.cmml"><msub id="S3.I1.i3.p1.1.m1.2.2.3" xref="S3.I1.i3.p1.1.m1.2.2.3.cmml"><mi id="S3.I1.i3.p1.1.m1.2.2.3.2" xref="S3.I1.i3.p1.1.m1.2.2.3.2.cmml">β</mi><mi id="S3.I1.i3.p1.1.m1.2.2.3.3" xref="S3.I1.i3.p1.1.m1.2.2.3.3.cmml">k</mi></msub><mo id="S3.I1.i3.p1.1.m1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.I1.i3.p1.1.m1.2.2.4.cmml">:=</mo><mrow id="S3.I1.i3.p1.1.m1.2.2.5" xref="S3.I1.i3.p1.1.m1.2.2.5.cmml"><mi id="S3.I1.i3.p1.1.m1.2.2.5.2" mathvariant="normal" xref="S3.I1.i3.p1.1.m1.2.2.5.2.cmml">∠</mi><mo id="S3.I1.i3.p1.1.m1.2.2.5.1" xref="S3.I1.i3.p1.1.m1.2.2.5.1.cmml">⁢</mo><mi id="S3.I1.i3.p1.1.m1.2.2.5.3" xref="S3.I1.i3.p1.1.m1.2.2.5.3.cmml">x</mi><mo id="S3.I1.i3.p1.1.m1.2.2.5.1a" xref="S3.I1.i3.p1.1.m1.2.2.5.1.cmml">⁢</mo><msub id="S3.I1.i3.p1.1.m1.2.2.5.4" xref="S3.I1.i3.p1.1.m1.2.2.5.4.cmml"><mi id="S3.I1.i3.p1.1.m1.2.2.5.4.2" xref="S3.I1.i3.p1.1.m1.2.2.5.4.2.cmml">v</mi><mi id="S3.I1.i3.p1.1.m1.2.2.5.4.3" xref="S3.I1.i3.p1.1.m1.2.2.5.4.3.cmml">k</mi></msub><mo id="S3.I1.i3.p1.1.m1.2.2.5.1b" xref="S3.I1.i3.p1.1.m1.2.2.5.1.cmml">⁢</mo><msub id="S3.I1.i3.p1.1.m1.2.2.5.5" xref="S3.I1.i3.p1.1.m1.2.2.5.5.cmml"><mi id="S3.I1.i3.p1.1.m1.2.2.5.5.2" xref="S3.I1.i3.p1.1.m1.2.2.5.5.2.cmml">v</mi><mrow id="S3.I1.i3.p1.1.m1.2.2.5.5.3" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.cmml"><mi id="S3.I1.i3.p1.1.m1.2.2.5.5.3.2" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.2.cmml">k</mi><mo id="S3.I1.i3.p1.1.m1.2.2.5.5.3.1" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.1.cmml">+</mo><mn id="S3.I1.i3.p1.1.m1.2.2.5.5.3.3" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S3.I1.i3.p1.1.m1.2.2.6" xref="S3.I1.i3.p1.1.m1.2.2.6.cmml">∈</mo><mrow id="S3.I1.i3.p1.1.m1.2.2.1.1" xref="S3.I1.i3.p1.1.m1.2.2.1.2.cmml"><mo id="S3.I1.i3.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S3.I1.i3.p1.1.m1.2.2.1.2.cmml">(</mo><mrow id="S3.I1.i3.p1.1.m1.2.2.1.1.1" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1.cmml"><mo id="S3.I1.i3.p1.1.m1.2.2.1.1.1a" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1.cmml">−</mo><mi id="S3.I1.i3.p1.1.m1.2.2.1.1.1.2" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1.2.cmml">π</mi></mrow><mo id="S3.I1.i3.p1.1.m1.2.2.1.1.3" xref="S3.I1.i3.p1.1.m1.2.2.1.2.cmml">,</mo><mi id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">π</mi><mo id="S3.I1.i3.p1.1.m1.2.2.1.1.4" stretchy="false" xref="S3.I1.i3.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.2b"><apply id="S3.I1.i3.p1.1.m1.2.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2"><and id="S3.I1.i3.p1.1.m1.2.2a.cmml" xref="S3.I1.i3.p1.1.m1.2.2"></and><apply id="S3.I1.i3.p1.1.m1.2.2b.cmml" xref="S3.I1.i3.p1.1.m1.2.2"><csymbol cd="latexml" id="S3.I1.i3.p1.1.m1.2.2.4.cmml" xref="S3.I1.i3.p1.1.m1.2.2.4">assign</csymbol><apply id="S3.I1.i3.p1.1.m1.2.2.3.cmml" xref="S3.I1.i3.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.2.2.3.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.3">subscript</csymbol><ci id="S3.I1.i3.p1.1.m1.2.2.3.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.3.2">𝛽</ci><ci id="S3.I1.i3.p1.1.m1.2.2.3.3.cmml" xref="S3.I1.i3.p1.1.m1.2.2.3.3">𝑘</ci></apply><apply id="S3.I1.i3.p1.1.m1.2.2.5.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5"><times id="S3.I1.i3.p1.1.m1.2.2.5.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.1"></times><ci id="S3.I1.i3.p1.1.m1.2.2.5.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.2">∠</ci><ci id="S3.I1.i3.p1.1.m1.2.2.5.3.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.3">𝑥</ci><apply id="S3.I1.i3.p1.1.m1.2.2.5.4.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.4"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.2.2.5.4.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.4">subscript</csymbol><ci id="S3.I1.i3.p1.1.m1.2.2.5.4.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.4.2">𝑣</ci><ci id="S3.I1.i3.p1.1.m1.2.2.5.4.3.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.4.3">𝑘</ci></apply><apply id="S3.I1.i3.p1.1.m1.2.2.5.5.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.2.2.5.5.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5">subscript</csymbol><ci id="S3.I1.i3.p1.1.m1.2.2.5.5.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5.2">𝑣</ci><apply id="S3.I1.i3.p1.1.m1.2.2.5.5.3.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3"><plus id="S3.I1.i3.p1.1.m1.2.2.5.5.3.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.1"></plus><ci id="S3.I1.i3.p1.1.m1.2.2.5.5.3.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.2">𝑘</ci><cn id="S3.I1.i3.p1.1.m1.2.2.5.5.3.3.cmml" type="integer" xref="S3.I1.i3.p1.1.m1.2.2.5.5.3.3">1</cn></apply></apply></apply></apply><apply id="S3.I1.i3.p1.1.m1.2.2c.cmml" xref="S3.I1.i3.p1.1.m1.2.2"><in id="S3.I1.i3.p1.1.m1.2.2.6.cmml" xref="S3.I1.i3.p1.1.m1.2.2.6"></in><share href="https://arxiv.org/html/2503.13001v1#S3.I1.i3.p1.1.m1.2.2.5.cmml" id="S3.I1.i3.p1.1.m1.2.2d.cmml" xref="S3.I1.i3.p1.1.m1.2.2"></share><interval closure="open" id="S3.I1.i3.p1.1.m1.2.2.1.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.1.1"><apply id="S3.I1.i3.p1.1.m1.2.2.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1"><minus id="S3.I1.i3.p1.1.m1.2.2.1.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1"></minus><ci id="S3.I1.i3.p1.1.m1.2.2.1.1.1.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2.1.1.1.2">𝜋</ci></apply><ci id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">𝜋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.2c">\beta_{k}:=\angle xv_{k}v_{k+1}\in(-\pi,\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.2d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := ∠ italic_x italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ∈ ( - italic_π , italic_π )</annotation></semantics></math>. As with <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.2.m2.1"><semantics id="S3.I1.i3.p1.2.m2.1a"><msub id="S3.I1.i3.p1.2.m2.1.1" xref="S3.I1.i3.p1.2.m2.1.1.cmml"><mi id="S3.I1.i3.p1.2.m2.1.1.2" xref="S3.I1.i3.p1.2.m2.1.1.2.cmml">α</mi><mi id="S3.I1.i3.p1.2.m2.1.1.3" xref="S3.I1.i3.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m2.1b"><apply id="S3.I1.i3.p1.2.m2.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.1.1.2.cmml" xref="S3.I1.i3.p1.2.m2.1.1.2">𝛼</ci><ci id="S3.I1.i3.p1.2.m2.1.1.3.cmml" xref="S3.I1.i3.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m2.1c">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the sign of <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.3.m3.1"><semantics id="S3.I1.i3.p1.3.m3.1a"><msub id="S3.I1.i3.p1.3.m3.1.1" xref="S3.I1.i3.p1.3.m3.1.1.cmml"><mi id="S3.I1.i3.p1.3.m3.1.1.2" xref="S3.I1.i3.p1.3.m3.1.1.2.cmml">β</mi><mi id="S3.I1.i3.p1.3.m3.1.1.3" xref="S3.I1.i3.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.3.m3.1b"><apply id="S3.I1.i3.p1.3.m3.1.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.3.m3.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.3.m3.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.1.1.2">𝛽</ci><ci id="S3.I1.i3.p1.3.m3.1.1.3.cmml" xref="S3.I1.i3.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.3.m3.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.3.m3.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is related to the <math alttext="P" class="ltx_Math" display="inline" id="S3.I1.i3.p1.4.m4.1"><semantics id="S3.I1.i3.p1.4.m4.1a"><mi id="S3.I1.i3.p1.4.m4.1.1" xref="S3.I1.i3.p1.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.4.m4.1b"><ci id="S3.I1.i3.p1.4.m4.1.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.4.m4.1d">italic_P</annotation></semantics></math>-side of <math alttext="e_{k+1}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.5.m5.1"><semantics id="S3.I1.i3.p1.5.m5.1a"><msub id="S3.I1.i3.p1.5.m5.1.1" xref="S3.I1.i3.p1.5.m5.1.1.cmml"><mi id="S3.I1.i3.p1.5.m5.1.1.2" xref="S3.I1.i3.p1.5.m5.1.1.2.cmml">e</mi><mrow id="S3.I1.i3.p1.5.m5.1.1.3" xref="S3.I1.i3.p1.5.m5.1.1.3.cmml"><mi id="S3.I1.i3.p1.5.m5.1.1.3.2" xref="S3.I1.i3.p1.5.m5.1.1.3.2.cmml">k</mi><mo id="S3.I1.i3.p1.5.m5.1.1.3.1" xref="S3.I1.i3.p1.5.m5.1.1.3.1.cmml">+</mo><mn id="S3.I1.i3.p1.5.m5.1.1.3.3" xref="S3.I1.i3.p1.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.5.m5.1b"><apply id="S3.I1.i3.p1.5.m5.1.1.cmml" xref="S3.I1.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.5.m5.1.1.1.cmml" xref="S3.I1.i3.p1.5.m5.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.5.m5.1.1.2.cmml" xref="S3.I1.i3.p1.5.m5.1.1.2">𝑒</ci><apply id="S3.I1.i3.p1.5.m5.1.1.3.cmml" xref="S3.I1.i3.p1.5.m5.1.1.3"><plus id="S3.I1.i3.p1.5.m5.1.1.3.1.cmml" xref="S3.I1.i3.p1.5.m5.1.1.3.1"></plus><ci id="S3.I1.i3.p1.5.m5.1.1.3.2.cmml" xref="S3.I1.i3.p1.5.m5.1.1.3.2">𝑘</ci><cn id="S3.I1.i3.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S3.I1.i3.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.5.m5.1c">e_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.5.m5.1d">italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, as <math alttext="P" class="ltx_Math" display="inline" id="S3.I1.i3.p1.6.m6.1"><semantics id="S3.I1.i3.p1.6.m6.1a"><mi id="S3.I1.i3.p1.6.m6.1.1" xref="S3.I1.i3.p1.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.6.m6.1b"><ci id="S3.I1.i3.p1.6.m6.1.1.cmml" xref="S3.I1.i3.p1.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.6.m6.1d">italic_P</annotation></semantics></math> is always on the left hand side:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E14"> <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="\beta_{k}\begin{cases}<0,\quad x\in H^{k+1}_{+}\\ >0,\quad x\in H^{k+1}_{-}\end{cases}" class="ltx_Math" display="block" id="S3.E14.m1.2"><semantics id="S3.E14.m1.2a"><mrow id="S3.E14.m1.2.3" xref="S3.E14.m1.2.3.cmml"><msub id="S3.E14.m1.2.3.2" xref="S3.E14.m1.2.3.2.cmml"><mi id="S3.E14.m1.2.3.2.2" xref="S3.E14.m1.2.3.2.2.cmml">β</mi><mi id="S3.E14.m1.2.3.2.3" xref="S3.E14.m1.2.3.2.3.cmml">k</mi></msub><mo id="S3.E14.m1.2.3.1" xref="S3.E14.m1.2.3.1.cmml">⁢</mo><mrow id="S3.E14.m1.2.2" xref="S3.E14.m1.2.3.3.1.cmml"><mo id="S3.E14.m1.2.2.3" xref="S3.E14.m1.2.3.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S3.E14.m1.2.2.2" rowspacing="0pt" xref="S3.E14.m1.2.3.3.1.cmml"><mtr id="S3.E14.m1.2.2.2a" xref="S3.E14.m1.2.3.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E14.m1.2.2.2b" xref="S3.E14.m1.2.3.3.1.cmml"><mrow id="S3.E14.m1.1.1.1.1.1.1.4" xref="S3.E14.m1.1.1.1.1.1.1.5.cmml"><mrow id="S3.E14.m1.1.1.1.1.1.1.3.1" xref="S3.E14.m1.1.1.1.1.1.1.3.1.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.3.1.2" xref="S3.E14.m1.1.1.1.1.1.1.3.1.2.cmml"></mi><mo id="S3.E14.m1.1.1.1.1.1.1.3.1.1" xref="S3.E14.m1.1.1.1.1.1.1.3.1.1.cmml"><</mo><mn id="S3.E14.m1.1.1.1.1.1.1.1" xref="S3.E14.m1.1.1.1.1.1.1.1.cmml">0</mn></mrow><mo id="S3.E14.m1.1.1.1.1.1.1.4.3" rspace="1.167em" xref="S3.E14.m1.1.1.1.1.1.1.5a.cmml">,</mo><mrow id="S3.E14.m1.1.1.1.1.1.1.4.2" xref="S3.E14.m1.1.1.1.1.1.1.4.2.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.2" xref="S3.E14.m1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S3.E14.m1.1.1.1.1.1.1.4.2.1" xref="S3.E14.m1.1.1.1.1.1.1.4.2.1.cmml">∈</mo><msubsup id="S3.E14.m1.1.1.1.1.1.1.4.2.2" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.2" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.2.cmml">H</mi><mo id="S3.E14.m1.1.1.1.1.1.1.4.2.2.3" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.3.cmml">+</mo><mrow id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.2" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.2.cmml">k</mi><mo 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xref="S3.E14.m1.2.2.2.2.1.1.2.2.cmml"><mi id="S3.E14.m1.2.2.2.2.1.1.2.2.2" xref="S3.E14.m1.2.2.2.2.1.1.2.2.2.cmml">x</mi><mo id="S3.E14.m1.2.2.2.2.1.1.2.2.1" xref="S3.E14.m1.2.2.2.2.1.1.2.2.1.cmml">∈</mo><msubsup id="S3.E14.m1.2.2.2.2.1.1.2.2.3" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.cmml"><mi id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.2" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.2.cmml">H</mi><mo id="S3.E14.m1.2.2.2.2.1.1.2.2.3.3" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.3.cmml">−</mo><mrow id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.cmml"><mi id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.2" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.2.cmml">k</mi><mo id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.1" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.1.cmml">+</mo><mn id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.3" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow></mtd><mtd id="S3.E14.m1.2.2.2f" xref="S3.E14.m1.2.3.3.1.1.cmml"></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E14.m1.2b"><apply id="S3.E14.m1.2.3.cmml" xref="S3.E14.m1.2.3"><times id="S3.E14.m1.2.3.1.cmml" xref="S3.E14.m1.2.3.1"></times><apply id="S3.E14.m1.2.3.2.cmml" xref="S3.E14.m1.2.3.2"><csymbol cd="ambiguous" id="S3.E14.m1.2.3.2.1.cmml" xref="S3.E14.m1.2.3.2">subscript</csymbol><ci id="S3.E14.m1.2.3.2.2.cmml" xref="S3.E14.m1.2.3.2.2">𝛽</ci><ci id="S3.E14.m1.2.3.2.3.cmml" xref="S3.E14.m1.2.3.2.3">𝑘</ci></apply><apply id="S3.E14.m1.2.3.3.1.cmml" xref="S3.E14.m1.2.2"><csymbol cd="latexml" id="S3.E14.m1.2.3.3.1.1.cmml" xref="S3.E14.m1.2.2.3">cases</csymbol><apply id="S3.E14.m1.1.1.1.1.1.1.5.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.E14.m1.1.1.1.1.1.1.5a.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.3">formulae-sequence</csymbol><apply id="S3.E14.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.3.1"><lt id="S3.E14.m1.1.1.1.1.1.1.3.1.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.3.1.1"></lt><csymbol cd="latexml" id="S3.E14.m1.1.1.1.1.1.1.3.1.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.3.1.2">absent</csymbol><cn id="S3.E14.m1.1.1.1.1.1.1.1.cmml" type="integer" xref="S3.E14.m1.1.1.1.1.1.1.1">0</cn></apply><apply id="S3.E14.m1.1.1.1.1.1.1.4.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2"><in id="S3.E14.m1.1.1.1.1.1.1.4.2.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.1"></in><ci id="S3.E14.m1.1.1.1.1.1.1.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.2">𝑥</ci><apply id="S3.E14.m1.1.1.1.1.1.1.4.2.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2"><csymbol cd="ambiguous" id="S3.E14.m1.1.1.1.1.1.1.4.2.2.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2">subscript</csymbol><apply id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2"><csymbol cd="ambiguous" id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2">superscript</csymbol><ci id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.2">𝐻</ci><apply id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3"><plus id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.1.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.1"></plus><ci id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.2.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.2">𝑘</ci><cn id="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.3.cmml" type="integer" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.2.3.3">1</cn></apply></apply><plus id="S3.E14.m1.1.1.1.1.1.1.4.2.2.3.cmml" xref="S3.E14.m1.1.1.1.1.1.1.4.2.2.3"></plus></apply></apply></apply><ci id="S3.E14.m1.2.3.3.1.3a.cmml" xref="S3.E14.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S3.E14.m1.2.3.3.1.3.cmml" xref="S3.E14.m1.2.2.3">otherwise</mtext></ci><apply id="S3.E14.m1.2.2.2.2.1.1.3.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.E14.m1.2.2.2.2.1.1.3a.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E14.m1.2.2.2.2.1.1.1.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.1.1"><gt id="S3.E14.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.1.1.1"></gt><csymbol cd="latexml" id="S3.E14.m1.2.2.2.2.1.1.1.1.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.1.1.2">absent</csymbol><cn id="S3.E14.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.E14.m1.2.2.2.2.1.1.1.1.3">0</cn></apply><apply id="S3.E14.m1.2.2.2.2.1.1.2.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2"><in id="S3.E14.m1.2.2.2.2.1.1.2.2.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.1"></in><ci id="S3.E14.m1.2.2.2.2.1.1.2.2.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.2">𝑥</ci><apply id="S3.E14.m1.2.2.2.2.1.1.2.2.3.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E14.m1.2.2.2.2.1.1.2.2.3.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3">subscript</csymbol><apply id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3">superscript</csymbol><ci id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.2">𝐻</ci><apply id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3"><plus id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.1.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.1"></plus><ci id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.2.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.2">𝑘</ci><cn id="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.3.cmml" type="integer" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.2.3.3">1</cn></apply></apply><minus id="S3.E14.m1.2.2.2.2.1.1.2.2.3.3.cmml" xref="S3.E14.m1.2.2.2.2.1.1.2.2.3.3"></minus></apply></apply></apply><ci id="S3.E14.m1.2.3.3.1.5a.cmml" xref="S3.E14.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S3.E14.m1.2.3.3.1.5.cmml" xref="S3.E14.m1.2.2.3">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E14.m1.2c">\beta_{k}\begin{cases}<0,\quad x\in H^{k+1}_{+}\\ >0,\quad x\in H^{k+1}_{-}\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.E14.m1.2d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { start_ROW start_CELL < 0 , italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL > 0 , italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW</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">(14)</span></td> </tr></tbody> </table> </div> </li> </ul> <p class="ltx_p" id="S3.SS1.SSS1.5.p5.9">Note that <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.1.m1.1"><semantics id="S3.SS1.SSS1.5.p5.1.m1.1a"><msub id="S3.SS1.SSS1.5.p5.1.m1.1.1" xref="S3.SS1.SSS1.5.p5.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.5.p5.1.m1.1.1.2" xref="S3.SS1.SSS1.5.p5.1.m1.1.1.2.cmml">α</mi><mi id="S3.SS1.SSS1.5.p5.1.m1.1.1.3" xref="S3.SS1.SSS1.5.p5.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.1.m1.1b"><apply id="S3.SS1.SSS1.5.p5.1.m1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.5.p5.1.m1.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.5.p5.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.1.m1.1.1.2">𝛼</ci><ci id="S3.SS1.SSS1.5.p5.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.1.m1.1c">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.2.m2.1"><semantics id="S3.SS1.SSS1.5.p5.2.m2.1a"><msub id="S3.SS1.SSS1.5.p5.2.m2.1.1" xref="S3.SS1.SSS1.5.p5.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.5.p5.2.m2.1.1.2" xref="S3.SS1.SSS1.5.p5.2.m2.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.5.p5.2.m2.1.1.3" xref="S3.SS1.SSS1.5.p5.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.2.m2.1b"><apply id="S3.SS1.SSS1.5.p5.2.m2.1.1.cmml" xref="S3.SS1.SSS1.5.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.5.p5.2.m2.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.5.p5.2.m2.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.2.m2.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.5.p5.2.m2.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.2.m2.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> are measured with values in <math alttext="(-\pi,\pi)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.3.m3.2"><semantics id="S3.SS1.SSS1.5.p5.3.m3.2a"><mrow id="S3.SS1.SSS1.5.p5.3.m3.2.2.1" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.2.cmml"><mo id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.2" stretchy="false" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.2.cmml">(</mo><mrow id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.cmml"><mo id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1a" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.cmml">−</mo><mi id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.2" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.2.cmml">π</mi></mrow><mo id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.3" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.2.cmml">,</mo><mi id="S3.SS1.SSS1.5.p5.3.m3.1.1" xref="S3.SS1.SSS1.5.p5.3.m3.1.1.cmml">π</mi><mo id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.4" stretchy="false" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.3.m3.2b"><interval closure="open" id="S3.SS1.SSS1.5.p5.3.m3.2.2.2.cmml" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1"><apply id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.cmml" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1"><minus id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1"></minus><ci id="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.3.m3.2.2.1.1.2">𝜋</ci></apply><ci id="S3.SS1.SSS1.5.p5.3.m3.1.1.cmml" xref="S3.SS1.SSS1.5.p5.3.m3.1.1">𝜋</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.3.m3.2c">(-\pi,\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.3.m3.2d">( - italic_π , italic_π )</annotation></semantics></math>, but <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.4.m4.1"><semantics id="S3.SS1.SSS1.5.p5.4.m4.1a"><msub id="S3.SS1.SSS1.5.p5.4.m4.1.1" xref="S3.SS1.SSS1.5.p5.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.5.p5.4.m4.1.1.2" xref="S3.SS1.SSS1.5.p5.4.m4.1.1.2.cmml">η</mi><mi id="S3.SS1.SSS1.5.p5.4.m4.1.1.3" xref="S3.SS1.SSS1.5.p5.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.4.m4.1b"><apply id="S3.SS1.SSS1.5.p5.4.m4.1.1.cmml" xref="S3.SS1.SSS1.5.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.5.p5.4.m4.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.5.p5.4.m4.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.4.m4.1.1.2">𝜂</ci><ci id="S3.SS1.SSS1.5.p5.4.m4.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.4.m4.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.4.m4.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with values in <math alttext="(0,2\pi)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.5.m5.2"><semantics id="S3.SS1.SSS1.5.p5.5.m5.2a"><mrow id="S3.SS1.SSS1.5.p5.5.m5.2.2.1" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.2.cmml"><mo id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.2" stretchy="false" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.2.cmml">(</mo><mn id="S3.SS1.SSS1.5.p5.5.m5.1.1" xref="S3.SS1.SSS1.5.p5.5.m5.1.1.cmml">0</mn><mo id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.3" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.2.cmml">,</mo><mrow id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.cmml"><mn id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.2" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.2.cmml">2</mn><mo id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.1" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.3" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.3.cmml">π</mi></mrow><mo id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.4" stretchy="false" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.5.m5.2b"><interval closure="open" id="S3.SS1.SSS1.5.p5.5.m5.2.2.2.cmml" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1"><cn id="S3.SS1.SSS1.5.p5.5.m5.1.1.cmml" type="integer" xref="S3.SS1.SSS1.5.p5.5.m5.1.1">0</cn><apply id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.cmml" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1"><times id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.1"></times><cn id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.2.cmml" type="integer" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.2">2</cn><ci id="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.5.m5.2.2.1.1.3">𝜋</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.5.m5.2c">(0,2\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.5.m5.2d">( 0 , 2 italic_π )</annotation></semantics></math>. Moreover, <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.6.m6.1"><semantics id="S3.SS1.SSS1.5.p5.6.m6.1a"><msub id="S3.SS1.SSS1.5.p5.6.m6.1.1" xref="S3.SS1.SSS1.5.p5.6.m6.1.1.cmml"><mi id="S3.SS1.SSS1.5.p5.6.m6.1.1.2" xref="S3.SS1.SSS1.5.p5.6.m6.1.1.2.cmml">α</mi><mi id="S3.SS1.SSS1.5.p5.6.m6.1.1.3" xref="S3.SS1.SSS1.5.p5.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.6.m6.1b"><apply id="S3.SS1.SSS1.5.p5.6.m6.1.1.cmml" xref="S3.SS1.SSS1.5.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.5.p5.6.m6.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.5.p5.6.m6.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.6.m6.1.1.2">𝛼</ci><ci id="S3.SS1.SSS1.5.p5.6.m6.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.6.m6.1c">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.6.m6.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.7.m7.1"><semantics id="S3.SS1.SSS1.5.p5.7.m7.1a"><msub id="S3.SS1.SSS1.5.p5.7.m7.1.1" xref="S3.SS1.SSS1.5.p5.7.m7.1.1.cmml"><mi id="S3.SS1.SSS1.5.p5.7.m7.1.1.2" xref="S3.SS1.SSS1.5.p5.7.m7.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.5.p5.7.m7.1.1.3" xref="S3.SS1.SSS1.5.p5.7.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.7.m7.1b"><apply id="S3.SS1.SSS1.5.p5.7.m7.1.1.cmml" xref="S3.SS1.SSS1.5.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.5.p5.7.m7.1.1.1.cmml" xref="S3.SS1.SSS1.5.p5.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.5.p5.7.m7.1.1.2.cmml" xref="S3.SS1.SSS1.5.p5.7.m7.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.5.p5.7.m7.1.1.3.cmml" xref="S3.SS1.SSS1.5.p5.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.7.m7.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.7.m7.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> are non-zero since <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.8.m8.1"><semantics id="S3.SS1.SSS1.5.p5.8.m8.1a"><mi id="S3.SS1.SSS1.5.p5.8.m8.1.1" xref="S3.SS1.SSS1.5.p5.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.8.m8.1b"><ci id="S3.SS1.SSS1.5.p5.8.m8.1.1.cmml" xref="S3.SS1.SSS1.5.p5.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.8.m8.1d">italic_x</annotation></semantics></math> is in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.5.p5.9.m9.1"><semantics id="S3.SS1.SSS1.5.p5.9.m9.1a"><mi id="S3.SS1.SSS1.5.p5.9.m9.1.1" xref="S3.SS1.SSS1.5.p5.9.m9.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.5.p5.9.m9.1b"><ci id="S3.SS1.SSS1.5.p5.9.m9.1.1.cmml" xref="S3.SS1.SSS1.5.p5.9.m9.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.5.p5.9.m9.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.5.p5.9.m9.1d">italic_P</annotation></semantics></math>-general position.</p> </div> <div class="ltx_para" id="S3.SS1.SSS1.6.p6"> <p class="ltx_p" id="S3.SS1.SSS1.6.p6.6">To determine the frequency of occurrences of the cases <span class="ltx_text ltx_font_typewriter" id="S3.SS1.SSS1.6.p6.6.1">+-concave</span> and <span class="ltx_text ltx_font_typewriter" id="S3.SS1.SSS1.6.p6.6.2">-+convex</span>, we consider, for each vertex-edge pair <math alttext="(v_{k},e_{k+1})" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.1.m1.2"><semantics id="S3.SS1.SSS1.6.p6.1.m1.2a"><mrow id="S3.SS1.SSS1.6.p6.1.m1.2.2.2" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.3.cmml"><mo id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.3" stretchy="false" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.3.cmml">(</mo><msub id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.2" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.3" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.4" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.cmml"><mi id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.2" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.2.cmml">e</mi><mrow id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.cmml"><mi id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.2" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.1" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.3" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.5" stretchy="false" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.1.m1.2b"><interval closure="open" id="S3.SS1.SSS1.6.p6.1.m1.2.2.3.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2"><apply id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.1.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.2.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.2">𝑒</ci><apply id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3"><plus id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.1.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.1"></plus><ci id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.2.cmml" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.6.p6.1.m1.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.1.m1.2c">(v_{k},e_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.1.m1.2d">( italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, the triangle defined by the points <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.2.m2.1"><semantics id="S3.SS1.SSS1.6.p6.2.m2.1a"><mi id="S3.SS1.SSS1.6.p6.2.m2.1.1" xref="S3.SS1.SSS1.6.p6.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.2.m2.1b"><ci id="S3.SS1.SSS1.6.p6.2.m2.1.1.cmml" xref="S3.SS1.SSS1.6.p6.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.2.m2.1d">italic_x</annotation></semantics></math>, <math alttext="v_{k-1}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.3.m3.1"><semantics id="S3.SS1.SSS1.6.p6.3.m3.1a"><msub id="S3.SS1.SSS1.6.p6.3.m3.1.1" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.cmml"><mi id="S3.SS1.SSS1.6.p6.3.m3.1.1.2" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.2.cmml">v</mi><mrow id="S3.SS1.SSS1.6.p6.3.m3.1.1.3" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.cmml"><mi id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.2" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.1" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.1.cmml">−</mo><mn id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.3" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.3.m3.1b"><apply id="S3.SS1.SSS1.6.p6.3.m3.1.1.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.6.p6.3.m3.1.1.1.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.6.p6.3.m3.1.1.2.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.2">𝑣</ci><apply id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3"><minus id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.1.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.1"></minus><ci id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.2.cmml" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.2">𝑘</ci><cn id="S3.SS1.SSS1.6.p6.3.m3.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS1.6.p6.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.3.m3.1c">v_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.3.m3.1d">italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.4.m4.1"><semantics id="S3.SS1.SSS1.6.p6.4.m4.1a"><msub id="S3.SS1.SSS1.6.p6.4.m4.1.1" xref="S3.SS1.SSS1.6.p6.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.6.p6.4.m4.1.1.2" xref="S3.SS1.SSS1.6.p6.4.m4.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.6.p6.4.m4.1.1.3" xref="S3.SS1.SSS1.6.p6.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.4.m4.1b"><apply id="S3.SS1.SSS1.6.p6.4.m4.1.1.cmml" xref="S3.SS1.SSS1.6.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.6.p6.4.m4.1.1.1.cmml" xref="S3.SS1.SSS1.6.p6.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.6.p6.4.m4.1.1.2.cmml" xref="S3.SS1.SSS1.6.p6.4.m4.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.6.p6.4.m4.1.1.3.cmml" xref="S3.SS1.SSS1.6.p6.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.4.m4.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.4.m4.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, see <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F5" title="Figure 5 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 5</span></a>. These three points are not collinear since <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.5.m5.1"><semantics id="S3.SS1.SSS1.6.p6.5.m5.1a"><mi id="S3.SS1.SSS1.6.p6.5.m5.1.1" xref="S3.SS1.SSS1.6.p6.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.5.m5.1b"><ci id="S3.SS1.SSS1.6.p6.5.m5.1.1.cmml" xref="S3.SS1.SSS1.6.p6.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.5.m5.1d">italic_x</annotation></semantics></math> is in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.6.p6.6.m6.1"><semantics id="S3.SS1.SSS1.6.p6.6.m6.1a"><mi id="S3.SS1.SSS1.6.p6.6.m6.1.1" xref="S3.SS1.SSS1.6.p6.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.6.p6.6.m6.1b"><ci id="S3.SS1.SSS1.6.p6.6.m6.1.1.cmml" xref="S3.SS1.SSS1.6.p6.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.6.m6.1d">italic_P</annotation></semantics></math>-general position. Therefore, the sum of the triangle’s interior angles satisfies the relation</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="\absolutevalue{\alpha_{k}}+\absolutevalue{\beta_{k-1}}+\delta_{k}=\pi," class="ltx_Math" display="block" id="S3.Ex17.m1.3"><semantics id="S3.Ex17.m1.3a"><mrow id="S3.Ex17.m1.3.3.1" xref="S3.Ex17.m1.3.3.1.1.cmml"><mrow id="S3.Ex17.m1.3.3.1.1" xref="S3.Ex17.m1.3.3.1.1.cmml"><mrow id="S3.Ex17.m1.3.3.1.1.2" xref="S3.Ex17.m1.3.3.1.1.2.cmml"><mrow id="S3.Ex17.m1.1.1.3" xref="S3.Ex17.m1.1.1.2.cmml"><mo id="S3.Ex17.m1.1.1.3.1" xref="S3.Ex17.m1.1.1.2.1.cmml">|</mo><msub id="S3.Ex17.m1.1.1.1.1.1" xref="S3.Ex17.m1.1.1.1.1.1.cmml"><mi id="S3.Ex17.m1.1.1.1.1.1.2" xref="S3.Ex17.m1.1.1.1.1.1.2.cmml">α</mi><mi id="S3.Ex17.m1.1.1.1.1.1.3" xref="S3.Ex17.m1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex17.m1.1.1.3.2" xref="S3.Ex17.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex17.m1.3.3.1.1.2.1" xref="S3.Ex17.m1.3.3.1.1.2.1.cmml">+</mo><mrow id="S3.Ex17.m1.2.2.3" xref="S3.Ex17.m1.2.2.2.cmml"><mo id="S3.Ex17.m1.2.2.3.1" xref="S3.Ex17.m1.2.2.2.1.cmml">|</mo><msub id="S3.Ex17.m1.2.2.1.1.1" xref="S3.Ex17.m1.2.2.1.1.1.cmml"><mi id="S3.Ex17.m1.2.2.1.1.1.2" xref="S3.Ex17.m1.2.2.1.1.1.2.cmml">β</mi><mrow id="S3.Ex17.m1.2.2.1.1.1.3" xref="S3.Ex17.m1.2.2.1.1.1.3.cmml"><mi id="S3.Ex17.m1.2.2.1.1.1.3.2" xref="S3.Ex17.m1.2.2.1.1.1.3.2.cmml">k</mi><mo id="S3.Ex17.m1.2.2.1.1.1.3.1" xref="S3.Ex17.m1.2.2.1.1.1.3.1.cmml">−</mo><mn id="S3.Ex17.m1.2.2.1.1.1.3.3" xref="S3.Ex17.m1.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex17.m1.2.2.3.2" xref="S3.Ex17.m1.2.2.2.1.cmml">|</mo></mrow><mo id="S3.Ex17.m1.3.3.1.1.2.1a" xref="S3.Ex17.m1.3.3.1.1.2.1.cmml">+</mo><msub id="S3.Ex17.m1.3.3.1.1.2.2" xref="S3.Ex17.m1.3.3.1.1.2.2.cmml"><mi id="S3.Ex17.m1.3.3.1.1.2.2.2" xref="S3.Ex17.m1.3.3.1.1.2.2.2.cmml">δ</mi><mi id="S3.Ex17.m1.3.3.1.1.2.2.3" xref="S3.Ex17.m1.3.3.1.1.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.Ex17.m1.3.3.1.1.1" xref="S3.Ex17.m1.3.3.1.1.1.cmml">=</mo><mi id="S3.Ex17.m1.3.3.1.1.3" xref="S3.Ex17.m1.3.3.1.1.3.cmml">π</mi></mrow><mo id="S3.Ex17.m1.3.3.1.2" xref="S3.Ex17.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex17.m1.3b"><apply id="S3.Ex17.m1.3.3.1.1.cmml" xref="S3.Ex17.m1.3.3.1"><eq id="S3.Ex17.m1.3.3.1.1.1.cmml" xref="S3.Ex17.m1.3.3.1.1.1"></eq><apply id="S3.Ex17.m1.3.3.1.1.2.cmml" xref="S3.Ex17.m1.3.3.1.1.2"><plus id="S3.Ex17.m1.3.3.1.1.2.1.cmml" xref="S3.Ex17.m1.3.3.1.1.2.1"></plus><apply id="S3.Ex17.m1.1.1.2.cmml" xref="S3.Ex17.m1.1.1.3"><abs id="S3.Ex17.m1.1.1.2.1.cmml" xref="S3.Ex17.m1.1.1.3.1"></abs><apply id="S3.Ex17.m1.1.1.1.1.1.cmml" xref="S3.Ex17.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex17.m1.1.1.1.1.1.1.cmml" xref="S3.Ex17.m1.1.1.1.1.1">subscript</csymbol><ci 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xref="S3.Ex17.m1.3.3.1.1.2.2">subscript</csymbol><ci id="S3.Ex17.m1.3.3.1.1.2.2.2.cmml" xref="S3.Ex17.m1.3.3.1.1.2.2.2">𝛿</ci><ci id="S3.Ex17.m1.3.3.1.1.2.2.3.cmml" xref="S3.Ex17.m1.3.3.1.1.2.2.3">𝑘</ci></apply></apply><ci id="S3.Ex17.m1.3.3.1.1.3.cmml" xref="S3.Ex17.m1.3.3.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m1.3c">\absolutevalue{\alpha_{k}}+\absolutevalue{\beta_{k-1}}+\delta_{k}=\pi,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m1.3d">| start_ARG italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | + | start_ARG italic_β start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_ARG | + italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.6.p6.7">where <math alttext="\delta_{k}:=\absolutevalue{\angle v_{k-1}v_{k}x}\in(0,\pi)" 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xref="S3.SS1.SSS1.6.p6.7.m1.3.4"></share><interval closure="open" id="S3.SS1.SSS1.6.p6.7.m1.3.4.5.1.cmml" xref="S3.SS1.SSS1.6.p6.7.m1.3.4.5.2"><cn id="S3.SS1.SSS1.6.p6.7.m1.2.2.cmml" type="integer" xref="S3.SS1.SSS1.6.p6.7.m1.2.2">0</cn><ci id="S3.SS1.SSS1.6.p6.7.m1.3.3.cmml" xref="S3.SS1.SSS1.6.p6.7.m1.3.3">𝜋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.6.p6.7.m1.3c">\delta_{k}:=\absolutevalue{\angle v_{k-1}v_{k}x}\in(0,\pi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.6.p6.7.m1.3d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := | start_ARG ∠ italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_x end_ARG | ∈ ( 0 , italic_π )</annotation></semantics></math>. Using the properties (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E11" title="Equation 11 ‣ 1st item ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">11</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E14" title="Equation 14 ‣ 3rd item ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">14</span></a>), this is equivalent to</p> <table class="ltx_equation ltx_eqn_table" id="S3.E15"> <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="\begin{rcases}\alpha_{k}-\beta_{k-1},&x\in H^{k}_{+}\\ \beta_{k-1}-\alpha_{k},&x\in 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cd="ambiguous" id="S3.E15.m1.4.4.4.4.2.1.3.1.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3">subscript</csymbol><apply id="S3.E15.m1.4.4.4.4.2.1.3.2.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3"><csymbol cd="ambiguous" id="S3.E15.m1.4.4.4.4.2.1.3.2.1.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3">superscript</csymbol><ci id="S3.E15.m1.4.4.4.4.2.1.3.2.2.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3.2.2">𝐻</ci><ci id="S3.E15.m1.4.4.4.4.2.1.3.2.3.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3.2.3">𝑘</ci></apply><minus id="S3.E15.m1.4.4.4.4.2.1.3.3.cmml" xref="S3.E15.m1.4.4.4.4.2.1.3.3"></minus></apply></apply></apply><apply id="S3.E15.m1.4.5.2.3.cmml" xref="S3.E15.m1.4.5.2.3"><csymbol cd="ambiguous" id="S3.E15.m1.4.5.2.3.1.cmml" xref="S3.E15.m1.4.5.2.3">subscript</csymbol><ci id="S3.E15.m1.4.5.2.3.2.cmml" xref="S3.E15.m1.4.5.2.3.2">𝛿</ci><ci id="S3.E15.m1.4.5.2.3.3.cmml" xref="S3.E15.m1.4.5.2.3.3">𝑘</ci></apply></apply><ci id="S3.E15.m1.4.5.3.cmml" xref="S3.E15.m1.4.5.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E15.m1.4c">\begin{rcases}\alpha_{k}-\beta_{k-1},&x\in H^{k}_{+}\\ \beta_{k-1}-\alpha_{k},&x\in H^{k}_{-}\end{rcases}+\delta_{k}=\pi</annotation><annotation encoding="application/x-llamapun" id="S3.E15.m1.4d">start_ROW start_CELL italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , end_CELL start_CELL italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_β start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , end_CELL start_CELL italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_CELL end_ROW } + italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_π</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">(15)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.SSS1.7.p7"> <p class="ltx_p" id="S3.SS1.SSS1.7.p7.13">From the relations of the angles involving vertex <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.1.m1.1"><semantics id="S3.SS1.SSS1.7.p7.1.m1.1a"><msub id="S3.SS1.SSS1.7.p7.1.m1.1.1" xref="S3.SS1.SSS1.7.p7.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.1.m1.1.1.2" xref="S3.SS1.SSS1.7.p7.1.m1.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS1.7.p7.1.m1.1.1.3" xref="S3.SS1.SSS1.7.p7.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.1.m1.1b"><apply id="S3.SS1.SSS1.7.p7.1.m1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.1.m1.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.1.m1.1.1.2">𝑣</ci><ci id="S3.SS1.SSS1.7.p7.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.1.m1.1c">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.1.m1.1d">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, we will now express <math alttext="\delta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.2.m2.1"><semantics id="S3.SS1.SSS1.7.p7.2.m2.1a"><msub id="S3.SS1.SSS1.7.p7.2.m2.1.1" xref="S3.SS1.SSS1.7.p7.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.2.m2.1.1.2" xref="S3.SS1.SSS1.7.p7.2.m2.1.1.2.cmml">δ</mi><mi id="S3.SS1.SSS1.7.p7.2.m2.1.1.3" xref="S3.SS1.SSS1.7.p7.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.2.m2.1b"><apply id="S3.SS1.SSS1.7.p7.2.m2.1.1.cmml" xref="S3.SS1.SSS1.7.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.2.m2.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.2.m2.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.2.m2.1.1.2">𝛿</ci><ci id="S3.SS1.SSS1.7.p7.2.m2.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.2.m2.1c">\delta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> in terms of <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.3.m3.1"><semantics id="S3.SS1.SSS1.7.p7.3.m3.1a"><msub id="S3.SS1.SSS1.7.p7.3.m3.1.1" xref="S3.SS1.SSS1.7.p7.3.m3.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.3.m3.1.1.2" xref="S3.SS1.SSS1.7.p7.3.m3.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.3.m3.1.1.3" xref="S3.SS1.SSS1.7.p7.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.3.m3.1b"><apply id="S3.SS1.SSS1.7.p7.3.m3.1.1.cmml" xref="S3.SS1.SSS1.7.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.3.m3.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.3.m3.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.3.m3.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.3.m3.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.3.m3.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.3.m3.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.4.m4.1"><semantics id="S3.SS1.SSS1.7.p7.4.m4.1a"><msub id="S3.SS1.SSS1.7.p7.4.m4.1.1" xref="S3.SS1.SSS1.7.p7.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.4.m4.1.1.2" xref="S3.SS1.SSS1.7.p7.4.m4.1.1.2.cmml">η</mi><mi id="S3.SS1.SSS1.7.p7.4.m4.1.1.3" xref="S3.SS1.SSS1.7.p7.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.4.m4.1b"><apply id="S3.SS1.SSS1.7.p7.4.m4.1.1.cmml" xref="S3.SS1.SSS1.7.p7.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.4.m4.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.4.m4.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.4.m4.1.1.2">𝜂</ci><ci id="S3.SS1.SSS1.7.p7.4.m4.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.4.m4.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.4.m4.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Since the orientation of <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.5.m5.1"><semantics id="S3.SS1.SSS1.7.p7.5.m5.1a"><msub id="S3.SS1.SSS1.7.p7.5.m5.1.1" xref="S3.SS1.SSS1.7.p7.5.m5.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.5.m5.1.1.2" xref="S3.SS1.SSS1.7.p7.5.m5.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.5.m5.1.1.3" xref="S3.SS1.SSS1.7.p7.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.5.m5.1b"><apply id="S3.SS1.SSS1.7.p7.5.m5.1.1.cmml" xref="S3.SS1.SSS1.7.p7.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.5.m5.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.5.m5.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.5.m5.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.5.m5.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.5.m5.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.5.m5.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, and the overlap of the angles may change depending on the convexity of the corner, and on the relative position of <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.6.m6.1"><semantics id="S3.SS1.SSS1.7.p7.6.m6.1a"><mi id="S3.SS1.SSS1.7.p7.6.m6.1.1" xref="S3.SS1.SSS1.7.p7.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.6.m6.1b"><ci id="S3.SS1.SSS1.7.p7.6.m6.1.1.cmml" xref="S3.SS1.SSS1.7.p7.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.6.m6.1d">italic_x</annotation></semantics></math> with respect to <math alttext="e_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.7.m7.1"><semantics id="S3.SS1.SSS1.7.p7.7.m7.1a"><msub id="S3.SS1.SSS1.7.p7.7.m7.1.1" xref="S3.SS1.SSS1.7.p7.7.m7.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.7.m7.1.1.2" xref="S3.SS1.SSS1.7.p7.7.m7.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS1.7.p7.7.m7.1.1.3" xref="S3.SS1.SSS1.7.p7.7.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.7.m7.1b"><apply id="S3.SS1.SSS1.7.p7.7.m7.1.1.cmml" xref="S3.SS1.SSS1.7.p7.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.7.m7.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.7.m7.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.7.m7.1.1.2">𝑒</ci><ci id="S3.SS1.SSS1.7.p7.7.m7.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.7.m7.1c">e_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{k+1}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.8.m8.1"><semantics id="S3.SS1.SSS1.7.p7.8.m8.1a"><msub id="S3.SS1.SSS1.7.p7.8.m8.1.1" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.8.m8.1.1.2" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.2.cmml">e</mi><mrow id="S3.SS1.SSS1.7.p7.8.m8.1.1.3" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.cmml"><mi id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.2" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.1" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.1.cmml">+</mo><mn id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.3" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.8.m8.1b"><apply id="S3.SS1.SSS1.7.p7.8.m8.1.1.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.8.m8.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.8.m8.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.2">𝑒</ci><apply id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3"><plus id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.1.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.1"></plus><ci id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.2.cmml" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.2">𝑘</ci><cn id="S3.SS1.SSS1.7.p7.8.m8.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS1.7.p7.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.8.m8.1c">e_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.8.m8.1d">italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we need to consider the eight different cases from <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.T1" title="Table 1 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Table 1</span></a> separately. If we consider, for example, the case <math alttext="k=\texttt{-+convex}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.9.m9.1"><semantics id="S3.SS1.SSS1.7.p7.9.m9.1a"><mrow id="S3.SS1.SSS1.7.p7.9.m9.1.1" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.9.m9.1.1.2" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.2.cmml">k</mi><mo id="S3.SS1.SSS1.7.p7.9.m9.1.1.1" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.1.cmml">=</mo><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.7.p7.9.m9.1.1.3" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.3a.cmml">-+convex</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.9.m9.1b"><apply id="S3.SS1.SSS1.7.p7.9.m9.1.1.cmml" xref="S3.SS1.SSS1.7.p7.9.m9.1.1"><eq id="S3.SS1.SSS1.7.p7.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.1"></eq><ci id="S3.SS1.SSS1.7.p7.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.2">𝑘</ci><ci id="S3.SS1.SSS1.7.p7.9.m9.1.1.3a.cmml" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S3.SS1.SSS1.7.p7.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.9.m9.1.1.3">-+convex</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.9.m9.1c">k=\texttt{-+convex}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.9.m9.1d">italic_k = -+convex</annotation></semantics></math> in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F5" title="Figure 5 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 5</span></a>, we have <math alttext="\delta_{k}+\eta_{k}=|\beta_{k}|" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.10.m10.1"><semantics id="S3.SS1.SSS1.7.p7.10.m10.1a"><mrow id="S3.SS1.SSS1.7.p7.10.m10.1.1" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.cmml"><mrow id="S3.SS1.SSS1.7.p7.10.m10.1.1.3" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.cmml"><msub id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.2" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.2.cmml">δ</mi><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.3" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.1" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.1.cmml">+</mo><msub id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.2" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.2.cmml">η</mi><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.3" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.3.cmml">k</mi></msub></mrow><mo id="S3.SS1.SSS1.7.p7.10.m10.1.1.2" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.2.cmml">=</mo><mrow id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.2.cmml"><mo id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.2.1.cmml">|</mo><msub id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.2" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.3" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.10.m10.1b"><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1"><eq id="S3.SS1.SSS1.7.p7.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.2"></eq><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3"><plus id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.1"></plus><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.2">𝛿</ci><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.2">𝜂</ci><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.3.3.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1"><abs id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.2"></abs><apply id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.10.m10.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.10.m10.1c">\delta_{k}+\eta_{k}=|\beta_{k}|</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.10.m10.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = | italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math>. Applying (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E14" title="Equation 14 ‣ 3rd item ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">14</span></a>), we get <math alttext="\delta_{k}=-\eta_{k}-\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.11.m11.1"><semantics id="S3.SS1.SSS1.7.p7.11.m11.1a"><mrow id="S3.SS1.SSS1.7.p7.11.m11.1.1" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.cmml"><msub id="S3.SS1.SSS1.7.p7.11.m11.1.1.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2.cmml"><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2.2.cmml">δ</mi><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.3" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.7.p7.11.m11.1.1.1" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS1.7.p7.11.m11.1.1.3" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.cmml"><mrow id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.cmml"><mo id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2a" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.cmml">−</mo><msub id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.cmml"><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.2.cmml">η</mi><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.3" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.1" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.1.cmml">−</mo><msub id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.2" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.3" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.11.m11.1b"><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1"><eq id="S3.SS1.SSS1.7.p7.11.m11.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.1"></eq><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2.2">𝛿</ci><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.2.3.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3"><minus id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.1"></minus><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2"><minus id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2"></minus><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.2">𝜂</ci><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.2.2.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.7.p7.11.m11.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.11.m11.1c">\delta_{k}=-\eta_{k}-\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.11.m11.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, which, when substituted into (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E15" title="Equation 15 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">15</span></a>), implies <math alttext="\beta_{k-1}-\beta_{k}=\alpha_{k}+\eta_{k}+\pi" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.12.m12.1"><semantics id="S3.SS1.SSS1.7.p7.12.m12.1a"><mrow id="S3.SS1.SSS1.7.p7.12.m12.1.1" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.cmml"><mrow id="S3.SS1.SSS1.7.p7.12.m12.1.1.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.cmml"><msub id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.cmml"><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.2.cmml">β</mi><mrow id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.cmml"><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.2.cmml">k</mi><mo id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.1" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.1.cmml">−</mo><mn id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.1" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.1.cmml">−</mo><msub id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.cmml"><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.3.cmml">k</mi></msub></mrow><mo id="S3.SS1.SSS1.7.p7.12.m12.1.1.1" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS1.7.p7.12.m12.1.1.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.cmml"><msub id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.2.cmml">α</mi><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1.cmml">+</mo><msub id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.cmml"><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.2" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.2.cmml">η</mi><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.3" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.3.cmml">k</mi></msub><mo id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1a" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1.cmml">+</mo><mi id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.4" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.4.cmml">π</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.12.m12.1b"><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1"><eq id="S3.SS1.SSS1.7.p7.12.m12.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.1"></eq><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2"><minus id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.1"></minus><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.2">𝛽</ci><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3"><minus id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.1"></minus><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.2.3.3">1</cn></apply></apply><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.2.3.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3"><plus id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.1"></plus><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.2">𝛼</ci><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.2.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.2">𝜂</ci><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.3.3">𝑘</ci></apply><ci id="S3.SS1.SSS1.7.p7.12.m12.1.1.3.4.cmml" xref="S3.SS1.SSS1.7.p7.12.m12.1.1.3.4">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.12.m12.1c">\beta_{k-1}-\beta_{k}=\alpha_{k}+\eta_{k}+\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.12.m12.1d">italic_β start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_π</annotation></semantics></math>. In general, the difference between two consecutive <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.7.p7.13.m13.1"><semantics id="S3.SS1.SSS1.7.p7.13.m13.1a"><msub id="S3.SS1.SSS1.7.p7.13.m13.1.1" xref="S3.SS1.SSS1.7.p7.13.m13.1.1.cmml"><mi id="S3.SS1.SSS1.7.p7.13.m13.1.1.2" xref="S3.SS1.SSS1.7.p7.13.m13.1.1.2.cmml">β</mi><mi id="S3.SS1.SSS1.7.p7.13.m13.1.1.3" xref="S3.SS1.SSS1.7.p7.13.m13.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.7.p7.13.m13.1b"><apply id="S3.SS1.SSS1.7.p7.13.m13.1.1.cmml" xref="S3.SS1.SSS1.7.p7.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.7.p7.13.m13.1.1.1.cmml" xref="S3.SS1.SSS1.7.p7.13.m13.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.7.p7.13.m13.1.1.2.cmml" xref="S3.SS1.SSS1.7.p7.13.m13.1.1.2">𝛽</ci><ci id="S3.SS1.SSS1.7.p7.13.m13.1.1.3.cmml" xref="S3.SS1.SSS1.7.p7.13.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.7.p7.13.m13.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.7.p7.13.m13.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can be summarised for all eight cases as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E16"> <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="\beta_{k-1}-\beta_{k}=\alpha_{k}+\eta_{k}-\pi+\begin{cases}\phantom{+}2\pi,% \quad&k=\texttt{-+convex}\\ -2\pi,\quad&k=\texttt{+-concave}\\ \phantom{-2}0,&\text{else}\end{cases}" class="ltx_Math" display="block" id="S3.E16.m1.6"><semantics id="S3.E16.m1.6a"><mrow id="S3.E16.m1.6.7" xref="S3.E16.m1.6.7.cmml"><mrow id="S3.E16.m1.6.7.2" xref="S3.E16.m1.6.7.2.cmml"><msub id="S3.E16.m1.6.7.2.2" xref="S3.E16.m1.6.7.2.2.cmml"><mi id="S3.E16.m1.6.7.2.2.2" xref="S3.E16.m1.6.7.2.2.2.cmml">β</mi><mrow 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id="S3.E16.m1.4.4.4.4.2.1.3.cmml" xref="S3.E16.m1.4.4.4.4.2.1.3">+-concave</mtext></ci></apply><cn id="S3.E16.m1.5.5.5.5.1.1.1.cmml" type="integer" xref="S3.E16.m1.5.5.5.5.1.1.1">0</cn><ci id="S3.E16.m1.6.6.6.6.2.1a.cmml" xref="S3.E16.m1.6.6.6.6.2.1"><mtext id="S3.E16.m1.6.6.6.6.2.1.cmml" xref="S3.E16.m1.6.6.6.6.2.1">else</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E16.m1.6c">\beta_{k-1}-\beta_{k}=\alpha_{k}+\eta_{k}-\pi+\begin{cases}\phantom{+}2\pi,% \quad&k=\texttt{-+convex}\\ -2\pi,\quad&k=\texttt{+-concave}\\ \phantom{-2}0,&\text{else}\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.E16.m1.6d">italic_β start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_π + { start_ROW start_CELL 2 italic_π , end_CELL start_CELL italic_k = -+convex end_CELL end_ROW start_ROW start_CELL - 2 italic_π , end_CELL start_CELL italic_k = +-concave end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL else end_CELL end_ROW</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">(16)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.SSS1.8.p8"> <p class="ltx_p" id="S3.SS1.SSS1.8.p8.2">Using properties (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E12" title="Equation 12 ‣ 1st item ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">12</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E13" title="Equation 13 ‣ 2nd item ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">13</span></a>) of the angles <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.8.p8.1.m1.1"><semantics id="S3.SS1.SSS1.8.p8.1.m1.1a"><msub id="S3.SS1.SSS1.8.p8.1.m1.1.1" xref="S3.SS1.SSS1.8.p8.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.8.p8.1.m1.1.1.2" xref="S3.SS1.SSS1.8.p8.1.m1.1.1.2.cmml">α</mi><mi id="S3.SS1.SSS1.8.p8.1.m1.1.1.3" xref="S3.SS1.SSS1.8.p8.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.8.p8.1.m1.1b"><apply id="S3.SS1.SSS1.8.p8.1.m1.1.1.cmml" xref="S3.SS1.SSS1.8.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.8.p8.1.m1.1.1.1.cmml" xref="S3.SS1.SSS1.8.p8.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.8.p8.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.8.p8.1.m1.1.1.2">𝛼</ci><ci id="S3.SS1.SSS1.8.p8.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.8.p8.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.8.p8.1.m1.1c">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.8.p8.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.8.p8.2.m2.1"><semantics id="S3.SS1.SSS1.8.p8.2.m2.1a"><msub id="S3.SS1.SSS1.8.p8.2.m2.1.1" xref="S3.SS1.SSS1.8.p8.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.8.p8.2.m2.1.1.2" xref="S3.SS1.SSS1.8.p8.2.m2.1.1.2.cmml">η</mi><mi id="S3.SS1.SSS1.8.p8.2.m2.1.1.3" xref="S3.SS1.SSS1.8.p8.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.8.p8.2.m2.1b"><apply id="S3.SS1.SSS1.8.p8.2.m2.1.1.cmml" xref="S3.SS1.SSS1.8.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.8.p8.2.m2.1.1.1.cmml" xref="S3.SS1.SSS1.8.p8.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.8.p8.2.m2.1.1.2.cmml" xref="S3.SS1.SSS1.8.p8.2.m2.1.1.2">𝜂</ci><ci id="S3.SS1.SSS1.8.p8.2.m2.1.1.3.cmml" xref="S3.SS1.SSS1.8.p8.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.8.p8.2.m2.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.8.p8.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E16" title="Equation 16 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">16</span></a>) directly implies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx5"> <tbody id="S3.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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xref="S3.Ex19.m1.2.2.1.1.1.2">+-concave</mtext></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m1.4c">\displaystyle=2\pi\cdot\mathds{1}_{P}(x)-2\pi+2\pi\absolutevalue{\{k:\texttt{-% +convex}\}}-2\pi\absolutevalue{\{k:\texttt{+-concave}\}},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m1.4d">= 2 italic_π ⋅ blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 2 italic_π + 2 italic_π | start_ARG { italic_k : -+convex } end_ARG | - 2 italic_π | start_ARG { italic_k : +-concave } end_ARG | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.8.p8.4">and thus</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="\absolutevalue{\{k:\texttt{+-concave}\}}-\absolutevalue{\{k:\texttt{-+convex}% \}}=\mathds{1}_{P}(x)-1." class="ltx_Math" display="block" id="S3.Ex20.m1.4"><semantics id="S3.Ex20.m1.4a"><mrow id="S3.Ex20.m1.4.4.1" xref="S3.Ex20.m1.4.4.1.1.cmml"><mrow id="S3.Ex20.m1.4.4.1.1" xref="S3.Ex20.m1.4.4.1.1.cmml"><mrow id="S3.Ex20.m1.4.4.1.1.2" xref="S3.Ex20.m1.4.4.1.1.2.cmml"><mrow id="S3.Ex20.m1.1.1.3" xref="S3.Ex20.m1.1.1.2.cmml"><mo id="S3.Ex20.m1.1.1.3.1" xref="S3.Ex20.m1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex20.m1.1.1.1.1.1.4" xref="S3.Ex20.m1.1.1.1.1.1.3.cmml"><mo id="S3.Ex20.m1.1.1.1.1.1.4.1" stretchy="false" xref="S3.Ex20.m1.1.1.1.1.1.3.1.cmml">{</mo><mi id="S3.Ex20.m1.1.1.1.1.1.1" xref="S3.Ex20.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S3.Ex20.m1.1.1.1.1.1.4.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex20.m1.1.1.1.1.1.3.1.cmml">:</mo><mtext class="ltx_mathvariant_monospace" id="S3.Ex20.m1.1.1.1.1.1.2" xref="S3.Ex20.m1.1.1.1.1.1.2a.cmml">+-concave</mtext><mo id="S3.Ex20.m1.1.1.1.1.1.4.3" stretchy="false" xref="S3.Ex20.m1.1.1.1.1.1.3.1.cmml">}</mo></mrow><mo id="S3.Ex20.m1.1.1.3.2" xref="S3.Ex20.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex20.m1.4.4.1.1.2.1" xref="S3.Ex20.m1.4.4.1.1.2.1.cmml">−</mo><mrow id="S3.Ex20.m1.2.2.3" xref="S3.Ex20.m1.2.2.2.cmml"><mo id="S3.Ex20.m1.2.2.3.1" xref="S3.Ex20.m1.2.2.2.1.cmml">|</mo><mrow id="S3.Ex20.m1.2.2.1.1.1.4" xref="S3.Ex20.m1.2.2.1.1.1.3.cmml"><mo id="S3.Ex20.m1.2.2.1.1.1.4.1" stretchy="false" xref="S3.Ex20.m1.2.2.1.1.1.3.1.cmml">{</mo><mi id="S3.Ex20.m1.2.2.1.1.1.1" xref="S3.Ex20.m1.2.2.1.1.1.1.cmml">k</mi><mo id="S3.Ex20.m1.2.2.1.1.1.4.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex20.m1.2.2.1.1.1.3.1.cmml">:</mo><mtext class="ltx_mathvariant_monospace" id="S3.Ex20.m1.2.2.1.1.1.2" xref="S3.Ex20.m1.2.2.1.1.1.2a.cmml">-+convex</mtext><mo id="S3.Ex20.m1.2.2.1.1.1.4.3" stretchy="false" xref="S3.Ex20.m1.2.2.1.1.1.3.1.cmml">}</mo></mrow><mo id="S3.Ex20.m1.2.2.3.2" xref="S3.Ex20.m1.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.Ex20.m1.4.4.1.1.1" xref="S3.Ex20.m1.4.4.1.1.1.cmml">=</mo><mrow id="S3.Ex20.m1.4.4.1.1.3" xref="S3.Ex20.m1.4.4.1.1.3.cmml"><mrow id="S3.Ex20.m1.4.4.1.1.3.2" xref="S3.Ex20.m1.4.4.1.1.3.2.cmml"><msub id="S3.Ex20.m1.4.4.1.1.3.2.2" xref="S3.Ex20.m1.4.4.1.1.3.2.2.cmml"><mn id="S3.Ex20.m1.4.4.1.1.3.2.2.2" xref="S3.Ex20.m1.4.4.1.1.3.2.2.2.cmml">𝟙</mn><mi id="S3.Ex20.m1.4.4.1.1.3.2.2.3" xref="S3.Ex20.m1.4.4.1.1.3.2.2.3.cmml">P</mi></msub><mo id="S3.Ex20.m1.4.4.1.1.3.2.1" xref="S3.Ex20.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.Ex20.m1.4.4.1.1.3.2.3.2" xref="S3.Ex20.m1.4.4.1.1.3.2.cmml"><mo id="S3.Ex20.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S3.Ex20.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S3.Ex20.m1.3.3" xref="S3.Ex20.m1.3.3.cmml">x</mi><mo id="S3.Ex20.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S3.Ex20.m1.4.4.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex20.m1.4.4.1.1.3.1" xref="S3.Ex20.m1.4.4.1.1.3.1.cmml">−</mo><mn id="S3.Ex20.m1.4.4.1.1.3.3" xref="S3.Ex20.m1.4.4.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S3.Ex20.m1.4.4.1.2" lspace="0em" xref="S3.Ex20.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex20.m1.4b"><apply id="S3.Ex20.m1.4.4.1.1.cmml" xref="S3.Ex20.m1.4.4.1"><eq id="S3.Ex20.m1.4.4.1.1.1.cmml" xref="S3.Ex20.m1.4.4.1.1.1"></eq><apply id="S3.Ex20.m1.4.4.1.1.2.cmml" xref="S3.Ex20.m1.4.4.1.1.2"><minus id="S3.Ex20.m1.4.4.1.1.2.1.cmml" xref="S3.Ex20.m1.4.4.1.1.2.1"></minus><apply id="S3.Ex20.m1.1.1.2.cmml" xref="S3.Ex20.m1.1.1.3"><abs id="S3.Ex20.m1.1.1.2.1.cmml" xref="S3.Ex20.m1.1.1.3.1"></abs><apply id="S3.Ex20.m1.1.1.1.1.1.3.cmml" xref="S3.Ex20.m1.1.1.1.1.1.4"><csymbol cd="latexml" id="S3.Ex20.m1.1.1.1.1.1.3.1.cmml" xref="S3.Ex20.m1.1.1.1.1.1.4.1">conditional-set</csymbol><ci id="S3.Ex20.m1.1.1.1.1.1.1.cmml" xref="S3.Ex20.m1.1.1.1.1.1.1">𝑘</ci><ci id="S3.Ex20.m1.1.1.1.1.1.2a.cmml" xref="S3.Ex20.m1.1.1.1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S3.Ex20.m1.1.1.1.1.1.2.cmml" xref="S3.Ex20.m1.1.1.1.1.1.2">+-concave</mtext></ci></apply></apply><apply id="S3.Ex20.m1.2.2.2.cmml" xref="S3.Ex20.m1.2.2.3"><abs id="S3.Ex20.m1.2.2.2.1.cmml" xref="S3.Ex20.m1.2.2.3.1"></abs><apply id="S3.Ex20.m1.2.2.1.1.1.3.cmml" xref="S3.Ex20.m1.2.2.1.1.1.4"><csymbol cd="latexml" id="S3.Ex20.m1.2.2.1.1.1.3.1.cmml" xref="S3.Ex20.m1.2.2.1.1.1.4.1">conditional-set</csymbol><ci id="S3.Ex20.m1.2.2.1.1.1.1.cmml" xref="S3.Ex20.m1.2.2.1.1.1.1">𝑘</ci><ci id="S3.Ex20.m1.2.2.1.1.1.2a.cmml" xref="S3.Ex20.m1.2.2.1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S3.Ex20.m1.2.2.1.1.1.2.cmml" xref="S3.Ex20.m1.2.2.1.1.1.2">-+convex</mtext></ci></apply></apply></apply><apply id="S3.Ex20.m1.4.4.1.1.3.cmml" xref="S3.Ex20.m1.4.4.1.1.3"><minus id="S3.Ex20.m1.4.4.1.1.3.1.cmml" xref="S3.Ex20.m1.4.4.1.1.3.1"></minus><apply id="S3.Ex20.m1.4.4.1.1.3.2.cmml" xref="S3.Ex20.m1.4.4.1.1.3.2"><times id="S3.Ex20.m1.4.4.1.1.3.2.1.cmml" xref="S3.Ex20.m1.4.4.1.1.3.2.1"></times><apply id="S3.Ex20.m1.4.4.1.1.3.2.2.cmml" xref="S3.Ex20.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.Ex20.m1.4.4.1.1.3.2.2.1.cmml" xref="S3.Ex20.m1.4.4.1.1.3.2.2">subscript</csymbol><cn id="S3.Ex20.m1.4.4.1.1.3.2.2.2.cmml" type="integer" xref="S3.Ex20.m1.4.4.1.1.3.2.2.2">1</cn><ci id="S3.Ex20.m1.4.4.1.1.3.2.2.3.cmml" xref="S3.Ex20.m1.4.4.1.1.3.2.2.3">𝑃</ci></apply><ci id="S3.Ex20.m1.3.3.cmml" xref="S3.Ex20.m1.3.3">𝑥</ci></apply><cn id="S3.Ex20.m1.4.4.1.1.3.3.cmml" type="integer" xref="S3.Ex20.m1.4.4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex20.m1.4c">\absolutevalue{\{k:\texttt{+-concave}\}}-\absolutevalue{\{k:\texttt{-+convex}% \}}=\mathds{1}_{P}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex20.m1.4d">| start_ARG { italic_k : +-concave } end_ARG | - | start_ARG { italic_k : -+convex } end_ARG | = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.8.p8.3">By (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E10" title="Equation 10 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">10</span></a>), this is precisely (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E7" title="Equation 7 ‣ Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">7</span></a>) with <math alttext="n_{h}(P)=0" class="ltx_Math" display="inline" id="S3.SS1.SSS1.8.p8.3.m1.1"><semantics id="S3.SS1.SSS1.8.p8.3.m1.1a"><mrow id="S3.SS1.SSS1.8.p8.3.m1.1.2" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.cmml"><mrow id="S3.SS1.SSS1.8.p8.3.m1.1.2.2" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.cmml"><msub id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.cmml"><mi id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.2" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.3" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.1" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.3.2" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.cmml"><mo id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS1.8.p8.3.m1.1.1" xref="S3.SS1.SSS1.8.p8.3.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.8.p8.3.m1.1.2.1" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS1.8.p8.3.m1.1.2.3" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.8.p8.3.m1.1b"><apply id="S3.SS1.SSS1.8.p8.3.m1.1.2.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2"><eq id="S3.SS1.SSS1.8.p8.3.m1.1.2.1.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.1"></eq><apply id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2"><times id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.1.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.1"></times><apply id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.1.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.2.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.3.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.2.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS1.8.p8.3.m1.1.1.cmml" xref="S3.SS1.SSS1.8.p8.3.m1.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS1.8.p8.3.m1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS1.8.p8.3.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.8.p8.3.m1.1c">n_{h}(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.8.p8.3.m1.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = 0</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S3.F5"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S3.F5.20"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.F5.4.4"> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.1.1.1" style="width:19.9pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.1.1.1.1"> <span class="ltx_p" id="S3.F5.1.1.1.1.1"><math alttext="H^{k}_{\bullet}" class="ltx_Math" display="inline" id="S3.F5.1.1.1.1.1.m1.1"><semantics id="S3.F5.1.1.1.1.1.m1.1a"><msubsup id="S3.F5.1.1.1.1.1.m1.1.1" xref="S3.F5.1.1.1.1.1.m1.1.1.cmml"><mi id="S3.F5.1.1.1.1.1.m1.1.1.2.2" xref="S3.F5.1.1.1.1.1.m1.1.1.2.2.cmml">H</mi><mo id="S3.F5.1.1.1.1.1.m1.1.1.3" xref="S3.F5.1.1.1.1.1.m1.1.1.3.cmml">∙</mo><mi id="S3.F5.1.1.1.1.1.m1.1.1.2.3" xref="S3.F5.1.1.1.1.1.m1.1.1.2.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.F5.1.1.1.1.1.m1.1b"><apply id="S3.F5.1.1.1.1.1.m1.1.1.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1">subscript</csymbol><apply id="S3.F5.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.1.1.1.1.1.m1.1.1.2.1.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1">superscript</csymbol><ci id="S3.F5.1.1.1.1.1.m1.1.1.2.2.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1.2.2">𝐻</ci><ci id="S3.F5.1.1.1.1.1.m1.1.1.2.3.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S3.F5.1.1.1.1.1.m1.1.1.3.cmml" xref="S3.F5.1.1.1.1.1.m1.1.1.3">∙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.1.1.1.1.1.m1.1c">H^{k}_{\bullet}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.1.1.1.1.1.m1.1d">italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT</annotation></semantics></math></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_r" id="S3.F5.2.2.2" style="width:22.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.2.2.2.1"> <span class="ltx_p" id="S3.F5.2.2.2.1.1"><math alttext="H^{k+1}_{\bullet}" class="ltx_Math" display="inline" id="S3.F5.2.2.2.1.1.m1.1"><semantics id="S3.F5.2.2.2.1.1.m1.1a"><msubsup id="S3.F5.2.2.2.1.1.m1.1.1" xref="S3.F5.2.2.2.1.1.m1.1.1.cmml"><mi id="S3.F5.2.2.2.1.1.m1.1.1.2.2" xref="S3.F5.2.2.2.1.1.m1.1.1.2.2.cmml">H</mi><mo id="S3.F5.2.2.2.1.1.m1.1.1.3" xref="S3.F5.2.2.2.1.1.m1.1.1.3.cmml">∙</mo><mrow id="S3.F5.2.2.2.1.1.m1.1.1.2.3" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.cmml"><mi id="S3.F5.2.2.2.1.1.m1.1.1.2.3.2" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.2.cmml">k</mi><mo id="S3.F5.2.2.2.1.1.m1.1.1.2.3.1" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.1.cmml">+</mo><mn id="S3.F5.2.2.2.1.1.m1.1.1.2.3.3" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.3.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.F5.2.2.2.1.1.m1.1b"><apply id="S3.F5.2.2.2.1.1.m1.1.1.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.2.2.2.1.1.m1.1.1.1.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1">subscript</csymbol><apply id="S3.F5.2.2.2.1.1.m1.1.1.2.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.2.2.2.1.1.m1.1.1.2.1.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1">superscript</csymbol><ci id="S3.F5.2.2.2.1.1.m1.1.1.2.2.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1.2.2">𝐻</ci><apply id="S3.F5.2.2.2.1.1.m1.1.1.2.3.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3"><plus id="S3.F5.2.2.2.1.1.m1.1.1.2.3.1.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.1"></plus><ci id="S3.F5.2.2.2.1.1.m1.1.1.2.3.2.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.2">𝑘</ci><cn id="S3.F5.2.2.2.1.1.m1.1.1.2.3.3.cmml" type="integer" xref="S3.F5.2.2.2.1.1.m1.1.1.2.3.3">1</cn></apply></apply><ci id="S3.F5.2.2.2.1.1.m1.1.1.3.cmml" xref="S3.F5.2.2.2.1.1.m1.1.1.3">∙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.2.2.2.1.1.m1.1c">H^{k+1}_{\bullet}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.2.2.2.1.1.m1.1d">italic_H start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT</annotation></semantics></math></span> </span> </td> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.3.3.3" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.3.3.3.1"> <span class="ltx_p" id="S3.F5.3.3.3.1.1">convex (<math alttext="\eta_{k}<\pi" class="ltx_Math" display="inline" id="S3.F5.3.3.3.1.1.m1.1"><semantics id="S3.F5.3.3.3.1.1.m1.1a"><mrow id="S3.F5.3.3.3.1.1.m1.1.1" xref="S3.F5.3.3.3.1.1.m1.1.1.cmml"><msub id="S3.F5.3.3.3.1.1.m1.1.1.2" xref="S3.F5.3.3.3.1.1.m1.1.1.2.cmml"><mi id="S3.F5.3.3.3.1.1.m1.1.1.2.2" xref="S3.F5.3.3.3.1.1.m1.1.1.2.2.cmml">η</mi><mi id="S3.F5.3.3.3.1.1.m1.1.1.2.3" xref="S3.F5.3.3.3.1.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S3.F5.3.3.3.1.1.m1.1.1.1" xref="S3.F5.3.3.3.1.1.m1.1.1.1.cmml"><</mo><mi id="S3.F5.3.3.3.1.1.m1.1.1.3" xref="S3.F5.3.3.3.1.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.3.3.3.1.1.m1.1b"><apply id="S3.F5.3.3.3.1.1.m1.1.1.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1"><lt id="S3.F5.3.3.3.1.1.m1.1.1.1.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.1"></lt><apply id="S3.F5.3.3.3.1.1.m1.1.1.2.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.F5.3.3.3.1.1.m1.1.1.2.1.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.F5.3.3.3.1.1.m1.1.1.2.2.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.2.2">𝜂</ci><ci id="S3.F5.3.3.3.1.1.m1.1.1.2.3.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S3.F5.3.3.3.1.1.m1.1.1.3.cmml" xref="S3.F5.3.3.3.1.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.3.3.3.1.1.m1.1c">\eta_{k}<\pi</annotation><annotation encoding="application/x-llamapun" id="S3.F5.3.3.3.1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_π</annotation></semantics></math>)</span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F5.4.4.4" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.4.4.4.1"> <span class="ltx_p" id="S3.F5.4.4.4.1.1">concave (<math alttext="\eta_{k}>\pi" class="ltx_Math" display="inline" id="S3.F5.4.4.4.1.1.m1.1"><semantics id="S3.F5.4.4.4.1.1.m1.1a"><mrow id="S3.F5.4.4.4.1.1.m1.1.1" xref="S3.F5.4.4.4.1.1.m1.1.1.cmml"><msub id="S3.F5.4.4.4.1.1.m1.1.1.2" xref="S3.F5.4.4.4.1.1.m1.1.1.2.cmml"><mi id="S3.F5.4.4.4.1.1.m1.1.1.2.2" xref="S3.F5.4.4.4.1.1.m1.1.1.2.2.cmml">η</mi><mi id="S3.F5.4.4.4.1.1.m1.1.1.2.3" xref="S3.F5.4.4.4.1.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S3.F5.4.4.4.1.1.m1.1.1.1" xref="S3.F5.4.4.4.1.1.m1.1.1.1.cmml">></mo><mi id="S3.F5.4.4.4.1.1.m1.1.1.3" xref="S3.F5.4.4.4.1.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.4.4.4.1.1.m1.1b"><apply id="S3.F5.4.4.4.1.1.m1.1.1.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1"><gt id="S3.F5.4.4.4.1.1.m1.1.1.1.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.1"></gt><apply id="S3.F5.4.4.4.1.1.m1.1.1.2.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.F5.4.4.4.1.1.m1.1.1.2.1.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.F5.4.4.4.1.1.m1.1.1.2.2.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.2.2">𝜂</ci><ci id="S3.F5.4.4.4.1.1.m1.1.1.2.3.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S3.F5.4.4.4.1.1.m1.1.1.3.cmml" xref="S3.F5.4.4.4.1.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.4.4.4.1.1.m1.1c">\eta_{k}>\pi</annotation><annotation encoding="application/x-llamapun" id="S3.F5.4.4.4.1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > italic_π</annotation></semantics></math>)</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.F5.8.8"> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row ltx_border_t" id="S3.F5.5.5.1" style="width:19.9pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.5.5.1.1"> <span class="ltx_p" id="S3.F5.5.5.1.1.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.F5.5.5.1.1.1.m1.1"><semantics id="S3.F5.5.5.1.1.1.m1.1a"><mo id="S3.F5.5.5.1.1.1.m1.1.1" mathsize="207%" xref="S3.F5.5.5.1.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.F5.5.5.1.1.1.m1.1b"><plus id="S3.F5.5.5.1.1.1.m1.1.1.cmml" xref="S3.F5.5.5.1.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.5.5.1.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.F5.5.5.1.1.1.m1.1d">+</annotation></semantics></math></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_r ltx_border_t" id="S3.F5.6.6.2" style="width:22.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.6.6.2.1"> <span class="ltx_p" id="S3.F5.6.6.2.1.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.F5.6.6.2.1.1.m1.1"><semantics id="S3.F5.6.6.2.1.1.m1.1a"><mo id="S3.F5.6.6.2.1.1.m1.1.1" mathsize="207%" xref="S3.F5.6.6.2.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.F5.6.6.2.1.1.m1.1b"><plus id="S3.F5.6.6.2.1.1.m1.1.1.cmml" xref="S3.F5.6.6.2.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.6.6.2.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.F5.6.6.2.1.1.m1.1d">+</annotation></semantics></math></span> </span> </td> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row ltx_border_t" id="S3.F5.7.7.3" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.7.7.3.1"> <span class="ltx_p" id="S3.F5.7.7.3.1.1"><foreignobject height="67.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="79.5pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="130" id="S3.F5.7.7.3.1.1.1.g1" src="x10.png" width="152"/></foreignobject></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" id="S3.F5.8.8.4" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.8.8.4.1"> <span class="ltx_p" id="S3.F5.8.8.4.1.1"><foreignobject height="67.2pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="93.2pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="129" id="S3.F5.8.8.4.1.1.1.g1" src="x11.png" width="179"/></foreignobject></span> </span> </td> </tr> <tr class="ltx_tr" id="S3.F5.12.12"> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.9.9.1" style="width:19.9pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.9.9.1.1"> <span class="ltx_p" id="S3.F5.9.9.1.1.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.F5.9.9.1.1.1.m1.1"><semantics id="S3.F5.9.9.1.1.1.m1.1a"><mo id="S3.F5.9.9.1.1.1.m1.1.1" mathsize="207%" xref="S3.F5.9.9.1.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.F5.9.9.1.1.1.m1.1b"><plus id="S3.F5.9.9.1.1.1.m1.1.1.cmml" xref="S3.F5.9.9.1.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.9.9.1.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.F5.9.9.1.1.1.m1.1d">+</annotation></semantics></math></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_r" id="S3.F5.10.10.2" style="width:22.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.10.10.2.1"> <span class="ltx_p" id="S3.F5.10.10.2.1.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.F5.10.10.2.1.1.m1.1"><semantics id="S3.F5.10.10.2.1.1.m1.1a"><mo id="S3.F5.10.10.2.1.1.m1.1.1" mathsize="207%" xref="S3.F5.10.10.2.1.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.F5.10.10.2.1.1.m1.1b"><minus id="S3.F5.10.10.2.1.1.m1.1.1.cmml" xref="S3.F5.10.10.2.1.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.10.10.2.1.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.F5.10.10.2.1.1.m1.1d">-</annotation></semantics></math></span> </span> </td> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.11.11.3" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.11.11.3.1"> <span class="ltx_p" id="S3.F5.11.11.3.1.1"><foreignobject height="65.8pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="83.1pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="127" id="S3.F5.11.11.3.1.1.1.g1" src="x12.png" width="160"/></foreignobject></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F5.12.12.4" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.12.12.4.1"> <span class="ltx_p" id="S3.F5.12.12.4.1.1"><foreignobject height="63.6pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="78.8pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="122" id="S3.F5.12.12.4.1.1.1.g1" src="x13.png" width="151"/></foreignobject></span> </span> </td> </tr> <tr class="ltx_tr" id="S3.F5.16.16"> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.13.13.1" style="width:19.9pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.13.13.1.1"> <span class="ltx_p" id="S3.F5.13.13.1.1.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.F5.13.13.1.1.1.m1.1"><semantics id="S3.F5.13.13.1.1.1.m1.1a"><mo id="S3.F5.13.13.1.1.1.m1.1.1" mathsize="207%" xref="S3.F5.13.13.1.1.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.F5.13.13.1.1.1.m1.1b"><minus id="S3.F5.13.13.1.1.1.m1.1.1.cmml" xref="S3.F5.13.13.1.1.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.13.13.1.1.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.F5.13.13.1.1.1.m1.1d">-</annotation></semantics></math></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_r" id="S3.F5.14.14.2" style="width:22.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.14.14.2.1"> <span class="ltx_p" id="S3.F5.14.14.2.1.1"><math alttext="+" class="ltx_Math" display="inline" id="S3.F5.14.14.2.1.1.m1.1"><semantics id="S3.F5.14.14.2.1.1.m1.1a"><mo id="S3.F5.14.14.2.1.1.m1.1.1" mathsize="207%" xref="S3.F5.14.14.2.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.F5.14.14.2.1.1.m1.1b"><plus id="S3.F5.14.14.2.1.1.m1.1.1.cmml" xref="S3.F5.14.14.2.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.14.14.2.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.F5.14.14.2.1.1.m1.1d">+</annotation></semantics></math></span> </span> </td> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.15.15.3" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.15.15.3.1"> <span class="ltx_p" id="S3.F5.15.15.3.1.1"><foreignobject height="68.7pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="79.5pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="132" id="S3.F5.15.15.3.1.1.1.g1" src="x14.png" width="153"/></foreignobject></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F5.16.16.4" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.16.16.4.1"> <span class="ltx_p" id="S3.F5.16.16.4.1.1"><foreignobject height="70.8pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="63.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="136" id="S3.F5.16.16.4.1.1.1.g1" src="x15.png" width="122"/></foreignobject></span> </span> </td> </tr> <tr class="ltx_tr" id="S3.F5.20.20"> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.17.17.1" style="width:19.9pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.17.17.1.1"> <span class="ltx_p" id="S3.F5.17.17.1.1.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.F5.17.17.1.1.1.m1.1"><semantics id="S3.F5.17.17.1.1.1.m1.1a"><mo id="S3.F5.17.17.1.1.1.m1.1.1" mathsize="207%" xref="S3.F5.17.17.1.1.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.F5.17.17.1.1.1.m1.1b"><minus id="S3.F5.17.17.1.1.1.m1.1.1.cmml" xref="S3.F5.17.17.1.1.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.17.17.1.1.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.F5.17.17.1.1.1.m1.1d">-</annotation></semantics></math></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle ltx_border_r" id="S3.F5.18.18.2" style="width:22.8pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.18.18.2.1"> <span class="ltx_p" id="S3.F5.18.18.2.1.1"><math alttext="-" class="ltx_Math" display="inline" id="S3.F5.18.18.2.1.1.m1.1"><semantics id="S3.F5.18.18.2.1.1.m1.1a"><mo id="S3.F5.18.18.2.1.1.m1.1.1" mathsize="207%" xref="S3.F5.18.18.2.1.1.m1.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S3.F5.18.18.2.1.1.m1.1b"><minus id="S3.F5.18.18.2.1.1.m1.1.1.cmml" xref="S3.F5.18.18.2.1.1.m1.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.18.18.2.1.1.m1.1c">-</annotation><annotation encoding="application/x-llamapun" id="S3.F5.18.18.2.1.1.m1.1d">-</annotation></semantics></math></span> </span> </td> <th class="ltx_td ltx_align_justify ltx_align_middle ltx_th ltx_th_row" id="S3.F5.19.19.3" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.19.19.3.1"> <span class="ltx_p" id="S3.F5.19.19.3.1.1"><foreignobject height="67.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="83.1pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="130" id="S3.F5.19.19.3.1.1.1.g1" src="x16.png" width="160"/></foreignobject></span> </span> </th> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F5.20.20.4" style="width:128.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F5.20.20.4.1"> <span class="ltx_p" id="S3.F5.20.20.4.1.1"><foreignobject height="68.7pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="86.7pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="132" id="S3.F5.20.20.4.1.1.1.g1" src="x17.png" width="167"/></foreignobject></span> </span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Angles that are relevant for the vertex-edge pair <math alttext="(v_{k},e_{k+1})" class="ltx_Math" display="inline" id="S3.F5.28.m1.2"><semantics id="S3.F5.28.m1.2b"><mrow id="S3.F5.28.m1.2.2.2" xref="S3.F5.28.m1.2.2.3.cmml"><mo id="S3.F5.28.m1.2.2.2.3" stretchy="false" xref="S3.F5.28.m1.2.2.3.cmml">(</mo><msub id="S3.F5.28.m1.1.1.1.1" xref="S3.F5.28.m1.1.1.1.1.cmml"><mi id="S3.F5.28.m1.1.1.1.1.2" xref="S3.F5.28.m1.1.1.1.1.2.cmml">v</mi><mi id="S3.F5.28.m1.1.1.1.1.3" xref="S3.F5.28.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.F5.28.m1.2.2.2.4" xref="S3.F5.28.m1.2.2.3.cmml">,</mo><msub id="S3.F5.28.m1.2.2.2.2" xref="S3.F5.28.m1.2.2.2.2.cmml"><mi id="S3.F5.28.m1.2.2.2.2.2" xref="S3.F5.28.m1.2.2.2.2.2.cmml">e</mi><mrow id="S3.F5.28.m1.2.2.2.2.3" xref="S3.F5.28.m1.2.2.2.2.3.cmml"><mi id="S3.F5.28.m1.2.2.2.2.3.2" xref="S3.F5.28.m1.2.2.2.2.3.2.cmml">k</mi><mo id="S3.F5.28.m1.2.2.2.2.3.1" xref="S3.F5.28.m1.2.2.2.2.3.1.cmml">+</mo><mn id="S3.F5.28.m1.2.2.2.2.3.3" xref="S3.F5.28.m1.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.F5.28.m1.2.2.2.5" stretchy="false" xref="S3.F5.28.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.28.m1.2c"><interval closure="open" id="S3.F5.28.m1.2.2.3.cmml" xref="S3.F5.28.m1.2.2.2"><apply id="S3.F5.28.m1.1.1.1.1.cmml" xref="S3.F5.28.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F5.28.m1.1.1.1.1.1.cmml" xref="S3.F5.28.m1.1.1.1.1">subscript</csymbol><ci id="S3.F5.28.m1.1.1.1.1.2.cmml" xref="S3.F5.28.m1.1.1.1.1.2">𝑣</ci><ci id="S3.F5.28.m1.1.1.1.1.3.cmml" xref="S3.F5.28.m1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.F5.28.m1.2.2.2.2.cmml" xref="S3.F5.28.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.F5.28.m1.2.2.2.2.1.cmml" xref="S3.F5.28.m1.2.2.2.2">subscript</csymbol><ci id="S3.F5.28.m1.2.2.2.2.2.cmml" xref="S3.F5.28.m1.2.2.2.2.2">𝑒</ci><apply id="S3.F5.28.m1.2.2.2.2.3.cmml" xref="S3.F5.28.m1.2.2.2.2.3"><plus id="S3.F5.28.m1.2.2.2.2.3.1.cmml" xref="S3.F5.28.m1.2.2.2.2.3.1"></plus><ci id="S3.F5.28.m1.2.2.2.2.3.2.cmml" xref="S3.F5.28.m1.2.2.2.2.3.2">𝑘</ci><cn id="S3.F5.28.m1.2.2.2.2.3.3.cmml" type="integer" xref="S3.F5.28.m1.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.28.m1.2d">(v_{k},e_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S3.F5.28.m1.2e">( italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, depending on the relative position of <math alttext="x" class="ltx_Math" display="inline" id="S3.F5.29.m2.1"><semantics id="S3.F5.29.m2.1b"><mi id="S3.F5.29.m2.1.1" xref="S3.F5.29.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.F5.29.m2.1c"><ci id="S3.F5.29.m2.1.1.cmml" xref="S3.F5.29.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.29.m2.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.F5.29.m2.1e">italic_x</annotation></semantics></math> with respect to the edges incident with <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.F5.30.m3.1"><semantics id="S3.F5.30.m3.1b"><msub id="S3.F5.30.m3.1.1" xref="S3.F5.30.m3.1.1.cmml"><mi id="S3.F5.30.m3.1.1.2" xref="S3.F5.30.m3.1.1.2.cmml">v</mi><mi id="S3.F5.30.m3.1.1.3" xref="S3.F5.30.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F5.30.m3.1c"><apply id="S3.F5.30.m3.1.1.cmml" xref="S3.F5.30.m3.1.1"><csymbol cd="ambiguous" id="S3.F5.30.m3.1.1.1.cmml" xref="S3.F5.30.m3.1.1">subscript</csymbol><ci id="S3.F5.30.m3.1.1.2.cmml" xref="S3.F5.30.m3.1.1.2">𝑣</ci><ci id="S3.F5.30.m3.1.1.3.cmml" xref="S3.F5.30.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.30.m3.1d">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.30.m3.1e">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and on the convexity of the corner. Due to the choice of the cyclic order on <math alttext="V(P)" class="ltx_Math" display="inline" id="S3.F5.31.m4.1"><semantics id="S3.F5.31.m4.1b"><mrow id="S3.F5.31.m4.1.2" xref="S3.F5.31.m4.1.2.cmml"><mi id="S3.F5.31.m4.1.2.2" xref="S3.F5.31.m4.1.2.2.cmml">V</mi><mo id="S3.F5.31.m4.1.2.1" xref="S3.F5.31.m4.1.2.1.cmml">⁢</mo><mrow id="S3.F5.31.m4.1.2.3.2" xref="S3.F5.31.m4.1.2.cmml"><mo id="S3.F5.31.m4.1.2.3.2.1" stretchy="false" xref="S3.F5.31.m4.1.2.cmml">(</mo><mi id="S3.F5.31.m4.1.1" xref="S3.F5.31.m4.1.1.cmml">P</mi><mo id="S3.F5.31.m4.1.2.3.2.2" stretchy="false" xref="S3.F5.31.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.31.m4.1c"><apply id="S3.F5.31.m4.1.2.cmml" xref="S3.F5.31.m4.1.2"><times id="S3.F5.31.m4.1.2.1.cmml" xref="S3.F5.31.m4.1.2.1"></times><ci id="S3.F5.31.m4.1.2.2.cmml" xref="S3.F5.31.m4.1.2.2">𝑉</ci><ci id="S3.F5.31.m4.1.1.cmml" xref="S3.F5.31.m4.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.31.m4.1d">V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.F5.31.m4.1e">italic_V ( italic_P )</annotation></semantics></math>, the polygon (shown in grey) is always on the left hand side when traversing <math alttext="e_{k}" class="ltx_Math" display="inline" id="S3.F5.32.m5.1"><semantics id="S3.F5.32.m5.1b"><msub id="S3.F5.32.m5.1.1" xref="S3.F5.32.m5.1.1.cmml"><mi id="S3.F5.32.m5.1.1.2" xref="S3.F5.32.m5.1.1.2.cmml">e</mi><mi id="S3.F5.32.m5.1.1.3" xref="S3.F5.32.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F5.32.m5.1c"><apply id="S3.F5.32.m5.1.1.cmml" xref="S3.F5.32.m5.1.1"><csymbol cd="ambiguous" id="S3.F5.32.m5.1.1.1.cmml" xref="S3.F5.32.m5.1.1">subscript</csymbol><ci id="S3.F5.32.m5.1.1.2.cmml" xref="S3.F5.32.m5.1.1.2">𝑒</ci><ci id="S3.F5.32.m5.1.1.3.cmml" xref="S3.F5.32.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.32.m5.1d">e_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.32.m5.1e">italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="v_{k-1}" class="ltx_Math" display="inline" id="S3.F5.33.m6.1"><semantics id="S3.F5.33.m6.1b"><msub id="S3.F5.33.m6.1.1" xref="S3.F5.33.m6.1.1.cmml"><mi id="S3.F5.33.m6.1.1.2" xref="S3.F5.33.m6.1.1.2.cmml">v</mi><mrow id="S3.F5.33.m6.1.1.3" xref="S3.F5.33.m6.1.1.3.cmml"><mi id="S3.F5.33.m6.1.1.3.2" xref="S3.F5.33.m6.1.1.3.2.cmml">k</mi><mo id="S3.F5.33.m6.1.1.3.1" xref="S3.F5.33.m6.1.1.3.1.cmml">−</mo><mn id="S3.F5.33.m6.1.1.3.3" xref="S3.F5.33.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F5.33.m6.1c"><apply id="S3.F5.33.m6.1.1.cmml" xref="S3.F5.33.m6.1.1"><csymbol cd="ambiguous" id="S3.F5.33.m6.1.1.1.cmml" xref="S3.F5.33.m6.1.1">subscript</csymbol><ci id="S3.F5.33.m6.1.1.2.cmml" xref="S3.F5.33.m6.1.1.2">𝑣</ci><apply id="S3.F5.33.m6.1.1.3.cmml" xref="S3.F5.33.m6.1.1.3"><minus id="S3.F5.33.m6.1.1.3.1.cmml" xref="S3.F5.33.m6.1.1.3.1"></minus><ci id="S3.F5.33.m6.1.1.3.2.cmml" xref="S3.F5.33.m6.1.1.3.2">𝑘</ci><cn id="S3.F5.33.m6.1.1.3.3.cmml" type="integer" xref="S3.F5.33.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.33.m6.1d">v_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.33.m6.1e">italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v_{k}" class="ltx_Math" display="inline" id="S3.F5.34.m7.1"><semantics id="S3.F5.34.m7.1b"><msub id="S3.F5.34.m7.1.1" xref="S3.F5.34.m7.1.1.cmml"><mi id="S3.F5.34.m7.1.1.2" xref="S3.F5.34.m7.1.1.2.cmml">v</mi><mi id="S3.F5.34.m7.1.1.3" xref="S3.F5.34.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F5.34.m7.1c"><apply id="S3.F5.34.m7.1.1.cmml" xref="S3.F5.34.m7.1.1"><csymbol cd="ambiguous" id="S3.F5.34.m7.1.1.1.cmml" xref="S3.F5.34.m7.1.1">subscript</csymbol><ci id="S3.F5.34.m7.1.1.2.cmml" xref="S3.F5.34.m7.1.1.2">𝑣</ci><ci id="S3.F5.34.m7.1.1.3.cmml" xref="S3.F5.34.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.34.m7.1d">v_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.34.m7.1e">italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Cases marked with ✗ are the ones that occur in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E10" title="Equation 10 ‣ Proof. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">10</span></a>).</figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS1.9.p9"> <p class="ltx_p" id="S3.SS1.SSS1.9.p9.15">Now, consider the case <math alttext="P=\overline{\operatorname*{out}(\gamma)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.1.m1.2"><semantics id="S3.SS1.SSS1.9.p9.1.m1.2a"><mrow id="S3.SS1.SSS1.9.p9.1.m1.2.3" xref="S3.SS1.SSS1.9.p9.1.m1.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.1.m1.2.3.2" xref="S3.SS1.SSS1.9.p9.1.m1.2.3.2.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.1.m1.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS1.9.p9.1.m1.2.3.1.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.1.m1.2.2" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.4" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.1.m1.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS1.9.p9.1.m1.1.1.1.1.cmml">out</mo><mrow id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.4.1" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.4.1.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.2" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.2.cmml">γ</mi><mo id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.4.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.1.m1.2.2.3" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.1.m1.2b"><apply id="S3.SS1.SSS1.9.p9.1.m1.2.3.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.3"><eq id="S3.SS1.SSS1.9.p9.1.m1.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.3.1"></eq><ci id="S3.SS1.SSS1.9.p9.1.m1.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.3.2">𝑃</ci><apply id="S3.SS1.SSS1.9.p9.1.m1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.2"><ci id="S3.SS1.SSS1.9.p9.1.m1.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.3">¯</ci><apply id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.4"><ci id="S3.SS1.SSS1.9.p9.1.m1.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.1.1.1.1">out</ci><ci id="S3.SS1.SSS1.9.p9.1.m1.2.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.1.m1.2.2.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.1.m1.2c">P=\overline{\operatorname*{out}(\gamma)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.1.m1.2d">italic_P = over¯ start_ARG roman_out ( italic_γ ) end_ARG</annotation></semantics></math>, i.e. <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.2.m2.1"><semantics id="S3.SS1.SSS1.9.p9.2.m2.1a"><mi id="S3.SS1.SSS1.9.p9.2.m2.1.1" xref="S3.SS1.SSS1.9.p9.2.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.2.m2.1b"><ci id="S3.SS1.SSS1.9.p9.2.m2.1.1.cmml" xref="S3.SS1.SSS1.9.p9.2.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.2.m2.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.2.m2.1d">italic_γ</annotation></semantics></math> is a hole and <math alttext="n_{h}(P)=1" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.3.m3.1"><semantics id="S3.SS1.SSS1.9.p9.3.m3.1a"><mrow id="S3.SS1.SSS1.9.p9.3.m3.1.2" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.3.m3.1.2.2" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.cmml"><msub id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.cmml"><mi id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.2" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.3" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.1" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.3.2" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.cmml"><mo id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.3.m3.1.1" xref="S3.SS1.SSS1.9.p9.3.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.3.m3.1.2.1" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS1.9.p9.3.m3.1.2.3" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.3.m3.1b"><apply id="S3.SS1.SSS1.9.p9.3.m3.1.2.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2"><eq id="S3.SS1.SSS1.9.p9.3.m3.1.2.1.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.1"></eq><apply id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2"><times id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.1"></times><apply id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.2.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS1.9.p9.3.m3.1.1.cmml" xref="S3.SS1.SSS1.9.p9.3.m3.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS1.9.p9.3.m3.1.2.3.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.3.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.3.m3.1c">n_{h}(P)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = 1</annotation></semantics></math>. Then, <math alttext="P^{c}:=\overline{\mathds{R}^{2}\setminus P}=\overline{\operatorname*{int}(% \gamma)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.4.m4.2"><semantics id="S3.SS1.SSS1.9.p9.4.m4.2a"><mrow id="S3.SS1.SSS1.9.p9.4.m4.2.3" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.cmml"><msup id="S3.SS1.SSS1.9.p9.4.m4.2.3.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2.cmml"><mi id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.3" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS1.9.p9.4.m4.2.3.3" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.3.cmml">:=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.4.m4.2.3.4" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.cmml"><mrow id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.cmml"><msup id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.cmml"><mi id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.2.cmml">ℝ</mi><mn id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.3" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.3.cmml">2</mn></msup><mo id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.1" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.1.cmml">∖</mo><mi id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.3" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.3.cmml">P</mi></mrow><mo id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.1" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.1.cmml">¯</mo></mover><mo id="S3.SS1.SSS1.9.p9.4.m4.2.3.5" rspace="0.1389em" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.5.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.4.m4.2.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.4" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.4.m4.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS1.9.p9.4.m4.1.1.1.1.cmml">int</mo><mrow id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.4.1" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.4.1.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.2" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.2.cmml">γ</mi><mo id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.4.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.4.m4.2.2.3" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.4.m4.2b"><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3"><and id="S3.SS1.SSS1.9.p9.4.m4.2.3a.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3"></and><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3b.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3"><csymbol cd="latexml" id="S3.SS1.SSS1.9.p9.4.m4.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.3">assign</csymbol><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.4.m4.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.2.3">𝑐</ci></apply><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4"><ci id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.1.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.1">¯</ci><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2"><setdiff id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.1.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.1"></setdiff><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.2">ℝ</ci><cn id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.3.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.2.3">2</cn></apply><ci id="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.4.2.3">𝑃</ci></apply></apply></apply><apply id="S3.SS1.SSS1.9.p9.4.m4.2.3c.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3"><eq id="S3.SS1.SSS1.9.p9.4.m4.2.3.5.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS1.9.p9.4.m4.2.3.4.cmml" id="S3.SS1.SSS1.9.p9.4.m4.2.3d.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.3"></share><apply id="S3.SS1.SSS1.9.p9.4.m4.2.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.2"><ci id="S3.SS1.SSS1.9.p9.4.m4.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.3">¯</ci><apply id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.4"><ci id="S3.SS1.SSS1.9.p9.4.m4.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.1.1.1.1">int</ci><ci id="S3.SS1.SSS1.9.p9.4.m4.2.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.4.m4.2.2.2.2">𝛾</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.4.m4.2c">P^{c}:=\overline{\mathds{R}^{2}\setminus P}=\overline{\operatorname*{int}(% \gamma)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.4.m4.2d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT := over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_P end_ARG = over¯ start_ARG roman_int ( italic_γ ) end_ARG</annotation></semantics></math> is a polygon with <math alttext="\partial P^{c}=\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.5.m5.1"><semantics id="S3.SS1.SSS1.9.p9.5.m5.1a"><mrow id="S3.SS1.SSS1.9.p9.5.m5.1.1" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.cmml"><mrow id="S3.SS1.SSS1.9.p9.5.m5.1.1.2" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.cmml"><mo id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.1" rspace="0em" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.1.cmml">∂</mo><msup id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.cmml"><mi id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.2" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.3" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.3.cmml">c</mi></msup></mrow><mo id="S3.SS1.SSS1.9.p9.5.m5.1.1.1" rspace="0.1389em" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS1.9.p9.5.m5.1.1.3" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3.cmml"><mo id="S3.SS1.SSS1.9.p9.5.m5.1.1.3.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3.1.cmml">∂</mo><mi id="S3.SS1.SSS1.9.p9.5.m5.1.1.3.2" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3.2.cmml">P</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.5.m5.1b"><apply id="S3.SS1.SSS1.9.p9.5.m5.1.1.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1"><eq id="S3.SS1.SSS1.9.p9.5.m5.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.1"></eq><apply id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2"><partialdiff id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.1"></partialdiff><apply id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.2.2.3">𝑐</ci></apply></apply><apply id="S3.SS1.SSS1.9.p9.5.m5.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3"><partialdiff id="S3.SS1.SSS1.9.p9.5.m5.1.1.3.1.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3.1"></partialdiff><ci id="S3.SS1.SSS1.9.p9.5.m5.1.1.3.2.cmml" xref="S3.SS1.SSS1.9.p9.5.m5.1.1.3.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.5.m5.1c">\partial P^{c}=\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.5.m5.1d">∂ italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = ∂ italic_P</annotation></semantics></math>, i.e. <math alttext="V(P^{c})=V(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.6.m6.2"><semantics id="S3.SS1.SSS1.9.p9.6.m6.2a"><mrow id="S3.SS1.SSS1.9.p9.6.m6.2.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.6.m6.2.2.1" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.cmml"><mi id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.3" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.3.cmml">V</mi><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.3" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.3.cmml">c</mi></msup><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS1.9.p9.6.m6.2.2.3" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.1" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.3.2" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.6.m6.1.1" xref="S3.SS1.SSS1.9.p9.6.m6.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.6.m6.2b"><apply id="S3.SS1.SSS1.9.p9.6.m6.2.2.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2"><eq id="S3.SS1.SSS1.9.p9.6.m6.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.2"></eq><apply id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1"><times id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.2.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.2"></times><ci id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.3.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.3">𝑉</ci><apply id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.1.1.1.1.3">𝑐</ci></apply></apply><apply id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3"><times id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.1"></times><ci id="S3.SS1.SSS1.9.p9.6.m6.2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.2.2.3.2">𝑉</ci><ci id="S3.SS1.SSS1.9.p9.6.m6.1.1.cmml" xref="S3.SS1.SSS1.9.p9.6.m6.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.6.m6.2c">V(P^{c})=V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.6.m6.2d">italic_V ( italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) = italic_V ( italic_P )</annotation></semantics></math> and <math alttext="E(P^{c})=E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.7.m7.2"><semantics id="S3.SS1.SSS1.9.p9.7.m7.2a"><mrow id="S3.SS1.SSS1.9.p9.7.m7.2.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.7.m7.2.2.1" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.cmml"><mi id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.3" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.3.cmml">E</mi><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.3" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.3.cmml">c</mi></msup><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS1.9.p9.7.m7.2.2.3" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.2.cmml">E</mi><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.1" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.3.2" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.7.m7.1.1" xref="S3.SS1.SSS1.9.p9.7.m7.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.7.m7.2b"><apply id="S3.SS1.SSS1.9.p9.7.m7.2.2.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2"><eq id="S3.SS1.SSS1.9.p9.7.m7.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.2"></eq><apply id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1"><times id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.2.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.2"></times><ci id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.3.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.3">𝐸</ci><apply id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.1.1.1.1.3">𝑐</ci></apply></apply><apply id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3"><times id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.1"></times><ci id="S3.SS1.SSS1.9.p9.7.m7.2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.2.2.3.2">𝐸</ci><ci id="S3.SS1.SSS1.9.p9.7.m7.1.1.cmml" xref="S3.SS1.SSS1.9.p9.7.m7.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.7.m7.2c">E(P^{c})=E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.7.m7.2d">italic_E ( italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) = italic_E ( italic_P )</annotation></semantics></math>. By the definition of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.8.m8.1"><semantics id="S3.SS1.SSS1.9.p9.8.m8.1a"><mi id="S3.SS1.SSS1.9.p9.8.m8.1.1" xref="S3.SS1.SSS1.9.p9.8.m8.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.8.m8.1b"><ci id="S3.SS1.SSS1.9.p9.8.m8.1.1.cmml" xref="S3.SS1.SSS1.9.p9.8.m8.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.8.m8.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.8.m8.1d">italic_P</annotation></semantics></math>-sides, we have <math alttext="Q_{P^{c}}^{v}=\overline{\mathds{R}^{2}\setminus Q^{v}_{P}}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.9.m9.1"><semantics id="S3.SS1.SSS1.9.p9.9.m9.1a"><mrow id="S3.SS1.SSS1.9.p9.9.m9.1.1" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.cmml"><msubsup id="S3.SS1.SSS1.9.p9.9.m9.1.1.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.cmml"><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.2.cmml">Q</mi><msup id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.3.cmml">c</mi></msup><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.3.cmml">v</mi></msubsup><mo id="S3.SS1.SSS1.9.p9.9.m9.1.1.1" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.1.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.9.m9.1.1.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.cmml"><mrow id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.cmml"><msup id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.cmml"><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.2.cmml">ℝ</mi><mn id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.1" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.1.cmml">∖</mo><msubsup id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.2" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.3.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.3" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.3.cmml">v</mi></msubsup></mrow><mo id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.1" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.9.m9.1b"><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1"><eq id="S3.SS1.SSS1.9.p9.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.1"></eq><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.2">𝑄</ci><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.2.3.3">𝑐</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.2.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.2.3">𝑣</ci></apply><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3"><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.1">¯</ci><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2"><setdiff id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.1"></setdiff><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.2">ℝ</ci><cn id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.3.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.2.3">2</cn></apply><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3">subscript</csymbol><apply id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.2">𝑄</ci><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.2.3">𝑣</ci></apply><ci id="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.9.m9.1.1.3.2.3.3">𝑃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.9.m9.1c">Q_{P^{c}}^{v}=\overline{\mathds{R}^{2}\setminus Q^{v}_{P}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.9.m9.1d">italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT = over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> and <math alttext="H_{P^{c}}^{e}=\overline{\mathds{R}^{2}\setminus H^{e}_{P}}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.10.m10.1"><semantics id="S3.SS1.SSS1.9.p9.10.m10.1a"><mrow id="S3.SS1.SSS1.9.p9.10.m10.1.1" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.cmml"><msubsup id="S3.SS1.SSS1.9.p9.10.m10.1.1.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.cmml"><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.2.cmml">H</mi><msup id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.3.cmml">c</mi></msup><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.3.cmml">e</mi></msubsup><mo id="S3.SS1.SSS1.9.p9.10.m10.1.1.1" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.1.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.10.m10.1.1.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.cmml"><mrow id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.cmml"><msup id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.cmml"><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.2.cmml">ℝ</mi><mn id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.1" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.1.cmml">∖</mo><msubsup id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.2" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.2.cmml">H</mi><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.3.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.3" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.3.cmml">e</mi></msubsup></mrow><mo id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.1" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.10.m10.1b"><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1"><eq id="S3.SS1.SSS1.9.p9.10.m10.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.1"></eq><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.2">𝐻</ci><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.2.3.3">𝑐</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.2.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.2.3">𝑒</ci></apply><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3"><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.1">¯</ci><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2"><setdiff id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.1"></setdiff><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.2">ℝ</ci><cn id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.3.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.2.3">2</cn></apply><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3">subscript</csymbol><apply id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.2">𝐻</ci><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.2.3">𝑒</ci></apply><ci id="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.10.m10.1.1.3.2.3.3">𝑃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.10.m10.1c">H_{P^{c}}^{e}=\overline{\mathds{R}^{2}\setminus H^{e}_{P}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.10.m10.1d">italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT = over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. Since <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.11.m11.1"><semantics id="S3.SS1.SSS1.9.p9.11.m11.1a"><mi id="S3.SS1.SSS1.9.p9.11.m11.1.1" xref="S3.SS1.SSS1.9.p9.11.m11.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.11.m11.1b"><ci id="S3.SS1.SSS1.9.p9.11.m11.1.1.cmml" xref="S3.SS1.SSS1.9.p9.11.m11.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.11.m11.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.11.m11.1d">italic_x</annotation></semantics></math> is assumed to be in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.12.m12.1"><semantics id="S3.SS1.SSS1.9.p9.12.m12.1a"><mi id="S3.SS1.SSS1.9.p9.12.m12.1.1" xref="S3.SS1.SSS1.9.p9.12.m12.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.12.m12.1b"><ci id="S3.SS1.SSS1.9.p9.12.m12.1.1.cmml" xref="S3.SS1.SSS1.9.p9.12.m12.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.12.m12.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.12.m12.1d">italic_P</annotation></semantics></math>-general position, this implies <math alttext="\mathds{1}_{Q^{v}_{P}}(x)=1-\mathds{1}_{Q_{P^{c}}^{v}}(x)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.13.m13.2"><semantics id="S3.SS1.SSS1.9.p9.13.m13.2a"><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.cmml"><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.cmml"><msub id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.cmml"><mn id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.3.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.3.cmml">v</mi></msubsup></msub><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.1" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.cmml"><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.13.m13.1.1" xref="S3.SS1.SSS1.9.p9.13.m13.1.1.cmml">x</mi><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.1" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.cmml"><mn id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.1" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.1.cmml">−</mo><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.cmml"><msub id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.cmml"><mn id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.2.cmml">Q</mi><msup id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.3.cmml">c</mi></msup><mi id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.3" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.3.cmml">v</mi></msubsup></msub><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.1" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.3.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.cmml"><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.13.m13.2.2" xref="S3.SS1.SSS1.9.p9.13.m13.2.2.cmml">x</mi><mo id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.13.m13.2b"><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3"><eq id="S3.SS1.SSS1.9.p9.13.m13.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.1"></eq><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2"><times id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.1"></times><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2">subscript</csymbol><cn id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.2.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.2">1</cn><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3">subscript</csymbol><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.2">𝑄</ci><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.2.3">𝑣</ci></apply><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.2.2.3.3">𝑃</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.13.m13.1.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.1.1">𝑥</ci></apply><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3"><minus id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.1"></minus><cn id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.2">1</cn><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3"><times id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.1"></times><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2">subscript</csymbol><cn id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.2.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.2">1</cn><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3">superscript</csymbol><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3">subscript</csymbol><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.2">𝑄</ci><apply id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.2.3.3">𝑐</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.3.3.3.2.3.3">𝑣</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.13.m13.2.2.cmml" xref="S3.SS1.SSS1.9.p9.13.m13.2.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.13.m13.2c">\mathds{1}_{Q^{v}_{P}}(x)=1-\mathds{1}_{Q_{P^{c}}^{v}}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.13.m13.2d">blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = 1 - blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> and <math alttext="\mathds{1}_{H^{e}_{P}}(x)=1-\mathds{1}_{H_{P^{c}}^{e}}(x)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.14.m14.2"><semantics id="S3.SS1.SSS1.9.p9.14.m14.2a"><mrow id="S3.SS1.SSS1.9.p9.14.m14.2.3" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.cmml"><mrow id="S3.SS1.SSS1.9.p9.14.m14.2.3.2" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.cmml"><msub id="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.cmml"><mn id="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.2" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3.cmml"><mi id="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3.2.2" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3.2.2.cmml">H</mi><mi id="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3.3" xref="S3.SS1.SSS1.9.p9.14.m14.2.3.2.2.3.3.cmml">P</mi><mi 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xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.1"></times><apply id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2">subscript</csymbol><cn id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.2.cmml" type="integer" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.2">1</cn><apply id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.3.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.3.3.3.2.3.3">𝑐</ci></apply></apply><ci id="S3.SS1.SSS1.9.p9.15.m15.2.2.cmml" xref="S3.SS1.SSS1.9.p9.15.m15.2.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.15.m15.2c">\mathds{1}_{P}(x)=1-\mathds{1}_{P^{c}}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.15.m15.2d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) = 1 - blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. Therefore,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx6"> <tbody id="S3.Ex21"><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 v_{P}(x)-e_{P}(x)" class="ltx_Math" display="inline" id="S3.Ex21.m1.2"><semantics id="S3.Ex21.m1.2a"><mrow id="S3.Ex21.m1.2.3" xref="S3.Ex21.m1.2.3.cmml"><mrow id="S3.Ex21.m1.2.3.2" xref="S3.Ex21.m1.2.3.2.cmml"><msub id="S3.Ex21.m1.2.3.2.2" xref="S3.Ex21.m1.2.3.2.2.cmml"><mi id="S3.Ex21.m1.2.3.2.2.2" xref="S3.Ex21.m1.2.3.2.2.2.cmml">v</mi><mi id="S3.Ex21.m1.2.3.2.2.3" xref="S3.Ex21.m1.2.3.2.2.3.cmml">P</mi></msub><mo id="S3.Ex21.m1.2.3.2.1" xref="S3.Ex21.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex21.m1.2.3.2.3.2" xref="S3.Ex21.m1.2.3.2.cmml"><mo id="S3.Ex21.m1.2.3.2.3.2.1" stretchy="false" xref="S3.Ex21.m1.2.3.2.cmml">(</mo><mi id="S3.Ex21.m1.1.1" xref="S3.Ex21.m1.1.1.cmml">x</mi><mo id="S3.Ex21.m1.2.3.2.3.2.2" stretchy="false" xref="S3.Ex21.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex21.m1.2.3.1" xref="S3.Ex21.m1.2.3.1.cmml">−</mo><mrow id="S3.Ex21.m1.2.3.3" xref="S3.Ex21.m1.2.3.3.cmml"><msub id="S3.Ex21.m1.2.3.3.2" xref="S3.Ex21.m1.2.3.3.2.cmml"><mi id="S3.Ex21.m1.2.3.3.2.2" xref="S3.Ex21.m1.2.3.3.2.2.cmml">e</mi><mi id="S3.Ex21.m1.2.3.3.2.3" xref="S3.Ex21.m1.2.3.3.2.3.cmml">P</mi></msub><mo id="S3.Ex21.m1.2.3.3.1" xref="S3.Ex21.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.Ex21.m1.2.3.3.3.2" xref="S3.Ex21.m1.2.3.3.cmml"><mo id="S3.Ex21.m1.2.3.3.3.2.1" stretchy="false" xref="S3.Ex21.m1.2.3.3.cmml">(</mo><mi id="S3.Ex21.m1.2.2" xref="S3.Ex21.m1.2.2.cmml">x</mi><mo id="S3.Ex21.m1.2.3.3.3.2.2" stretchy="false" xref="S3.Ex21.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex21.m1.2b"><apply id="S3.Ex21.m1.2.3.cmml" xref="S3.Ex21.m1.2.3"><minus id="S3.Ex21.m1.2.3.1.cmml" xref="S3.Ex21.m1.2.3.1"></minus><apply id="S3.Ex21.m1.2.3.2.cmml" xref="S3.Ex21.m1.2.3.2"><times id="S3.Ex21.m1.2.3.2.1.cmml" xref="S3.Ex21.m1.2.3.2.1"></times><apply id="S3.Ex21.m1.2.3.2.2.cmml" xref="S3.Ex21.m1.2.3.2.2"><csymbol cd="ambiguous" id="S3.Ex21.m1.2.3.2.2.1.cmml" xref="S3.Ex21.m1.2.3.2.2">subscript</csymbol><ci id="S3.Ex21.m1.2.3.2.2.2.cmml" xref="S3.Ex21.m1.2.3.2.2.2">𝑣</ci><ci id="S3.Ex21.m1.2.3.2.2.3.cmml" xref="S3.Ex21.m1.2.3.2.2.3">𝑃</ci></apply><ci id="S3.Ex21.m1.1.1.cmml" xref="S3.Ex21.m1.1.1">𝑥</ci></apply><apply id="S3.Ex21.m1.2.3.3.cmml" xref="S3.Ex21.m1.2.3.3"><times id="S3.Ex21.m1.2.3.3.1.cmml" xref="S3.Ex21.m1.2.3.3.1"></times><apply id="S3.Ex21.m1.2.3.3.2.cmml" xref="S3.Ex21.m1.2.3.3.2"><csymbol cd="ambiguous" id="S3.Ex21.m1.2.3.3.2.1.cmml" xref="S3.Ex21.m1.2.3.3.2">subscript</csymbol><ci id="S3.Ex21.m1.2.3.3.2.2.cmml" xref="S3.Ex21.m1.2.3.3.2.2">𝑒</ci><ci id="S3.Ex21.m1.2.3.3.2.3.cmml" xref="S3.Ex21.m1.2.3.3.2.3">𝑃</ci></apply><ci id="S3.Ex21.m1.2.2.cmml" xref="S3.Ex21.m1.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex21.m1.2c">\displaystyle v_{P}(x)-e_{P}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex21.m1.2d">italic_v start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}% \mathds{1}_{H^{e}_{P}}(x)" class="ltx_Math" display="inline" id="S3.Ex21.m2.4"><semantics id="S3.Ex21.m2.4a"><mrow id="S3.Ex21.m2.4.5" xref="S3.Ex21.m2.4.5.cmml"><mi id="S3.Ex21.m2.4.5.2" xref="S3.Ex21.m2.4.5.2.cmml"></mi><mo id="S3.Ex21.m2.4.5.1" xref="S3.Ex21.m2.4.5.1.cmml">=</mo><mrow id="S3.Ex21.m2.4.5.3" xref="S3.Ex21.m2.4.5.3.cmml"><mrow id="S3.Ex21.m2.4.5.3.2" xref="S3.Ex21.m2.4.5.3.2.cmml"><mstyle displaystyle="true" 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italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex22"><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=\sum_{v\in V(P)}(1-\mathds{1}_{Q_{P^{c}}^{v}}(x))-\sum_{e\in E_{% b}(P)}(1-\mathds{1}_{H_{P^{c}}^{e}}(x))" class="ltx_Math" display="inline" 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start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT ( 1 - blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex23"><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=-\Big{(}\sum_{v\in V(P)}\mathds{1}_{Q_{P^{c}}^{v}}(x)-\sum_{e\in E% _{b}(P)}\mathds{1}_{H_{P^{c}}^{e}}(x)\Big{)}" class="ltx_Math" display="inline" id="S3.Ex23.m1.5"><semantics id="S3.Ex23.m1.5a"><mrow 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xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.1"></times><apply id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.1.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2">subscript</csymbol><cn id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.2.cmml" type="integer" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.2">1</cn><apply id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.1.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3">superscript</csymbol><apply id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.1.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3">subscript</csymbol><ci id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.2.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.2">𝐻</ci><apply id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3"><csymbol cd="ambiguous" id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.1.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3">superscript</csymbol><ci id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.2.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.2">𝑃</ci><ci id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.3.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.2.3.3">𝑐</ci></apply></apply><ci id="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.3.cmml" xref="S3.Ex23.m1.5.5.1.1.1.1.3.2.2.3.3">𝑒</ci></apply></apply><ci id="S3.Ex23.m1.4.4.cmml" xref="S3.Ex23.m1.4.4">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex23.m1.5c">\displaystyle=-\Big{(}\sum_{v\in V(P)}\mathds{1}_{Q_{P^{c}}^{v}}(x)-\sum_{e\in E% _{b}(P)}\mathds{1}_{H_{P^{c}}^{e}}(x)\Big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex23.m1.5d">= - ( ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.9.p9.18">where we have used that <math alttext="|V(P)|=|E(P)|" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.16.m1.4"><semantics id="S3.SS1.SSS1.9.p9.16.m1.4a"><mrow id="S3.SS1.SSS1.9.p9.16.m1.4.4" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.cmml"><mrow id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.2.cmml"><mo id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.cmml"><mi id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.2" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.2.cmml">V</mi><mo id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.3.2" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.cmml"><mo id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.16.m1.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.3" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.2.1.cmml">|</mo></mrow><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.3" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.3.cmml">=</mo><mrow id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.2.cmml"><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.2.1.cmml">|</mo><mrow id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.cmml"><mi id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.2" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.2.cmml">E</mi><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.1" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.3.2" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.cmml"><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.3.2.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.16.m1.2.2" xref="S3.SS1.SSS1.9.p9.16.m1.2.2.cmml">P</mi><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.3.2.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.3" stretchy="false" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.16.m1.4b"><apply id="S3.SS1.SSS1.9.p9.16.m1.4.4.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4"><eq id="S3.SS1.SSS1.9.p9.16.m1.4.4.3.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.3"></eq><apply id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.2.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1"><abs id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.2.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.2"></abs><apply id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1"><times id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.1"></times><ci id="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.3.3.1.1.1.2">𝑉</ci><ci id="S3.SS1.SSS1.9.p9.16.m1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.1.1">𝑃</ci></apply></apply><apply id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.2.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1"><abs id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.2.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.2"></abs><apply id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1"><times id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.1"></times><ci id="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.4.4.2.1.1.2">𝐸</ci><ci id="S3.SS1.SSS1.9.p9.16.m1.2.2.cmml" xref="S3.SS1.SSS1.9.p9.16.m1.2.2">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.16.m1.4c">|V(P)|=|E(P)|</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.16.m1.4d">| italic_V ( italic_P ) | = | italic_E ( italic_P ) |</annotation></semantics></math> in the last step. Since <math alttext="P^{c}=\overline{\operatorname*{int}(\gamma)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.17.m2.2"><semantics id="S3.SS1.SSS1.9.p9.17.m2.2a"><mrow id="S3.SS1.SSS1.9.p9.17.m2.2.3" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.cmml"><msup id="S3.SS1.SSS1.9.p9.17.m2.2.3.2" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2.cmml"><mi id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.2" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.3" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS1.9.p9.17.m2.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.1.cmml">=</mo><mover accent="true" id="S3.SS1.SSS1.9.p9.17.m2.2.2" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.cmml"><mrow id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.4" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.17.m2.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS1.9.p9.17.m2.1.1.1.1.cmml">int</mo><mrow id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.4.1" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.4.1.1" stretchy="false" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.2" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.2.cmml">γ</mi><mo id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.4.1.2" stretchy="false" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.9.p9.17.m2.2.2.3" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.17.m2.2b"><apply id="S3.SS1.SSS1.9.p9.17.m2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3"><eq id="S3.SS1.SSS1.9.p9.17.m2.2.3.1.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.1"></eq><apply id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.1.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.2.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.17.m2.2.3.2.3.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.3.2.3">𝑐</ci></apply><apply id="S3.SS1.SSS1.9.p9.17.m2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.2"><ci id="S3.SS1.SSS1.9.p9.17.m2.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.3">¯</ci><apply id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.3.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.4"><ci id="S3.SS1.SSS1.9.p9.17.m2.1.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.1.1.1.1">int</ci><ci id="S3.SS1.SSS1.9.p9.17.m2.2.2.2.2.cmml" xref="S3.SS1.SSS1.9.p9.17.m2.2.2.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.17.m2.2c">P^{c}=\overline{\operatorname*{int}(\gamma)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.17.m2.2d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = over¯ start_ARG roman_int ( italic_γ ) end_ARG</annotation></semantics></math>, we can apply the previous case to <math alttext="P^{c}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.9.p9.18.m3.1"><semantics id="S3.SS1.SSS1.9.p9.18.m3.1a"><msup id="S3.SS1.SSS1.9.p9.18.m3.1.1" xref="S3.SS1.SSS1.9.p9.18.m3.1.1.cmml"><mi id="S3.SS1.SSS1.9.p9.18.m3.1.1.2" xref="S3.SS1.SSS1.9.p9.18.m3.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS1.9.p9.18.m3.1.1.3" xref="S3.SS1.SSS1.9.p9.18.m3.1.1.3.cmml">c</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.9.p9.18.m3.1b"><apply id="S3.SS1.SSS1.9.p9.18.m3.1.1.cmml" xref="S3.SS1.SSS1.9.p9.18.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.9.p9.18.m3.1.1.1.cmml" xref="S3.SS1.SSS1.9.p9.18.m3.1.1">superscript</csymbol><ci id="S3.SS1.SSS1.9.p9.18.m3.1.1.2.cmml" xref="S3.SS1.SSS1.9.p9.18.m3.1.1.2">𝑃</ci><ci id="S3.SS1.SSS1.9.p9.18.m3.1.1.3.cmml" xref="S3.SS1.SSS1.9.p9.18.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.9.p9.18.m3.1c">P^{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.9.p9.18.m3.1d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math>, and get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx7"> <tbody id="S3.Ex24"><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 v_{P}(x)-e_{P}(x)-n_{h}(P)=1-\mathds{1}_{P^{c}}(x)-1=\mathds{1}_% {P}(x)-1." class="ltx_Math" display="inline" id="S3.Ex24.m1.6"><semantics id="S3.Ex24.m1.6a"><mrow id="S3.Ex24.m1.6.6.1" xref="S3.Ex24.m1.6.6.1.1.cmml"><mrow id="S3.Ex24.m1.6.6.1.1" xref="S3.Ex24.m1.6.6.1.1.cmml"><mrow id="S3.Ex24.m1.6.6.1.1.2" xref="S3.Ex24.m1.6.6.1.1.2.cmml"><mrow id="S3.Ex24.m1.6.6.1.1.2.2" xref="S3.Ex24.m1.6.6.1.1.2.2.cmml"><msub id="S3.Ex24.m1.6.6.1.1.2.2.2" xref="S3.Ex24.m1.6.6.1.1.2.2.2.cmml"><mi id="S3.Ex24.m1.6.6.1.1.2.2.2.2" xref="S3.Ex24.m1.6.6.1.1.2.2.2.2.cmml">v</mi><mi id="S3.Ex24.m1.6.6.1.1.2.2.2.3" xref="S3.Ex24.m1.6.6.1.1.2.2.2.3.cmml">P</mi></msub><mo id="S3.Ex24.m1.6.6.1.1.2.2.1" xref="S3.Ex24.m1.6.6.1.1.2.2.1.cmml">⁢</mo><mrow id="S3.Ex24.m1.6.6.1.1.2.2.3.2" xref="S3.Ex24.m1.6.6.1.1.2.2.cmml"><mo id="S3.Ex24.m1.6.6.1.1.2.2.3.2.1" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.2.cmml">(</mo><mi id="S3.Ex24.m1.1.1" xref="S3.Ex24.m1.1.1.cmml">x</mi><mo id="S3.Ex24.m1.6.6.1.1.2.2.3.2.2" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.1.2.1" xref="S3.Ex24.m1.6.6.1.1.2.1.cmml">−</mo><mrow id="S3.Ex24.m1.6.6.1.1.2.3" xref="S3.Ex24.m1.6.6.1.1.2.3.cmml"><msub id="S3.Ex24.m1.6.6.1.1.2.3.2" xref="S3.Ex24.m1.6.6.1.1.2.3.2.cmml"><mi id="S3.Ex24.m1.6.6.1.1.2.3.2.2" xref="S3.Ex24.m1.6.6.1.1.2.3.2.2.cmml">e</mi><mi id="S3.Ex24.m1.6.6.1.1.2.3.2.3" xref="S3.Ex24.m1.6.6.1.1.2.3.2.3.cmml">P</mi></msub><mo id="S3.Ex24.m1.6.6.1.1.2.3.1" xref="S3.Ex24.m1.6.6.1.1.2.3.1.cmml">⁢</mo><mrow id="S3.Ex24.m1.6.6.1.1.2.3.3.2" xref="S3.Ex24.m1.6.6.1.1.2.3.cmml"><mo id="S3.Ex24.m1.6.6.1.1.2.3.3.2.1" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.3.cmml">(</mo><mi id="S3.Ex24.m1.2.2" xref="S3.Ex24.m1.2.2.cmml">x</mi><mo id="S3.Ex24.m1.6.6.1.1.2.3.3.2.2" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.1.2.1a" xref="S3.Ex24.m1.6.6.1.1.2.1.cmml">−</mo><mrow id="S3.Ex24.m1.6.6.1.1.2.4" xref="S3.Ex24.m1.6.6.1.1.2.4.cmml"><msub id="S3.Ex24.m1.6.6.1.1.2.4.2" xref="S3.Ex24.m1.6.6.1.1.2.4.2.cmml"><mi id="S3.Ex24.m1.6.6.1.1.2.4.2.2" xref="S3.Ex24.m1.6.6.1.1.2.4.2.2.cmml">n</mi><mi id="S3.Ex24.m1.6.6.1.1.2.4.2.3" xref="S3.Ex24.m1.6.6.1.1.2.4.2.3.cmml">h</mi></msub><mo id="S3.Ex24.m1.6.6.1.1.2.4.1" xref="S3.Ex24.m1.6.6.1.1.2.4.1.cmml">⁢</mo><mrow id="S3.Ex24.m1.6.6.1.1.2.4.3.2" xref="S3.Ex24.m1.6.6.1.1.2.4.cmml"><mo id="S3.Ex24.m1.6.6.1.1.2.4.3.2.1" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.4.cmml">(</mo><mi id="S3.Ex24.m1.3.3" xref="S3.Ex24.m1.3.3.cmml">P</mi><mo id="S3.Ex24.m1.6.6.1.1.2.4.3.2.2" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.2.4.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.1.3" xref="S3.Ex24.m1.6.6.1.1.3.cmml">=</mo><mrow id="S3.Ex24.m1.6.6.1.1.4" xref="S3.Ex24.m1.6.6.1.1.4.cmml"><mn id="S3.Ex24.m1.6.6.1.1.4.2" xref="S3.Ex24.m1.6.6.1.1.4.2.cmml">1</mn><mo id="S3.Ex24.m1.6.6.1.1.4.1" xref="S3.Ex24.m1.6.6.1.1.4.1.cmml">−</mo><mrow id="S3.Ex24.m1.6.6.1.1.4.3" xref="S3.Ex24.m1.6.6.1.1.4.3.cmml"><msub id="S3.Ex24.m1.6.6.1.1.4.3.2" xref="S3.Ex24.m1.6.6.1.1.4.3.2.cmml"><mn id="S3.Ex24.m1.6.6.1.1.4.3.2.2" xref="S3.Ex24.m1.6.6.1.1.4.3.2.2.cmml">𝟙</mn><msup id="S3.Ex24.m1.6.6.1.1.4.3.2.3" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3.cmml"><mi id="S3.Ex24.m1.6.6.1.1.4.3.2.3.2" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3.2.cmml">P</mi><mi id="S3.Ex24.m1.6.6.1.1.4.3.2.3.3" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3.3.cmml">c</mi></msup></msub><mo id="S3.Ex24.m1.6.6.1.1.4.3.1" xref="S3.Ex24.m1.6.6.1.1.4.3.1.cmml">⁢</mo><mrow id="S3.Ex24.m1.6.6.1.1.4.3.3.2" xref="S3.Ex24.m1.6.6.1.1.4.3.cmml"><mo id="S3.Ex24.m1.6.6.1.1.4.3.3.2.1" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.4.3.cmml">(</mo><mi id="S3.Ex24.m1.4.4" xref="S3.Ex24.m1.4.4.cmml">x</mi><mo id="S3.Ex24.m1.6.6.1.1.4.3.3.2.2" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.4.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.1.4.1a" xref="S3.Ex24.m1.6.6.1.1.4.1.cmml">−</mo><mn id="S3.Ex24.m1.6.6.1.1.4.4" xref="S3.Ex24.m1.6.6.1.1.4.4.cmml">1</mn></mrow><mo id="S3.Ex24.m1.6.6.1.1.5" xref="S3.Ex24.m1.6.6.1.1.5.cmml">=</mo><mrow id="S3.Ex24.m1.6.6.1.1.6" xref="S3.Ex24.m1.6.6.1.1.6.cmml"><mrow id="S3.Ex24.m1.6.6.1.1.6.2" xref="S3.Ex24.m1.6.6.1.1.6.2.cmml"><msub id="S3.Ex24.m1.6.6.1.1.6.2.2" xref="S3.Ex24.m1.6.6.1.1.6.2.2.cmml"><mn id="S3.Ex24.m1.6.6.1.1.6.2.2.2" xref="S3.Ex24.m1.6.6.1.1.6.2.2.2.cmml">𝟙</mn><mi id="S3.Ex24.m1.6.6.1.1.6.2.2.3" xref="S3.Ex24.m1.6.6.1.1.6.2.2.3.cmml">P</mi></msub><mo id="S3.Ex24.m1.6.6.1.1.6.2.1" xref="S3.Ex24.m1.6.6.1.1.6.2.1.cmml">⁢</mo><mrow id="S3.Ex24.m1.6.6.1.1.6.2.3.2" xref="S3.Ex24.m1.6.6.1.1.6.2.cmml"><mo id="S3.Ex24.m1.6.6.1.1.6.2.3.2.1" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.6.2.cmml">(</mo><mi id="S3.Ex24.m1.5.5" xref="S3.Ex24.m1.5.5.cmml">x</mi><mo id="S3.Ex24.m1.6.6.1.1.6.2.3.2.2" stretchy="false" xref="S3.Ex24.m1.6.6.1.1.6.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.1.6.1" xref="S3.Ex24.m1.6.6.1.1.6.1.cmml">−</mo><mn id="S3.Ex24.m1.6.6.1.1.6.3" xref="S3.Ex24.m1.6.6.1.1.6.3.cmml">1</mn></mrow></mrow><mo id="S3.Ex24.m1.6.6.1.2" lspace="0em" xref="S3.Ex24.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex24.m1.6b"><apply id="S3.Ex24.m1.6.6.1.1.cmml" xref="S3.Ex24.m1.6.6.1"><and id="S3.Ex24.m1.6.6.1.1a.cmml" xref="S3.Ex24.m1.6.6.1"></and><apply id="S3.Ex24.m1.6.6.1.1b.cmml" xref="S3.Ex24.m1.6.6.1"><eq id="S3.Ex24.m1.6.6.1.1.3.cmml" xref="S3.Ex24.m1.6.6.1.1.3"></eq><apply id="S3.Ex24.m1.6.6.1.1.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2"><minus id="S3.Ex24.m1.6.6.1.1.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.1"></minus><apply id="S3.Ex24.m1.6.6.1.1.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2"><times id="S3.Ex24.m1.6.6.1.1.2.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2.1"></times><apply id="S3.Ex24.m1.6.6.1.1.2.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.2.2.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2.2">subscript</csymbol><ci id="S3.Ex24.m1.6.6.1.1.2.2.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2.2.2">𝑣</ci><ci id="S3.Ex24.m1.6.6.1.1.2.2.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.2.2.2.3">𝑃</ci></apply><ci id="S3.Ex24.m1.1.1.cmml" xref="S3.Ex24.m1.1.1">𝑥</ci></apply><apply id="S3.Ex24.m1.6.6.1.1.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3"><times id="S3.Ex24.m1.6.6.1.1.2.3.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3.1"></times><apply id="S3.Ex24.m1.6.6.1.1.2.3.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3.2"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.2.3.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3.2">subscript</csymbol><ci id="S3.Ex24.m1.6.6.1.1.2.3.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3.2.2">𝑒</ci><ci id="S3.Ex24.m1.6.6.1.1.2.3.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.2.3.2.3">𝑃</ci></apply><ci id="S3.Ex24.m1.2.2.cmml" xref="S3.Ex24.m1.2.2">𝑥</ci></apply><apply id="S3.Ex24.m1.6.6.1.1.2.4.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4"><times id="S3.Ex24.m1.6.6.1.1.2.4.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4.1"></times><apply id="S3.Ex24.m1.6.6.1.1.2.4.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4.2"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.2.4.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4.2">subscript</csymbol><ci id="S3.Ex24.m1.6.6.1.1.2.4.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4.2.2">𝑛</ci><ci id="S3.Ex24.m1.6.6.1.1.2.4.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.2.4.2.3">ℎ</ci></apply><ci id="S3.Ex24.m1.3.3.cmml" xref="S3.Ex24.m1.3.3">𝑃</ci></apply></apply><apply id="S3.Ex24.m1.6.6.1.1.4.cmml" xref="S3.Ex24.m1.6.6.1.1.4"><minus id="S3.Ex24.m1.6.6.1.1.4.1.cmml" xref="S3.Ex24.m1.6.6.1.1.4.1"></minus><cn id="S3.Ex24.m1.6.6.1.1.4.2.cmml" type="integer" xref="S3.Ex24.m1.6.6.1.1.4.2">1</cn><apply id="S3.Ex24.m1.6.6.1.1.4.3.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3"><times id="S3.Ex24.m1.6.6.1.1.4.3.1.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.1"></times><apply id="S3.Ex24.m1.6.6.1.1.4.3.2.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.4.3.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2">subscript</csymbol><cn id="S3.Ex24.m1.6.6.1.1.4.3.2.2.cmml" type="integer" xref="S3.Ex24.m1.6.6.1.1.4.3.2.2">1</cn><apply id="S3.Ex24.m1.6.6.1.1.4.3.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.4.3.2.3.1.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3">superscript</csymbol><ci id="S3.Ex24.m1.6.6.1.1.4.3.2.3.2.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3.2">𝑃</ci><ci id="S3.Ex24.m1.6.6.1.1.4.3.2.3.3.cmml" xref="S3.Ex24.m1.6.6.1.1.4.3.2.3.3">𝑐</ci></apply></apply><ci id="S3.Ex24.m1.4.4.cmml" xref="S3.Ex24.m1.4.4">𝑥</ci></apply><cn id="S3.Ex24.m1.6.6.1.1.4.4.cmml" type="integer" xref="S3.Ex24.m1.6.6.1.1.4.4">1</cn></apply></apply><apply id="S3.Ex24.m1.6.6.1.1c.cmml" xref="S3.Ex24.m1.6.6.1"><eq id="S3.Ex24.m1.6.6.1.1.5.cmml" xref="S3.Ex24.m1.6.6.1.1.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S3.Ex24.m1.6.6.1.1.4.cmml" id="S3.Ex24.m1.6.6.1.1d.cmml" xref="S3.Ex24.m1.6.6.1"></share><apply id="S3.Ex24.m1.6.6.1.1.6.cmml" xref="S3.Ex24.m1.6.6.1.1.6"><minus id="S3.Ex24.m1.6.6.1.1.6.1.cmml" xref="S3.Ex24.m1.6.6.1.1.6.1"></minus><apply id="S3.Ex24.m1.6.6.1.1.6.2.cmml" xref="S3.Ex24.m1.6.6.1.1.6.2"><times id="S3.Ex24.m1.6.6.1.1.6.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.6.2.1"></times><apply id="S3.Ex24.m1.6.6.1.1.6.2.2.cmml" xref="S3.Ex24.m1.6.6.1.1.6.2.2"><csymbol cd="ambiguous" id="S3.Ex24.m1.6.6.1.1.6.2.2.1.cmml" xref="S3.Ex24.m1.6.6.1.1.6.2.2">subscript</csymbol><cn id="S3.Ex24.m1.6.6.1.1.6.2.2.2.cmml" type="integer" xref="S3.Ex24.m1.6.6.1.1.6.2.2.2">1</cn><ci id="S3.Ex24.m1.6.6.1.1.6.2.2.3.cmml" xref="S3.Ex24.m1.6.6.1.1.6.2.2.3">𝑃</ci></apply><ci id="S3.Ex24.m1.5.5.cmml" xref="S3.Ex24.m1.5.5">𝑥</ci></apply><cn id="S3.Ex24.m1.6.6.1.1.6.3.cmml" type="integer" xref="S3.Ex24.m1.6.6.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex24.m1.6c">\displaystyle v_{P}(x)-e_{P}(x)-n_{h}(P)=1-\mathds{1}_{P^{c}}(x)-1=\mathds{1}_% {P}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex24.m1.6d">italic_v start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_e start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = 1 - blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) - 1 = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS1.9.p9.19">This proves the remaining case of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E7" title="Equation 7 ‣ Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">7</span></a>). ∎</p> </div> </div> </section> <section class="ltx_subsubsection" id="S3.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.1.2 </span>Only arcs</h4> <div class="ltx_para" id="S3.SS1.SSS2.p1"> <p class="ltx_p" id="S3.SS1.SSS2.p1.1">In this section, I prove the following special case of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.4.4">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon whose boundary consists of <math alttext="n_{a}(P)\geq 1" 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.m2.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.cmml"><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.2.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.cmml"><msub id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.cmml"><mo id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem4.p1.2.2.m2.1.2.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.1.cmml">≥</mo><mn id="S3.Thmtheorem4.p1.2.2.m2.1.2.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2"><geq id="S3.Thmtheorem4.p1.2.2.m2.1.2.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.1"></geq><apply id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2"><times id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.1"></times><apply id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.2.2.3">𝑎</ci></apply><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1">𝑃</ci></apply><cn id="S3.Thmtheorem4.p1.2.2.m2.1.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.2.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">n_{a}(P)\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) ≥ 1</annotation></semantics></math> polygonal arcs. Let <math alttext="x\in\mathds{R}^{2}" 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">x</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">ℝ</mi><mn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.cmml">2</mn></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">𝑥</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">ℝ</ci><cn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> be an arbitrary point in <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">italic_P</annotation></semantics></math>-general position. Then, it holds that</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E17"> <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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)+\sum_{e\in E_{l}(P)}\mathds{1}_{H^{e% }_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P}}(x)-n_{a}(P)\\ =\mathds{1}_{P}(x)-1." class="ltx_Math" display="block" id="S3.E17.m1.40"><semantics id="S3.E17.m1.40a"><mtable displaystyle="true" id="S3.E17.m1.40.40.2" rowspacing="0pt"><mtr id="S3.E17.m1.40.40.2a"><mtd class="ltx_align_left" columnalign="left" id="S3.E17.m1.40.40.2b"><mrow id="S3.E17.m1.29.29.29.29.29"><mrow id="S3.E17.m1.29.29.29.29.29.30"><mrow id="S3.E17.m1.29.29.29.29.29.30.1"><munder id="S3.E17.m1.29.29.29.29.29.30.1.1"><mo id="S3.E17.m1.1.1.1.1.1.1" movablelimits="false" xref="S3.E17.m1.1.1.1.1.1.1.cmml">∑</mo><mrow id="S3.E17.m1.2.2.2.2.2.2.1" xref="S3.E17.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E17.m1.2.2.2.2.2.2.1.3" xref="S3.E17.m1.2.2.2.2.2.2.1.3.cmml">v</mi><mo id="S3.E17.m1.2.2.2.2.2.2.1.2" xref="S3.E17.m1.2.2.2.2.2.2.1.2.cmml">∈</mo><mrow id="S3.E17.m1.2.2.2.2.2.2.1.4" xref="S3.E17.m1.2.2.2.2.2.2.1.4.cmml"><mi id="S3.E17.m1.2.2.2.2.2.2.1.4.2" xref="S3.E17.m1.2.2.2.2.2.2.1.4.2.cmml">V</mi><mo id="S3.E17.m1.2.2.2.2.2.2.1.4.1" xref="S3.E17.m1.2.2.2.2.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.E17.m1.2.2.2.2.2.2.1.4.3.2" xref="S3.E17.m1.2.2.2.2.2.2.1.4.cmml"><mo id="S3.E17.m1.2.2.2.2.2.2.1.4.3.2.1" stretchy="false" xref="S3.E17.m1.2.2.2.2.2.2.1.4.cmml">(</mo><mi id="S3.E17.m1.2.2.2.2.2.2.1.1" xref="S3.E17.m1.2.2.2.2.2.2.1.1.cmml">P</mi><mo id="S3.E17.m1.2.2.2.2.2.2.1.4.3.2.2" stretchy="false" xref="S3.E17.m1.2.2.2.2.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E17.m1.29.29.29.29.29.30.1.2"><msub id="S3.E17.m1.29.29.29.29.29.30.1.2.2"><mn id="S3.E17.m1.3.3.3.3.3.3" xref="S3.E17.m1.3.3.3.3.3.3.cmml">𝟙</mn><msubsup 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E_{b}(P)}\mathds{1}_{H^{e}_{P}}(x)-n_{a}(P)\\ =\mathds{1}_{P}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.E17.m1.40d">start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) end_CELL end_ROW start_ROW start_CELL = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 . end_CELL end_ROW</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">(17)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.SSS2.p2"> <p class="ltx_p" id="S3.SS1.SSS2.p2.4">The boundary of any polygon <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.1.m1.1"><semantics id="S3.SS1.SSS2.p2.1.m1.1a"><mi id="S3.SS1.SSS2.p2.1.m1.1.1" xref="S3.SS1.SSS2.p2.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.1.m1.1b"><ci id="S3.SS1.SSS2.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS2.p2.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.1.m1.1d">italic_P</annotation></semantics></math> that satisfies the assumptions of the lemma contains no cycles and consequently no holes, i.e. <math alttext="n_{h}(P)=0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.2.m2.1"><semantics id="S3.SS1.SSS2.p2.2.m2.1a"><mrow id="S3.SS1.SSS2.p2.2.m2.1.2" xref="S3.SS1.SSS2.p2.2.m2.1.2.cmml"><mrow id="S3.SS1.SSS2.p2.2.m2.1.2.2" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.cmml"><msub id="S3.SS1.SSS2.p2.2.m2.1.2.2.2" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.2" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.3" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2.3.cmml">h</mi></msub><mo id="S3.SS1.SSS2.p2.2.m2.1.2.2.1" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p2.2.m2.1.2.2.3.2" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.cmml"><mo id="S3.SS1.SSS2.p2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.p2.2.m2.1.1" xref="S3.SS1.SSS2.p2.2.m2.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.p2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.p2.2.m2.1.2.1" xref="S3.SS1.SSS2.p2.2.m2.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.p2.2.m2.1.2.3" xref="S3.SS1.SSS2.p2.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.2.m2.1b"><apply id="S3.SS1.SSS2.p2.2.m2.1.2.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2"><eq id="S3.SS1.SSS2.p2.2.m2.1.2.1.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.1"></eq><apply id="S3.SS1.SSS2.p2.2.m2.1.2.2.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2"><times id="S3.SS1.SSS2.p2.2.m2.1.2.2.1.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.1"></times><apply id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.1.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.2.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.p2.2.m2.1.2.2.2.3.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.2.2.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS2.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS2.p2.2.m2.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.p2.2.m2.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.p2.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.2.m2.1c">n_{h}(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P ) = 0</annotation></semantics></math>. Moreover, as we will see in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.7</span></a>, we again have <math alttext="d(P)=0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.3.m3.1"><semantics id="S3.SS1.SSS2.p2.3.m3.1a"><mrow id="S3.SS1.SSS2.p2.3.m3.1.2" xref="S3.SS1.SSS2.p2.3.m3.1.2.cmml"><mrow id="S3.SS1.SSS2.p2.3.m3.1.2.2" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.cmml"><mi id="S3.SS1.SSS2.p2.3.m3.1.2.2.2" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.2.cmml">d</mi><mo id="S3.SS1.SSS2.p2.3.m3.1.2.2.1" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p2.3.m3.1.2.2.3.2" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.cmml"><mo id="S3.SS1.SSS2.p2.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.p2.3.m3.1.1" xref="S3.SS1.SSS2.p2.3.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.p2.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.p2.3.m3.1.2.1" xref="S3.SS1.SSS2.p2.3.m3.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.p2.3.m3.1.2.3" xref="S3.SS1.SSS2.p2.3.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.3.m3.1b"><apply id="S3.SS1.SSS2.p2.3.m3.1.2.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.2"><eq id="S3.SS1.SSS2.p2.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.2.1"></eq><apply id="S3.SS1.SSS2.p2.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.2.2"><times id="S3.SS1.SSS2.p2.3.m3.1.2.2.1.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.1"></times><ci id="S3.SS1.SSS2.p2.3.m3.1.2.2.2.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.2.2.2">𝑑</ci><ci id="S3.SS1.SSS2.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS2.p2.3.m3.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.p2.3.m3.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.p2.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.3.m3.1c">d(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.3.m3.1d">italic_d ( italic_P ) = 0</annotation></semantics></math>. This gives <math alttext="c(P)=1-n_{a}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.4.m4.2"><semantics id="S3.SS1.SSS2.p2.4.m4.2a"><mrow id="S3.SS1.SSS2.p2.4.m4.2.3" xref="S3.SS1.SSS2.p2.4.m4.2.3.cmml"><mrow id="S3.SS1.SSS2.p2.4.m4.2.3.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.2.3.2.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.2.cmml">c</mi><mo id="S3.SS1.SSS2.p2.4.m4.2.3.2.1" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p2.4.m4.2.3.2.3.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.cmml"><mo id="S3.SS1.SSS2.p2.4.m4.2.3.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.cmml">(</mo><mi id="S3.SS1.SSS2.p2.4.m4.1.1" xref="S3.SS1.SSS2.p2.4.m4.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.p2.4.m4.2.3.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.p2.4.m4.2.3.1" xref="S3.SS1.SSS2.p2.4.m4.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS2.p2.4.m4.2.3.3" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.cmml"><mn id="S3.SS1.SSS2.p2.4.m4.2.3.3.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS2.p2.4.m4.2.3.3.1" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.1.cmml">−</mo><mrow id="S3.SS1.SSS2.p2.4.m4.2.3.3.3" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.cmml"><msub id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.3" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.1" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.3.2" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.cmml"><mo id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.cmml">(</mo><mi id="S3.SS1.SSS2.p2.4.m4.2.2" xref="S3.SS1.SSS2.p2.4.m4.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.4.m4.2b"><apply id="S3.SS1.SSS2.p2.4.m4.2.3.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3"><eq id="S3.SS1.SSS2.p2.4.m4.2.3.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.1"></eq><apply id="S3.SS1.SSS2.p2.4.m4.2.3.2.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.2"><times id="S3.SS1.SSS2.p2.4.m4.2.3.2.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.1"></times><ci id="S3.SS1.SSS2.p2.4.m4.2.3.2.2.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.2.2">𝑐</ci><ci id="S3.SS1.SSS2.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS2.p2.4.m4.2.3.3.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3"><minus id="S3.SS1.SSS2.p2.4.m4.2.3.3.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.1"></minus><cn id="S3.SS1.SSS2.p2.4.m4.2.3.3.2.cmml" type="integer" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.2">1</cn><apply id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3"><times id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.1"></times><apply id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.1.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.2.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.2">𝑛</ci><ci id="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.3.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.3.3.3.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.p2.4.m4.2.2.cmml" xref="S3.SS1.SSS2.p2.4.m4.2.2">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.4.m4.2c">c(P)=1-n_{a}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.4.m4.2d">italic_c ( italic_P ) = 1 - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> and confirms that <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a> is a special case of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.p3"> <p class="ltx_p" id="S3.SS1.SSS2.p3.1">As the next lemma shows, it is sufficient to restrict attention to polygons that contain no lines as boundary components.</p> </div> <div class="ltx_theorem ltx_theorem_definition" 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">Definition 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.p1.12">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon, and let <math alttext="l\in E_{l}(P)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.m2.1a"><mrow id="S3.Thmtheorem5.p1.2.m2.1.2" xref="S3.Thmtheorem5.p1.2.m2.1.2.cmml"><mi id="S3.Thmtheorem5.p1.2.m2.1.2.2" xref="S3.Thmtheorem5.p1.2.m2.1.2.2.cmml">l</mi><mo id="S3.Thmtheorem5.p1.2.m2.1.2.1" xref="S3.Thmtheorem5.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem5.p1.2.m2.1.2.3" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.cmml"><msub id="S3.Thmtheorem5.p1.2.m2.1.2.3.2" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2.cmml"><mi id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.2" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2.2.cmml">E</mi><mi id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.3" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2.3.cmml">l</mi></msub><mo id="S3.Thmtheorem5.p1.2.m2.1.2.3.1" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.2.m2.1.2.3.3.2" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.cmml"><mo id="S3.Thmtheorem5.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem5.p1.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem5.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.m2.1.2.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2"><in id="S3.Thmtheorem5.p1.2.m2.1.2.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.1"></in><ci id="S3.Thmtheorem5.p1.2.m2.1.2.2.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.2">𝑙</ci><apply id="S3.Thmtheorem5.p1.2.m2.1.2.3.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3"><times id="S3.Thmtheorem5.p1.2.m2.1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.1"></times><apply id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.2.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2.2">𝐸</ci><ci id="S3.Thmtheorem5.p1.2.m2.1.2.3.2.3.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.2.3.2.3">𝑙</ci></apply><ci id="S3.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.m2.1c">l\in E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.m2.1d">italic_l ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> be a line. Choose two rays <math alttext="e_{1}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.m3.1a"><msub id="S3.Thmtheorem5.p1.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem5.p1.3.m3.1.1.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.3.m3.1.1.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.m3.1b"><apply id="S3.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.m3.1c">e_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.m3.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.m4.1a"><msub id="S3.Thmtheorem5.p1.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.cmml"><mi id="S3.Thmtheorem5.p1.4.m4.1.1.2" xref="S3.Thmtheorem5.p1.4.m4.1.1.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.4.m4.1.1.3" xref="S3.Thmtheorem5.p1.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.m4.1c">e_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.m4.1d">italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="e_{1}\cup e_{2}=l" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.m5.1a"><mrow id="S3.Thmtheorem5.p1.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.cmml"><mrow id="S3.Thmtheorem5.p1.5.m5.1.1.2" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.cmml"><msub id="S3.Thmtheorem5.p1.5.m5.1.1.2.2" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2.cmml"><mi id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.2" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.3" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S3.Thmtheorem5.p1.5.m5.1.1.2.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.1.cmml">∪</mo><msub id="S3.Thmtheorem5.p1.5.m5.1.1.2.3" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3.cmml"><mi id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.2" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.3" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S3.Thmtheorem5.p1.5.m5.1.1.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.1.cmml">=</mo><mi id="S3.Thmtheorem5.p1.5.m5.1.1.3" xref="S3.Thmtheorem5.p1.5.m5.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.m5.1b"><apply id="S3.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1"><eq id="S3.Thmtheorem5.p1.5.m5.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.1"></eq><apply id="S3.Thmtheorem5.p1.5.m5.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2"><union id="S3.Thmtheorem5.p1.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.1"></union><apply id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2">subscript</csymbol><ci id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.5.m5.1.1.2.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.2.3">1</cn></apply><apply id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3">subscript</csymbol><ci id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.5.m5.1.1.2.3.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.5.m5.1.1.2.3.3">2</cn></apply></apply><ci id="S3.Thmtheorem5.p1.5.m5.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.m5.1c">e_{1}\cup e_{2}=l</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.m5.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_l</annotation></semantics></math> and <math alttext="e_{1}\cap e_{2}=\{v\}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.6.m6.1"><semantics id="S3.Thmtheorem5.p1.6.m6.1a"><mrow id="S3.Thmtheorem5.p1.6.m6.1.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.cmml"><mrow id="S3.Thmtheorem5.p1.6.m6.1.2.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.cmml"><msub id="S3.Thmtheorem5.p1.6.m6.1.2.2.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2.cmml"><mi id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.3" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2.3.cmml">1</mn></msub><mo id="S3.Thmtheorem5.p1.6.m6.1.2.2.1" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.1.cmml">∩</mo><msub id="S3.Thmtheorem5.p1.6.m6.1.2.2.3" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3.cmml"><mi id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.3" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3.3.cmml">2</mn></msub></mrow><mo id="S3.Thmtheorem5.p1.6.m6.1.2.1" xref="S3.Thmtheorem5.p1.6.m6.1.2.1.cmml">=</mo><mrow id="S3.Thmtheorem5.p1.6.m6.1.2.3.2" xref="S3.Thmtheorem5.p1.6.m6.1.2.3.1.cmml"><mo id="S3.Thmtheorem5.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.6.m6.1.2.3.1.cmml">{</mo><mi id="S3.Thmtheorem5.p1.6.m6.1.1" xref="S3.Thmtheorem5.p1.6.m6.1.1.cmml">v</mi><mo id="S3.Thmtheorem5.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.6.m6.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.6.m6.1b"><apply id="S3.Thmtheorem5.p1.6.m6.1.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2"><eq id="S3.Thmtheorem5.p1.6.m6.1.2.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.1"></eq><apply id="S3.Thmtheorem5.p1.6.m6.1.2.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2"><intersect id="S3.Thmtheorem5.p1.6.m6.1.2.2.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.1"></intersect><apply id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.6.m6.1.2.2.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.2.3">1</cn></apply><apply id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3">subscript</csymbol><ci id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.2.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.6.m6.1.2.2.3.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.6.m6.1.2.2.3.3">2</cn></apply></apply><set id="S3.Thmtheorem5.p1.6.m6.1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.2.3.2"><ci id="S3.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem5.p1.6.m6.1.1">𝑣</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.m6.1c">e_{1}\cap e_{2}=\{v\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.m6.1d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { italic_v }</annotation></semantics></math> for some <math alttext="v\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.7.m7.1"><semantics id="S3.Thmtheorem5.p1.7.m7.1a"><mrow id="S3.Thmtheorem5.p1.7.m7.1.1" xref="S3.Thmtheorem5.p1.7.m7.1.1.cmml"><mi id="S3.Thmtheorem5.p1.7.m7.1.1.2" xref="S3.Thmtheorem5.p1.7.m7.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem5.p1.7.m7.1.1.1" xref="S3.Thmtheorem5.p1.7.m7.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem5.p1.7.m7.1.1.3" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.7.m7.1.1.3.2" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.2.cmml">ℝ</mi><mn id="S3.Thmtheorem5.p1.7.m7.1.1.3.3" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.7.m7.1b"><apply id="S3.Thmtheorem5.p1.7.m7.1.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1"><in id="S3.Thmtheorem5.p1.7.m7.1.1.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.1"></in><ci id="S3.Thmtheorem5.p1.7.m7.1.1.2.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.2">𝑣</ci><apply id="S3.Thmtheorem5.p1.7.m7.1.1.3.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.7.m7.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.7.m7.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.2">ℝ</ci><cn id="S3.Thmtheorem5.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.7.m7.1c">v\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.7.m7.1d">italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. The operation of redefining <math alttext="V(l):=\{v\}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.8.m8.2"><semantics id="S3.Thmtheorem5.p1.8.m8.2a"><mrow id="S3.Thmtheorem5.p1.8.m8.2.3" xref="S3.Thmtheorem5.p1.8.m8.2.3.cmml"><mrow id="S3.Thmtheorem5.p1.8.m8.2.3.2" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.cmml"><mi id="S3.Thmtheorem5.p1.8.m8.2.3.2.2" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.2.cmml">V</mi><mo id="S3.Thmtheorem5.p1.8.m8.2.3.2.1" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.8.m8.2.3.2.3.2" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.cmml"><mo id="S3.Thmtheorem5.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem5.p1.8.m8.1.1" xref="S3.Thmtheorem5.p1.8.m8.1.1.cmml">l</mi><mo id="S3.Thmtheorem5.p1.8.m8.2.3.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem5.p1.8.m8.2.3.1" rspace="0.278em" xref="S3.Thmtheorem5.p1.8.m8.2.3.1.cmml">:=</mo><mrow id="S3.Thmtheorem5.p1.8.m8.2.3.3.2" xref="S3.Thmtheorem5.p1.8.m8.2.3.3.1.cmml"><mo id="S3.Thmtheorem5.p1.8.m8.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.8.m8.2.3.3.1.cmml">{</mo><mi id="S3.Thmtheorem5.p1.8.m8.2.2" xref="S3.Thmtheorem5.p1.8.m8.2.2.cmml">v</mi><mo id="S3.Thmtheorem5.p1.8.m8.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.8.m8.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.8.m8.2b"><apply id="S3.Thmtheorem5.p1.8.m8.2.3.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.8.m8.2.3.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3.1">assign</csymbol><apply id="S3.Thmtheorem5.p1.8.m8.2.3.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3.2"><times id="S3.Thmtheorem5.p1.8.m8.2.3.2.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.1"></times><ci id="S3.Thmtheorem5.p1.8.m8.2.3.2.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3.2.2">𝑉</ci><ci id="S3.Thmtheorem5.p1.8.m8.1.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.1.1">𝑙</ci></apply><set id="S3.Thmtheorem5.p1.8.m8.2.3.3.1.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.3.3.2"><ci id="S3.Thmtheorem5.p1.8.m8.2.2.cmml" xref="S3.Thmtheorem5.p1.8.m8.2.2">𝑣</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.8.m8.2c">V(l):=\{v\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.8.m8.2d">italic_V ( italic_l ) := { italic_v }</annotation></semantics></math> and <math alttext="E(l):=\{e_{1},e_{2}\}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.9.m9.3"><semantics id="S3.Thmtheorem5.p1.9.m9.3a"><mrow id="S3.Thmtheorem5.p1.9.m9.3.3" xref="S3.Thmtheorem5.p1.9.m9.3.3.cmml"><mrow id="S3.Thmtheorem5.p1.9.m9.3.3.4" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.cmml"><mi id="S3.Thmtheorem5.p1.9.m9.3.3.4.2" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.2.cmml">E</mi><mo id="S3.Thmtheorem5.p1.9.m9.3.3.4.1" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.1.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.9.m9.3.3.4.3.2" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.cmml"><mo id="S3.Thmtheorem5.p1.9.m9.3.3.4.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.cmml">(</mo><mi id="S3.Thmtheorem5.p1.9.m9.1.1" xref="S3.Thmtheorem5.p1.9.m9.1.1.cmml">l</mi><mo id="S3.Thmtheorem5.p1.9.m9.3.3.4.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem5.p1.9.m9.3.3.3" rspace="0.278em" xref="S3.Thmtheorem5.p1.9.m9.3.3.3.cmml">:=</mo><mrow id="S3.Thmtheorem5.p1.9.m9.3.3.2.2" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.3.cmml"><mo id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.3.cmml">{</mo><msub id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.2" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.3" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.4" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.3.cmml">,</mo><msub id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.cmml"><mi id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.2" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.2.cmml">e</mi><mn id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.3" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.5" stretchy="false" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.9.m9.3b"><apply id="S3.Thmtheorem5.p1.9.m9.3.3.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.9.m9.3.3.3.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.3">assign</csymbol><apply id="S3.Thmtheorem5.p1.9.m9.3.3.4.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.4"><times id="S3.Thmtheorem5.p1.9.m9.3.3.4.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.1"></times><ci id="S3.Thmtheorem5.p1.9.m9.3.3.4.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.4.2">𝐸</ci><ci id="S3.Thmtheorem5.p1.9.m9.1.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.1.1">𝑙</ci></apply><set id="S3.Thmtheorem5.p1.9.m9.3.3.2.3.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2"><apply id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.9.m9.2.2.1.1.1.3">1</cn></apply><apply id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.1.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.2.cmml" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.2">𝑒</ci><cn id="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.9.m9.3.3.2.2.2.3">2</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.9.m9.3c">E(l):=\{e_{1},e_{2}\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.9.m9.3d">italic_E ( italic_l ) := { italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT }</annotation></semantics></math>, and updating the vertex and edge sets of <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.10.m10.1"><semantics id="S3.Thmtheorem5.p1.10.m10.1a"><mi id="S3.Thmtheorem5.p1.10.m10.1.1" xref="S3.Thmtheorem5.p1.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.10.m10.1b"><ci id="S3.Thmtheorem5.p1.10.m10.1.1.cmml" xref="S3.Thmtheorem5.p1.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.10.m10.1d">italic_P</annotation></semantics></math> accordingly, is called <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem5.p1.12.1">splitting</em> the line <math alttext="l" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.11.m11.1"><semantics id="S3.Thmtheorem5.p1.11.m11.1a"><mi id="S3.Thmtheorem5.p1.11.m11.1.1" xref="S3.Thmtheorem5.p1.11.m11.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.11.m11.1b"><ci id="S3.Thmtheorem5.p1.11.m11.1.1.cmml" xref="S3.Thmtheorem5.p1.11.m11.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.11.m11.1c">l</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.11.m11.1d">italic_l</annotation></semantics></math> at <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.12.m12.1"><semantics id="S3.Thmtheorem5.p1.12.m12.1a"><mi id="S3.Thmtheorem5.p1.12.m12.1.1" xref="S3.Thmtheorem5.p1.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.12.m12.1b"><ci id="S3.Thmtheorem5.p1.12.m12.1.1.cmml" xref="S3.Thmtheorem5.p1.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.12.m12.1d">italic_v</annotation></semantics></math>.</p> </div> </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.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem6.p1.1.1">For any polygon <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.1.m1.1d">italic_P</annotation></semantics></math>, both sides of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E17" title="Equation 17 ‣ Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">17</span></a>) are invariant under the operation of splitting a line.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS2.1.p1"> <p class="ltx_p" id="S3.SS1.SSS2.1.p1.13">Let <math alttext="l\in E_{l}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.1.m1.1"><semantics id="S3.SS1.SSS2.1.p1.1.m1.1a"><mrow id="S3.SS1.SSS2.1.p1.1.m1.1.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.cmml"><mi id="S3.SS1.SSS2.1.p1.1.m1.1.2.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.2.cmml">l</mi><mo id="S3.SS1.SSS2.1.p1.1.m1.1.2.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.1.p1.1.m1.1.2.3" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.cmml"><msub id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.cmml"><mi id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.2.cmml">E</mi><mi id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.3" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.3.cmml">l</mi></msub><mo id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.3.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.cmml"><mo id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.1.m1.1b"><apply id="S3.SS1.SSS2.1.p1.1.m1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2"><in id="S3.SS1.SSS2.1.p1.1.m1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.1"></in><ci id="S3.SS1.SSS2.1.p1.1.m1.1.2.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.2">𝑙</ci><apply id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3"><times id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.1"></times><apply id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.2">𝐸</ci><ci id="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.3.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.2.3.2.3">𝑙</ci></apply><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.1.m1.1c">l\in E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.1.m1.1d">italic_l ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> be a line that is split at <math alttext="v\in l" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.2.m2.1"><semantics id="S3.SS1.SSS2.1.p1.2.m2.1a"><mrow id="S3.SS1.SSS2.1.p1.2.m2.1.1" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.2.cmml">v</mi><mo id="S3.SS1.SSS2.1.p1.2.m2.1.1.1" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.3" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.2.m2.1b"><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1"><in id="S3.SS1.SSS2.1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.1"></in><ci id="S3.SS1.SSS2.1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.2">𝑣</ci><ci id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.2.m2.1c">v\in l</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.2.m2.1d">italic_v ∈ italic_l</annotation></semantics></math>. The right hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E17" title="Equation 17 ‣ Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">17</span></a>) is invariant under the operation of splitting a line because it does not depend on the specific choice of vertices and edges. Additionally, the number of arcs <math alttext="n_{a}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.3.m3.1"><semantics id="S3.SS1.SSS2.1.p1.3.m3.1a"><mrow id="S3.SS1.SSS2.1.p1.3.m3.1.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.cmml"><msub id="S3.SS1.SSS2.1.p1.3.m3.1.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2.cmml"><mi id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.3" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.1.p1.3.m3.1.2.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.3.m3.1.2.3.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.cmml"><mo id="S3.SS1.SSS2.1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.1.p1.3.m3.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.3.m3.1b"><apply id="S3.SS1.SSS2.1.p1.3.m3.1.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2"><times id="S3.SS1.SSS2.1.p1.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.1"></times><apply id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.1.p1.3.m3.1.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.1.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.3.m3.1c">n_{a}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.3.m3.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> and the set of line segments <math alttext="E_{b}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.4.m4.1"><semantics id="S3.SS1.SSS2.1.p1.4.m4.1a"><mrow id="S3.SS1.SSS2.1.p1.4.m4.1.2" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.cmml"><msub id="S3.SS1.SSS2.1.p1.4.m4.1.2.2" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2.cmml"><mi id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.2" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2.2.cmml">E</mi><mi id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.3" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.1.p1.4.m4.1.2.1" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.4.m4.1.2.3.2" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.cmml"><mo id="S3.SS1.SSS2.1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.1.p1.4.m4.1.1" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.4.m4.1b"><apply id="S3.SS1.SSS2.1.p1.4.m4.1.2.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2"><times id="S3.SS1.SSS2.1.p1.4.m4.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.1"></times><apply id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.1.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2.2">𝐸</ci><ci id="S3.SS1.SSS2.1.p1.4.m4.1.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.2.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.1.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.4.m4.1c">E_{b}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math> remain unchanged by this operation. The <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.5.m5.1"><semantics id="S3.SS1.SSS2.1.p1.5.m5.1a"><mi id="S3.SS1.SSS2.1.p1.5.m5.1.1" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.5.m5.1b"><ci id="S3.SS1.SSS2.1.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS2.1.p1.5.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.5.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.5.m5.1d">italic_P</annotation></semantics></math>-side of <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.6.m6.1"><semantics id="S3.SS1.SSS2.1.p1.6.m6.1a"><mi id="S3.SS1.SSS2.1.p1.6.m6.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.6.m6.1b"><ci id="S3.SS1.SSS2.1.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.6.m6.1d">italic_v</annotation></semantics></math> is a half-plane that agrees with the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.7.m7.1"><semantics id="S3.SS1.SSS2.1.p1.7.m7.1a"><mi id="S3.SS1.SSS2.1.p1.7.m7.1.1" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.7.m7.1b"><ci id="S3.SS1.SSS2.1.p1.7.m7.1.1.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.7.m7.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.7.m7.1d">italic_P</annotation></semantics></math>-side of <math alttext="l" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.8.m8.1"><semantics id="S3.SS1.SSS2.1.p1.8.m8.1a"><mi id="S3.SS1.SSS2.1.p1.8.m8.1.1" xref="S3.SS1.SSS2.1.p1.8.m8.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.8.m8.1b"><ci id="S3.SS1.SSS2.1.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS2.1.p1.8.m8.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.8.m8.1c">l</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.8.m8.1d">italic_l</annotation></semantics></math>, see <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F6" title="Figure 6 ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 6</span></a>. Therefore, <math alttext="\mathds{1}_{Q_{P}^{v}}=\mathds{1}_{H_{P}^{l}}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.9.m9.1"><semantics id="S3.SS1.SSS2.1.p1.9.m9.1a"><mrow id="S3.SS1.SSS2.1.p1.9.m9.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.cmml"><msub id="S3.SS1.SSS2.1.p1.9.m9.1.1.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.cmml"><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.2.cmml">Q</mi><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.3.cmml">P</mi><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.3.cmml">v</mi></msubsup></msub><mo id="S3.SS1.SSS2.1.p1.9.m9.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS2.1.p1.9.m9.1.1.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.cmml"><mn id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.2.cmml">𝟙</mn><msubsup id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.2.cmml">H</mi><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.3.cmml">P</mi><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.3.cmml">l</mi></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.9.m9.1b"><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1"><eq id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1"></eq><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2">subscript</csymbol><cn id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.2">1</cn><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3">superscript</csymbol><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.2">𝑄</ci><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.2.3">𝑃</ci></apply><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.2.3.3">𝑣</ci></apply></apply><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3">subscript</csymbol><cn id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.2">1</cn><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3">superscript</csymbol><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.2">𝐻</ci><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.2.3">𝑃</ci></apply><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.3.3.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.9.m9.1c">\mathds{1}_{Q_{P}^{v}}=\mathds{1}_{H_{P}^{l}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.9.m9.1d">blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and since the operation adds <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.10.m10.1"><semantics id="S3.SS1.SSS2.1.p1.10.m10.1a"><mi id="S3.SS1.SSS2.1.p1.10.m10.1.1" xref="S3.SS1.SSS2.1.p1.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.10.m10.1b"><ci id="S3.SS1.SSS2.1.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS2.1.p1.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.10.m10.1d">italic_v</annotation></semantics></math> to <math alttext="V(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.11.m11.1"><semantics id="S3.SS1.SSS2.1.p1.11.m11.1a"><mrow id="S3.SS1.SSS2.1.p1.11.m11.1.2" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.cmml"><mi id="S3.SS1.SSS2.1.p1.11.m11.1.2.2" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.2.cmml">V</mi><mo id="S3.SS1.SSS2.1.p1.11.m11.1.2.1" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.11.m11.1.2.3.2" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.cmml"><mo id="S3.SS1.SSS2.1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.1.p1.11.m11.1.1" xref="S3.SS1.SSS2.1.p1.11.m11.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.11.m11.1b"><apply id="S3.SS1.SSS2.1.p1.11.m11.1.2.cmml" xref="S3.SS1.SSS2.1.p1.11.m11.1.2"><times id="S3.SS1.SSS2.1.p1.11.m11.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.1"></times><ci id="S3.SS1.SSS2.1.p1.11.m11.1.2.2.cmml" xref="S3.SS1.SSS2.1.p1.11.m11.1.2.2">𝑉</ci><ci id="S3.SS1.SSS2.1.p1.11.m11.1.1.cmml" xref="S3.SS1.SSS2.1.p1.11.m11.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.11.m11.1c">V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.11.m11.1d">italic_V ( italic_P )</annotation></semantics></math> and removes <math alttext="l" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.12.m12.1"><semantics id="S3.SS1.SSS2.1.p1.12.m12.1a"><mi id="S3.SS1.SSS2.1.p1.12.m12.1.1" xref="S3.SS1.SSS2.1.p1.12.m12.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.12.m12.1b"><ci id="S3.SS1.SSS2.1.p1.12.m12.1.1.cmml" xref="S3.SS1.SSS2.1.p1.12.m12.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.12.m12.1c">l</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.12.m12.1d">italic_l</annotation></semantics></math> from <math alttext="E_{l}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.13.m13.1"><semantics id="S3.SS1.SSS2.1.p1.13.m13.1a"><mrow id="S3.SS1.SSS2.1.p1.13.m13.1.2" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.cmml"><msub id="S3.SS1.SSS2.1.p1.13.m13.1.2.2" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2.cmml"><mi id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.2" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2.2.cmml">E</mi><mi id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.3" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2.3.cmml">l</mi></msub><mo id="S3.SS1.SSS2.1.p1.13.m13.1.2.1" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.13.m13.1.2.3.2" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.cmml"><mo id="S3.SS1.SSS2.1.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.1.p1.13.m13.1.1" xref="S3.SS1.SSS2.1.p1.13.m13.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.1.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.13.m13.1b"><apply id="S3.SS1.SSS2.1.p1.13.m13.1.2.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2"><times id="S3.SS1.SSS2.1.p1.13.m13.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.1"></times><apply id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.1.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2.2">𝐸</ci><ci id="S3.SS1.SSS2.1.p1.13.m13.1.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.2.2.3">𝑙</ci></apply><ci id="S3.SS1.SSS2.1.p1.13.m13.1.1.cmml" xref="S3.SS1.SSS2.1.p1.13.m13.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.13.m13.1c">E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.13.m13.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>, it does not affect the left hand side. ∎</p> </div> </div> <figure class="ltx_figure" id="S3.F6"> <p class="ltx_p ltx_align_center" id="S3.F6.1"><span class="ltx_text" id="S3.F6.1.1"><foreignobject height="30.4pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="117.1pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="59" id="S3.F6.1.1.1.g1" src="x18.png" width="225"/></foreignobject></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>A line of a polygon is split by a vertex. The propagation of the <math alttext="P" class="ltx_Math" display="inline" id="S3.F6.3.m1.1"><semantics id="S3.F6.3.m1.1b"><mi id="S3.F6.3.m1.1.1" xref="S3.F6.3.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F6.3.m1.1c"><ci id="S3.F6.3.m1.1.1.cmml" xref="S3.F6.3.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.3.m1.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F6.3.m1.1e">italic_P</annotation></semantics></math>-sides of the line and vertex are indicated in dark grey.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS2.p4"> <p class="ltx_p" id="S3.SS1.SSS2.p4.1">The idea for the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a> is to carefully modify the polygon by adding auxiliary edges and vertices in such a way that its boundary becomes a polygonal cycle to which <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem3" title="Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.3</span></a> can be applied. To do so, we will cut off two rays and connect the remainder of the corresponding arcs with line segments.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.p5"> <p class="ltx_p" id="S3.SS1.SSS2.p5.1">The following lemma will ensure that the modified boundary components remain simple curves.</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.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.4.4">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.1.m1.1a"><mi id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.1b"><ci id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon. For each vertex <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem7.p1.2.2.m2.1.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.2.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.2.cmml">v</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.1.2.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem7.p1.2.2.m2.1.2.3" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.2.cmml">V</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.3.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.cmml"><mo id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem7.p1.2.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.1b"><apply id="S3.Thmtheorem7.p1.2.2.m2.1.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2"><in id="S3.Thmtheorem7.p1.2.2.m2.1.2.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.1"></in><ci id="S3.Thmtheorem7.p1.2.2.m2.1.2.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.2">𝑣</ci><apply id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3"><times id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.1"></times><ci id="S3.Thmtheorem7.p1.2.2.m2.1.2.3.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.2.3.2">𝑉</ci><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math>, there is at most one polygonal arc containing <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.3.m3.1"><semantics id="S3.Thmtheorem7.p1.3.3.m3.1a"><mi id="S3.Thmtheorem7.p1.3.3.m3.1.1" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.3.3.m3.1b"><ci id="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.3.m3.1d">italic_v</annotation></semantics></math> that is a boundary component of <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.4.4.m4.1"><semantics id="S3.Thmtheorem7.p1.4.4.m4.1a"><mi id="S3.Thmtheorem7.p1.4.4.m4.1.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.4.4.m4.1b"><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.4.m4.1d">italic_P</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS2.2.p1"> <p class="ltx_p" id="S3.SS1.SSS2.2.p1.50">Assume that <math alttext="\gamma_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.1.m1.1"><semantics id="S3.SS1.SSS2.2.p1.1.m1.1a"><msub id="S3.SS1.SSS2.2.p1.1.m1.1.1" xref="S3.SS1.SSS2.2.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.1.m1.1.1.2" xref="S3.SS1.SSS2.2.p1.1.m1.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.1.m1.1.1.3" xref="S3.SS1.SSS2.2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.1.m1.1b"><apply id="S3.SS1.SSS2.2.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.1.m1.1.1.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.1.m1.1c">\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.1.m1.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.2.m2.1"><semantics id="S3.SS1.SSS2.2.p1.2.m2.1a"><msub id="S3.SS1.SSS2.2.p1.2.m2.1.1" xref="S3.SS1.SSS2.2.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.2.m2.1.1.2" xref="S3.SS1.SSS2.2.p1.2.m2.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.2.m2.1.1.3" xref="S3.SS1.SSS2.2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.2.m2.1b"><apply id="S3.SS1.SSS2.2.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS2.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.2.m2.1.1.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.2.m2.1c">\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.2.m2.1d">italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are two polygonal arcs that intersect at a vertex <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.3.m3.1"><semantics id="S3.SS1.SSS2.2.p1.3.m3.1a"><mrow id="S3.SS1.SSS2.2.p1.3.m3.1.2" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.3.m3.1.2.2" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.2.cmml">v</mi><mo id="S3.SS1.SSS2.2.p1.3.m3.1.2.1" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.2.p1.3.m3.1.2.3" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.cmml"><mi id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.2" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.1" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.3.2" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.cmml"><mo id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS2.2.p1.3.m3.1.1" xref="S3.SS1.SSS2.2.p1.3.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.3.m3.1b"><apply id="S3.SS1.SSS2.2.p1.3.m3.1.2.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2"><in id="S3.SS1.SSS2.2.p1.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.1"></in><ci id="S3.SS1.SSS2.2.p1.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.2">𝑣</ci><apply id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3"><times id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.1.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.1"></times><ci id="S3.SS1.SSS2.2.p1.3.m3.1.2.3.2.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.2.3.2">𝑉</ci><ci id="S3.SS1.SSS2.2.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS2.2.p1.3.m3.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.3.m3.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.3.m3.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math>. Let <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.4.m4.1"><semantics id="S3.SS1.SSS2.2.p1.4.m4.1a"><mi id="S3.SS1.SSS2.2.p1.4.m4.1.1" xref="S3.SS1.SSS2.2.p1.4.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.4.m4.1b"><ci id="S3.SS1.SSS2.2.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS2.2.p1.4.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.4.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.4.m4.1d">italic_D</annotation></semantics></math> be a disk large enough such that <math alttext="V(P)\subset D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.5.m5.1"><semantics id="S3.SS1.SSS2.2.p1.5.m5.1a"><mrow id="S3.SS1.SSS2.2.p1.5.m5.1.2" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.cmml"><mrow id="S3.SS1.SSS2.2.p1.5.m5.1.2.2" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.2" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.2.cmml">V</mi><mo id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.1" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.3.2" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.cmml"><mo id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.2.p1.5.m5.1.1" xref="S3.SS1.SSS2.2.p1.5.m5.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.2.p1.5.m5.1.2.1" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.1.cmml">⊂</mo><mi id="S3.SS1.SSS2.2.p1.5.m5.1.2.3" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.5.m5.1b"><apply id="S3.SS1.SSS2.2.p1.5.m5.1.2.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2"><subset id="S3.SS1.SSS2.2.p1.5.m5.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.1"></subset><apply id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2"><times id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.1"></times><ci id="S3.SS1.SSS2.2.p1.5.m5.1.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.2.2">𝑉</ci><ci id="S3.SS1.SSS2.2.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.1">𝑃</ci></apply><ci id="S3.SS1.SSS2.2.p1.5.m5.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.5.m5.1.2.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.5.m5.1c">V(P)\subset D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.5.m5.1d">italic_V ( italic_P ) ⊂ italic_D</annotation></semantics></math>. Due to the convexity of the disk, all line segments are contained in <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.6.m6.1"><semantics id="S3.SS1.SSS2.2.p1.6.m6.1a"><mi id="S3.SS1.SSS2.2.p1.6.m6.1.1" xref="S3.SS1.SSS2.2.p1.6.m6.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.6.m6.1b"><ci id="S3.SS1.SSS2.2.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS2.2.p1.6.m6.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.6.m6.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.6.m6.1d">italic_D</annotation></semantics></math>. If <math alttext="e\in E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.7.m7.1"><semantics id="S3.SS1.SSS2.2.p1.7.m7.1a"><mrow id="S3.SS1.SSS2.2.p1.7.m7.1.2" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.7.m7.1.2.2" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.2.cmml">e</mi><mo id="S3.SS1.SSS2.2.p1.7.m7.1.2.1" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.2.p1.7.m7.1.2.3" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.cmml"><mi id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.2" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.2.cmml">E</mi><mo id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.1" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.3.2" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.cmml"><mo id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS2.2.p1.7.m7.1.1" xref="S3.SS1.SSS2.2.p1.7.m7.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.7.m7.1b"><apply id="S3.SS1.SSS2.2.p1.7.m7.1.2.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2"><in id="S3.SS1.SSS2.2.p1.7.m7.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.1"></in><ci id="S3.SS1.SSS2.2.p1.7.m7.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.2">𝑒</ci><apply id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3"><times id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.1.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.1"></times><ci id="S3.SS1.SSS2.2.p1.7.m7.1.2.3.2.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.2.3.2">𝐸</ci><ci id="S3.SS1.SSS2.2.p1.7.m7.1.1.cmml" xref="S3.SS1.SSS2.2.p1.7.m7.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.7.m7.1c">e\in E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.7.m7.1d">italic_e ∈ italic_E ( italic_P )</annotation></semantics></math> is a ray, <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.8.m8.1"><semantics id="S3.SS1.SSS2.2.p1.8.m8.1a"><mi id="S3.SS1.SSS2.2.p1.8.m8.1.1" xref="S3.SS1.SSS2.2.p1.8.m8.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.8.m8.1b"><ci id="S3.SS1.SSS2.2.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS2.2.p1.8.m8.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.8.m8.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.8.m8.1d">italic_e</annotation></semantics></math> intersects <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.9.m9.1"><semantics id="S3.SS1.SSS2.2.p1.9.m9.1a"><mrow id="S3.SS1.SSS2.2.p1.9.m9.1.1" xref="S3.SS1.SSS2.2.p1.9.m9.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.9.m9.1.1.1" rspace="0em" xref="S3.SS1.SSS2.2.p1.9.m9.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.9.m9.1.1.2" xref="S3.SS1.SSS2.2.p1.9.m9.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.9.m9.1b"><apply id="S3.SS1.SSS2.2.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS2.2.p1.9.m9.1.1"><partialdiff id="S3.SS1.SSS2.2.p1.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.9.m9.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.9.m9.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.9.m9.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.9.m9.1d">∂ italic_D</annotation></semantics></math>, since its vertex is contained in <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.10.m10.1"><semantics id="S3.SS1.SSS2.2.p1.10.m10.1a"><mi id="S3.SS1.SSS2.2.p1.10.m10.1.1" xref="S3.SS1.SSS2.2.p1.10.m10.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.10.m10.1b"><ci id="S3.SS1.SSS2.2.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS2.2.p1.10.m10.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.10.m10.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.10.m10.1d">italic_D</annotation></semantics></math> and <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.11.m11.1"><semantics id="S3.SS1.SSS2.2.p1.11.m11.1a"><mi id="S3.SS1.SSS2.2.p1.11.m11.1.1" xref="S3.SS1.SSS2.2.p1.11.m11.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.11.m11.1b"><ci id="S3.SS1.SSS2.2.p1.11.m11.1.1.cmml" xref="S3.SS1.SSS2.2.p1.11.m11.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.11.m11.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.11.m11.1d">italic_e</annotation></semantics></math> is unbounded. Moreover, there is exactly one such intersection because <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.12.m12.1"><semantics id="S3.SS1.SSS2.2.p1.12.m12.1a"><mi id="S3.SS1.SSS2.2.p1.12.m12.1.1" xref="S3.SS1.SSS2.2.p1.12.m12.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.12.m12.1b"><ci id="S3.SS1.SSS2.2.p1.12.m12.1.1.cmml" xref="S3.SS1.SSS2.2.p1.12.m12.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.12.m12.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.12.m12.1d">italic_e</annotation></semantics></math> is straight. Therefore, <math alttext="\gamma_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.13.m13.1"><semantics id="S3.SS1.SSS2.2.p1.13.m13.1a"><msub id="S3.SS1.SSS2.2.p1.13.m13.1.1" xref="S3.SS1.SSS2.2.p1.13.m13.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.13.m13.1.1.2" xref="S3.SS1.SSS2.2.p1.13.m13.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.13.m13.1.1.3" xref="S3.SS1.SSS2.2.p1.13.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.13.m13.1b"><apply id="S3.SS1.SSS2.2.p1.13.m13.1.1.cmml" xref="S3.SS1.SSS2.2.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.13.m13.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.13.m13.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.13.m13.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.13.m13.1.1.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.13.m13.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.13.m13.1c">\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.13.m13.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.14.m14.1"><semantics id="S3.SS1.SSS2.2.p1.14.m14.1a"><msub id="S3.SS1.SSS2.2.p1.14.m14.1.1" xref="S3.SS1.SSS2.2.p1.14.m14.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.14.m14.1.1.2" xref="S3.SS1.SSS2.2.p1.14.m14.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.14.m14.1.1.3" xref="S3.SS1.SSS2.2.p1.14.m14.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.14.m14.1b"><apply id="S3.SS1.SSS2.2.p1.14.m14.1.1.cmml" xref="S3.SS1.SSS2.2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.14.m14.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.14.m14.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.14.m14.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.14.m14.1.1.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.14.m14.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.14.m14.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.14.m14.1c">\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.14.m14.1d">italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> create four intersection points of <math alttext="\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.15.m15.1"><semantics id="S3.SS1.SSS2.2.p1.15.m15.1a"><mrow id="S3.SS1.SSS2.2.p1.15.m15.1.1" xref="S3.SS1.SSS2.2.p1.15.m15.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.15.m15.1.1.1" rspace="0em" xref="S3.SS1.SSS2.2.p1.15.m15.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.15.m15.1.1.2" xref="S3.SS1.SSS2.2.p1.15.m15.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.15.m15.1b"><apply id="S3.SS1.SSS2.2.p1.15.m15.1.1.cmml" xref="S3.SS1.SSS2.2.p1.15.m15.1.1"><partialdiff id="S3.SS1.SSS2.2.p1.15.m15.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.15.m15.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.15.m15.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.15.m15.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.15.m15.1c">\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.15.m15.1d">∂ italic_P</annotation></semantics></math> with <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.16.m16.1"><semantics id="S3.SS1.SSS2.2.p1.16.m16.1a"><mrow id="S3.SS1.SSS2.2.p1.16.m16.1.1" xref="S3.SS1.SSS2.2.p1.16.m16.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.16.m16.1.1.1" rspace="0em" xref="S3.SS1.SSS2.2.p1.16.m16.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.16.m16.1.1.2" xref="S3.SS1.SSS2.2.p1.16.m16.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.16.m16.1b"><apply id="S3.SS1.SSS2.2.p1.16.m16.1.1.cmml" xref="S3.SS1.SSS2.2.p1.16.m16.1.1"><partialdiff id="S3.SS1.SSS2.2.p1.16.m16.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.16.m16.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.16.m16.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.16.m16.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.16.m16.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.16.m16.1d">∂ italic_D</annotation></semantics></math>, and subdivide <math alttext="\mathds{R}^{2}\setminus D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.17.m17.1"><semantics id="S3.SS1.SSS2.2.p1.17.m17.1a"><mrow id="S3.SS1.SSS2.2.p1.17.m17.1.1" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.cmml"><msup id="S3.SS1.SSS2.2.p1.17.m17.1.1.2" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.2" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2.2.cmml">ℝ</mi><mn id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.3" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2.3.cmml">2</mn></msup><mo id="S3.SS1.SSS2.2.p1.17.m17.1.1.1" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.2.p1.17.m17.1.1.3" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.17.m17.1b"><apply id="S3.SS1.SSS2.2.p1.17.m17.1.1.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1"><setdiff id="S3.SS1.SSS2.2.p1.17.m17.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.1"></setdiff><apply id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2.2">ℝ</ci><cn id="S3.SS1.SSS2.2.p1.17.m17.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.2.3">2</cn></apply><ci id="S3.SS1.SSS2.2.p1.17.m17.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.17.m17.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.17.m17.1c">\mathds{R}^{2}\setminus D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.17.m17.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_D</annotation></semantics></math> into four sectors (see <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F7" title="Figure 7 ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 7</span></a>). Let <math alttext="y_{1},y_{2}\in P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.18.m18.2"><semantics id="S3.SS1.SSS2.2.p1.18.m18.2a"><mrow id="S3.SS1.SSS2.2.p1.18.m18.2.2" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.cmml"><mrow id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.3.cmml"><msub id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.2" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.2.cmml">y</mi><mn id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.3" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.2.cmml">y</mi><mn id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S3.SS1.SSS2.2.p1.18.m18.2.2.3" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.3.cmml">∈</mo><mi id="S3.SS1.SSS2.2.p1.18.m18.2.2.4" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.4.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.18.m18.2b"><apply id="S3.SS1.SSS2.2.p1.18.m18.2.2.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2"><in id="S3.SS1.SSS2.2.p1.18.m18.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.3"></in><list id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2"><apply id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.2">𝑦</ci><cn id="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.18.m18.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.2">𝑦</ci><cn id="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.2.2.2.3">2</cn></apply></list><ci id="S3.SS1.SSS2.2.p1.18.m18.2.2.4.cmml" xref="S3.SS1.SSS2.2.p1.18.m18.2.2.4">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.18.m18.2c">y_{1},y_{2}\in P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.18.m18.2d">italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_P</annotation></semantics></math> be points that are contained in different sectors <math alttext="S_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.19.m19.1"><semantics id="S3.SS1.SSS2.2.p1.19.m19.1a"><msub id="S3.SS1.SSS2.2.p1.19.m19.1.1" xref="S3.SS1.SSS2.2.p1.19.m19.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.19.m19.1.1.2" xref="S3.SS1.SSS2.2.p1.19.m19.1.1.2.cmml">S</mi><mn id="S3.SS1.SSS2.2.p1.19.m19.1.1.3" xref="S3.SS1.SSS2.2.p1.19.m19.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.19.m19.1b"><apply id="S3.SS1.SSS2.2.p1.19.m19.1.1.cmml" xref="S3.SS1.SSS2.2.p1.19.m19.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.19.m19.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.19.m19.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.19.m19.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.19.m19.1.1.2">𝑆</ci><cn id="S3.SS1.SSS2.2.p1.19.m19.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.19.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.19.m19.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.19.m19.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="S_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.20.m20.1"><semantics id="S3.SS1.SSS2.2.p1.20.m20.1a"><msub id="S3.SS1.SSS2.2.p1.20.m20.1.1" xref="S3.SS1.SSS2.2.p1.20.m20.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.20.m20.1.1.2" xref="S3.SS1.SSS2.2.p1.20.m20.1.1.2.cmml">S</mi><mn id="S3.SS1.SSS2.2.p1.20.m20.1.1.3" xref="S3.SS1.SSS2.2.p1.20.m20.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.20.m20.1b"><apply id="S3.SS1.SSS2.2.p1.20.m20.1.1.cmml" xref="S3.SS1.SSS2.2.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.20.m20.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.20.m20.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.20.m20.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.20.m20.1.1.2">𝑆</ci><cn id="S3.SS1.SSS2.2.p1.20.m20.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.20.m20.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.20.m20.1c">S_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.20.m20.1d">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Then, from the connectedness of <math alttext="\operatorname*{int}P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.21.m21.1"><semantics id="S3.SS1.SSS2.2.p1.21.m21.1a"><mrow id="S3.SS1.SSS2.2.p1.21.m21.1.1" xref="S3.SS1.SSS2.2.p1.21.m21.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.21.m21.1.1.1" rspace="0.167em" xref="S3.SS1.SSS2.2.p1.21.m21.1.1.1.cmml">int</mo><mi id="S3.SS1.SSS2.2.p1.21.m21.1.1.2" xref="S3.SS1.SSS2.2.p1.21.m21.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.21.m21.1b"><apply id="S3.SS1.SSS2.2.p1.21.m21.1.1.cmml" xref="S3.SS1.SSS2.2.p1.21.m21.1.1"><ci id="S3.SS1.SSS2.2.p1.21.m21.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.21.m21.1.1.1">int</ci><ci id="S3.SS1.SSS2.2.p1.21.m21.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.21.m21.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.21.m21.1c">\operatorname*{int}P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.21.m21.1d">roman_int italic_P</annotation></semantics></math>, it follows that there is a path <math alttext="\lambda\subset\operatorname*{int}P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.22.m22.1"><semantics id="S3.SS1.SSS2.2.p1.22.m22.1a"><mrow id="S3.SS1.SSS2.2.p1.22.m22.1.1" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.22.m22.1.1.2" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.2.cmml">λ</mi><mo id="S3.SS1.SSS2.2.p1.22.m22.1.1.1" rspace="0.1389em" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.1.cmml">⊂</mo><mrow id="S3.SS1.SSS2.2.p1.22.m22.1.1.3" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3.cmml"><mo id="S3.SS1.SSS2.2.p1.22.m22.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3.1.cmml">int</mo><mi id="S3.SS1.SSS2.2.p1.22.m22.1.1.3.2" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3.2.cmml">P</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.22.m22.1b"><apply id="S3.SS1.SSS2.2.p1.22.m22.1.1.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1"><subset id="S3.SS1.SSS2.2.p1.22.m22.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.1"></subset><ci id="S3.SS1.SSS2.2.p1.22.m22.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.2">𝜆</ci><apply id="S3.SS1.SSS2.2.p1.22.m22.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3"><ci id="S3.SS1.SSS2.2.p1.22.m22.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3.1">int</ci><ci id="S3.SS1.SSS2.2.p1.22.m22.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.22.m22.1.1.3.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.22.m22.1c">\lambda\subset\operatorname*{int}P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.22.m22.1d">italic_λ ⊂ roman_int italic_P</annotation></semantics></math> connecting <math alttext="y_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.23.m23.1"><semantics id="S3.SS1.SSS2.2.p1.23.m23.1a"><msub id="S3.SS1.SSS2.2.p1.23.m23.1.1" xref="S3.SS1.SSS2.2.p1.23.m23.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.23.m23.1.1.2" xref="S3.SS1.SSS2.2.p1.23.m23.1.1.2.cmml">y</mi><mn id="S3.SS1.SSS2.2.p1.23.m23.1.1.3" xref="S3.SS1.SSS2.2.p1.23.m23.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.23.m23.1b"><apply id="S3.SS1.SSS2.2.p1.23.m23.1.1.cmml" xref="S3.SS1.SSS2.2.p1.23.m23.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.23.m23.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.23.m23.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.23.m23.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.23.m23.1.1.2">𝑦</ci><cn id="S3.SS1.SSS2.2.p1.23.m23.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.23.m23.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.23.m23.1c">y_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.23.m23.1d">italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="y_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.24.m24.1"><semantics id="S3.SS1.SSS2.2.p1.24.m24.1a"><msub id="S3.SS1.SSS2.2.p1.24.m24.1.1" xref="S3.SS1.SSS2.2.p1.24.m24.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.24.m24.1.1.2" xref="S3.SS1.SSS2.2.p1.24.m24.1.1.2.cmml">y</mi><mn id="S3.SS1.SSS2.2.p1.24.m24.1.1.3" xref="S3.SS1.SSS2.2.p1.24.m24.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.24.m24.1b"><apply id="S3.SS1.SSS2.2.p1.24.m24.1.1.cmml" xref="S3.SS1.SSS2.2.p1.24.m24.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.24.m24.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.24.m24.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.24.m24.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.24.m24.1.1.2">𝑦</ci><cn id="S3.SS1.SSS2.2.p1.24.m24.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.24.m24.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.24.m24.1c">y_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.24.m24.1d">italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.25.m25.1"><semantics id="S3.SS1.SSS2.2.p1.25.m25.1a"><mi id="S3.SS1.SSS2.2.p1.25.m25.1.1" xref="S3.SS1.SSS2.2.p1.25.m25.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.25.m25.1b"><ci id="S3.SS1.SSS2.2.p1.25.m25.1.1.cmml" xref="S3.SS1.SSS2.2.p1.25.m25.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.25.m25.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.25.m25.1d">italic_λ</annotation></semantics></math> is not allowed to intersect <math alttext="\gamma_{i}\subset\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.26.m26.1"><semantics id="S3.SS1.SSS2.2.p1.26.m26.1a"><mrow id="S3.SS1.SSS2.2.p1.26.m26.1.1" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.cmml"><msub id="S3.SS1.SSS2.2.p1.26.m26.1.1.2" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.2" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2.2.cmml">γ</mi><mi id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.3" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.SSS2.2.p1.26.m26.1.1.1" rspace="0.1389em" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.1.cmml">⊂</mo><mrow id="S3.SS1.SSS2.2.p1.26.m26.1.1.3" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3.cmml"><mo id="S3.SS1.SSS2.2.p1.26.m26.1.1.3.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.26.m26.1.1.3.2" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3.2.cmml">P</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.26.m26.1b"><apply id="S3.SS1.SSS2.2.p1.26.m26.1.1.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1"><subset id="S3.SS1.SSS2.2.p1.26.m26.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.1"></subset><apply id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.2.p1.26.m26.1.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.SSS2.2.p1.26.m26.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3"><partialdiff id="S3.SS1.SSS2.2.p1.26.m26.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.26.m26.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.26.m26.1.1.3.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.26.m26.1c">\gamma_{i}\subset\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.26.m26.1d">italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊂ ∂ italic_P</annotation></semantics></math>, i.e., it must pass through <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.27.m27.1"><semantics id="S3.SS1.SSS2.2.p1.27.m27.1a"><mi id="S3.SS1.SSS2.2.p1.27.m27.1.1" xref="S3.SS1.SSS2.2.p1.27.m27.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.27.m27.1b"><ci id="S3.SS1.SSS2.2.p1.27.m27.1.1.cmml" xref="S3.SS1.SSS2.2.p1.27.m27.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.27.m27.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.27.m27.1d">italic_D</annotation></semantics></math>. Therefore, there are two points <math alttext="x_{1},x_{2}\in\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.28.m28.2"><semantics id="S3.SS1.SSS2.2.p1.28.m28.2a"><mrow id="S3.SS1.SSS2.2.p1.28.m28.2.2" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.cmml"><mrow id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.3.cmml"><msub id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.2" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.3" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S3.SS1.SSS2.2.p1.28.m28.2.2.3" rspace="0.1389em" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.3.cmml">∈</mo><mrow id="S3.SS1.SSS2.2.p1.28.m28.2.2.4" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4.cmml"><mo id="S3.SS1.SSS2.2.p1.28.m28.2.2.4.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.28.m28.2.2.4.2" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4.2.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.28.m28.2b"><apply id="S3.SS1.SSS2.2.p1.28.m28.2.2.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2"><in id="S3.SS1.SSS2.2.p1.28.m28.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.3"></in><list id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2"><apply id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.28.m28.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.2.2.2.3">2</cn></apply></list><apply id="S3.SS1.SSS2.2.p1.28.m28.2.2.4.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4"><partialdiff id="S3.SS1.SSS2.2.p1.28.m28.2.2.4.1.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.28.m28.2.2.4.2.cmml" xref="S3.SS1.SSS2.2.p1.28.m28.2.2.4.2">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.28.m28.2c">x_{1},x_{2}\in\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.28.m28.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ ∂ italic_D</annotation></semantics></math>, with <math alttext="x_{2}\in S_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.29.m29.1"><semantics id="S3.SS1.SSS2.2.p1.29.m29.1a"><mrow id="S3.SS1.SSS2.2.p1.29.m29.1.1" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.cmml"><msub id="S3.SS1.SSS2.2.p1.29.m29.1.1.2" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.2" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.3" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2.3.cmml">2</mn></msub><mo id="S3.SS1.SSS2.2.p1.29.m29.1.1.1" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.1.cmml">∈</mo><msub id="S3.SS1.SSS2.2.p1.29.m29.1.1.3" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3.cmml"><mi id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.2" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3.2.cmml">S</mi><mn id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.3" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.29.m29.1b"><apply id="S3.SS1.SSS2.2.p1.29.m29.1.1.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1"><in id="S3.SS1.SSS2.2.p1.29.m29.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.1"></in><apply id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.29.m29.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.2.3">2</cn></apply><apply id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3.2">𝑆</ci><cn id="S3.SS1.SSS2.2.p1.29.m29.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.29.m29.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.29.m29.1c">x_{2}\in S_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.29.m29.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{1}\not\in S_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.30.m30.1"><semantics id="S3.SS1.SSS2.2.p1.30.m30.1a"><mrow id="S3.SS1.SSS2.2.p1.30.m30.1.1" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.cmml"><msub id="S3.SS1.SSS2.2.p1.30.m30.1.1.2" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.2" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.3" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.2.p1.30.m30.1.1.1" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.1.cmml">∉</mo><msub id="S3.SS1.SSS2.2.p1.30.m30.1.1.3" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3.cmml"><mi id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.2" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3.2.cmml">S</mi><mn id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.3" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.30.m30.1b"><apply id="S3.SS1.SSS2.2.p1.30.m30.1.1.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1"><notin id="S3.SS1.SSS2.2.p1.30.m30.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.1"></notin><apply id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.30.m30.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.2.3">1</cn></apply><apply id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3.2">𝑆</ci><cn id="S3.SS1.SSS2.2.p1.30.m30.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.30.m30.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.30.m30.1c">x_{1}\not\in S_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.30.m30.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∉ italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, that are connected by a segment <math alttext="\lambda^{\prime}\subset\lambda" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.31.m31.1"><semantics id="S3.SS1.SSS2.2.p1.31.m31.1a"><mrow id="S3.SS1.SSS2.2.p1.31.m31.1.1" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.cmml"><msup id="S3.SS1.SSS2.2.p1.31.m31.1.1.2" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.2" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2.2.cmml">λ</mi><mo id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.3" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.2.p1.31.m31.1.1.1" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.1.cmml">⊂</mo><mi id="S3.SS1.SSS2.2.p1.31.m31.1.1.3" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.31.m31.1b"><apply id="S3.SS1.SSS2.2.p1.31.m31.1.1.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1"><subset id="S3.SS1.SSS2.2.p1.31.m31.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.1"></subset><apply id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2.2">𝜆</ci><ci id="S3.SS1.SSS2.2.p1.31.m31.1.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.2.3">′</ci></apply><ci id="S3.SS1.SSS2.2.p1.31.m31.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.31.m31.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.31.m31.1c">\lambda^{\prime}\subset\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.31.m31.1d">italic_λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_λ</annotation></semantics></math> completely contained within <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.32.m32.1"><semantics id="S3.SS1.SSS2.2.p1.32.m32.1a"><mi id="S3.SS1.SSS2.2.p1.32.m32.1.1" xref="S3.SS1.SSS2.2.p1.32.m32.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.32.m32.1b"><ci id="S3.SS1.SSS2.2.p1.32.m32.1.1.cmml" xref="S3.SS1.SSS2.2.p1.32.m32.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.32.m32.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.32.m32.1d">italic_D</annotation></semantics></math>. If <math alttext="r_{3},r_{4}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.33.m33.2"><semantics id="S3.SS1.SSS2.2.p1.33.m33.2a"><mrow id="S3.SS1.SSS2.2.p1.33.m33.2.2.2" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.3.cmml"><msub id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.2" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.2.cmml">r</mi><mn id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.3" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.3.cmml">3</mn></msub><mo id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.3" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.2.cmml">r</mi><mn id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.33.m33.2b"><list id="S3.SS1.SSS2.2.p1.33.m33.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2"><apply id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.2">𝑟</ci><cn id="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.33.m33.1.1.1.1.3">3</cn></apply><apply id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.2">𝑟</ci><cn id="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.33.m33.2.2.2.2.3">4</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.33.m33.2c">r_{3},r_{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.33.m33.2d">italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> are the rays that define the sector <math alttext="S_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.34.m34.1"><semantics id="S3.SS1.SSS2.2.p1.34.m34.1a"><msub id="S3.SS1.SSS2.2.p1.34.m34.1.1" xref="S3.SS1.SSS2.2.p1.34.m34.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.34.m34.1.1.2" xref="S3.SS1.SSS2.2.p1.34.m34.1.1.2.cmml">S</mi><mn id="S3.SS1.SSS2.2.p1.34.m34.1.1.3" xref="S3.SS1.SSS2.2.p1.34.m34.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.34.m34.1b"><apply id="S3.SS1.SSS2.2.p1.34.m34.1.1.cmml" xref="S3.SS1.SSS2.2.p1.34.m34.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.34.m34.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.34.m34.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.34.m34.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.34.m34.1.1.2">𝑆</ci><cn id="S3.SS1.SSS2.2.p1.34.m34.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.34.m34.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.34.m34.1c">S_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.34.m34.1d">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="x_{3},x_{4}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.35.m35.2"><semantics id="S3.SS1.SSS2.2.p1.35.m35.2a"><mrow id="S3.SS1.SSS2.2.p1.35.m35.2.2.2" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.3.cmml"><msub id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.2" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.3" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.3.cmml">3</mn></msub><mo id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.3" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.35.m35.2b"><list id="S3.SS1.SSS2.2.p1.35.m35.2.2.3.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2"><apply id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.35.m35.1.1.1.1.3">3</cn></apply><apply id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.35.m35.2.2.2.2.3">4</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.35.m35.2c">x_{3},x_{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.35.m35.2d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> be their respective intersections with <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.36.m36.1"><semantics id="S3.SS1.SSS2.2.p1.36.m36.1a"><mrow id="S3.SS1.SSS2.2.p1.36.m36.1.1" xref="S3.SS1.SSS2.2.p1.36.m36.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.36.m36.1.1.1" rspace="0em" xref="S3.SS1.SSS2.2.p1.36.m36.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.36.m36.1.1.2" xref="S3.SS1.SSS2.2.p1.36.m36.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.36.m36.1b"><apply id="S3.SS1.SSS2.2.p1.36.m36.1.1.cmml" xref="S3.SS1.SSS2.2.p1.36.m36.1.1"><partialdiff id="S3.SS1.SSS2.2.p1.36.m36.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.36.m36.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.36.m36.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.36.m36.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.36.m36.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.36.m36.1d">∂ italic_D</annotation></semantics></math>. Then, since <math alttext="v\in\gamma_{1}\cap\gamma_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.37.m37.1"><semantics id="S3.SS1.SSS2.2.p1.37.m37.1a"><mrow id="S3.SS1.SSS2.2.p1.37.m37.1.1" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.37.m37.1.1.2" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.2.cmml">v</mi><mo id="S3.SS1.SSS2.2.p1.37.m37.1.1.1" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.2.p1.37.m37.1.1.3" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.cmml"><msub id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.2" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.3" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.1" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.1.cmml">∩</mo><msub id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.2" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.2.cmml">γ</mi><mn id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.3" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.37.m37.1b"><apply id="S3.SS1.SSS2.2.p1.37.m37.1.1.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1"><in id="S3.SS1.SSS2.2.p1.37.m37.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.1"></in><ci id="S3.SS1.SSS2.2.p1.37.m37.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.2">𝑣</ci><apply id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3"><intersect id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.1"></intersect><apply id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.2.3">1</cn></apply><apply id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.2">𝛾</ci><cn id="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.37.m37.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.37.m37.1c">v\in\gamma_{1}\cap\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.37.m37.1d">italic_v ∈ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, there is also a path <math alttext="\gamma^{\prime}\subset\overline{D}\cap\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.38.m38.1"><semantics id="S3.SS1.SSS2.2.p1.38.m38.1a"><mrow id="S3.SS1.SSS2.2.p1.38.m38.1.1" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.cmml"><msup id="S3.SS1.SSS2.2.p1.38.m38.1.1.2" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.2" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.3" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.2.p1.38.m38.1.1.1" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.1.cmml">⊂</mo><mrow id="S3.SS1.SSS2.2.p1.38.m38.1.1.3" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.cmml"><mover accent="true" id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.2" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.2.cmml">D</mi><mo id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.1" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.1.cmml">¯</mo></mover><mo id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.1" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.1.cmml">∩</mo><mrow id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.cmml"><mo id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.1" lspace="0em" rspace="0em" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.2" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.2.cmml">P</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.38.m38.1b"><apply id="S3.SS1.SSS2.2.p1.38.m38.1.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1"><subset id="S3.SS1.SSS2.2.p1.38.m38.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.1"></subset><apply id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.2.p1.38.m38.1.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3"><intersect id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.1"></intersect><apply id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2"><ci id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.1">¯</ci><ci id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.2.2">𝐷</ci></apply><apply id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3"><partialdiff id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.2.p1.38.m38.1.1.3.3.2">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.38.m38.1c">\gamma^{\prime}\subset\overline{D}\cap\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.38.m38.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ over¯ start_ARG italic_D end_ARG ∩ ∂ italic_P</annotation></semantics></math> with endpoints <math alttext="x_{3}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.39.m39.1"><semantics id="S3.SS1.SSS2.2.p1.39.m39.1a"><msub id="S3.SS1.SSS2.2.p1.39.m39.1.1" xref="S3.SS1.SSS2.2.p1.39.m39.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.39.m39.1.1.2" xref="S3.SS1.SSS2.2.p1.39.m39.1.1.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.39.m39.1.1.3" xref="S3.SS1.SSS2.2.p1.39.m39.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.39.m39.1b"><apply id="S3.SS1.SSS2.2.p1.39.m39.1.1.cmml" xref="S3.SS1.SSS2.2.p1.39.m39.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.39.m39.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.39.m39.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.39.m39.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.39.m39.1.1.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.39.m39.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.39.m39.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.39.m39.1c">x_{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.39.m39.1d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{4}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.40.m40.1"><semantics id="S3.SS1.SSS2.2.p1.40.m40.1a"><msub id="S3.SS1.SSS2.2.p1.40.m40.1.1" xref="S3.SS1.SSS2.2.p1.40.m40.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.40.m40.1.1.2" xref="S3.SS1.SSS2.2.p1.40.m40.1.1.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.40.m40.1.1.3" xref="S3.SS1.SSS2.2.p1.40.m40.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.40.m40.1b"><apply id="S3.SS1.SSS2.2.p1.40.m40.1.1.cmml" xref="S3.SS1.SSS2.2.p1.40.m40.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.40.m40.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.40.m40.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.40.m40.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.40.m40.1.1.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.40.m40.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.40.m40.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.40.m40.1c">x_{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.40.m40.1d">italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>. By compactness of <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.41.m41.1"><semantics id="S3.SS1.SSS2.2.p1.41.m41.1a"><mi id="S3.SS1.SSS2.2.p1.41.m41.1.1" xref="S3.SS1.SSS2.2.p1.41.m41.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.41.m41.1b"><ci id="S3.SS1.SSS2.2.p1.41.m41.1.1.cmml" xref="S3.SS1.SSS2.2.p1.41.m41.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.41.m41.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.41.m41.1d">italic_λ</annotation></semantics></math>, we can assume that <math alttext="\lambda^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.42.m42.1"><semantics id="S3.SS1.SSS2.2.p1.42.m42.1a"><msup id="S3.SS1.SSS2.2.p1.42.m42.1.1" xref="S3.SS1.SSS2.2.p1.42.m42.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.42.m42.1.1.2" xref="S3.SS1.SSS2.2.p1.42.m42.1.1.2.cmml">λ</mi><mo id="S3.SS1.SSS2.2.p1.42.m42.1.1.3" xref="S3.SS1.SSS2.2.p1.42.m42.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.42.m42.1b"><apply id="S3.SS1.SSS2.2.p1.42.m42.1.1.cmml" xref="S3.SS1.SSS2.2.p1.42.m42.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.42.m42.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.42.m42.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.42.m42.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.42.m42.1.1.2">𝜆</ci><ci id="S3.SS1.SSS2.2.p1.42.m42.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.42.m42.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.42.m42.1c">\lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.42.m42.1d">italic_λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> consists of finitely many line segments, i.e. <math alttext="\lambda^{\prime}\cup\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.43.m43.1"><semantics id="S3.SS1.SSS2.2.p1.43.m43.1a"><mrow id="S3.SS1.SSS2.2.p1.43.m43.1.1" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.cmml"><msup id="S3.SS1.SSS2.2.p1.43.m43.1.1.2" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2.cmml"><mi id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.2" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2.2.cmml">λ</mi><mo id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.3" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.2.p1.43.m43.1.1.1" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.1.cmml">∪</mo><msup id="S3.SS1.SSS2.2.p1.43.m43.1.1.3" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3.cmml"><mi id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.2" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3.2.cmml">γ</mi><mo id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.3" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.43.m43.1b"><apply id="S3.SS1.SSS2.2.p1.43.m43.1.1.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1"><union id="S3.SS1.SSS2.2.p1.43.m43.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.1"></union><apply id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.1.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.2.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2.2">𝜆</ci><ci id="S3.SS1.SSS2.2.p1.43.m43.1.1.2.3.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.1.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.2.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS2.2.p1.43.m43.1.1.3.3.cmml" xref="S3.SS1.SSS2.2.p1.43.m43.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.43.m43.1c">\lambda^{\prime}\cup\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.43.m43.1d">italic_λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> forms a planar graph, embedded in <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.44.m44.1"><semantics id="S3.SS1.SSS2.2.p1.44.m44.1a"><mi id="S3.SS1.SSS2.2.p1.44.m44.1.1" xref="S3.SS1.SSS2.2.p1.44.m44.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.44.m44.1b"><ci id="S3.SS1.SSS2.2.p1.44.m44.1.1.cmml" xref="S3.SS1.SSS2.2.p1.44.m44.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.44.m44.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.44.m44.1d">italic_D</annotation></semantics></math>. Since the intersection points occur in the order <math alttext="x_{1},x_{3},x_{2},x_{4}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.45.m45.4"><semantics id="S3.SS1.SSS2.2.p1.45.m45.4a"><mrow id="S3.SS1.SSS2.2.p1.45.m45.4.4.4" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.5.cmml"><msub id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.2" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.3" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.5" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.5.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.2" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.3" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.3.cmml">3</mn></msub><mo id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.6" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.5.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.cmml"><mi id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.2" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.3" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.3.cmml">2</mn></msub><mo id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.7" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.5.cmml">,</mo><msub id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.cmml"><mi id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.2" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.2.cmml">x</mi><mn id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.3" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.45.m45.4b"><list id="S3.SS1.SSS2.2.p1.45.m45.4.4.5.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4"><apply id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.45.m45.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.45.m45.2.2.2.2.3">3</cn></apply><apply id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.1.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.2.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.45.m45.3.3.3.3.3">2</cn></apply><apply id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.1.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4">subscript</csymbol><ci id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.2.cmml" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.2">𝑥</ci><cn id="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.3.cmml" type="integer" xref="S3.SS1.SSS2.2.p1.45.m45.4.4.4.4.3">4</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.45.m45.4c">x_{1},x_{3},x_{2},x_{4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.45.m45.4d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> cyclically on <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.46.m46.1"><semantics id="S3.SS1.SSS2.2.p1.46.m46.1a"><mrow id="S3.SS1.SSS2.2.p1.46.m46.1.1" xref="S3.SS1.SSS2.2.p1.46.m46.1.1.cmml"><mo id="S3.SS1.SSS2.2.p1.46.m46.1.1.1" rspace="0em" xref="S3.SS1.SSS2.2.p1.46.m46.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.2.p1.46.m46.1.1.2" xref="S3.SS1.SSS2.2.p1.46.m46.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.46.m46.1b"><apply id="S3.SS1.SSS2.2.p1.46.m46.1.1.cmml" xref="S3.SS1.SSS2.2.p1.46.m46.1.1"><partialdiff id="S3.SS1.SSS2.2.p1.46.m46.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.46.m46.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.2.p1.46.m46.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.46.m46.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.46.m46.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.46.m46.1d">∂ italic_D</annotation></semantics></math>, <math alttext="\lambda^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.47.m47.1"><semantics id="S3.SS1.SSS2.2.p1.47.m47.1a"><msup id="S3.SS1.SSS2.2.p1.47.m47.1.1" xref="S3.SS1.SSS2.2.p1.47.m47.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.47.m47.1.1.2" xref="S3.SS1.SSS2.2.p1.47.m47.1.1.2.cmml">λ</mi><mo id="S3.SS1.SSS2.2.p1.47.m47.1.1.3" xref="S3.SS1.SSS2.2.p1.47.m47.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.47.m47.1b"><apply id="S3.SS1.SSS2.2.p1.47.m47.1.1.cmml" xref="S3.SS1.SSS2.2.p1.47.m47.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.47.m47.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.47.m47.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.47.m47.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.47.m47.1.1.2">𝜆</ci><ci id="S3.SS1.SSS2.2.p1.47.m47.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.47.m47.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.47.m47.1c">\lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.47.m47.1d">italic_λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.48.m48.1"><semantics id="S3.SS1.SSS2.2.p1.48.m48.1a"><msup id="S3.SS1.SSS2.2.p1.48.m48.1.1" xref="S3.SS1.SSS2.2.p1.48.m48.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.48.m48.1.1.2" xref="S3.SS1.SSS2.2.p1.48.m48.1.1.2.cmml">γ</mi><mo id="S3.SS1.SSS2.2.p1.48.m48.1.1.3" xref="S3.SS1.SSS2.2.p1.48.m48.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.48.m48.1b"><apply id="S3.SS1.SSS2.2.p1.48.m48.1.1.cmml" xref="S3.SS1.SSS2.2.p1.48.m48.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.2.p1.48.m48.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.48.m48.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.2.p1.48.m48.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.48.m48.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.2.p1.48.m48.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.48.m48.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.48.m48.1c">\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.48.m48.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> must intersect, see e.g. <span class="ltx_ERROR undefined" id="S3.SS1.SSS2.2.p1.50.1">\citet</span>[Ch. 71]schrijver2003. Thus, there is a point <math alttext="x\in\lambda" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.49.m49.1"><semantics id="S3.SS1.SSS2.2.p1.49.m49.1a"><mrow id="S3.SS1.SSS2.2.p1.49.m49.1.1" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.cmml"><mi id="S3.SS1.SSS2.2.p1.49.m49.1.1.2" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS2.2.p1.49.m49.1.1.1" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.1.cmml">∈</mo><mi id="S3.SS1.SSS2.2.p1.49.m49.1.1.3" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.49.m49.1b"><apply id="S3.SS1.SSS2.2.p1.49.m49.1.1.cmml" xref="S3.SS1.SSS2.2.p1.49.m49.1.1"><in id="S3.SS1.SSS2.2.p1.49.m49.1.1.1.cmml" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.1"></in><ci id="S3.SS1.SSS2.2.p1.49.m49.1.1.2.cmml" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.2">𝑥</ci><ci id="S3.SS1.SSS2.2.p1.49.m49.1.1.3.cmml" xref="S3.SS1.SSS2.2.p1.49.m49.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.49.m49.1c">x\in\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.49.m49.1d">italic_x ∈ italic_λ</annotation></semantics></math> that lies on the boundary of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p1.50.m50.1"><semantics id="S3.SS1.SSS2.2.p1.50.m50.1a"><mi id="S3.SS1.SSS2.2.p1.50.m50.1.1" xref="S3.SS1.SSS2.2.p1.50.m50.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p1.50.m50.1b"><ci id="S3.SS1.SSS2.2.p1.50.m50.1.1.cmml" xref="S3.SS1.SSS2.2.p1.50.m50.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p1.50.m50.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p1.50.m50.1d">italic_P</annotation></semantics></math>, which is a contradiction. ∎</p> </div> </div> <figure class="ltx_figure" id="S3.F7"> <p class="ltx_p ltx_align_center" id="S3.F7.1"><span class="ltx_text" id="S3.F7.1.1"><foreignobject height="80.2pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="140.2pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="154" id="S3.F7.1.1.1.g1" src="x19.png" width="270"/></foreignobject></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Two arcs subdivide <math alttext="\mathds{R}^{2}\setminus D" class="ltx_Math" display="inline" id="S3.F7.8.m1.1"><semantics id="S3.F7.8.m1.1b"><mrow id="S3.F7.8.m1.1.1" xref="S3.F7.8.m1.1.1.cmml"><msup id="S3.F7.8.m1.1.1.2" xref="S3.F7.8.m1.1.1.2.cmml"><mi id="S3.F7.8.m1.1.1.2.2" xref="S3.F7.8.m1.1.1.2.2.cmml">ℝ</mi><mn id="S3.F7.8.m1.1.1.2.3" xref="S3.F7.8.m1.1.1.2.3.cmml">2</mn></msup><mo id="S3.F7.8.m1.1.1.1" xref="S3.F7.8.m1.1.1.1.cmml">∖</mo><mi id="S3.F7.8.m1.1.1.3" xref="S3.F7.8.m1.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.8.m1.1c"><apply id="S3.F7.8.m1.1.1.cmml" xref="S3.F7.8.m1.1.1"><setdiff id="S3.F7.8.m1.1.1.1.cmml" xref="S3.F7.8.m1.1.1.1"></setdiff><apply id="S3.F7.8.m1.1.1.2.cmml" xref="S3.F7.8.m1.1.1.2"><csymbol cd="ambiguous" id="S3.F7.8.m1.1.1.2.1.cmml" xref="S3.F7.8.m1.1.1.2">superscript</csymbol><ci id="S3.F7.8.m1.1.1.2.2.cmml" xref="S3.F7.8.m1.1.1.2.2">ℝ</ci><cn id="S3.F7.8.m1.1.1.2.3.cmml" type="integer" xref="S3.F7.8.m1.1.1.2.3">2</cn></apply><ci id="S3.F7.8.m1.1.1.3.cmml" xref="S3.F7.8.m1.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.8.m1.1d">\mathds{R}^{2}\setminus D</annotation><annotation encoding="application/x-llamapun" id="S3.F7.8.m1.1e">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_D</annotation></semantics></math> into four sectors. <math alttext="y_{1},y_{2}\in P" class="ltx_Math" display="inline" id="S3.F7.9.m2.2"><semantics id="S3.F7.9.m2.2b"><mrow id="S3.F7.9.m2.2.2" xref="S3.F7.9.m2.2.2.cmml"><mrow id="S3.F7.9.m2.2.2.2.2" xref="S3.F7.9.m2.2.2.2.3.cmml"><msub id="S3.F7.9.m2.1.1.1.1.1" xref="S3.F7.9.m2.1.1.1.1.1.cmml"><mi id="S3.F7.9.m2.1.1.1.1.1.2" xref="S3.F7.9.m2.1.1.1.1.1.2.cmml">y</mi><mn id="S3.F7.9.m2.1.1.1.1.1.3" xref="S3.F7.9.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.F7.9.m2.2.2.2.2.3" xref="S3.F7.9.m2.2.2.2.3.cmml">,</mo><msub id="S3.F7.9.m2.2.2.2.2.2" xref="S3.F7.9.m2.2.2.2.2.2.cmml"><mi id="S3.F7.9.m2.2.2.2.2.2.2" xref="S3.F7.9.m2.2.2.2.2.2.2.cmml">y</mi><mn id="S3.F7.9.m2.2.2.2.2.2.3" xref="S3.F7.9.m2.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S3.F7.9.m2.2.2.3" xref="S3.F7.9.m2.2.2.3.cmml">∈</mo><mi id="S3.F7.9.m2.2.2.4" xref="S3.F7.9.m2.2.2.4.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.9.m2.2c"><apply id="S3.F7.9.m2.2.2.cmml" xref="S3.F7.9.m2.2.2"><in id="S3.F7.9.m2.2.2.3.cmml" xref="S3.F7.9.m2.2.2.3"></in><list id="S3.F7.9.m2.2.2.2.3.cmml" xref="S3.F7.9.m2.2.2.2.2"><apply id="S3.F7.9.m2.1.1.1.1.1.cmml" xref="S3.F7.9.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F7.9.m2.1.1.1.1.1.1.cmml" xref="S3.F7.9.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.F7.9.m2.1.1.1.1.1.2.cmml" xref="S3.F7.9.m2.1.1.1.1.1.2">𝑦</ci><cn id="S3.F7.9.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.F7.9.m2.1.1.1.1.1.3">1</cn></apply><apply id="S3.F7.9.m2.2.2.2.2.2.cmml" xref="S3.F7.9.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.F7.9.m2.2.2.2.2.2.1.cmml" xref="S3.F7.9.m2.2.2.2.2.2">subscript</csymbol><ci id="S3.F7.9.m2.2.2.2.2.2.2.cmml" xref="S3.F7.9.m2.2.2.2.2.2.2">𝑦</ci><cn id="S3.F7.9.m2.2.2.2.2.2.3.cmml" type="integer" xref="S3.F7.9.m2.2.2.2.2.2.3">2</cn></apply></list><ci id="S3.F7.9.m2.2.2.4.cmml" xref="S3.F7.9.m2.2.2.4">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.9.m2.2d">y_{1},y_{2}\in P</annotation><annotation encoding="application/x-llamapun" id="S3.F7.9.m2.2e">italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_P</annotation></semantics></math> are contained in different sectors and are connected by a path <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.F7.10.m3.1"><semantics id="S3.F7.10.m3.1b"><mi id="S3.F7.10.m3.1.1" xref="S3.F7.10.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.F7.10.m3.1c"><ci id="S3.F7.10.m3.1.1.cmml" xref="S3.F7.10.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.10.m3.1d">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.F7.10.m3.1e">italic_λ</annotation></semantics></math> which enters and leaves each sector only through <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.F7.11.m4.1"><semantics id="S3.F7.11.m4.1b"><mrow id="S3.F7.11.m4.1.1" xref="S3.F7.11.m4.1.1.cmml"><mo id="S3.F7.11.m4.1.1.1" rspace="0em" xref="S3.F7.11.m4.1.1.1.cmml">∂</mo><mi id="S3.F7.11.m4.1.1.2" xref="S3.F7.11.m4.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.11.m4.1c"><apply id="S3.F7.11.m4.1.1.cmml" xref="S3.F7.11.m4.1.1"><partialdiff id="S3.F7.11.m4.1.1.1.cmml" xref="S3.F7.11.m4.1.1.1"></partialdiff><ci id="S3.F7.11.m4.1.1.2.cmml" xref="S3.F7.11.m4.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.11.m4.1d">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.F7.11.m4.1e">∂ italic_D</annotation></semantics></math>. If the arcs share a vertex in <math alttext="D" class="ltx_Math" display="inline" id="S3.F7.12.m5.1"><semantics id="S3.F7.12.m5.1b"><mi id="S3.F7.12.m5.1.1" xref="S3.F7.12.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.F7.12.m5.1c"><ci id="S3.F7.12.m5.1.1.cmml" xref="S3.F7.12.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.12.m5.1d">D</annotation><annotation encoding="application/x-llamapun" id="S3.F7.12.m5.1e">italic_D</annotation></semantics></math>, then <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.F7.13.m6.1"><semantics id="S3.F7.13.m6.1b"><mi id="S3.F7.13.m6.1.1" xref="S3.F7.13.m6.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.F7.13.m6.1c"><ci id="S3.F7.13.m6.1.1.cmml" xref="S3.F7.13.m6.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.13.m6.1d">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.F7.13.m6.1e">italic_λ</annotation></semantics></math> must intersect one of the arcs.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS2.p6"> <p class="ltx_p" id="S3.SS1.SSS2.p6.2">Next, I will constructively show that, for fixed <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p6.1.m1.1"><semantics id="S3.SS1.SSS2.p6.1.m1.1a"><mi id="S3.SS1.SSS2.p6.1.m1.1.1" xref="S3.SS1.SSS2.p6.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p6.1.m1.1b"><ci id="S3.SS1.SSS2.p6.1.m1.1.1.cmml" xref="S3.SS1.SSS2.p6.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p6.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p6.1.m1.1d">italic_x</annotation></semantics></math>, the number of polygonal arcs in <math alttext="\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p6.2.m2.1"><semantics id="S3.SS1.SSS2.p6.2.m2.1a"><mrow id="S3.SS1.SSS2.p6.2.m2.1.1" xref="S3.SS1.SSS2.p6.2.m2.1.1.cmml"><mo id="S3.SS1.SSS2.p6.2.m2.1.1.1" rspace="0em" xref="S3.SS1.SSS2.p6.2.m2.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.p6.2.m2.1.1.2" xref="S3.SS1.SSS2.p6.2.m2.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p6.2.m2.1b"><apply id="S3.SS1.SSS2.p6.2.m2.1.1.cmml" xref="S3.SS1.SSS2.p6.2.m2.1.1"><partialdiff id="S3.SS1.SSS2.p6.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.p6.2.m2.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.p6.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.p6.2.m2.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p6.2.m2.1c">\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p6.2.m2.1d">∂ italic_P</annotation></semantics></math> can be reduced without affecting (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E17" title="Equation 17 ‣ Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">17</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.10.10">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.1.m1.1"><semantics id="S3.Thmtheorem8.p1.1.1.m1.1a"><mi id="S3.Thmtheorem8.p1.1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.1.m1.1b"><ci id="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon whose boundary consists of <math alttext="n_{a}(P)\geq 1" 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.m2.1.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.cmml"><mrow id="S3.Thmtheorem8.p1.2.2.m2.1.2.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.cmml"><msub id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.3.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.cmml"><mo id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem8.p1.2.2.m2.1.2.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.1.cmml">≥</mo><mn id="S3.Thmtheorem8.p1.2.2.m2.1.2.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.2.m2.1b"><apply id="S3.Thmtheorem8.p1.2.2.m2.1.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2"><geq id="S3.Thmtheorem8.p1.2.2.m2.1.2.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.1"></geq><apply id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2"><times id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.1"></times><apply id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.3.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.2.2.3">𝑎</ci></apply><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1">𝑃</ci></apply><cn id="S3.Thmtheorem8.p1.2.2.m2.1.2.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.2.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.2.m2.1c">n_{a}(P)\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) ≥ 1</annotation></semantics></math> polygonal arcs without lines, and let <math alttext="x\in\mathds{R}^{2}" 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.m3.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mn id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1"><in id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2">𝑥</ci><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2">ℝ</ci><cn id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.3.m3.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> be in <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.4.m4.1"><semantics id="S3.Thmtheorem8.p1.4.4.m4.1a"><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.4.m4.1b"><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.4.m4.1d">italic_P</annotation></semantics></math>-general position. Then, there is a polygon <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.5.5.m5.1"><semantics id="S3.Thmtheorem8.p1.5.5.m5.1a"><msup id="S3.Thmtheorem8.p1.5.5.m5.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.5.5.m5.1b"><apply id="S3.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1">superscript</csymbol><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.5.5.m5.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.5.5.m5.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, also without lines and with <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.6.6.m6.1"><semantics id="S3.Thmtheorem8.p1.6.6.m6.1a"><mi id="S3.Thmtheorem8.p1.6.6.m6.1.1" xref="S3.Thmtheorem8.p1.6.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.6.6.m6.1b"><ci id="S3.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem8.p1.6.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.6.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.6.6.m6.1d">italic_x</annotation></semantics></math> in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.7.7.m7.1"><semantics id="S3.Thmtheorem8.p1.7.7.m7.1a"><msup id="S3.Thmtheorem8.p1.7.7.m7.1.1" xref="S3.Thmtheorem8.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem8.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem8.p1.7.7.m7.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.7.7.m7.1.1.3" xref="S3.Thmtheorem8.p1.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.7.7.m7.1b"><apply id="S3.Thmtheorem8.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem8.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem8.p1.7.7.m7.1.1">superscript</csymbol><ci id="S3.Thmtheorem8.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem8.p1.7.7.m7.1.1.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem8.p1.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.7.7.m7.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.7.7.m7.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position, such that <math alttext="\partial P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.8.8.m8.1"><semantics id="S3.Thmtheorem8.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem8.p1.8.8.m8.1.1" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.cmml"><mo id="S3.Thmtheorem8.p1.8.8.m8.1.1.1" rspace="0em" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.1.cmml">∂</mo><msup id="S3.Thmtheorem8.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2.cmml"><mi id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.2" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.3" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.8.8.m8.1b"><apply id="S3.Thmtheorem8.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1"><partialdiff id="S3.Thmtheorem8.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.1"></partialdiff><apply id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.8.8.m8.1.1.2.3.cmml" xref="S3.Thmtheorem8.p1.8.8.m8.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.8.8.m8.1c">\partial P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.8.8.m8.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> consists of <math alttext="n_{a}(P^{\prime})=n_{a}(P)-1" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.9.9.m9.2"><semantics id="S3.Thmtheorem8.p1.9.9.m9.2a"><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.cmml"><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2.1" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.cmml"><msub id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.cmml"><mi id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.2.cmml">n</mi><mi id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.3.cmml">a</mi></msub><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.2" stretchy="false" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.cmml"><mi id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.3" stretchy="false" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.2.cmml">=</mo><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.cmml"><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.cmml"><msub id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.cmml"><mi id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.2.cmml">n</mi><mi id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.3.cmml">a</mi></msub><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.1" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.3.2" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.cmml"><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem8.p1.9.9.m9.1.1" xref="S3.Thmtheorem8.p1.9.9.m9.1.1.cmml">P</mi><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.1" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.1.cmml">−</mo><mn id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.3" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.9.9.m9.2b"><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2"><eq id="S3.Thmtheorem8.p1.9.9.m9.2.2.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.2"></eq><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1"><times id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.2"></times><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3">subscript</csymbol><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.2">𝑛</ci><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.3.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.3.3">𝑎</ci></apply><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3"><minus id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.1"></minus><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2"><times id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.1"></times><apply id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2">subscript</csymbol><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.2.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.2">𝑛</ci><ci id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.3.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.2.2.3">𝑎</ci></apply><ci id="S3.Thmtheorem8.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem8.p1.9.9.m9.1.1">𝑃</ci></apply><cn id="S3.Thmtheorem8.p1.9.9.m9.2.2.3.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.9.9.m9.2.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.9.9.m9.2c">n_{a}(P^{\prime})=n_{a}(P)-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.9.9.m9.2d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - 1</annotation></semantics></math> arcs and both sides of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E17" title="Equation 17 ‣ Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">17</span></a>) are preserved at <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.10.10.m10.1"><semantics id="S3.Thmtheorem8.p1.10.10.m10.1a"><mi id="S3.Thmtheorem8.p1.10.10.m10.1.1" xref="S3.Thmtheorem8.p1.10.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.10.10.m10.1b"><ci id="S3.Thmtheorem8.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem8.p1.10.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.10.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.10.10.m10.1d">italic_x</annotation></semantics></math>. That is,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E18"> <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="\mathds{1}_{P}(x)-1=\mathds{1}_{P^{\prime}}(x)-1," class="ltx_Math" display="block" id="S3.E18.m1.3"><semantics id="S3.E18.m1.3a"><mrow id="S3.E18.m1.3.3.1" xref="S3.E18.m1.3.3.1.1.cmml"><mrow id="S3.E18.m1.3.3.1.1" xref="S3.E18.m1.3.3.1.1.cmml"><mrow id="S3.E18.m1.3.3.1.1.2" xref="S3.E18.m1.3.3.1.1.2.cmml"><mrow id="S3.E18.m1.3.3.1.1.2.2" xref="S3.E18.m1.3.3.1.1.2.2.cmml"><msub id="S3.E18.m1.3.3.1.1.2.2.2" xref="S3.E18.m1.3.3.1.1.2.2.2.cmml"><mn id="S3.E18.m1.3.3.1.1.2.2.2.2" xref="S3.E18.m1.3.3.1.1.2.2.2.2.cmml">𝟙</mn><mi id="S3.E18.m1.3.3.1.1.2.2.2.3" xref="S3.E18.m1.3.3.1.1.2.2.2.3.cmml">P</mi></msub><mo id="S3.E18.m1.3.3.1.1.2.2.1" xref="S3.E18.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S3.E18.m1.3.3.1.1.2.2.3.2" xref="S3.E18.m1.3.3.1.1.2.2.cmml"><mo id="S3.E18.m1.3.3.1.1.2.2.3.2.1" stretchy="false" xref="S3.E18.m1.3.3.1.1.2.2.cmml">(</mo><mi id="S3.E18.m1.1.1" xref="S3.E18.m1.1.1.cmml">x</mi><mo id="S3.E18.m1.3.3.1.1.2.2.3.2.2" stretchy="false" xref="S3.E18.m1.3.3.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.E18.m1.3.3.1.1.2.1" xref="S3.E18.m1.3.3.1.1.2.1.cmml">−</mo><mn id="S3.E18.m1.3.3.1.1.2.3" xref="S3.E18.m1.3.3.1.1.2.3.cmml">1</mn></mrow><mo id="S3.E18.m1.3.3.1.1.1" xref="S3.E18.m1.3.3.1.1.1.cmml">=</mo><mrow id="S3.E18.m1.3.3.1.1.3" xref="S3.E18.m1.3.3.1.1.3.cmml"><mrow id="S3.E18.m1.3.3.1.1.3.2" xref="S3.E18.m1.3.3.1.1.3.2.cmml"><msub id="S3.E18.m1.3.3.1.1.3.2.2" xref="S3.E18.m1.3.3.1.1.3.2.2.cmml"><mn id="S3.E18.m1.3.3.1.1.3.2.2.2" xref="S3.E18.m1.3.3.1.1.3.2.2.2.cmml">𝟙</mn><msup id="S3.E18.m1.3.3.1.1.3.2.2.3" xref="S3.E18.m1.3.3.1.1.3.2.2.3.cmml"><mi id="S3.E18.m1.3.3.1.1.3.2.2.3.2" xref="S3.E18.m1.3.3.1.1.3.2.2.3.2.cmml">P</mi><mo id="S3.E18.m1.3.3.1.1.3.2.2.3.3" xref="S3.E18.m1.3.3.1.1.3.2.2.3.3.cmml">′</mo></msup></msub><mo id="S3.E18.m1.3.3.1.1.3.2.1" xref="S3.E18.m1.3.3.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E18.m1.3.3.1.1.3.2.3.2" xref="S3.E18.m1.3.3.1.1.3.2.cmml"><mo id="S3.E18.m1.3.3.1.1.3.2.3.2.1" stretchy="false" xref="S3.E18.m1.3.3.1.1.3.2.cmml">(</mo><mi id="S3.E18.m1.2.2" xref="S3.E18.m1.2.2.cmml">x</mi><mo id="S3.E18.m1.3.3.1.1.3.2.3.2.2" stretchy="false" xref="S3.E18.m1.3.3.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S3.E18.m1.3.3.1.1.3.1" xref="S3.E18.m1.3.3.1.1.3.1.cmml">−</mo><mn id="S3.E18.m1.3.3.1.1.3.3" xref="S3.E18.m1.3.3.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S3.E18.m1.3.3.1.2" xref="S3.E18.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E18.m1.3b"><apply id="S3.E18.m1.3.3.1.1.cmml" xref="S3.E18.m1.3.3.1"><eq id="S3.E18.m1.3.3.1.1.1.cmml" xref="S3.E18.m1.3.3.1.1.1"></eq><apply id="S3.E18.m1.3.3.1.1.2.cmml" xref="S3.E18.m1.3.3.1.1.2"><minus id="S3.E18.m1.3.3.1.1.2.1.cmml" xref="S3.E18.m1.3.3.1.1.2.1"></minus><apply id="S3.E18.m1.3.3.1.1.2.2.cmml" xref="S3.E18.m1.3.3.1.1.2.2"><times id="S3.E18.m1.3.3.1.1.2.2.1.cmml" xref="S3.E18.m1.3.3.1.1.2.2.1"></times><apply id="S3.E18.m1.3.3.1.1.2.2.2.cmml" xref="S3.E18.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E18.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.E18.m1.3.3.1.1.2.2.2">subscript</csymbol><cn id="S3.E18.m1.3.3.1.1.2.2.2.2.cmml" type="integer" xref="S3.E18.m1.3.3.1.1.2.2.2.2">1</cn><ci id="S3.E18.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.E18.m1.3.3.1.1.2.2.2.3">𝑃</ci></apply><ci id="S3.E18.m1.1.1.cmml" xref="S3.E18.m1.1.1">𝑥</ci></apply><cn id="S3.E18.m1.3.3.1.1.2.3.cmml" type="integer" xref="S3.E18.m1.3.3.1.1.2.3">1</cn></apply><apply id="S3.E18.m1.3.3.1.1.3.cmml" xref="S3.E18.m1.3.3.1.1.3"><minus id="S3.E18.m1.3.3.1.1.3.1.cmml" xref="S3.E18.m1.3.3.1.1.3.1"></minus><apply id="S3.E18.m1.3.3.1.1.3.2.cmml" xref="S3.E18.m1.3.3.1.1.3.2"><times id="S3.E18.m1.3.3.1.1.3.2.1.cmml" xref="S3.E18.m1.3.3.1.1.3.2.1"></times><apply id="S3.E18.m1.3.3.1.1.3.2.2.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E18.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2">subscript</csymbol><cn id="S3.E18.m1.3.3.1.1.3.2.2.2.cmml" type="integer" xref="S3.E18.m1.3.3.1.1.3.2.2.2">1</cn><apply id="S3.E18.m1.3.3.1.1.3.2.2.3.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.E18.m1.3.3.1.1.3.2.2.3.1.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2.3">superscript</csymbol><ci id="S3.E18.m1.3.3.1.1.3.2.2.3.2.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2.3.2">𝑃</ci><ci id="S3.E18.m1.3.3.1.1.3.2.2.3.3.cmml" xref="S3.E18.m1.3.3.1.1.3.2.2.3.3">′</ci></apply></apply><ci id="S3.E18.m1.2.2.cmml" xref="S3.E18.m1.2.2">𝑥</ci></apply><cn id="S3.E18.m1.3.3.1.1.3.3.cmml" type="integer" xref="S3.E18.m1.3.3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E18.m1.3c">\mathds{1}_{P}(x)-1=\mathds{1}_{P^{\prime}}(x)-1,</annotation><annotation encoding="application/x-llamapun" id="S3.E18.m1.3d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 = blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) - 1 ,</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">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem8.p1.14"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.14.1">and,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E19"> <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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)\\ =\sum_{v\in V(P^{\prime})}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_{b}(% P^{\prime})}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P^{\prime})." class="ltx_Math" display="block" id="S3.E19.m1.47"><semantics id="S3.E19.m1.47a"><mtable displaystyle="true" id="S3.E19.m1.47.47.2" rowspacing="0pt"><mtr id="S3.E19.m1.47.47.2a"><mtd class="ltx_align_left" columnalign="left" id="S3.E19.m1.47.47.2b"><mrow id="S3.E19.m1.21.21.21.21.21"><mrow id="S3.E19.m1.21.21.21.21.21.22"><munder id="S3.E19.m1.21.21.21.21.21.22.1"><mo id="S3.E19.m1.1.1.1.1.1.1" movablelimits="false" xref="S3.E19.m1.1.1.1.1.1.1.cmml">∑</mo><mrow id="S3.E19.m1.2.2.2.2.2.2.1" xref="S3.E19.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E19.m1.2.2.2.2.2.2.1.3" xref="S3.E19.m1.2.2.2.2.2.2.1.3.cmml">v</mi><mo id="S3.E19.m1.2.2.2.2.2.2.1.2" xref="S3.E19.m1.2.2.2.2.2.2.1.2.cmml">∈</mo><mrow id="S3.E19.m1.2.2.2.2.2.2.1.4" xref="S3.E19.m1.2.2.2.2.2.2.1.4.cmml"><mi id="S3.E19.m1.2.2.2.2.2.2.1.4.2" xref="S3.E19.m1.2.2.2.2.2.2.1.4.2.cmml">V</mi><mo id="S3.E19.m1.2.2.2.2.2.2.1.4.1" xref="S3.E19.m1.2.2.2.2.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.E19.m1.2.2.2.2.2.2.1.4.3.2" xref="S3.E19.m1.2.2.2.2.2.2.1.4.cmml"><mo id="S3.E19.m1.2.2.2.2.2.2.1.4.3.2.1" stretchy="false" xref="S3.E19.m1.2.2.2.2.2.2.1.4.cmml">(</mo><mi id="S3.E19.m1.2.2.2.2.2.2.1.1" xref="S3.E19.m1.2.2.2.2.2.2.1.1.cmml">P</mi><mo id="S3.E19.m1.2.2.2.2.2.2.1.4.3.2.2" stretchy="false" xref="S3.E19.m1.2.2.2.2.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E19.m1.21.21.21.21.21.22.2"><msub id="S3.E19.m1.21.21.21.21.21.22.2.2"><mn id="S3.E19.m1.3.3.3.3.3.3" xref="S3.E19.m1.3.3.3.3.3.3.cmml">𝟙</mn><msubsup id="S3.E19.m1.4.4.4.4.4.4.1" xref="S3.E19.m1.4.4.4.4.4.4.1.cmml"><mi id="S3.E19.m1.4.4.4.4.4.4.1.2.2" xref="S3.E19.m1.4.4.4.4.4.4.1.2.2.cmml">Q</mi><mi id="S3.E19.m1.4.4.4.4.4.4.1.3" xref="S3.E19.m1.4.4.4.4.4.4.1.3.cmml">P</mi><mi id="S3.E19.m1.4.4.4.4.4.4.1.2.3" xref="S3.E19.m1.4.4.4.4.4.4.1.2.3.cmml">v</mi></msubsup></msub><mo id="S3.E19.m1.21.21.21.21.21.22.2.1" xref="S3.E19.m1.46.46.1.1.1.cmml">⁢</mo><mrow id="S3.E19.m1.21.21.21.21.21.22.2.3"><mo id="S3.E19.m1.5.5.5.5.5.5" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">(</mo><mi id="S3.E19.m1.6.6.6.6.6.6" xref="S3.E19.m1.6.6.6.6.6.6.cmml">x</mi><mo id="S3.E19.m1.7.7.7.7.7.7" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E19.m1.8.8.8.8.8.8" rspace="0.055em" xref="S3.E19.m1.8.8.8.8.8.8.cmml">−</mo><mrow id="S3.E19.m1.21.21.21.21.21.23"><munder id="S3.E19.m1.21.21.21.21.21.23.1"><mo id="S3.E19.m1.9.9.9.9.9.9" movablelimits="false" xref="S3.E19.m1.9.9.9.9.9.9.cmml">∑</mo><mrow id="S3.E19.m1.10.10.10.10.10.10.1" xref="S3.E19.m1.10.10.10.10.10.10.1.cmml"><mi id="S3.E19.m1.10.10.10.10.10.10.1.3" xref="S3.E19.m1.10.10.10.10.10.10.1.3.cmml">e</mi><mo id="S3.E19.m1.10.10.10.10.10.10.1.2" xref="S3.E19.m1.10.10.10.10.10.10.1.2.cmml">∈</mo><mrow id="S3.E19.m1.10.10.10.10.10.10.1.4" xref="S3.E19.m1.10.10.10.10.10.10.1.4.cmml"><msub id="S3.E19.m1.10.10.10.10.10.10.1.4.2" xref="S3.E19.m1.10.10.10.10.10.10.1.4.2.cmml"><mi id="S3.E19.m1.10.10.10.10.10.10.1.4.2.2" xref="S3.E19.m1.10.10.10.10.10.10.1.4.2.2.cmml">E</mi><mi id="S3.E19.m1.10.10.10.10.10.10.1.4.2.3" xref="S3.E19.m1.10.10.10.10.10.10.1.4.2.3.cmml">b</mi></msub><mo id="S3.E19.m1.10.10.10.10.10.10.1.4.1" xref="S3.E19.m1.10.10.10.10.10.10.1.4.1.cmml">⁢</mo><mrow id="S3.E19.m1.10.10.10.10.10.10.1.4.3.2" xref="S3.E19.m1.10.10.10.10.10.10.1.4.cmml"><mo id="S3.E19.m1.10.10.10.10.10.10.1.4.3.2.1" stretchy="false" xref="S3.E19.m1.10.10.10.10.10.10.1.4.cmml">(</mo><mi id="S3.E19.m1.10.10.10.10.10.10.1.1" xref="S3.E19.m1.10.10.10.10.10.10.1.1.cmml">P</mi><mo id="S3.E19.m1.10.10.10.10.10.10.1.4.3.2.2" stretchy="false" xref="S3.E19.m1.10.10.10.10.10.10.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E19.m1.21.21.21.21.21.23.2"><msub id="S3.E19.m1.21.21.21.21.21.23.2.2"><mn id="S3.E19.m1.11.11.11.11.11.11" xref="S3.E19.m1.11.11.11.11.11.11.cmml">𝟙</mn><msubsup id="S3.E19.m1.12.12.12.12.12.12.1" xref="S3.E19.m1.12.12.12.12.12.12.1.cmml"><mi id="S3.E19.m1.12.12.12.12.12.12.1.2.2" xref="S3.E19.m1.12.12.12.12.12.12.1.2.2.cmml">H</mi><mi id="S3.E19.m1.12.12.12.12.12.12.1.3" xref="S3.E19.m1.12.12.12.12.12.12.1.3.cmml">P</mi><mi id="S3.E19.m1.12.12.12.12.12.12.1.2.3" xref="S3.E19.m1.12.12.12.12.12.12.1.2.3.cmml">e</mi></msubsup></msub><mo id="S3.E19.m1.21.21.21.21.21.23.2.1" xref="S3.E19.m1.46.46.1.1.1.cmml">⁢</mo><mrow id="S3.E19.m1.21.21.21.21.21.23.2.3"><mo id="S3.E19.m1.13.13.13.13.13.13" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">(</mo><mi id="S3.E19.m1.14.14.14.14.14.14" xref="S3.E19.m1.14.14.14.14.14.14.cmml">x</mi><mo id="S3.E19.m1.15.15.15.15.15.15" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E19.m1.8.8.8.8.8.8a" xref="S3.E19.m1.8.8.8.8.8.8.cmml">−</mo><mrow id="S3.E19.m1.21.21.21.21.21.24"><msub id="S3.E19.m1.21.21.21.21.21.24.2"><mi id="S3.E19.m1.17.17.17.17.17.17" xref="S3.E19.m1.17.17.17.17.17.17.cmml">n</mi><mi id="S3.E19.m1.18.18.18.18.18.18.1" xref="S3.E19.m1.18.18.18.18.18.18.1.cmml">a</mi></msub><mo id="S3.E19.m1.21.21.21.21.21.24.1" xref="S3.E19.m1.46.46.1.1.1.cmml">⁢</mo><mrow id="S3.E19.m1.21.21.21.21.21.24.3"><mo id="S3.E19.m1.19.19.19.19.19.19" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">(</mo><mi id="S3.E19.m1.20.20.20.20.20.20" xref="S3.E19.m1.20.20.20.20.20.20.cmml">P</mi><mo id="S3.E19.m1.21.21.21.21.21.21" stretchy="false" xref="S3.E19.m1.46.46.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E19.m1.47.47.2c"><mtd class="ltx_align_right" columnalign="right" id="S3.E19.m1.47.47.2d"><mrow id="S3.E19.m1.47.47.2.46.25.25.25"><mrow id="S3.E19.m1.47.47.2.46.25.25.25.1"><mi id="S3.E19.m1.47.47.2.46.25.25.25.1.2" xref="S3.E19.m1.46.46.1.1.1.cmml"></mi><mo id="S3.E19.m1.22.22.22.1.1.1" rspace="0.111em" xref="S3.E19.m1.22.22.22.1.1.1.cmml">=</mo><mrow id="S3.E19.m1.47.47.2.46.25.25.25.1.1"><mrow id="S3.E19.m1.47.47.2.46.25.25.25.1.1.2"><munder id="S3.E19.m1.47.47.2.46.25.25.25.1.1.2.1"><mo id="S3.E19.m1.23.23.23.2.2.2" movablelimits="false" xref="S3.E19.m1.23.23.23.2.2.2.cmml">∑</mo><mrow id="S3.E19.m1.24.24.24.3.3.3.1" xref="S3.E19.m1.24.24.24.3.3.3.1.cmml"><mi id="S3.E19.m1.24.24.24.3.3.3.1.3" xref="S3.E19.m1.24.24.24.3.3.3.1.3.cmml">v</mi><mo id="S3.E19.m1.24.24.24.3.3.3.1.2" xref="S3.E19.m1.24.24.24.3.3.3.1.2.cmml">∈</mo><mrow id="S3.E19.m1.24.24.24.3.3.3.1.1" xref="S3.E19.m1.24.24.24.3.3.3.1.1.cmml"><mi id="S3.E19.m1.24.24.24.3.3.3.1.1.3" xref="S3.E19.m1.24.24.24.3.3.3.1.1.3.cmml">V</mi><mo id="S3.E19.m1.24.24.24.3.3.3.1.1.2" xref="S3.E19.m1.24.24.24.3.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.cmml"><mo id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.2" stretchy="false" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.cmml">(</mo><msup id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.cmml"><mi id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.2" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.2.cmml">P</mi><mo id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.3" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.3" stretchy="false" xref="S3.E19.m1.24.24.24.3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E19.m1.47.47.2.46.25.25.25.1.1.2.2"><msub id="S3.E19.m1.47.47.2.46.25.25.25.1.1.2.2.2"><mn id="S3.E19.m1.25.25.25.4.4.4" xref="S3.E19.m1.25.25.25.4.4.4.cmml">𝟙</mn><msubsup id="S3.E19.m1.26.26.26.5.5.5.1" xref="S3.E19.m1.26.26.26.5.5.5.1.cmml"><mi id="S3.E19.m1.26.26.26.5.5.5.1.2.2" xref="S3.E19.m1.26.26.26.5.5.5.1.2.2.cmml">Q</mi><msup id="S3.E19.m1.26.26.26.5.5.5.1.3" xref="S3.E19.m1.26.26.26.5.5.5.1.3.cmml"><mi id="S3.E19.m1.26.26.26.5.5.5.1.3.2" 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end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) end_CELL end_ROW start_ROW start_CELL = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) . end_CELL end_ROW</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">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem8.p1.13"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.13.3">Moreover, if <math alttext="n_{a}(P)=1" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.11.1.m1.1"><semantics id="S3.Thmtheorem8.p1.11.1.m1.1a"><mrow id="S3.Thmtheorem8.p1.11.1.m1.1.2" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.cmml"><mrow id="S3.Thmtheorem8.p1.11.1.m1.1.2.2" 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xref="S3.Thmtheorem8.p1.11.1.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.11.1.m1.1b"><apply id="S3.Thmtheorem8.p1.11.1.m1.1.2.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2"><eq id="S3.Thmtheorem8.p1.11.1.m1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.1"></eq><apply id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2"><times id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.1.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2.1"></times><apply id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.1.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.2.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.2">𝑛</ci><ci id="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.3.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.2.2.3">𝑎</ci></apply><ci id="S3.Thmtheorem8.p1.11.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.11.1.m1.1.1">𝑃</ci></apply><cn id="S3.Thmtheorem8.p1.11.1.m1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.11.1.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.11.1.m1.1c">n_{a}(P)=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.11.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = 1</annotation></semantics></math>, then <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.12.2.m2.1"><semantics id="S3.Thmtheorem8.p1.12.2.m2.1a"><msup id="S3.Thmtheorem8.p1.12.2.m2.1.1" xref="S3.Thmtheorem8.p1.12.2.m2.1.1.cmml"><mi id="S3.Thmtheorem8.p1.12.2.m2.1.1.2" xref="S3.Thmtheorem8.p1.12.2.m2.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.12.2.m2.1.1.3" xref="S3.Thmtheorem8.p1.12.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.12.2.m2.1b"><apply id="S3.Thmtheorem8.p1.12.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.12.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.12.2.m2.1.1.1.cmml" xref="S3.Thmtheorem8.p1.12.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem8.p1.12.2.m2.1.1.2.cmml" xref="S3.Thmtheorem8.p1.12.2.m2.1.1.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.12.2.m2.1.1.3.cmml" xref="S3.Thmtheorem8.p1.12.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.12.2.m2.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.12.2.m2.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is bounded and <math alttext="\partial P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.13.3.m3.1"><semantics id="S3.Thmtheorem8.p1.13.3.m3.1a"><mrow id="S3.Thmtheorem8.p1.13.3.m3.1.1" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.cmml"><mo id="S3.Thmtheorem8.p1.13.3.m3.1.1.1" rspace="0em" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.1.cmml">∂</mo><msup id="S3.Thmtheorem8.p1.13.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.2" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2.2.cmml">P</mi><mo id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.3" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.13.3.m3.1b"><apply id="S3.Thmtheorem8.p1.13.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1"><partialdiff id="S3.Thmtheorem8.p1.13.3.m3.1.1.1.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.1"></partialdiff><apply id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2.2">𝑃</ci><ci id="S3.Thmtheorem8.p1.13.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem8.p1.13.3.m3.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.13.3.m3.1c">\partial P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.13.3.m3.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a cycle.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS2.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS2.3.p1"> <p class="ltx_p" id="S3.SS1.SSS2.3.p1.3">Let <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p1.1.m1.1"><semantics id="S3.SS1.SSS2.3.p1.1.m1.1a"><mi id="S3.SS1.SSS2.3.p1.1.m1.1.1" xref="S3.SS1.SSS2.3.p1.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p1.1.m1.1b"><ci id="S3.SS1.SSS2.3.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS2.3.p1.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p1.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p1.1.m1.1d">italic_D</annotation></semantics></math> be a disk, centered at <math alttext="0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p1.2.m2.1"><semantics id="S3.SS1.SSS2.3.p1.2.m2.1a"><mn id="S3.SS1.SSS2.3.p1.2.m2.1.1" xref="S3.SS1.SSS2.3.p1.2.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p1.2.m2.1b"><cn id="S3.SS1.SSS2.3.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS1.SSS2.3.p1.2.m2.1.1">0</cn></annotation-xml></semantics></math> and large enough such that <math alttext="\{x\}\cup V(P)\subset D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p1.3.m3.2"><semantics id="S3.SS1.SSS2.3.p1.3.m3.2a"><mrow id="S3.SS1.SSS2.3.p1.3.m3.2.3" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.cmml"><mrow id="S3.SS1.SSS2.3.p1.3.m3.2.3.2" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.cmml"><mrow id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.2" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.1.cmml"><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.1.cmml">{</mo><mi id="S3.SS1.SSS2.3.p1.3.m3.1.1" xref="S3.SS1.SSS2.3.p1.3.m3.1.1.cmml">x</mi><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.2.2" stretchy="false" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.1.cmml">}</mo></mrow><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.1" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.1.cmml">∪</mo><mrow id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.cmml"><mi id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.2" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.1" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.3.2" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.cmml"><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.cmml">(</mo><mi id="S3.SS1.SSS2.3.p1.3.m3.2.2" xref="S3.SS1.SSS2.3.p1.3.m3.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.SSS2.3.p1.3.m3.2.3.1" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.1.cmml">⊂</mo><mi id="S3.SS1.SSS2.3.p1.3.m3.2.3.3" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p1.3.m3.2b"><apply id="S3.SS1.SSS2.3.p1.3.m3.2.3.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3"><subset id="S3.SS1.SSS2.3.p1.3.m3.2.3.1.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.1"></subset><apply id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2"><union id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.1.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.1"></union><set id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.1.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.2.2"><ci id="S3.SS1.SSS2.3.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.1.1">𝑥</ci></set><apply id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3"><times id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.1.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.1"></times><ci id="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.2.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.2.3.2">𝑉</ci><ci id="S3.SS1.SSS2.3.p1.3.m3.2.2.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.2">𝑃</ci></apply></apply><ci id="S3.SS1.SSS2.3.p1.3.m3.2.3.3.cmml" xref="S3.SS1.SSS2.3.p1.3.m3.2.3.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p1.3.m3.2c">\{x\}\cup V(P)\subset D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p1.3.m3.2d">{ italic_x } ∪ italic_V ( italic_P ) ⊂ italic_D</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.4.p2"> <p class="ltx_p" id="S3.SS1.SSS2.4.p2.10">Convexity of <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.1.m1.1"><semantics id="S3.SS1.SSS2.4.p2.1.m1.1a"><mi id="S3.SS1.SSS2.4.p2.1.m1.1.1" xref="S3.SS1.SSS2.4.p2.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.1.m1.1b"><ci id="S3.SS1.SSS2.4.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS2.4.p2.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.1.m1.1d">italic_D</annotation></semantics></math> implies that all line segments are contained in <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.2.m2.1"><semantics id="S3.SS1.SSS2.4.p2.2.m2.1a"><mi id="S3.SS1.SSS2.4.p2.2.m2.1.1" xref="S3.SS1.SSS2.4.p2.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.2.m2.1b"><ci id="S3.SS1.SSS2.4.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS2.4.p2.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.2.m2.1d">italic_D</annotation></semantics></math>. If <math alttext="e\in E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.3.m3.1"><semantics id="S3.SS1.SSS2.4.p2.3.m3.1a"><mrow id="S3.SS1.SSS2.4.p2.3.m3.1.2" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.cmml"><mi id="S3.SS1.SSS2.4.p2.3.m3.1.2.2" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.2.cmml">e</mi><mo id="S3.SS1.SSS2.4.p2.3.m3.1.2.1" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.4.p2.3.m3.1.2.3" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.cmml"><mi id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.2" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.2.cmml">E</mi><mo id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.1" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.3.2" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.cmml"><mo id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS2.4.p2.3.m3.1.1" xref="S3.SS1.SSS2.4.p2.3.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.3.m3.1b"><apply id="S3.SS1.SSS2.4.p2.3.m3.1.2.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2"><in id="S3.SS1.SSS2.4.p2.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.1"></in><ci id="S3.SS1.SSS2.4.p2.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.2">𝑒</ci><apply id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3"><times id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.1.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.1"></times><ci id="S3.SS1.SSS2.4.p2.3.m3.1.2.3.2.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.2.3.2">𝐸</ci><ci id="S3.SS1.SSS2.4.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS2.4.p2.3.m3.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.3.m3.1c">e\in E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.3.m3.1d">italic_e ∈ italic_E ( italic_P )</annotation></semantics></math> is a ray, <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.4.m4.1"><semantics id="S3.SS1.SSS2.4.p2.4.m4.1a"><mi id="S3.SS1.SSS2.4.p2.4.m4.1.1" xref="S3.SS1.SSS2.4.p2.4.m4.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.4.m4.1b"><ci id="S3.SS1.SSS2.4.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS2.4.p2.4.m4.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.4.m4.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.4.m4.1d">italic_e</annotation></semantics></math> intersects <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.5.m5.1"><semantics id="S3.SS1.SSS2.4.p2.5.m5.1a"><mrow id="S3.SS1.SSS2.4.p2.5.m5.1.1" xref="S3.SS1.SSS2.4.p2.5.m5.1.1.cmml"><mo id="S3.SS1.SSS2.4.p2.5.m5.1.1.1" rspace="0em" xref="S3.SS1.SSS2.4.p2.5.m5.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.4.p2.5.m5.1.1.2" xref="S3.SS1.SSS2.4.p2.5.m5.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.5.m5.1b"><apply id="S3.SS1.SSS2.4.p2.5.m5.1.1.cmml" xref="S3.SS1.SSS2.4.p2.5.m5.1.1"><partialdiff id="S3.SS1.SSS2.4.p2.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.4.p2.5.m5.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.4.p2.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.4.p2.5.m5.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.5.m5.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.5.m5.1d">∂ italic_D</annotation></semantics></math>, since its vertex is contained in <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.6.m6.1"><semantics id="S3.SS1.SSS2.4.p2.6.m6.1a"><mi id="S3.SS1.SSS2.4.p2.6.m6.1.1" xref="S3.SS1.SSS2.4.p2.6.m6.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.6.m6.1b"><ci id="S3.SS1.SSS2.4.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS2.4.p2.6.m6.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.6.m6.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.6.m6.1d">italic_D</annotation></semantics></math> and <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.7.m7.1"><semantics id="S3.SS1.SSS2.4.p2.7.m7.1a"><mi id="S3.SS1.SSS2.4.p2.7.m7.1.1" xref="S3.SS1.SSS2.4.p2.7.m7.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.7.m7.1b"><ci id="S3.SS1.SSS2.4.p2.7.m7.1.1.cmml" xref="S3.SS1.SSS2.4.p2.7.m7.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.7.m7.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.7.m7.1d">italic_e</annotation></semantics></math> is unbounded. Moreover, the rays intersect <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.8.m8.1"><semantics id="S3.SS1.SSS2.4.p2.8.m8.1a"><mrow id="S3.SS1.SSS2.4.p2.8.m8.1.1" xref="S3.SS1.SSS2.4.p2.8.m8.1.1.cmml"><mo id="S3.SS1.SSS2.4.p2.8.m8.1.1.1" rspace="0em" xref="S3.SS1.SSS2.4.p2.8.m8.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.4.p2.8.m8.1.1.2" xref="S3.SS1.SSS2.4.p2.8.m8.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.8.m8.1b"><apply id="S3.SS1.SSS2.4.p2.8.m8.1.1.cmml" xref="S3.SS1.SSS2.4.p2.8.m8.1.1"><partialdiff id="S3.SS1.SSS2.4.p2.8.m8.1.1.1.cmml" xref="S3.SS1.SSS2.4.p2.8.m8.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.4.p2.8.m8.1.1.2.cmml" xref="S3.SS1.SSS2.4.p2.8.m8.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.8.m8.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.8.m8.1d">∂ italic_D</annotation></semantics></math> non-tangentially, so they subdivide <math alttext="\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.9.m9.1"><semantics id="S3.SS1.SSS2.4.p2.9.m9.1a"><mrow id="S3.SS1.SSS2.4.p2.9.m9.1.1" xref="S3.SS1.SSS2.4.p2.9.m9.1.1.cmml"><mo id="S3.SS1.SSS2.4.p2.9.m9.1.1.1" rspace="0em" xref="S3.SS1.SSS2.4.p2.9.m9.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.4.p2.9.m9.1.1.2" xref="S3.SS1.SSS2.4.p2.9.m9.1.1.2.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.9.m9.1b"><apply id="S3.SS1.SSS2.4.p2.9.m9.1.1.cmml" xref="S3.SS1.SSS2.4.p2.9.m9.1.1"><partialdiff id="S3.SS1.SSS2.4.p2.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.4.p2.9.m9.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.4.p2.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.4.p2.9.m9.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.9.m9.1c">\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.9.m9.1d">∂ italic_D</annotation></semantics></math> into circular arcs whose interior is alternately contained in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p2.10.m10.1"><semantics id="S3.SS1.SSS2.4.p2.10.m10.1a"><mi id="S3.SS1.SSS2.4.p2.10.m10.1.1" xref="S3.SS1.SSS2.4.p2.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p2.10.m10.1b"><ci id="S3.SS1.SSS2.4.p2.10.m10.1.1.cmml" xref="S3.SS1.SSS2.4.p2.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p2.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p2.10.m10.1d">italic_P</annotation></semantics></math> and in its complement.</p> </div> <figure class="ltx_figure" id="S3.F8"> <table class="ltx_tabular ltx_align_middle" id="S3.F8.2"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.F8.2.2"> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F8.1.1.1" style="width:162.2pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F8.1.1.1.1"> <span class="ltx_p" id="S3.F8.1.1.1.1.1"><foreignobject height="94.7pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="114.9pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="182" id="S3.F8.1.1.1.1.1.1.g1" src="x20.png" width="221"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F8.2.2.2" style="width:170.7pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F8.2.2.2.1"> <span class="ltx_p" id="S3.F8.2.2.2.1.1"><foreignobject height="125.7pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="104.1pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="241" id="S3.F8.2.2.2.1.1.1.g1" src="x21.png" width="200"/></foreignobject></span> </span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>The circular arc <math alttext="\eta\subset P" class="ltx_Math" display="inline" id="S3.F8.10.m1.1"><semantics id="S3.F8.10.m1.1b"><mrow id="S3.F8.10.m1.1.1" xref="S3.F8.10.m1.1.1.cmml"><mi id="S3.F8.10.m1.1.1.2" xref="S3.F8.10.m1.1.1.2.cmml">η</mi><mo id="S3.F8.10.m1.1.1.1" xref="S3.F8.10.m1.1.1.1.cmml">⊂</mo><mi id="S3.F8.10.m1.1.1.3" xref="S3.F8.10.m1.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F8.10.m1.1c"><apply id="S3.F8.10.m1.1.1.cmml" xref="S3.F8.10.m1.1.1"><subset id="S3.F8.10.m1.1.1.1.cmml" xref="S3.F8.10.m1.1.1.1"></subset><ci id="S3.F8.10.m1.1.1.2.cmml" xref="S3.F8.10.m1.1.1.2">𝜂</ci><ci id="S3.F8.10.m1.1.1.3.cmml" xref="S3.F8.10.m1.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.10.m1.1d">\eta\subset P</annotation><annotation encoding="application/x-llamapun" id="S3.F8.10.m1.1e">italic_η ⊂ italic_P</annotation></semantics></math> between <math alttext="a" class="ltx_Math" display="inline" id="S3.F8.11.m2.1"><semantics id="S3.F8.11.m2.1b"><mi id="S3.F8.11.m2.1.1" xref="S3.F8.11.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.F8.11.m2.1c"><ci id="S3.F8.11.m2.1.1.cmml" xref="S3.F8.11.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.11.m2.1d">a</annotation><annotation encoding="application/x-llamapun" id="S3.F8.11.m2.1e">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S3.F8.12.m3.1"><semantics id="S3.F8.12.m3.1b"><mi id="S3.F8.12.m3.1.1" xref="S3.F8.12.m3.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.F8.12.m3.1c"><ci id="S3.F8.12.m3.1.1.cmml" xref="S3.F8.12.m3.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.12.m3.1d">b</annotation><annotation encoding="application/x-llamapun" id="S3.F8.12.m3.1e">italic_b</annotation></semantics></math> can be approximated by a path <math alttext="\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.F8.13.m4.1"><semantics id="S3.F8.13.m4.1b"><msup id="S3.F8.13.m4.1.1" xref="S3.F8.13.m4.1.1.cmml"><mi id="S3.F8.13.m4.1.1.2" xref="S3.F8.13.m4.1.1.2.cmml">γ</mi><mo id="S3.F8.13.m4.1.1.3" xref="S3.F8.13.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.F8.13.m4.1c"><apply id="S3.F8.13.m4.1.1.cmml" xref="S3.F8.13.m4.1.1"><csymbol cd="ambiguous" id="S3.F8.13.m4.1.1.1.cmml" xref="S3.F8.13.m4.1.1">superscript</csymbol><ci id="S3.F8.13.m4.1.1.2.cmml" xref="S3.F8.13.m4.1.1.2">𝛾</ci><ci id="S3.F8.13.m4.1.1.3.cmml" xref="S3.F8.13.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.13.m4.1d">\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.F8.13.m4.1e">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> (dark grey) of three line segments that are contained in tangents. This path can be used to cut off an unbounded area <math alttext="K^{\prime}\subset P" class="ltx_Math" display="inline" id="S3.F8.14.m5.1"><semantics id="S3.F8.14.m5.1b"><mrow id="S3.F8.14.m5.1.1" xref="S3.F8.14.m5.1.1.cmml"><msup id="S3.F8.14.m5.1.1.2" xref="S3.F8.14.m5.1.1.2.cmml"><mi id="S3.F8.14.m5.1.1.2.2" xref="S3.F8.14.m5.1.1.2.2.cmml">K</mi><mo id="S3.F8.14.m5.1.1.2.3" xref="S3.F8.14.m5.1.1.2.3.cmml">′</mo></msup><mo id="S3.F8.14.m5.1.1.1" xref="S3.F8.14.m5.1.1.1.cmml">⊂</mo><mi id="S3.F8.14.m5.1.1.3" xref="S3.F8.14.m5.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F8.14.m5.1c"><apply id="S3.F8.14.m5.1.1.cmml" xref="S3.F8.14.m5.1.1"><subset id="S3.F8.14.m5.1.1.1.cmml" xref="S3.F8.14.m5.1.1.1"></subset><apply id="S3.F8.14.m5.1.1.2.cmml" xref="S3.F8.14.m5.1.1.2"><csymbol cd="ambiguous" id="S3.F8.14.m5.1.1.2.1.cmml" xref="S3.F8.14.m5.1.1.2">superscript</csymbol><ci id="S3.F8.14.m5.1.1.2.2.cmml" xref="S3.F8.14.m5.1.1.2.2">𝐾</ci><ci id="S3.F8.14.m5.1.1.2.3.cmml" xref="S3.F8.14.m5.1.1.2.3">′</ci></apply><ci id="S3.F8.14.m5.1.1.3.cmml" xref="S3.F8.14.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.14.m5.1d">K^{\prime}\subset P</annotation><annotation encoding="application/x-llamapun" id="S3.F8.14.m5.1e">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_P</annotation></semantics></math> (hatched) from <math alttext="P" class="ltx_Math" display="inline" id="S3.F8.15.m6.1"><semantics id="S3.F8.15.m6.1b"><mi id="S3.F8.15.m6.1.1" xref="S3.F8.15.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F8.15.m6.1c"><ci id="S3.F8.15.m6.1.1.cmml" xref="S3.F8.15.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.15.m6.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F8.15.m6.1e">italic_P</annotation></semantics></math> to form a new polygon <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.F8.16.m7.1"><semantics id="S3.F8.16.m7.1b"><msup id="S3.F8.16.m7.1.1" xref="S3.F8.16.m7.1.1.cmml"><mi id="S3.F8.16.m7.1.1.2" xref="S3.F8.16.m7.1.1.2.cmml">P</mi><mo id="S3.F8.16.m7.1.1.3" xref="S3.F8.16.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.F8.16.m7.1c"><apply id="S3.F8.16.m7.1.1.cmml" xref="S3.F8.16.m7.1.1"><csymbol cd="ambiguous" id="S3.F8.16.m7.1.1.1.cmml" xref="S3.F8.16.m7.1.1">superscript</csymbol><ci id="S3.F8.16.m7.1.1.2.cmml" xref="S3.F8.16.m7.1.1.2">𝑃</ci><ci id="S3.F8.16.m7.1.1.3.cmml" xref="S3.F8.16.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F8.16.m7.1d">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.F8.16.m7.1e">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> (dotted).</figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS2.5.p3"> <p class="ltx_p" id="S3.SS1.SSS2.5.p3.20">Let <math alttext="a,b\in\partial P\cap\partial D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.1.m1.2"><semantics id="S3.SS1.SSS2.5.p3.1.m1.2a"><mrow id="S3.SS1.SSS2.5.p3.1.m1.2.3" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.cmml"><mrow id="S3.SS1.SSS2.5.p3.1.m1.2.3.2.2" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.2.1.cmml"><mi id="S3.SS1.SSS2.5.p3.1.m1.1.1" xref="S3.SS1.SSS2.5.p3.1.m1.1.1.cmml">a</mi><mo id="S3.SS1.SSS2.5.p3.1.m1.2.3.2.2.1" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.2.1.cmml">,</mo><mi id="S3.SS1.SSS2.5.p3.1.m1.2.2" xref="S3.SS1.SSS2.5.p3.1.m1.2.2.cmml">b</mi></mrow><mo id="S3.SS1.SSS2.5.p3.1.m1.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.5.p3.1.m1.2.3.3" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.cmml"><mrow id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.cmml"><mo id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.1.cmml">∂</mo><mi id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.2" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.2.cmml">P</mi></mrow><mo id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.1" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.1.cmml">∩</mo><mrow id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.cmml"><mo id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.1" lspace="0em" rspace="0em" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.1.cmml">∂</mo><mi id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.2" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.2.cmml">D</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.1.m1.2b"><apply id="S3.SS1.SSS2.5.p3.1.m1.2.3.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3"><in id="S3.SS1.SSS2.5.p3.1.m1.2.3.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.1"></in><list id="S3.SS1.SSS2.5.p3.1.m1.2.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.2.2"><ci id="S3.SS1.SSS2.5.p3.1.m1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.1.1">𝑎</ci><ci id="S3.SS1.SSS2.5.p3.1.m1.2.2.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.2">𝑏</ci></list><apply id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3"><intersect id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.1"></intersect><apply id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2"><partialdiff id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.1"></partialdiff><ci id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.2.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.2.2">𝑃</ci></apply><apply id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3"><partialdiff id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.1.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.1"></partialdiff><ci id="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.2.cmml" xref="S3.SS1.SSS2.5.p3.1.m1.2.3.3.3.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.1.m1.2c">a,b\in\partial P\cap\partial D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.1.m1.2d">italic_a , italic_b ∈ ∂ italic_P ∩ ∂ italic_D</annotation></semantics></math> be the intersection points of two distinct rays <math alttext="e_{a},e_{b}\in E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.2.m2.3"><semantics id="S3.SS1.SSS2.5.p3.2.m2.3a"><mrow id="S3.SS1.SSS2.5.p3.2.m2.3.3" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.cmml"><mrow id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.3.cmml"><msub id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.2" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.3" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.3" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.cmml"><mi id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.2" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.3" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.3.cmml">b</mi></msub></mrow><mo id="S3.SS1.SSS2.5.p3.2.m2.3.3.3" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.3.cmml">∈</mo><mrow id="S3.SS1.SSS2.5.p3.2.m2.3.3.4" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.cmml"><mi id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.2" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.2.cmml">E</mi><mo id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.1" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.3.2" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.cmml"><mo id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.3.2.1" stretchy="false" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.cmml">(</mo><mi id="S3.SS1.SSS2.5.p3.2.m2.1.1" xref="S3.SS1.SSS2.5.p3.2.m2.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.3.2.2" stretchy="false" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.2.m2.3b"><apply id="S3.SS1.SSS2.5.p3.2.m2.3.3.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3"><in id="S3.SS1.SSS2.5.p3.2.m2.3.3.3.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.3"></in><list id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.3.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2"><apply id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.2.2.1.1.1.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.1.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.2.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.3.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.2.2.2.3">𝑏</ci></apply></list><apply id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4"><times id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.1.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.1"></times><ci id="S3.SS1.SSS2.5.p3.2.m2.3.3.4.2.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.3.3.4.2">𝐸</ci><ci id="S3.SS1.SSS2.5.p3.2.m2.1.1.cmml" xref="S3.SS1.SSS2.5.p3.2.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.2.m2.3c">e_{a},e_{b}\in E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.2.m2.3d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ italic_E ( italic_P )</annotation></semantics></math> such that the circular arc <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.3.m3.1"><semantics id="S3.SS1.SSS2.5.p3.3.m3.1a"><mi id="S3.SS1.SSS2.5.p3.3.m3.1.1" xref="S3.SS1.SSS2.5.p3.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.3.m3.1b"><ci id="S3.SS1.SSS2.5.p3.3.m3.1.1.cmml" xref="S3.SS1.SSS2.5.p3.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.3.m3.1d">italic_η</annotation></semantics></math> from <math alttext="a" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.4.m4.1"><semantics id="S3.SS1.SSS2.5.p3.4.m4.1a"><mi id="S3.SS1.SSS2.5.p3.4.m4.1.1" xref="S3.SS1.SSS2.5.p3.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.4.m4.1b"><ci id="S3.SS1.SSS2.5.p3.4.m4.1.1.cmml" xref="S3.SS1.SSS2.5.p3.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.4.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.4.m4.1d">italic_a</annotation></semantics></math> to <math alttext="b" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.5.m5.1"><semantics id="S3.SS1.SSS2.5.p3.5.m5.1a"><mi id="S3.SS1.SSS2.5.p3.5.m5.1.1" xref="S3.SS1.SSS2.5.p3.5.m5.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.5.m5.1b"><ci id="S3.SS1.SSS2.5.p3.5.m5.1.1.cmml" xref="S3.SS1.SSS2.5.p3.5.m5.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.5.m5.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.5.m5.1d">italic_b</annotation></semantics></math> in clockwise direction is contained in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.6.m6.1"><semantics id="S3.SS1.SSS2.5.p3.6.m6.1a"><mi id="S3.SS1.SSS2.5.p3.6.m6.1.1" xref="S3.SS1.SSS2.5.p3.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.6.m6.1b"><ci id="S3.SS1.SSS2.5.p3.6.m6.1.1.cmml" xref="S3.SS1.SSS2.5.p3.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.6.m6.1d">italic_P</annotation></semantics></math>, cf. <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F8" title="Figure 8 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 8</span></a>. We subdivide these rays into a line segment <math alttext="s_{a}:=e_{a}\cap\overline{D}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.7.m7.1"><semantics id="S3.SS1.SSS2.5.p3.7.m7.1a"><mrow id="S3.SS1.SSS2.5.p3.7.m7.1.1" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.cmml"><msub id="S3.SS1.SSS2.5.p3.7.m7.1.1.2" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2.cmml"><mi id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.2" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2.2.cmml">s</mi><mi id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.3" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.5.p3.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.1.cmml">:=</mo><mrow id="S3.SS1.SSS2.5.p3.7.m7.1.1.3" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.cmml"><msub id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.2" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.3" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.1" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.1.cmml">∩</mo><mover accent="true" id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.2" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.2.cmml">D</mi><mo id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.1" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.7.m7.1b"><apply id="S3.SS1.SSS2.5.p3.7.m7.1.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.5.p3.7.m7.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.2.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2.2">𝑠</ci><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.2.3.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3"><intersect id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.1"></intersect><apply id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3"><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.1">¯</ci><ci id="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.5.p3.7.m7.1.1.3.3.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.7.m7.1c">s_{a}:=e_{a}\cap\overline{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.7.m7.1d">italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT := italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_D end_ARG</annotation></semantics></math>, resp. <math alttext="s_{b}:=e_{b}\cap\overline{D}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.8.m8.1"><semantics id="S3.SS1.SSS2.5.p3.8.m8.1a"><mrow id="S3.SS1.SSS2.5.p3.8.m8.1.1" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.cmml"><msub id="S3.SS1.SSS2.5.p3.8.m8.1.1.2" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2.cmml"><mi id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.2" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2.2.cmml">s</mi><mi id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.3" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.5.p3.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.1.cmml">:=</mo><mrow id="S3.SS1.SSS2.5.p3.8.m8.1.1.3" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.cmml"><msub id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.2" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.3" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.1" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.1.cmml">∩</mo><mover accent="true" id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.2" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.2.cmml">D</mi><mo id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.1" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.8.m8.1b"><apply id="S3.SS1.SSS2.5.p3.8.m8.1.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.5.p3.8.m8.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.2.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2.2">𝑠</ci><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.2.3.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.2.3">𝑏</ci></apply><apply id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3"><intersect id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.1"></intersect><apply id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.2.3">𝑏</ci></apply><apply id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3"><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.1">¯</ci><ci id="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.5.p3.8.m8.1.1.3.3.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.8.m8.1c">s_{b}:=e_{b}\cap\overline{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.8.m8.1d">italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT := italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∩ over¯ start_ARG italic_D end_ARG</annotation></semantics></math>, and a ray <math alttext="e_{a}^{\prime}:=e_{a}\setminus D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.9.m9.1"><semantics id="S3.SS1.SSS2.5.p3.9.m9.1a"><mrow id="S3.SS1.SSS2.5.p3.9.m9.1.1" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.cmml"><msubsup id="S3.SS1.SSS2.5.p3.9.m9.1.1.2" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.cmml"><mi id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.2" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.3" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.3.cmml">a</mi><mo id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.3" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.SSS2.5.p3.9.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.1.cmml">:=</mo><mrow id="S3.SS1.SSS2.5.p3.9.m9.1.1.3" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.cmml"><msub id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.2" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.3" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.1" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.1.cmml">∖</mo><mi id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.3" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.9.m9.1b"><apply id="S3.SS1.SSS2.5.p3.9.m9.1.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.5.p3.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.2.3.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3"><setdiff id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.1"></setdiff><apply id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.5.p3.9.m9.1.1.3.3.cmml" xref="S3.SS1.SSS2.5.p3.9.m9.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.9.m9.1c">e_{a}^{\prime}:=e_{a}\setminus D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.9.m9.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∖ italic_D</annotation></semantics></math>, resp. <math alttext="e_{b}^{\prime}:=e_{b}\setminus D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.10.m10.1"><semantics id="S3.SS1.SSS2.5.p3.10.m10.1a"><mrow id="S3.SS1.SSS2.5.p3.10.m10.1.1" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.cmml"><msubsup id="S3.SS1.SSS2.5.p3.10.m10.1.1.2" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.cmml"><mi id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.2" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.3" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.3.cmml">b</mi><mo id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.3" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.SSS2.5.p3.10.m10.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.1.cmml">:=</mo><mrow id="S3.SS1.SSS2.5.p3.10.m10.1.1.3" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.cmml"><msub id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.2" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.3" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.1" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.1.cmml">∖</mo><mi id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.3" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.10.m10.1b"><apply id="S3.SS1.SSS2.5.p3.10.m10.1.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.5.p3.10.m10.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.2.3.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3"><setdiff id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.1"></setdiff><apply id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.2">𝑒</ci><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.5.p3.10.m10.1.1.3.3.cmml" xref="S3.SS1.SSS2.5.p3.10.m10.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.10.m10.1c">e_{b}^{\prime}:=e_{b}\setminus D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.10.m10.1d">italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∖ italic_D</annotation></semantics></math>. Define <math alttext="a^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.11.m11.1"><semantics id="S3.SS1.SSS2.5.p3.11.m11.1a"><msup id="S3.SS1.SSS2.5.p3.11.m11.1.1" xref="S3.SS1.SSS2.5.p3.11.m11.1.1.cmml"><mi id="S3.SS1.SSS2.5.p3.11.m11.1.1.2" xref="S3.SS1.SSS2.5.p3.11.m11.1.1.2.cmml">a</mi><mo id="S3.SS1.SSS2.5.p3.11.m11.1.1.3" xref="S3.SS1.SSS2.5.p3.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.11.m11.1b"><apply id="S3.SS1.SSS2.5.p3.11.m11.1.1.cmml" xref="S3.SS1.SSS2.5.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.11.m11.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.11.m11.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.5.p3.11.m11.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.11.m11.1.1.2">𝑎</ci><ci id="S3.SS1.SSS2.5.p3.11.m11.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.11.m11.1c">a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.11.m11.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="b^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.12.m12.1"><semantics id="S3.SS1.SSS2.5.p3.12.m12.1a"><msup id="S3.SS1.SSS2.5.p3.12.m12.1.1" xref="S3.SS1.SSS2.5.p3.12.m12.1.1.cmml"><mi id="S3.SS1.SSS2.5.p3.12.m12.1.1.2" xref="S3.SS1.SSS2.5.p3.12.m12.1.1.2.cmml">b</mi><mo id="S3.SS1.SSS2.5.p3.12.m12.1.1.3" xref="S3.SS1.SSS2.5.p3.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.12.m12.1b"><apply id="S3.SS1.SSS2.5.p3.12.m12.1.1.cmml" xref="S3.SS1.SSS2.5.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.5.p3.12.m12.1.1.1.cmml" xref="S3.SS1.SSS2.5.p3.12.m12.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.5.p3.12.m12.1.1.2.cmml" xref="S3.SS1.SSS2.5.p3.12.m12.1.1.2">𝑏</ci><ci id="S3.SS1.SSS2.5.p3.12.m12.1.1.3.cmml" xref="S3.SS1.SSS2.5.p3.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.12.m12.1c">b^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.12.m12.1d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> as the intersection points of the tangents to <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.13.m13.1"><semantics id="S3.SS1.SSS2.5.p3.13.m13.1a"><mi id="S3.SS1.SSS2.5.p3.13.m13.1.1" xref="S3.SS1.SSS2.5.p3.13.m13.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.13.m13.1b"><ci id="S3.SS1.SSS2.5.p3.13.m13.1.1.cmml" xref="S3.SS1.SSS2.5.p3.13.m13.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.13.m13.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.13.m13.1d">italic_D</annotation></semantics></math> at <math alttext="a" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.14.m14.1"><semantics id="S3.SS1.SSS2.5.p3.14.m14.1a"><mi id="S3.SS1.SSS2.5.p3.14.m14.1.1" xref="S3.SS1.SSS2.5.p3.14.m14.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.14.m14.1b"><ci id="S3.SS1.SSS2.5.p3.14.m14.1.1.cmml" xref="S3.SS1.SSS2.5.p3.14.m14.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.14.m14.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.14.m14.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.15.m15.1"><semantics id="S3.SS1.SSS2.5.p3.15.m15.1a"><mi id="S3.SS1.SSS2.5.p3.15.m15.1.1" xref="S3.SS1.SSS2.5.p3.15.m15.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.15.m15.1b"><ci id="S3.SS1.SSS2.5.p3.15.m15.1.1.cmml" xref="S3.SS1.SSS2.5.p3.15.m15.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.15.m15.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.15.m15.1d">italic_b</annotation></semantics></math> with the tangent at the midpoint <math alttext="y" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.16.m16.1"><semantics id="S3.SS1.SSS2.5.p3.16.m16.1a"><mi id="S3.SS1.SSS2.5.p3.16.m16.1.1" xref="S3.SS1.SSS2.5.p3.16.m16.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.16.m16.1b"><ci id="S3.SS1.SSS2.5.p3.16.m16.1.1.cmml" xref="S3.SS1.SSS2.5.p3.16.m16.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.16.m16.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.16.m16.1d">italic_y</annotation></semantics></math> of <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.17.m17.1"><semantics id="S3.SS1.SSS2.5.p3.17.m17.1a"><mi id="S3.SS1.SSS2.5.p3.17.m17.1.1" xref="S3.SS1.SSS2.5.p3.17.m17.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.17.m17.1b"><ci id="S3.SS1.SSS2.5.p3.17.m17.1.1.cmml" xref="S3.SS1.SSS2.5.p3.17.m17.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.17.m17.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.17.m17.1d">italic_η</annotation></semantics></math> (they do intersect because <math alttext="y" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.18.m18.1"><semantics id="S3.SS1.SSS2.5.p3.18.m18.1a"><mi id="S3.SS1.SSS2.5.p3.18.m18.1.1" xref="S3.SS1.SSS2.5.p3.18.m18.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.18.m18.1b"><ci id="S3.SS1.SSS2.5.p3.18.m18.1.1.cmml" xref="S3.SS1.SSS2.5.p3.18.m18.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.18.m18.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.18.m18.1d">italic_y</annotation></semantics></math> cannot be the negative of <math alttext="a" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.19.m19.1"><semantics id="S3.SS1.SSS2.5.p3.19.m19.1a"><mi id="S3.SS1.SSS2.5.p3.19.m19.1.1" xref="S3.SS1.SSS2.5.p3.19.m19.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.19.m19.1b"><ci id="S3.SS1.SSS2.5.p3.19.m19.1.1.cmml" xref="S3.SS1.SSS2.5.p3.19.m19.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.19.m19.1c">a</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.19.m19.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p3.20.m20.1"><semantics id="S3.SS1.SSS2.5.p3.20.m20.1a"><mi id="S3.SS1.SSS2.5.p3.20.m20.1.1" xref="S3.SS1.SSS2.5.p3.20.m20.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p3.20.m20.1b"><ci id="S3.SS1.SSS2.5.p3.20.m20.1.1.cmml" xref="S3.SS1.SSS2.5.p3.20.m20.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p3.20.m20.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p3.20.m20.1d">italic_b</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.6.p4"> <p class="ltx_p" id="S3.SS1.SSS2.6.p4.23">Define <math alttext="\gamma^{\prime}:=\overline{aa^{\prime}}\cup\overline{a^{\prime}b^{\prime}}\cup% \overline{b^{\prime}b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.1.m1.1"><semantics id="S3.SS1.SSS2.6.p4.1.m1.1a"><mrow id="S3.SS1.SSS2.6.p4.1.m1.1.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.cmml"><msup id="S3.SS1.SSS2.6.p4.1.m1.1.1.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.1.cmml">:=</mo><mrow id="S3.SS1.SSS2.6.p4.1.m1.1.1.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.cmml"><mover accent="true" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.cmml"><mrow id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.2.cmml">a</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.1.cmml">⁢</mo><msup id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.2.cmml">a</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.1.cmml">¯</mo></mover><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1.cmml">∪</mo><mover accent="true" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.cmml"><mrow id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.cmml"><msup id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.2.cmml">a</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.1.cmml">⁢</mo><msup id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.2.cmml">b</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.1.cmml">¯</mo></mover><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1a" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1.cmml">∪</mo><mover accent="true" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.cmml"><mrow id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.cmml"><msup id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.2" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.2.cmml">b</mi><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.3" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.3.cmml">b</mi></mrow><mo id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.1" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.1.cmml">¯</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.1.m1.1b"><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.6.p4.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3"><union id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.1"></union><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2"><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.1">¯</ci><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2"><times id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.1"></times><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.2">𝑎</ci><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.2">𝑎</ci><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.2.2.3.3">′</ci></apply></apply></apply><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3"><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.1">¯</ci><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2"><times id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.1"></times><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.2">𝑎</ci><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.2.3">′</ci></apply><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.2">𝑏</ci><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.3.2.3.3">′</ci></apply></apply></apply><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4"><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.1">¯</ci><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2"><times id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.1"></times><apply id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.2">𝑏</ci><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.2.3">′</ci></apply><ci id="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.3.cmml" xref="S3.SS1.SSS2.6.p4.1.m1.1.1.3.4.2.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.1.m1.1c">\gamma^{\prime}:=\overline{aa^{\prime}}\cup\overline{a^{\prime}b^{\prime}}\cup% \overline{b^{\prime}b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.1.m1.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := over¯ start_ARG italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ∪ over¯ start_ARG italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ∪ over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG</annotation></semantics></math>, and <math alttext="K" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.2.m2.1"><semantics id="S3.SS1.SSS2.6.p4.2.m2.1a"><mi id="S3.SS1.SSS2.6.p4.2.m2.1.1" xref="S3.SS1.SSS2.6.p4.2.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.2.m2.1b"><ci id="S3.SS1.SSS2.6.p4.2.m2.1.1.cmml" xref="S3.SS1.SSS2.6.p4.2.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.2.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.2.m2.1d">italic_K</annotation></semantics></math> as the connected component of <math alttext="P\setminus D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.3.m3.1"><semantics id="S3.SS1.SSS2.6.p4.3.m3.1a"><mrow id="S3.SS1.SSS2.6.p4.3.m3.1.1" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.3.m3.1.1.2" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.3.m3.1.1.1" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.6.p4.3.m3.1.1.3" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.3.m3.1b"><apply id="S3.SS1.SSS2.6.p4.3.m3.1.1.cmml" xref="S3.SS1.SSS2.6.p4.3.m3.1.1"><setdiff id="S3.SS1.SSS2.6.p4.3.m3.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.1"></setdiff><ci id="S3.SS1.SSS2.6.p4.3.m3.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.6.p4.3.m3.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.3.m3.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.3.m3.1c">P\setminus D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.3.m3.1d">italic_P ∖ italic_D</annotation></semantics></math> between the rays <math alttext="e_{a}^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.4.m4.1"><semantics id="S3.SS1.SSS2.6.p4.4.m4.1a"><msubsup id="S3.SS1.SSS2.6.p4.4.m4.1.1" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.2" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.3" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.2.3.cmml">a</mi><mo id="S3.SS1.SSS2.6.p4.4.m4.1.1.3" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.4.m4.1b"><apply id="S3.SS1.SSS2.6.p4.4.m4.1.1.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1">superscript</csymbol><apply id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.4.m4.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.6.p4.4.m4.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.4.m4.1c">e_{a}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.4.m4.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="e_{b}^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.5.m5.1"><semantics id="S3.SS1.SSS2.6.p4.5.m5.1a"><msubsup id="S3.SS1.SSS2.6.p4.5.m5.1.1" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.2" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.3" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.2.3.cmml">b</mi><mo id="S3.SS1.SSS2.6.p4.5.m5.1.1.3" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.5.m5.1b"><apply id="S3.SS1.SSS2.6.p4.5.m5.1.1.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1">superscript</csymbol><apply id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.5.m5.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.6.p4.5.m5.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.5.m5.1c">e_{b}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.5.m5.1d">italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, due to the choice of <math alttext="y" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.6.m6.1"><semantics id="S3.SS1.SSS2.6.p4.6.m6.1a"><mi id="S3.SS1.SSS2.6.p4.6.m6.1.1" xref="S3.SS1.SSS2.6.p4.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.6.m6.1b"><ci id="S3.SS1.SSS2.6.p4.6.m6.1.1.cmml" xref="S3.SS1.SSS2.6.p4.6.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.6.m6.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.6.m6.1d">italic_y</annotation></semantics></math> as the midpoint of <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.7.m7.1"><semantics id="S3.SS1.SSS2.6.p4.7.m7.1a"><mi id="S3.SS1.SSS2.6.p4.7.m7.1.1" xref="S3.SS1.SSS2.6.p4.7.m7.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.7.m7.1b"><ci id="S3.SS1.SSS2.6.p4.7.m7.1.1.cmml" xref="S3.SS1.SSS2.6.p4.7.m7.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.7.m7.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.7.m7.1d">italic_η</annotation></semantics></math>, we have that <math alttext="\gamma^{\prime}\subset K" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.8.m8.1"><semantics id="S3.SS1.SSS2.6.p4.8.m8.1a"><mrow id="S3.SS1.SSS2.6.p4.8.m8.1.1" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.cmml"><msup id="S3.SS1.SSS2.6.p4.8.m8.1.1.2" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.2" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.3" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.6.p4.8.m8.1.1.1" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.1.cmml">⊂</mo><mi id="S3.SS1.SSS2.6.p4.8.m8.1.1.3" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.8.m8.1b"><apply id="S3.SS1.SSS2.6.p4.8.m8.1.1.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1"><subset id="S3.SS1.SSS2.6.p4.8.m8.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.1"></subset><apply id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.6.p4.8.m8.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.2.3">′</ci></apply><ci id="S3.SS1.SSS2.6.p4.8.m8.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.8.m8.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.8.m8.1c">\gamma^{\prime}\subset K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.8.m8.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ italic_K</annotation></semantics></math>, but <math alttext="e\cap K=\emptyset" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.9.m9.1"><semantics id="S3.SS1.SSS2.6.p4.9.m9.1a"><mrow id="S3.SS1.SSS2.6.p4.9.m9.1.1" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.cmml"><mrow id="S3.SS1.SSS2.6.p4.9.m9.1.1.2" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.2" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.2.cmml">e</mi><mo id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.1" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.3" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.3.cmml">K</mi></mrow><mo id="S3.SS1.SSS2.6.p4.9.m9.1.1.1" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.1.cmml">=</mo><mi id="S3.SS1.SSS2.6.p4.9.m9.1.1.3" mathvariant="normal" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.9.m9.1b"><apply id="S3.SS1.SSS2.6.p4.9.m9.1.1.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1"><eq id="S3.SS1.SSS2.6.p4.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.1"></eq><apply id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2"><intersect id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.1"></intersect><ci id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.9.m9.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.2.3">𝐾</ci></apply><emptyset id="S3.SS1.SSS2.6.p4.9.m9.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.9.m9.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.9.m9.1c">e\cap K=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.9.m9.1d">italic_e ∩ italic_K = ∅</annotation></semantics></math> for all edges <math alttext="e\in E(P)\setminus\{e_{a},e_{b}\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.10.m10.3"><semantics id="S3.SS1.SSS2.6.p4.10.m10.3a"><mrow id="S3.SS1.SSS2.6.p4.10.m10.3.3" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.cmml"><mi id="S3.SS1.SSS2.6.p4.10.m10.3.3.4" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.4.cmml">e</mi><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.3" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.3.cmml">∈</mo><mrow id="S3.SS1.SSS2.6.p4.10.m10.3.3.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.cmml"><mrow id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.cmml"><mi id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.2.cmml">E</mi><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.1" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.3.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.cmml"><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.cmml">(</mo><mi id="S3.SS1.SSS2.6.p4.10.m10.1.1" xref="S3.SS1.SSS2.6.p4.10.m10.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.3" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.3.cmml">∖</mo><mrow id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.3.cmml"><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.3" stretchy="false" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.3.cmml">{</mo><msub id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.2" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.3" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.4" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.2" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.3" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.5" stretchy="false" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.10.m10.3b"><apply id="S3.SS1.SSS2.6.p4.10.m10.3.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3"><in id="S3.SS1.SSS2.6.p4.10.m10.3.3.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.3"></in><ci id="S3.SS1.SSS2.6.p4.10.m10.3.3.4.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.4">𝑒</ci><apply id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2"><setdiff id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.3"></setdiff><apply id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4"><times id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.1.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.1"></times><ci id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.2.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.4.2">𝐸</ci><ci id="S3.SS1.SSS2.6.p4.10.m10.1.1.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.1.1">𝑃</ci></apply><set id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2"><apply id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.2.2.1.1.1.1.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.10.m10.3.3.2.2.2.2.3">𝑏</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.10.m10.3c">e\in E(P)\setminus\{e_{a},e_{b}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.10.m10.3d">italic_e ∈ italic_E ( italic_P ) ∖ { italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT }</annotation></semantics></math> and for <math alttext="e\in\{s_{a},s_{b}\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.11.m11.2"><semantics id="S3.SS1.SSS2.6.p4.11.m11.2a"><mrow id="S3.SS1.SSS2.6.p4.11.m11.2.2" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.11.m11.2.2.4" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.4.cmml">e</mi><mo id="S3.SS1.SSS2.6.p4.11.m11.2.2.3" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.3.cmml">∈</mo><mrow id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.3.cmml"><mo id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.3" stretchy="false" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.3.cmml">{</mo><msub id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.2" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.2.cmml">s</mi><mi id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.3" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.4" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.2" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.2.cmml">s</mi><mi id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.3" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.5" stretchy="false" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.11.m11.2b"><apply id="S3.SS1.SSS2.6.p4.11.m11.2.2.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2"><in id="S3.SS1.SSS2.6.p4.11.m11.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.3"></in><ci id="S3.SS1.SSS2.6.p4.11.m11.2.2.4.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.4">𝑒</ci><set id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2"><apply id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.2">𝑠</ci><ci id="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.1.1.1.1.1.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.2">𝑠</ci><ci id="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.11.m11.2.2.2.2.2.3">𝑏</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.11.m11.2c">e\in\{s_{a},s_{b}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.11.m11.2d">italic_e ∈ { italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT }</annotation></semantics></math>. Let <math alttext="K^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.12.m12.1"><semantics id="S3.SS1.SSS2.6.p4.12.m12.1a"><msup id="S3.SS1.SSS2.6.p4.12.m12.1.1" xref="S3.SS1.SSS2.6.p4.12.m12.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.12.m12.1.1.2" xref="S3.SS1.SSS2.6.p4.12.m12.1.1.2.cmml">K</mi><mo id="S3.SS1.SSS2.6.p4.12.m12.1.1.3" xref="S3.SS1.SSS2.6.p4.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.12.m12.1b"><apply id="S3.SS1.SSS2.6.p4.12.m12.1.1.cmml" xref="S3.SS1.SSS2.6.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.12.m12.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.12.m12.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.12.m12.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.12.m12.1.1.2">𝐾</ci><ci id="S3.SS1.SSS2.6.p4.12.m12.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.12.m12.1c">K^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.12.m12.1d">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, be the connected component of <math alttext="\mathds{R}^{2}\setminus(e_{a}^{\prime}\cup e_{b}^{\prime}\cup\gamma^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.13.m13.1"><semantics id="S3.SS1.SSS2.6.p4.13.m13.1a"><mrow id="S3.SS1.SSS2.6.p4.13.m13.1.1" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.cmml"><msup id="S3.SS1.SSS2.6.p4.13.m13.1.1.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3.cmml"><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3.3.cmml">2</mn></msup><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.2.cmml">∖</mo><mrow id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.cmml"><msubsup id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.3.cmml">a</mi><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1.cmml">∪</mo><msubsup id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.cmml"><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.3.cmml">b</mi><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.3.cmml">′</mo></msubsup><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1a" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1.cmml">∪</mo><msup id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.cmml"><mi id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.2" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.2.cmml">γ</mi><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.3" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.13.m13.1b"><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1"><setdiff id="S3.SS1.SSS2.6.p4.13.m13.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.2"></setdiff><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3.2">ℝ</ci><cn id="S3.SS1.SSS2.6.p4.13.m13.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.3.3">2</cn></apply><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1"><union id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.1"></union><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3">superscript</csymbol><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.3.3">′</ci></apply><apply id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.1.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.2.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.2">𝛾</ci><ci id="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.3.cmml" xref="S3.SS1.SSS2.6.p4.13.m13.1.1.1.1.1.4.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.13.m13.1c">\mathds{R}^{2}\setminus(e_{a}^{\prime}\cup e_{b}^{\prime}\cup\gamma^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.13.m13.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ ( italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> that is contained in <math alttext="K" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.14.m14.1"><semantics id="S3.SS1.SSS2.6.p4.14.m14.1a"><mi id="S3.SS1.SSS2.6.p4.14.m14.1.1" xref="S3.SS1.SSS2.6.p4.14.m14.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.14.m14.1b"><ci id="S3.SS1.SSS2.6.p4.14.m14.1.1.cmml" xref="S3.SS1.SSS2.6.p4.14.m14.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.14.m14.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.14.m14.1d">italic_K</annotation></semantics></math>, and define <math alttext="P^{\prime}:=\overline{P\setminus K^{\prime}}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.15.m15.1"><semantics id="S3.SS1.SSS2.6.p4.15.m15.1a"><mrow id="S3.SS1.SSS2.6.p4.15.m15.1.1" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.cmml"><msup id="S3.SS1.SSS2.6.p4.15.m15.1.1.2" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.2" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.3" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.6.p4.15.m15.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.1.cmml">:=</mo><mover accent="true" id="S3.SS1.SSS2.6.p4.15.m15.1.1.3" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.cmml"><mrow id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.2" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.1" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.1.cmml">∖</mo><msup id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.cmml"><mi id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.2" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.2.cmml">K</mi><mo id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.3" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.1" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.15.m15.1b"><apply id="S3.SS1.SSS2.6.p4.15.m15.1.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1"><csymbol cd="latexml" id="S3.SS1.SSS2.6.p4.15.m15.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.1">assign</csymbol><apply id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3"><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.1">¯</ci><apply id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2"><setdiff id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.1"></setdiff><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.2">𝑃</ci><apply id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.2.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.2">𝐾</ci><ci id="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.3.cmml" xref="S3.SS1.SSS2.6.p4.15.m15.1.1.3.2.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.15.m15.1c">P^{\prime}:=\overline{P\setminus K^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.15.m15.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := over¯ start_ARG italic_P ∖ italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>. Then, <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.16.m16.1"><semantics id="S3.SS1.SSS2.6.p4.16.m16.1a"><msup id="S3.SS1.SSS2.6.p4.16.m16.1.1" xref="S3.SS1.SSS2.6.p4.16.m16.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.16.m16.1.1.2" xref="S3.SS1.SSS2.6.p4.16.m16.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.16.m16.1.1.3" xref="S3.SS1.SSS2.6.p4.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.16.m16.1b"><apply id="S3.SS1.SSS2.6.p4.16.m16.1.1.cmml" xref="S3.SS1.SSS2.6.p4.16.m16.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.16.m16.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.16.m16.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.16.m16.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.16.m16.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.6.p4.16.m16.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.16.m16.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.16.m16.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is again a closed set with connected interior. For the connectedness, consider any two points in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.17.m17.1"><semantics id="S3.SS1.SSS2.6.p4.17.m17.1a"><msup id="S3.SS1.SSS2.6.p4.17.m17.1.1" xref="S3.SS1.SSS2.6.p4.17.m17.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.17.m17.1.1.2" xref="S3.SS1.SSS2.6.p4.17.m17.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.17.m17.1.1.3" xref="S3.SS1.SSS2.6.p4.17.m17.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.17.m17.1b"><apply id="S3.SS1.SSS2.6.p4.17.m17.1.1.cmml" xref="S3.SS1.SSS2.6.p4.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.17.m17.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.17.m17.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.17.m17.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.17.m17.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.6.p4.17.m17.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.17.m17.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.17.m17.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.17.m17.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. These can be connected by a path in the interior of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.18.m18.1"><semantics id="S3.SS1.SSS2.6.p4.18.m18.1a"><mi id="S3.SS1.SSS2.6.p4.18.m18.1.1" xref="S3.SS1.SSS2.6.p4.18.m18.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.18.m18.1b"><ci id="S3.SS1.SSS2.6.p4.18.m18.1.1.cmml" xref="S3.SS1.SSS2.6.p4.18.m18.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.18.m18.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.18.m18.1d">italic_P</annotation></semantics></math> that can only enter or leave <math alttext="K^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.19.m19.1"><semantics id="S3.SS1.SSS2.6.p4.19.m19.1a"><msup id="S3.SS1.SSS2.6.p4.19.m19.1.1" xref="S3.SS1.SSS2.6.p4.19.m19.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.19.m19.1.1.2" xref="S3.SS1.SSS2.6.p4.19.m19.1.1.2.cmml">K</mi><mo id="S3.SS1.SSS2.6.p4.19.m19.1.1.3" xref="S3.SS1.SSS2.6.p4.19.m19.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.19.m19.1b"><apply id="S3.SS1.SSS2.6.p4.19.m19.1.1.cmml" xref="S3.SS1.SSS2.6.p4.19.m19.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.19.m19.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.19.m19.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.19.m19.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.19.m19.1.1.2">𝐾</ci><ci id="S3.SS1.SSS2.6.p4.19.m19.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.19.m19.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.19.m19.1c">K^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.19.m19.1d">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> through <math alttext="\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.20.m20.1"><semantics id="S3.SS1.SSS2.6.p4.20.m20.1a"><msup id="S3.SS1.SSS2.6.p4.20.m20.1.1" xref="S3.SS1.SSS2.6.p4.20.m20.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.20.m20.1.1.2" xref="S3.SS1.SSS2.6.p4.20.m20.1.1.2.cmml">γ</mi><mo id="S3.SS1.SSS2.6.p4.20.m20.1.1.3" xref="S3.SS1.SSS2.6.p4.20.m20.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.20.m20.1b"><apply id="S3.SS1.SSS2.6.p4.20.m20.1.1.cmml" xref="S3.SS1.SSS2.6.p4.20.m20.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.20.m20.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.20.m20.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.20.m20.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.20.m20.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.6.p4.20.m20.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.20.m20.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.20.m20.1c">\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.20.m20.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, the segments of the path that run through <math alttext="K^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.21.m21.1"><semantics id="S3.SS1.SSS2.6.p4.21.m21.1a"><msup id="S3.SS1.SSS2.6.p4.21.m21.1.1" xref="S3.SS1.SSS2.6.p4.21.m21.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.21.m21.1.1.2" xref="S3.SS1.SSS2.6.p4.21.m21.1.1.2.cmml">K</mi><mo id="S3.SS1.SSS2.6.p4.21.m21.1.1.3" xref="S3.SS1.SSS2.6.p4.21.m21.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.21.m21.1b"><apply id="S3.SS1.SSS2.6.p4.21.m21.1.1.cmml" xref="S3.SS1.SSS2.6.p4.21.m21.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.21.m21.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.21.m21.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.21.m21.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.21.m21.1.1.2">𝐾</ci><ci id="S3.SS1.SSS2.6.p4.21.m21.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.21.m21.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.21.m21.1c">K^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.21.m21.1d">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> can be shortened via a path in the interior of <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.22.m22.1"><semantics id="S3.SS1.SSS2.6.p4.22.m22.1a"><msup id="S3.SS1.SSS2.6.p4.22.m22.1.1" xref="S3.SS1.SSS2.6.p4.22.m22.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.22.m22.1.1.2" xref="S3.SS1.SSS2.6.p4.22.m22.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.6.p4.22.m22.1.1.3" xref="S3.SS1.SSS2.6.p4.22.m22.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.22.m22.1b"><apply id="S3.SS1.SSS2.6.p4.22.m22.1.1.cmml" xref="S3.SS1.SSS2.6.p4.22.m22.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.22.m22.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.22.m22.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.22.m22.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.22.m22.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.6.p4.22.m22.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.22.m22.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.22.m22.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.22.m22.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> close to <math alttext="\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p4.23.m23.1"><semantics id="S3.SS1.SSS2.6.p4.23.m23.1a"><msup id="S3.SS1.SSS2.6.p4.23.m23.1.1" xref="S3.SS1.SSS2.6.p4.23.m23.1.1.cmml"><mi id="S3.SS1.SSS2.6.p4.23.m23.1.1.2" xref="S3.SS1.SSS2.6.p4.23.m23.1.1.2.cmml">γ</mi><mo id="S3.SS1.SSS2.6.p4.23.m23.1.1.3" xref="S3.SS1.SSS2.6.p4.23.m23.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p4.23.m23.1b"><apply id="S3.SS1.SSS2.6.p4.23.m23.1.1.cmml" xref="S3.SS1.SSS2.6.p4.23.m23.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p4.23.m23.1.1.1.cmml" xref="S3.SS1.SSS2.6.p4.23.m23.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.6.p4.23.m23.1.1.2.cmml" xref="S3.SS1.SSS2.6.p4.23.m23.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.6.p4.23.m23.1.1.3.cmml" xref="S3.SS1.SSS2.6.p4.23.m23.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p4.23.m23.1c">\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p4.23.m23.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.7.p5"> <p class="ltx_p" id="S3.SS1.SSS2.7.p5.2">Since <math alttext="K^{\prime}\cap D=\emptyset" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p5.1.m1.1"><semantics id="S3.SS1.SSS2.7.p5.1.m1.1a"><mrow id="S3.SS1.SSS2.7.p5.1.m1.1.1" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.cmml"><mrow id="S3.SS1.SSS2.7.p5.1.m1.1.1.2" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.cmml"><msup id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.cmml"><mi id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.2" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.2.cmml">K</mi><mo id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.3" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.1" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.3" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS2.7.p5.1.m1.1.1.1" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.1.cmml">=</mo><mi id="S3.SS1.SSS2.7.p5.1.m1.1.1.3" mathvariant="normal" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p5.1.m1.1b"><apply id="S3.SS1.SSS2.7.p5.1.m1.1.1.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1"><eq id="S3.SS1.SSS2.7.p5.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.1"></eq><apply id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2"><intersect id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.1"></intersect><apply id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.2">𝐾</ci><ci id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.2.3">′</ci></apply><ci id="S3.SS1.SSS2.7.p5.1.m1.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.2.3">𝐷</ci></apply><emptyset id="S3.SS1.SSS2.7.p5.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p5.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p5.1.m1.1c">K^{\prime}\cap D=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p5.1.m1.1d">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_D = ∅</annotation></semantics></math>, we have <math alttext="P^{\prime}\cap D=P\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p5.2.m2.1"><semantics id="S3.SS1.SSS2.7.p5.2.m2.1a"><mrow id="S3.SS1.SSS2.7.p5.2.m2.1.1" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.cmml"><mrow id="S3.SS1.SSS2.7.p5.2.m2.1.1.2" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.cmml"><msup id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.cmml"><mi id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.2" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.3" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.1" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.3" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS2.7.p5.2.m2.1.1.1" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS2.7.p5.2.m2.1.1.3" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.cmml"><mi id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.2" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.1" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.3" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p5.2.m2.1b"><apply id="S3.SS1.SSS2.7.p5.2.m2.1.1.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1"><eq id="S3.SS1.SSS2.7.p5.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.1"></eq><apply id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2"><intersect id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.1"></intersect><apply id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.2.3">′</ci></apply><ci id="S3.SS1.SSS2.7.p5.2.m2.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3"><intersect id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.1"></intersect><ci id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS2.7.p5.2.m2.1.1.3.3.cmml" xref="S3.SS1.SSS2.7.p5.2.m2.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p5.2.m2.1c">P^{\prime}\cap D=P\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p5.2.m2.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_D = italic_P ∩ italic_D</annotation></semantics></math>, verifying (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E18" title="Equation 18 ‣ Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">18</span></a>).</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.8.p6"> <p class="ltx_p" id="S3.SS1.SSS2.8.p6.5">Let <math alttext="\gamma_{a}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.1.m1.1"><semantics id="S3.SS1.SSS2.8.p6.1.m1.1a"><msub id="S3.SS1.SSS2.8.p6.1.m1.1.1" xref="S3.SS1.SSS2.8.p6.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.1.m1.1.1.2" xref="S3.SS1.SSS2.8.p6.1.m1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS2.8.p6.1.m1.1.1.3" xref="S3.SS1.SSS2.8.p6.1.m1.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.1.m1.1b"><apply id="S3.SS1.SSS2.8.p6.1.m1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p6.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.1.m1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.8.p6.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.1.m1.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.1.m1.1c">\gamma_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.1.m1.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma_{b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.2.m2.1"><semantics id="S3.SS1.SSS2.8.p6.2.m2.1a"><msub id="S3.SS1.SSS2.8.p6.2.m2.1.1" xref="S3.SS1.SSS2.8.p6.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.2.m2.1.1.2" xref="S3.SS1.SSS2.8.p6.2.m2.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS2.8.p6.2.m2.1.1.3" xref="S3.SS1.SSS2.8.p6.2.m2.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.2.m2.1b"><apply id="S3.SS1.SSS2.8.p6.2.m2.1.1.cmml" xref="S3.SS1.SSS2.8.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p6.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.2.m2.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.8.p6.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.2.m2.1c">\gamma_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.2.m2.1d">italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> be the polygonal arcs of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.3.m3.1"><semantics id="S3.SS1.SSS2.8.p6.3.m3.1a"><mi id="S3.SS1.SSS2.8.p6.3.m3.1.1" xref="S3.SS1.SSS2.8.p6.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.3.m3.1b"><ci id="S3.SS1.SSS2.8.p6.3.m3.1.1.cmml" xref="S3.SS1.SSS2.8.p6.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.3.m3.1d">italic_P</annotation></semantics></math> that contain the rays <math alttext="e_{a}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.4.m4.1"><semantics id="S3.SS1.SSS2.8.p6.4.m4.1a"><msub id="S3.SS1.SSS2.8.p6.4.m4.1.1" xref="S3.SS1.SSS2.8.p6.4.m4.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.4.m4.1.1.2" xref="S3.SS1.SSS2.8.p6.4.m4.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS2.8.p6.4.m4.1.1.3" xref="S3.SS1.SSS2.8.p6.4.m4.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.4.m4.1b"><apply id="S3.SS1.SSS2.8.p6.4.m4.1.1.cmml" xref="S3.SS1.SSS2.8.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p6.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.4.m4.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.8.p6.4.m4.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.4.m4.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.4.m4.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.4.m4.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.5.m5.1"><semantics id="S3.SS1.SSS2.8.p6.5.m5.1a"><msub id="S3.SS1.SSS2.8.p6.5.m5.1.1" xref="S3.SS1.SSS2.8.p6.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.5.m5.1.1.2" xref="S3.SS1.SSS2.8.p6.5.m5.1.1.2.cmml">e</mi><mi id="S3.SS1.SSS2.8.p6.5.m5.1.1.3" xref="S3.SS1.SSS2.8.p6.5.m5.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.5.m5.1b"><apply id="S3.SS1.SSS2.8.p6.5.m5.1.1.cmml" xref="S3.SS1.SSS2.8.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p6.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.5.m5.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.8.p6.5.m5.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.5.m5.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.5.m5.1c">e_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.5.m5.1d">italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. With</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex25"> <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="\gamma:=(\gamma_{a}\cup\gamma_{b})\setminus(e_{a}\cup e_{b})\cup s_{a}\cup s_{% b}\cup\gamma^{\prime}," class="ltx_Math" display="block" id="S3.Ex25.m1.1"><semantics id="S3.Ex25.m1.1a"><mrow id="S3.Ex25.m1.1.1.1" xref="S3.Ex25.m1.1.1.1.1.cmml"><mrow id="S3.Ex25.m1.1.1.1.1" xref="S3.Ex25.m1.1.1.1.1.cmml"><mi id="S3.Ex25.m1.1.1.1.1.4" xref="S3.Ex25.m1.1.1.1.1.4.cmml">γ</mi><mo id="S3.Ex25.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex25.m1.1.1.1.1.3.cmml">:=</mo><mrow id="S3.Ex25.m1.1.1.1.1.2" xref="S3.Ex25.m1.1.1.1.1.2.cmml"><mrow id="S3.Ex25.m1.1.1.1.1.2.2" xref="S3.Ex25.m1.1.1.1.1.2.2.cmml"><mrow id="S3.Ex25.m1.1.1.1.1.1.1.1.1" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex25.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">γ</mi><mi id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2.3" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.1.cmml">∪</mo><msub id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">γ</mi><mi id="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">b</mi></msub></mrow><mo id="S3.Ex25.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex25.m1.1.1.1.1.2.2.3" xref="S3.Ex25.m1.1.1.1.1.2.2.3.cmml">∖</mo><mrow id="S3.Ex25.m1.1.1.1.1.2.2.2.1" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.cmml"><mo id="S3.Ex25.m1.1.1.1.1.2.2.2.1.2" stretchy="false" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.cmml">(</mo><mrow id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.cmml"><msub id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2.cmml"><mi id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2.2" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2.2.cmml">e</mi><mi id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2.3" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.2.3.cmml">a</mi></msub><mo id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.1" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.1.cmml">∪</mo><msub id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3.cmml"><mi id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3.2" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3.2.cmml">e</mi><mi id="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3.3" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.3.3.cmml">b</mi></msub></mrow><mo id="S3.Ex25.m1.1.1.1.1.2.2.2.1.3" stretchy="false" xref="S3.Ex25.m1.1.1.1.1.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex25.m1.1.1.1.1.2.3" xref="S3.Ex25.m1.1.1.1.1.2.3.cmml">∪</mo><msub id="S3.Ex25.m1.1.1.1.1.2.4" xref="S3.Ex25.m1.1.1.1.1.2.4.cmml"><mi id="S3.Ex25.m1.1.1.1.1.2.4.2" xref="S3.Ex25.m1.1.1.1.1.2.4.2.cmml">s</mi><mi id="S3.Ex25.m1.1.1.1.1.2.4.3" xref="S3.Ex25.m1.1.1.1.1.2.4.3.cmml">a</mi></msub><mo id="S3.Ex25.m1.1.1.1.1.2.3a" xref="S3.Ex25.m1.1.1.1.1.2.3.cmml">∪</mo><msub id="S3.Ex25.m1.1.1.1.1.2.5" xref="S3.Ex25.m1.1.1.1.1.2.5.cmml"><mi id="S3.Ex25.m1.1.1.1.1.2.5.2" xref="S3.Ex25.m1.1.1.1.1.2.5.2.cmml">s</mi><mi id="S3.Ex25.m1.1.1.1.1.2.5.3" xref="S3.Ex25.m1.1.1.1.1.2.5.3.cmml">b</mi></msub><mo id="S3.Ex25.m1.1.1.1.1.2.3b" xref="S3.Ex25.m1.1.1.1.1.2.3.cmml">∪</mo><msup id="S3.Ex25.m1.1.1.1.1.2.6" xref="S3.Ex25.m1.1.1.1.1.2.6.cmml"><mi id="S3.Ex25.m1.1.1.1.1.2.6.2" xref="S3.Ex25.m1.1.1.1.1.2.6.2.cmml">γ</mi><mo id="S3.Ex25.m1.1.1.1.1.2.6.3" xref="S3.Ex25.m1.1.1.1.1.2.6.3.cmml">′</mo></msup></mrow></mrow><mo 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xref="S3.Ex25.m1.1.1.1.1.2.4.3">𝑎</ci></apply><apply id="S3.Ex25.m1.1.1.1.1.2.5.cmml" xref="S3.Ex25.m1.1.1.1.1.2.5"><csymbol cd="ambiguous" id="S3.Ex25.m1.1.1.1.1.2.5.1.cmml" xref="S3.Ex25.m1.1.1.1.1.2.5">subscript</csymbol><ci id="S3.Ex25.m1.1.1.1.1.2.5.2.cmml" xref="S3.Ex25.m1.1.1.1.1.2.5.2">𝑠</ci><ci id="S3.Ex25.m1.1.1.1.1.2.5.3.cmml" xref="S3.Ex25.m1.1.1.1.1.2.5.3">𝑏</ci></apply><apply id="S3.Ex25.m1.1.1.1.1.2.6.cmml" xref="S3.Ex25.m1.1.1.1.1.2.6"><csymbol cd="ambiguous" id="S3.Ex25.m1.1.1.1.1.2.6.1.cmml" xref="S3.Ex25.m1.1.1.1.1.2.6">superscript</csymbol><ci id="S3.Ex25.m1.1.1.1.1.2.6.2.cmml" xref="S3.Ex25.m1.1.1.1.1.2.6.2">𝛾</ci><ci id="S3.Ex25.m1.1.1.1.1.2.6.3.cmml" xref="S3.Ex25.m1.1.1.1.1.2.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex25.m1.1c">\gamma:=(\gamma_{a}\cup\gamma_{b})\setminus(e_{a}\cup e_{b})\cup s_{a}\cup s_{% b}\cup\gamma^{\prime},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex25.m1.1d">italic_γ := ( italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∪ italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) ∖ ( italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∪ italic_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) ∪ italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∪ italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∪ italic_γ 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.SS1.SSS2.8.p6.6">we can describe the boundary of <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.6.m1.1"><semantics id="S3.SS1.SSS2.8.p6.6.m1.1a"><msup id="S3.SS1.SSS2.8.p6.6.m1.1.1" xref="S3.SS1.SSS2.8.p6.6.m1.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.6.m1.1.1.2" xref="S3.SS1.SSS2.8.p6.6.m1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.8.p6.6.m1.1.1.3" xref="S3.SS1.SSS2.8.p6.6.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.6.m1.1b"><apply id="S3.SS1.SSS2.8.p6.6.m1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.6.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.6.m1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.6.m1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.8.p6.6.m1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.6.m1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.8.p6.6.m1.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.6.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.6.m1.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.6.m1.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E20"> <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="\partial P^{\prime}=(\partial P\setminus(\gamma_{a}\cup\gamma_{b}))\cup\gamma." class="ltx_Math" display="block" id="S3.E20.m1.1"><semantics id="S3.E20.m1.1a"><mrow id="S3.E20.m1.1.1.1" xref="S3.E20.m1.1.1.1.1.cmml"><mrow id="S3.E20.m1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.cmml"><mrow id="S3.E20.m1.1.1.1.1.3" xref="S3.E20.m1.1.1.1.1.3.cmml"><mo id="S3.E20.m1.1.1.1.1.3.1" rspace="0em" xref="S3.E20.m1.1.1.1.1.3.1.cmml">∂</mo><msup id="S3.E20.m1.1.1.1.1.3.2" xref="S3.E20.m1.1.1.1.1.3.2.cmml"><mi id="S3.E20.m1.1.1.1.1.3.2.2" xref="S3.E20.m1.1.1.1.1.3.2.2.cmml">P</mi><mo id="S3.E20.m1.1.1.1.1.3.2.3" xref="S3.E20.m1.1.1.1.1.3.2.3.cmml">′</mo></msup></mrow><mo id="S3.E20.m1.1.1.1.1.2" xref="S3.E20.m1.1.1.1.1.2.cmml">=</mo><mrow id="S3.E20.m1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.cmml"><mrow id="S3.E20.m1.1.1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E20.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E20.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E20.m1.1.1.1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.E20.m1.1.1.1.1.1.1.1.1.3" xref="S3.E20.m1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S3.E20.m1.1.1.1.1.1.1.1.1.3.1" lspace="0em" rspace="0em" xref="S3.E20.m1.1.1.1.1.1.1.1.1.3.1.cmml">∂</mo><mi id="S3.E20.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.E20.m1.1.1.1.1.1.1.1.1.3.2.cmml">P</mi></mrow><mo id="S3.E20.m1.1.1.1.1.1.1.1.1.2" xref="S3.E20.m1.1.1.1.1.1.1.1.1.2.cmml">∖</mo><mrow id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">γ</mi><mi id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">a</mi></msub><mo 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id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.3.2">𝛾</ci><ci id="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.E20.m1.1.1.1.1.1.1.1.1.1.1.1.3.3">𝑏</ci></apply></apply></apply><ci id="S3.E20.m1.1.1.1.1.1.3.cmml" xref="S3.E20.m1.1.1.1.1.1.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E20.m1.1c">\partial P^{\prime}=(\partial P\setminus(\gamma_{a}\cup\gamma_{b}))\cup\gamma.</annotation><annotation encoding="application/x-llamapun" id="S3.E20.m1.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( ∂ italic_P ∖ ( italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∪ italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) ) ∪ italic_γ .</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">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.8.p6.9">Before addressing (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E19" title="Equation 19 ‣ Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">19</span></a>), we prove that <math alttext="n_{a}(P^{\prime})=n_{a}(P)-1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.7.m1.2"><semantics id="S3.SS1.SSS2.8.p6.7.m1.2a"><mrow id="S3.SS1.SSS2.8.p6.7.m1.2.2" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.cmml"><mrow id="S3.SS1.SSS2.8.p6.7.m1.2.2.1" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.1.cmml"><msub id="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3.cmml"><mi id="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3.2" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3.2.cmml">n</mi><mi id="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3.3" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.1.3.3.cmml">a</mi></msub><mo 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xref="S3.SS1.SSS2.8.p6.7.m1.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.8.p6.7.m1.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p6.7.m1.2.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.7.m1.2c">n_{a}(P^{\prime})=n_{a}(P)-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.7.m1.2d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - 1</annotation></semantics></math> and that <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.8.m2.1"><semantics id="S3.SS1.SSS2.8.p6.8.m2.1a"><msup id="S3.SS1.SSS2.8.p6.8.m2.1.1" xref="S3.SS1.SSS2.8.p6.8.m2.1.1.cmml"><mi id="S3.SS1.SSS2.8.p6.8.m2.1.1.2" xref="S3.SS1.SSS2.8.p6.8.m2.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.8.p6.8.m2.1.1.3" xref="S3.SS1.SSS2.8.p6.8.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.8.m2.1b"><apply id="S3.SS1.SSS2.8.p6.8.m2.1.1.cmml" xref="S3.SS1.SSS2.8.p6.8.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.8.m2.1.1.1.cmml" xref="S3.SS1.SSS2.8.p6.8.m2.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.8.p6.8.m2.1.1.2.cmml" xref="S3.SS1.SSS2.8.p6.8.m2.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.8.p6.8.m2.1.1.3.cmml" xref="S3.SS1.SSS2.8.p6.8.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.8.m2.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.8.m2.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is bounded with its boundary being a cycle if <math alttext="n_{a}(P)=1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p6.9.m3.1"><semantics id="S3.SS1.SSS2.8.p6.9.m3.1a"><mrow id="S3.SS1.SSS2.8.p6.9.m3.1.2" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.cmml"><mrow id="S3.SS1.SSS2.8.p6.9.m3.1.2.2" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.cmml"><msub id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.2" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.3" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.1" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.3.2" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.cmml"><mo id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.8.p6.9.m3.1.1" xref="S3.SS1.SSS2.8.p6.9.m3.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.8.p6.9.m3.1.2.1" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.8.p6.9.m3.1.2.3" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p6.9.m3.1b"><apply id="S3.SS1.SSS2.8.p6.9.m3.1.2.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2"><eq id="S3.SS1.SSS2.8.p6.9.m3.1.2.1.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.1"></eq><apply id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2"><times id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.1.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.1"></times><apply id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.1.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.2.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.3.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.8.p6.9.m3.1.1.cmml" xref="S3.SS1.SSS2.8.p6.9.m3.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.8.p6.9.m3.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p6.9.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p6.9.m3.1c">n_{a}(P)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p6.9.m3.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.9.p7"> <p class="ltx_p" id="S3.SS1.SSS2.9.p7.17"><span class="ltx_text ltx_font_bold" id="S3.SS1.SSS2.9.p7.17.1">Case 1:</span> Assume that <math alttext="n_{a}(P)=1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.1.m1.1"><semantics id="S3.SS1.SSS2.9.p7.1.m1.1a"><mrow id="S3.SS1.SSS2.9.p7.1.m1.1.2" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.cmml"><mrow id="S3.SS1.SSS2.9.p7.1.m1.1.2.2" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.cmml"><msub id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.2" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.3" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.1" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.3.2" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.cmml"><mo id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.9.p7.1.m1.1.1" xref="S3.SS1.SSS2.9.p7.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.9.p7.1.m1.1.2.1" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.9.p7.1.m1.1.2.3" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.1.m1.1b"><apply id="S3.SS1.SSS2.9.p7.1.m1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2"><eq id="S3.SS1.SSS2.9.p7.1.m1.1.2.1.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.1"></eq><apply id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2"><times id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.1"></times><apply id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.2.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.3.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.9.p7.1.m1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.1.m1.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.9.p7.1.m1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.9.p7.1.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.1.m1.1c">n_{a}(P)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = 1</annotation></semantics></math>. <br class="ltx_break"/>In that case, <math alttext="\gamma_{a}=\gamma_{b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.2.m2.1"><semantics id="S3.SS1.SSS2.9.p7.2.m2.1a"><mrow id="S3.SS1.SSS2.9.p7.2.m2.1.1" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.cmml"><msub id="S3.SS1.SSS2.9.p7.2.m2.1.1.2" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2.cmml"><mi id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.2" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2.2.cmml">γ</mi><mi id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.3" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.9.p7.2.m2.1.1.1" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS2.9.p7.2.m2.1.1.3" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3.cmml"><mi id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.2" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3.2.cmml">γ</mi><mi id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.3" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3.3.cmml">b</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.2.m2.1b"><apply id="S3.SS1.SSS2.9.p7.2.m2.1.1.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1"><eq id="S3.SS1.SSS2.9.p7.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.1"></eq><apply id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.1.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.2.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.9.p7.2.m2.1.1.2.3.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.1.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.2.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS2.9.p7.2.m2.1.1.3.3.cmml" xref="S3.SS1.SSS2.9.p7.2.m2.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.2.m2.1c">\gamma_{a}=\gamma_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.2.m2.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.3.m3.1"><semantics id="S3.SS1.SSS2.9.p7.3.m3.1a"><mi id="S3.SS1.SSS2.9.p7.3.m3.1.1" xref="S3.SS1.SSS2.9.p7.3.m3.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.3.m3.1b"><ci id="S3.SS1.SSS2.9.p7.3.m3.1.1.cmml" xref="S3.SS1.SSS2.9.p7.3.m3.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.3.m3.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.3.m3.1d">italic_γ</annotation></semantics></math> becomes a polygonal cycle. Since <math alttext="\partial P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.4.m4.1"><semantics id="S3.SS1.SSS2.9.p7.4.m4.1a"><mrow id="S3.SS1.SSS2.9.p7.4.m4.1.1" xref="S3.SS1.SSS2.9.p7.4.m4.1.1.cmml"><mo id="S3.SS1.SSS2.9.p7.4.m4.1.1.1" rspace="0em" xref="S3.SS1.SSS2.9.p7.4.m4.1.1.1.cmml">∂</mo><mi id="S3.SS1.SSS2.9.p7.4.m4.1.1.2" xref="S3.SS1.SSS2.9.p7.4.m4.1.1.2.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.4.m4.1b"><apply id="S3.SS1.SSS2.9.p7.4.m4.1.1.cmml" xref="S3.SS1.SSS2.9.p7.4.m4.1.1"><partialdiff id="S3.SS1.SSS2.9.p7.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.4.m4.1.1.1"></partialdiff><ci id="S3.SS1.SSS2.9.p7.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.4.m4.1.1.2">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.4.m4.1c">\partial P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.4.m4.1d">∂ italic_P</annotation></semantics></math> consists only of arcs, and <math alttext="\gamma_{a}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.5.m5.1"><semantics id="S3.SS1.SSS2.9.p7.5.m5.1a"><msub id="S3.SS1.SSS2.9.p7.5.m5.1.1" xref="S3.SS1.SSS2.9.p7.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.5.m5.1.1.2" xref="S3.SS1.SSS2.9.p7.5.m5.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS2.9.p7.5.m5.1.1.3" xref="S3.SS1.SSS2.9.p7.5.m5.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.5.m5.1b"><apply id="S3.SS1.SSS2.9.p7.5.m5.1.1.cmml" xref="S3.SS1.SSS2.9.p7.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.5.m5.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.9.p7.5.m5.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.5.m5.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.5.m5.1c">\gamma_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.5.m5.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is the only arc, we get from (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E20" title="Equation 20 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">20</span></a>) that <math alttext="\partial P^{\prime}=\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.6.m6.1"><semantics id="S3.SS1.SSS2.9.p7.6.m6.1a"><mrow id="S3.SS1.SSS2.9.p7.6.m6.1.1" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.cmml"><mrow id="S3.SS1.SSS2.9.p7.6.m6.1.1.2" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.cmml"><mo id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.1" rspace="0em" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.1.cmml">∂</mo><msup id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.cmml"><mi id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.2" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.3" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.9.p7.6.m6.1.1.1" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.1.cmml">=</mo><mi id="S3.SS1.SSS2.9.p7.6.m6.1.1.3" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.6.m6.1b"><apply id="S3.SS1.SSS2.9.p7.6.m6.1.1.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1"><eq id="S3.SS1.SSS2.9.p7.6.m6.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.1"></eq><apply id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2"><partialdiff id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.1.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.1"></partialdiff><apply id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.2.2.3">′</ci></apply></apply><ci id="S3.SS1.SSS2.9.p7.6.m6.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.6.m6.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.6.m6.1c">\partial P^{\prime}=\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.6.m6.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_γ</annotation></semantics></math>, and <math alttext="n_{a}(P^{\prime})=n_{a}(P)-1=0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.7.m7.2"><semantics id="S3.SS1.SSS2.9.p7.7.m7.2a"><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.cmml"><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2.1" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.cmml"><msub id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.cmml"><mi id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.2.cmml">n</mi><mi id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.3.cmml">=</mo><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2.4" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.cmml"><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.cmml"><msub id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.cmml"><mi id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.1" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.3.2" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.cmml"><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.cmml">(</mo><mi id="S3.SS1.SSS2.9.p7.7.m7.1.1" xref="S3.SS1.SSS2.9.p7.7.m7.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.1" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.1.cmml">−</mo><mn id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.3" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.9.p7.7.m7.2.2.5" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.5.cmml">=</mo><mn id="S3.SS1.SSS2.9.p7.7.m7.2.2.6" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.7.m7.2b"><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2"><and id="S3.SS1.SSS2.9.p7.7.m7.2.2a.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2"></and><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2b.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2"><eq id="S3.SS1.SSS2.9.p7.7.m7.2.2.3.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.3"></eq><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1"><times id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.2"></times><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.2">𝑛</ci><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.3.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.3.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4"><minus id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.1"></minus><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2"><times id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.1"></times><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.2.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.3.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.9.p7.7.m7.1.1.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.9.p7.7.m7.2.2.4.3.cmml" type="integer" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.4.3">1</cn></apply></apply><apply id="S3.SS1.SSS2.9.p7.7.m7.2.2c.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2"><eq id="S3.SS1.SSS2.9.p7.7.m7.2.2.5.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS2.9.p7.7.m7.2.2.4.cmml" id="S3.SS1.SSS2.9.p7.7.m7.2.2d.cmml" xref="S3.SS1.SSS2.9.p7.7.m7.2.2"></share><cn id="S3.SS1.SSS2.9.p7.7.m7.2.2.6.cmml" type="integer" xref="S3.SS1.SSS2.9.p7.7.m7.2.2.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.7.m7.2c">n_{a}(P^{\prime})=n_{a}(P)-1=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.7.m7.2d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - 1 = 0</annotation></semantics></math>. Moreover, for any <math alttext="y^{\prime}\in\eta\setminus\{a,b,y\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.8.m8.3"><semantics id="S3.SS1.SSS2.9.p7.8.m8.3a"><mrow id="S3.SS1.SSS2.9.p7.8.m8.3.4" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.cmml"><msup id="S3.SS1.SSS2.9.p7.8.m8.3.4.2" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2.cmml"><mi id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.2" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2.2.cmml">y</mi><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.3" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.1" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.9.p7.8.m8.3.4.3" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.cmml"><mi id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.2" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.2.cmml">η</mi><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.1" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.1.cmml">∖</mo><mrow id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml">{</mo><mi id="S3.SS1.SSS2.9.p7.8.m8.1.1" xref="S3.SS1.SSS2.9.p7.8.m8.1.1.cmml">a</mi><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2.2" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml">,</mo><mi id="S3.SS1.SSS2.9.p7.8.m8.2.2" xref="S3.SS1.SSS2.9.p7.8.m8.2.2.cmml">b</mi><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2.3" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml">,</mo><mi id="S3.SS1.SSS2.9.p7.8.m8.3.3" xref="S3.SS1.SSS2.9.p7.8.m8.3.3.cmml">y</mi><mo id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2.4" stretchy="false" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.8.m8.3b"><apply id="S3.SS1.SSS2.9.p7.8.m8.3.4.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4"><in id="S3.SS1.SSS2.9.p7.8.m8.3.4.1.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.1"></in><apply id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.1.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.2.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2.2">𝑦</ci><ci id="S3.SS1.SSS2.9.p7.8.m8.3.4.2.3.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.2.3">′</ci></apply><apply id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3"><setdiff id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.1.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.1"></setdiff><ci id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.2.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.2">𝜂</ci><set id="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.1.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.4.3.3.2"><ci id="S3.SS1.SSS2.9.p7.8.m8.1.1.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.1.1">𝑎</ci><ci id="S3.SS1.SSS2.9.p7.8.m8.2.2.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.2.2">𝑏</ci><ci id="S3.SS1.SSS2.9.p7.8.m8.3.3.cmml" xref="S3.SS1.SSS2.9.p7.8.m8.3.3">𝑦</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.8.m8.3c">y^{\prime}\in\eta\setminus\{a,b,y\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.8.m8.3d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_η ∖ { italic_a , italic_b , italic_y }</annotation></semantics></math>, the ray <math alttext="\{\rho y^{\prime}:\rho\in[1,\infty)\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.9.m9.4"><semantics id="S3.SS1.SSS2.9.p7.9.m9.4a"><mrow id="S3.SS1.SSS2.9.p7.9.m9.4.4.2" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.3.cmml"><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.3" stretchy="false" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.3.1.cmml">{</mo><mrow id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.2" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.2.cmml">ρ</mi><mo id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.1" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.1.cmml">⁢</mo><msup id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.cmml"><mi id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.2" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.2.cmml">y</mi><mo id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.3" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.4" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.3.1.cmml">:</mo><mrow id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.cmml"><mi id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.2" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.2.cmml">ρ</mi><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.1" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.2" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.1.cmml">[</mo><mn id="S3.SS1.SSS2.9.p7.9.m9.1.1" xref="S3.SS1.SSS2.9.p7.9.m9.1.1.cmml">1</mn><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.2.2" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.1.cmml">,</mo><mi id="S3.SS1.SSS2.9.p7.9.m9.2.2" mathvariant="normal" xref="S3.SS1.SSS2.9.p7.9.m9.2.2.cmml">∞</mi><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.2.3" stretchy="false" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.5" stretchy="false" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.9.m9.4b"><apply id="S3.SS1.SSS2.9.p7.9.m9.4.4.3.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2"><csymbol cd="latexml" id="S3.SS1.SSS2.9.p7.9.m9.4.4.3.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.3">conditional-set</csymbol><apply id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1"><times id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.1"></times><ci id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.2">𝜌</ci><apply id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.2.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.2">𝑦</ci><ci id="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.3.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.3.3.1.1.3.3">′</ci></apply></apply><apply id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2"><in id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.1"></in><ci id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.2.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.2">𝜌</ci><interval closure="closed-open" id="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.1.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.4.4.2.2.3.2"><cn id="S3.SS1.SSS2.9.p7.9.m9.1.1.cmml" type="integer" xref="S3.SS1.SSS2.9.p7.9.m9.1.1">1</cn><infinity id="S3.SS1.SSS2.9.p7.9.m9.2.2.cmml" xref="S3.SS1.SSS2.9.p7.9.m9.2.2"></infinity></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.9.m9.4c">\{\rho y^{\prime}:\rho\in[1,\infty)\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.9.m9.4d">{ italic_ρ italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_ρ ∈ [ 1 , ∞ ) }</annotation></semantics></math> intersects <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.10.m10.1"><semantics id="S3.SS1.SSS2.9.p7.10.m10.1a"><mi id="S3.SS1.SSS2.9.p7.10.m10.1.1" xref="S3.SS1.SSS2.9.p7.10.m10.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.10.m10.1b"><ci id="S3.SS1.SSS2.9.p7.10.m10.1.1.cmml" xref="S3.SS1.SSS2.9.p7.10.m10.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.10.m10.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.10.m10.1d">italic_γ</annotation></semantics></math> for exactly one <math alttext="\rho>0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.11.m11.1"><semantics id="S3.SS1.SSS2.9.p7.11.m11.1a"><mrow id="S3.SS1.SSS2.9.p7.11.m11.1.1" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.11.m11.1.1.2" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.2.cmml">ρ</mi><mo id="S3.SS1.SSS2.9.p7.11.m11.1.1.1" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.1.cmml">></mo><mn id="S3.SS1.SSS2.9.p7.11.m11.1.1.3" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.11.m11.1b"><apply id="S3.SS1.SSS2.9.p7.11.m11.1.1.cmml" xref="S3.SS1.SSS2.9.p7.11.m11.1.1"><gt id="S3.SS1.SSS2.9.p7.11.m11.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.1"></gt><ci id="S3.SS1.SSS2.9.p7.11.m11.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.2">𝜌</ci><cn id="S3.SS1.SSS2.9.p7.11.m11.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.9.p7.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.11.m11.1c">\rho>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.11.m11.1d">italic_ρ > 0</annotation></semantics></math>, since <math alttext="\gamma\setminus D=\gamma^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.12.m12.1"><semantics id="S3.SS1.SSS2.9.p7.12.m12.1a"><mrow id="S3.SS1.SSS2.9.p7.12.m12.1.1" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.cmml"><mrow id="S3.SS1.SSS2.9.p7.12.m12.1.1.2" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.cmml"><mi id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.2" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.1" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.1.cmml">∖</mo><mi id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.3" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS2.9.p7.12.m12.1.1.1" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.1.cmml">=</mo><msup id="S3.SS1.SSS2.9.p7.12.m12.1.1.3" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3.cmml"><mi id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.2" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3.2.cmml">γ</mi><mo id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.3" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.12.m12.1b"><apply id="S3.SS1.SSS2.9.p7.12.m12.1.1.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1"><eq id="S3.SS1.SSS2.9.p7.12.m12.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.1"></eq><apply id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2"><setdiff id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.1.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.1"></setdiff><ci id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.2.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.9.p7.12.m12.1.1.2.3.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.1.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.2.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS2.9.p7.12.m12.1.1.3.3.cmml" xref="S3.SS1.SSS2.9.p7.12.m12.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.12.m12.1c">\gamma\setminus D=\gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.12.m12.1d">italic_γ ∖ italic_D = italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus, an initial segment of that ray, and in particular <math alttext="y^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.13.m13.1"><semantics id="S3.SS1.SSS2.9.p7.13.m13.1a"><msup id="S3.SS1.SSS2.9.p7.13.m13.1.1" xref="S3.SS1.SSS2.9.p7.13.m13.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.13.m13.1.1.2" xref="S3.SS1.SSS2.9.p7.13.m13.1.1.2.cmml">y</mi><mo id="S3.SS1.SSS2.9.p7.13.m13.1.1.3" xref="S3.SS1.SSS2.9.p7.13.m13.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.13.m13.1b"><apply id="S3.SS1.SSS2.9.p7.13.m13.1.1.cmml" xref="S3.SS1.SSS2.9.p7.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.13.m13.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.13.m13.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.13.m13.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.13.m13.1.1.2">𝑦</ci><ci id="S3.SS1.SSS2.9.p7.13.m13.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.13.m13.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.13.m13.1c">y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.13.m13.1d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, is contained in <math alttext="\operatorname*{int}(\gamma)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.14.m14.2"><semantics id="S3.SS1.SSS2.9.p7.14.m14.2a"><mrow id="S3.SS1.SSS2.9.p7.14.m14.2.3.2" xref="S3.SS1.SSS2.9.p7.14.m14.2.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.14.m14.1.1" rspace="0em" xref="S3.SS1.SSS2.9.p7.14.m14.1.1.cmml">int</mo><mrow id="S3.SS1.SSS2.9.p7.14.m14.2.3.2.1" xref="S3.SS1.SSS2.9.p7.14.m14.2.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.14.m14.2.3.2.1.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.14.m14.2.3.1.cmml">(</mo><mi id="S3.SS1.SSS2.9.p7.14.m14.2.2" xref="S3.SS1.SSS2.9.p7.14.m14.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.9.p7.14.m14.2.3.2.1.2" stretchy="false" xref="S3.SS1.SSS2.9.p7.14.m14.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.14.m14.2b"><apply id="S3.SS1.SSS2.9.p7.14.m14.2.3.1.cmml" xref="S3.SS1.SSS2.9.p7.14.m14.2.3.2"><ci id="S3.SS1.SSS2.9.p7.14.m14.1.1.cmml" xref="S3.SS1.SSS2.9.p7.14.m14.1.1">int</ci><ci id="S3.SS1.SSS2.9.p7.14.m14.2.2.cmml" xref="S3.SS1.SSS2.9.p7.14.m14.2.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.14.m14.2c">\operatorname*{int}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.14.m14.2d">roman_int ( italic_γ )</annotation></semantics></math>. Since <math alttext="y^{\prime}\in P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.15.m15.1"><semantics id="S3.SS1.SSS2.9.p7.15.m15.1a"><mrow id="S3.SS1.SSS2.9.p7.15.m15.1.1" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.cmml"><msup id="S3.SS1.SSS2.9.p7.15.m15.1.1.2" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2.cmml"><mi id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.2" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2.2.cmml">y</mi><mo id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.3" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.9.p7.15.m15.1.1.1" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS2.9.p7.15.m15.1.1.3" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3.cmml"><mi id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.2" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.3" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.15.m15.1b"><apply id="S3.SS1.SSS2.9.p7.15.m15.1.1.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1"><in id="S3.SS1.SSS2.9.p7.15.m15.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.1"></in><apply id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.1.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.2.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2.2">𝑦</ci><ci id="S3.SS1.SSS2.9.p7.15.m15.1.1.2.3.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.1.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.2.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS2.9.p7.15.m15.1.1.3.3.cmml" xref="S3.SS1.SSS2.9.p7.15.m15.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.15.m15.1c">y^{\prime}\in P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.15.m15.1d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, we have <math alttext="P^{\prime}\subseteq\operatorname*{int}(\gamma)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.16.m16.2"><semantics id="S3.SS1.SSS2.9.p7.16.m16.2a"><mrow id="S3.SS1.SSS2.9.p7.16.m16.2.3" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.cmml"><msup id="S3.SS1.SSS2.9.p7.16.m16.2.3.2" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2.cmml"><mi id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.2" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.3" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.9.p7.16.m16.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.1.cmml">⊆</mo><mrow id="S3.SS1.SSS2.9.p7.16.m16.2.3.3.2" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.16.m16.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS2.9.p7.16.m16.1.1.cmml">int</mo><mrow id="S3.SS1.SSS2.9.p7.16.m16.2.3.3.2.1" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.3.1.cmml"><mo id="S3.SS1.SSS2.9.p7.16.m16.2.3.3.2.1.1" stretchy="false" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.3.1.cmml">(</mo><mi id="S3.SS1.SSS2.9.p7.16.m16.2.2" xref="S3.SS1.SSS2.9.p7.16.m16.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.9.p7.16.m16.2.3.3.2.1.2" stretchy="false" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.16.m16.2b"><apply id="S3.SS1.SSS2.9.p7.16.m16.2.3.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3"><subset id="S3.SS1.SSS2.9.p7.16.m16.2.3.1.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.1"></subset><apply id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.1.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.2.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2.2">𝑃</ci><ci id="S3.SS1.SSS2.9.p7.16.m16.2.3.2.3.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.9.p7.16.m16.2.3.3.1.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.3.3.2"><ci id="S3.SS1.SSS2.9.p7.16.m16.1.1.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.1.1">int</ci><ci id="S3.SS1.SSS2.9.p7.16.m16.2.2.cmml" xref="S3.SS1.SSS2.9.p7.16.m16.2.2">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.16.m16.2c">P^{\prime}\subseteq\operatorname*{int}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.16.m16.2d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ roman_int ( italic_γ )</annotation></semantics></math>, thus <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.9.p7.17.m17.1"><semantics id="S3.SS1.SSS2.9.p7.17.m17.1a"><msup id="S3.SS1.SSS2.9.p7.17.m17.1.1" xref="S3.SS1.SSS2.9.p7.17.m17.1.1.cmml"><mi id="S3.SS1.SSS2.9.p7.17.m17.1.1.2" xref="S3.SS1.SSS2.9.p7.17.m17.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.9.p7.17.m17.1.1.3" xref="S3.SS1.SSS2.9.p7.17.m17.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.9.p7.17.m17.1b"><apply id="S3.SS1.SSS2.9.p7.17.m17.1.1.cmml" xref="S3.SS1.SSS2.9.p7.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.9.p7.17.m17.1.1.1.cmml" xref="S3.SS1.SSS2.9.p7.17.m17.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.9.p7.17.m17.1.1.2.cmml" xref="S3.SS1.SSS2.9.p7.17.m17.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.9.p7.17.m17.1.1.3.cmml" xref="S3.SS1.SSS2.9.p7.17.m17.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.9.p7.17.m17.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.9.p7.17.m17.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is bounded.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.10.p8"> <p class="ltx_p" id="S3.SS1.SSS2.10.p8.9"><span class="ltx_text ltx_font_bold" id="S3.SS1.SSS2.10.p8.9.1">Case 2:</span> Assume that <math alttext="n_{a}(P)\geq 2" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.1.m1.1"><semantics id="S3.SS1.SSS2.10.p8.1.m1.1a"><mrow id="S3.SS1.SSS2.10.p8.1.m1.1.2" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.cmml"><mrow id="S3.SS1.SSS2.10.p8.1.m1.1.2.2" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.cmml"><msub id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.2" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.3" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.1" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.3.2" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.cmml"><mo id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.10.p8.1.m1.1.1" xref="S3.SS1.SSS2.10.p8.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.10.p8.1.m1.1.2.1" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.1.cmml">≥</mo><mn id="S3.SS1.SSS2.10.p8.1.m1.1.2.3" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.1.m1.1b"><apply id="S3.SS1.SSS2.10.p8.1.m1.1.2.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2"><geq id="S3.SS1.SSS2.10.p8.1.m1.1.2.1.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.1"></geq><apply id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2"><times id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.1.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.1"></times><apply id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.1.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.2.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.3.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.10.p8.1.m1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.1.m1.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.10.p8.1.m1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.10.p8.1.m1.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.1.m1.1c">n_{a}(P)\geq 2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) ≥ 2</annotation></semantics></math>. <br class="ltx_break"/>Assume for the sake of contradiction that <math alttext="\gamma_{a}=\gamma_{b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.2.m2.1"><semantics id="S3.SS1.SSS2.10.p8.2.m2.1a"><mrow id="S3.SS1.SSS2.10.p8.2.m2.1.1" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.cmml"><msub id="S3.SS1.SSS2.10.p8.2.m2.1.1.2" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2.cmml"><mi id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.2" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.3" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.2.m2.1.1.1" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS2.10.p8.2.m2.1.1.3" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3.cmml"><mi id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.2" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.3" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3.3.cmml">b</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.2.m2.1b"><apply id="S3.SS1.SSS2.10.p8.2.m2.1.1.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1"><eq id="S3.SS1.SSS2.10.p8.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.1"></eq><apply id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.1.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.2.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.2.m2.1.1.2.3.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.1.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.2.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.2.m2.1.1.3.3.cmml" xref="S3.SS1.SSS2.10.p8.2.m2.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.2.m2.1c">\gamma_{a}=\gamma_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.2.m2.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, i.e. <math alttext="\gamma_{a}\cap\partial D=\{a,b\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.3.m3.2"><semantics id="S3.SS1.SSS2.10.p8.3.m3.2a"><mrow id="S3.SS1.SSS2.10.p8.3.m3.2.3" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.cmml"><mrow id="S3.SS1.SSS2.10.p8.3.m3.2.3.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.cmml"><msub id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.cmml"><mi id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.3" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.1" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.1.cmml">∩</mo><mrow id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.cmml"><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.1" lspace="0em" rspace="0em" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.1.cmml">∂</mo><mi id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.2.cmml">D</mi></mrow></mrow><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.1" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS2.10.p8.3.m3.2.3.3.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.3.1.cmml"><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.3.1.cmml">{</mo><mi id="S3.SS1.SSS2.10.p8.3.m3.1.1" xref="S3.SS1.SSS2.10.p8.3.m3.1.1.cmml">a</mi><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.3.2.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.SS1.SSS2.10.p8.3.m3.2.2" xref="S3.SS1.SSS2.10.p8.3.m3.2.2.cmml">b</mi><mo id="S3.SS1.SSS2.10.p8.3.m3.2.3.3.2.3" stretchy="false" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.3.m3.2b"><apply id="S3.SS1.SSS2.10.p8.3.m3.2.3.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3"><eq id="S3.SS1.SSS2.10.p8.3.m3.2.3.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.1"></eq><apply id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2"><intersect id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.1"></intersect><apply id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.2.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.3.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3"><partialdiff id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.1"></partialdiff><ci id="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.2.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.2.3.2">𝐷</ci></apply></apply><set id="S3.SS1.SSS2.10.p8.3.m3.2.3.3.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.3.3.2"><ci id="S3.SS1.SSS2.10.p8.3.m3.1.1.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.1.1">𝑎</ci><ci id="S3.SS1.SSS2.10.p8.3.m3.2.2.cmml" xref="S3.SS1.SSS2.10.p8.3.m3.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.3.m3.2c">\gamma_{a}\cap\partial D=\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.3.m3.2d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∩ ∂ italic_D = { italic_a , italic_b }</annotation></semantics></math>. Then, as before, <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.4.m4.1"><semantics id="S3.SS1.SSS2.10.p8.4.m4.1a"><mi id="S3.SS1.SSS2.10.p8.4.m4.1.1" xref="S3.SS1.SSS2.10.p8.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.4.m4.1b"><ci id="S3.SS1.SSS2.10.p8.4.m4.1.1.cmml" xref="S3.SS1.SSS2.10.p8.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.4.m4.1d">italic_γ</annotation></semantics></math> becomes a cycle with <math alttext="P^{\prime}\subset\operatorname*{int}(\gamma)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.5.m5.2"><semantics id="S3.SS1.SSS2.10.p8.5.m5.2a"><mrow id="S3.SS1.SSS2.10.p8.5.m5.2.3" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.cmml"><msup id="S3.SS1.SSS2.10.p8.5.m5.2.3.2" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2.cmml"><mi id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.2" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.3" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.10.p8.5.m5.2.3.1" rspace="0.1389em" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.1.cmml">⊂</mo><mrow id="S3.SS1.SSS2.10.p8.5.m5.2.3.3.2" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.3.1.cmml"><mo id="S3.SS1.SSS2.10.p8.5.m5.1.1" lspace="0.1389em" rspace="0em" xref="S3.SS1.SSS2.10.p8.5.m5.1.1.cmml">int</mo><mrow id="S3.SS1.SSS2.10.p8.5.m5.2.3.3.2.1" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.3.1.cmml"><mo id="S3.SS1.SSS2.10.p8.5.m5.2.3.3.2.1.1" stretchy="false" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.3.1.cmml">(</mo><mi id="S3.SS1.SSS2.10.p8.5.m5.2.2" xref="S3.SS1.SSS2.10.p8.5.m5.2.2.cmml">γ</mi><mo id="S3.SS1.SSS2.10.p8.5.m5.2.3.3.2.1.2" stretchy="false" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.5.m5.2b"><apply id="S3.SS1.SSS2.10.p8.5.m5.2.3.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3"><subset id="S3.SS1.SSS2.10.p8.5.m5.2.3.1.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.1"></subset><apply id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.1.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.2.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2.2">𝑃</ci><ci id="S3.SS1.SSS2.10.p8.5.m5.2.3.2.3.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.10.p8.5.m5.2.3.3.1.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.3.3.2"><ci id="S3.SS1.SSS2.10.p8.5.m5.1.1.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.1.1">int</ci><ci id="S3.SS1.SSS2.10.p8.5.m5.2.2.cmml" xref="S3.SS1.SSS2.10.p8.5.m5.2.2">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.5.m5.2c">P^{\prime}\subset\operatorname*{int}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.5.m5.2d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ roman_int ( italic_γ )</annotation></semantics></math>. This contradicts the presence of additional polygonal arcs other than <math alttext="\gamma_{a}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.6.m6.1"><semantics id="S3.SS1.SSS2.10.p8.6.m6.1a"><msub id="S3.SS1.SSS2.10.p8.6.m6.1.1" xref="S3.SS1.SSS2.10.p8.6.m6.1.1.cmml"><mi id="S3.SS1.SSS2.10.p8.6.m6.1.1.2" xref="S3.SS1.SSS2.10.p8.6.m6.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.6.m6.1.1.3" xref="S3.SS1.SSS2.10.p8.6.m6.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.6.m6.1b"><apply id="S3.SS1.SSS2.10.p8.6.m6.1.1.cmml" xref="S3.SS1.SSS2.10.p8.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.6.m6.1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.6.m6.1.1.2.cmml" xref="S3.SS1.SSS2.10.p8.6.m6.1.1.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.6.m6.1.1.3.cmml" xref="S3.SS1.SSS2.10.p8.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.6.m6.1c">\gamma_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.6.m6.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, as arcs are unbounded by definition. Thus, <math alttext="\gamma_{a}\neq\gamma_{b}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.7.m7.1"><semantics id="S3.SS1.SSS2.10.p8.7.m7.1a"><mrow id="S3.SS1.SSS2.10.p8.7.m7.1.1" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.cmml"><msub id="S3.SS1.SSS2.10.p8.7.m7.1.1.2" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2.cmml"><mi id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.2" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.3" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.7.m7.1.1.1" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.1.cmml">≠</mo><msub id="S3.SS1.SSS2.10.p8.7.m7.1.1.3" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3.cmml"><mi id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.2" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3.2.cmml">γ</mi><mi id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.3" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3.3.cmml">b</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.7.m7.1b"><apply id="S3.SS1.SSS2.10.p8.7.m7.1.1.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1"><neq id="S3.SS1.SSS2.10.p8.7.m7.1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.1"></neq><apply id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.1.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.2.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.7.m7.1.1.2.3.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.2.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.1.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.2.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS2.10.p8.7.m7.1.1.3.3.cmml" xref="S3.SS1.SSS2.10.p8.7.m7.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.7.m7.1c">\gamma_{a}\neq\gamma_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.7.m7.1d">italic_γ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ≠ italic_γ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.8.m8.1"><semantics id="S3.SS1.SSS2.10.p8.8.m8.1a"><mi id="S3.SS1.SSS2.10.p8.8.m8.1.1" xref="S3.SS1.SSS2.10.p8.8.m8.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.8.m8.1b"><ci id="S3.SS1.SSS2.10.p8.8.m8.1.1.cmml" xref="S3.SS1.SSS2.10.p8.8.m8.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.8.m8.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.8.m8.1d">italic_γ</annotation></semantics></math> is a polygonal arc, due to <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.7</span></a>. Together with (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E20" title="Equation 20 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">20</span></a>), this implies <math alttext="n_{a}(P^{\prime})=n_{a}(P)-1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.10.p8.9.m9.2"><semantics id="S3.SS1.SSS2.10.p8.9.m9.2a"><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.cmml"><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2.1" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.cmml"><msub id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.cmml"><mi id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.2.cmml">n</mi><mi id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.cmml"><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.cmml"><msub id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.cmml"><mi id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.2.cmml">n</mi><mi id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.3.cmml">a</mi></msub><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.1" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.3.2" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.cmml"><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.cmml">(</mo><mi id="S3.SS1.SSS2.10.p8.9.m9.1.1" xref="S3.SS1.SSS2.10.p8.9.m9.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.1" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.1.cmml">−</mo><mn id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.3" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.10.p8.9.m9.2b"><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2"><eq id="S3.SS1.SSS2.10.p8.9.m9.2.2.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.2"></eq><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1"><times id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.2"></times><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.2">𝑛</ci><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.3.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.3.3">𝑎</ci></apply><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3"><minus id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.1"></minus><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2"><times id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.1"></times><apply id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.2.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.2">𝑛</ci><ci id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.3.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.2.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS2.10.p8.9.m9.1.1.cmml" xref="S3.SS1.SSS2.10.p8.9.m9.1.1">𝑃</ci></apply><cn id="S3.SS1.SSS2.10.p8.9.m9.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS2.10.p8.9.m9.2.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.10.p8.9.m9.2c">n_{a}(P^{\prime})=n_{a}(P)-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.10.p8.9.m9.2d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.11.p9"> <p class="ltx_p" id="S3.SS1.SSS2.11.p9.1">To prove (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E19" title="Equation 19 ‣ Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">19</span></a>), note that in either case, <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.11.p9.1.m1.1"><semantics id="S3.SS1.SSS2.11.p9.1.m1.1a"><msup id="S3.SS1.SSS2.11.p9.1.m1.1.1" xref="S3.SS1.SSS2.11.p9.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.11.p9.1.m1.1.1.2" xref="S3.SS1.SSS2.11.p9.1.m1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.11.p9.1.m1.1.1.3" xref="S3.SS1.SSS2.11.p9.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.11.p9.1.m1.1b"><apply id="S3.SS1.SSS2.11.p9.1.m1.1.1.cmml" xref="S3.SS1.SSS2.11.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.11.p9.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.11.p9.1.m1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.11.p9.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.11.p9.1.m1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.11.p9.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.11.p9.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.11.p9.1.m1.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.11.p9.1.m1.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a polygon with</p> <table class="ltx_equation ltx_eqn_table" id="S3.E21"> <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(P^{\prime})=V(P)\cup\{a,b,a^{\prime},b^{\prime}\}," class="ltx_Math" display="block" id="S3.E21.m1.4"><semantics id="S3.E21.m1.4a"><mrow id="S3.E21.m1.4.4.1" xref="S3.E21.m1.4.4.1.1.cmml"><mrow id="S3.E21.m1.4.4.1.1" xref="S3.E21.m1.4.4.1.1.cmml"><mrow id="S3.E21.m1.4.4.1.1.1" xref="S3.E21.m1.4.4.1.1.1.cmml"><mi id="S3.E21.m1.4.4.1.1.1.3" xref="S3.E21.m1.4.4.1.1.1.3.cmml">V</mi><mo id="S3.E21.m1.4.4.1.1.1.2" xref="S3.E21.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.E21.m1.4.4.1.1.1.1.1" xref="S3.E21.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S3.E21.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.E21.m1.4.4.1.1.1.1.1.1.cmml">(</mo><msup id="S3.E21.m1.4.4.1.1.1.1.1.1" xref="S3.E21.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.E21.m1.4.4.1.1.1.1.1.1.2" xref="S3.E21.m1.4.4.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.E21.m1.4.4.1.1.1.1.1.1.3" xref="S3.E21.m1.4.4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.E21.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S3.E21.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E21.m1.4.4.1.1.4" xref="S3.E21.m1.4.4.1.1.4.cmml">=</mo><mrow id="S3.E21.m1.4.4.1.1.3" xref="S3.E21.m1.4.4.1.1.3.cmml"><mrow id="S3.E21.m1.4.4.1.1.3.4" xref="S3.E21.m1.4.4.1.1.3.4.cmml"><mi id="S3.E21.m1.4.4.1.1.3.4.2" xref="S3.E21.m1.4.4.1.1.3.4.2.cmml">V</mi><mo id="S3.E21.m1.4.4.1.1.3.4.1" xref="S3.E21.m1.4.4.1.1.3.4.1.cmml">⁢</mo><mrow id="S3.E21.m1.4.4.1.1.3.4.3.2" xref="S3.E21.m1.4.4.1.1.3.4.cmml"><mo id="S3.E21.m1.4.4.1.1.3.4.3.2.1" stretchy="false" xref="S3.E21.m1.4.4.1.1.3.4.cmml">(</mo><mi id="S3.E21.m1.1.1" xref="S3.E21.m1.1.1.cmml">P</mi><mo id="S3.E21.m1.4.4.1.1.3.4.3.2.2" stretchy="false" xref="S3.E21.m1.4.4.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S3.E21.m1.4.4.1.1.3.3" xref="S3.E21.m1.4.4.1.1.3.3.cmml">∪</mo><mrow id="S3.E21.m1.4.4.1.1.3.2.2" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml"><mo id="S3.E21.m1.4.4.1.1.3.2.2.3" stretchy="false" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml">{</mo><mi id="S3.E21.m1.2.2" xref="S3.E21.m1.2.2.cmml">a</mi><mo id="S3.E21.m1.4.4.1.1.3.2.2.4" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml">,</mo><mi id="S3.E21.m1.3.3" xref="S3.E21.m1.3.3.cmml">b</mi><mo id="S3.E21.m1.4.4.1.1.3.2.2.5" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml">,</mo><msup id="S3.E21.m1.4.4.1.1.2.1.1.1" xref="S3.E21.m1.4.4.1.1.2.1.1.1.cmml"><mi id="S3.E21.m1.4.4.1.1.2.1.1.1.2" xref="S3.E21.m1.4.4.1.1.2.1.1.1.2.cmml">a</mi><mo id="S3.E21.m1.4.4.1.1.2.1.1.1.3" xref="S3.E21.m1.4.4.1.1.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.E21.m1.4.4.1.1.3.2.2.6" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml">,</mo><msup id="S3.E21.m1.4.4.1.1.3.2.2.2" xref="S3.E21.m1.4.4.1.1.3.2.2.2.cmml"><mi id="S3.E21.m1.4.4.1.1.3.2.2.2.2" xref="S3.E21.m1.4.4.1.1.3.2.2.2.2.cmml">b</mi><mo id="S3.E21.m1.4.4.1.1.3.2.2.2.3" xref="S3.E21.m1.4.4.1.1.3.2.2.2.3.cmml">′</mo></msup><mo id="S3.E21.m1.4.4.1.1.3.2.2.7" stretchy="false" xref="S3.E21.m1.4.4.1.1.3.2.3.cmml">}</mo></mrow></mrow></mrow><mo id="S3.E21.m1.4.4.1.2" xref="S3.E21.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E21.m1.4b"><apply id="S3.E21.m1.4.4.1.1.cmml" xref="S3.E21.m1.4.4.1"><eq id="S3.E21.m1.4.4.1.1.4.cmml" xref="S3.E21.m1.4.4.1.1.4"></eq><apply id="S3.E21.m1.4.4.1.1.1.cmml" xref="S3.E21.m1.4.4.1.1.1"><times id="S3.E21.m1.4.4.1.1.1.2.cmml" xref="S3.E21.m1.4.4.1.1.1.2"></times><ci id="S3.E21.m1.4.4.1.1.1.3.cmml" xref="S3.E21.m1.4.4.1.1.1.3">𝑉</ci><apply id="S3.E21.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.E21.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E21.m1.4.4.1.1.1.1.1.1.1.cmml" 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xref="S3.E21.m1.4.4.1.1.2.1.1.1">superscript</csymbol><ci id="S3.E21.m1.4.4.1.1.2.1.1.1.2.cmml" xref="S3.E21.m1.4.4.1.1.2.1.1.1.2">𝑎</ci><ci id="S3.E21.m1.4.4.1.1.2.1.1.1.3.cmml" xref="S3.E21.m1.4.4.1.1.2.1.1.1.3">′</ci></apply><apply id="S3.E21.m1.4.4.1.1.3.2.2.2.cmml" xref="S3.E21.m1.4.4.1.1.3.2.2.2"><csymbol cd="ambiguous" id="S3.E21.m1.4.4.1.1.3.2.2.2.1.cmml" xref="S3.E21.m1.4.4.1.1.3.2.2.2">superscript</csymbol><ci id="S3.E21.m1.4.4.1.1.3.2.2.2.2.cmml" xref="S3.E21.m1.4.4.1.1.3.2.2.2.2">𝑏</ci><ci id="S3.E21.m1.4.4.1.1.3.2.2.2.3.cmml" xref="S3.E21.m1.4.4.1.1.3.2.2.2.3">′</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E21.m1.4c">V(P^{\prime})=V(P)\cup\{a,b,a^{\prime},b^{\prime}\},</annotation><annotation encoding="application/x-llamapun" id="S3.E21.m1.4d">italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_V ( italic_P ) ∪ { italic_a , italic_b , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ 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">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.11.p9.8">and</p> <table class="ltx_equation ltx_eqn_table" id="S3.E22"> <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="E_{b}(P^{\prime})=E_{b}(P)\cup\{s_{a},\overline{aa^{\prime}},\overline{a^{% \prime}b^{\prime}},\overline{b^{\prime}b},s_{b}\}." class="ltx_Math" display="block" id="S3.E22.m1.5"><semantics id="S3.E22.m1.5a"><mrow id="S3.E22.m1.5.5.1" xref="S3.E22.m1.5.5.1.1.cmml"><mrow id="S3.E22.m1.5.5.1.1" xref="S3.E22.m1.5.5.1.1.cmml"><mrow id="S3.E22.m1.5.5.1.1.1" xref="S3.E22.m1.5.5.1.1.1.cmml"><msub 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG , italic_s start_POSTSUBSCRIPT italic_b 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">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.11.p9.7">Since <math alttext="s_{a}\subset e_{a}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.11.p9.2.m1.1"><semantics id="S3.SS1.SSS2.11.p9.2.m1.1a"><mrow id="S3.SS1.SSS2.11.p9.2.m1.1.1" xref="S3.SS1.SSS2.11.p9.2.m1.1.1.cmml"><msub id="S3.SS1.SSS2.11.p9.2.m1.1.1.2" xref="S3.SS1.SSS2.11.p9.2.m1.1.1.2.cmml"><mi id="S3.SS1.SSS2.11.p9.2.m1.1.1.2.2" xref="S3.SS1.SSS2.11.p9.2.m1.1.1.2.2.cmml">s</mi><mi id="S3.SS1.SSS2.11.p9.2.m1.1.1.2.3" 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xref="S3.SS1.SSS2.11.p9.4.m3.3.3.2.3">𝑏</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.11.p9.4.m3.3c">\overline{aa^{\prime}},\overline{a^{\prime}b^{\prime}},\overline{b^{\prime}b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.11.p9.4.m3.3d">over¯ start_ARG italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG</annotation></semantics></math> are tangential to <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.11.p9.5.m4.1"><semantics id="S3.SS1.SSS2.11.p9.5.m4.1a"><mi id="S3.SS1.SSS2.11.p9.5.m4.1.1" xref="S3.SS1.SSS2.11.p9.5.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.11.p9.5.m4.1b"><ci id="S3.SS1.SSS2.11.p9.5.m4.1.1.cmml" xref="S3.SS1.SSS2.11.p9.5.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.11.p9.5.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.11.p9.5.m4.1d">italic_D</annotation></semantics></math>, <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS2.11.p9.6.m5.1"><semantics id="S3.SS1.SSS2.11.p9.6.m5.1a"><mi id="S3.SS1.SSS2.11.p9.6.m5.1.1" xref="S3.SS1.SSS2.11.p9.6.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.11.p9.6.m5.1b"><ci id="S3.SS1.SSS2.11.p9.6.m5.1.1.cmml" xref="S3.SS1.SSS2.11.p9.6.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.11.p9.6.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.11.p9.6.m5.1d">italic_x</annotation></semantics></math> is also in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.11.p9.7.m6.1"><semantics id="S3.SS1.SSS2.11.p9.7.m6.1a"><msup id="S3.SS1.SSS2.11.p9.7.m6.1.1" xref="S3.SS1.SSS2.11.p9.7.m6.1.1.cmml"><mi id="S3.SS1.SSS2.11.p9.7.m6.1.1.2" xref="S3.SS1.SSS2.11.p9.7.m6.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.11.p9.7.m6.1.1.3" xref="S3.SS1.SSS2.11.p9.7.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.11.p9.7.m6.1b"><apply id="S3.SS1.SSS2.11.p9.7.m6.1.1.cmml" xref="S3.SS1.SSS2.11.p9.7.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.11.p9.7.m6.1.1.1.cmml" xref="S3.SS1.SSS2.11.p9.7.m6.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.11.p9.7.m6.1.1.2.cmml" xref="S3.SS1.SSS2.11.p9.7.m6.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.11.p9.7.m6.1.1.3.cmml" xref="S3.SS1.SSS2.11.p9.7.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.11.p9.7.m6.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.11.p9.7.m6.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.12.p10"> <p class="ltx_p" id="S3.SS1.SSS2.12.p10.4">The definition of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.1.m1.1"><semantics id="S3.SS1.SSS2.12.p10.1.m1.1a"><mi id="S3.SS1.SSS2.12.p10.1.m1.1.1" xref="S3.SS1.SSS2.12.p10.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.1.m1.1b"><ci id="S3.SS1.SSS2.12.p10.1.m1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.1.m1.1d">italic_P</annotation></semantics></math>-sides is local in the sense that they are fully determined by any neighbourhood of the vertex or edge under consideration. Since <math alttext="V(P),E_{b}(P)\subset D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.2.m2.4"><semantics id="S3.SS1.SSS2.12.p10.2.m2.4a"><mrow id="S3.SS1.SSS2.12.p10.2.m2.4.4" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.cmml"><mrow id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.3.cmml"><mrow id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.cmml"><mi id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.2" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.2.cmml">V</mi><mo id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.1" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.3.2" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.cmml"><mo id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.cmml">(</mo><mi id="S3.SS1.SSS2.12.p10.2.m2.1.1" xref="S3.SS1.SSS2.12.p10.2.m2.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.3" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.3.cmml">,</mo><mrow id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.cmml"><msub id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.2" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.2.cmml">E</mi><mi id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.3" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.3.cmml">b</mi></msub><mo id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.1" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.3.2" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.cmml"><mo id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.12.p10.2.m2.2.2" xref="S3.SS1.SSS2.12.p10.2.m2.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.SSS2.12.p10.2.m2.4.4.3" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.3.cmml">⊂</mo><mi id="S3.SS1.SSS2.12.p10.2.m2.4.4.4" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.4.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.2.m2.4b"><apply id="S3.SS1.SSS2.12.p10.2.m2.4.4.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4"><subset id="S3.SS1.SSS2.12.p10.2.m2.4.4.3.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.3"></subset><list id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.3.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2"><apply id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1"><times id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.1"></times><ci id="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.2.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.3.3.1.1.1.2">𝑉</ci><ci id="S3.SS1.SSS2.12.p10.2.m2.1.1.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2"><times id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.1"></times><apply id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.2">𝐸</ci><ci id="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.2.2.2.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS2.12.p10.2.m2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.2.2">𝑃</ci></apply></list><ci id="S3.SS1.SSS2.12.p10.2.m2.4.4.4.cmml" xref="S3.SS1.SSS2.12.p10.2.m2.4.4.4">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.2.m2.4c">V(P),E_{b}(P)\subset D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.2.m2.4d">italic_V ( italic_P ) , italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) ⊂ italic_D</annotation></semantics></math>, the disk <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.3.m3.1"><semantics id="S3.SS1.SSS2.12.p10.3.m3.1a"><mi id="S3.SS1.SSS2.12.p10.3.m3.1.1" xref="S3.SS1.SSS2.12.p10.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.3.m3.1b"><ci id="S3.SS1.SSS2.12.p10.3.m3.1.1.cmml" xref="S3.SS1.SSS2.12.p10.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.3.m3.1d">italic_D</annotation></semantics></math> is a neighbourhood of all vertices and line segments. Therefore, <math alttext="P^{\prime}\cap D=P\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.4.m4.1"><semantics id="S3.SS1.SSS2.12.p10.4.m4.1a"><mrow id="S3.SS1.SSS2.12.p10.4.m4.1.1" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.cmml"><mrow id="S3.SS1.SSS2.12.p10.4.m4.1.1.2" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.cmml"><msup id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.cmml"><mi id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.2" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.3" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.1" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.3" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS2.12.p10.4.m4.1.1.1" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS2.12.p10.4.m4.1.1.3" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.cmml"><mi id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.2" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.1" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.3" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.4.m4.1b"><apply id="S3.SS1.SSS2.12.p10.4.m4.1.1.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1"><eq id="S3.SS1.SSS2.12.p10.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.1"></eq><apply id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2"><intersect id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.1.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.1"></intersect><apply id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.3.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.2.3">′</ci></apply><ci id="S3.SS1.SSS2.12.p10.4.m4.1.1.2.3.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3"><intersect id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.1.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.1"></intersect><ci id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.2.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS2.12.p10.4.m4.1.1.3.3.cmml" xref="S3.SS1.SSS2.12.p10.4.m4.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.4.m4.1c">P^{\prime}\cap D=P\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.4.m4.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_D = italic_P ∩ italic_D</annotation></semantics></math> implies that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E23"> <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="Q^{v}_{P^{\prime}}=Q^{v}_{P}\quad\forall v\in V(P)\quad\text{and}\quad H^{e}_{% P^{\prime}}=H^{e}_{P}\quad\forall e\in E_{b}(P)." class="ltx_Math" display="block" id="S3.E23.m1.4"><semantics id="S3.E23.m1.4a"><mrow id="S3.E23.m1.4.4.1"><mrow id="S3.E23.m1.4.4.1.1.2" xref="S3.E23.m1.4.4.1.1.3.cmml"><mrow id="S3.E23.m1.4.4.1.1.1.1" xref="S3.E23.m1.4.4.1.1.1.1.cmml"><msubsup id="S3.E23.m1.4.4.1.1.1.1.2" xref="S3.E23.m1.4.4.1.1.1.1.2.cmml"><mi id="S3.E23.m1.4.4.1.1.1.1.2.2.2" xref="S3.E23.m1.4.4.1.1.1.1.2.2.2.cmml">Q</mi><msup id="S3.E23.m1.4.4.1.1.1.1.2.3" xref="S3.E23.m1.4.4.1.1.1.1.2.3.cmml"><mi id="S3.E23.m1.4.4.1.1.1.1.2.3.2" xref="S3.E23.m1.4.4.1.1.1.1.2.3.2.cmml">P</mi><mo id="S3.E23.m1.4.4.1.1.1.1.2.3.3" xref="S3.E23.m1.4.4.1.1.1.1.2.3.3.cmml">′</mo></msup><mi id="S3.E23.m1.4.4.1.1.1.1.2.2.3" xref="S3.E23.m1.4.4.1.1.1.1.2.2.3.cmml">v</mi></msubsup><mo id="S3.E23.m1.4.4.1.1.1.1.1" xref="S3.E23.m1.4.4.1.1.1.1.1.cmml">=</mo><msubsup id="S3.E23.m1.4.4.1.1.1.1.3" xref="S3.E23.m1.4.4.1.1.1.1.3.cmml"><mi id="S3.E23.m1.4.4.1.1.1.1.3.2.2" xref="S3.E23.m1.4.4.1.1.1.1.3.2.2.cmml">Q</mi><mi id="S3.E23.m1.4.4.1.1.1.1.3.3" xref="S3.E23.m1.4.4.1.1.1.1.3.3.cmml">P</mi><mi id="S3.E23.m1.4.4.1.1.1.1.3.2.3" xref="S3.E23.m1.4.4.1.1.1.1.3.2.3.cmml">v</mi></msubsup></mrow><mspace id="S3.E23.m1.4.4.1.1.2.3" width="1.167em" xref="S3.E23.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S3.E23.m1.4.4.1.1.2.2.2" xref="S3.E23.m1.4.4.1.1.2.2.3.cmml"><mrow id="S3.E23.m1.4.4.1.1.2.2.1.1" xref="S3.E23.m1.4.4.1.1.2.2.1.1.cmml"><mrow id="S3.E23.m1.4.4.1.1.2.2.1.1.3" xref="S3.E23.m1.4.4.1.1.2.2.1.1.3.cmml"><mo id="S3.E23.m1.4.4.1.1.2.2.1.1.3.1" rspace="0.167em" xref="S3.E23.m1.4.4.1.1.2.2.1.1.3.1.cmml">∀</mo><mi id="S3.E23.m1.4.4.1.1.2.2.1.1.3.2" 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id="S3.E23.m1.3.3" xref="S3.E23.m1.3.3a.cmml">and</mtext></mrow></mrow><mspace id="S3.E23.m1.4.4.1.1.2.2.2.3" width="1em" xref="S3.E23.m1.4.4.1.1.2.2.3a.cmml"></mspace><mrow id="S3.E23.m1.4.4.1.1.2.2.2.2.2" xref="S3.E23.m1.4.4.1.1.2.2.2.2.3.cmml"><mrow id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.cmml"><msubsup id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.cmml"><mi id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.2.2" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.2.2.cmml">H</mi><msup id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3.cmml"><mi id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3.2" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3.2.cmml">P</mi><mo id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3.3" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.3.3.cmml">′</mo></msup><mi id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.2.3" xref="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.2.2.3.cmml">e</mi></msubsup><mo id="S3.E23.m1.4.4.1.1.2.2.2.2.1.1.1" 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xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.1" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.3" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.3.cmml">b</mi></mrow><mo id="S3.SS1.SSS2.12.p10.5.m1.3.3.1" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.1.cmml">¯</mo></mover><mo id="S3.SS1.SSS2.12.p10.5.m1.3.4.3.2.4" stretchy="false" xref="S3.SS1.SSS2.12.p10.5.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.5.m1.3b"><apply id="S3.SS1.SSS2.12.p10.5.m1.3.4.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.4"><in id="S3.SS1.SSS2.12.p10.5.m1.3.4.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.4.1"></in><ci id="S3.SS1.SSS2.12.p10.5.m1.3.4.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.4.2">𝑒</ci><set id="S3.SS1.SSS2.12.p10.5.m1.3.4.3.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.4.3.2"><apply id="S3.SS1.SSS2.12.p10.5.m1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1"><ci id="S3.SS1.SSS2.12.p10.5.m1.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.1">¯</ci><apply id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2"><times id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.1"></times><ci id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.2">𝑎</ci><apply id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.2">𝑎</ci><ci id="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.1.1.2.3.3">′</ci></apply></apply></apply><apply id="S3.SS1.SSS2.12.p10.5.m1.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2"><ci id="S3.SS1.SSS2.12.p10.5.m1.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.1">¯</ci><apply id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2"><times id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.1"></times><apply id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.2">𝑎</ci><ci id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.2.3">′</ci></apply><apply id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.2">𝑏</ci><ci id="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.2.2.2.3.3">′</ci></apply></apply></apply><apply id="S3.SS1.SSS2.12.p10.5.m1.3.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3"><ci id="S3.SS1.SSS2.12.p10.5.m1.3.3.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.1">¯</ci><apply id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2"><times id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.1"></times><apply id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.1.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.2">𝑏</ci><ci id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.2.3">′</ci></apply><ci id="S3.SS1.SSS2.12.p10.5.m1.3.3.2.3.cmml" xref="S3.SS1.SSS2.12.p10.5.m1.3.3.2.3">𝑏</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.5.m1.3c">e\in\{\overline{aa^{\prime}},\overline{a^{\prime}b^{\prime}},\overline{b^{% \prime}b}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.5.m1.3d">italic_e ∈ { over¯ start_ARG italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG }</annotation></semantics></math> is tangent to <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.6.m2.1"><semantics id="S3.SS1.SSS2.12.p10.6.m2.1a"><mi id="S3.SS1.SSS2.12.p10.6.m2.1.1" xref="S3.SS1.SSS2.12.p10.6.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.6.m2.1b"><ci id="S3.SS1.SSS2.12.p10.6.m2.1.1.cmml" xref="S3.SS1.SSS2.12.p10.6.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.6.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.6.m2.1d">italic_D</annotation></semantics></math> with <math alttext="D\subset H^{e}_{P^{\prime}}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.7.m3.1"><semantics id="S3.SS1.SSS2.12.p10.7.m3.1a"><mrow id="S3.SS1.SSS2.12.p10.7.m3.1.1" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.cmml"><mi id="S3.SS1.SSS2.12.p10.7.m3.1.1.2" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.2.cmml">D</mi><mo id="S3.SS1.SSS2.12.p10.7.m3.1.1.1" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.1.cmml">⊂</mo><msubsup id="S3.SS1.SSS2.12.p10.7.m3.1.1.3" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.cmml"><mi id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.2" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.2.cmml">H</mi><msup id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.2" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.2.cmml">P</mi><mo id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.3" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.3.cmml">′</mo></msup><mi id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.3" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.3.cmml">e</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.7.m3.1b"><apply id="S3.SS1.SSS2.12.p10.7.m3.1.1.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1"><subset id="S3.SS1.SSS2.12.p10.7.m3.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.1"></subset><ci id="S3.SS1.SSS2.12.p10.7.m3.1.1.2.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.2">𝐷</ci><apply id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.1.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3">subscript</csymbol><apply id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.2">𝐻</ci><ci id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.2.3">𝑒</ci></apply><apply id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3">superscript</csymbol><ci id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.2">𝑃</ci><ci id="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.3.cmml" xref="S3.SS1.SSS2.12.p10.7.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.7.m3.1c">D\subset H^{e}_{P^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.7.m3.1d">italic_D ⊂ italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="x\in D" class="ltx_Math" display="inline" id="S3.SS1.SSS2.12.p10.8.m4.1"><semantics id="S3.SS1.SSS2.12.p10.8.m4.1a"><mrow id="S3.SS1.SSS2.12.p10.8.m4.1.1" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.cmml"><mi id="S3.SS1.SSS2.12.p10.8.m4.1.1.2" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS2.12.p10.8.m4.1.1.1" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.1.cmml">∈</mo><mi id="S3.SS1.SSS2.12.p10.8.m4.1.1.3" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.12.p10.8.m4.1b"><apply id="S3.SS1.SSS2.12.p10.8.m4.1.1.cmml" xref="S3.SS1.SSS2.12.p10.8.m4.1.1"><in id="S3.SS1.SSS2.12.p10.8.m4.1.1.1.cmml" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.1"></in><ci id="S3.SS1.SSS2.12.p10.8.m4.1.1.2.cmml" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.2">𝑥</ci><ci id="S3.SS1.SSS2.12.p10.8.m4.1.1.3.cmml" xref="S3.SS1.SSS2.12.p10.8.m4.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.12.p10.8.m4.1c">x\in D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.12.p10.8.m4.1d">italic_x ∈ italic_D</annotation></semantics></math>, we get that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E24"> <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\in H^{e}_{P^{\prime}}\text{ for }e\in\{\overline{aa^{\prime}},\overline{a^{% \prime}b^{\prime}},\overline{b^{\prime}b}\}\quad\text{and}\quad x\in Q^{v}_{P^% {\prime}}\text{ for }v\in\{a^{\prime},b^{\prime}\}." class="ltx_Math" display="block" id="S3.E24.m1.6"><semantics id="S3.E24.m1.6a"><mrow id="S3.E24.m1.6.6.1"><mrow id="S3.E24.m1.6.6.1.1.2" xref="S3.E24.m1.6.6.1.1.3.cmml"><mrow id="S3.E24.m1.6.6.1.1.1.1" xref="S3.E24.m1.6.6.1.1.1.1.cmml"><mi id="S3.E24.m1.6.6.1.1.1.1.2" xref="S3.E24.m1.6.6.1.1.1.1.2.cmml">x</mi><mo id="S3.E24.m1.6.6.1.1.1.1.3" xref="S3.E24.m1.6.6.1.1.1.1.3.cmml">∈</mo><mrow id="S3.E24.m1.6.6.1.1.1.1.4" xref="S3.E24.m1.6.6.1.1.1.1.4.cmml"><msubsup id="S3.E24.m1.6.6.1.1.1.1.4.2" xref="S3.E24.m1.6.6.1.1.1.1.4.2.cmml"><mi id="S3.E24.m1.6.6.1.1.1.1.4.2.2.2" xref="S3.E24.m1.6.6.1.1.1.1.4.2.2.2.cmml">H</mi><msup id="S3.E24.m1.6.6.1.1.1.1.4.2.3" xref="S3.E24.m1.6.6.1.1.1.1.4.2.3.cmml"><mi id="S3.E24.m1.6.6.1.1.1.1.4.2.3.2" xref="S3.E24.m1.6.6.1.1.1.1.4.2.3.2.cmml">P</mi><mo id="S3.E24.m1.6.6.1.1.1.1.4.2.3.3" xref="S3.E24.m1.6.6.1.1.1.1.4.2.3.3.cmml">′</mo></msup><mi id="S3.E24.m1.6.6.1.1.1.1.4.2.2.3" xref="S3.E24.m1.6.6.1.1.1.1.4.2.2.3.cmml">e</mi></msubsup><mo id="S3.E24.m1.6.6.1.1.1.1.4.1" xref="S3.E24.m1.6.6.1.1.1.1.4.1.cmml">⁢</mo><mtext id="S3.E24.m1.6.6.1.1.1.1.4.3" xref="S3.E24.m1.6.6.1.1.1.1.4.3a.cmml"> for </mtext><mo id="S3.E24.m1.6.6.1.1.1.1.4.1a" xref="S3.E24.m1.6.6.1.1.1.1.4.1.cmml">⁢</mo><mi id="S3.E24.m1.6.6.1.1.1.1.4.4" xref="S3.E24.m1.6.6.1.1.1.1.4.4.cmml">e</mi></mrow><mo id="S3.E24.m1.6.6.1.1.1.1.5" xref="S3.E24.m1.6.6.1.1.1.1.5.cmml">∈</mo><mrow id="S3.E24.m1.6.6.1.1.1.1.6.2" xref="S3.E24.m1.6.6.1.1.1.1.6.1.cmml"><mo id="S3.E24.m1.6.6.1.1.1.1.6.2.1" stretchy="false" xref="S3.E24.m1.6.6.1.1.1.1.6.1.cmml">{</mo><mover accent="true" id="S3.E24.m1.1.1" xref="S3.E24.m1.1.1.cmml"><mrow id="S3.E24.m1.1.1.2" xref="S3.E24.m1.1.1.2.cmml"><mi id="S3.E24.m1.1.1.2.2" xref="S3.E24.m1.1.1.2.2.cmml">a</mi><mo id="S3.E24.m1.1.1.2.1" xref="S3.E24.m1.1.1.2.1.cmml">⁢</mo><msup id="S3.E24.m1.1.1.2.3" xref="S3.E24.m1.1.1.2.3.cmml"><mi id="S3.E24.m1.1.1.2.3.2" xref="S3.E24.m1.1.1.2.3.2.cmml">a</mi><mo id="S3.E24.m1.1.1.2.3.3" xref="S3.E24.m1.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.E24.m1.1.1.1" xref="S3.E24.m1.1.1.1.cmml">¯</mo></mover><mo id="S3.E24.m1.6.6.1.1.1.1.6.2.2" xref="S3.E24.m1.6.6.1.1.1.1.6.1.cmml">,</mo><mover accent="true" id="S3.E24.m1.2.2" xref="S3.E24.m1.2.2.cmml"><mrow id="S3.E24.m1.2.2.2" xref="S3.E24.m1.2.2.2.cmml"><msup id="S3.E24.m1.2.2.2.2" xref="S3.E24.m1.2.2.2.2.cmml"><mi id="S3.E24.m1.2.2.2.2.2" xref="S3.E24.m1.2.2.2.2.2.cmml">a</mi><mo id="S3.E24.m1.2.2.2.2.3" xref="S3.E24.m1.2.2.2.2.3.cmml">′</mo></msup><mo id="S3.E24.m1.2.2.2.1" xref="S3.E24.m1.2.2.2.1.cmml">⁢</mo><msup id="S3.E24.m1.2.2.2.3" xref="S3.E24.m1.2.2.2.3.cmml"><mi id="S3.E24.m1.2.2.2.3.2" xref="S3.E24.m1.2.2.2.3.2.cmml">b</mi><mo id="S3.E24.m1.2.2.2.3.3" xref="S3.E24.m1.2.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.E24.m1.2.2.1" xref="S3.E24.m1.2.2.1.cmml">¯</mo></mover><mo id="S3.E24.m1.6.6.1.1.1.1.6.2.3" xref="S3.E24.m1.6.6.1.1.1.1.6.1.cmml">,</mo><mover accent="true" id="S3.E24.m1.3.3" xref="S3.E24.m1.3.3.cmml"><mrow id="S3.E24.m1.3.3.2" xref="S3.E24.m1.3.3.2.cmml"><msup id="S3.E24.m1.3.3.2.2" xref="S3.E24.m1.3.3.2.2.cmml"><mi id="S3.E24.m1.3.3.2.2.2" xref="S3.E24.m1.3.3.2.2.2.cmml">b</mi><mo id="S3.E24.m1.3.3.2.2.3" xref="S3.E24.m1.3.3.2.2.3.cmml">′</mo></msup><mo id="S3.E24.m1.3.3.2.1" xref="S3.E24.m1.3.3.2.1.cmml">⁢</mo><mi id="S3.E24.m1.3.3.2.3" xref="S3.E24.m1.3.3.2.3.cmml">b</mi></mrow><mo id="S3.E24.m1.3.3.1" xref="S3.E24.m1.3.3.1.cmml">¯</mo></mover><mo id="S3.E24.m1.6.6.1.1.1.1.6.2.4" stretchy="false" xref="S3.E24.m1.6.6.1.1.1.1.6.1.cmml">}</mo></mrow></mrow><mspace id="S3.E24.m1.6.6.1.1.2.3" width="1em" xref="S3.E24.m1.6.6.1.1.3a.cmml"></mspace><mrow id="S3.E24.m1.6.6.1.1.2.2" xref="S3.E24.m1.6.6.1.1.2.2.cmml"><mrow id="S3.E24.m1.6.6.1.1.2.2.4.2" xref="S3.E24.m1.6.6.1.1.2.2.4.1.cmml"><mtext id="S3.E24.m1.4.4" xref="S3.E24.m1.4.4a.cmml">and</mtext><mspace id="S3.E24.m1.6.6.1.1.2.2.4.2.1" width="1em" xref="S3.E24.m1.6.6.1.1.2.2.4.1.cmml"></mspace><mi id="S3.E24.m1.5.5" xref="S3.E24.m1.5.5.cmml">x</mi></mrow><mo id="S3.E24.m1.6.6.1.1.2.2.5" xref="S3.E24.m1.6.6.1.1.2.2.5.cmml">∈</mo><mrow id="S3.E24.m1.6.6.1.1.2.2.6" xref="S3.E24.m1.6.6.1.1.2.2.6.cmml"><msubsup id="S3.E24.m1.6.6.1.1.2.2.6.2" xref="S3.E24.m1.6.6.1.1.2.2.6.2.cmml"><mi id="S3.E24.m1.6.6.1.1.2.2.6.2.2.2" xref="S3.E24.m1.6.6.1.1.2.2.6.2.2.2.cmml">Q</mi><msup id="S3.E24.m1.6.6.1.1.2.2.6.2.3" xref="S3.E24.m1.6.6.1.1.2.2.6.2.3.cmml"><mi id="S3.E24.m1.6.6.1.1.2.2.6.2.3.2" xref="S3.E24.m1.6.6.1.1.2.2.6.2.3.2.cmml">P</mi><mo id="S3.E24.m1.6.6.1.1.2.2.6.2.3.3" xref="S3.E24.m1.6.6.1.1.2.2.6.2.3.3.cmml">′</mo></msup><mi id="S3.E24.m1.6.6.1.1.2.2.6.2.2.3" xref="S3.E24.m1.6.6.1.1.2.2.6.2.2.3.cmml">v</mi></msubsup><mo id="S3.E24.m1.6.6.1.1.2.2.6.1" xref="S3.E24.m1.6.6.1.1.2.2.6.1.cmml">⁢</mo><mtext id="S3.E24.m1.6.6.1.1.2.2.6.3" xref="S3.E24.m1.6.6.1.1.2.2.6.3a.cmml"> for </mtext><mo id="S3.E24.m1.6.6.1.1.2.2.6.1a" xref="S3.E24.m1.6.6.1.1.2.2.6.1.cmml">⁢</mo><mi id="S3.E24.m1.6.6.1.1.2.2.6.4" xref="S3.E24.m1.6.6.1.1.2.2.6.4.cmml">v</mi></mrow><mo id="S3.E24.m1.6.6.1.1.2.2.7" xref="S3.E24.m1.6.6.1.1.2.2.7.cmml">∈</mo><mrow id="S3.E24.m1.6.6.1.1.2.2.2.2" xref="S3.E24.m1.6.6.1.1.2.2.2.3.cmml"><mo id="S3.E24.m1.6.6.1.1.2.2.2.2.3" stretchy="false" xref="S3.E24.m1.6.6.1.1.2.2.2.3.cmml">{</mo><msup id="S3.E24.m1.6.6.1.1.2.2.1.1.1" xref="S3.E24.m1.6.6.1.1.2.2.1.1.1.cmml"><mi id="S3.E24.m1.6.6.1.1.2.2.1.1.1.2" xref="S3.E24.m1.6.6.1.1.2.2.1.1.1.2.cmml">a</mi><mo id="S3.E24.m1.6.6.1.1.2.2.1.1.1.3" xref="S3.E24.m1.6.6.1.1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.E24.m1.6.6.1.1.2.2.2.2.4" xref="S3.E24.m1.6.6.1.1.2.2.2.3.cmml">,</mo><msup id="S3.E24.m1.6.6.1.1.2.2.2.2.2" xref="S3.E24.m1.6.6.1.1.2.2.2.2.2.cmml"><mi id="S3.E24.m1.6.6.1.1.2.2.2.2.2.2" xref="S3.E24.m1.6.6.1.1.2.2.2.2.2.2.cmml">b</mi><mo id="S3.E24.m1.6.6.1.1.2.2.2.2.2.3" xref="S3.E24.m1.6.6.1.1.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S3.E24.m1.6.6.1.1.2.2.2.2.5" stretchy="false" xref="S3.E24.m1.6.6.1.1.2.2.2.3.cmml">}</mo></mrow></mrow></mrow><mo id="S3.E24.m1.6.6.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E24.m1.6b"><apply id="S3.E24.m1.6.6.1.1.3.cmml" xref="S3.E24.m1.6.6.1.1.2"><csymbol cd="ambiguous" id="S3.E24.m1.6.6.1.1.3a.cmml" xref="S3.E24.m1.6.6.1.1.2.3">formulae-sequence</csymbol><apply 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over¯ start_ARG italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG } and italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT for italic_v ∈ { italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ 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">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.12.p10.9">This, together with</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S3.Ex26.m1.2.2.1.1.2.2.3.3.3.2.2.2">𝑏</ci><ci id="S3.Ex26.m1.2.2.1.1.2.2.3.3.3.2.2.3.cmml" xref="S3.Ex26.m1.2.2.1.1.2.2.3.3.3.2.2.3">′</ci></apply><ci id="S3.Ex26.m1.2.2.1.1.2.2.3.3.3.2.3.cmml" xref="S3.Ex26.m1.2.2.1.1.2.2.3.3.3.2.3">𝑏</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex26.m1.2c">Q^{a}_{P^{\prime}}=H_{P^{\prime}}^{s_{a}}\cap H_{P^{\prime}}^{\overline{aa^{% \prime}}}\quad\text{and}\quad Q^{b}_{P^{\prime}}=H_{P^{\prime}}^{s_{b}}\cap H_% {P^{\prime}}^{\overline{b^{\prime}b}},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex26.m1.2d">italic_Q start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over¯ start_ARG italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG end_POSTSUPERSCRIPT and italic_Q start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG 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.SS1.SSS2.12.p10.10">yields that</p> <table 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xref="S3.E25.m1.2.2.1.1.7.3.3.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E25.m1.2c">x\in Q^{a}_{P^{\prime}}\Leftrightarrow x\in H^{s_{a}}_{P^{\prime}}\quad\text{% and}\quad x\in Q^{b}_{P^{\prime}}\Leftrightarrow x\in H^{s_{b}}_{P^{\prime}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E25.m1.2d">italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⇔ italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and italic_x ∈ italic_Q start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⇔ italic_x ∈ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 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">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.12.p10.11">Using (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E21" title="Equation 21 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">21</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E22" title="Equation 22 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">22</span></a>), we split the sums on the right hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E19" title="Equation 19 ‣ Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">19</span></a>)</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx8"> <tbody id="S3.Ex27"><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\sum_{v\in V(P^{\prime})}" class="ltx_Math" display="inline" id="S3.Ex27.m1.1"><semantics id="S3.Ex27.m1.1a"><mstyle displaystyle="true" id="S3.Ex27.m1.1.2" xref="S3.Ex27.m1.1.2.cmml"><munder id="S3.Ex27.m1.1.2a" xref="S3.Ex27.m1.1.2.cmml"><mo id="S3.Ex27.m1.1.2.2" movablelimits="false" 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xref="S3.Ex27.m1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex27.m1.1c">\displaystyle\sum_{v\in V(P^{\prime})}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex27.m1.1d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hskip 10.00002pt\mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_% {b}(P^{\prime})}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P^{\prime})" class="ltx_Math" display="inline" id="S3.Ex27.m2.4"><semantics id="S3.Ex27.m2.4a"><mrow id="S3.Ex27.m2.4.4" xref="S3.Ex27.m2.4.4.cmml"><mrow id="S3.Ex27.m2.4.4.3" xref="S3.Ex27.m2.4.4.3.cmml"><msub id="S3.Ex27.m2.4.4.3.2" xref="S3.Ex27.m2.4.4.3.2.cmml"><mn id="S3.Ex27.m2.4.4.3.2.2" xref="S3.Ex27.m2.4.4.3.2.2.cmml">𝟙</mn><msubsup id="S3.Ex27.m2.4.4.3.2.3" 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xref="S3.Ex27.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.Ex27.m2.4.4.1.1.1.1.1.cmml" xref="S3.Ex27.m2.4.4.1.1.1">superscript</csymbol><ci id="S3.Ex27.m2.4.4.1.1.1.1.2.cmml" xref="S3.Ex27.m2.4.4.1.1.1.1.2">𝑃</ci><ci id="S3.Ex27.m2.4.4.1.1.1.1.3.cmml" xref="S3.Ex27.m2.4.4.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex27.m2.4c">\displaystyle\hskip 10.00002pt\mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_% {b}(P^{\prime})}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex27.m2.4d">blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex28"><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=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)+\sum_{v\in\{a% ,b\}}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)+\sum_{v\in\{a^{\prime},b^{\prime}\}}% \mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P^{% \prime}}}(x)" class="ltx_Math" display="inline" id="S3.Ex28.m1.10"><semantics 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id="S3.Ex28.m1.10.11.3.3.2.2.3.3.3.cmml" xref="S3.Ex28.m1.10.11.3.3.2.2.3.3.3">′</ci></apply></apply></apply><ci id="S3.Ex28.m1.10.10.cmml" xref="S3.Ex28.m1.10.10">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex28.m1.10c">\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)+\sum_{v\in\{a% ,b\}}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)+\sum_{v\in\{a^{\prime},b^{\prime}\}}% \mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P^{% \prime}}}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex28.m1.10d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_v ∈ { italic_a , italic_b } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_v ∈ { italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex29"><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\quad-\sum_{e\in\{s_{a},s_{b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x% )-\sum_{e\in\{\overline{aa^{\prime}},\overline{a^{\prime}b^{\prime}},\overline% {b^{\prime}b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P)+1" class="ltx_Math" display="inline" id="S3.Ex29.m1.8"><semantics id="S3.Ex29.m1.8a"><mrow id="S3.Ex29.m1.8.9" xref="S3.Ex29.m1.8.9.cmml"><mrow id="S3.Ex29.m1.8.9.2" xref="S3.Ex29.m1.8.9.2.cmml"><mrow id="S3.Ex29.m1.8.9.2.2" xref="S3.Ex29.m1.8.9.2.2.cmml"><mo id="S3.Ex29.m1.8.9.2.2a" xref="S3.Ex29.m1.8.9.2.2.cmml">−</mo><mrow id="S3.Ex29.m1.8.9.2.2.2" xref="S3.Ex29.m1.8.9.2.2.2.cmml"><mstyle displaystyle="true" id="S3.Ex29.m1.8.9.2.2.2.1" xref="S3.Ex29.m1.8.9.2.2.2.1.cmml"><munder 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id="S3.Ex29.m1.8.9.2.3.2.2.3.3.3.cmml" xref="S3.Ex29.m1.8.9.2.3.2.2.3.3.3">′</ci></apply></apply></apply><ci id="S3.Ex29.m1.7.7.cmml" xref="S3.Ex29.m1.7.7">𝑥</ci></apply></apply><apply id="S3.Ex29.m1.8.9.2.4.cmml" xref="S3.Ex29.m1.8.9.2.4"><times id="S3.Ex29.m1.8.9.2.4.1.cmml" xref="S3.Ex29.m1.8.9.2.4.1"></times><apply id="S3.Ex29.m1.8.9.2.4.2.cmml" xref="S3.Ex29.m1.8.9.2.4.2"><csymbol cd="ambiguous" id="S3.Ex29.m1.8.9.2.4.2.1.cmml" xref="S3.Ex29.m1.8.9.2.4.2">subscript</csymbol><ci id="S3.Ex29.m1.8.9.2.4.2.2.cmml" xref="S3.Ex29.m1.8.9.2.4.2.2">𝑛</ci><ci id="S3.Ex29.m1.8.9.2.4.2.3.cmml" xref="S3.Ex29.m1.8.9.2.4.2.3">𝑎</ci></apply><ci id="S3.Ex29.m1.8.8.cmml" xref="S3.Ex29.m1.8.8">𝑃</ci></apply></apply><cn id="S3.Ex29.m1.8.9.3.cmml" type="integer" xref="S3.Ex29.m1.8.9.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex29.m1.8c">\displaystyle\quad-\sum_{e\in\{s_{a},s_{b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x% )-\sum_{e\in\{\overline{aa^{\prime}},\overline{a^{\prime}b^{\prime}},\overline% {b^{\prime}b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P)+1</annotation><annotation encoding="application/x-llamapun" id="S3.Ex29.m1.8d">- ∑ start_POSTSUBSCRIPT italic_e ∈ { italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ { over¯ start_ARG italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG , over¯ start_ARG italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b end_ARG } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) + 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.12.p10.12">Applying first (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E23" title="Equation 23 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">23</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E24" title="Equation 24 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">24</span></a>), and then (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E25" title="Equation 25 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">25</span></a>), gives</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx9"> <tbody id="S3.Ex30"><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\sum_{v\in V(P^{\prime})}" class="ltx_Math" display="inline" id="S3.Ex30.m1.1"><semantics id="S3.Ex30.m1.1a"><mstyle displaystyle="true" id="S3.Ex30.m1.1.2" xref="S3.Ex30.m1.1.2.cmml"><munder id="S3.Ex30.m1.1.2a" xref="S3.Ex30.m1.1.2.cmml"><mo id="S3.Ex30.m1.1.2.2" movablelimits="false" xref="S3.Ex30.m1.1.2.2.cmml">∑</mo><mrow id="S3.Ex30.m1.1.1.1" xref="S3.Ex30.m1.1.1.1.cmml"><mi id="S3.Ex30.m1.1.1.1.3" xref="S3.Ex30.m1.1.1.1.3.cmml">v</mi><mo id="S3.Ex30.m1.1.1.1.2" xref="S3.Ex30.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex30.m1.1.1.1.1" xref="S3.Ex30.m1.1.1.1.1.cmml"><mi id="S3.Ex30.m1.1.1.1.1.3" xref="S3.Ex30.m1.1.1.1.1.3.cmml">V</mi><mo id="S3.Ex30.m1.1.1.1.1.2" xref="S3.Ex30.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex30.m1.1.1.1.1.1.1" xref="S3.Ex30.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex30.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex30.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Ex30.m1.1.1.1.1.1.1.1" xref="S3.Ex30.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex30.m1.1.1.1.1.1.1.1.2" xref="S3.Ex30.m1.1.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.Ex30.m1.1.1.1.1.1.1.1.3" xref="S3.Ex30.m1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Ex30.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex30.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><annotation-xml encoding="MathML-Content" id="S3.Ex30.m1.1b"><apply id="S3.Ex30.m1.1.2.cmml" 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encoding="application/x-tex" id="S3.Ex30.m1.1c">\displaystyle\sum_{v\in V(P^{\prime})}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex30.m1.1d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hskip 10.00002pt\mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_% {b}(P^{\prime})}\mathds{1}_{H^{e}_{P^{\prime}}}(x)-n_{a}(P^{\prime})" class="ltx_Math" display="inline" id="S3.Ex30.m2.4"><semantics id="S3.Ex30.m2.4a"><mrow id="S3.Ex30.m2.4.4" xref="S3.Ex30.m2.4.4.cmml"><mrow id="S3.Ex30.m2.4.4.3" xref="S3.Ex30.m2.4.4.3.cmml"><msub id="S3.Ex30.m2.4.4.3.2" xref="S3.Ex30.m2.4.4.3.2.cmml"><mn id="S3.Ex30.m2.4.4.3.2.2" xref="S3.Ex30.m2.4.4.3.2.2.cmml">𝟙</mn><msubsup id="S3.Ex30.m2.4.4.3.2.3" xref="S3.Ex30.m2.4.4.3.2.3.cmml"><mi id="S3.Ex30.m2.4.4.3.2.3.2.2" 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blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex31"><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=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}% \mathds{1}_{H^{e}_{P}}(x)+\sum_{v\in\{a,b\}}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)% -\sum_{e\in\{s_{a},s_{b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x)" class="ltx_Math" display="inline" id="S3.Ex31.m1.10"><semantics id="S3.Ex31.m1.10a"><mrow id="S3.Ex31.m1.10.11" 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xref="S3.Ex31.m1.10.11.3.3.2.2.3.2.2">𝐻</ci><ci id="S3.Ex31.m1.10.11.3.3.2.2.3.2.3.cmml" xref="S3.Ex31.m1.10.11.3.3.2.2.3.2.3">𝑒</ci></apply><apply id="S3.Ex31.m1.10.11.3.3.2.2.3.3.cmml" xref="S3.Ex31.m1.10.11.3.3.2.2.3.3"><csymbol cd="ambiguous" id="S3.Ex31.m1.10.11.3.3.2.2.3.3.1.cmml" xref="S3.Ex31.m1.10.11.3.3.2.2.3.3">superscript</csymbol><ci id="S3.Ex31.m1.10.11.3.3.2.2.3.3.2.cmml" xref="S3.Ex31.m1.10.11.3.3.2.2.3.3.2">𝑃</ci><ci id="S3.Ex31.m1.10.11.3.3.2.2.3.3.3.cmml" xref="S3.Ex31.m1.10.11.3.3.2.2.3.3.3">′</ci></apply></apply></apply><ci id="S3.Ex31.m1.10.10.cmml" xref="S3.Ex31.m1.10.10">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex31.m1.10c">\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}% \mathds{1}_{H^{e}_{P}}(x)+\sum_{v\in\{a,b\}}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)% -\sum_{e\in\{s_{a},s_{b}\}}\mathds{1}_{H^{e}_{P^{\prime}}}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex31.m1.10d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT italic_v ∈ { italic_a , italic_b } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ { italic_s start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT } end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex32"><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\quad+2-3-n_{a}(P)+1" class="ltx_Math" display="inline" id="S3.Ex32.m1.1"><semantics id="S3.Ex32.m1.1a"><mrow id="S3.Ex32.m1.1.2" xref="S3.Ex32.m1.1.2.cmml"><mrow id="S3.Ex32.m1.1.2.2" xref="S3.Ex32.m1.1.2.2.cmml"><mrow id="S3.Ex32.m1.1.2.2.2" xref="S3.Ex32.m1.1.2.2.2.cmml"><mo id="S3.Ex32.m1.1.2.2.2a" xref="S3.Ex32.m1.1.2.2.2.cmml">+</mo><mn id="S3.Ex32.m1.1.2.2.2.2" xref="S3.Ex32.m1.1.2.2.2.2.cmml">2</mn></mrow><mo id="S3.Ex32.m1.1.2.2.1" xref="S3.Ex32.m1.1.2.2.1.cmml">−</mo><mn id="S3.Ex32.m1.1.2.2.3" xref="S3.Ex32.m1.1.2.2.3.cmml">3</mn><mo id="S3.Ex32.m1.1.2.2.1a" xref="S3.Ex32.m1.1.2.2.1.cmml">−</mo><mrow id="S3.Ex32.m1.1.2.2.4" xref="S3.Ex32.m1.1.2.2.4.cmml"><msub id="S3.Ex32.m1.1.2.2.4.2" xref="S3.Ex32.m1.1.2.2.4.2.cmml"><mi id="S3.Ex32.m1.1.2.2.4.2.2" xref="S3.Ex32.m1.1.2.2.4.2.2.cmml">n</mi><mi id="S3.Ex32.m1.1.2.2.4.2.3" xref="S3.Ex32.m1.1.2.2.4.2.3.cmml">a</mi></msub><mo id="S3.Ex32.m1.1.2.2.4.1" xref="S3.Ex32.m1.1.2.2.4.1.cmml">⁢</mo><mrow id="S3.Ex32.m1.1.2.2.4.3.2" xref="S3.Ex32.m1.1.2.2.4.cmml"><mo id="S3.Ex32.m1.1.2.2.4.3.2.1" stretchy="false" xref="S3.Ex32.m1.1.2.2.4.cmml">(</mo><mi id="S3.Ex32.m1.1.1" xref="S3.Ex32.m1.1.1.cmml">P</mi><mo id="S3.Ex32.m1.1.2.2.4.3.2.2" stretchy="false" xref="S3.Ex32.m1.1.2.2.4.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex32.m1.1.2.1" xref="S3.Ex32.m1.1.2.1.cmml">+</mo><mn id="S3.Ex32.m1.1.2.3" xref="S3.Ex32.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex32.m1.1b"><apply id="S3.Ex32.m1.1.2.cmml" xref="S3.Ex32.m1.1.2"><plus id="S3.Ex32.m1.1.2.1.cmml" xref="S3.Ex32.m1.1.2.1"></plus><apply id="S3.Ex32.m1.1.2.2.cmml" xref="S3.Ex32.m1.1.2.2"><minus id="S3.Ex32.m1.1.2.2.1.cmml" xref="S3.Ex32.m1.1.2.2.1"></minus><apply id="S3.Ex32.m1.1.2.2.2.cmml" xref="S3.Ex32.m1.1.2.2.2"><plus id="S3.Ex32.m1.1.2.2.2.1.cmml" xref="S3.Ex32.m1.1.2.2.2"></plus><cn id="S3.Ex32.m1.1.2.2.2.2.cmml" type="integer" xref="S3.Ex32.m1.1.2.2.2.2">2</cn></apply><cn id="S3.Ex32.m1.1.2.2.3.cmml" type="integer" xref="S3.Ex32.m1.1.2.2.3">3</cn><apply id="S3.Ex32.m1.1.2.2.4.cmml" xref="S3.Ex32.m1.1.2.2.4"><times id="S3.Ex32.m1.1.2.2.4.1.cmml" xref="S3.Ex32.m1.1.2.2.4.1"></times><apply id="S3.Ex32.m1.1.2.2.4.2.cmml" xref="S3.Ex32.m1.1.2.2.4.2"><csymbol cd="ambiguous" id="S3.Ex32.m1.1.2.2.4.2.1.cmml" xref="S3.Ex32.m1.1.2.2.4.2">subscript</csymbol><ci id="S3.Ex32.m1.1.2.2.4.2.2.cmml" xref="S3.Ex32.m1.1.2.2.4.2.2">𝑛</ci><ci id="S3.Ex32.m1.1.2.2.4.2.3.cmml" xref="S3.Ex32.m1.1.2.2.4.2.3">𝑎</ci></apply><ci id="S3.Ex32.m1.1.1.cmml" xref="S3.Ex32.m1.1.1">𝑃</ci></apply></apply><cn id="S3.Ex32.m1.1.2.3.cmml" type="integer" xref="S3.Ex32.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex32.m1.1c">\displaystyle\quad+2-3-n_{a}(P)+1</annotation><annotation encoding="application/x-llamapun" id="S3.Ex32.m1.1d">+ 2 - 3 - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) + 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex33"><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=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}% \mathds{1}_{H^{e}_{P}}(x)-n_{a}(P)," 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xref="S3.Ex33.m1.6.6.1.1.3.4.2">subscript</csymbol><ci id="S3.Ex33.m1.6.6.1.1.3.4.2.2.cmml" xref="S3.Ex33.m1.6.6.1.1.3.4.2.2">𝑛</ci><ci id="S3.Ex33.m1.6.6.1.1.3.4.2.3.cmml" xref="S3.Ex33.m1.6.6.1.1.3.4.2.3">𝑎</ci></apply><ci id="S3.Ex33.m1.5.5.cmml" xref="S3.Ex33.m1.5.5">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex33.m1.6c">\displaystyle=\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}% \mathds{1}_{H^{e}_{P}}(x)-n_{a}(P),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex33.m1.6d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.12.p10.13">which is (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E19" title="Equation 19 ‣ Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">19</span></a>). ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" 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">Corollary 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.9.9">Let <math alttext="P" 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">P</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">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon whose boundary consists of <math alttext="n_{a}(P)\geq 1" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.2.m2.1"><semantics id="S3.Thmtheorem9.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem9.p1.2.2.m2.1.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.cmml"><mrow id="S3.Thmtheorem9.p1.2.2.m2.1.2.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.cmml"><msub id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.cmml"><mi id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.3" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.3.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.cmml"><mo id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem9.p1.2.2.m2.1.2.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.1.cmml">≥</mo><mn id="S3.Thmtheorem9.p1.2.2.m2.1.2.3" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.2.m2.1b"><apply id="S3.Thmtheorem9.p1.2.2.m2.1.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2"><geq id="S3.Thmtheorem9.p1.2.2.m2.1.2.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.1"></geq><apply id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2"><times id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.1"></times><apply id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.3.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.2.2.3">𝑎</ci></apply><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1">𝑃</ci></apply><cn id="S3.Thmtheorem9.p1.2.2.m2.1.2.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.2.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.2.m2.1c">n_{a}(P)\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) ≥ 1</annotation></semantics></math> polygonal arcs without lines. For any <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.3.m3.1"><semantics id="S3.Thmtheorem9.p1.3.3.m3.1a"><mrow 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">x</mi><mo id="S3.Thmtheorem9.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mn id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml">2</mn></msup></mrow><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"><in id="S3.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.2">𝑥</ci><apply id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2">ℝ</ci><cn id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.3.m3.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.3.m3.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.4.4.m4.1"><semantics id="S3.Thmtheorem9.p1.4.4.m4.1a"><mi id="S3.Thmtheorem9.p1.4.4.m4.1.1" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.4.4.m4.1b"><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.4.m4.1d">italic_P</annotation></semantics></math>-general position, there is a bounded polygon <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.5.5.m5.1"><semantics id="S3.Thmtheorem9.p1.5.5.m5.1a"><msup id="S3.Thmtheorem9.p1.5.5.m5.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.5.5.m5.1b"><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1">superscript</csymbol><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2">𝑃</ci><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.5.5.m5.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.5.5.m5.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\partial P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.6.6.m6.1"><semantics id="S3.Thmtheorem9.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem9.p1.6.6.m6.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.1.1.1" rspace="0em" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.1.cmml">∂</mo><msup id="S3.Thmtheorem9.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2.2.cmml">P</mi><mo id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.3" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.6.6.m6.1b"><apply id="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1"><partialdiff id="S3.Thmtheorem9.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.1"></partialdiff><apply id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2.2">𝑃</ci><ci id="S3.Thmtheorem9.p1.6.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.6.6.m6.1c">\partial P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.6.6.m6.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a cycle, such that <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.7.7.m7.1"><semantics id="S3.Thmtheorem9.p1.7.7.m7.1a"><mi id="S3.Thmtheorem9.p1.7.7.m7.1.1" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.7.7.m7.1b"><ci id="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.7.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.7.7.m7.1d">italic_x</annotation></semantics></math> is in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.8.8.m8.1"><semantics id="S3.Thmtheorem9.p1.8.8.m8.1a"><msup id="S3.Thmtheorem9.p1.8.8.m8.1.1" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem9.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem9.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.8.8.m8.1b"><apply id="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1">superscript</csymbol><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2">𝑃</ci><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.8.8.m8.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.8.8.m8.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position, and such that both sides of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E17" title="Equation 17 ‣ Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">17</span></a>) are preserved at <math alttext="x" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.9.9.m9.1"><semantics id="S3.Thmtheorem9.p1.9.9.m9.1a"><mi id="S3.Thmtheorem9.p1.9.9.m9.1.1" xref="S3.Thmtheorem9.p1.9.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.9.9.m9.1b"><ci id="S3.Thmtheorem9.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem9.p1.9.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.9.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.9.9.m9.1d">italic_x</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS2.13"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS2.13.p1"> <p class="ltx_p" id="S3.SS1.SSS2.13.p1.1">This follows directly from <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem8" title="Lemma 3.8. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.8</span></a> by induction. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS2.p7"> <p class="ltx_p" id="S3.SS1.SSS2.p7.1">We are now in position to prove <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a>.</p> </div> <div class="ltx_proof" id="S3.SS1.SSS2.14"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a>.</h6> <div class="ltx_para" id="S3.SS1.SSS2.14.p1"> <p class="ltx_p" id="S3.SS1.SSS2.14.p1.1">By <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem6" title="Lemma 3.6. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.6</span></a>, we can assume that <math alttext="E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.1.m1.1"><semantics id="S3.SS1.SSS2.14.p1.1.m1.1a"><mrow id="S3.SS1.SSS2.14.p1.1.m1.1.2" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.cmml"><mi id="S3.SS1.SSS2.14.p1.1.m1.1.2.2" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.2.cmml">E</mi><mo id="S3.SS1.SSS2.14.p1.1.m1.1.2.1" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.14.p1.1.m1.1.2.3.2" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.cmml"><mo id="S3.SS1.SSS2.14.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.14.p1.1.m1.1.1" xref="S3.SS1.SSS2.14.p1.1.m1.1.1.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.1.m1.1b"><apply id="S3.SS1.SSS2.14.p1.1.m1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.1.m1.1.2"><times id="S3.SS1.SSS2.14.p1.1.m1.1.2.1.cmml" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.1"></times><ci id="S3.SS1.SSS2.14.p1.1.m1.1.2.2.cmml" xref="S3.SS1.SSS2.14.p1.1.m1.1.2.2">𝐸</ci><ci id="S3.SS1.SSS2.14.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.1.m1.1c">E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.1.m1.1d">italic_E ( italic_P )</annotation></semantics></math> contains no lines, otherwise, we split them into two rays. Therefore, we have to prove</p> <table class="ltx_equation ltx_eqn_table" id="S3.E26"> <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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)=\mathds{1}_{P}(x)-1." class="ltx_Math" display="block" id="S3.E26.m1.7"><semantics id="S3.E26.m1.7a"><mrow id="S3.E26.m1.7.7.1" xref="S3.E26.m1.7.7.1.1.cmml"><mrow id="S3.E26.m1.7.7.1.1" xref="S3.E26.m1.7.7.1.1.cmml"><mrow id="S3.E26.m1.7.7.1.1.2" xref="S3.E26.m1.7.7.1.1.2.cmml"><mrow id="S3.E26.m1.7.7.1.1.2.2" xref="S3.E26.m1.7.7.1.1.2.2.cmml"><munder id="S3.E26.m1.7.7.1.1.2.2.1" xref="S3.E26.m1.7.7.1.1.2.2.1.cmml"><mo id="S3.E26.m1.7.7.1.1.2.2.1.2" movablelimits="false" xref="S3.E26.m1.7.7.1.1.2.2.1.2.cmml">∑</mo><mrow id="S3.E26.m1.1.1.1" xref="S3.E26.m1.1.1.1.cmml"><mi id="S3.E26.m1.1.1.1.3" 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id="S3.E26.m1.7.7.1.1.2.3.2.2.3.2.1.cmml" xref="S3.E26.m1.7.7.1.1.2.3.2.2.3">superscript</csymbol><ci id="S3.E26.m1.7.7.1.1.2.3.2.2.3.2.2.cmml" xref="S3.E26.m1.7.7.1.1.2.3.2.2.3.2.2">𝐻</ci><ci id="S3.E26.m1.7.7.1.1.2.3.2.2.3.2.3.cmml" xref="S3.E26.m1.7.7.1.1.2.3.2.2.3.2.3">𝑒</ci></apply><ci id="S3.E26.m1.7.7.1.1.2.3.2.2.3.3.cmml" xref="S3.E26.m1.7.7.1.1.2.3.2.2.3.3">𝑃</ci></apply></apply><ci id="S3.E26.m1.4.4.cmml" xref="S3.E26.m1.4.4">𝑥</ci></apply></apply><apply id="S3.E26.m1.7.7.1.1.2.4.cmml" xref="S3.E26.m1.7.7.1.1.2.4"><times id="S3.E26.m1.7.7.1.1.2.4.1.cmml" xref="S3.E26.m1.7.7.1.1.2.4.1"></times><apply id="S3.E26.m1.7.7.1.1.2.4.2.cmml" xref="S3.E26.m1.7.7.1.1.2.4.2"><csymbol cd="ambiguous" id="S3.E26.m1.7.7.1.1.2.4.2.1.cmml" xref="S3.E26.m1.7.7.1.1.2.4.2">subscript</csymbol><ci id="S3.E26.m1.7.7.1.1.2.4.2.2.cmml" xref="S3.E26.m1.7.7.1.1.2.4.2.2">𝑛</ci><ci id="S3.E26.m1.7.7.1.1.2.4.2.3.cmml" xref="S3.E26.m1.7.7.1.1.2.4.2.3">𝑎</ci></apply><ci id="S3.E26.m1.5.5.cmml" xref="S3.E26.m1.5.5">𝑃</ci></apply></apply><apply id="S3.E26.m1.7.7.1.1.3.cmml" xref="S3.E26.m1.7.7.1.1.3"><minus id="S3.E26.m1.7.7.1.1.3.1.cmml" xref="S3.E26.m1.7.7.1.1.3.1"></minus><apply id="S3.E26.m1.7.7.1.1.3.2.cmml" xref="S3.E26.m1.7.7.1.1.3.2"><times id="S3.E26.m1.7.7.1.1.3.2.1.cmml" xref="S3.E26.m1.7.7.1.1.3.2.1"></times><apply id="S3.E26.m1.7.7.1.1.3.2.2.cmml" xref="S3.E26.m1.7.7.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E26.m1.7.7.1.1.3.2.2.1.cmml" xref="S3.E26.m1.7.7.1.1.3.2.2">subscript</csymbol><cn id="S3.E26.m1.7.7.1.1.3.2.2.2.cmml" type="integer" xref="S3.E26.m1.7.7.1.1.3.2.2.2">1</cn><ci id="S3.E26.m1.7.7.1.1.3.2.2.3.cmml" xref="S3.E26.m1.7.7.1.1.3.2.2.3">𝑃</ci></apply><ci id="S3.E26.m1.6.6.cmml" xref="S3.E26.m1.6.6">𝑥</ci></apply><cn id="S3.E26.m1.7.7.1.1.3.3.cmml" type="integer" xref="S3.E26.m1.7.7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E26.m1.7c">\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)=\mathds{1}_{P}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.E26.m1.7d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 .</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">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.14.p1.6">Let <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.2.m1.1"><semantics id="S3.SS1.SSS2.14.p1.2.m1.1a"><mrow id="S3.SS1.SSS2.14.p1.2.m1.1.1" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.2.m1.1.1.2" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS2.14.p1.2.m1.1.1.1" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS2.14.p1.2.m1.1.1.3" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3.cmml"><mi id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.2" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.3" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.2.m1.1b"><apply id="S3.SS1.SSS2.14.p1.2.m1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1"><in id="S3.SS1.SSS2.14.p1.2.m1.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.1"></in><ci id="S3.SS1.SSS2.14.p1.2.m1.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.2">𝑥</ci><apply id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.1.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.2.cmml" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3.2">ℝ</ci><cn id="S3.SS1.SSS2.14.p1.2.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.14.p1.2.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.2.m1.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.2.m1.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.3.m2.1"><semantics id="S3.SS1.SSS2.14.p1.3.m2.1a"><mi id="S3.SS1.SSS2.14.p1.3.m2.1.1" xref="S3.SS1.SSS2.14.p1.3.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.3.m2.1b"><ci id="S3.SS1.SSS2.14.p1.3.m2.1.1.cmml" xref="S3.SS1.SSS2.14.p1.3.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.3.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.3.m2.1d">italic_P</annotation></semantics></math>-general position be fixed. Then, we can apply <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem9" title="Corollary 3.9. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.9</span></a>, and get a bounded polygon <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.4.m3.1"><semantics id="S3.SS1.SSS2.14.p1.4.m3.1a"><msup id="S3.SS1.SSS2.14.p1.4.m3.1.1" xref="S3.SS1.SSS2.14.p1.4.m3.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.4.m3.1.1.2" xref="S3.SS1.SSS2.14.p1.4.m3.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.4.m3.1.1.3" xref="S3.SS1.SSS2.14.p1.4.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.4.m3.1b"><apply id="S3.SS1.SSS2.14.p1.4.m3.1.1.cmml" xref="S3.SS1.SSS2.14.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.4.m3.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.4.m3.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.4.m3.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.4.m3.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.4.m3.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.4.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.4.m3.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.4.m3.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> whose boundary is a cycle, and with <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.5.m4.1"><semantics id="S3.SS1.SSS2.14.p1.5.m4.1a"><mi id="S3.SS1.SSS2.14.p1.5.m4.1.1" xref="S3.SS1.SSS2.14.p1.5.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.5.m4.1b"><ci id="S3.SS1.SSS2.14.p1.5.m4.1.1.cmml" xref="S3.SS1.SSS2.14.p1.5.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.5.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.5.m4.1d">italic_x</annotation></semantics></math> in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.6.m5.1"><semantics id="S3.SS1.SSS2.14.p1.6.m5.1a"><msup id="S3.SS1.SSS2.14.p1.6.m5.1.1" xref="S3.SS1.SSS2.14.p1.6.m5.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.6.m5.1.1.2" xref="S3.SS1.SSS2.14.p1.6.m5.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.6.m5.1.1.3" xref="S3.SS1.SSS2.14.p1.6.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.6.m5.1b"><apply id="S3.SS1.SSS2.14.p1.6.m5.1.1.cmml" xref="S3.SS1.SSS2.14.p1.6.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.6.m5.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.6.m5.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.6.m5.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.6.m5.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.6.m5.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.6.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.6.m5.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.6.m5.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position, such that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E27"> <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="\mathds{1}_{P}(x)-1=\mathds{1}_{P^{\prime}}(x)-1," class="ltx_Math" display="block" id="S3.E27.m1.3"><semantics id="S3.E27.m1.3a"><mrow id="S3.E27.m1.3.3.1" xref="S3.E27.m1.3.3.1.1.cmml"><mrow id="S3.E27.m1.3.3.1.1" xref="S3.E27.m1.3.3.1.1.cmml"><mrow id="S3.E27.m1.3.3.1.1.2" xref="S3.E27.m1.3.3.1.1.2.cmml"><mrow id="S3.E27.m1.3.3.1.1.2.2" xref="S3.E27.m1.3.3.1.1.2.2.cmml"><msub id="S3.E27.m1.3.3.1.1.2.2.2" xref="S3.E27.m1.3.3.1.1.2.2.2.cmml"><mn id="S3.E27.m1.3.3.1.1.2.2.2.2" xref="S3.E27.m1.3.3.1.1.2.2.2.2.cmml">𝟙</mn><mi id="S3.E27.m1.3.3.1.1.2.2.2.3" xref="S3.E27.m1.3.3.1.1.2.2.2.3.cmml">P</mi></msub><mo id="S3.E27.m1.3.3.1.1.2.2.1" xref="S3.E27.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S3.E27.m1.3.3.1.1.2.2.3.2" xref="S3.E27.m1.3.3.1.1.2.2.cmml"><mo id="S3.E27.m1.3.3.1.1.2.2.3.2.1" stretchy="false" xref="S3.E27.m1.3.3.1.1.2.2.cmml">(</mo><mi id="S3.E27.m1.1.1" xref="S3.E27.m1.1.1.cmml">x</mi><mo id="S3.E27.m1.3.3.1.1.2.2.3.2.2" stretchy="false" xref="S3.E27.m1.3.3.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.E27.m1.3.3.1.1.2.1" xref="S3.E27.m1.3.3.1.1.2.1.cmml">−</mo><mn id="S3.E27.m1.3.3.1.1.2.3" xref="S3.E27.m1.3.3.1.1.2.3.cmml">1</mn></mrow><mo id="S3.E27.m1.3.3.1.1.1" xref="S3.E27.m1.3.3.1.1.1.cmml">=</mo><mrow id="S3.E27.m1.3.3.1.1.3" xref="S3.E27.m1.3.3.1.1.3.cmml"><mrow id="S3.E27.m1.3.3.1.1.3.2" xref="S3.E27.m1.3.3.1.1.3.2.cmml"><msub id="S3.E27.m1.3.3.1.1.3.2.2" xref="S3.E27.m1.3.3.1.1.3.2.2.cmml"><mn id="S3.E27.m1.3.3.1.1.3.2.2.2" xref="S3.E27.m1.3.3.1.1.3.2.2.2.cmml">𝟙</mn><msup id="S3.E27.m1.3.3.1.1.3.2.2.3" xref="S3.E27.m1.3.3.1.1.3.2.2.3.cmml"><mi id="S3.E27.m1.3.3.1.1.3.2.2.3.2" xref="S3.E27.m1.3.3.1.1.3.2.2.3.2.cmml">P</mi><mo id="S3.E27.m1.3.3.1.1.3.2.2.3.3" xref="S3.E27.m1.3.3.1.1.3.2.2.3.3.cmml">′</mo></msup></msub><mo id="S3.E27.m1.3.3.1.1.3.2.1" xref="S3.E27.m1.3.3.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E27.m1.3.3.1.1.3.2.3.2" xref="S3.E27.m1.3.3.1.1.3.2.cmml"><mo id="S3.E27.m1.3.3.1.1.3.2.3.2.1" stretchy="false" xref="S3.E27.m1.3.3.1.1.3.2.cmml">(</mo><mi id="S3.E27.m1.2.2" xref="S3.E27.m1.2.2.cmml">x</mi><mo id="S3.E27.m1.3.3.1.1.3.2.3.2.2" stretchy="false" xref="S3.E27.m1.3.3.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S3.E27.m1.3.3.1.1.3.1" xref="S3.E27.m1.3.3.1.1.3.1.cmml">−</mo><mn id="S3.E27.m1.3.3.1.1.3.3" xref="S3.E27.m1.3.3.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S3.E27.m1.3.3.1.2" xref="S3.E27.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E27.m1.3b"><apply id="S3.E27.m1.3.3.1.1.cmml" xref="S3.E27.m1.3.3.1"><eq id="S3.E27.m1.3.3.1.1.1.cmml" xref="S3.E27.m1.3.3.1.1.1"></eq><apply id="S3.E27.m1.3.3.1.1.2.cmml" xref="S3.E27.m1.3.3.1.1.2"><minus id="S3.E27.m1.3.3.1.1.2.1.cmml" xref="S3.E27.m1.3.3.1.1.2.1"></minus><apply id="S3.E27.m1.3.3.1.1.2.2.cmml" xref="S3.E27.m1.3.3.1.1.2.2"><times id="S3.E27.m1.3.3.1.1.2.2.1.cmml" xref="S3.E27.m1.3.3.1.1.2.2.1"></times><apply id="S3.E27.m1.3.3.1.1.2.2.2.cmml" xref="S3.E27.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E27.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.E27.m1.3.3.1.1.2.2.2">subscript</csymbol><cn id="S3.E27.m1.3.3.1.1.2.2.2.2.cmml" type="integer" xref="S3.E27.m1.3.3.1.1.2.2.2.2">1</cn><ci id="S3.E27.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.E27.m1.3.3.1.1.2.2.2.3">𝑃</ci></apply><ci id="S3.E27.m1.1.1.cmml" xref="S3.E27.m1.1.1">𝑥</ci></apply><cn id="S3.E27.m1.3.3.1.1.2.3.cmml" type="integer" xref="S3.E27.m1.3.3.1.1.2.3">1</cn></apply><apply id="S3.E27.m1.3.3.1.1.3.cmml" xref="S3.E27.m1.3.3.1.1.3"><minus id="S3.E27.m1.3.3.1.1.3.1.cmml" xref="S3.E27.m1.3.3.1.1.3.1"></minus><apply id="S3.E27.m1.3.3.1.1.3.2.cmml" xref="S3.E27.m1.3.3.1.1.3.2"><times id="S3.E27.m1.3.3.1.1.3.2.1.cmml" xref="S3.E27.m1.3.3.1.1.3.2.1"></times><apply id="S3.E27.m1.3.3.1.1.3.2.2.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E27.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2">subscript</csymbol><cn id="S3.E27.m1.3.3.1.1.3.2.2.2.cmml" type="integer" xref="S3.E27.m1.3.3.1.1.3.2.2.2">1</cn><apply id="S3.E27.m1.3.3.1.1.3.2.2.3.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.E27.m1.3.3.1.1.3.2.2.3.1.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2.3">superscript</csymbol><ci id="S3.E27.m1.3.3.1.1.3.2.2.3.2.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2.3.2">𝑃</ci><ci id="S3.E27.m1.3.3.1.1.3.2.2.3.3.cmml" xref="S3.E27.m1.3.3.1.1.3.2.2.3.3">′</ci></apply></apply><ci id="S3.E27.m1.2.2.cmml" xref="S3.E27.m1.2.2">𝑥</ci></apply><cn id="S3.E27.m1.3.3.1.1.3.3.cmml" type="integer" xref="S3.E27.m1.3.3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E27.m1.3c">\mathds{1}_{P}(x)-1=\mathds{1}_{P^{\prime}}(x)-1,</annotation><annotation encoding="application/x-llamapun" id="S3.E27.m1.3d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) - 1 = blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) - 1 ,</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">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.14.p1.13">and</p> <table class="ltx_equation ltx_eqn_table" id="S3.E28"> <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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)\\ =\sum_{v\in V(P^{\prime})}\mathds{1}_{Q^{v}_{P^{\prime}}}(x)-\sum_{e\in E_{b}(% P^{\prime})}\mathds{1}_{H^{e}_{P^{\prime}}}(x)." class="ltx_Math" display="block" id="S3.E28.m1.40"><semantics id="S3.E28.m1.40a"><mtable displaystyle="true" id="S3.E28.m1.40.40.2" rowspacing="0pt"><mtr id="S3.E28.m1.40.40.2a"><mtd class="ltx_align_left" columnalign="left" id="S3.E28.m1.40.40.2b"><mrow id="S3.E28.m1.21.21.21.21.21"><mrow id="S3.E28.m1.21.21.21.21.21.22"><munder id="S3.E28.m1.21.21.21.21.21.22.1"><mo id="S3.E28.m1.1.1.1.1.1.1" movablelimits="false" xref="S3.E28.m1.1.1.1.1.1.1.cmml">∑</mo><mrow id="S3.E28.m1.2.2.2.2.2.2.1" xref="S3.E28.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E28.m1.2.2.2.2.2.2.1.3" xref="S3.E28.m1.2.2.2.2.2.2.1.3.cmml">v</mi><mo id="S3.E28.m1.2.2.2.2.2.2.1.2" xref="S3.E28.m1.2.2.2.2.2.2.1.2.cmml">∈</mo><mrow id="S3.E28.m1.2.2.2.2.2.2.1.4" xref="S3.E28.m1.2.2.2.2.2.2.1.4.cmml"><mi id="S3.E28.m1.2.2.2.2.2.2.1.4.2" xref="S3.E28.m1.2.2.2.2.2.2.1.4.2.cmml">V</mi><mo id="S3.E28.m1.2.2.2.2.2.2.1.4.1" xref="S3.E28.m1.2.2.2.2.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.E28.m1.2.2.2.2.2.2.1.4.3.2" xref="S3.E28.m1.2.2.2.2.2.2.1.4.cmml"><mo id="S3.E28.m1.2.2.2.2.2.2.1.4.3.2.1" stretchy="false" xref="S3.E28.m1.2.2.2.2.2.2.1.4.cmml">(</mo><mi id="S3.E28.m1.2.2.2.2.2.2.1.1" xref="S3.E28.m1.2.2.2.2.2.2.1.1.cmml">P</mi><mo id="S3.E28.m1.2.2.2.2.2.2.1.4.3.2.2" stretchy="false" xref="S3.E28.m1.2.2.2.2.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E28.m1.21.21.21.21.21.22.2"><msub id="S3.E28.m1.21.21.21.21.21.22.2.2"><mn id="S3.E28.m1.3.3.3.3.3.3" xref="S3.E28.m1.3.3.3.3.3.3.cmml">𝟙</mn><msubsup id="S3.E28.m1.4.4.4.4.4.4.1" xref="S3.E28.m1.4.4.4.4.4.4.1.cmml"><mi id="S3.E28.m1.4.4.4.4.4.4.1.2.2" xref="S3.E28.m1.4.4.4.4.4.4.1.2.2.cmml">Q</mi><mi id="S3.E28.m1.4.4.4.4.4.4.1.3" xref="S3.E28.m1.4.4.4.4.4.4.1.3.cmml">P</mi><mi id="S3.E28.m1.4.4.4.4.4.4.1.2.3" xref="S3.E28.m1.4.4.4.4.4.4.1.2.3.cmml">v</mi></msubsup></msub><mo id="S3.E28.m1.21.21.21.21.21.22.2.1" xref="S3.E28.m1.39.39.1.1.1.cmml">⁢</mo><mrow id="S3.E28.m1.21.21.21.21.21.22.2.3"><mo id="S3.E28.m1.5.5.5.5.5.5" stretchy="false" xref="S3.E28.m1.39.39.1.1.1.cmml">(</mo><mi id="S3.E28.m1.6.6.6.6.6.6" xref="S3.E28.m1.6.6.6.6.6.6.cmml">x</mi><mo id="S3.E28.m1.7.7.7.7.7.7" stretchy="false" xref="S3.E28.m1.39.39.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E28.m1.8.8.8.8.8.8" rspace="0.055em" xref="S3.E28.m1.8.8.8.8.8.8.cmml">−</mo><mrow id="S3.E28.m1.21.21.21.21.21.23"><munder id="S3.E28.m1.21.21.21.21.21.23.1"><mo id="S3.E28.m1.9.9.9.9.9.9" movablelimits="false" xref="S3.E28.m1.9.9.9.9.9.9.cmml">∑</mo><mrow id="S3.E28.m1.10.10.10.10.10.10.1" xref="S3.E28.m1.10.10.10.10.10.10.1.cmml"><mi id="S3.E28.m1.10.10.10.10.10.10.1.3" xref="S3.E28.m1.10.10.10.10.10.10.1.3.cmml">e</mi><mo id="S3.E28.m1.10.10.10.10.10.10.1.2" xref="S3.E28.m1.10.10.10.10.10.10.1.2.cmml">∈</mo><mrow id="S3.E28.m1.10.10.10.10.10.10.1.4" xref="S3.E28.m1.10.10.10.10.10.10.1.4.cmml"><msub id="S3.E28.m1.10.10.10.10.10.10.1.4.2" xref="S3.E28.m1.10.10.10.10.10.10.1.4.2.cmml"><mi id="S3.E28.m1.10.10.10.10.10.10.1.4.2.2" xref="S3.E28.m1.10.10.10.10.10.10.1.4.2.2.cmml">E</mi><mi id="S3.E28.m1.10.10.10.10.10.10.1.4.2.3" xref="S3.E28.m1.10.10.10.10.10.10.1.4.2.3.cmml">b</mi></msub><mo id="S3.E28.m1.10.10.10.10.10.10.1.4.1" xref="S3.E28.m1.10.10.10.10.10.10.1.4.1.cmml">⁢</mo><mrow id="S3.E28.m1.10.10.10.10.10.10.1.4.3.2" xref="S3.E28.m1.10.10.10.10.10.10.1.4.cmml"><mo id="S3.E28.m1.10.10.10.10.10.10.1.4.3.2.1" stretchy="false" xref="S3.E28.m1.10.10.10.10.10.10.1.4.cmml">(</mo><mi id="S3.E28.m1.10.10.10.10.10.10.1.1" xref="S3.E28.m1.10.10.10.10.10.10.1.1.cmml">P</mi><mo id="S3.E28.m1.10.10.10.10.10.10.1.4.3.2.2" 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) end_CELL end_ROW start_ROW start_CELL = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) . end_CELL end_ROW</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">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.14.p1.9">Since <math alttext="\partial P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.7.m1.1"><semantics id="S3.SS1.SSS2.14.p1.7.m1.1a"><mrow id="S3.SS1.SSS2.14.p1.7.m1.1.1" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.cmml"><mo id="S3.SS1.SSS2.14.p1.7.m1.1.1.1" rspace="0em" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.1.cmml">∂</mo><msup id="S3.SS1.SSS2.14.p1.7.m1.1.1.2" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2.cmml"><mi id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.2" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.3" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.7.m1.1b"><apply id="S3.SS1.SSS2.14.p1.7.m1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1"><partialdiff id="S3.SS1.SSS2.14.p1.7.m1.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.1"></partialdiff><apply id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.1.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.2.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.7.m1.1.1.2.3.cmml" xref="S3.SS1.SSS2.14.p1.7.m1.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.7.m1.1c">\partial P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.7.m1.1d">∂ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a single polygonal cycle, and <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.8.m2.1"><semantics id="S3.SS1.SSS2.14.p1.8.m2.1a"><mi id="S3.SS1.SSS2.14.p1.8.m2.1.1" xref="S3.SS1.SSS2.14.p1.8.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.8.m2.1b"><ci id="S3.SS1.SSS2.14.p1.8.m2.1.1.cmml" xref="S3.SS1.SSS2.14.p1.8.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.8.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.8.m2.1d">italic_x</annotation></semantics></math> is in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.9.m3.1"><semantics id="S3.SS1.SSS2.14.p1.9.m3.1a"><msup id="S3.SS1.SSS2.14.p1.9.m3.1.1" xref="S3.SS1.SSS2.14.p1.9.m3.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.9.m3.1.1.2" xref="S3.SS1.SSS2.14.p1.9.m3.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.9.m3.1.1.3" xref="S3.SS1.SSS2.14.p1.9.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.9.m3.1b"><apply id="S3.SS1.SSS2.14.p1.9.m3.1.1.cmml" xref="S3.SS1.SSS2.14.p1.9.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.9.m3.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.9.m3.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.9.m3.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.9.m3.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.9.m3.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.9.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.9.m3.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.9.m3.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position, we can apply <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem3" title="Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.3</span></a> to the right hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E28" title="Equation 28 ‣ Proof of 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">28</span></a>), and get</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex34"> <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_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)=n_{h}(P^{\prime})+\mathds{1}_{P^{\prime}}(x)-1." class="ltx_Math" display="block" id="S3.Ex34.m1.7"><semantics id="S3.Ex34.m1.7a"><mrow id="S3.Ex34.m1.7.7.1" xref="S3.Ex34.m1.7.7.1.1.cmml"><mrow id="S3.Ex34.m1.7.7.1.1" xref="S3.Ex34.m1.7.7.1.1.cmml"><mrow id="S3.Ex34.m1.7.7.1.1.3" xref="S3.Ex34.m1.7.7.1.1.3.cmml"><mrow id="S3.Ex34.m1.7.7.1.1.3.2" xref="S3.Ex34.m1.7.7.1.1.3.2.cmml"><munder id="S3.Ex34.m1.7.7.1.1.3.2.1" xref="S3.Ex34.m1.7.7.1.1.3.2.1.cmml"><mo id="S3.Ex34.m1.7.7.1.1.3.2.1.2" movablelimits="false" xref="S3.Ex34.m1.7.7.1.1.3.2.1.2.cmml">∑</mo><mrow id="S3.Ex34.m1.1.1.1" xref="S3.Ex34.m1.1.1.1.cmml"><mi id="S3.Ex34.m1.1.1.1.3" xref="S3.Ex34.m1.1.1.1.3.cmml">v</mi><mo id="S3.Ex34.m1.1.1.1.2" xref="S3.Ex34.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex34.m1.1.1.1.4" xref="S3.Ex34.m1.1.1.1.4.cmml"><mi id="S3.Ex34.m1.1.1.1.4.2" xref="S3.Ex34.m1.1.1.1.4.2.cmml">V</mi><mo id="S3.Ex34.m1.1.1.1.4.1" xref="S3.Ex34.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.1.1.1.4.3.2" xref="S3.Ex34.m1.1.1.1.4.cmml"><mo id="S3.Ex34.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex34.m1.1.1.1.4.cmml">(</mo><mi id="S3.Ex34.m1.1.1.1.1" xref="S3.Ex34.m1.1.1.1.1.cmml">P</mi><mo id="S3.Ex34.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex34.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex34.m1.7.7.1.1.3.2.2" xref="S3.Ex34.m1.7.7.1.1.3.2.2.cmml"><msub id="S3.Ex34.m1.7.7.1.1.3.2.2.2" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.cmml"><mn id="S3.Ex34.m1.7.7.1.1.3.2.2.2.2" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex34.m1.7.7.1.1.3.2.2.2.3" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.cmml"><mi id="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.2.2" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.2.2.cmml">Q</mi><mi id="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.3" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.3.cmml">P</mi><mi id="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.2.3" xref="S3.Ex34.m1.7.7.1.1.3.2.2.2.3.2.3.cmml">v</mi></msubsup></msub><mo id="S3.Ex34.m1.7.7.1.1.3.2.2.1" xref="S3.Ex34.m1.7.7.1.1.3.2.2.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.7.7.1.1.3.2.2.3.2" xref="S3.Ex34.m1.7.7.1.1.3.2.2.cmml"><mo id="S3.Ex34.m1.7.7.1.1.3.2.2.3.2.1" stretchy="false" xref="S3.Ex34.m1.7.7.1.1.3.2.2.cmml">(</mo><mi id="S3.Ex34.m1.3.3" xref="S3.Ex34.m1.3.3.cmml">x</mi><mo id="S3.Ex34.m1.7.7.1.1.3.2.2.3.2.2" stretchy="false" xref="S3.Ex34.m1.7.7.1.1.3.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex34.m1.7.7.1.1.3.1" rspace="0.055em" xref="S3.Ex34.m1.7.7.1.1.3.1.cmml">−</mo><mrow id="S3.Ex34.m1.7.7.1.1.3.3" xref="S3.Ex34.m1.7.7.1.1.3.3.cmml"><munder id="S3.Ex34.m1.7.7.1.1.3.3.1" xref="S3.Ex34.m1.7.7.1.1.3.3.1.cmml"><mo id="S3.Ex34.m1.7.7.1.1.3.3.1.2" movablelimits="false" xref="S3.Ex34.m1.7.7.1.1.3.3.1.2.cmml">∑</mo><mrow id="S3.Ex34.m1.2.2.1" xref="S3.Ex34.m1.2.2.1.cmml"><mi id="S3.Ex34.m1.2.2.1.3" xref="S3.Ex34.m1.2.2.1.3.cmml">e</mi><mo id="S3.Ex34.m1.2.2.1.2" xref="S3.Ex34.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex34.m1.2.2.1.4" xref="S3.Ex34.m1.2.2.1.4.cmml"><msub id="S3.Ex34.m1.2.2.1.4.2" xref="S3.Ex34.m1.2.2.1.4.2.cmml"><mi id="S3.Ex34.m1.2.2.1.4.2.2" xref="S3.Ex34.m1.2.2.1.4.2.2.cmml">E</mi><mi id="S3.Ex34.m1.2.2.1.4.2.3" xref="S3.Ex34.m1.2.2.1.4.2.3.cmml">b</mi></msub><mo id="S3.Ex34.m1.2.2.1.4.1" xref="S3.Ex34.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.2.2.1.4.3.2" xref="S3.Ex34.m1.2.2.1.4.cmml"><mo id="S3.Ex34.m1.2.2.1.4.3.2.1" stretchy="false" xref="S3.Ex34.m1.2.2.1.4.cmml">(</mo><mi id="S3.Ex34.m1.2.2.1.1" xref="S3.Ex34.m1.2.2.1.1.cmml">P</mi><mo id="S3.Ex34.m1.2.2.1.4.3.2.2" stretchy="false" xref="S3.Ex34.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex34.m1.7.7.1.1.3.3.2" xref="S3.Ex34.m1.7.7.1.1.3.3.2.cmml"><msub id="S3.Ex34.m1.7.7.1.1.3.3.2.2" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.cmml"><mn id="S3.Ex34.m1.7.7.1.1.3.3.2.2.2" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex34.m1.7.7.1.1.3.3.2.2.3" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.cmml"><mi id="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.2.2" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.2.2.cmml">H</mi><mi id="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.3" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.3.cmml">P</mi><mi id="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.2.3" xref="S3.Ex34.m1.7.7.1.1.3.3.2.2.3.2.3.cmml">e</mi></msubsup></msub><mo id="S3.Ex34.m1.7.7.1.1.3.3.2.1" xref="S3.Ex34.m1.7.7.1.1.3.3.2.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.7.7.1.1.3.3.2.3.2" xref="S3.Ex34.m1.7.7.1.1.3.3.2.cmml"><mo id="S3.Ex34.m1.7.7.1.1.3.3.2.3.2.1" stretchy="false" xref="S3.Ex34.m1.7.7.1.1.3.3.2.cmml">(</mo><mi id="S3.Ex34.m1.4.4" xref="S3.Ex34.m1.4.4.cmml">x</mi><mo id="S3.Ex34.m1.7.7.1.1.3.3.2.3.2.2" stretchy="false" xref="S3.Ex34.m1.7.7.1.1.3.3.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex34.m1.7.7.1.1.3.1a" xref="S3.Ex34.m1.7.7.1.1.3.1.cmml">−</mo><mrow id="S3.Ex34.m1.7.7.1.1.3.4" xref="S3.Ex34.m1.7.7.1.1.3.4.cmml"><msub id="S3.Ex34.m1.7.7.1.1.3.4.2" xref="S3.Ex34.m1.7.7.1.1.3.4.2.cmml"><mi id="S3.Ex34.m1.7.7.1.1.3.4.2.2" xref="S3.Ex34.m1.7.7.1.1.3.4.2.2.cmml">n</mi><mi id="S3.Ex34.m1.7.7.1.1.3.4.2.3" xref="S3.Ex34.m1.7.7.1.1.3.4.2.3.cmml">a</mi></msub><mo id="S3.Ex34.m1.7.7.1.1.3.4.1" 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xref="S3.Ex34.m1.7.7.1.1.1.1.3.2.2">1</cn><apply id="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.cmml" xref="S3.Ex34.m1.7.7.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.1.cmml" xref="S3.Ex34.m1.7.7.1.1.1.1.3.2.3">superscript</csymbol><ci id="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.2.cmml" xref="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.2">𝑃</ci><ci id="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.3.cmml" xref="S3.Ex34.m1.7.7.1.1.1.1.3.2.3.3">′</ci></apply></apply><ci id="S3.Ex34.m1.6.6.cmml" xref="S3.Ex34.m1.6.6">𝑥</ci></apply></apply><cn id="S3.Ex34.m1.7.7.1.1.1.3.cmml" type="integer" xref="S3.Ex34.m1.7.7.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex34.m1.7c">\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}(x)-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e% }_{P}}(x)-n_{a}(P)=n_{h}(P^{\prime})+\mathds{1}_{P^{\prime}}(x)-1.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex34.m1.7d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) - 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.14.p1.12">Since <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.10.m1.1"><semantics id="S3.SS1.SSS2.14.p1.10.m1.1a"><msup id="S3.SS1.SSS2.14.p1.10.m1.1.1" xref="S3.SS1.SSS2.14.p1.10.m1.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.10.m1.1.1.2" xref="S3.SS1.SSS2.14.p1.10.m1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.10.m1.1.1.3" xref="S3.SS1.SSS2.14.p1.10.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.10.m1.1b"><apply id="S3.SS1.SSS2.14.p1.10.m1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.10.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.10.m1.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.10.m1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.10.m1.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.10.m1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.10.m1.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.10.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.10.m1.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.10.m1.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is bounded and has only one boundary component, <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.11.m2.1"><semantics id="S3.SS1.SSS2.14.p1.11.m2.1a"><msup id="S3.SS1.SSS2.14.p1.11.m2.1.1" xref="S3.SS1.SSS2.14.p1.11.m2.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.11.m2.1.1.2" xref="S3.SS1.SSS2.14.p1.11.m2.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.11.m2.1.1.3" xref="S3.SS1.SSS2.14.p1.11.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.11.m2.1b"><apply id="S3.SS1.SSS2.14.p1.11.m2.1.1.cmml" xref="S3.SS1.SSS2.14.p1.11.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.11.m2.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.11.m2.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.11.m2.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.11.m2.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.11.m2.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.11.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.11.m2.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.11.m2.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> has no holes, i.e. <math alttext="n_{h}(P^{\prime})=0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.14.p1.12.m3.1"><semantics id="S3.SS1.SSS2.14.p1.12.m3.1a"><mrow id="S3.SS1.SSS2.14.p1.12.m3.1.1" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.cmml"><mrow id="S3.SS1.SSS2.14.p1.12.m3.1.1.1" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.cmml"><msub id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.cmml"><mi id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.2" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.2.cmml">n</mi><mi id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.3" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.3.cmml">h</mi></msub><mo id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.2" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.14.p1.12.m3.1.1.2" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.2.cmml">=</mo><mn id="S3.SS1.SSS2.14.p1.12.m3.1.1.3" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.14.p1.12.m3.1b"><apply id="S3.SS1.SSS2.14.p1.12.m3.1.1.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1"><eq id="S3.SS1.SSS2.14.p1.12.m3.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.2"></eq><apply id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1"><times id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.2"></times><apply id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.2">𝑛</ci><ci id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.3.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.3.3">ℎ</ci></apply><apply id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.1.1.1.1.3">′</ci></apply></apply><cn id="S3.SS1.SSS2.14.p1.12.m3.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.14.p1.12.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.14.p1.12.m3.1c">n_{h}(P^{\prime})=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.14.p1.12.m3.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>. Together with (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E27" title="Equation 27 ‣ Proof of 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">27</span></a>), this proves (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E26" title="Equation 26 ‣ Proof of 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">26</span></a>), and thus <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a>. ∎</p> </div> </div> </section> <section class="ltx_subsubsection" id="S3.SS1.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.1.3 </span>General case</h4> <div class="ltx_para" id="S3.SS1.SSS3.p1"> <p class="ltx_p" id="S3.SS1.SSS3.p1.1">Finally, the remaining cases of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a> can be proved using the results from the previous sections. First of all, the next lemma notes that all cycles of a polygon must describe holes except for possibly one.</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">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.1.1.m1.1"><semantics id="S3.Thmtheorem10.p1.1.1.m1.1a"><mi id="S3.Thmtheorem10.p1.1.1.m1.1.1" xref="S3.Thmtheorem10.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.1.1.m1.1b"><ci id="S3.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon and let <math alttext="\gamma_{1}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.2.2.m2.1"><semantics id="S3.Thmtheorem10.p1.2.2.m2.1a"><msub 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">γ</mi><mn id="S3.Thmtheorem10.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3.cmml">1</mn></msub><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"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1">subscript</csymbol><ci id="S3.Thmtheorem10.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.2">𝛾</ci><cn id="S3.Thmtheorem10.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem10.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.2.2.m2.1c">\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.2.2.m2.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma_{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.3.3.m3.1"><semantics id="S3.Thmtheorem10.p1.3.3.m3.1a"><msub id="S3.Thmtheorem10.p1.3.3.m3.1.1" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem10.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.2.cmml">γ</mi><mn id="S3.Thmtheorem10.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.3.3.m3.1b"><apply id="S3.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem10.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem10.p1.3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem10.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.2">𝛾</ci><cn id="S3.Thmtheorem10.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheorem10.p1.3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.3.3.m3.1c">\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.3.3.m3.1d">italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be distinct polygonal cycles that are boundary components of <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.4.4.m4.1"><semantics id="S3.Thmtheorem10.p1.4.4.m4.1a"><mi id="S3.Thmtheorem10.p1.4.4.m4.1.1" xref="S3.Thmtheorem10.p1.4.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.4.4.m4.1b"><ci id="S3.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem10.p1.4.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.4.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.4.4.m4.1d">italic_P</annotation></semantics></math>. Then, at least one <math alttext="\gamma_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.5.5.m5.1"><semantics id="S3.Thmtheorem10.p1.5.5.m5.1a"><msub id="S3.Thmtheorem10.p1.5.5.m5.1.1" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem10.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.2.cmml">γ</mi><mi id="S3.Thmtheorem10.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.5.5.m5.1b"><apply id="S3.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.1.1">subscript</csymbol><ci id="S3.Thmtheorem10.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.2">𝛾</ci><ci id="S3.Thmtheorem10.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem10.p1.5.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.5.5.m5.1c">\gamma_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.5.5.m5.1d">italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a hole.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS3.1.p1"> <p class="ltx_p" id="S3.SS1.SSS3.1.p1.10">Suppose that neither <math alttext="\gamma_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.1.m1.1"><semantics id="S3.SS1.SSS3.1.p1.1.m1.1a"><msub id="S3.SS1.SSS3.1.p1.1.m1.1.1" xref="S3.SS1.SSS3.1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS3.1.p1.1.m1.1.1.2" xref="S3.SS1.SSS3.1.p1.1.m1.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS3.1.p1.1.m1.1.1.3" xref="S3.SS1.SSS3.1.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.1.m1.1b"><apply id="S3.SS1.SSS3.1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.1.m1.1.1.2">𝛾</ci><cn id="S3.SS1.SSS3.1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.1.m1.1c">\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.1.m1.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> nor <math alttext="\gamma_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.2.m2.1"><semantics id="S3.SS1.SSS3.1.p1.2.m2.1a"><msub id="S3.SS1.SSS3.1.p1.2.m2.1.1" xref="S3.SS1.SSS3.1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS3.1.p1.2.m2.1.1.2" xref="S3.SS1.SSS3.1.p1.2.m2.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS3.1.p1.2.m2.1.1.3" xref="S3.SS1.SSS3.1.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.2.m2.1b"><apply id="S3.SS1.SSS3.1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.2.m2.1.1.2">𝛾</ci><cn id="S3.SS1.SSS3.1.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.2.m2.1c">\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.2.m2.1d">italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is a hole. Then, <math alttext="P\subseteq C_{i}:=\overline{\operatorname*{int}(\gamma_{i})}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.3.m3.2"><semantics id="S3.SS1.SSS3.1.p1.3.m3.2a"><mrow id="S3.SS1.SSS3.1.p1.3.m3.2.3" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.cmml"><mi id="S3.SS1.SSS3.1.p1.3.m3.2.3.2" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.1.p1.3.m3.2.3.3" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.3.cmml">⊆</mo><msub id="S3.SS1.SSS3.1.p1.3.m3.2.3.4" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4.cmml"><mi id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.2" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4.2.cmml">C</mi><mi id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.3" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4.3.cmml">i</mi></msub><mo id="S3.SS1.SSS3.1.p1.3.m3.2.3.5" lspace="0.278em" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.5.cmml">:=</mo><mover accent="true" id="S3.SS1.SSS3.1.p1.3.m3.2.2" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.cmml"><mrow id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.3.cmml"><mo id="S3.SS1.SSS3.1.p1.3.m3.1.1.1.1" lspace="0.111em" rspace="0em" xref="S3.SS1.SSS3.1.p1.3.m3.1.1.1.1.cmml">int</mo><mrow id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.3.cmml"><mo id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.2" stretchy="false" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.3.cmml">(</mo><msub id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.cmml"><mi id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.2" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.3" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.3" stretchy="false" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.1.p1.3.m3.2.2.3" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.3.m3.2b"><apply id="S3.SS1.SSS3.1.p1.3.m3.2.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3"><and id="S3.SS1.SSS3.1.p1.3.m3.2.3a.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3"></and><apply id="S3.SS1.SSS3.1.p1.3.m3.2.3b.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3"><subset id="S3.SS1.SSS3.1.p1.3.m3.2.3.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.3"></subset><ci id="S3.SS1.SSS3.1.p1.3.m3.2.3.2.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.2">𝑃</ci><apply id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.1.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.2.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4.2">𝐶</ci><ci id="S3.SS1.SSS3.1.p1.3.m3.2.3.4.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.4.3">𝑖</ci></apply></apply><apply id="S3.SS1.SSS3.1.p1.3.m3.2.3c.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3"><csymbol cd="latexml" id="S3.SS1.SSS3.1.p1.3.m3.2.3.5.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3.5">assign</csymbol><share href="https://arxiv.org/html/2503.13001v1#S3.SS1.SSS3.1.p1.3.m3.2.3.4.cmml" id="S3.SS1.SSS3.1.p1.3.m3.2.3d.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.3"></share><apply id="S3.SS1.SSS3.1.p1.3.m3.2.2.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2"><ci id="S3.SS1.SSS3.1.p1.3.m3.2.2.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.3">¯</ci><apply id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2"><ci id="S3.SS1.SSS3.1.p1.3.m3.1.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.1.1.1.1">int</ci><apply id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.3.m3.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.3.m3.2c">P\subseteq C_{i}:=\overline{\operatorname*{int}(\gamma_{i})}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.3.m3.2d">italic_P ⊆ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := over¯ start_ARG roman_int ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math> for <math alttext="i=1,2" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.4.m4.2"><semantics id="S3.SS1.SSS3.1.p1.4.m4.2a"><mrow id="S3.SS1.SSS3.1.p1.4.m4.2.3" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.cmml"><mi id="S3.SS1.SSS3.1.p1.4.m4.2.3.2" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.2.cmml">i</mi><mo id="S3.SS1.SSS3.1.p1.4.m4.2.3.1" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS3.1.p1.4.m4.2.3.3.2" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.3.1.cmml"><mn id="S3.SS1.SSS3.1.p1.4.m4.1.1" xref="S3.SS1.SSS3.1.p1.4.m4.1.1.cmml">1</mn><mo id="S3.SS1.SSS3.1.p1.4.m4.2.3.3.2.1" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.3.1.cmml">,</mo><mn id="S3.SS1.SSS3.1.p1.4.m4.2.2" xref="S3.SS1.SSS3.1.p1.4.m4.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.4.m4.2b"><apply id="S3.SS1.SSS3.1.p1.4.m4.2.3.cmml" xref="S3.SS1.SSS3.1.p1.4.m4.2.3"><eq id="S3.SS1.SSS3.1.p1.4.m4.2.3.1.cmml" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.1"></eq><ci id="S3.SS1.SSS3.1.p1.4.m4.2.3.2.cmml" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.2">𝑖</ci><list id="S3.SS1.SSS3.1.p1.4.m4.2.3.3.1.cmml" xref="S3.SS1.SSS3.1.p1.4.m4.2.3.3.2"><cn id="S3.SS1.SSS3.1.p1.4.m4.1.1.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.4.m4.1.1">1</cn><cn id="S3.SS1.SSS3.1.p1.4.m4.2.2.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.4.m4.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.4.m4.2c">i=1,2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.4.m4.2d">italic_i = 1 , 2</annotation></semantics></math>. In particular, <math alttext="\gamma_{2}\subset C_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.5.m5.1"><semantics id="S3.SS1.SSS3.1.p1.5.m5.1a"><mrow id="S3.SS1.SSS3.1.p1.5.m5.1.1" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.cmml"><msub id="S3.SS1.SSS3.1.p1.5.m5.1.1.2" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2.cmml"><mi id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.2" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2.2.cmml">γ</mi><mn id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.3" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2.3.cmml">2</mn></msub><mo id="S3.SS1.SSS3.1.p1.5.m5.1.1.1" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.1.cmml">⊂</mo><msub id="S3.SS1.SSS3.1.p1.5.m5.1.1.3" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3.cmml"><mi id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.2" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.3" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.5.m5.1b"><apply id="S3.SS1.SSS3.1.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1"><subset id="S3.SS1.SSS3.1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.1"></subset><apply id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2.2">𝛾</ci><cn id="S3.SS1.SSS3.1.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.2.3">2</cn></apply><apply id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.1.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.2.cmml" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.5.m5.1c">\gamma_{2}\subset C_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.5.m5.1d">italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊂ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and since <math alttext="C_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.6.m6.1"><semantics id="S3.SS1.SSS3.1.p1.6.m6.1a"><msub id="S3.SS1.SSS3.1.p1.6.m6.1.1" xref="S3.SS1.SSS3.1.p1.6.m6.1.1.cmml"><mi id="S3.SS1.SSS3.1.p1.6.m6.1.1.2" xref="S3.SS1.SSS3.1.p1.6.m6.1.1.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.6.m6.1.1.3" xref="S3.SS1.SSS3.1.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.6.m6.1b"><apply id="S3.SS1.SSS3.1.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS3.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.6.m6.1.1.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.6.m6.1c">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.6.m6.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is homeomorphic to a closed disk, it is simply connected and therefore also <math alttext="C_{2}\subseteq C_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.7.m7.1"><semantics id="S3.SS1.SSS3.1.p1.7.m7.1a"><mrow id="S3.SS1.SSS3.1.p1.7.m7.1.1" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.cmml"><msub id="S3.SS1.SSS3.1.p1.7.m7.1.1.2" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2.cmml"><mi id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.2" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.3" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2.3.cmml">2</mn></msub><mo id="S3.SS1.SSS3.1.p1.7.m7.1.1.1" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.1.cmml">⊆</mo><msub id="S3.SS1.SSS3.1.p1.7.m7.1.1.3" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3.cmml"><mi id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.2" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.3" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.7.m7.1b"><apply id="S3.SS1.SSS3.1.p1.7.m7.1.1.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1"><subset id="S3.SS1.SSS3.1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.1"></subset><apply id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.1.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.2.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.2.3">2</cn></apply><apply id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.1.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.2.cmml" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.7.m7.1c">C_{2}\subseteq C_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.7.m7.1d">italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊆ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Identically, one can show <math alttext="C_{1}\subseteq C_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.8.m8.1"><semantics id="S3.SS1.SSS3.1.p1.8.m8.1a"><mrow id="S3.SS1.SSS3.1.p1.8.m8.1.1" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.cmml"><msub id="S3.SS1.SSS3.1.p1.8.m8.1.1.2" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2.cmml"><mi id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.2" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.3" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS3.1.p1.8.m8.1.1.1" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.1.cmml">⊆</mo><msub id="S3.SS1.SSS3.1.p1.8.m8.1.1.3" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3.cmml"><mi id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.2" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.3" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.8.m8.1b"><apply id="S3.SS1.SSS3.1.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1"><subset id="S3.SS1.SSS3.1.p1.8.m8.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.1"></subset><apply id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.1.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.2.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.2.3">1</cn></apply><apply id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.1.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.2.cmml" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.8.m8.1c">C_{1}\subseteq C_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.8.m8.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. This means that <math alttext="C_{1}=C_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.9.m9.1"><semantics id="S3.SS1.SSS3.1.p1.9.m9.1a"><mrow id="S3.SS1.SSS3.1.p1.9.m9.1.1" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.cmml"><msub id="S3.SS1.SSS3.1.p1.9.m9.1.1.2" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2.cmml"><mi id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.2" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.3" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS3.1.p1.9.m9.1.1.1" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS3.1.p1.9.m9.1.1.3" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3.cmml"><mi id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.2" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3.2.cmml">C</mi><mn id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.3" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.9.m9.1b"><apply id="S3.SS1.SSS3.1.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1"><eq id="S3.SS1.SSS3.1.p1.9.m9.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.1"></eq><apply id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.2.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.9.m9.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.2.3">1</cn></apply><apply id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.2.cmml" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3.2">𝐶</ci><cn id="S3.SS1.SSS3.1.p1.9.m9.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.9.m9.1c">C_{1}=C_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.9.m9.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and thus <math alttext="\gamma_{1}=\gamma_{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.1.p1.10.m10.1"><semantics id="S3.SS1.SSS3.1.p1.10.m10.1a"><mrow id="S3.SS1.SSS3.1.p1.10.m10.1.1" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.cmml"><msub id="S3.SS1.SSS3.1.p1.10.m10.1.1.2" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2.cmml"><mi id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.2" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2.2.cmml">γ</mi><mn id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.3" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS3.1.p1.10.m10.1.1.1" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS3.1.p1.10.m10.1.1.3" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3.cmml"><mi id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.2" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3.2.cmml">γ</mi><mn id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.3" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.1.p1.10.m10.1b"><apply id="S3.SS1.SSS3.1.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1"><eq id="S3.SS1.SSS3.1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.1"></eq><apply id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.1.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.2.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2.2">𝛾</ci><cn id="S3.SS1.SSS3.1.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.2.3">1</cn></apply><apply id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3.2">𝛾</ci><cn id="S3.SS1.SSS3.1.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.1.p1.10.m10.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.1.p1.10.m10.1c">\gamma_{1}=\gamma_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.1.p1.10.m10.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which is a contradiction. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS3.p2"> <p class="ltx_p" id="S3.SS1.SSS3.p2.2">For a general polygon <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.p2.1.m1.1"><semantics id="S3.SS1.SSS3.p2.1.m1.1a"><mi id="S3.SS1.SSS3.p2.1.m1.1.1" xref="S3.SS1.SSS3.p2.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.p2.1.m1.1b"><ci id="S3.SS1.SSS3.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS3.p2.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.p2.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.p2.1.m1.1d">italic_P</annotation></semantics></math>, multiple boundary components may intersect in a common vertex. The following lemma provides a way to describe the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.p2.2.m2.1"><semantics id="S3.SS1.SSS3.p2.2.m2.1a"><mi id="S3.SS1.SSS3.p2.2.m2.1.1" xref="S3.SS1.SSS3.p2.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.p2.2.m2.1b"><ci id="S3.SS1.SSS3.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS3.p2.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.p2.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.p2.2.m2.1d">italic_P</annotation></semantics></math>-side of a vertex in terms of simpler components.</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 class="ltx_p" id="S3.Thmtheorem11.p1.14"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem11.p1.14.14">Let <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.1.1.m1.1"><semantics id="S3.Thmtheorem11.p1.1.1.m1.1a"><mi id="S3.Thmtheorem11.p1.1.1.m1.1.1" xref="S3.Thmtheorem11.p1.1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.1.1.m1.1b"><ci id="S3.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.1.1.m1.1d">italic_P</annotation></semantics></math> be a polygon, and <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.2.2.m2.1"><semantics id="S3.Thmtheorem11.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem11.p1.2.2.m2.1.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.cmml"><mi id="S3.Thmtheorem11.p1.2.2.m2.1.2.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.2.cmml">v</mi><mo id="S3.Thmtheorem11.p1.2.2.m2.1.2.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem11.p1.2.2.m2.1.2.3" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.cmml"><mi id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.2.cmml">V</mi><mo id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.3.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.cmml"><mo id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem11.p1.2.2.m2.1.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml">P</mi><mo id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.2.2.m2.1b"><apply id="S3.Thmtheorem11.p1.2.2.m2.1.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2"><in id="S3.Thmtheorem11.p1.2.2.m2.1.2.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.1"></in><ci id="S3.Thmtheorem11.p1.2.2.m2.1.2.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.2">𝑣</ci><apply id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3"><times id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.1"></times><ci id="S3.Thmtheorem11.p1.2.2.m2.1.2.3.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.2.3.2">𝑉</ci><ci id="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.2.2.m2.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.2.2.m2.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math>. Let <math alttext="\gamma_{1},\dots,\gamma_{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.3.3.m3.3"><semantics id="S3.Thmtheorem11.p1.3.3.m3.3a"><mrow id="S3.Thmtheorem11.p1.3.3.m3.3.3.2" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.3.cmml"><msub id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2.cmml">γ</mi><mn id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.3" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.3.cmml">,</mo><mi id="S3.Thmtheorem11.p1.3.3.m3.1.1" mathvariant="normal" xref="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml">…</mi><mo id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.4" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.3.cmml">,</mo><msub id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.cmml"><mi id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.2" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.2.cmml">γ</mi><mi id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.3" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.3.3.m3.3b"><list id="S3.Thmtheorem11.p1.3.3.m3.3.3.3.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2"><apply id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2">𝛾</ci><cn id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3">1</cn></apply><ci id="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.1.1">…</ci><apply id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2">subscript</csymbol><ci id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.2.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.2">𝛾</ci><ci id="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.3.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.3.3.2.2.3">𝑑</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.3.3.m3.3c">\gamma_{1},\dots,\gamma_{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.3.3.m3.3d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> be the <math alttext="d\in\mathds{N}" 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">d</mi><mo id="S3.Thmtheorem11.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem11.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.3.cmml">ℕ</mi></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"><in id="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.2">𝑑</ci><ci id="S3.Thmtheorem11.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.4.4.m4.1c">d\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.4.4.m4.1d">italic_d ∈ blackboard_N</annotation></semantics></math> boundary components of <math alttext="P" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.5.5.m5.1"><semantics id="S3.Thmtheorem11.p1.5.5.m5.1a"><mi id="S3.Thmtheorem11.p1.5.5.m5.1.1" xref="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.5.5.m5.1b"><ci id="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.5.5.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.5.5.m5.1d">italic_P</annotation></semantics></math> that contain <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.6.6.m6.1"><semantics id="S3.Thmtheorem11.p1.6.6.m6.1a"><mi id="S3.Thmtheorem11.p1.6.6.m6.1.1" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.6.6.m6.1b"><ci id="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.6.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.6.6.m6.1d">italic_v</annotation></semantics></math>, i.e. <math alttext="v\in V(\gamma_{k})" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.7.7.m7.1"><semantics id="S3.Thmtheorem11.p1.7.7.m7.1a"><mrow id="S3.Thmtheorem11.p1.7.7.m7.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem11.p1.7.7.m7.1.1.3" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.3.cmml">v</mi><mo id="S3.Thmtheorem11.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.2.cmml">∈</mo><mrow id="S3.Thmtheorem11.p1.7.7.m7.1.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.cmml"><mi id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.3" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.3.cmml">V</mi><mo id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.2" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.2" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.2.cmml">γ</mi><mi id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.3" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.7.7.m7.1b"><apply id="S3.Thmtheorem11.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1"><in id="S3.Thmtheorem11.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.2"></in><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.3">𝑣</ci><apply id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1"><times id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.2.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.2"></times><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.3.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.3">𝑉</ci><apply id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.2">𝛾</ci><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.7.7.m7.1c">v\in V(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.7.7.m7.1d">italic_v ∈ italic_V ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> for all <math alttext="k\in[d]" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.8.8.m8.1"><semantics id="S3.Thmtheorem11.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem11.p1.8.8.m8.1.2" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.cmml"><mi id="S3.Thmtheorem11.p1.8.8.m8.1.2.2" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.2.cmml">k</mi><mo id="S3.Thmtheorem11.p1.8.8.m8.1.2.1" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem11.p1.8.8.m8.1.2.3.2" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.3.1.cmml"><mo id="S3.Thmtheorem11.p1.8.8.m8.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.3.1.1.cmml">[</mo><mi id="S3.Thmtheorem11.p1.8.8.m8.1.1" xref="S3.Thmtheorem11.p1.8.8.m8.1.1.cmml">d</mi><mo id="S3.Thmtheorem11.p1.8.8.m8.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.8.8.m8.1b"><apply id="S3.Thmtheorem11.p1.8.8.m8.1.2.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.2"><in id="S3.Thmtheorem11.p1.8.8.m8.1.2.1.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.1"></in><ci id="S3.Thmtheorem11.p1.8.8.m8.1.2.2.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.2">𝑘</ci><apply id="S3.Thmtheorem11.p1.8.8.m8.1.2.3.1.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.3.2"><csymbol cd="latexml" id="S3.Thmtheorem11.p1.8.8.m8.1.2.3.1.1.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem11.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem11.p1.8.8.m8.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.8.8.m8.1c">k\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.8.8.m8.1d">italic_k ∈ [ italic_d ]</annotation></semantics></math>. Assume that there are polygons <math alttext="P_{1},\dots,P_{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.9.9.m9.3"><semantics id="S3.Thmtheorem11.p1.9.9.m9.3a"><mrow id="S3.Thmtheorem11.p1.9.9.m9.3.3.2" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.3.cmml"><msub id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.2" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.2.cmml">P</mi><mn id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.3" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.3" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.3.cmml">,</mo><mi id="S3.Thmtheorem11.p1.9.9.m9.1.1" mathvariant="normal" xref="S3.Thmtheorem11.p1.9.9.m9.1.1.cmml">…</mi><mo id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.4" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.3.cmml">,</mo><msub id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.cmml"><mi id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.2" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.2.cmml">P</mi><mi id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.3" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.9.9.m9.3b"><list id="S3.Thmtheorem11.p1.9.9.m9.3.3.3.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2"><apply id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.2">𝑃</ci><cn id="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.9.9.m9.2.2.1.1.3">1</cn></apply><ci id="S3.Thmtheorem11.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.1.1">…</ci><apply id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.1.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2">subscript</csymbol><ci id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.2.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.2">𝑃</ci><ci id="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.3.cmml" xref="S3.Thmtheorem11.p1.9.9.m9.3.3.2.2.3">𝑑</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.9.9.m9.3c">P_{1},\dots,P_{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.9.9.m9.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.10.10.m10.1"><semantics id="S3.Thmtheorem11.p1.10.10.m10.1a"><msub id="S3.Thmtheorem11.p1.10.10.m10.1.1" xref="S3.Thmtheorem11.p1.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem11.p1.10.10.m10.1.1.2" xref="S3.Thmtheorem11.p1.10.10.m10.1.1.2.cmml">γ</mi><mi id="S3.Thmtheorem11.p1.10.10.m10.1.1.3" xref="S3.Thmtheorem11.p1.10.10.m10.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.10.10.m10.1b"><apply id="S3.Thmtheorem11.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem11.p1.10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem11.p1.10.10.m10.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem11.p1.10.10.m10.1.1.2">𝛾</ci><ci id="S3.Thmtheorem11.p1.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem11.p1.10.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.10.10.m10.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.10.10.m10.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is the only boundary component of <math alttext="P_{k}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.11.11.m11.1"><semantics id="S3.Thmtheorem11.p1.11.11.m11.1a"><msub id="S3.Thmtheorem11.p1.11.11.m11.1.1" xref="S3.Thmtheorem11.p1.11.11.m11.1.1.cmml"><mi id="S3.Thmtheorem11.p1.11.11.m11.1.1.2" xref="S3.Thmtheorem11.p1.11.11.m11.1.1.2.cmml">P</mi><mi id="S3.Thmtheorem11.p1.11.11.m11.1.1.3" xref="S3.Thmtheorem11.p1.11.11.m11.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.11.11.m11.1b"><apply id="S3.Thmtheorem11.p1.11.11.m11.1.1.cmml" xref="S3.Thmtheorem11.p1.11.11.m11.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem11.p1.11.11.m11.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem11.p1.11.11.m11.1.1.2">𝑃</ci><ci id="S3.Thmtheorem11.p1.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem11.p1.11.11.m11.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.11.11.m11.1c">P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.11.11.m11.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> that contains <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.12.12.m12.1"><semantics id="S3.Thmtheorem11.p1.12.12.m12.1a"><mi id="S3.Thmtheorem11.p1.12.12.m12.1.1" xref="S3.Thmtheorem11.p1.12.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.12.12.m12.1b"><ci id="S3.Thmtheorem11.p1.12.12.m12.1.1.cmml" xref="S3.Thmtheorem11.p1.12.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.12.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.12.12.m12.1d">italic_v</annotation></semantics></math>. If <math alttext="P\subseteq P_{k}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.13.13.m13.1"><semantics id="S3.Thmtheorem11.p1.13.13.m13.1a"><mrow id="S3.Thmtheorem11.p1.13.13.m13.1.1" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.cmml"><mi id="S3.Thmtheorem11.p1.13.13.m13.1.1.2" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.2.cmml">P</mi><mo id="S3.Thmtheorem11.p1.13.13.m13.1.1.1" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.1.cmml">⊆</mo><msub id="S3.Thmtheorem11.p1.13.13.m13.1.1.3" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3.cmml"><mi id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.2" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3.2.cmml">P</mi><mi id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.3" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.13.13.m13.1b"><apply id="S3.Thmtheorem11.p1.13.13.m13.1.1.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1"><subset id="S3.Thmtheorem11.p1.13.13.m13.1.1.1.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.1"></subset><ci id="S3.Thmtheorem11.p1.13.13.m13.1.1.2.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.2">𝑃</ci><apply id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.1.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.2.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3.2">𝑃</ci><ci id="S3.Thmtheorem11.p1.13.13.m13.1.1.3.3.cmml" xref="S3.Thmtheorem11.p1.13.13.m13.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.13.13.m13.1c">P\subseteq P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.13.13.m13.1d">italic_P ⊆ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="k\in[d]" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.14.14.m14.1"><semantics id="S3.Thmtheorem11.p1.14.14.m14.1a"><mrow id="S3.Thmtheorem11.p1.14.14.m14.1.2" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.cmml"><mi id="S3.Thmtheorem11.p1.14.14.m14.1.2.2" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.2.cmml">k</mi><mo id="S3.Thmtheorem11.p1.14.14.m14.1.2.1" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.1.cmml">∈</mo><mrow id="S3.Thmtheorem11.p1.14.14.m14.1.2.3.2" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.3.1.cmml"><mo id="S3.Thmtheorem11.p1.14.14.m14.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.3.1.1.cmml">[</mo><mi id="S3.Thmtheorem11.p1.14.14.m14.1.1" xref="S3.Thmtheorem11.p1.14.14.m14.1.1.cmml">d</mi><mo id="S3.Thmtheorem11.p1.14.14.m14.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.14.14.m14.1b"><apply id="S3.Thmtheorem11.p1.14.14.m14.1.2.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.2"><in id="S3.Thmtheorem11.p1.14.14.m14.1.2.1.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.1"></in><ci id="S3.Thmtheorem11.p1.14.14.m14.1.2.2.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.2">𝑘</ci><apply id="S3.Thmtheorem11.p1.14.14.m14.1.2.3.1.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.3.2"><csymbol cd="latexml" id="S3.Thmtheorem11.p1.14.14.m14.1.2.3.1.1.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem11.p1.14.14.m14.1.1.cmml" xref="S3.Thmtheorem11.p1.14.14.m14.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.14.14.m14.1c">k\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.14.14.m14.1d">italic_k ∈ [ italic_d ]</annotation></semantics></math>, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E29"> <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_{k\in[d]}\mathds{1}_{Q_{P_{k}}^{v}}=d-1+\mathds{1}_{Q^{v}_{P}}." class="ltx_Math" display="block" id="S3.E29.m1.2"><semantics id="S3.E29.m1.2a"><mrow id="S3.E29.m1.2.2.1" xref="S3.E29.m1.2.2.1.1.cmml"><mrow id="S3.E29.m1.2.2.1.1" xref="S3.E29.m1.2.2.1.1.cmml"><mrow id="S3.E29.m1.2.2.1.1.2" xref="S3.E29.m1.2.2.1.1.2.cmml"><munder id="S3.E29.m1.2.2.1.1.2.1" xref="S3.E29.m1.2.2.1.1.2.1.cmml"><mo id="S3.E29.m1.2.2.1.1.2.1.2" movablelimits="false" xref="S3.E29.m1.2.2.1.1.2.1.2.cmml">∑</mo><mrow id="S3.E29.m1.1.1.1" xref="S3.E29.m1.1.1.1.cmml"><mi id="S3.E29.m1.1.1.1.3" xref="S3.E29.m1.1.1.1.3.cmml">k</mi><mo id="S3.E29.m1.1.1.1.2" xref="S3.E29.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E29.m1.1.1.1.4.2" xref="S3.E29.m1.1.1.1.4.1.cmml"><mo id="S3.E29.m1.1.1.1.4.2.1" stretchy="false" xref="S3.E29.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S3.E29.m1.1.1.1.1" xref="S3.E29.m1.1.1.1.1.cmml">d</mi><mo id="S3.E29.m1.1.1.1.4.2.2" stretchy="false" xref="S3.E29.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><msub id="S3.E29.m1.2.2.1.1.2.2" xref="S3.E29.m1.2.2.1.1.2.2.cmml"><mn id="S3.E29.m1.2.2.1.1.2.2.2" xref="S3.E29.m1.2.2.1.1.2.2.2.cmml">𝟙</mn><msubsup id="S3.E29.m1.2.2.1.1.2.2.3" xref="S3.E29.m1.2.2.1.1.2.2.3.cmml"><mi id="S3.E29.m1.2.2.1.1.2.2.3.2.2" xref="S3.E29.m1.2.2.1.1.2.2.3.2.2.cmml">Q</mi><msub id="S3.E29.m1.2.2.1.1.2.2.3.2.3" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3.cmml"><mi id="S3.E29.m1.2.2.1.1.2.2.3.2.3.2" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3.2.cmml">P</mi><mi id="S3.E29.m1.2.2.1.1.2.2.3.2.3.3" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3.3.cmml">k</mi></msub><mi id="S3.E29.m1.2.2.1.1.2.2.3.3" xref="S3.E29.m1.2.2.1.1.2.2.3.3.cmml">v</mi></msubsup></msub></mrow><mo id="S3.E29.m1.2.2.1.1.1" xref="S3.E29.m1.2.2.1.1.1.cmml">=</mo><mrow id="S3.E29.m1.2.2.1.1.3" xref="S3.E29.m1.2.2.1.1.3.cmml"><mrow id="S3.E29.m1.2.2.1.1.3.2" xref="S3.E29.m1.2.2.1.1.3.2.cmml"><mi id="S3.E29.m1.2.2.1.1.3.2.2" xref="S3.E29.m1.2.2.1.1.3.2.2.cmml">d</mi><mo id="S3.E29.m1.2.2.1.1.3.2.1" xref="S3.E29.m1.2.2.1.1.3.2.1.cmml">−</mo><mn id="S3.E29.m1.2.2.1.1.3.2.3" xref="S3.E29.m1.2.2.1.1.3.2.3.cmml">1</mn></mrow><mo id="S3.E29.m1.2.2.1.1.3.1" xref="S3.E29.m1.2.2.1.1.3.1.cmml">+</mo><msub id="S3.E29.m1.2.2.1.1.3.3" xref="S3.E29.m1.2.2.1.1.3.3.cmml"><mn id="S3.E29.m1.2.2.1.1.3.3.2" xref="S3.E29.m1.2.2.1.1.3.3.2.cmml">𝟙</mn><msubsup id="S3.E29.m1.2.2.1.1.3.3.3" xref="S3.E29.m1.2.2.1.1.3.3.3.cmml"><mi id="S3.E29.m1.2.2.1.1.3.3.3.2.2" xref="S3.E29.m1.2.2.1.1.3.3.3.2.2.cmml">Q</mi><mi id="S3.E29.m1.2.2.1.1.3.3.3.3" xref="S3.E29.m1.2.2.1.1.3.3.3.3.cmml">P</mi><mi id="S3.E29.m1.2.2.1.1.3.3.3.2.3" xref="S3.E29.m1.2.2.1.1.3.3.3.2.3.cmml">v</mi></msubsup></msub></mrow></mrow><mo id="S3.E29.m1.2.2.1.2" lspace="0em" xref="S3.E29.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E29.m1.2b"><apply id="S3.E29.m1.2.2.1.1.cmml" xref="S3.E29.m1.2.2.1"><eq id="S3.E29.m1.2.2.1.1.1.cmml" xref="S3.E29.m1.2.2.1.1.1"></eq><apply 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xref="S3.E29.m1.2.2.1.1.2.2.2">1</cn><apply id="S3.E29.m1.2.2.1.1.2.2.3.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.2.2.3.1.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3">superscript</csymbol><apply id="S3.E29.m1.2.2.1.1.2.2.3.2.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.2.2.3.2.1.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3">subscript</csymbol><ci id="S3.E29.m1.2.2.1.1.2.2.3.2.2.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.2.2">𝑄</ci><apply id="S3.E29.m1.2.2.1.1.2.2.3.2.3.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.2.2.3.2.3.1.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3">subscript</csymbol><ci id="S3.E29.m1.2.2.1.1.2.2.3.2.3.2.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3.2">𝑃</ci><ci id="S3.E29.m1.2.2.1.1.2.2.3.2.3.3.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.2.3.3">𝑘</ci></apply></apply><ci id="S3.E29.m1.2.2.1.1.2.2.3.3.cmml" xref="S3.E29.m1.2.2.1.1.2.2.3.3">𝑣</ci></apply></apply></apply><apply id="S3.E29.m1.2.2.1.1.3.cmml" xref="S3.E29.m1.2.2.1.1.3"><plus id="S3.E29.m1.2.2.1.1.3.1.cmml" xref="S3.E29.m1.2.2.1.1.3.1"></plus><apply id="S3.E29.m1.2.2.1.1.3.2.cmml" xref="S3.E29.m1.2.2.1.1.3.2"><minus id="S3.E29.m1.2.2.1.1.3.2.1.cmml" xref="S3.E29.m1.2.2.1.1.3.2.1"></minus><ci id="S3.E29.m1.2.2.1.1.3.2.2.cmml" xref="S3.E29.m1.2.2.1.1.3.2.2">𝑑</ci><cn id="S3.E29.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S3.E29.m1.2.2.1.1.3.2.3">1</cn></apply><apply id="S3.E29.m1.2.2.1.1.3.3.cmml" xref="S3.E29.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.3.3.1.cmml" xref="S3.E29.m1.2.2.1.1.3.3">subscript</csymbol><cn id="S3.E29.m1.2.2.1.1.3.3.2.cmml" type="integer" xref="S3.E29.m1.2.2.1.1.3.3.2">1</cn><apply id="S3.E29.m1.2.2.1.1.3.3.3.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.3.3.3.1.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3">subscript</csymbol><apply id="S3.E29.m1.2.2.1.1.3.3.3.2.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.E29.m1.2.2.1.1.3.3.3.2.1.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3">superscript</csymbol><ci id="S3.E29.m1.2.2.1.1.3.3.3.2.2.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3.2.2">𝑄</ci><ci id="S3.E29.m1.2.2.1.1.3.3.3.2.3.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3.2.3">𝑣</ci></apply><ci id="S3.E29.m1.2.2.1.1.3.3.3.3.cmml" xref="S3.E29.m1.2.2.1.1.3.3.3.3">𝑃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E29.m1.2c">\sum_{k\in[d]}\mathds{1}_{Q_{P_{k}}^{v}}=d-1+\mathds{1}_{Q^{v}_{P}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E29.m1.2d">∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_d ] end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_d - 1 + blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT 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">(29)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS3.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <figure class="ltx_figure" id="S3.F9"> <table class="ltx_tabular ltx_align_middle" id="S3.F9.4"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.F9.4.4"> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F9.1.1.1" style="width:96.7pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F9.1.1.1.1"> <span class="ltx_p" id="S3.F9.1.1.1.1.1"><foreignobject height="63.6pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="63.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="123" id="S3.F9.1.1.1.1.1.1.g1" src="x22.png" width="122"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F9.2.2.2" style="width:74.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F9.2.2.2.1"> <span class="ltx_p" id="S3.F9.2.2.2.1.1"><foreignobject height="41.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="41.2pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="80" id="S3.F9.2.2.2.1.1.1.g1" src="x23.png" width="80"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F9.3.3.3" style="width:74.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F9.3.3.3.1"> <span class="ltx_p" id="S3.F9.3.3.3.1.1"><foreignobject height="47.0pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="44.1pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="90" id="S3.F9.3.3.3.1.1.1.g1" src="x24.png" width="85"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S3.F9.4.4.4" style="width:74.0pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.F9.4.4.4.1"> <span class="ltx_p" id="S3.F9.4.4.4.1.1"><foreignobject height="44.1pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="42.6pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="85" id="S3.F9.4.4.4.1.1.1.g1" src="x25.png" width="82"/></foreignobject></span> </span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 9: </span> Three boundary components <math alttext="\gamma_{1},\gamma_{2},\gamma_{3}" class="ltx_Math" display="inline" id="S3.F9.14.m1.3"><semantics id="S3.F9.14.m1.3b"><mrow id="S3.F9.14.m1.3.3.3" xref="S3.F9.14.m1.3.3.4.cmml"><msub id="S3.F9.14.m1.1.1.1.1" xref="S3.F9.14.m1.1.1.1.1.cmml"><mi id="S3.F9.14.m1.1.1.1.1.2" xref="S3.F9.14.m1.1.1.1.1.2.cmml">γ</mi><mn id="S3.F9.14.m1.1.1.1.1.3" xref="S3.F9.14.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.F9.14.m1.3.3.3.4" xref="S3.F9.14.m1.3.3.4.cmml">,</mo><msub id="S3.F9.14.m1.2.2.2.2" xref="S3.F9.14.m1.2.2.2.2.cmml"><mi id="S3.F9.14.m1.2.2.2.2.2" xref="S3.F9.14.m1.2.2.2.2.2.cmml">γ</mi><mn id="S3.F9.14.m1.2.2.2.2.3" xref="S3.F9.14.m1.2.2.2.2.3.cmml">2</mn></msub><mo id="S3.F9.14.m1.3.3.3.5" xref="S3.F9.14.m1.3.3.4.cmml">,</mo><msub id="S3.F9.14.m1.3.3.3.3" xref="S3.F9.14.m1.3.3.3.3.cmml"><mi id="S3.F9.14.m1.3.3.3.3.2" xref="S3.F9.14.m1.3.3.3.3.2.cmml">γ</mi><mn id="S3.F9.14.m1.3.3.3.3.3" xref="S3.F9.14.m1.3.3.3.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F9.14.m1.3c"><list id="S3.F9.14.m1.3.3.4.cmml" xref="S3.F9.14.m1.3.3.3"><apply id="S3.F9.14.m1.1.1.1.1.cmml" xref="S3.F9.14.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.F9.14.m1.1.1.1.1.1.cmml" xref="S3.F9.14.m1.1.1.1.1">subscript</csymbol><ci id="S3.F9.14.m1.1.1.1.1.2.cmml" xref="S3.F9.14.m1.1.1.1.1.2">𝛾</ci><cn id="S3.F9.14.m1.1.1.1.1.3.cmml" type="integer" xref="S3.F9.14.m1.1.1.1.1.3">1</cn></apply><apply id="S3.F9.14.m1.2.2.2.2.cmml" xref="S3.F9.14.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.F9.14.m1.2.2.2.2.1.cmml" xref="S3.F9.14.m1.2.2.2.2">subscript</csymbol><ci id="S3.F9.14.m1.2.2.2.2.2.cmml" xref="S3.F9.14.m1.2.2.2.2.2">𝛾</ci><cn id="S3.F9.14.m1.2.2.2.2.3.cmml" type="integer" xref="S3.F9.14.m1.2.2.2.2.3">2</cn></apply><apply id="S3.F9.14.m1.3.3.3.3.cmml" xref="S3.F9.14.m1.3.3.3.3"><csymbol cd="ambiguous" id="S3.F9.14.m1.3.3.3.3.1.cmml" xref="S3.F9.14.m1.3.3.3.3">subscript</csymbol><ci id="S3.F9.14.m1.3.3.3.3.2.cmml" xref="S3.F9.14.m1.3.3.3.3.2">𝛾</ci><cn id="S3.F9.14.m1.3.3.3.3.3.cmml" type="integer" xref="S3.F9.14.m1.3.3.3.3.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.14.m1.3d">\gamma_{1},\gamma_{2},\gamma_{3}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.14.m1.3e">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="P" class="ltx_Math" display="inline" id="S3.F9.15.m2.1"><semantics id="S3.F9.15.m2.1b"><mi id="S3.F9.15.m2.1.1" xref="S3.F9.15.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F9.15.m2.1c"><ci id="S3.F9.15.m2.1.1.cmml" xref="S3.F9.15.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.15.m2.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F9.15.m2.1e">italic_P</annotation></semantics></math> (grey) intersect at vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.F9.16.m3.1"><semantics id="S3.F9.16.m3.1b"><mi id="S3.F9.16.m3.1.1" xref="S3.F9.16.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.F9.16.m3.1c"><ci id="S3.F9.16.m3.1.1.cmml" xref="S3.F9.16.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.16.m3.1d">v</annotation><annotation encoding="application/x-llamapun" id="S3.F9.16.m3.1e">italic_v</annotation></semantics></math>, thus the <math alttext="P" class="ltx_Math" display="inline" id="S3.F9.17.m4.1"><semantics id="S3.F9.17.m4.1b"><mi id="S3.F9.17.m4.1.1" xref="S3.F9.17.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F9.17.m4.1c"><ci id="S3.F9.17.m4.1.1.cmml" xref="S3.F9.17.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.17.m4.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F9.17.m4.1e">italic_P</annotation></semantics></math>-side <math alttext="Q^{v}_{P}" class="ltx_Math" display="inline" id="S3.F9.18.m5.1"><semantics id="S3.F9.18.m5.1b"><msubsup id="S3.F9.18.m5.1.1" xref="S3.F9.18.m5.1.1.cmml"><mi id="S3.F9.18.m5.1.1.2.2" xref="S3.F9.18.m5.1.1.2.2.cmml">Q</mi><mi id="S3.F9.18.m5.1.1.3" xref="S3.F9.18.m5.1.1.3.cmml">P</mi><mi id="S3.F9.18.m5.1.1.2.3" xref="S3.F9.18.m5.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.F9.18.m5.1c"><apply id="S3.F9.18.m5.1.1.cmml" xref="S3.F9.18.m5.1.1"><csymbol cd="ambiguous" id="S3.F9.18.m5.1.1.1.cmml" xref="S3.F9.18.m5.1.1">subscript</csymbol><apply id="S3.F9.18.m5.1.1.2.cmml" xref="S3.F9.18.m5.1.1"><csymbol cd="ambiguous" id="S3.F9.18.m5.1.1.2.1.cmml" xref="S3.F9.18.m5.1.1">superscript</csymbol><ci id="S3.F9.18.m5.1.1.2.2.cmml" xref="S3.F9.18.m5.1.1.2.2">𝑄</ci><ci id="S3.F9.18.m5.1.1.2.3.cmml" xref="S3.F9.18.m5.1.1.2.3">𝑣</ci></apply><ci id="S3.F9.18.m5.1.1.3.cmml" xref="S3.F9.18.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.18.m5.1d">Q^{v}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.18.m5.1e">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> (dotted), as well as its complement, are split into three circular sectors. In the interior of the dashed circle, <math alttext="Q^{v}_{P}" class="ltx_Math" display="inline" id="S3.F9.19.m6.1"><semantics id="S3.F9.19.m6.1b"><msubsup id="S3.F9.19.m6.1.1" xref="S3.F9.19.m6.1.1.cmml"><mi id="S3.F9.19.m6.1.1.2.2" xref="S3.F9.19.m6.1.1.2.2.cmml">Q</mi><mi id="S3.F9.19.m6.1.1.3" xref="S3.F9.19.m6.1.1.3.cmml">P</mi><mi id="S3.F9.19.m6.1.1.2.3" xref="S3.F9.19.m6.1.1.2.3.cmml">v</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.F9.19.m6.1c"><apply id="S3.F9.19.m6.1.1.cmml" xref="S3.F9.19.m6.1.1"><csymbol cd="ambiguous" id="S3.F9.19.m6.1.1.1.cmml" xref="S3.F9.19.m6.1.1">subscript</csymbol><apply id="S3.F9.19.m6.1.1.2.cmml" xref="S3.F9.19.m6.1.1"><csymbol cd="ambiguous" id="S3.F9.19.m6.1.1.2.1.cmml" xref="S3.F9.19.m6.1.1">superscript</csymbol><ci id="S3.F9.19.m6.1.1.2.2.cmml" xref="S3.F9.19.m6.1.1.2.2">𝑄</ci><ci id="S3.F9.19.m6.1.1.2.3.cmml" xref="S3.F9.19.m6.1.1.2.3">𝑣</ci></apply><ci id="S3.F9.19.m6.1.1.3.cmml" xref="S3.F9.19.m6.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.19.m6.1d">Q^{v}_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.19.m6.1e">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P" class="ltx_Math" display="inline" id="S3.F9.20.m7.1"><semantics id="S3.F9.20.m7.1b"><mi id="S3.F9.20.m7.1.1" xref="S3.F9.20.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.F9.20.m7.1c"><ci id="S3.F9.20.m7.1.1.cmml" xref="S3.F9.20.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.20.m7.1d">P</annotation><annotation encoding="application/x-llamapun" id="S3.F9.20.m7.1e">italic_P</annotation></semantics></math> agree. Since <math alttext="P\subset P_{k}" class="ltx_Math" display="inline" id="S3.F9.21.m8.1"><semantics id="S3.F9.21.m8.1b"><mrow id="S3.F9.21.m8.1.1" xref="S3.F9.21.m8.1.1.cmml"><mi id="S3.F9.21.m8.1.1.2" xref="S3.F9.21.m8.1.1.2.cmml">P</mi><mo id="S3.F9.21.m8.1.1.1" xref="S3.F9.21.m8.1.1.1.cmml">⊂</mo><msub id="S3.F9.21.m8.1.1.3" xref="S3.F9.21.m8.1.1.3.cmml"><mi id="S3.F9.21.m8.1.1.3.2" xref="S3.F9.21.m8.1.1.3.2.cmml">P</mi><mi id="S3.F9.21.m8.1.1.3.3" xref="S3.F9.21.m8.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F9.21.m8.1c"><apply id="S3.F9.21.m8.1.1.cmml" xref="S3.F9.21.m8.1.1"><subset id="S3.F9.21.m8.1.1.1.cmml" xref="S3.F9.21.m8.1.1.1"></subset><ci id="S3.F9.21.m8.1.1.2.cmml" xref="S3.F9.21.m8.1.1.2">𝑃</ci><apply id="S3.F9.21.m8.1.1.3.cmml" xref="S3.F9.21.m8.1.1.3"><csymbol cd="ambiguous" id="S3.F9.21.m8.1.1.3.1.cmml" xref="S3.F9.21.m8.1.1.3">subscript</csymbol><ci id="S3.F9.21.m8.1.1.3.2.cmml" xref="S3.F9.21.m8.1.1.3.2">𝑃</ci><ci id="S3.F9.21.m8.1.1.3.3.cmml" xref="S3.F9.21.m8.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.21.m8.1d">P\subset P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.21.m8.1e">italic_P ⊂ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the two edges of component <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.F9.22.m9.1"><semantics id="S3.F9.22.m9.1b"><msub id="S3.F9.22.m9.1.1" xref="S3.F9.22.m9.1.1.cmml"><mi id="S3.F9.22.m9.1.1.2" xref="S3.F9.22.m9.1.1.2.cmml">γ</mi><mi id="S3.F9.22.m9.1.1.3" xref="S3.F9.22.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F9.22.m9.1c"><apply id="S3.F9.22.m9.1.1.cmml" xref="S3.F9.22.m9.1.1"><csymbol cd="ambiguous" id="S3.F9.22.m9.1.1.1.cmml" xref="S3.F9.22.m9.1.1">subscript</csymbol><ci id="S3.F9.22.m9.1.1.2.cmml" xref="S3.F9.22.m9.1.1.2">𝛾</ci><ci id="S3.F9.22.m9.1.1.3.cmml" xref="S3.F9.22.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F9.22.m9.1d">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.F9.22.m9.1e">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> must enclose a sector of the complement. </figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS3.2.p1"> <p class="ltx_p" id="S3.SS1.SSS3.2.p1.12">By the definition of the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.1.m1.1"><semantics id="S3.SS1.SSS3.2.p1.1.m1.1a"><mi id="S3.SS1.SSS3.2.p1.1.m1.1.1" xref="S3.SS1.SSS3.2.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.1.m1.1b"><ci id="S3.SS1.SSS3.2.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS3.2.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.1.m1.1d">italic_P</annotation></semantics></math>-side, it is sufficient to consider (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E29" title="Equation 29 ‣ Lemma 3.11. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">29</span></a>) on a disk around <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.2.m2.1"><semantics id="S3.SS1.SSS3.2.p1.2.m2.1a"><mi id="S3.SS1.SSS3.2.p1.2.m2.1.1" xref="S3.SS1.SSS3.2.p1.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.2.m2.1b"><ci id="S3.SS1.SSS3.2.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS3.2.p1.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.2.m2.1d">italic_v</annotation></semantics></math>. Let <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.3.m3.1"><semantics id="S3.SS1.SSS3.2.p1.3.m3.1a"><mi id="S3.SS1.SSS3.2.p1.3.m3.1.1" xref="S3.SS1.SSS3.2.p1.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.3.m3.1b"><ci id="S3.SS1.SSS3.2.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS3.2.p1.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.3.m3.1d">italic_D</annotation></semantics></math> be a small disk centered at <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.4.m4.1"><semantics id="S3.SS1.SSS3.2.p1.4.m4.1a"><mi id="S3.SS1.SSS3.2.p1.4.m4.1.1" xref="S3.SS1.SSS3.2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.4.m4.1b"><ci id="S3.SS1.SSS3.2.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS3.2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.4.m4.1d">italic_v</annotation></semantics></math> such that <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.5.m5.1"><semantics id="S3.SS1.SSS3.2.p1.5.m5.1a"><mi id="S3.SS1.SSS3.2.p1.5.m5.1.1" xref="S3.SS1.SSS3.2.p1.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.5.m5.1b"><ci id="S3.SS1.SSS3.2.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS3.2.p1.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.5.m5.1d">italic_D</annotation></semantics></math> intersects only those edges of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.6.m6.1"><semantics id="S3.SS1.SSS3.2.p1.6.m6.1a"><mi id="S3.SS1.SSS3.2.p1.6.m6.1.1" xref="S3.SS1.SSS3.2.p1.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.6.m6.1b"><ci id="S3.SS1.SSS3.2.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS3.2.p1.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.6.m6.1d">italic_P</annotation></semantics></math> and <math alttext="P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.7.m7.1"><semantics id="S3.SS1.SSS3.2.p1.7.m7.1a"><msub id="S3.SS1.SSS3.2.p1.7.m7.1.1" xref="S3.SS1.SSS3.2.p1.7.m7.1.1.cmml"><mi id="S3.SS1.SSS3.2.p1.7.m7.1.1.2" xref="S3.SS1.SSS3.2.p1.7.m7.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS3.2.p1.7.m7.1.1.3" xref="S3.SS1.SSS3.2.p1.7.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.7.m7.1b"><apply id="S3.SS1.SSS3.2.p1.7.m7.1.1.cmml" xref="S3.SS1.SSS3.2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.7.m7.1.1.1.cmml" xref="S3.SS1.SSS3.2.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.2.p1.7.m7.1.1.2.cmml" xref="S3.SS1.SSS3.2.p1.7.m7.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.2.p1.7.m7.1.1.3.cmml" xref="S3.SS1.SSS3.2.p1.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.7.m7.1c">P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.7.m7.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="k\in[d]" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.8.m8.1"><semantics id="S3.SS1.SSS3.2.p1.8.m8.1a"><mrow id="S3.SS1.SSS3.2.p1.8.m8.1.2" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.cmml"><mi id="S3.SS1.SSS3.2.p1.8.m8.1.2.2" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS3.2.p1.8.m8.1.2.1" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.2.p1.8.m8.1.2.3.2" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.3.1.cmml"><mo id="S3.SS1.SSS3.2.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.3.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.2.p1.8.m8.1.1" xref="S3.SS1.SSS3.2.p1.8.m8.1.1.cmml">d</mi><mo id="S3.SS1.SSS3.2.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.8.m8.1b"><apply id="S3.SS1.SSS3.2.p1.8.m8.1.2.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.2"><in id="S3.SS1.SSS3.2.p1.8.m8.1.2.1.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.1"></in><ci id="S3.SS1.SSS3.2.p1.8.m8.1.2.2.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.2">𝑘</ci><apply id="S3.SS1.SSS3.2.p1.8.m8.1.2.3.1.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.3.2"><csymbol cd="latexml" id="S3.SS1.SSS3.2.p1.8.m8.1.2.3.1.1.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.2.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS3.2.p1.8.m8.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.8.m8.1c">k\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.8.m8.1d">italic_k ∈ [ italic_d ]</annotation></semantics></math>, that contain <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.9.m9.1"><semantics id="S3.SS1.SSS3.2.p1.9.m9.1a"><mi id="S3.SS1.SSS3.2.p1.9.m9.1.1" xref="S3.SS1.SSS3.2.p1.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.9.m9.1b"><ci id="S3.SS1.SSS3.2.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS3.2.p1.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.9.m9.1d">italic_v</annotation></semantics></math>, cf. <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F9" title="Figure 9 ‣ Proof. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 9</span></a>. Then, by the local properties of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.10.m10.1"><semantics id="S3.SS1.SSS3.2.p1.10.m10.1a"><mi id="S3.SS1.SSS3.2.p1.10.m10.1.1" xref="S3.SS1.SSS3.2.p1.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.10.m10.1b"><ci id="S3.SS1.SSS3.2.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS3.2.p1.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.10.m10.1d">italic_P</annotation></semantics></math>-sides, we have <math alttext="Q^{v}_{P}\cap D=P\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.11.m11.1"><semantics id="S3.SS1.SSS3.2.p1.11.m11.1a"><mrow id="S3.SS1.SSS3.2.p1.11.m11.1.1" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.cmml"><mrow id="S3.SS1.SSS3.2.p1.11.m11.1.1.2" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.cmml"><msubsup id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.cmml"><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.2" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.2.cmml">Q</mi><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.3" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.3.cmml">P</mi><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.3" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.3.cmml">v</mi></msubsup><mo id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.1" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.3" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS3.2.p1.11.m11.1.1.1" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS3.2.p1.11.m11.1.1.3" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.cmml"><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.2" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.1" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.3" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.11.m11.1b"><apply id="S3.SS1.SSS3.2.p1.11.m11.1.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1"><eq id="S3.SS1.SSS3.2.p1.11.m11.1.1.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.1"></eq><apply id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2"><intersect id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.1"></intersect><apply id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2">subscript</csymbol><apply id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.2.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.2">𝑄</ci><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.3.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.2.3">𝑣</ci></apply><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.2.3">𝑃</ci></apply><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.2.3.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3"><intersect id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.1.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.1"></intersect><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.2.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.2.p1.11.m11.1.1.3.3.cmml" xref="S3.SS1.SSS3.2.p1.11.m11.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.11.m11.1c">Q^{v}_{P}\cap D=P\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.11.m11.1d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ∩ italic_D = italic_P ∩ italic_D</annotation></semantics></math> and <math alttext="Q^{v}_{P_{k}}\cap D=P_{k}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.2.p1.12.m12.1"><semantics id="S3.SS1.SSS3.2.p1.12.m12.1a"><mrow id="S3.SS1.SSS3.2.p1.12.m12.1.1" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.cmml"><mrow id="S3.SS1.SSS3.2.p1.12.m12.1.1.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.cmml"><msubsup id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.cmml"><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.2.cmml">Q</mi><msub id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.cmml"><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.3.cmml">k</mi></msub><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.3.cmml">v</mi></msubsup><mo id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.1" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS3.2.p1.12.m12.1.1.1" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS3.2.p1.12.m12.1.1.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.cmml"><msub id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.cmml"><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.2" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.1" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.3" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.2.p1.12.m12.1b"><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1"><eq id="S3.SS1.SSS3.2.p1.12.m12.1.1.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.1"></eq><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2"><intersect id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.1"></intersect><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2">subscript</csymbol><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2">superscript</csymbol><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.2">𝑄</ci><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.2.3">𝑣</ci></apply><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3">subscript</csymbol><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.2">𝑃</ci><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.2.3.3">𝑘</ci></apply></apply><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.2.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3"><intersect id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.1"></intersect><apply id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.1.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.2.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.2">𝑃</ci><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.2.p1.12.m12.1.1.3.3.cmml" xref="S3.SS1.SSS3.2.p1.12.m12.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.2.p1.12.m12.1c">Q^{v}_{P_{k}}\cap D=P_{k}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.2.p1.12.m12.1d">italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ italic_D = italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ italic_D</annotation></semantics></math>. Thus, it suffices to show that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E30"> <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_{k\in[d]}\mathds{1}_{P_{k}}(x)=d-1+\mathds{1}_{P}(x)\quad\forall x\in D." class="ltx_Math" display="block" id="S3.E30.m1.4"><semantics id="S3.E30.m1.4a"><mrow id="S3.E30.m1.4.4.1"><mrow id="S3.E30.m1.4.4.1.1.2" xref="S3.E30.m1.4.4.1.1.3.cmml"><mrow id="S3.E30.m1.4.4.1.1.1.1" xref="S3.E30.m1.4.4.1.1.1.1.cmml"><mrow id="S3.E30.m1.4.4.1.1.1.1.2" xref="S3.E30.m1.4.4.1.1.1.1.2.cmml"><munder id="S3.E30.m1.4.4.1.1.1.1.2.1" xref="S3.E30.m1.4.4.1.1.1.1.2.1.cmml"><mo id="S3.E30.m1.4.4.1.1.1.1.2.1.2" movablelimits="false" xref="S3.E30.m1.4.4.1.1.1.1.2.1.2.cmml">∑</mo><mrow id="S3.E30.m1.1.1.1" xref="S3.E30.m1.1.1.1.cmml"><mi id="S3.E30.m1.1.1.1.3" xref="S3.E30.m1.1.1.1.3.cmml">k</mi><mo id="S3.E30.m1.1.1.1.2" xref="S3.E30.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E30.m1.1.1.1.4.2" xref="S3.E30.m1.1.1.1.4.1.cmml"><mo id="S3.E30.m1.1.1.1.4.2.1" stretchy="false" xref="S3.E30.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S3.E30.m1.1.1.1.1" xref="S3.E30.m1.1.1.1.1.cmml">d</mi><mo id="S3.E30.m1.1.1.1.4.2.2" stretchy="false" xref="S3.E30.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="S3.E30.m1.4.4.1.1.1.1.2.2" xref="S3.E30.m1.4.4.1.1.1.1.2.2.cmml"><msub id="S3.E30.m1.4.4.1.1.1.1.2.2.2" xref="S3.E30.m1.4.4.1.1.1.1.2.2.2.cmml"><mn id="S3.E30.m1.4.4.1.1.1.1.2.2.2.2" xref="S3.E30.m1.4.4.1.1.1.1.2.2.2.2.cmml">𝟙</mn><msub id="S3.E30.m1.4.4.1.1.1.1.2.2.2.3" xref="S3.E30.m1.4.4.1.1.1.1.2.2.2.3.cmml"><mi id="S3.E30.m1.4.4.1.1.1.1.2.2.2.3.2" xref="S3.E30.m1.4.4.1.1.1.1.2.2.2.3.2.cmml">P</mi><mi id="S3.E30.m1.4.4.1.1.1.1.2.2.2.3.3" xref="S3.E30.m1.4.4.1.1.1.1.2.2.2.3.3.cmml">k</mi></msub></msub><mo id="S3.E30.m1.4.4.1.1.1.1.2.2.1" xref="S3.E30.m1.4.4.1.1.1.1.2.2.1.cmml">⁢</mo><mrow 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id="S3.E30.m1.4.4.1.1.1.1.3.3.2" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2.cmml"><mn id="S3.E30.m1.4.4.1.1.1.1.3.3.2.2" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2.2.cmml">𝟙</mn><mi id="S3.E30.m1.4.4.1.1.1.1.3.3.2.3" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2.3.cmml">P</mi></msub><mo id="S3.E30.m1.4.4.1.1.1.1.3.3.1" xref="S3.E30.m1.4.4.1.1.1.1.3.3.1.cmml">⁢</mo><mrow id="S3.E30.m1.4.4.1.1.1.1.3.3.3.2" xref="S3.E30.m1.4.4.1.1.1.1.3.3.cmml"><mo id="S3.E30.m1.4.4.1.1.1.1.3.3.3.2.1" stretchy="false" xref="S3.E30.m1.4.4.1.1.1.1.3.3.cmml">(</mo><mi id="S3.E30.m1.3.3" xref="S3.E30.m1.3.3.cmml">x</mi><mo id="S3.E30.m1.4.4.1.1.1.1.3.3.3.2.2" stretchy="false" xref="S3.E30.m1.4.4.1.1.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mspace id="S3.E30.m1.4.4.1.1.2.3" width="1.167em" xref="S3.E30.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S3.E30.m1.4.4.1.1.2.2" xref="S3.E30.m1.4.4.1.1.2.2.cmml"><mrow id="S3.E30.m1.4.4.1.1.2.2.2" xref="S3.E30.m1.4.4.1.1.2.2.2.cmml"><mo id="S3.E30.m1.4.4.1.1.2.2.2.1" rspace="0.167em" xref="S3.E30.m1.4.4.1.1.2.2.2.1.cmml">∀</mo><mi id="S3.E30.m1.4.4.1.1.2.2.2.2" xref="S3.E30.m1.4.4.1.1.2.2.2.2.cmml">x</mi></mrow><mo id="S3.E30.m1.4.4.1.1.2.2.1" xref="S3.E30.m1.4.4.1.1.2.2.1.cmml">∈</mo><mi id="S3.E30.m1.4.4.1.1.2.2.3" xref="S3.E30.m1.4.4.1.1.2.2.3.cmml">D</mi></mrow></mrow><mo id="S3.E30.m1.4.4.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E30.m1.4b"><apply id="S3.E30.m1.4.4.1.1.3.cmml" xref="S3.E30.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S3.E30.m1.4.4.1.1.3a.cmml" xref="S3.E30.m1.4.4.1.1.2.3">formulae-sequence</csymbol><apply id="S3.E30.m1.4.4.1.1.1.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1"><eq id="S3.E30.m1.4.4.1.1.1.1.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1.1"></eq><apply id="S3.E30.m1.4.4.1.1.1.1.2.cmml" xref="S3.E30.m1.4.4.1.1.1.1.2"><apply id="S3.E30.m1.4.4.1.1.1.1.2.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S3.E30.m1.4.4.1.1.1.1.2.1.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1.2.1">subscript</csymbol><sum 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id="S3.E30.m1.4.4.1.1.1.1.3.3.cmml" xref="S3.E30.m1.4.4.1.1.1.1.3.3"><times id="S3.E30.m1.4.4.1.1.1.1.3.3.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1.3.3.1"></times><apply id="S3.E30.m1.4.4.1.1.1.1.3.3.2.cmml" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.E30.m1.4.4.1.1.1.1.3.3.2.1.cmml" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2">subscript</csymbol><cn id="S3.E30.m1.4.4.1.1.1.1.3.3.2.2.cmml" type="integer" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2.2">1</cn><ci id="S3.E30.m1.4.4.1.1.1.1.3.3.2.3.cmml" xref="S3.E30.m1.4.4.1.1.1.1.3.3.2.3">𝑃</ci></apply><ci id="S3.E30.m1.3.3.cmml" xref="S3.E30.m1.3.3">𝑥</ci></apply></apply></apply><apply id="S3.E30.m1.4.4.1.1.2.2.cmml" xref="S3.E30.m1.4.4.1.1.2.2"><in id="S3.E30.m1.4.4.1.1.2.2.1.cmml" xref="S3.E30.m1.4.4.1.1.2.2.1"></in><apply id="S3.E30.m1.4.4.1.1.2.2.2.cmml" xref="S3.E30.m1.4.4.1.1.2.2.2"><csymbol cd="latexml" id="S3.E30.m1.4.4.1.1.2.2.2.1.cmml" xref="S3.E30.m1.4.4.1.1.2.2.2.1">for-all</csymbol><ci id="S3.E30.m1.4.4.1.1.2.2.2.2.cmml" xref="S3.E30.m1.4.4.1.1.2.2.2.2">𝑥</ci></apply><ci id="S3.E30.m1.4.4.1.1.2.2.3.cmml" xref="S3.E30.m1.4.4.1.1.2.2.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E30.m1.4c">\sum_{k\in[d]}\mathds{1}_{P_{k}}(x)=d-1+\mathds{1}_{P}(x)\quad\forall x\in D.</annotation><annotation encoding="application/x-llamapun" id="S3.E30.m1.4d">∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_d ] end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = italic_d - 1 + blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_x ) ∀ italic_x ∈ italic_D .</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">(30)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.SSS3.3.p2"> <p class="ltx_p" id="S3.SS1.SSS3.3.p2.18">The edges of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.1.m1.1"><semantics id="S3.SS1.SSS3.3.p2.1.m1.1a"><mi id="S3.SS1.SSS3.3.p2.1.m1.1.1" xref="S3.SS1.SSS3.3.p2.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.1.m1.1b"><ci id="S3.SS1.SSS3.3.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.1.m1.1d">italic_P</annotation></semantics></math> incident with <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.2.m2.1"><semantics id="S3.SS1.SSS3.3.p2.2.m2.1a"><mi id="S3.SS1.SSS3.3.p2.2.m2.1.1" xref="S3.SS1.SSS3.3.p2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.2.m2.1b"><ci id="S3.SS1.SSS3.3.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS3.3.p2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.2.m2.1d">italic_v</annotation></semantics></math> subdivide <math alttext="D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.3.m3.1"><semantics id="S3.SS1.SSS3.3.p2.3.m3.1a"><mi id="S3.SS1.SSS3.3.p2.3.m3.1.1" xref="S3.SS1.SSS3.3.p2.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.3.m3.1b"><ci id="S3.SS1.SSS3.3.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS3.3.p2.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.3.m3.1d">italic_D</annotation></semantics></math> into circular sectors whose interior is alternately contained in <math alttext="P\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.4.m4.1"><semantics id="S3.SS1.SSS3.3.p2.4.m4.1a"><mrow id="S3.SS1.SSS3.3.p2.4.m4.1.1" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.4.m4.1.1.2" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.3.p2.4.m4.1.1.1" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.4.m4.1.1.3" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.4.m4.1b"><apply id="S3.SS1.SSS3.3.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS3.3.p2.4.m4.1.1"><intersect id="S3.SS1.SSS3.3.p2.4.m4.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.1"></intersect><ci id="S3.SS1.SSS3.3.p2.4.m4.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.4.m4.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.4.m4.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.4.m4.1c">P\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.4.m4.1d">italic_P ∩ italic_D</annotation></semantics></math> and <math alttext="P^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.5.m5.1"><semantics id="S3.SS1.SSS3.3.p2.5.m5.1a"><mrow id="S3.SS1.SSS3.3.p2.5.m5.1.1" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.cmml"><msup id="S3.SS1.SSS3.3.p2.5.m5.1.1.2" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.2" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.3" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS3.3.p2.5.m5.1.1.1" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.5.m5.1.1.3" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.5.m5.1b"><apply id="S3.SS1.SSS3.3.p2.5.m5.1.1.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1"><intersect id="S3.SS1.SSS3.3.p2.5.m5.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.5.m5.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.5.m5.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.5.m5.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.5.m5.1c">P^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.5.m5.1d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math>. There are exactly two incident edges per <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.6.m6.1"><semantics id="S3.SS1.SSS3.3.p2.6.m6.1a"><msub id="S3.SS1.SSS3.3.p2.6.m6.1.1" xref="S3.SS1.SSS3.3.p2.6.m6.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.6.m6.1.1.2" xref="S3.SS1.SSS3.3.p2.6.m6.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.3.p2.6.m6.1.1.3" xref="S3.SS1.SSS3.3.p2.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.6.m6.1b"><apply id="S3.SS1.SSS3.3.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS3.3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.6.m6.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.6.m6.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.6.m6.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.3.p2.6.m6.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.6.m6.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.6.m6.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., there are <math alttext="2d" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.7.m7.1"><semantics id="S3.SS1.SSS3.3.p2.7.m7.1a"><mrow id="S3.SS1.SSS3.3.p2.7.m7.1.1" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.cmml"><mn id="S3.SS1.SSS3.3.p2.7.m7.1.1.2" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.2.cmml">2</mn><mo id="S3.SS1.SSS3.3.p2.7.m7.1.1.1" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS3.3.p2.7.m7.1.1.3" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.7.m7.1b"><apply id="S3.SS1.SSS3.3.p2.7.m7.1.1.cmml" xref="S3.SS1.SSS3.3.p2.7.m7.1.1"><times id="S3.SS1.SSS3.3.p2.7.m7.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.1"></times><cn id="S3.SS1.SSS3.3.p2.7.m7.1.1.2.cmml" type="integer" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.2">2</cn><ci id="S3.SS1.SSS3.3.p2.7.m7.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.7.m7.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.7.m7.1c">2d</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.7.m7.1d">2 italic_d</annotation></semantics></math> sectors. Additionally, the edges of <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.8.m8.1"><semantics id="S3.SS1.SSS3.3.p2.8.m8.1a"><msub id="S3.SS1.SSS3.3.p2.8.m8.1.1" xref="S3.SS1.SSS3.3.p2.8.m8.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.8.m8.1.1.2" xref="S3.SS1.SSS3.3.p2.8.m8.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.3.p2.8.m8.1.1.3" xref="S3.SS1.SSS3.3.p2.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.8.m8.1b"><apply id="S3.SS1.SSS3.3.p2.8.m8.1.1.cmml" xref="S3.SS1.SSS3.3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.8.m8.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.8.m8.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.8.m8.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.8.m8.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.3.p2.8.m8.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.8.m8.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.8.m8.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> define two sectors, namely <math alttext="P_{k}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.9.m9.1"><semantics id="S3.SS1.SSS3.3.p2.9.m9.1a"><mrow id="S3.SS1.SSS3.3.p2.9.m9.1.1" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.cmml"><msub id="S3.SS1.SSS3.3.p2.9.m9.1.1.2" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.2" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.3" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.3.p2.9.m9.1.1.1" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.9.m9.1.1.3" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.9.m9.1b"><apply id="S3.SS1.SSS3.3.p2.9.m9.1.1.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1"><intersect id="S3.SS1.SSS3.3.p2.9.m9.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.9.m9.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.3.p2.9.m9.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.9.m9.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.9.m9.1c">P_{k}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.9.m9.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ italic_D</annotation></semantics></math> and <math alttext="P_{k}^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.10.m10.1"><semantics id="S3.SS1.SSS3.3.p2.10.m10.1a"><mrow id="S3.SS1.SSS3.3.p2.10.m10.1.1" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.cmml"><msubsup id="S3.SS1.SSS3.3.p2.10.m10.1.1.2" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.2" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.3" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.3.cmml">k</mi><mi id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.3" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.3.cmml">c</mi></msubsup><mo id="S3.SS1.SSS3.3.p2.10.m10.1.1.1" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.10.m10.1.1.3" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.10.m10.1b"><apply id="S3.SS1.SSS3.3.p2.10.m10.1.1.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1"><intersect id="S3.SS1.SSS3.3.p2.10.m10.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.3.p2.10.m10.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.10.m10.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.10.m10.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.10.m10.1c">P_{k}^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.10.m10.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math>, because <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.11.m11.1"><semantics id="S3.SS1.SSS3.3.p2.11.m11.1a"><msub id="S3.SS1.SSS3.3.p2.11.m11.1.1" xref="S3.SS1.SSS3.3.p2.11.m11.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.11.m11.1.1.2" xref="S3.SS1.SSS3.3.p2.11.m11.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.3.p2.11.m11.1.1.3" xref="S3.SS1.SSS3.3.p2.11.m11.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.11.m11.1b"><apply id="S3.SS1.SSS3.3.p2.11.m11.1.1.cmml" xref="S3.SS1.SSS3.3.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.11.m11.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.11.m11.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.11.m11.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.11.m11.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.3.p2.11.m11.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.11.m11.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.11.m11.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.11.m11.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is the only boundary component of <math alttext="P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.12.m12.1"><semantics id="S3.SS1.SSS3.3.p2.12.m12.1a"><msub id="S3.SS1.SSS3.3.p2.12.m12.1.1" xref="S3.SS1.SSS3.3.p2.12.m12.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.12.m12.1.1.2" xref="S3.SS1.SSS3.3.p2.12.m12.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.12.m12.1.1.3" xref="S3.SS1.SSS3.3.p2.12.m12.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.12.m12.1b"><apply id="S3.SS1.SSS3.3.p2.12.m12.1.1.cmml" xref="S3.SS1.SSS3.3.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.12.m12.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.12.m12.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.12.m12.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.12.m12.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.12.m12.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.12.m12.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.12.m12.1c">P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.12.m12.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> containing <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.13.m13.1"><semantics id="S3.SS1.SSS3.3.p2.13.m13.1a"><mi id="S3.SS1.SSS3.3.p2.13.m13.1.1" xref="S3.SS1.SSS3.3.p2.13.m13.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.13.m13.1b"><ci id="S3.SS1.SSS3.3.p2.13.m13.1.1.cmml" xref="S3.SS1.SSS3.3.p2.13.m13.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.13.m13.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.13.m13.1d">italic_v</annotation></semantics></math>. Since <math alttext="P\subseteq P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.14.m14.1"><semantics id="S3.SS1.SSS3.3.p2.14.m14.1a"><mrow id="S3.SS1.SSS3.3.p2.14.m14.1.1" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.cmml"><mi id="S3.SS1.SSS3.3.p2.14.m14.1.1.2" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.3.p2.14.m14.1.1.1" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.1.cmml">⊆</mo><msub id="S3.SS1.SSS3.3.p2.14.m14.1.1.3" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3.cmml"><mi id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.2" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.3" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.14.m14.1b"><apply id="S3.SS1.SSS3.3.p2.14.m14.1.1.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1"><subset id="S3.SS1.SSS3.3.p2.14.m14.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.1"></subset><ci id="S3.SS1.SSS3.3.p2.14.m14.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.2">𝑃</ci><apply id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.1.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.2.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.14.m14.1.1.3.3.cmml" xref="S3.SS1.SSS3.3.p2.14.m14.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.14.m14.1c">P\subseteq P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.14.m14.1d">italic_P ⊆ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, it follows that <math alttext="P_{k}^{c}\cap D\subseteq P^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.15.m15.1"><semantics id="S3.SS1.SSS3.3.p2.15.m15.1a"><mrow id="S3.SS1.SSS3.3.p2.15.m15.1.1" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.cmml"><mrow id="S3.SS1.SSS3.3.p2.15.m15.1.1.2" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.cmml"><msubsup id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.cmml"><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.2" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.3.cmml">k</mi><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.3.cmml">c</mi></msubsup><mo id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.1" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.3.cmml">D</mi></mrow><mo id="S3.SS1.SSS3.3.p2.15.m15.1.1.1" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.1.cmml">⊆</mo><mrow id="S3.SS1.SSS3.3.p2.15.m15.1.1.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.cmml"><msup id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.cmml"><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.2" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.1" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.3" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.15.m15.1b"><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1"><subset id="S3.SS1.SSS3.3.p2.15.m15.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.1"></subset><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2"><intersect id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.1"></intersect><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2">superscript</csymbol><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.2.3">𝐷</ci></apply><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3"><intersect id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.1"></intersect><apply id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.1.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.2.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.15.m15.1.1.3.3.cmml" xref="S3.SS1.SSS3.3.p2.15.m15.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.15.m15.1c">P_{k}^{c}\cap D\subseteq P^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.15.m15.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D ⊆ italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math>. Thus, <math alttext="P_{k}^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.16.m16.1"><semantics id="S3.SS1.SSS3.3.p2.16.m16.1a"><mrow id="S3.SS1.SSS3.3.p2.16.m16.1.1" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.cmml"><msubsup id="S3.SS1.SSS3.3.p2.16.m16.1.1.2" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.2" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.3" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.3.cmml">k</mi><mi id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.3" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.3.cmml">c</mi></msubsup><mo id="S3.SS1.SSS3.3.p2.16.m16.1.1.1" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.16.m16.1.1.3" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.16.m16.1b"><apply id="S3.SS1.SSS3.3.p2.16.m16.1.1.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1"><intersect id="S3.SS1.SSS3.3.p2.16.m16.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.3.p2.16.m16.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.16.m16.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.16.m16.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.16.m16.1c">P_{k}^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.16.m16.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math> equals one of the sectors of <math alttext="P^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.17.m17.1"><semantics id="S3.SS1.SSS3.3.p2.17.m17.1a"><mrow id="S3.SS1.SSS3.3.p2.17.m17.1.1" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.cmml"><msup id="S3.SS1.SSS3.3.p2.17.m17.1.1.2" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.2" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.3" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS3.3.p2.17.m17.1.1.1" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.17.m17.1.1.3" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.17.m17.1b"><apply id="S3.SS1.SSS3.3.p2.17.m17.1.1.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1"><intersect id="S3.SS1.SSS3.3.p2.17.m17.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.17.m17.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.3.p2.17.m17.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.17.m17.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.17.m17.1c">P^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.17.m17.1d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math>, and <math alttext="P_{k}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.3.p2.18.m18.1"><semantics id="S3.SS1.SSS3.3.p2.18.m18.1a"><mrow id="S3.SS1.SSS3.3.p2.18.m18.1.1" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.cmml"><msub id="S3.SS1.SSS3.3.p2.18.m18.1.1.2" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2.cmml"><mi id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.2" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.3" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.3.p2.18.m18.1.1.1" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.3.p2.18.m18.1.1.3" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.3.p2.18.m18.1b"><apply id="S3.SS1.SSS3.3.p2.18.m18.1.1.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1"><intersect id="S3.SS1.SSS3.3.p2.18.m18.1.1.1.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.1"></intersect><apply id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.1.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.2.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.3.p2.18.m18.1.1.2.3.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.3.p2.18.m18.1.1.3.cmml" xref="S3.SS1.SSS3.3.p2.18.m18.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.3.p2.18.m18.1c">P_{k}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.3.p2.18.m18.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ italic_D</annotation></semantics></math> contains all the other sectors.</p> </div> <div class="ltx_para" id="S3.SS1.SSS3.4.p3"> <p class="ltx_p" id="S3.SS1.SSS3.4.p3.13">Given that the <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.1.m1.1"><semantics id="S3.SS1.SSS3.4.p3.1.m1.1a"><msub id="S3.SS1.SSS3.4.p3.1.m1.1.1" xref="S3.SS1.SSS3.4.p3.1.m1.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.1.m1.1.1.2" xref="S3.SS1.SSS3.4.p3.1.m1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.4.p3.1.m1.1.1.3" xref="S3.SS1.SSS3.4.p3.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.1.m1.1b"><apply id="S3.SS1.SSS3.4.p3.1.m1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.1.m1.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.1.m1.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.1.m1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.4.p3.1.m1.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.1.m1.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.1.m1.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> are disjoint (except at <math alttext="v" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.2.m2.1"><semantics id="S3.SS1.SSS3.4.p3.2.m2.1a"><mi id="S3.SS1.SSS3.4.p3.2.m2.1.1" xref="S3.SS1.SSS3.4.p3.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.2.m2.1b"><ci id="S3.SS1.SSS3.4.p3.2.m2.1.1.cmml" xref="S3.SS1.SSS3.4.p3.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.2.m2.1d">italic_v</annotation></semantics></math>), each of the <math alttext="d" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.3.m3.1"><semantics id="S3.SS1.SSS3.4.p3.3.m3.1a"><mi id="S3.SS1.SSS3.4.p3.3.m3.1.1" xref="S3.SS1.SSS3.4.p3.3.m3.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.3.m3.1b"><ci id="S3.SS1.SSS3.4.p3.3.m3.1.1.cmml" xref="S3.SS1.SSS3.4.p3.3.m3.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.3.m3.1c">d</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.3.m3.1d">italic_d</annotation></semantics></math> sectors belonging to <math alttext="P^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.4.m4.1"><semantics id="S3.SS1.SSS3.4.p3.4.m4.1a"><mrow id="S3.SS1.SSS3.4.p3.4.m4.1.1" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.cmml"><msup id="S3.SS1.SSS3.4.p3.4.m4.1.1.2" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2.cmml"><mi id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.2" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.3" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS3.4.p3.4.m4.1.1.1" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.4.p3.4.m4.1.1.3" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.4.m4.1b"><apply id="S3.SS1.SSS3.4.p3.4.m4.1.1.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1"><intersect id="S3.SS1.SSS3.4.p3.4.m4.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.1"></intersect><apply id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.1.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.2.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.4.m4.1.1.2.3.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.4.p3.4.m4.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.4.m4.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.4.m4.1c">P^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.4.m4.1d">italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math> equals <math alttext="P_{k}^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.5.m5.1"><semantics id="S3.SS1.SSS3.4.p3.5.m5.1a"><mrow id="S3.SS1.SSS3.4.p3.5.m5.1.1" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.cmml"><msubsup id="S3.SS1.SSS3.4.p3.5.m5.1.1.2" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.cmml"><mi id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.2" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.3" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.3.cmml">k</mi><mi id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.3" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.3.cmml">c</mi></msubsup><mo id="S3.SS1.SSS3.4.p3.5.m5.1.1.1" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS3.4.p3.5.m5.1.1.3" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.5.m5.1b"><apply id="S3.SS1.SSS3.4.p3.5.m5.1.1.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1"><intersect id="S3.SS1.SSS3.4.p3.5.m5.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.1"></intersect><apply id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.1.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2">superscript</csymbol><apply id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.2.3">𝑘</ci></apply><ci id="S3.SS1.SSS3.4.p3.5.m5.1.1.2.3.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.4.p3.5.m5.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.5.m5.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.5.m5.1c">P_{k}^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.5.m5.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math> for exactly one <math alttext="k\in[d]" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.6.m6.1"><semantics id="S3.SS1.SSS3.4.p3.6.m6.1a"><mrow id="S3.SS1.SSS3.4.p3.6.m6.1.2" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.cmml"><mi id="S3.SS1.SSS3.4.p3.6.m6.1.2.2" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS3.4.p3.6.m6.1.2.1" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.6.m6.1.2.3.2" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.3.1.cmml"><mo id="S3.SS1.SSS3.4.p3.6.m6.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.4.p3.6.m6.1.1" xref="S3.SS1.SSS3.4.p3.6.m6.1.1.cmml">d</mi><mo id="S3.SS1.SSS3.4.p3.6.m6.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.6.m6.1b"><apply id="S3.SS1.SSS3.4.p3.6.m6.1.2.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.2"><in id="S3.SS1.SSS3.4.p3.6.m6.1.2.1.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.1"></in><ci id="S3.SS1.SSS3.4.p3.6.m6.1.2.2.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.2">𝑘</ci><apply id="S3.SS1.SSS3.4.p3.6.m6.1.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.3.2"><csymbol cd="latexml" id="S3.SS1.SSS3.4.p3.6.m6.1.2.3.1.1.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.4.p3.6.m6.1.1.cmml" xref="S3.SS1.SSS3.4.p3.6.m6.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.6.m6.1c">k\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.6.m6.1d">italic_k ∈ [ italic_d ]</annotation></semantics></math> and is contained in <math alttext="P_{l}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.7.m7.1"><semantics id="S3.SS1.SSS3.4.p3.7.m7.1a"><msub id="S3.SS1.SSS3.4.p3.7.m7.1.1" xref="S3.SS1.SSS3.4.p3.7.m7.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.7.m7.1.1.2" xref="S3.SS1.SSS3.4.p3.7.m7.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.7.m7.1.1.3" xref="S3.SS1.SSS3.4.p3.7.m7.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.7.m7.1b"><apply id="S3.SS1.SSS3.4.p3.7.m7.1.1.cmml" xref="S3.SS1.SSS3.4.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.7.m7.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.7.m7.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.7.m7.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.7.m7.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.7.m7.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.7.m7.1c">P_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.7.m7.1d">italic_P start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="l\in[d]\setminus\{k\}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.8.m8.2"><semantics id="S3.SS1.SSS3.4.p3.8.m8.2a"><mrow id="S3.SS1.SSS3.4.p3.8.m8.2.3" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.cmml"><mi id="S3.SS1.SSS3.4.p3.8.m8.2.3.2" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.2.cmml">l</mi><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.1" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.8.m8.2.3.3" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.cmml"><mrow id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.2" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.1.cmml"><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.4.p3.8.m8.1.1" xref="S3.SS1.SSS3.4.p3.8.m8.1.1.cmml">d</mi><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.1.1.cmml">]</mo></mrow><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.1" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.1.cmml">∖</mo><mrow id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.2" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.1.cmml"><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.1.cmml">{</mo><mi id="S3.SS1.SSS3.4.p3.8.m8.2.2" xref="S3.SS1.SSS3.4.p3.8.m8.2.2.cmml">k</mi><mo id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.8.m8.2b"><apply id="S3.SS1.SSS3.4.p3.8.m8.2.3.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3"><in id="S3.SS1.SSS3.4.p3.8.m8.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.1"></in><ci id="S3.SS1.SSS3.4.p3.8.m8.2.3.2.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.2">𝑙</ci><apply id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3"><setdiff id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.1"></setdiff><apply id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.2"><csymbol cd="latexml" id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.1.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.2.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.4.p3.8.m8.1.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.1.1">𝑑</ci></apply><set id="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.1.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.3.3.3.2"><ci id="S3.SS1.SSS3.4.p3.8.m8.2.2.cmml" xref="S3.SS1.SSS3.4.p3.8.m8.2.2">𝑘</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.8.m8.2c">l\in[d]\setminus\{k\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.8.m8.2d">italic_l ∈ [ italic_d ] ∖ { italic_k }</annotation></semantics></math>. Therefore, <math alttext="\sum_{k\in[d]}\mathds{1}_{P_{k}}(x)=d-1" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.9.m9.2"><semantics id="S3.SS1.SSS3.4.p3.9.m9.2a"><mrow id="S3.SS1.SSS3.4.p3.9.m9.2.3" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.cmml"><mrow id="S3.SS1.SSS3.4.p3.9.m9.2.3.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.cmml"><msub id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.cmml"><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.2.cmml">∑</mo><mrow id="S3.SS1.SSS3.4.p3.9.m9.1.1.1" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.3" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.3.cmml">k</mi><mo id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.2" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.2" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.1.cmml"><mo id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.1" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.1.cmml">d</mi><mo id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mrow id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.cmml"><msub id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.cmml"><mn id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.2.cmml">𝟙</mn><msub id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.cmml"><mi id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.3" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.3.cmml">k</mi></msub></msub><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.1" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.3.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.cmml"><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.cmml">(</mo><mi id="S3.SS1.SSS3.4.p3.9.m9.2.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.2.cmml">x</mi><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.1" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS3.4.p3.9.m9.2.3.3" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.cmml"><mi id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.2" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.2.cmml">d</mi><mo id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.1" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.1.cmml">−</mo><mn id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.3" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.9.m9.2b"><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3"><eq id="S3.SS1.SSS3.4.p3.9.m9.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.1"></eq><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2"><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1">subscript</csymbol><sum id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.1.2"></sum><apply id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1"><in id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.2"></in><ci id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.3">𝑘</ci><apply id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.2"><csymbol cd="latexml" id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.1.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.4.p3.9.m9.1.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.1.1.1.1">𝑑</ci></apply></apply></apply><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2"><times id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.1"></times><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2">subscript</csymbol><cn id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.2.cmml" type="integer" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.2">1</cn><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.3.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.2.2.2.3.3">𝑘</ci></apply></apply><ci id="S3.SS1.SSS3.4.p3.9.m9.2.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.2">𝑥</ci></apply></apply><apply id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3"><minus id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.1.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.1"></minus><ci id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.2.cmml" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.2">𝑑</ci><cn id="S3.SS1.SSS3.4.p3.9.m9.2.3.3.3.cmml" type="integer" xref="S3.SS1.SSS3.4.p3.9.m9.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.9.m9.2c">\sum_{k\in[d]}\mathds{1}_{P_{k}}(x)=d-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.9.m9.2d">∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_d ] end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = italic_d - 1</annotation></semantics></math> for all <math alttext="x\in P^{c}\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.10.m10.1"><semantics id="S3.SS1.SSS3.4.p3.10.m10.1a"><mrow id="S3.SS1.SSS3.4.p3.10.m10.1.1" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.10.m10.1.1.2" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS3.4.p3.10.m10.1.1.1" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.10.m10.1.1.3" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.cmml"><msup id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.cmml"><mi id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.2" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.3" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.3.cmml">c</mi></msup><mo id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.1" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.3" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.10.m10.1b"><apply id="S3.SS1.SSS3.4.p3.10.m10.1.1.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1"><in id="S3.SS1.SSS3.4.p3.10.m10.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.1"></in><ci id="S3.SS1.SSS3.4.p3.10.m10.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.2">𝑥</ci><apply id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3"><intersect id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.1"></intersect><apply id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.3.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.2.3">𝑐</ci></apply><ci id="S3.SS1.SSS3.4.p3.10.m10.1.1.3.3.cmml" xref="S3.SS1.SSS3.4.p3.10.m10.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.10.m10.1c">x\in P^{c}\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.10.m10.1d">italic_x ∈ italic_P start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∩ italic_D</annotation></semantics></math>. Since <math alttext="P\subseteq P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.11.m11.1"><semantics id="S3.SS1.SSS3.4.p3.11.m11.1a"><mrow id="S3.SS1.SSS3.4.p3.11.m11.1.1" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.11.m11.1.1.2" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.4.p3.11.m11.1.1.1" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.1.cmml">⊆</mo><msub id="S3.SS1.SSS3.4.p3.11.m11.1.1.3" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3.cmml"><mi id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.2" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.3" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.11.m11.1b"><apply id="S3.SS1.SSS3.4.p3.11.m11.1.1.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1"><subset id="S3.SS1.SSS3.4.p3.11.m11.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.1"></subset><ci id="S3.SS1.SSS3.4.p3.11.m11.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.2">𝑃</ci><apply id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.1.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.2.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.11.m11.1.1.3.3.cmml" xref="S3.SS1.SSS3.4.p3.11.m11.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.11.m11.1c">P\subseteq P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.11.m11.1d">italic_P ⊆ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, it directly follows that <math alttext="\sum_{k\in[d]}\mathds{1}_{P_{k}}(x)=d" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.12.m12.2"><semantics id="S3.SS1.SSS3.4.p3.12.m12.2a"><mrow id="S3.SS1.SSS3.4.p3.12.m12.2.3" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.cmml"><mrow id="S3.SS1.SSS3.4.p3.12.m12.2.3.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.cmml"><msub id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.cmml"><mo id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.2.cmml">∑</mo><mrow id="S3.SS1.SSS3.4.p3.12.m12.1.1.1" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.3" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.3.cmml">k</mi><mo id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.2" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.2" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.1.cmml"><mo id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.1" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.1.cmml">d</mi><mo id="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.12.m12.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mrow id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.cmml"><msub id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.cmml"><mn id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.2.cmml">𝟙</mn><msub id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.cmml"><mi id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.3" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.3.cmml">k</mi></msub></msub><mo id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.1" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.3.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.cmml"><mo id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.cmml">(</mo><mi id="S3.SS1.SSS3.4.p3.12.m12.2.2" xref="S3.SS1.SSS3.4.p3.12.m12.2.2.cmml">x</mi><mo id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.SSS3.4.p3.12.m12.2.3.1" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.1.cmml">=</mo><mi id="S3.SS1.SSS3.4.p3.12.m12.2.3.3" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.12.m12.2b"><apply id="S3.SS1.SSS3.4.p3.12.m12.2.3.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3"><eq id="S3.SS1.SSS3.4.p3.12.m12.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.1"></eq><apply id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2"><apply id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.1.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1">subscript</csymbol><sum id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.1.2.cmml" 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id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.1.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2">subscript</csymbol><cn id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.2.cmml" type="integer" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.2">1</cn><apply id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.1.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3">subscript</csymbol><ci id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.2.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.3.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.2.2.2.3.3">𝑘</ci></apply></apply><ci id="S3.SS1.SSS3.4.p3.12.m12.2.2.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.2">𝑥</ci></apply></apply><ci id="S3.SS1.SSS3.4.p3.12.m12.2.3.3.cmml" xref="S3.SS1.SSS3.4.p3.12.m12.2.3.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.12.m12.2c">\sum_{k\in[d]}\mathds{1}_{P_{k}}(x)=d</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.12.m12.2d">∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_d ] end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = italic_d</annotation></semantics></math> for all <math alttext="x\in P\cap D" class="ltx_Math" display="inline" id="S3.SS1.SSS3.4.p3.13.m13.1"><semantics id="S3.SS1.SSS3.4.p3.13.m13.1a"><mrow id="S3.SS1.SSS3.4.p3.13.m13.1.1" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.cmml"><mi id="S3.SS1.SSS3.4.p3.13.m13.1.1.2" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS3.4.p3.13.m13.1.1.1" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.4.p3.13.m13.1.1.3" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.cmml"><mi id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.2" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.1" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.1.cmml">∩</mo><mi id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.3" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.3.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.4.p3.13.m13.1b"><apply id="S3.SS1.SSS3.4.p3.13.m13.1.1.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1"><in id="S3.SS1.SSS3.4.p3.13.m13.1.1.1.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.1"></in><ci id="S3.SS1.SSS3.4.p3.13.m13.1.1.2.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.2">𝑥</ci><apply id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3"><intersect id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.1.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.1"></intersect><ci id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.2.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.4.p3.13.m13.1.1.3.3.cmml" xref="S3.SS1.SSS3.4.p3.13.m13.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.4.p3.13.m13.1c">x\in P\cap D</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.4.p3.13.m13.1d">italic_x ∈ italic_P ∩ italic_D</annotation></semantics></math>, which proves (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E30" title="Equation 30 ‣ Proof. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">30</span></a>). ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS3.p3"> <p class="ltx_p" id="S3.SS1.SSS3.p3.1">Now that the necessary preliminaries are in place, the proof of the main lemma follows.</p> </div> <div class="ltx_proof" id="S3.SS1.SSS3.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</h6> <div class="ltx_para" id="S3.SS1.SSS3.5.p1"> <p class="ltx_p" id="S3.SS1.SSS3.5.p1.4">To simplify notation, the <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.1.m1.1"><semantics id="S3.SS1.SSS3.5.p1.1.m1.1a"><mi id="S3.SS1.SSS3.5.p1.1.m1.1.1" xref="S3.SS1.SSS3.5.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.1.m1.1b"><ci id="S3.SS1.SSS3.5.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.1.m1.1d">italic_P</annotation></semantics></math>-dependency of the number of holes is dropped by defining <math alttext="h:=n_{h}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.2.m2.1"><semantics id="S3.SS1.SSS3.5.p1.2.m2.1a"><mrow id="S3.SS1.SSS3.5.p1.2.m2.1.2" 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stretchy="false" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.2.m2.1b"><apply id="S3.SS1.SSS3.5.p1.2.m2.1.2.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.2.m2.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.1">assign</csymbol><ci id="S3.SS1.SSS3.5.p1.2.m2.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.2">ℎ</ci><apply id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3"><times id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.1"></times><apply id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.1.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.2">𝑛</ci><ci id="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.3.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.2.3.2.3">ℎ</ci></apply><ci id="S3.SS1.SSS3.5.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.2.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.2.m2.1c">h:=n_{h}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.2.m2.1d">italic_h := italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>. Let <math alttext="\gamma_{1},\dots,\gamma_{h}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.3.m3.3"><semantics id="S3.SS1.SSS3.5.p1.3.m3.3a"><mrow id="S3.SS1.SSS3.5.p1.3.m3.3.3.2" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.3.cmml"><msub id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.2" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.2.cmml">γ</mi><mn id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.3" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.3" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS3.5.p1.3.m3.1.1" mathvariant="normal" xref="S3.SS1.SSS3.5.p1.3.m3.1.1.cmml">…</mi><mo id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.4" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.3.cmml">,</mo><msub id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.cmml"><mi id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.2" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.3" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.3.cmml">h</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.3.m3.3b"><list id="S3.SS1.SSS3.5.p1.3.m3.3.3.3.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2"><apply id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.2">𝛾</ci><cn id="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.3.m3.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS3.5.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.1.1">…</ci><apply id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.3.m3.3.3.2.2.3">ℎ</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.3.m3.3c">\gamma_{1},\dots,\gamma_{h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.3.m3.3d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> be the boundary components of <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.4.m4.1"><semantics id="S3.SS1.SSS3.5.p1.4.m4.1a"><mi id="S3.SS1.SSS3.5.p1.4.m4.1.1" xref="S3.SS1.SSS3.5.p1.4.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.4.m4.1b"><ci id="S3.SS1.SSS3.5.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.4.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.4.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.4.m4.1d">italic_P</annotation></semantics></math> that describe holes. Then,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex35"> <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="P^{\prime}:=P\cup\bigcup_{k\in[h]}\operatorname*{int}(\gamma_{k})" class="ltx_Math" display="block" id="S3.Ex35.m1.3"><semantics id="S3.Ex35.m1.3a"><mrow id="S3.Ex35.m1.3.3" xref="S3.Ex35.m1.3.3.cmml"><msup id="S3.Ex35.m1.3.3.3" xref="S3.Ex35.m1.3.3.3.cmml"><mi id="S3.Ex35.m1.3.3.3.2" xref="S3.Ex35.m1.3.3.3.2.cmml">P</mi><mo id="S3.Ex35.m1.3.3.3.3" xref="S3.Ex35.m1.3.3.3.3.cmml">′</mo></msup><mo id="S3.Ex35.m1.3.3.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex35.m1.3.3.2.cmml">:=</mo><mrow id="S3.Ex35.m1.3.3.1" xref="S3.Ex35.m1.3.3.1.cmml"><mi id="S3.Ex35.m1.3.3.1.3" xref="S3.Ex35.m1.3.3.1.3.cmml">P</mi><mo id="S3.Ex35.m1.3.3.1.2" rspace="0.055em" xref="S3.Ex35.m1.3.3.1.2.cmml">∪</mo><mrow id="S3.Ex35.m1.3.3.1.1" xref="S3.Ex35.m1.3.3.1.1.cmml"><munder id="S3.Ex35.m1.3.3.1.1.2" xref="S3.Ex35.m1.3.3.1.1.2.cmml"><mo id="S3.Ex35.m1.3.3.1.1.2.2" movablelimits="false" xref="S3.Ex35.m1.3.3.1.1.2.2.cmml">⋃</mo><mrow id="S3.Ex35.m1.1.1.1" xref="S3.Ex35.m1.1.1.1.cmml"><mi id="S3.Ex35.m1.1.1.1.3" xref="S3.Ex35.m1.1.1.1.3.cmml">k</mi><mo id="S3.Ex35.m1.1.1.1.2" xref="S3.Ex35.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex35.m1.1.1.1.4.2" xref="S3.Ex35.m1.1.1.1.4.1.cmml"><mo id="S3.Ex35.m1.1.1.1.4.2.1" stretchy="false" xref="S3.Ex35.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S3.Ex35.m1.1.1.1.1" xref="S3.Ex35.m1.1.1.1.1.cmml">h</mi><mo id="S3.Ex35.m1.1.1.1.4.2.2" stretchy="false" xref="S3.Ex35.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="S3.Ex35.m1.3.3.1.1.1.1" xref="S3.Ex35.m1.3.3.1.1.1.2.cmml"><mo id="S3.Ex35.m1.2.2" lspace="0em" rspace="0em" xref="S3.Ex35.m1.2.2.cmml">int</mo><mrow id="S3.Ex35.m1.3.3.1.1.1.1.1" xref="S3.Ex35.m1.3.3.1.1.1.2.cmml"><mo id="S3.Ex35.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.Ex35.m1.3.3.1.1.1.2.cmml">(</mo><msub id="S3.Ex35.m1.3.3.1.1.1.1.1.1" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex35.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1.2.cmml">γ</mi><mi id="S3.Ex35.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex35.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.Ex35.m1.3.3.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex35.m1.3b"><apply id="S3.Ex35.m1.3.3.cmml" xref="S3.Ex35.m1.3.3"><csymbol cd="latexml" id="S3.Ex35.m1.3.3.2.cmml" xref="S3.Ex35.m1.3.3.2">assign</csymbol><apply id="S3.Ex35.m1.3.3.3.cmml" xref="S3.Ex35.m1.3.3.3"><csymbol cd="ambiguous" id="S3.Ex35.m1.3.3.3.1.cmml" xref="S3.Ex35.m1.3.3.3">superscript</csymbol><ci id="S3.Ex35.m1.3.3.3.2.cmml" xref="S3.Ex35.m1.3.3.3.2">𝑃</ci><ci id="S3.Ex35.m1.3.3.3.3.cmml" xref="S3.Ex35.m1.3.3.3.3">′</ci></apply><apply id="S3.Ex35.m1.3.3.1.cmml" xref="S3.Ex35.m1.3.3.1"><union id="S3.Ex35.m1.3.3.1.2.cmml" xref="S3.Ex35.m1.3.3.1.2"></union><ci id="S3.Ex35.m1.3.3.1.3.cmml" xref="S3.Ex35.m1.3.3.1.3">𝑃</ci><apply id="S3.Ex35.m1.3.3.1.1.cmml" xref="S3.Ex35.m1.3.3.1.1"><apply id="S3.Ex35.m1.3.3.1.1.2.cmml" xref="S3.Ex35.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.Ex35.m1.3.3.1.1.2.1.cmml" xref="S3.Ex35.m1.3.3.1.1.2">subscript</csymbol><union id="S3.Ex35.m1.3.3.1.1.2.2.cmml" xref="S3.Ex35.m1.3.3.1.1.2.2"></union><apply id="S3.Ex35.m1.1.1.1.cmml" xref="S3.Ex35.m1.1.1.1"><in id="S3.Ex35.m1.1.1.1.2.cmml" xref="S3.Ex35.m1.1.1.1.2"></in><ci id="S3.Ex35.m1.1.1.1.3.cmml" xref="S3.Ex35.m1.1.1.1.3">𝑘</ci><apply id="S3.Ex35.m1.1.1.1.4.1.cmml" xref="S3.Ex35.m1.1.1.1.4.2"><csymbol cd="latexml" id="S3.Ex35.m1.1.1.1.4.1.1.cmml" xref="S3.Ex35.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.Ex35.m1.1.1.1.1.cmml" xref="S3.Ex35.m1.1.1.1.1">ℎ</ci></apply></apply></apply><apply id="S3.Ex35.m1.3.3.1.1.1.2.cmml" xref="S3.Ex35.m1.3.3.1.1.1.1"><ci id="S3.Ex35.m1.2.2.cmml" xref="S3.Ex35.m1.2.2">int</ci><apply id="S3.Ex35.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex35.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex35.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1.2">𝛾</ci><ci id="S3.Ex35.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S3.Ex35.m1.3.3.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex35.m1.3c">P^{\prime}:=P\cup\bigcup_{k\in[h]}\operatorname*{int}(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex35.m1.3d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_P ∪ ⋃ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT roman_int ( italic_γ start_POSTSUBSCRIPT italic_k 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.SS1.SSS3.5.p1.38">is a polygon that satisfies</p> <table class="ltx_equation ltx_eqn_table" id="S3.E31"> <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="\mathds{1}_{P^{\prime}}-1=-n_{a}(P)+\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^% {\prime}}^{v}}+\sum_{e\in E_{l}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}-% \sum_{e\in E_{b}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}" class="ltx_Math" display="block" id="S3.E31.m1.4"><semantics id="S3.E31.m1.4a"><mrow id="S3.E31.m1.4.5" xref="S3.E31.m1.4.5.cmml"><mrow id="S3.E31.m1.4.5.2" xref="S3.E31.m1.4.5.2.cmml"><msub id="S3.E31.m1.4.5.2.2" xref="S3.E31.m1.4.5.2.2.cmml"><mn id="S3.E31.m1.4.5.2.2.2" xref="S3.E31.m1.4.5.2.2.2.cmml">𝟙</mn><msup id="S3.E31.m1.4.5.2.2.3" xref="S3.E31.m1.4.5.2.2.3.cmml"><mi id="S3.E31.m1.4.5.2.2.3.2" xref="S3.E31.m1.4.5.2.2.3.2.cmml">P</mi><mo id="S3.E31.m1.4.5.2.2.3.3" xref="S3.E31.m1.4.5.2.2.3.3.cmml">′</mo></msup></msub><mo id="S3.E31.m1.4.5.2.1" xref="S3.E31.m1.4.5.2.1.cmml">−</mo><mn id="S3.E31.m1.4.5.2.3" xref="S3.E31.m1.4.5.2.3.cmml">1</mn></mrow><mo id="S3.E31.m1.4.5.1" xref="S3.E31.m1.4.5.1.cmml">=</mo><mrow id="S3.E31.m1.4.5.3" xref="S3.E31.m1.4.5.3.cmml"><mrow id="S3.E31.m1.4.5.3.2" xref="S3.E31.m1.4.5.3.2.cmml"><mrow id="S3.E31.m1.4.5.3.2.2" xref="S3.E31.m1.4.5.3.2.2.cmml"><mo id="S3.E31.m1.4.5.3.2.2a" xref="S3.E31.m1.4.5.3.2.2.cmml">−</mo><mrow id="S3.E31.m1.4.5.3.2.2.2" xref="S3.E31.m1.4.5.3.2.2.2.cmml"><msub id="S3.E31.m1.4.5.3.2.2.2.2" xref="S3.E31.m1.4.5.3.2.2.2.2.cmml"><mi id="S3.E31.m1.4.5.3.2.2.2.2.2" xref="S3.E31.m1.4.5.3.2.2.2.2.2.cmml">n</mi><mi id="S3.E31.m1.4.5.3.2.2.2.2.3" xref="S3.E31.m1.4.5.3.2.2.2.2.3.cmml">a</mi></msub><mo id="S3.E31.m1.4.5.3.2.2.2.1" xref="S3.E31.m1.4.5.3.2.2.2.1.cmml">⁢</mo><mrow id="S3.E31.m1.4.5.3.2.2.2.3.2" xref="S3.E31.m1.4.5.3.2.2.2.cmml"><mo id="S3.E31.m1.4.5.3.2.2.2.3.2.1" stretchy="false" xref="S3.E31.m1.4.5.3.2.2.2.cmml">(</mo><mi id="S3.E31.m1.4.4" xref="S3.E31.m1.4.4.cmml">P</mi><mo id="S3.E31.m1.4.5.3.2.2.2.3.2.2" stretchy="false" xref="S3.E31.m1.4.5.3.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E31.m1.4.5.3.2.1" rspace="0.055em" xref="S3.E31.m1.4.5.3.2.1.cmml">+</mo><mrow id="S3.E31.m1.4.5.3.2.3" xref="S3.E31.m1.4.5.3.2.3.cmml"><munder id="S3.E31.m1.4.5.3.2.3.1" xref="S3.E31.m1.4.5.3.2.3.1.cmml"><mo id="S3.E31.m1.4.5.3.2.3.1.2" movablelimits="false" xref="S3.E31.m1.4.5.3.2.3.1.2.cmml">∑</mo><mrow id="S3.E31.m1.1.1.1" xref="S3.E31.m1.1.1.1.cmml"><mi id="S3.E31.m1.1.1.1.3" xref="S3.E31.m1.1.1.1.3.cmml">v</mi><mo id="S3.E31.m1.1.1.1.2" xref="S3.E31.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E31.m1.1.1.1.1" xref="S3.E31.m1.1.1.1.1.cmml"><mi id="S3.E31.m1.1.1.1.1.3" xref="S3.E31.m1.1.1.1.1.3.cmml">V</mi><mo id="S3.E31.m1.1.1.1.1.2" xref="S3.E31.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E31.m1.1.1.1.1.1.1" xref="S3.E31.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.E31.m1.1.1.1.1.1.1.2" 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id="S3.E31.m1.4.5.3.3.2.3.2.3.cmml" xref="S3.E31.m1.4.5.3.3.2.3.2.3"><csymbol cd="ambiguous" id="S3.E31.m1.4.5.3.3.2.3.2.3.1.cmml" xref="S3.E31.m1.4.5.3.3.2.3.2.3">superscript</csymbol><ci id="S3.E31.m1.4.5.3.3.2.3.2.3.2.cmml" xref="S3.E31.m1.4.5.3.3.2.3.2.3.2">𝑃</ci><ci id="S3.E31.m1.4.5.3.3.2.3.2.3.3.cmml" xref="S3.E31.m1.4.5.3.3.2.3.2.3.3">′</ci></apply></apply><ci id="S3.E31.m1.4.5.3.3.2.3.3.cmml" xref="S3.E31.m1.4.5.3.3.2.3.3">𝑒</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E31.m1.4c">\mathds{1}_{P^{\prime}}-1=-n_{a}(P)+\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^% {\prime}}^{v}}+\sum_{e\in E_{l}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}-% \sum_{e\in E_{b}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}</annotation><annotation encoding="application/x-llamapun" id="S3.E31.m1.4d">blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - 1 = - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) + ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT 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">(31)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS3.5.p1.7">for all <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.5.m1.1"><semantics id="S3.SS1.SSS3.5.p1.5.m1.1a"><mi id="S3.SS1.SSS3.5.p1.5.m1.1.1" xref="S3.SS1.SSS3.5.p1.5.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.5.m1.1b"><ci id="S3.SS1.SSS3.5.p1.5.m1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.5.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.5.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.5.m1.1d">italic_x</annotation></semantics></math> in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.6.m2.1"><semantics id="S3.SS1.SSS3.5.p1.6.m2.1a"><msup id="S3.SS1.SSS3.5.p1.6.m2.1.1" xref="S3.SS1.SSS3.5.p1.6.m2.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.6.m2.1.1.2" xref="S3.SS1.SSS3.5.p1.6.m2.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.6.m2.1.1.3" xref="S3.SS1.SSS3.5.p1.6.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.6.m2.1b"><apply id="S3.SS1.SSS3.5.p1.6.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.6.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.6.m2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.6.m2.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.6.m2.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.6.m2.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.6.m2.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.6.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.6.m2.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.6.m2.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position. Indeed, that <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.7.m3.1"><semantics id="S3.SS1.SSS3.5.p1.7.m3.1a"><msup id="S3.SS1.SSS3.5.p1.7.m3.1.1" xref="S3.SS1.SSS3.5.p1.7.m3.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.7.m3.1.1.2" xref="S3.SS1.SSS3.5.p1.7.m3.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.7.m3.1.1.3" xref="S3.SS1.SSS3.5.p1.7.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.7.m3.1b"><apply id="S3.SS1.SSS3.5.p1.7.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.7.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.7.m3.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.7.m3.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.7.m3.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.7.m3.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.7.m3.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.7.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.7.m3.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.7.m3.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a polygon is trivial, and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E31" title="Equation 31 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">31</span></a>) can be checked for the following cases separately:</p> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.4">Case: <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mi id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><ci id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">italic_P</annotation></semantics></math> is unbounded and contains some arcs as boundary components. <br class="ltx_break"/>Since <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mi id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><ci id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">italic_P</annotation></semantics></math> is unbounded, any polygonal cycle of its boundary must be a hole. This means that all boundary components that are left in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><msup id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml">P</mi><mo id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.3.m3.1.1.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2">𝑃</ci><ci id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are polygonal arcs. With <math alttext="n_{a}(P)=n_{a}(P^{\prime})" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.2"><semantics id="S3.I2.i1.p1.4.m4.2a"><mrow id="S3.I2.i1.p1.4.m4.2.2" xref="S3.I2.i1.p1.4.m4.2.2.cmml"><mrow id="S3.I2.i1.p1.4.m4.2.2.3" xref="S3.I2.i1.p1.4.m4.2.2.3.cmml"><msub id="S3.I2.i1.p1.4.m4.2.2.3.2" xref="S3.I2.i1.p1.4.m4.2.2.3.2.cmml"><mi id="S3.I2.i1.p1.4.m4.2.2.3.2.2" xref="S3.I2.i1.p1.4.m4.2.2.3.2.2.cmml">n</mi><mi id="S3.I2.i1.p1.4.m4.2.2.3.2.3" xref="S3.I2.i1.p1.4.m4.2.2.3.2.3.cmml">a</mi></msub><mo id="S3.I2.i1.p1.4.m4.2.2.3.1" xref="S3.I2.i1.p1.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="S3.I2.i1.p1.4.m4.2.2.3.3.2" xref="S3.I2.i1.p1.4.m4.2.2.3.cmml"><mo id="S3.I2.i1.p1.4.m4.2.2.3.3.2.1" stretchy="false" xref="S3.I2.i1.p1.4.m4.2.2.3.cmml">(</mo><mi id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml">P</mi><mo id="S3.I2.i1.p1.4.m4.2.2.3.3.2.2" stretchy="false" xref="S3.I2.i1.p1.4.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I2.i1.p1.4.m4.2.2.2" xref="S3.I2.i1.p1.4.m4.2.2.2.cmml">=</mo><mrow id="S3.I2.i1.p1.4.m4.2.2.1" xref="S3.I2.i1.p1.4.m4.2.2.1.cmml"><msub id="S3.I2.i1.p1.4.m4.2.2.1.3" xref="S3.I2.i1.p1.4.m4.2.2.1.3.cmml"><mi id="S3.I2.i1.p1.4.m4.2.2.1.3.2" xref="S3.I2.i1.p1.4.m4.2.2.1.3.2.cmml">n</mi><mi id="S3.I2.i1.p1.4.m4.2.2.1.3.3" xref="S3.I2.i1.p1.4.m4.2.2.1.3.3.cmml">a</mi></msub><mo id="S3.I2.i1.p1.4.m4.2.2.1.2" xref="S3.I2.i1.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S3.I2.i1.p1.4.m4.2.2.1.1.1" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="S3.I2.i1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><msup id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.2" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.3" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I2.i1.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.2b"><apply id="S3.I2.i1.p1.4.m4.2.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2"><eq id="S3.I2.i1.p1.4.m4.2.2.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.2"></eq><apply id="S3.I2.i1.p1.4.m4.2.2.3.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3"><times id="S3.I2.i1.p1.4.m4.2.2.3.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3.1"></times><apply id="S3.I2.i1.p1.4.m4.2.2.3.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.4.m4.2.2.3.2.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3.2">subscript</csymbol><ci id="S3.I2.i1.p1.4.m4.2.2.3.2.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3.2.2">𝑛</ci><ci id="S3.I2.i1.p1.4.m4.2.2.3.2.3.cmml" xref="S3.I2.i1.p1.4.m4.2.2.3.2.3">𝑎</ci></apply><ci id="S3.I2.i1.p1.4.m4.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1">𝑃</ci></apply><apply id="S3.I2.i1.p1.4.m4.2.2.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1"><times id="S3.I2.i1.p1.4.m4.2.2.1.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.2"></times><apply id="S3.I2.i1.p1.4.m4.2.2.1.3.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.4.m4.2.2.1.3.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.3">subscript</csymbol><ci id="S3.I2.i1.p1.4.m4.2.2.1.3.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.3.2">𝑛</ci><ci id="S3.I2.i1.p1.4.m4.2.2.1.3.3.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.3.3">𝑎</ci></apply><apply id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S3.I2.i1.p1.4.m4.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.2c">n_{a}(P)=n_{a}(P^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.2d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.4</span></a> gives (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E31" title="Equation 31 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">31</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.5">Case: <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><mi id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><ci id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">italic_P</annotation></semantics></math> is unbounded but does not contain any arcs in the boundary. <br class="ltx_break"/>In this case, all boundary components are holes and <math alttext="n_{a}(P)=0" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><mrow id="S3.I2.i2.p1.2.m2.1.2" xref="S3.I2.i2.p1.2.m2.1.2.cmml"><mrow id="S3.I2.i2.p1.2.m2.1.2.2" xref="S3.I2.i2.p1.2.m2.1.2.2.cmml"><msub id="S3.I2.i2.p1.2.m2.1.2.2.2" xref="S3.I2.i2.p1.2.m2.1.2.2.2.cmml"><mi id="S3.I2.i2.p1.2.m2.1.2.2.2.2" xref="S3.I2.i2.p1.2.m2.1.2.2.2.2.cmml">n</mi><mi id="S3.I2.i2.p1.2.m2.1.2.2.2.3" xref="S3.I2.i2.p1.2.m2.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.I2.i2.p1.2.m2.1.2.2.1" xref="S3.I2.i2.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.I2.i2.p1.2.m2.1.2.2.3.2" xref="S3.I2.i2.p1.2.m2.1.2.2.cmml"><mo id="S3.I2.i2.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.2.2.cmml">(</mo><mi id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml">P</mi><mo id="S3.I2.i2.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I2.i2.p1.2.m2.1.2.1" xref="S3.I2.i2.p1.2.m2.1.2.1.cmml">=</mo><mn id="S3.I2.i2.p1.2.m2.1.2.3" xref="S3.I2.i2.p1.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2"><eq id="S3.I2.i2.p1.2.m2.1.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2.1"></eq><apply id="S3.I2.i2.p1.2.m2.1.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2"><times id="S3.I2.i2.p1.2.m2.1.2.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.1"></times><apply id="S3.I2.i2.p1.2.m2.1.2.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.2.2.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.2">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.2.2.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.2.2">𝑛</ci><ci id="S3.I2.i2.p1.2.m2.1.2.2.2.3.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.2.3">𝑎</ci></apply><ci id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">𝑃</ci></apply><cn id="S3.I2.i2.p1.2.m2.1.2.3.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">n_{a}(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = 0</annotation></semantics></math>. Therefore, we have <math alttext="P^{\prime}=\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><mrow id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><msup id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2.2" xref="S3.I2.i2.p1.3.m3.1.1.2.2.cmml">P</mi><mo id="S3.I2.i2.p1.3.m3.1.1.2.3" xref="S3.I2.i2.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.I2.i2.p1.3.m3.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.cmml">=</mo><msup id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.3.2" xref="S3.I2.i2.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mn id="S3.I2.i2.p1.3.m3.1.1.3.3" xref="S3.I2.i2.p1.3.m3.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><eq id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1"></eq><apply id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">superscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.2">𝑃</ci><ci id="S3.I2.i2.p1.3.m3.1.1.2.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.3">′</ci></apply><apply id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">superscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3.2">ℝ</ci><cn id="S3.I2.i2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">P^{\prime}=\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="V(P^{\prime})=E(P^{\prime})=\emptyset" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.2"><semantics id="S3.I2.i2.p1.4.m4.2a"><mrow id="S3.I2.i2.p1.4.m4.2.2" xref="S3.I2.i2.p1.4.m4.2.2.cmml"><mrow id="S3.I2.i2.p1.4.m4.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.1.3" xref="S3.I2.i2.p1.4.m4.1.1.1.3.cmml">V</mi><mo id="S3.I2.i2.p1.4.m4.1.1.1.2" xref="S3.I2.i2.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i2.p1.4.m4.1.1.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.2" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.3" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I2.i2.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.i2.p1.4.m4.2.2.4" xref="S3.I2.i2.p1.4.m4.2.2.4.cmml">=</mo><mrow id="S3.I2.i2.p1.4.m4.2.2.2" xref="S3.I2.i2.p1.4.m4.2.2.2.cmml"><mi id="S3.I2.i2.p1.4.m4.2.2.2.3" xref="S3.I2.i2.p1.4.m4.2.2.2.3.cmml">E</mi><mo id="S3.I2.i2.p1.4.m4.2.2.2.2" xref="S3.I2.i2.p1.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="S3.I2.i2.p1.4.m4.2.2.2.1.1" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.cmml"><mo id="S3.I2.i2.p1.4.m4.2.2.2.1.1.2" stretchy="false" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.cmml">(</mo><msup id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.cmml"><mi id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.2" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.2.cmml">P</mi><mo id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.3" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S3.I2.i2.p1.4.m4.2.2.2.1.1.3" stretchy="false" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.i2.p1.4.m4.2.2.5" xref="S3.I2.i2.p1.4.m4.2.2.5.cmml">=</mo><mi id="S3.I2.i2.p1.4.m4.2.2.6" mathvariant="normal" xref="S3.I2.i2.p1.4.m4.2.2.6.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.2b"><apply id="S3.I2.i2.p1.4.m4.2.2.cmml" xref="S3.I2.i2.p1.4.m4.2.2"><and id="S3.I2.i2.p1.4.m4.2.2a.cmml" xref="S3.I2.i2.p1.4.m4.2.2"></and><apply id="S3.I2.i2.p1.4.m4.2.2b.cmml" xref="S3.I2.i2.p1.4.m4.2.2"><eq id="S3.I2.i2.p1.4.m4.2.2.4.cmml" xref="S3.I2.i2.p1.4.m4.2.2.4"></eq><apply id="S3.I2.i2.p1.4.m4.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1"><times id="S3.I2.i2.p1.4.m4.1.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.2"></times><ci id="S3.I2.i2.p1.4.m4.1.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.3">𝑉</ci><apply id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.I2.i2.p1.4.m4.2.2.2.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2"><times id="S3.I2.i2.p1.4.m4.2.2.2.2.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.2"></times><ci id="S3.I2.i2.p1.4.m4.2.2.2.3.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.3">𝐸</ci><apply id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1">superscript</csymbol><ci id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.2">𝑃</ci><ci id="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.2.2.2.1.1.1.3">′</ci></apply></apply></apply><apply id="S3.I2.i2.p1.4.m4.2.2c.cmml" xref="S3.I2.i2.p1.4.m4.2.2"><eq id="S3.I2.i2.p1.4.m4.2.2.5.cmml" xref="S3.I2.i2.p1.4.m4.2.2.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S3.I2.i2.p1.4.m4.2.2.2.cmml" id="S3.I2.i2.p1.4.m4.2.2d.cmml" xref="S3.I2.i2.p1.4.m4.2.2"></share><emptyset id="S3.I2.i2.p1.4.m4.2.2.6.cmml" xref="S3.I2.i2.p1.4.m4.2.2.6"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.2c">V(P^{\prime})=E(P^{\prime})=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.2d">italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_E ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∅</annotation></semantics></math> and <math alttext="\mathds{1}_{P^{\prime}}=1" class="ltx_Math" display="inline" id="S3.I2.i2.p1.5.m5.1"><semantics id="S3.I2.i2.p1.5.m5.1a"><mrow id="S3.I2.i2.p1.5.m5.1.1" xref="S3.I2.i2.p1.5.m5.1.1.cmml"><msub id="S3.I2.i2.p1.5.m5.1.1.2" xref="S3.I2.i2.p1.5.m5.1.1.2.cmml"><mn id="S3.I2.i2.p1.5.m5.1.1.2.2" xref="S3.I2.i2.p1.5.m5.1.1.2.2.cmml">𝟙</mn><msup id="S3.I2.i2.p1.5.m5.1.1.2.3" xref="S3.I2.i2.p1.5.m5.1.1.2.3.cmml"><mi id="S3.I2.i2.p1.5.m5.1.1.2.3.2" xref="S3.I2.i2.p1.5.m5.1.1.2.3.2.cmml">P</mi><mo id="S3.I2.i2.p1.5.m5.1.1.2.3.3" xref="S3.I2.i2.p1.5.m5.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.I2.i2.p1.5.m5.1.1.1" xref="S3.I2.i2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S3.I2.i2.p1.5.m5.1.1.3" xref="S3.I2.i2.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.5.m5.1b"><apply id="S3.I2.i2.p1.5.m5.1.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1"><eq id="S3.I2.i2.p1.5.m5.1.1.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1.1"></eq><apply id="S3.I2.i2.p1.5.m5.1.1.2.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.5.m5.1.1.2.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2">subscript</csymbol><cn id="S3.I2.i2.p1.5.m5.1.1.2.2.cmml" type="integer" xref="S3.I2.i2.p1.5.m5.1.1.2.2">1</cn><apply id="S3.I2.i2.p1.5.m5.1.1.2.3.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.5.m5.1.1.2.3.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2.3">superscript</csymbol><ci id="S3.I2.i2.p1.5.m5.1.1.2.3.2.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2.3.2">𝑃</ci><ci id="S3.I2.i2.p1.5.m5.1.1.2.3.3.cmml" xref="S3.I2.i2.p1.5.m5.1.1.2.3.3">′</ci></apply></apply><cn id="S3.I2.i2.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.5.m5.1c">\mathds{1}_{P^{\prime}}=1</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.5.m5.1d">blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math>, verifying (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E31" title="Equation 31 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">31</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S3.I2.i3.p1"> <p class="ltx_p" id="S3.I2.i3.p1.9">Case: <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i3.p1.1.m1.1"><semantics id="S3.I2.i3.p1.1.m1.1a"><mi id="S3.I2.i3.p1.1.m1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.1.m1.1b"><ci id="S3.I2.i3.p1.1.m1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.1.m1.1d">italic_P</annotation></semantics></math> is bounded. <br class="ltx_break"/>As <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i3.p1.2.m2.1"><semantics id="S3.I2.i3.p1.2.m2.1a"><mi id="S3.I2.i3.p1.2.m2.1.1" xref="S3.I2.i3.p1.2.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.2.m2.1b"><ci id="S3.I2.i3.p1.2.m2.1.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.2.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.2.m2.1d">italic_P</annotation></semantics></math> cannot contain an arc as a boundary component, all boundary components are cycles. If all of those cycles would be holes, <math alttext="P" class="ltx_Math" display="inline" id="S3.I2.i3.p1.3.m3.1"><semantics id="S3.I2.i3.p1.3.m3.1a"><mi id="S3.I2.i3.p1.3.m3.1.1" xref="S3.I2.i3.p1.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.3.m3.1b"><ci id="S3.I2.i3.p1.3.m3.1.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.3.m3.1d">italic_P</annotation></semantics></math> would be unbounded, i.e. there must be at least one cycle <math alttext="\gamma_{o}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.4.m4.1"><semantics id="S3.I2.i3.p1.4.m4.1a"><msub id="S3.I2.i3.p1.4.m4.1.1" xref="S3.I2.i3.p1.4.m4.1.1.cmml"><mi id="S3.I2.i3.p1.4.m4.1.1.2" xref="S3.I2.i3.p1.4.m4.1.1.2.cmml">γ</mi><mi id="S3.I2.i3.p1.4.m4.1.1.3" xref="S3.I2.i3.p1.4.m4.1.1.3.cmml">o</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.4.m4.1b"><apply id="S3.I2.i3.p1.4.m4.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.4.m4.1.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S3.I2.i3.p1.4.m4.1.1.2.cmml" xref="S3.I2.i3.p1.4.m4.1.1.2">𝛾</ci><ci id="S3.I2.i3.p1.4.m4.1.1.3.cmml" xref="S3.I2.i3.p1.4.m4.1.1.3">𝑜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.4.m4.1c">\gamma_{o}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.4.m4.1d">italic_γ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="P\subseteq\overline{\operatorname*{int}(\gamma_{o})}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.5.m5.2"><semantics id="S3.I2.i3.p1.5.m5.2a"><mrow id="S3.I2.i3.p1.5.m5.2.3" xref="S3.I2.i3.p1.5.m5.2.3.cmml"><mi id="S3.I2.i3.p1.5.m5.2.3.2" xref="S3.I2.i3.p1.5.m5.2.3.2.cmml">P</mi><mo id="S3.I2.i3.p1.5.m5.2.3.1" rspace="0.1389em" xref="S3.I2.i3.p1.5.m5.2.3.1.cmml">⊆</mo><mover accent="true" id="S3.I2.i3.p1.5.m5.2.2" xref="S3.I2.i3.p1.5.m5.2.2.cmml"><mrow id="S3.I2.i3.p1.5.m5.2.2.2.2" xref="S3.I2.i3.p1.5.m5.2.2.2.3.cmml"><mo id="S3.I2.i3.p1.5.m5.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.I2.i3.p1.5.m5.1.1.1.1.cmml">int</mo><mrow id="S3.I2.i3.p1.5.m5.2.2.2.2.1" xref="S3.I2.i3.p1.5.m5.2.2.2.3.cmml"><mo id="S3.I2.i3.p1.5.m5.2.2.2.2.1.2" stretchy="false" xref="S3.I2.i3.p1.5.m5.2.2.2.3.cmml">(</mo><msub id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.cmml"><mi id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.2" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.2.cmml">γ</mi><mi id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.3" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.3.cmml">o</mi></msub><mo id="S3.I2.i3.p1.5.m5.2.2.2.2.1.3" stretchy="false" xref="S3.I2.i3.p1.5.m5.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.5.m5.2.2.3" xref="S3.I2.i3.p1.5.m5.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.5.m5.2b"><apply id="S3.I2.i3.p1.5.m5.2.3.cmml" xref="S3.I2.i3.p1.5.m5.2.3"><subset id="S3.I2.i3.p1.5.m5.2.3.1.cmml" xref="S3.I2.i3.p1.5.m5.2.3.1"></subset><ci id="S3.I2.i3.p1.5.m5.2.3.2.cmml" xref="S3.I2.i3.p1.5.m5.2.3.2">𝑃</ci><apply id="S3.I2.i3.p1.5.m5.2.2.cmml" xref="S3.I2.i3.p1.5.m5.2.2"><ci id="S3.I2.i3.p1.5.m5.2.2.3.cmml" xref="S3.I2.i3.p1.5.m5.2.2.3">¯</ci><apply id="S3.I2.i3.p1.5.m5.2.2.2.3.cmml" xref="S3.I2.i3.p1.5.m5.2.2.2.2"><ci id="S3.I2.i3.p1.5.m5.1.1.1.1.cmml" xref="S3.I2.i3.p1.5.m5.1.1.1.1">int</ci><apply id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.cmml" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.1.cmml" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1">subscript</csymbol><ci id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.2.cmml" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.3.cmml" xref="S3.I2.i3.p1.5.m5.2.2.2.2.1.1.3">𝑜</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.5.m5.2c">P\subseteq\overline{\operatorname*{int}(\gamma_{o})}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.5.m5.2d">italic_P ⊆ over¯ start_ARG roman_int ( italic_γ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math>. Due to <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.10</span></a>, all other cycles must be holes. Therefore, <math alttext="P^{\prime}=\overline{\operatorname*{int}(\gamma_{o})}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.6.m6.2"><semantics id="S3.I2.i3.p1.6.m6.2a"><mrow id="S3.I2.i3.p1.6.m6.2.3" xref="S3.I2.i3.p1.6.m6.2.3.cmml"><msup id="S3.I2.i3.p1.6.m6.2.3.2" xref="S3.I2.i3.p1.6.m6.2.3.2.cmml"><mi id="S3.I2.i3.p1.6.m6.2.3.2.2" xref="S3.I2.i3.p1.6.m6.2.3.2.2.cmml">P</mi><mo id="S3.I2.i3.p1.6.m6.2.3.2.3" xref="S3.I2.i3.p1.6.m6.2.3.2.3.cmml">′</mo></msup><mo id="S3.I2.i3.p1.6.m6.2.3.1" rspace="0.1389em" xref="S3.I2.i3.p1.6.m6.2.3.1.cmml">=</mo><mover accent="true" id="S3.I2.i3.p1.6.m6.2.2" xref="S3.I2.i3.p1.6.m6.2.2.cmml"><mrow id="S3.I2.i3.p1.6.m6.2.2.2.2" xref="S3.I2.i3.p1.6.m6.2.2.2.3.cmml"><mo id="S3.I2.i3.p1.6.m6.1.1.1.1" lspace="0.1389em" rspace="0em" xref="S3.I2.i3.p1.6.m6.1.1.1.1.cmml">int</mo><mrow id="S3.I2.i3.p1.6.m6.2.2.2.2.1" xref="S3.I2.i3.p1.6.m6.2.2.2.3.cmml"><mo id="S3.I2.i3.p1.6.m6.2.2.2.2.1.2" stretchy="false" xref="S3.I2.i3.p1.6.m6.2.2.2.3.cmml">(</mo><msub id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.cmml"><mi id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.2" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.2.cmml">γ</mi><mi id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.3" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.3.cmml">o</mi></msub><mo id="S3.I2.i3.p1.6.m6.2.2.2.2.1.3" stretchy="false" xref="S3.I2.i3.p1.6.m6.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.6.m6.2.2.3" xref="S3.I2.i3.p1.6.m6.2.2.3.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.6.m6.2b"><apply id="S3.I2.i3.p1.6.m6.2.3.cmml" xref="S3.I2.i3.p1.6.m6.2.3"><eq id="S3.I2.i3.p1.6.m6.2.3.1.cmml" xref="S3.I2.i3.p1.6.m6.2.3.1"></eq><apply id="S3.I2.i3.p1.6.m6.2.3.2.cmml" xref="S3.I2.i3.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="S3.I2.i3.p1.6.m6.2.3.2.1.cmml" xref="S3.I2.i3.p1.6.m6.2.3.2">superscript</csymbol><ci id="S3.I2.i3.p1.6.m6.2.3.2.2.cmml" xref="S3.I2.i3.p1.6.m6.2.3.2.2">𝑃</ci><ci id="S3.I2.i3.p1.6.m6.2.3.2.3.cmml" xref="S3.I2.i3.p1.6.m6.2.3.2.3">′</ci></apply><apply id="S3.I2.i3.p1.6.m6.2.2.cmml" xref="S3.I2.i3.p1.6.m6.2.2"><ci id="S3.I2.i3.p1.6.m6.2.2.3.cmml" xref="S3.I2.i3.p1.6.m6.2.2.3">¯</ci><apply id="S3.I2.i3.p1.6.m6.2.2.2.3.cmml" xref="S3.I2.i3.p1.6.m6.2.2.2.2"><ci id="S3.I2.i3.p1.6.m6.1.1.1.1.cmml" xref="S3.I2.i3.p1.6.m6.1.1.1.1">int</ci><apply id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.cmml" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.1.cmml" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1">subscript</csymbol><ci id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.2.cmml" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.3.cmml" xref="S3.I2.i3.p1.6.m6.2.2.2.2.1.1.3">𝑜</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.6.m6.2c">P^{\prime}=\overline{\operatorname*{int}(\gamma_{o})}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.6.m6.2d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = over¯ start_ARG roman_int ( italic_γ start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math>, and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E31" title="Equation 31 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">31</span></a>) follows from <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem3" title="Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.3</span></a> since <math alttext="n_{a}(P)=0" class="ltx_Math" display="inline" id="S3.I2.i3.p1.7.m7.1"><semantics id="S3.I2.i3.p1.7.m7.1a"><mrow id="S3.I2.i3.p1.7.m7.1.2" xref="S3.I2.i3.p1.7.m7.1.2.cmml"><mrow id="S3.I2.i3.p1.7.m7.1.2.2" xref="S3.I2.i3.p1.7.m7.1.2.2.cmml"><msub id="S3.I2.i3.p1.7.m7.1.2.2.2" xref="S3.I2.i3.p1.7.m7.1.2.2.2.cmml"><mi id="S3.I2.i3.p1.7.m7.1.2.2.2.2" xref="S3.I2.i3.p1.7.m7.1.2.2.2.2.cmml">n</mi><mi id="S3.I2.i3.p1.7.m7.1.2.2.2.3" xref="S3.I2.i3.p1.7.m7.1.2.2.2.3.cmml">a</mi></msub><mo id="S3.I2.i3.p1.7.m7.1.2.2.1" xref="S3.I2.i3.p1.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S3.I2.i3.p1.7.m7.1.2.2.3.2" xref="S3.I2.i3.p1.7.m7.1.2.2.cmml"><mo id="S3.I2.i3.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="S3.I2.i3.p1.7.m7.1.2.2.cmml">(</mo><mi id="S3.I2.i3.p1.7.m7.1.1" xref="S3.I2.i3.p1.7.m7.1.1.cmml">P</mi><mo id="S3.I2.i3.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="S3.I2.i3.p1.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.7.m7.1.2.1" xref="S3.I2.i3.p1.7.m7.1.2.1.cmml">=</mo><mn id="S3.I2.i3.p1.7.m7.1.2.3" xref="S3.I2.i3.p1.7.m7.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.7.m7.1b"><apply id="S3.I2.i3.p1.7.m7.1.2.cmml" xref="S3.I2.i3.p1.7.m7.1.2"><eq id="S3.I2.i3.p1.7.m7.1.2.1.cmml" xref="S3.I2.i3.p1.7.m7.1.2.1"></eq><apply id="S3.I2.i3.p1.7.m7.1.2.2.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2"><times id="S3.I2.i3.p1.7.m7.1.2.2.1.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2.1"></times><apply id="S3.I2.i3.p1.7.m7.1.2.2.2.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i3.p1.7.m7.1.2.2.2.1.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2.2">subscript</csymbol><ci id="S3.I2.i3.p1.7.m7.1.2.2.2.2.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2.2.2">𝑛</ci><ci id="S3.I2.i3.p1.7.m7.1.2.2.2.3.cmml" xref="S3.I2.i3.p1.7.m7.1.2.2.2.3">𝑎</ci></apply><ci id="S3.I2.i3.p1.7.m7.1.1.cmml" xref="S3.I2.i3.p1.7.m7.1.1">𝑃</ci></apply><cn id="S3.I2.i3.p1.7.m7.1.2.3.cmml" type="integer" xref="S3.I2.i3.p1.7.m7.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.7.m7.1c">n_{a}(P)=0</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.7.m7.1d">italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) = 0</annotation></semantics></math>, <math alttext="n_{h}(P^{\prime})=0" class="ltx_Math" display="inline" id="S3.I2.i3.p1.8.m8.1"><semantics id="S3.I2.i3.p1.8.m8.1a"><mrow id="S3.I2.i3.p1.8.m8.1.1" xref="S3.I2.i3.p1.8.m8.1.1.cmml"><mrow id="S3.I2.i3.p1.8.m8.1.1.1" xref="S3.I2.i3.p1.8.m8.1.1.1.cmml"><msub id="S3.I2.i3.p1.8.m8.1.1.1.3" xref="S3.I2.i3.p1.8.m8.1.1.1.3.cmml"><mi id="S3.I2.i3.p1.8.m8.1.1.1.3.2" xref="S3.I2.i3.p1.8.m8.1.1.1.3.2.cmml">n</mi><mi id="S3.I2.i3.p1.8.m8.1.1.1.3.3" xref="S3.I2.i3.p1.8.m8.1.1.1.3.3.cmml">h</mi></msub><mo id="S3.I2.i3.p1.8.m8.1.1.1.2" xref="S3.I2.i3.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i3.p1.8.m8.1.1.1.1.1" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="S3.I2.i3.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.2" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.3" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I2.i3.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.8.m8.1.1.2" xref="S3.I2.i3.p1.8.m8.1.1.2.cmml">=</mo><mn id="S3.I2.i3.p1.8.m8.1.1.3" xref="S3.I2.i3.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.8.m8.1b"><apply id="S3.I2.i3.p1.8.m8.1.1.cmml" xref="S3.I2.i3.p1.8.m8.1.1"><eq id="S3.I2.i3.p1.8.m8.1.1.2.cmml" xref="S3.I2.i3.p1.8.m8.1.1.2"></eq><apply id="S3.I2.i3.p1.8.m8.1.1.1.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1"><times id="S3.I2.i3.p1.8.m8.1.1.1.2.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.2"></times><apply id="S3.I2.i3.p1.8.m8.1.1.1.3.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i3.p1.8.m8.1.1.1.3.1.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.3">subscript</csymbol><ci id="S3.I2.i3.p1.8.m8.1.1.1.3.2.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.3.2">𝑛</ci><ci id="S3.I2.i3.p1.8.m8.1.1.1.3.3.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.3.3">ℎ</ci></apply><apply id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S3.I2.i3.p1.8.m8.1.1.1.1.1.1.3">′</ci></apply></apply><cn id="S3.I2.i3.p1.8.m8.1.1.3.cmml" type="integer" xref="S3.I2.i3.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.8.m8.1c">n_{h}(P^{\prime})=0</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.8.m8.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>, and <math alttext="E_{l}(P^{\prime})=\emptyset" class="ltx_Math" display="inline" id="S3.I2.i3.p1.9.m9.1"><semantics id="S3.I2.i3.p1.9.m9.1a"><mrow id="S3.I2.i3.p1.9.m9.1.1" xref="S3.I2.i3.p1.9.m9.1.1.cmml"><mrow id="S3.I2.i3.p1.9.m9.1.1.1" xref="S3.I2.i3.p1.9.m9.1.1.1.cmml"><msub id="S3.I2.i3.p1.9.m9.1.1.1.3" xref="S3.I2.i3.p1.9.m9.1.1.1.3.cmml"><mi id="S3.I2.i3.p1.9.m9.1.1.1.3.2" xref="S3.I2.i3.p1.9.m9.1.1.1.3.2.cmml">E</mi><mi id="S3.I2.i3.p1.9.m9.1.1.1.3.3" xref="S3.I2.i3.p1.9.m9.1.1.1.3.3.cmml">l</mi></msub><mo id="S3.I2.i3.p1.9.m9.1.1.1.2" xref="S3.I2.i3.p1.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i3.p1.9.m9.1.1.1.1.1" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.cmml"><mo id="S3.I2.i3.p1.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.2" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.3" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.I2.i3.p1.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.9.m9.1.1.2" xref="S3.I2.i3.p1.9.m9.1.1.2.cmml">=</mo><mi id="S3.I2.i3.p1.9.m9.1.1.3" mathvariant="normal" xref="S3.I2.i3.p1.9.m9.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.9.m9.1b"><apply id="S3.I2.i3.p1.9.m9.1.1.cmml" xref="S3.I2.i3.p1.9.m9.1.1"><eq id="S3.I2.i3.p1.9.m9.1.1.2.cmml" xref="S3.I2.i3.p1.9.m9.1.1.2"></eq><apply id="S3.I2.i3.p1.9.m9.1.1.1.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1"><times id="S3.I2.i3.p1.9.m9.1.1.1.2.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.2"></times><apply id="S3.I2.i3.p1.9.m9.1.1.1.3.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i3.p1.9.m9.1.1.1.3.1.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.3">subscript</csymbol><ci id="S3.I2.i3.p1.9.m9.1.1.1.3.2.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.3.2">𝐸</ci><ci id="S3.I2.i3.p1.9.m9.1.1.1.3.3.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.3.3">𝑙</ci></apply><apply id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1">superscript</csymbol><ci id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.I2.i3.p1.9.m9.1.1.1.1.1.1.3">′</ci></apply></apply><emptyset id="S3.I2.i3.p1.9.m9.1.1.3.cmml" xref="S3.I2.i3.p1.9.m9.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.9.m9.1c">E_{l}(P^{\prime})=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.9.m9.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∅</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS1.SSS3.5.p1.16">To keep the notation as simple as possible, we consider all following equations involving indicator functions to be restricted to <math alttext="x\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.8.m1.1"><semantics id="S3.SS1.SSS3.5.p1.8.m1.1a"><mrow id="S3.SS1.SSS3.5.p1.8.m1.1.1" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.8.m1.1.1.2" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.2.cmml">x</mi><mo id="S3.SS1.SSS3.5.p1.8.m1.1.1.1" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.1.cmml">∈</mo><msup id="S3.SS1.SSS3.5.p1.8.m1.1.1.3" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.2" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3.2.cmml">ℝ</mi><mn id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.3" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.8.m1.1b"><apply id="S3.SS1.SSS3.5.p1.8.m1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1"><in id="S3.SS1.SSS3.5.p1.8.m1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.1"></in><ci id="S3.SS1.SSS3.5.p1.8.m1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.2">𝑥</ci><apply id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3.2">ℝ</ci><cn id="S3.SS1.SSS3.5.p1.8.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.8.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.8.m1.1c">x\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.8.m1.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.9.m2.1"><semantics id="S3.SS1.SSS3.5.p1.9.m2.1a"><mi id="S3.SS1.SSS3.5.p1.9.m2.1.1" xref="S3.SS1.SSS3.5.p1.9.m2.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.9.m2.1b"><ci id="S3.SS1.SSS3.5.p1.9.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.9.m2.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.9.m2.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.9.m2.1d">italic_P</annotation></semantics></math>-general position. These <math alttext="x" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.10.m3.1"><semantics id="S3.SS1.SSS3.5.p1.10.m3.1a"><mi id="S3.SS1.SSS3.5.p1.10.m3.1.1" xref="S3.SS1.SSS3.5.p1.10.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.10.m3.1b"><ci id="S3.SS1.SSS3.5.p1.10.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.10.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.10.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.10.m3.1d">italic_x</annotation></semantics></math> are then also in <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.11.m4.1"><semantics id="S3.SS1.SSS3.5.p1.11.m4.1a"><msup id="S3.SS1.SSS3.5.p1.11.m4.1.1" xref="S3.SS1.SSS3.5.p1.11.m4.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.11.m4.1.1.2" xref="S3.SS1.SSS3.5.p1.11.m4.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.11.m4.1.1.3" xref="S3.SS1.SSS3.5.p1.11.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.11.m4.1b"><apply id="S3.SS1.SSS3.5.p1.11.m4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.11.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.11.m4.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.11.m4.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.11.m4.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.11.m4.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.11.m4.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.11.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.11.m4.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.11.m4.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-general position because <math alttext="E(P^{\prime})\subset E(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.12.m5.2"><semantics id="S3.SS1.SSS3.5.p1.12.m5.2a"><mrow id="S3.SS1.SSS3.5.p1.12.m5.2.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.cmml"><mrow id="S3.SS1.SSS3.5.p1.12.m5.2.2.1" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.cmml"><mi id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.3" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.3.cmml">E</mi><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.2.cmml">⊂</mo><mrow id="S3.SS1.SSS3.5.p1.12.m5.2.2.3" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.cmml"><mi id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.2.cmml">E</mi><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.1" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.3.2" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS3.5.p1.12.m5.1.1" xref="S3.SS1.SSS3.5.p1.12.m5.1.1.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.12.m5.2b"><apply id="S3.SS1.SSS3.5.p1.12.m5.2.2.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2"><subset id="S3.SS1.SSS3.5.p1.12.m5.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.2"></subset><apply id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1"><times id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.2.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.2"></times><ci id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.3.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.3">𝐸</ci><apply id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3"><times id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.1"></times><ci id="S3.SS1.SSS3.5.p1.12.m5.2.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.2.2.3.2">𝐸</ci><ci id="S3.SS1.SSS3.5.p1.12.m5.1.1.cmml" xref="S3.SS1.SSS3.5.p1.12.m5.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.12.m5.2c">E(P^{\prime})\subset E(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.12.m5.2d">italic_E ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊂ italic_E ( italic_P )</annotation></semantics></math>. We have, <math alttext="P=P^{\prime}\setminus\bigcup_{k\in[h]}\operatorname*{int}(\gamma_{k})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.13.m6.3"><semantics id="S3.SS1.SSS3.5.p1.13.m6.3a"><mrow id="S3.SS1.SSS3.5.p1.13.m6.3.3" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.cmml"><mi id="S3.SS1.SSS3.5.p1.13.m6.3.3.3" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.3.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.2" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.2.cmml">=</mo><mrow id="S3.SS1.SSS3.5.p1.13.m6.3.3.1" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.cmml"><msup id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.2" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.3" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.3.cmml">′</mo></msup><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.2" rspace="0.055em" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.2.cmml">∖</mo><mrow id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.cmml"><msub id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.cmml"><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.2" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.2.cmml">⋃</mo><mrow id="S3.SS1.SSS3.5.p1.13.m6.1.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.3" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.3.cmml">k</mi><mo id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.2" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.2" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.1.cmml"><mo id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.1.cmml">h</mi><mo id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mrow id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.2.cmml"><mo id="S3.SS1.SSS3.5.p1.13.m6.2.2" lspace="0em" rspace="0em" xref="S3.SS1.SSS3.5.p1.13.m6.2.2.cmml">int</mo><mrow id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.2.cmml"><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.2.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.13.m6.3b"><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3"><eq id="S3.SS1.SSS3.5.p1.13.m6.3.3.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.2"></eq><ci id="S3.SS1.SSS3.5.p1.13.m6.3.3.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.3">𝑃</ci><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1"><setdiff id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.2"></setdiff><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.3.3">′</ci></apply><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1"><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2">subscript</csymbol><union id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.2.2"></union><apply id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1"><in id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.2"></in><ci id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.3">𝑘</ci><apply id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.2"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.5.p1.13.m6.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.1.1.1.1">ℎ</ci></apply></apply></apply><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1"><ci id="S3.SS1.SSS3.5.p1.13.m6.2.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.2.2">int</ci><apply id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.13.m6.3.3.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.13.m6.3c">P=P^{\prime}\setminus\bigcup_{k\in[h]}\operatorname*{int}(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.13.m6.3d">italic_P = italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∖ ⋃ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT roman_int ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. Since <math alttext="\operatorname*{int}(\gamma_{k})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.14.m7.2"><semantics id="S3.SS1.SSS3.5.p1.14.m7.2a"><mrow id="S3.SS1.SSS3.5.p1.14.m7.2.2.1" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.2.cmml"><mo id="S3.SS1.SSS3.5.p1.14.m7.1.1" rspace="0em" xref="S3.SS1.SSS3.5.p1.14.m7.1.1.cmml">int</mo><mrow id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.2.cmml"><mo id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.2.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.2" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.3" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.14.m7.2b"><apply id="S3.SS1.SSS3.5.p1.14.m7.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1"><ci id="S3.SS1.SSS3.5.p1.14.m7.1.1.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.1.1">int</ci><apply id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.14.m7.2.2.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.14.m7.2c">\operatorname*{int}(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.14.m7.2d">roman_int ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\operatorname*{int}(\gamma_{l})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.15.m8.2"><semantics id="S3.SS1.SSS3.5.p1.15.m8.2a"><mrow id="S3.SS1.SSS3.5.p1.15.m8.2.2.1" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.2.cmml"><mo id="S3.SS1.SSS3.5.p1.15.m8.1.1" rspace="0em" xref="S3.SS1.SSS3.5.p1.15.m8.1.1.cmml">int</mo><mrow id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.2.cmml"><mo id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.2.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.2" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.3" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.3.cmml">l</mi></msub><mo id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.15.m8.2b"><apply id="S3.SS1.SSS3.5.p1.15.m8.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1"><ci id="S3.SS1.SSS3.5.p1.15.m8.1.1.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.1.1">int</ci><apply id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.15.m8.2.2.1.1.1.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.15.m8.2c">\operatorname*{int}(\gamma_{l})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.15.m8.2d">roman_int ( italic_γ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math> are disjoint for two holes with <math alttext="k\neq l" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.16.m9.1"><semantics id="S3.SS1.SSS3.5.p1.16.m9.1a"><mrow id="S3.SS1.SSS3.5.p1.16.m9.1.1" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.16.m9.1.1.2" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.2.cmml">k</mi><mo id="S3.SS1.SSS3.5.p1.16.m9.1.1.1" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.1.cmml">≠</mo><mi id="S3.SS1.SSS3.5.p1.16.m9.1.1.3" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.16.m9.1b"><apply id="S3.SS1.SSS3.5.p1.16.m9.1.1.cmml" xref="S3.SS1.SSS3.5.p1.16.m9.1.1"><neq id="S3.SS1.SSS3.5.p1.16.m9.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.1"></neq><ci id="S3.SS1.SSS3.5.p1.16.m9.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.2">𝑘</ci><ci id="S3.SS1.SSS3.5.p1.16.m9.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.16.m9.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.16.m9.1c">k\neq l</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.16.m9.1d">italic_k ≠ italic_l</annotation></semantics></math>, we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx10"> <tbody id="S3.Ex36"><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\mathds{1}_{P}-1" class="ltx_Math" display="inline" id="S3.Ex36.m1.1"><semantics id="S3.Ex36.m1.1a"><mrow id="S3.Ex36.m1.1.1" xref="S3.Ex36.m1.1.1.cmml"><msub id="S3.Ex36.m1.1.1.2" xref="S3.Ex36.m1.1.1.2.cmml"><mn id="S3.Ex36.m1.1.1.2.2" xref="S3.Ex36.m1.1.1.2.2.cmml">𝟙</mn><mi id="S3.Ex36.m1.1.1.2.3" xref="S3.Ex36.m1.1.1.2.3.cmml">P</mi></msub><mo id="S3.Ex36.m1.1.1.1" xref="S3.Ex36.m1.1.1.1.cmml">−</mo><mn id="S3.Ex36.m1.1.1.3" xref="S3.Ex36.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex36.m1.1b"><apply id="S3.Ex36.m1.1.1.cmml" xref="S3.Ex36.m1.1.1"><minus id="S3.Ex36.m1.1.1.1.cmml" xref="S3.Ex36.m1.1.1.1"></minus><apply id="S3.Ex36.m1.1.1.2.cmml" xref="S3.Ex36.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex36.m1.1.1.2.1.cmml" xref="S3.Ex36.m1.1.1.2">subscript</csymbol><cn id="S3.Ex36.m1.1.1.2.2.cmml" type="integer" xref="S3.Ex36.m1.1.1.2.2">1</cn><ci id="S3.Ex36.m1.1.1.2.3.cmml" xref="S3.Ex36.m1.1.1.2.3">𝑃</ci></apply><cn id="S3.Ex36.m1.1.1.3.cmml" type="integer" xref="S3.Ex36.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex36.m1.1c">\displaystyle\mathds{1}_{P}-1</annotation><annotation encoding="application/x-llamapun" id="S3.Ex36.m1.1d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathds{1}_{P^{\prime}}-\sum_{k\in[h]}\mathds{1}_{\operatorname*% {int}(\gamma_{k})}-1=\mathds{1}_{P^{\prime}}-1+\sum_{k\in[h]}(\mathds{1}_{% \overline{\operatorname*{out}(\gamma_{k})}}-1)" class="ltx_Math" display="inline" id="S3.Ex36.m2.7"><semantics id="S3.Ex36.m2.7a"><mrow id="S3.Ex36.m2.7.7" xref="S3.Ex36.m2.7.7.cmml"><mi id="S3.Ex36.m2.7.7.3" xref="S3.Ex36.m2.7.7.3.cmml"></mi><mo id="S3.Ex36.m2.7.7.4" xref="S3.Ex36.m2.7.7.4.cmml">=</mo><mrow id="S3.Ex36.m2.7.7.5" xref="S3.Ex36.m2.7.7.5.cmml"><msub id="S3.Ex36.m2.7.7.5.2" xref="S3.Ex36.m2.7.7.5.2.cmml"><mn id="S3.Ex36.m2.7.7.5.2.2" xref="S3.Ex36.m2.7.7.5.2.2.cmml">𝟙</mn><msup id="S3.Ex36.m2.7.7.5.2.3" xref="S3.Ex36.m2.7.7.5.2.3.cmml"><mi id="S3.Ex36.m2.7.7.5.2.3.2" xref="S3.Ex36.m2.7.7.5.2.3.2.cmml">P</mi><mo id="S3.Ex36.m2.7.7.5.2.3.3" xref="S3.Ex36.m2.7.7.5.2.3.3.cmml">′</mo></msup></msub><mo id="S3.Ex36.m2.7.7.5.1" xref="S3.Ex36.m2.7.7.5.1.cmml">−</mo><mrow id="S3.Ex36.m2.7.7.5.3" xref="S3.Ex36.m2.7.7.5.3.cmml"><mstyle displaystyle="true" id="S3.Ex36.m2.7.7.5.3.1" xref="S3.Ex36.m2.7.7.5.3.1.cmml"><munder id="S3.Ex36.m2.7.7.5.3.1a" xref="S3.Ex36.m2.7.7.5.3.1.cmml"><mo id="S3.Ex36.m2.7.7.5.3.1.2" movablelimits="false" xref="S3.Ex36.m2.7.7.5.3.1.2.cmml">∑</mo><mrow 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xref="S3.Ex36.m2.6.6.2.2.2.2"><ci id="S3.Ex36.m2.5.5.1.1.1.1.cmml" xref="S3.Ex36.m2.5.5.1.1.1.1">out</ci><apply id="S3.Ex36.m2.6.6.2.2.2.2.1.1.cmml" xref="S3.Ex36.m2.6.6.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex36.m2.6.6.2.2.2.2.1.1.1.cmml" xref="S3.Ex36.m2.6.6.2.2.2.2.1.1">subscript</csymbol><ci id="S3.Ex36.m2.6.6.2.2.2.2.1.1.2.cmml" xref="S3.Ex36.m2.6.6.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.Ex36.m2.6.6.2.2.2.2.1.1.3.cmml" xref="S3.Ex36.m2.6.6.2.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3.Ex36.m2.7.7.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex36.m2.7.7.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex36.m2.7c">\displaystyle=\mathds{1}_{P^{\prime}}-\sum_{k\in[h]}\mathds{1}_{\operatorname*% {int}(\gamma_{k})}-1=\mathds{1}_{P^{\prime}}-1+\sum_{k\in[h]}(\mathds{1}_{% \overline{\operatorname*{out}(\gamma_{k})}}-1)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex36.m2.7d">= blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT roman_int ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT - 1 = blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - 1 + ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT ( blackboard_1 start_POSTSUBSCRIPT over¯ start_ARG roman_out ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG end_POSTSUBSCRIPT - 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex37"><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=-n_{a}(P)+\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^{\prime}}^{v% }}+\sum_{e\in E_{l}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}-\sum_{e\in E_{% b}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}" class="ltx_Math" display="inline" id="S3.Ex37.m1.4"><semantics id="S3.Ex37.m1.4a"><mrow id="S3.Ex37.m1.4.5" xref="S3.Ex37.m1.4.5.cmml"><mi id="S3.Ex37.m1.4.5.2" xref="S3.Ex37.m1.4.5.2.cmml"></mi><mo id="S3.Ex37.m1.4.5.1" xref="S3.Ex37.m1.4.5.1.cmml">=</mo><mrow id="S3.Ex37.m1.4.5.3" xref="S3.Ex37.m1.4.5.3.cmml"><mrow id="S3.Ex37.m1.4.5.3.2" xref="S3.Ex37.m1.4.5.3.2.cmml"><mrow id="S3.Ex37.m1.4.5.3.2.2" xref="S3.Ex37.m1.4.5.3.2.2.cmml"><mo id="S3.Ex37.m1.4.5.3.2.2a" xref="S3.Ex37.m1.4.5.3.2.2.cmml">−</mo><mrow id="S3.Ex37.m1.4.5.3.2.2.2" xref="S3.Ex37.m1.4.5.3.2.2.2.cmml"><msub id="S3.Ex37.m1.4.5.3.2.2.2.2" xref="S3.Ex37.m1.4.5.3.2.2.2.2.cmml"><mi id="S3.Ex37.m1.4.5.3.2.2.2.2.2" xref="S3.Ex37.m1.4.5.3.2.2.2.2.2.cmml">n</mi><mi id="S3.Ex37.m1.4.5.3.2.2.2.2.3" xref="S3.Ex37.m1.4.5.3.2.2.2.2.3.cmml">a</mi></msub><mo 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b}(P^{\prime})}\mathds{1}_{H_{P^{\prime}}^{e}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex37.m1.4d">= - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) + ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex38"><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\qquad+\sum_{k\in[h]}(\mathds{1}_{\overline{\operatorname*{out}(% \gamma_{k})}}-1)," class="ltx_Math" display="inline" id="S3.Ex38.m1.4"><semantics id="S3.Ex38.m1.4a"><mrow id="S3.Ex38.m1.4.4.1" xref="S3.Ex38.m1.4.4.1.1.cmml"><mrow id="S3.Ex38.m1.4.4.1.1" xref="S3.Ex38.m1.4.4.1.1.cmml"><mo id="S3.Ex38.m1.4.4.1.1a" xref="S3.Ex38.m1.4.4.1.1.cmml">+</mo><mrow id="S3.Ex38.m1.4.4.1.1.1" 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xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex38.m1.4.4.1.1.1.1.1.1" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.cmml"><msub id="S3.Ex38.m1.4.4.1.1.1.1.1.1.2" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.cmml"><mn id="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.2" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.2.cmml">𝟙</mn><mover accent="true" id="S3.Ex38.m1.3.3.2" xref="S3.Ex38.m1.3.3.2.cmml"><mrow id="S3.Ex38.m1.3.3.2.2.2.2" xref="S3.Ex38.m1.3.3.2.2.2.3.cmml"><mo id="S3.Ex38.m1.2.2.1.1.1.1" rspace="0em" xref="S3.Ex38.m1.2.2.1.1.1.1.cmml">out</mo><mrow id="S3.Ex38.m1.3.3.2.2.2.2.1" xref="S3.Ex38.m1.3.3.2.2.2.3.cmml"><mo id="S3.Ex38.m1.3.3.2.2.2.2.1.2" stretchy="false" xref="S3.Ex38.m1.3.3.2.2.2.3.cmml">(</mo><msub id="S3.Ex38.m1.3.3.2.2.2.2.1.1" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1.cmml"><mi id="S3.Ex38.m1.3.3.2.2.2.2.1.1.2" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1.2.cmml">γ</mi><mi id="S3.Ex38.m1.3.3.2.2.2.2.1.1.3" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S3.Ex38.m1.3.3.2.2.2.2.1.3" stretchy="false" xref="S3.Ex38.m1.3.3.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex38.m1.3.3.2.3" xref="S3.Ex38.m1.3.3.2.3.cmml">¯</mo></mover></msub><mo id="S3.Ex38.m1.4.4.1.1.1.1.1.1.1" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.Ex38.m1.4.4.1.1.1.1.1.1.3" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Ex38.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex38.m1.4.4.1.2" xref="S3.Ex38.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex38.m1.4b"><apply id="S3.Ex38.m1.4.4.1.1.cmml" xref="S3.Ex38.m1.4.4.1"><plus id="S3.Ex38.m1.4.4.1.1.2.cmml" xref="S3.Ex38.m1.4.4.1"></plus><apply id="S3.Ex38.m1.4.4.1.1.1.cmml" xref="S3.Ex38.m1.4.4.1.1.1"><apply id="S3.Ex38.m1.4.4.1.1.1.2.cmml" xref="S3.Ex38.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex38.m1.4.4.1.1.1.2.1.cmml" xref="S3.Ex38.m1.4.4.1.1.1.2">subscript</csymbol><sum id="S3.Ex38.m1.4.4.1.1.1.2.2.cmml" xref="S3.Ex38.m1.4.4.1.1.1.2.2"></sum><apply id="S3.Ex38.m1.1.1.1.cmml" xref="S3.Ex38.m1.1.1.1"><in id="S3.Ex38.m1.1.1.1.2.cmml" xref="S3.Ex38.m1.1.1.1.2"></in><ci id="S3.Ex38.m1.1.1.1.3.cmml" xref="S3.Ex38.m1.1.1.1.3">𝑘</ci><apply id="S3.Ex38.m1.1.1.1.4.1.cmml" xref="S3.Ex38.m1.1.1.1.4.2"><csymbol cd="latexml" id="S3.Ex38.m1.1.1.1.4.1.1.cmml" xref="S3.Ex38.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.Ex38.m1.1.1.1.1.cmml" xref="S3.Ex38.m1.1.1.1.1">ℎ</ci></apply></apply></apply><apply id="S3.Ex38.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex38.m1.4.4.1.1.1.1.1"><minus id="S3.Ex38.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.1"></minus><apply id="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.2">subscript</csymbol><cn id="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.2.2">1</cn><apply id="S3.Ex38.m1.3.3.2.cmml" xref="S3.Ex38.m1.3.3.2"><ci id="S3.Ex38.m1.3.3.2.3.cmml" xref="S3.Ex38.m1.3.3.2.3">¯</ci><apply id="S3.Ex38.m1.3.3.2.2.2.3.cmml" xref="S3.Ex38.m1.3.3.2.2.2.2"><ci id="S3.Ex38.m1.2.2.1.1.1.1.cmml" xref="S3.Ex38.m1.2.2.1.1.1.1">out</ci><apply id="S3.Ex38.m1.3.3.2.2.2.2.1.1.cmml" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex38.m1.3.3.2.2.2.2.1.1.1.cmml" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1">subscript</csymbol><ci id="S3.Ex38.m1.3.3.2.2.2.2.1.1.2.cmml" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.Ex38.m1.3.3.2.2.2.2.1.1.3.cmml" xref="S3.Ex38.m1.3.3.2.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3.Ex38.m1.4.4.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex38.m1.4.4.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex38.m1.4c">\displaystyle\qquad+\sum_{k\in[h]}(\mathds{1}_{\overline{\operatorname*{out}(% \gamma_{k})}}-1),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex38.m1.4d">+ ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT ( blackboard_1 start_POSTSUBSCRIPT over¯ start_ARG roman_out ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG 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="S3.SS1.SSS3.5.p1.26">where we have expressed <math alttext="\mathds{1}_{P^{\prime}}-1" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.17.m1.1"><semantics id="S3.SS1.SSS3.5.p1.17.m1.1a"><mrow id="S3.SS1.SSS3.5.p1.17.m1.1.1" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.cmml"><msub id="S3.SS1.SSS3.5.p1.17.m1.1.1.2" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.cmml"><mn id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.2" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.2.cmml">𝟙</mn><msup id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.cmml"><mi id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.2" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.3" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S3.SS1.SSS3.5.p1.17.m1.1.1.1" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS3.5.p1.17.m1.1.1.3" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.17.m1.1b"><apply id="S3.SS1.SSS3.5.p1.17.m1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1"><minus id="S3.SS1.SSS3.5.p1.17.m1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.1"></minus><apply id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2">subscript</csymbol><cn id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.2">1</cn><apply id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.3.cmml" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.2.3.3">′</ci></apply></apply><cn id="S3.SS1.SSS3.5.p1.17.m1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.17.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.17.m1.1c">\mathds{1}_{P^{\prime}}-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.17.m1.1d">blackboard_1 start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - 1</annotation></semantics></math> using (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E31" title="Equation 31 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">31</span></a>). <math alttext="P_{k}:=\overline{\operatorname*{out}{\gamma_{k}}}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.18.m2.1"><semantics id="S3.SS1.SSS3.5.p1.18.m2.1a"><mrow id="S3.SS1.SSS3.5.p1.18.m2.1.1" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.cmml"><msub id="S3.SS1.SSS3.5.p1.18.m2.1.1.2" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2.cmml"><mi id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.2" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.3" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.18.m2.1.1.1" lspace="0.278em" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.1.cmml">:=</mo><mover accent="true" id="S3.SS1.SSS3.5.p1.18.m2.1.1.3" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.cmml"><mrow id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.cmml"><mo id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.1" lspace="0.111em" rspace="0.167em" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.1.cmml">out</mo><msub id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.cmml"><mi id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.2" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.3" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.1" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.18.m2.1b"><apply id="S3.SS1.SSS3.5.p1.18.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.18.m2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.1">assign</csymbol><apply id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.2.3.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.2.3">𝑘</ci></apply><apply id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3"><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.1">¯</ci><apply id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2"><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.1">out</ci><apply id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.18.m2.1.1.3.2.2.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.18.m2.1c">P_{k}:=\overline{\operatorname*{out}{\gamma_{k}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.18.m2.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := over¯ start_ARG roman_out italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> are polygons with <math alttext="\partial P_{k}=\gamma_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.19.m3.1"><semantics id="S3.SS1.SSS3.5.p1.19.m3.1a"><mrow id="S3.SS1.SSS3.5.p1.19.m3.1.1" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.cmml"><mrow id="S3.SS1.SSS3.5.p1.19.m3.1.1.2" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.cmml"><mo id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.1" rspace="0em" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.1.cmml">∂</mo><msub id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.cmml"><mi id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.2" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.3" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.3.cmml">k</mi></msub></mrow><mo id="S3.SS1.SSS3.5.p1.19.m3.1.1.1" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.1.cmml">=</mo><msub id="S3.SS1.SSS3.5.p1.19.m3.1.1.3" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.2" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.3" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.19.m3.1b"><apply id="S3.SS1.SSS3.5.p1.19.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1"><eq id="S3.SS1.SSS3.5.p1.19.m3.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.1"></eq><apply id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2"><partialdiff id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.1"></partialdiff><apply id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.2.2.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.19.m3.1.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.19.m3.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.19.m3.1c">\partial P_{k}=\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.19.m3.1d">∂ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="n_{h}(P_{k})=1" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.20.m4.1"><semantics id="S3.SS1.SSS3.5.p1.20.m4.1a"><mrow id="S3.SS1.SSS3.5.p1.20.m4.1.1" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.cmml"><mrow id="S3.SS1.SSS3.5.p1.20.m4.1.1.1" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.cmml"><msub id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.2" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.2.cmml">n</mi><mi id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.3" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.3.cmml">h</mi></msub><mo id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.2" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.20.m4.1.1.2" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.2.cmml">=</mo><mn id="S3.SS1.SSS3.5.p1.20.m4.1.1.3" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.20.m4.1b"><apply id="S3.SS1.SSS3.5.p1.20.m4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1"><eq id="S3.SS1.SSS3.5.p1.20.m4.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.2"></eq><apply id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1"><times id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.2"></times><apply id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.2">𝑛</ci><ci id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.3.3">ℎ</ci></apply><apply id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="S3.SS1.SSS3.5.p1.20.m4.1.1.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.20.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.20.m4.1c">n_{h}(P_{k})=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.20.m4.1d">italic_n start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = 1</annotation></semantics></math>. Note, that the <math alttext="P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.21.m5.1"><semantics id="S3.SS1.SSS3.5.p1.21.m5.1a"><msub id="S3.SS1.SSS3.5.p1.21.m5.1.1" xref="S3.SS1.SSS3.5.p1.21.m5.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.21.m5.1.1.2" xref="S3.SS1.SSS3.5.p1.21.m5.1.1.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.21.m5.1.1.3" xref="S3.SS1.SSS3.5.p1.21.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.21.m5.1b"><apply id="S3.SS1.SSS3.5.p1.21.m5.1.1.cmml" xref="S3.SS1.SSS3.5.p1.21.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.21.m5.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.21.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.21.m5.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.21.m5.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.21.m5.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.21.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.21.m5.1c">P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.21.m5.1d">italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>-side of an edge <math alttext="e\in E(\gamma_{k})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.22.m6.1"><semantics id="S3.SS1.SSS3.5.p1.22.m6.1a"><mrow id="S3.SS1.SSS3.5.p1.22.m6.1.1" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.22.m6.1.1.3" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.3.cmml">e</mi><mo id="S3.SS1.SSS3.5.p1.22.m6.1.1.2" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.22.m6.1.1.1" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.3" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.3.cmml">E</mi><mo id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.2" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.22.m6.1b"><apply id="S3.SS1.SSS3.5.p1.22.m6.1.1.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1"><in id="S3.SS1.SSS3.5.p1.22.m6.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.2"></in><ci id="S3.SS1.SSS3.5.p1.22.m6.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.3">𝑒</ci><apply id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1"><times id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.2"></times><ci id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.3">𝐸</ci><apply id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.22.m6.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.22.m6.1c">e\in E(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.22.m6.1d">italic_e ∈ italic_E ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> agrees with its <math alttext="P" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.23.m7.1"><semantics id="S3.SS1.SSS3.5.p1.23.m7.1a"><mi id="S3.SS1.SSS3.5.p1.23.m7.1.1" xref="S3.SS1.SSS3.5.p1.23.m7.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.23.m7.1b"><ci id="S3.SS1.SSS3.5.p1.23.m7.1.1.cmml" xref="S3.SS1.SSS3.5.p1.23.m7.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.23.m7.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.23.m7.1d">italic_P</annotation></semantics></math>-side, and the same holds for the <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.24.m8.1"><semantics id="S3.SS1.SSS3.5.p1.24.m8.1a"><msup id="S3.SS1.SSS3.5.p1.24.m8.1.1" xref="S3.SS1.SSS3.5.p1.24.m8.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.24.m8.1.1.2" xref="S3.SS1.SSS3.5.p1.24.m8.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.24.m8.1.1.3" xref="S3.SS1.SSS3.5.p1.24.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.24.m8.1b"><apply id="S3.SS1.SSS3.5.p1.24.m8.1.1.cmml" xref="S3.SS1.SSS3.5.p1.24.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.24.m8.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.24.m8.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.24.m8.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.24.m8.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.24.m8.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.24.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.24.m8.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.24.m8.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-sides of <math alttext="e\in E(P^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.25.m9.1"><semantics id="S3.SS1.SSS3.5.p1.25.m9.1a"><mrow id="S3.SS1.SSS3.5.p1.25.m9.1.1" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.25.m9.1.1.3" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.3.cmml">e</mi><mo id="S3.SS1.SSS3.5.p1.25.m9.1.1.2" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.2.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.25.m9.1.1.1" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.3" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.3.cmml">E</mi><mo id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.2" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.25.m9.1b"><apply id="S3.SS1.SSS3.5.p1.25.m9.1.1.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1"><in id="S3.SS1.SSS3.5.p1.25.m9.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.2"></in><ci id="S3.SS1.SSS3.5.p1.25.m9.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.3">𝑒</ci><apply id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1"><times id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.2"></times><ci id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.3">𝐸</ci><apply id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.25.m9.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.25.m9.1c">e\in E(P^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.25.m9.1d">italic_e ∈ italic_E ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then, expressing the summands <math alttext="\mathds{1}_{\overline{\operatorname*{out}(\gamma_{k})}}-1" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.26.m10.2"><semantics id="S3.SS1.SSS3.5.p1.26.m10.2a"><mrow id="S3.SS1.SSS3.5.p1.26.m10.2.3" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.cmml"><msub id="S3.SS1.SSS3.5.p1.26.m10.2.3.2" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.2.cmml"><mn id="S3.SS1.SSS3.5.p1.26.m10.2.3.2.2" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.2.2.cmml">𝟙</mn><mover accent="true" id="S3.SS1.SSS3.5.p1.26.m10.2.2.2" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.cmml"><mrow id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.26.m10.1.1.1.1.1.1" rspace="0em" xref="S3.SS1.SSS3.5.p1.26.m10.1.1.1.1.1.1.cmml">out</mo><mrow id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.3.cmml">(</mo><msub id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.2" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.2.cmml">γ</mi><mi id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.3" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.3" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.3.cmml">¯</mo></mover></msub><mo id="S3.SS1.SSS3.5.p1.26.m10.2.3.1" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.1.cmml">−</mo><mn id="S3.SS1.SSS3.5.p1.26.m10.2.3.3" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.26.m10.2b"><apply id="S3.SS1.SSS3.5.p1.26.m10.2.3.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.3"><minus id="S3.SS1.SSS3.5.p1.26.m10.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.1"></minus><apply id="S3.SS1.SSS3.5.p1.26.m10.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.26.m10.2.3.2.1.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.2">subscript</csymbol><cn id="S3.SS1.SSS3.5.p1.26.m10.2.3.2.2.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.2.2">1</cn><apply id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2"><ci id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.3">¯</ci><apply id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2"><ci id="S3.SS1.SSS3.5.p1.26.m10.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.1.1.1.1.1.1">out</ci><apply id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.26.m10.2.2.2.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3.SS1.SSS3.5.p1.26.m10.2.3.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.26.m10.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.26.m10.2c">\mathds{1}_{\overline{\operatorname*{out}(\gamma_{k})}}-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.26.m10.2d">blackboard_1 start_POSTSUBSCRIPT over¯ start_ARG roman_out ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_ARG end_POSTSUBSCRIPT - 1</annotation></semantics></math> using <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem3" title="Lemma 3.3. ‣ 3.1.1 Only one cycle ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.3</span></a>, yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx11"> <tbody id="S3.Ex39"><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\mathds{1}_{P}-1" class="ltx_Math" display="inline" id="S3.Ex39.m1.1"><semantics id="S3.Ex39.m1.1a"><mrow id="S3.Ex39.m1.1.1" xref="S3.Ex39.m1.1.1.cmml"><msub id="S3.Ex39.m1.1.1.2" xref="S3.Ex39.m1.1.1.2.cmml"><mn id="S3.Ex39.m1.1.1.2.2" xref="S3.Ex39.m1.1.1.2.2.cmml">𝟙</mn><mi id="S3.Ex39.m1.1.1.2.3" xref="S3.Ex39.m1.1.1.2.3.cmml">P</mi></msub><mo id="S3.Ex39.m1.1.1.1" xref="S3.Ex39.m1.1.1.1.cmml">−</mo><mn id="S3.Ex39.m1.1.1.3" xref="S3.Ex39.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex39.m1.1b"><apply id="S3.Ex39.m1.1.1.cmml" xref="S3.Ex39.m1.1.1"><minus id="S3.Ex39.m1.1.1.1.cmml" xref="S3.Ex39.m1.1.1.1"></minus><apply id="S3.Ex39.m1.1.1.2.cmml" xref="S3.Ex39.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex39.m1.1.1.2.1.cmml" xref="S3.Ex39.m1.1.1.2">subscript</csymbol><cn id="S3.Ex39.m1.1.1.2.2.cmml" type="integer" xref="S3.Ex39.m1.1.1.2.2">1</cn><ci id="S3.Ex39.m1.1.1.2.3.cmml" xref="S3.Ex39.m1.1.1.2.3">𝑃</ci></apply><cn id="S3.Ex39.m1.1.1.3.cmml" type="integer" xref="S3.Ex39.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex39.m1.1c">\displaystyle\mathds{1}_{P}-1</annotation><annotation encoding="application/x-llamapun" id="S3.Ex39.m1.1d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-n_{a}(P)+\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^{\prime}}^{v% }}+\sum_{e\in E_{l}(P^{\prime})}\mathds{1}_{H^{e}_{P}}-\sum_{e\in E_{b}(P^{% \prime})}\mathds{1}_{H^{e}_{P}}" class="ltx_Math" display="inline" id="S3.Ex39.m2.4"><semantics id="S3.Ex39.m2.4a"><mrow id="S3.Ex39.m2.4.5" xref="S3.Ex39.m2.4.5.cmml"><mi id="S3.Ex39.m2.4.5.2" xref="S3.Ex39.m2.4.5.2.cmml"></mi><mo id="S3.Ex39.m2.4.5.1" xref="S3.Ex39.m2.4.5.1.cmml">=</mo><mrow id="S3.Ex39.m2.4.5.3" xref="S3.Ex39.m2.4.5.3.cmml"><mrow id="S3.Ex39.m2.4.5.3.2" xref="S3.Ex39.m2.4.5.3.2.cmml"><mrow id="S3.Ex39.m2.4.5.3.2.2" xref="S3.Ex39.m2.4.5.3.2.2.cmml"><mo id="S3.Ex39.m2.4.5.3.2.2a" xref="S3.Ex39.m2.4.5.3.2.2.cmml">−</mo><mrow id="S3.Ex39.m2.4.5.3.2.2.2" xref="S3.Ex39.m2.4.5.3.2.2.2.cmml"><msub id="S3.Ex39.m2.4.5.3.2.2.2.2" xref="S3.Ex39.m2.4.5.3.2.2.2.2.cmml"><mi id="S3.Ex39.m2.4.5.3.2.2.2.2.2" xref="S3.Ex39.m2.4.5.3.2.2.2.2.2.cmml">n</mi><mi id="S3.Ex39.m2.4.5.3.2.2.2.2.3" xref="S3.Ex39.m2.4.5.3.2.2.2.2.3.cmml">a</mi></msub><mo id="S3.Ex39.m2.4.5.3.2.2.2.1" xref="S3.Ex39.m2.4.5.3.2.2.2.1.cmml">⁢</mo><mrow id="S3.Ex39.m2.4.5.3.2.2.2.3.2" xref="S3.Ex39.m2.4.5.3.2.2.2.cmml"><mo id="S3.Ex39.m2.4.5.3.2.2.2.3.2.1" stretchy="false" 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell 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id="S3.Ex40.m1.4c">\displaystyle\qquad+\sum_{k\in[h]}\big{(}\sum_{v\in V(\gamma_{k})}\mathds{1}_{% Q_{P_{k}}^{v}}-\sum_{e\in E(\gamma_{k})}\mathds{1}_{H^{e}_{P}}-1\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex40.m1.4d">+ ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT - 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex41"><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=-n_{a}(P)-h+\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^{\prime}}^% {v}}+\sum_{e\in E_{l}(P)}\mathds{1}_{H^{e}_{P}}-\sum_{e\in E_{b}(P)}\mathds{1}% _{H^{e}_{P}}" class="ltx_Math" display="inline" id="S3.Ex41.m1.4"><semantics id="S3.Ex41.m1.4a"><mrow id="S3.Ex41.m1.4.5" xref="S3.Ex41.m1.4.5.cmml"><mi id="S3.Ex41.m1.4.5.2" xref="S3.Ex41.m1.4.5.2.cmml"></mi><mo id="S3.Ex41.m1.4.5.1" xref="S3.Ex41.m1.4.5.1.cmml">=</mo><mrow id="S3.Ex41.m1.4.5.3" xref="S3.Ex41.m1.4.5.3.cmml"><mrow id="S3.Ex41.m1.4.5.3.2" xref="S3.Ex41.m1.4.5.3.2.cmml"><mrow id="S3.Ex41.m1.4.5.3.2.2" xref="S3.Ex41.m1.4.5.3.2.2.cmml"><mrow id="S3.Ex41.m1.4.5.3.2.2.2" xref="S3.Ex41.m1.4.5.3.2.2.2.cmml"><mo id="S3.Ex41.m1.4.5.3.2.2.2a" 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E32"><tr class="ltx_equation ltx_eqn_row 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id="S3.E32.m1.3d">+ ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT 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">(32)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS3.5.p1.29">where we used that <math alttext="E_{b}(P)=E_{b}(P^{\prime})\cup\bigcup_{k\in[h]}E(\gamma_{k})" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.27.m1.4"><semantics id="S3.SS1.SSS3.5.p1.27.m1.4a"><mrow id="S3.SS1.SSS3.5.p1.27.m1.4.4" 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xref="S3.SS1.SSS3.5.p1.27.m1.4.4.4.2.2">𝐸</ci><ci id="S3.SS1.SSS3.5.p1.27.m1.4.4.4.2.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.4.2.3">𝑏</ci></apply><ci id="S3.SS1.SSS3.5.p1.27.m1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.2.2">𝑃</ci></apply><apply id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2"><union id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.3"></union><apply id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1"><times id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.2"></times><apply id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.2">𝐸</ci><ci id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.3.3">𝑏</ci></apply><apply id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.3.3.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2"><apply id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2">subscript</csymbol><union id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.2.2"></union><apply id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1"><in id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1.2"></in><ci id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1.3">𝑘</ci><apply id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.4.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1.4.2"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.5.p1.27.m1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.1.1.1.1">ℎ</ci></apply></apply></apply><apply id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1"><times id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.2"></times><ci id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.3">𝐸</ci><apply id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.2">𝛾</ci><ci id="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.27.m1.4.4.2.2.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.27.m1.4c">E_{b}(P)=E_{b}(P^{\prime})\cup\bigcup_{k\in[h]}E(\gamma_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.27.m1.4d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) = italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∪ ⋃ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT italic_E ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is a disjoint union, and <math alttext="E_{l}(P^{\prime})=E_{l}(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.28.m2.2"><semantics id="S3.SS1.SSS3.5.p1.28.m2.2a"><mrow id="S3.SS1.SSS3.5.p1.28.m2.2.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.cmml"><mrow id="S3.SS1.SSS3.5.p1.28.m2.2.2.1" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.cmml"><msub id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.2.cmml">E</mi><mi id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.3" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.3.cmml">l</mi></msub><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS3.5.p1.28.m2.2.2.3" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.cmml"><msub id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.cmml"><mi id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.2.cmml">E</mi><mi id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.3" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.3.cmml">l</mi></msub><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.1" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.3.2" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS3.5.p1.28.m2.1.1" xref="S3.SS1.SSS3.5.p1.28.m2.1.1.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.28.m2.2b"><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2"><eq id="S3.SS1.SSS3.5.p1.28.m2.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.2"></eq><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1"><times id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.2"></times><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.2">𝐸</ci><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.3.3">𝑙</ci></apply><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3"><times id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.1"></times><apply id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.2">𝐸</ci><ci id="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.3.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.2.2.3.2.3">𝑙</ci></apply><ci id="S3.SS1.SSS3.5.p1.28.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.28.m2.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.28.m2.2c">E_{l}(P^{\prime})=E_{l}(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.28.m2.2d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P )</annotation></semantics></math>. With <math alttext="K(v):=\{R\in\{P^{\prime},P_{1},\dots,P_{h}\}:v\in V(R)\}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.29.m3.5"><semantics id="S3.SS1.SSS3.5.p1.29.m3.5a"><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.cmml"><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.4" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.2.cmml">K</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.1" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.3.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.cmml"><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.cmml">(</mo><mi id="S3.SS1.SSS3.5.p1.29.m3.1.1" xref="S3.SS1.SSS3.5.p1.29.m3.1.1.cmml">v</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.3.2.2" rspace="0.278em" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.3" rspace="0.278em" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.3.cmml">:=</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.3" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.1.cmml">{</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.5" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.5.cmml">R</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.4" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.4.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml"><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.4" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml">{</mo><msup id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.2" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.3" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.5" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml">,</mo><msub id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.2" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.2.cmml">P</mi><mn id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.3" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.6" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml">,</mo><mi id="S3.SS1.SSS3.5.p1.29.m3.2.2" mathvariant="normal" xref="S3.SS1.SSS3.5.p1.29.m3.2.2.cmml">…</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.7" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml">,</mo><msub id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.2" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.3" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.3.cmml">h</mi></msub><mo id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.8" rspace="0.278em" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml">}</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.4" rspace="0.278em" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.1.cmml">:</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.2.cmml">v</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.1" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.cmml"><mi id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.1" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.3.2" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.cmml">(</mo><mi id="S3.SS1.SSS3.5.p1.29.m3.3.3" xref="S3.SS1.SSS3.5.p1.29.m3.3.3.cmml">R</mi><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.5" stretchy="false" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.29.m3.5b"><apply id="S3.SS1.SSS3.5.p1.29.m3.5.5.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.29.m3.5.5.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.3">assign</csymbol><apply id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4"><times id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.1"></times><ci id="S3.SS1.SSS3.5.p1.29.m3.5.5.4.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.4.2">𝐾</ci><ci id="S3.SS1.SSS3.5.p1.29.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.1.1">𝑣</ci></apply><apply id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.3">conditional-set</csymbol><apply id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1"><in id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.4.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.4"></in><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.5.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.5">𝑅</ci><set id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.4.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3"><apply id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.1.1.1.3">′</ci></apply><apply id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.2">𝑃</ci><cn id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.2.2.2.3">1</cn></apply><ci id="S3.SS1.SSS3.5.p1.29.m3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.2.2">…</ci><apply id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.4.4.1.1.1.3.3.3.3">ℎ</ci></apply></set></apply><apply id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2"><in id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.1"></in><ci id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.2">𝑣</ci><apply id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3"><times id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.1"></times><ci id="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.5.5.2.2.2.3.2">𝑉</ci><ci id="S3.SS1.SSS3.5.p1.29.m3.3.3.cmml" xref="S3.SS1.SSS3.5.p1.29.m3.3.3">𝑅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.29.m3.5c">K(v):=\{R\in\{P^{\prime},P_{1},\dots,P_{h}\}:v\in V(R)\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.29.m3.5d">italic_K ( italic_v ) := { italic_R ∈ { italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT } : italic_v ∈ italic_V ( italic_R ) }</annotation></semantics></math>, we can write</p> <table class="ltx_equation ltx_eqn_table" id="S3.E33"> <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_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^{\prime}}^{v}}+\sum_{k\in[h]}\sum_{v% \in V(\gamma_{k})}\mathds{1}_{Q_{P_{k}}^{v}}=\sum_{v\in V(P)}\sum_{R\in K(v)}% \mathds{1}_{Q_{R}^{v}}." class="ltx_Math" display="block" id="S3.E33.m1.6"><semantics id="S3.E33.m1.6a"><mrow id="S3.E33.m1.6.6.1" xref="S3.E33.m1.6.6.1.1.cmml"><mrow id="S3.E33.m1.6.6.1.1" xref="S3.E33.m1.6.6.1.1.cmml"><mrow id="S3.E33.m1.6.6.1.1.2" xref="S3.E33.m1.6.6.1.1.2.cmml"><mrow id="S3.E33.m1.6.6.1.1.2.2" xref="S3.E33.m1.6.6.1.1.2.2.cmml"><munder id="S3.E33.m1.6.6.1.1.2.2.1" xref="S3.E33.m1.6.6.1.1.2.2.1.cmml"><mo id="S3.E33.m1.6.6.1.1.2.2.1.2" movablelimits="false" xref="S3.E33.m1.6.6.1.1.2.2.1.2.cmml">∑</mo><mrow id="S3.E33.m1.1.1.1" xref="S3.E33.m1.1.1.1.cmml"><mi id="S3.E33.m1.1.1.1.3" xref="S3.E33.m1.1.1.1.3.cmml">v</mi><mo id="S3.E33.m1.1.1.1.2" xref="S3.E33.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E33.m1.1.1.1.1" xref="S3.E33.m1.1.1.1.1.cmml"><mi id="S3.E33.m1.1.1.1.1.3" xref="S3.E33.m1.1.1.1.1.3.cmml">V</mi><mo id="S3.E33.m1.1.1.1.1.2" xref="S3.E33.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E33.m1.1.1.1.1.1.1" xref="S3.E33.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.E33.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E33.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.E33.m1.1.1.1.1.1.1.1" xref="S3.E33.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.E33.m1.1.1.1.1.1.1.1.2" xref="S3.E33.m1.1.1.1.1.1.1.1.2.cmml">P</mi><mo id="S3.E33.m1.1.1.1.1.1.1.1.3" xref="S3.E33.m1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.E33.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E33.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><msub id="S3.E33.m1.6.6.1.1.2.2.2" xref="S3.E33.m1.6.6.1.1.2.2.2.cmml"><mn id="S3.E33.m1.6.6.1.1.2.2.2.2" xref="S3.E33.m1.6.6.1.1.2.2.2.2.cmml">𝟙</mn><msubsup id="S3.E33.m1.6.6.1.1.2.2.2.3" xref="S3.E33.m1.6.6.1.1.2.2.2.3.cmml"><mi id="S3.E33.m1.6.6.1.1.2.2.2.3.2.2" xref="S3.E33.m1.6.6.1.1.2.2.2.3.2.2.cmml">Q</mi><msup id="S3.E33.m1.6.6.1.1.2.2.2.3.2.3" xref="S3.E33.m1.6.6.1.1.2.2.2.3.2.3.cmml"><mi id="S3.E33.m1.6.6.1.1.2.2.2.3.2.3.2" xref="S3.E33.m1.6.6.1.1.2.2.2.3.2.3.2.cmml">P</mi><mo id="S3.E33.m1.6.6.1.1.2.2.2.3.2.3.3" xref="S3.E33.m1.6.6.1.1.2.2.2.3.2.3.3.cmml">′</mo></msup><mi id="S3.E33.m1.6.6.1.1.2.2.2.3.3" xref="S3.E33.m1.6.6.1.1.2.2.2.3.3.cmml">v</mi></msubsup></msub></mrow><mo id="S3.E33.m1.6.6.1.1.2.1" rspace="0.055em" xref="S3.E33.m1.6.6.1.1.2.1.cmml">+</mo><mrow id="S3.E33.m1.6.6.1.1.2.3" xref="S3.E33.m1.6.6.1.1.2.3.cmml"><munder id="S3.E33.m1.6.6.1.1.2.3.1" xref="S3.E33.m1.6.6.1.1.2.3.1.cmml"><mo id="S3.E33.m1.6.6.1.1.2.3.1.2" movablelimits="false" rspace="0em" xref="S3.E33.m1.6.6.1.1.2.3.1.2.cmml">∑</mo><mrow id="S3.E33.m1.2.2.1" xref="S3.E33.m1.2.2.1.cmml"><mi id="S3.E33.m1.2.2.1.3" xref="S3.E33.m1.2.2.1.3.cmml">k</mi><mo id="S3.E33.m1.2.2.1.2" xref="S3.E33.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.E33.m1.2.2.1.4.2" xref="S3.E33.m1.2.2.1.4.1.cmml"><mo id="S3.E33.m1.2.2.1.4.2.1" stretchy="false" xref="S3.E33.m1.2.2.1.4.1.1.cmml">[</mo><mi id="S3.E33.m1.2.2.1.1" xref="S3.E33.m1.2.2.1.1.cmml">h</mi><mo id="S3.E33.m1.2.2.1.4.2.2" stretchy="false" xref="S3.E33.m1.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="S3.E33.m1.6.6.1.1.2.3.2" xref="S3.E33.m1.6.6.1.1.2.3.2.cmml"><munder id="S3.E33.m1.6.6.1.1.2.3.2.1" xref="S3.E33.m1.6.6.1.1.2.3.2.1.cmml"><mo id="S3.E33.m1.6.6.1.1.2.3.2.1.2" movablelimits="false" xref="S3.E33.m1.6.6.1.1.2.3.2.1.2.cmml">∑</mo><mrow id="S3.E33.m1.3.3.1" xref="S3.E33.m1.3.3.1.cmml"><mi id="S3.E33.m1.3.3.1.3" xref="S3.E33.m1.3.3.1.3.cmml">v</mi><mo id="S3.E33.m1.3.3.1.2" xref="S3.E33.m1.3.3.1.2.cmml">∈</mo><mrow id="S3.E33.m1.3.3.1.1" xref="S3.E33.m1.3.3.1.1.cmml"><mi id="S3.E33.m1.3.3.1.1.3" xref="S3.E33.m1.3.3.1.1.3.cmml">V</mi><mo id="S3.E33.m1.3.3.1.1.2" xref="S3.E33.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.E33.m1.3.3.1.1.1.1" xref="S3.E33.m1.3.3.1.1.1.1.1.cmml"><mo id="S3.E33.m1.3.3.1.1.1.1.2" stretchy="false" xref="S3.E33.m1.3.3.1.1.1.1.1.cmml">(</mo><msub id="S3.E33.m1.3.3.1.1.1.1.1" xref="S3.E33.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.E33.m1.3.3.1.1.1.1.1.2" xref="S3.E33.m1.3.3.1.1.1.1.1.2.cmml">γ</mi><mi id="S3.E33.m1.3.3.1.1.1.1.1.3" xref="S3.E33.m1.3.3.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.E33.m1.3.3.1.1.1.1.3" stretchy="false" xref="S3.E33.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><msub id="S3.E33.m1.6.6.1.1.2.3.2.2" xref="S3.E33.m1.6.6.1.1.2.3.2.2.cmml"><mn id="S3.E33.m1.6.6.1.1.2.3.2.2.2" xref="S3.E33.m1.6.6.1.1.2.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.E33.m1.6.6.1.1.2.3.2.2.3" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.cmml"><mi id="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.2" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.2.cmml">Q</mi><msub id="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3.cmml"><mi id="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3.2" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3.2.cmml">P</mi><mi id="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3.3" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.2.3.3.cmml">k</mi></msub><mi id="S3.E33.m1.6.6.1.1.2.3.2.2.3.3" xref="S3.E33.m1.6.6.1.1.2.3.2.2.3.3.cmml">v</mi></msubsup></msub></mrow></mrow></mrow><mo id="S3.E33.m1.6.6.1.1.1" rspace="0.111em" xref="S3.E33.m1.6.6.1.1.1.cmml">=</mo><mrow id="S3.E33.m1.6.6.1.1.3" xref="S3.E33.m1.6.6.1.1.3.cmml"><munder id="S3.E33.m1.6.6.1.1.3.1" xref="S3.E33.m1.6.6.1.1.3.1.cmml"><mo id="S3.E33.m1.6.6.1.1.3.1.2" movablelimits="false" rspace="0em" xref="S3.E33.m1.6.6.1.1.3.1.2.cmml">∑</mo><mrow id="S3.E33.m1.4.4.1" xref="S3.E33.m1.4.4.1.cmml"><mi id="S3.E33.m1.4.4.1.3" xref="S3.E33.m1.4.4.1.3.cmml">v</mi><mo id="S3.E33.m1.4.4.1.2" xref="S3.E33.m1.4.4.1.2.cmml">∈</mo><mrow id="S3.E33.m1.4.4.1.4" xref="S3.E33.m1.4.4.1.4.cmml"><mi id="S3.E33.m1.4.4.1.4.2" xref="S3.E33.m1.4.4.1.4.2.cmml">V</mi><mo id="S3.E33.m1.4.4.1.4.1" xref="S3.E33.m1.4.4.1.4.1.cmml">⁢</mo><mrow id="S3.E33.m1.4.4.1.4.3.2" xref="S3.E33.m1.4.4.1.4.cmml"><mo id="S3.E33.m1.4.4.1.4.3.2.1" stretchy="false" xref="S3.E33.m1.4.4.1.4.cmml">(</mo><mi id="S3.E33.m1.4.4.1.1" xref="S3.E33.m1.4.4.1.1.cmml">P</mi><mo id="S3.E33.m1.4.4.1.4.3.2.2" stretchy="false" xref="S3.E33.m1.4.4.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.E33.m1.6.6.1.1.3.2" xref="S3.E33.m1.6.6.1.1.3.2.cmml"><munder id="S3.E33.m1.6.6.1.1.3.2.1" xref="S3.E33.m1.6.6.1.1.3.2.1.cmml"><mo id="S3.E33.m1.6.6.1.1.3.2.1.2" movablelimits="false" xref="S3.E33.m1.6.6.1.1.3.2.1.2.cmml">∑</mo><mrow id="S3.E33.m1.5.5.1" xref="S3.E33.m1.5.5.1.cmml"><mi id="S3.E33.m1.5.5.1.3" xref="S3.E33.m1.5.5.1.3.cmml">R</mi><mo id="S3.E33.m1.5.5.1.2" xref="S3.E33.m1.5.5.1.2.cmml">∈</mo><mrow id="S3.E33.m1.5.5.1.4" xref="S3.E33.m1.5.5.1.4.cmml"><mi id="S3.E33.m1.5.5.1.4.2" xref="S3.E33.m1.5.5.1.4.2.cmml">K</mi><mo id="S3.E33.m1.5.5.1.4.1" xref="S3.E33.m1.5.5.1.4.1.cmml">⁢</mo><mrow id="S3.E33.m1.5.5.1.4.3.2" xref="S3.E33.m1.5.5.1.4.cmml"><mo id="S3.E33.m1.5.5.1.4.3.2.1" stretchy="false" xref="S3.E33.m1.5.5.1.4.cmml">(</mo><mi id="S3.E33.m1.5.5.1.1" xref="S3.E33.m1.5.5.1.1.cmml">v</mi><mo 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xref="S3.E33.m1.5.5.1.3">𝑅</ci><apply id="S3.E33.m1.5.5.1.4.cmml" xref="S3.E33.m1.5.5.1.4"><times id="S3.E33.m1.5.5.1.4.1.cmml" xref="S3.E33.m1.5.5.1.4.1"></times><ci id="S3.E33.m1.5.5.1.4.2.cmml" xref="S3.E33.m1.5.5.1.4.2">𝐾</ci><ci id="S3.E33.m1.5.5.1.1.cmml" xref="S3.E33.m1.5.5.1.1">𝑣</ci></apply></apply></apply><apply id="S3.E33.m1.6.6.1.1.3.2.2.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E33.m1.6.6.1.1.3.2.2.1.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2">subscript</csymbol><cn id="S3.E33.m1.6.6.1.1.3.2.2.2.cmml" type="integer" xref="S3.E33.m1.6.6.1.1.3.2.2.2">1</cn><apply id="S3.E33.m1.6.6.1.1.3.2.2.3.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.E33.m1.6.6.1.1.3.2.2.3.1.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3">superscript</csymbol><apply id="S3.E33.m1.6.6.1.1.3.2.2.3.2.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.E33.m1.6.6.1.1.3.2.2.3.2.1.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3">subscript</csymbol><ci id="S3.E33.m1.6.6.1.1.3.2.2.3.2.2.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3.2.2">𝑄</ci><ci id="S3.E33.m1.6.6.1.1.3.2.2.3.2.3.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3.2.3">𝑅</ci></apply><ci id="S3.E33.m1.6.6.1.1.3.2.2.3.3.cmml" xref="S3.E33.m1.6.6.1.1.3.2.2.3.3">𝑣</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E33.m1.6c">\sum_{v\in V(P^{\prime})}\mathds{1}_{Q_{P^{\prime}}^{v}}+\sum_{k\in[h]}\sum_{v% \in V(\gamma_{k})}\mathds{1}_{Q_{P_{k}}^{v}}=\sum_{v\in V(P)}\sum_{R\in K(v)}% \mathds{1}_{Q_{R}^{v}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E33.m1.6d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_k ∈ [ italic_h ] end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_R ∈ italic_K ( italic_v ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT 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">(33)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS3.5.p1.34">Since <math alttext="P\subseteq P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.30.m1.1"><semantics id="S3.SS1.SSS3.5.p1.30.m1.1a"><mrow id="S3.SS1.SSS3.5.p1.30.m1.1.1" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.30.m1.1.1.2" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.30.m1.1.1.1" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.1.cmml">⊆</mo><msup id="S3.SS1.SSS3.5.p1.30.m1.1.1.3" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.2" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.3" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.30.m1.1b"><apply id="S3.SS1.SSS3.5.p1.30.m1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1"><subset id="S3.SS1.SSS3.5.p1.30.m1.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.1"></subset><ci id="S3.SS1.SSS3.5.p1.30.m1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.2">𝑃</ci><apply id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.30.m1.1.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.30.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.30.m1.1c">P\subseteq P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.30.m1.1d">italic_P ⊆ italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="P\subseteq P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.31.m2.1"><semantics id="S3.SS1.SSS3.5.p1.31.m2.1a"><mrow id="S3.SS1.SSS3.5.p1.31.m2.1.1" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.31.m2.1.1.2" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.31.m2.1.1.1" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.1.cmml">⊆</mo><msub id="S3.SS1.SSS3.5.p1.31.m2.1.1.3" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3.cmml"><mi id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.2" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3.2.cmml">P</mi><mi id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.3" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.31.m2.1b"><apply id="S3.SS1.SSS3.5.p1.31.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1"><subset id="S3.SS1.SSS3.5.p1.31.m2.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.1"></subset><ci id="S3.SS1.SSS3.5.p1.31.m2.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.2">𝑃</ci><apply id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.1.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.2.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.31.m2.1.1.3.3.cmml" xref="S3.SS1.SSS3.5.p1.31.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.31.m2.1c">P\subseteq P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.31.m2.1d">italic_P ⊆ italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="k\in[h]" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.32.m3.1"><semantics id="S3.SS1.SSS3.5.p1.32.m3.1a"><mrow id="S3.SS1.SSS3.5.p1.32.m3.1.2" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.cmml"><mi id="S3.SS1.SSS3.5.p1.32.m3.1.2.2" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS3.5.p1.32.m3.1.2.1" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.32.m3.1.2.3.2" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.3.1.cmml"><mo id="S3.SS1.SSS3.5.p1.32.m3.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.3.1.1.cmml">[</mo><mi id="S3.SS1.SSS3.5.p1.32.m3.1.1" xref="S3.SS1.SSS3.5.p1.32.m3.1.1.cmml">h</mi><mo id="S3.SS1.SSS3.5.p1.32.m3.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.32.m3.1b"><apply id="S3.SS1.SSS3.5.p1.32.m3.1.2.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.2"><in id="S3.SS1.SSS3.5.p1.32.m3.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.1"></in><ci id="S3.SS1.SSS3.5.p1.32.m3.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.2">𝑘</ci><apply id="S3.SS1.SSS3.5.p1.32.m3.1.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.3.2"><csymbol cd="latexml" id="S3.SS1.SSS3.5.p1.32.m3.1.2.3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS3.5.p1.32.m3.1.1.cmml" xref="S3.SS1.SSS3.5.p1.32.m3.1.1">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.32.m3.1c">k\in[h]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.32.m3.1d">italic_k ∈ [ italic_h ]</annotation></semantics></math>, and because <math alttext="P^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.33.m4.1"><semantics id="S3.SS1.SSS3.5.p1.33.m4.1a"><msup id="S3.SS1.SSS3.5.p1.33.m4.1.1" xref="S3.SS1.SSS3.5.p1.33.m4.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.33.m4.1.1.2" xref="S3.SS1.SSS3.5.p1.33.m4.1.1.2.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.33.m4.1.1.3" xref="S3.SS1.SSS3.5.p1.33.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.33.m4.1b"><apply id="S3.SS1.SSS3.5.p1.33.m4.1.1.cmml" xref="S3.SS1.SSS3.5.p1.33.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.33.m4.1.1.1.cmml" xref="S3.SS1.SSS3.5.p1.33.m4.1.1">superscript</csymbol><ci id="S3.SS1.SSS3.5.p1.33.m4.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.33.m4.1.1.2">𝑃</ci><ci id="S3.SS1.SSS3.5.p1.33.m4.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.33.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.33.m4.1c">P^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.33.m4.1d">italic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> has at most one boundary component containing <math alttext="v\in V(P)" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.34.m5.1"><semantics id="S3.SS1.SSS3.5.p1.34.m5.1a"><mrow id="S3.SS1.SSS3.5.p1.34.m5.1.2" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.cmml"><mi id="S3.SS1.SSS3.5.p1.34.m5.1.2.2" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.2.cmml">v</mi><mo id="S3.SS1.SSS3.5.p1.34.m5.1.2.1" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS3.5.p1.34.m5.1.2.3" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.cmml"><mi id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.2" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.2.cmml">V</mi><mo id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.1" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.3.2" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.cmml"><mo id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.cmml">(</mo><mi id="S3.SS1.SSS3.5.p1.34.m5.1.1" xref="S3.SS1.SSS3.5.p1.34.m5.1.1.cmml">P</mi><mo id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.34.m5.1b"><apply id="S3.SS1.SSS3.5.p1.34.m5.1.2.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2"><in id="S3.SS1.SSS3.5.p1.34.m5.1.2.1.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.1"></in><ci id="S3.SS1.SSS3.5.p1.34.m5.1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.2">𝑣</ci><apply id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3"><times id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.1.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.1"></times><ci id="S3.SS1.SSS3.5.p1.34.m5.1.2.3.2.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.2.3.2">𝑉</ci><ci id="S3.SS1.SSS3.5.p1.34.m5.1.1.cmml" xref="S3.SS1.SSS3.5.p1.34.m5.1.1">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.34.m5.1c">v\in V(P)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.34.m5.1d">italic_v ∈ italic_V ( italic_P )</annotation></semantics></math> by <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.7</span></a> and <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem10" title="Lemma 3.10. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.10</span></a>, we can apply <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem11" title="Lemma 3.11. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.11</span></a> to (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E33" title="Equation 33 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">33</span></a>) to get</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex42"> <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_{v\in V(P)}\sum_{R\in K(v)}\mathds{1}_{Q_{R}^{v}}=\sum_{v\in V(P)}(% \absolutevalue{K(v)}-1+\mathds{1}_{Q^{v}_{P}})=d(P)+\sum_{v\in V(P)}\mathds{1}% _{Q^{v}_{P}}." class="ltx_Math" display="block" id="S3.Ex42.m1.7"><semantics id="S3.Ex42.m1.7a"><mrow id="S3.Ex42.m1.7.7.1" xref="S3.Ex42.m1.7.7.1.1.cmml"><mrow id="S3.Ex42.m1.7.7.1.1" xref="S3.Ex42.m1.7.7.1.1.cmml"><mrow id="S3.Ex42.m1.7.7.1.1.3" xref="S3.Ex42.m1.7.7.1.1.3.cmml"><munder id="S3.Ex42.m1.7.7.1.1.3.1" xref="S3.Ex42.m1.7.7.1.1.3.1.cmml"><mo id="S3.Ex42.m1.7.7.1.1.3.1.2" movablelimits="false" xref="S3.Ex42.m1.7.7.1.1.3.1.2.cmml">∑</mo><mrow id="S3.Ex42.m1.2.2.1" xref="S3.Ex42.m1.2.2.1.cmml"><mi id="S3.Ex42.m1.2.2.1.3" xref="S3.Ex42.m1.2.2.1.3.cmml">v</mi><mo id="S3.Ex42.m1.2.2.1.2" xref="S3.Ex42.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex42.m1.2.2.1.4" xref="S3.Ex42.m1.2.2.1.4.cmml"><mi id="S3.Ex42.m1.2.2.1.4.2" xref="S3.Ex42.m1.2.2.1.4.2.cmml">V</mi><mo id="S3.Ex42.m1.2.2.1.4.1" xref="S3.Ex42.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.Ex42.m1.2.2.1.4.3.2" xref="S3.Ex42.m1.2.2.1.4.cmml"><mo id="S3.Ex42.m1.2.2.1.4.3.2.1" stretchy="false" xref="S3.Ex42.m1.2.2.1.4.cmml">(</mo><mi id="S3.Ex42.m1.2.2.1.1" xref="S3.Ex42.m1.2.2.1.1.cmml">P</mi><mo id="S3.Ex42.m1.2.2.1.4.3.2.2" stretchy="false" xref="S3.Ex42.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex42.m1.7.7.1.1.3.2" xref="S3.Ex42.m1.7.7.1.1.3.2.cmml"><munder id="S3.Ex42.m1.7.7.1.1.3.2.1" xref="S3.Ex42.m1.7.7.1.1.3.2.1.cmml"><mo id="S3.Ex42.m1.7.7.1.1.3.2.1.2" lspace="0.167em" movablelimits="false" xref="S3.Ex42.m1.7.7.1.1.3.2.1.2.cmml">∑</mo><mrow id="S3.Ex42.m1.3.3.1" xref="S3.Ex42.m1.3.3.1.cmml"><mi id="S3.Ex42.m1.3.3.1.3" xref="S3.Ex42.m1.3.3.1.3.cmml">R</mi><mo id="S3.Ex42.m1.3.3.1.2" xref="S3.Ex42.m1.3.3.1.2.cmml">∈</mo><mrow id="S3.Ex42.m1.3.3.1.4" xref="S3.Ex42.m1.3.3.1.4.cmml"><mi id="S3.Ex42.m1.3.3.1.4.2" xref="S3.Ex42.m1.3.3.1.4.2.cmml">K</mi><mo id="S3.Ex42.m1.3.3.1.4.1" xref="S3.Ex42.m1.3.3.1.4.1.cmml">⁢</mo><mrow id="S3.Ex42.m1.3.3.1.4.3.2" xref="S3.Ex42.m1.3.3.1.4.cmml"><mo id="S3.Ex42.m1.3.3.1.4.3.2.1" stretchy="false" xref="S3.Ex42.m1.3.3.1.4.cmml">(</mo><mi id="S3.Ex42.m1.3.3.1.1" xref="S3.Ex42.m1.3.3.1.1.cmml">v</mi><mo id="S3.Ex42.m1.3.3.1.4.3.2.2" stretchy="false" xref="S3.Ex42.m1.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></munder><msub id="S3.Ex42.m1.7.7.1.1.3.2.2" xref="S3.Ex42.m1.7.7.1.1.3.2.2.cmml"><mn id="S3.Ex42.m1.7.7.1.1.3.2.2.2" xref="S3.Ex42.m1.7.7.1.1.3.2.2.2.cmml">𝟙</mn><msubsup id="S3.Ex42.m1.7.7.1.1.3.2.2.3" xref="S3.Ex42.m1.7.7.1.1.3.2.2.3.cmml"><mi id="S3.Ex42.m1.7.7.1.1.3.2.2.3.2.2" xref="S3.Ex42.m1.7.7.1.1.3.2.2.3.2.2.cmml">Q</mi><mi id="S3.Ex42.m1.7.7.1.1.3.2.2.3.2.3" xref="S3.Ex42.m1.7.7.1.1.3.2.2.3.2.3.cmml">R</mi><mi id="S3.Ex42.m1.7.7.1.1.3.2.2.3.3" xref="S3.Ex42.m1.7.7.1.1.3.2.2.3.3.cmml">v</mi></msubsup></msub></mrow></mrow><mo id="S3.Ex42.m1.7.7.1.1.4" rspace="0.111em" xref="S3.Ex42.m1.7.7.1.1.4.cmml">=</mo><mrow id="S3.Ex42.m1.7.7.1.1.1" xref="S3.Ex42.m1.7.7.1.1.1.cmml"><munder id="S3.Ex42.m1.7.7.1.1.1.2" xref="S3.Ex42.m1.7.7.1.1.1.2.cmml"><mo id="S3.Ex42.m1.7.7.1.1.1.2.2" movablelimits="false" rspace="0em" xref="S3.Ex42.m1.7.7.1.1.1.2.2.cmml">∑</mo><mrow id="S3.Ex42.m1.4.4.1" xref="S3.Ex42.m1.4.4.1.cmml"><mi id="S3.Ex42.m1.4.4.1.3" xref="S3.Ex42.m1.4.4.1.3.cmml">v</mi><mo id="S3.Ex42.m1.4.4.1.2" xref="S3.Ex42.m1.4.4.1.2.cmml">∈</mo><mrow id="S3.Ex42.m1.4.4.1.4" xref="S3.Ex42.m1.4.4.1.4.cmml"><mi id="S3.Ex42.m1.4.4.1.4.2" xref="S3.Ex42.m1.4.4.1.4.2.cmml">V</mi><mo id="S3.Ex42.m1.4.4.1.4.1" xref="S3.Ex42.m1.4.4.1.4.1.cmml">⁢</mo><mrow id="S3.Ex42.m1.4.4.1.4.3.2" xref="S3.Ex42.m1.4.4.1.4.cmml"><mo id="S3.Ex42.m1.4.4.1.4.3.2.1" stretchy="false" xref="S3.Ex42.m1.4.4.1.4.cmml">(</mo><mi id="S3.Ex42.m1.4.4.1.1" xref="S3.Ex42.m1.4.4.1.1.cmml">P</mi><mo id="S3.Ex42.m1.4.4.1.4.3.2.2" stretchy="false" xref="S3.Ex42.m1.4.4.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S3.Ex42.m1.7.7.1.1.1.1.1" xref="S3.Ex42.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S3.Ex42.m1.7.7.1.1.1.1.1.2" stretchy="false" 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_{Q^{v}_{P}}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex42.m1.7d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_R ∈ italic_K ( italic_v ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT ( | start_ARG italic_K ( italic_v ) end_ARG | - 1 + blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_d ( italic_P ) + ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT 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.SS1.SSS3.5.p1.35">Here, we have used that <math alttext="|K(v)|=\deg_{P}(v)/2" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.35.m1.4"><semantics id="S3.SS1.SSS3.5.p1.35.m1.4a"><mrow id="S3.SS1.SSS3.5.p1.35.m1.4.4" xref="S3.SS1.SSS3.5.p1.35.m1.4.4.cmml"><mrow id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.2.cmml"><mo id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.2" stretchy="false" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.cmml"><mi id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.2" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.2.cmml">K</mi><mo id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.1" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.3.2" xref="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.cmml"><mo id="S3.SS1.SSS3.5.p1.35.m1.3.3.1.1.1.3.2.1" stretchy="false" 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id="S3.SS1.SSS3.5.p1.35.m1.4.4.2.1.1.1.2.cmml" xref="S3.SS1.SSS3.5.p1.35.m1.4.4.2.1.1.1.2">degree</csymbol><ci id="S3.SS1.SSS3.5.p1.35.m1.4.4.2.1.1.1.3.cmml" xref="S3.SS1.SSS3.5.p1.35.m1.4.4.2.1.1.1.3">𝑃</ci></apply><ci id="S3.SS1.SSS3.5.p1.35.m1.2.2.cmml" xref="S3.SS1.SSS3.5.p1.35.m1.2.2">𝑣</ci></apply><cn id="S3.SS1.SSS3.5.p1.35.m1.4.4.2.3.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.35.m1.4.4.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.35.m1.4c">|K(v)|=\deg_{P}(v)/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.35.m1.4d">| italic_K ( italic_v ) | = roman_deg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_v ) / 2</annotation></semantics></math>, since every boundary component has two incident edges per vertex. Substituting this into (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E33" title="Equation 33 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">33</span></a>), together with (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S3.E32" title="Equation 32 ‣ Proof of 3.2. ‣ 3.1.3 General case ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">32</span></a>), gives</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex43"> <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="\mathds{1}_{P}-1=-n_{a}(P)-h+d(P)+\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}+\sum_% {e\in E_{l}(P)}\mathds{1}_{H^{e}_{P}}-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P% }}." class="ltx_Math" display="block" id="S3.Ex43.m1.6"><semantics id="S3.Ex43.m1.6a"><mrow id="S3.Ex43.m1.6.6.1" xref="S3.Ex43.m1.6.6.1.1.cmml"><mrow id="S3.Ex43.m1.6.6.1.1" xref="S3.Ex43.m1.6.6.1.1.cmml"><mrow id="S3.Ex43.m1.6.6.1.1.2" xref="S3.Ex43.m1.6.6.1.1.2.cmml"><msub id="S3.Ex43.m1.6.6.1.1.2.2" xref="S3.Ex43.m1.6.6.1.1.2.2.cmml"><mn id="S3.Ex43.m1.6.6.1.1.2.2.2" xref="S3.Ex43.m1.6.6.1.1.2.2.2.cmml">𝟙</mn><mi id="S3.Ex43.m1.6.6.1.1.2.2.3" xref="S3.Ex43.m1.6.6.1.1.2.2.3.cmml">P</mi></msub><mo id="S3.Ex43.m1.6.6.1.1.2.1" xref="S3.Ex43.m1.6.6.1.1.2.1.cmml">−</mo><mn id="S3.Ex43.m1.6.6.1.1.2.3" xref="S3.Ex43.m1.6.6.1.1.2.3.cmml">1</mn></mrow><mo id="S3.Ex43.m1.6.6.1.1.1" xref="S3.Ex43.m1.6.6.1.1.1.cmml">=</mo><mrow id="S3.Ex43.m1.6.6.1.1.3" xref="S3.Ex43.m1.6.6.1.1.3.cmml"><mrow id="S3.Ex43.m1.6.6.1.1.3.2" xref="S3.Ex43.m1.6.6.1.1.3.2.cmml"><mrow id="S3.Ex43.m1.6.6.1.1.3.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.cmml"><mrow id="S3.Ex43.m1.6.6.1.1.3.2.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.cmml"><mo id="S3.Ex43.m1.6.6.1.1.3.2.2.2a" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.cmml">−</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.cmml"><msub id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2.cmml"><mi id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2.2.cmml">n</mi><mi id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2.3" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.2.3.cmml">a</mi></msub><mo id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.1" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.3.2" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.cmml"><mo id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.3.2.1" stretchy="false" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.cmml">(</mo><mi id="S3.Ex43.m1.4.4" xref="S3.Ex43.m1.4.4.cmml">P</mi><mo id="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.3.2.2" stretchy="false" xref="S3.Ex43.m1.6.6.1.1.3.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex43.m1.6.6.1.1.3.2.2.1" xref="S3.Ex43.m1.6.6.1.1.3.2.2.1.cmml">−</mo><mi id="S3.Ex43.m1.6.6.1.1.3.2.2.3" xref="S3.Ex43.m1.6.6.1.1.3.2.2.3.cmml">h</mi></mrow><mo id="S3.Ex43.m1.6.6.1.1.3.2.1" xref="S3.Ex43.m1.6.6.1.1.3.2.1.cmml">+</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.3" xref="S3.Ex43.m1.6.6.1.1.3.2.3.cmml"><mi id="S3.Ex43.m1.6.6.1.1.3.2.3.2" xref="S3.Ex43.m1.6.6.1.1.3.2.3.2.cmml">d</mi><mo id="S3.Ex43.m1.6.6.1.1.3.2.3.1" xref="S3.Ex43.m1.6.6.1.1.3.2.3.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.3.3.2" xref="S3.Ex43.m1.6.6.1.1.3.2.3.cmml"><mo id="S3.Ex43.m1.6.6.1.1.3.2.3.3.2.1" stretchy="false" xref="S3.Ex43.m1.6.6.1.1.3.2.3.cmml">(</mo><mi id="S3.Ex43.m1.5.5" xref="S3.Ex43.m1.5.5.cmml">P</mi><mo id="S3.Ex43.m1.6.6.1.1.3.2.3.3.2.2" stretchy="false" xref="S3.Ex43.m1.6.6.1.1.3.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex43.m1.6.6.1.1.3.2.1a" rspace="0.055em" xref="S3.Ex43.m1.6.6.1.1.3.2.1.cmml">+</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.4" xref="S3.Ex43.m1.6.6.1.1.3.2.4.cmml"><munder id="S3.Ex43.m1.6.6.1.1.3.2.4.1" xref="S3.Ex43.m1.6.6.1.1.3.2.4.1.cmml"><mo id="S3.Ex43.m1.6.6.1.1.3.2.4.1.2" movablelimits="false" xref="S3.Ex43.m1.6.6.1.1.3.2.4.1.2.cmml">∑</mo><mrow id="S3.Ex43.m1.1.1.1" xref="S3.Ex43.m1.1.1.1.cmml"><mi id="S3.Ex43.m1.1.1.1.3" xref="S3.Ex43.m1.1.1.1.3.cmml">v</mi><mo id="S3.Ex43.m1.1.1.1.2" xref="S3.Ex43.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex43.m1.1.1.1.4" xref="S3.Ex43.m1.1.1.1.4.cmml"><mi id="S3.Ex43.m1.1.1.1.4.2" xref="S3.Ex43.m1.1.1.1.4.2.cmml">V</mi><mo id="S3.Ex43.m1.1.1.1.4.1" xref="S3.Ex43.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.1.1.1.4.3.2" xref="S3.Ex43.m1.1.1.1.4.cmml"><mo id="S3.Ex43.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex43.m1.1.1.1.4.cmml">(</mo><mi id="S3.Ex43.m1.1.1.1.1" xref="S3.Ex43.m1.1.1.1.1.cmml">P</mi><mo id="S3.Ex43.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex43.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><msub id="S3.Ex43.m1.6.6.1.1.3.2.4.2" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.cmml"><mn id="S3.Ex43.m1.6.6.1.1.3.2.4.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.2.cmml">𝟙</mn><msubsup id="S3.Ex43.m1.6.6.1.1.3.2.4.2.3" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.cmml"><mi id="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.2.2.cmml">Q</mi><mi id="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.3" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.3.cmml">P</mi><mi id="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.2.3" xref="S3.Ex43.m1.6.6.1.1.3.2.4.2.3.2.3.cmml">v</mi></msubsup></msub></mrow><mo id="S3.Ex43.m1.6.6.1.1.3.2.1b" rspace="0.055em" xref="S3.Ex43.m1.6.6.1.1.3.2.1.cmml">+</mo><mrow id="S3.Ex43.m1.6.6.1.1.3.2.5" xref="S3.Ex43.m1.6.6.1.1.3.2.5.cmml"><munder id="S3.Ex43.m1.6.6.1.1.3.2.5.1" xref="S3.Ex43.m1.6.6.1.1.3.2.5.1.cmml"><mo id="S3.Ex43.m1.6.6.1.1.3.2.5.1.2" movablelimits="false" xref="S3.Ex43.m1.6.6.1.1.3.2.5.1.2.cmml">∑</mo><mrow id="S3.Ex43.m1.2.2.1" xref="S3.Ex43.m1.2.2.1.cmml"><mi id="S3.Ex43.m1.2.2.1.3" xref="S3.Ex43.m1.2.2.1.3.cmml">e</mi><mo id="S3.Ex43.m1.2.2.1.2" xref="S3.Ex43.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex43.m1.2.2.1.4" xref="S3.Ex43.m1.2.2.1.4.cmml"><msub id="S3.Ex43.m1.2.2.1.4.2" xref="S3.Ex43.m1.2.2.1.4.2.cmml"><mi id="S3.Ex43.m1.2.2.1.4.2.2" xref="S3.Ex43.m1.2.2.1.4.2.2.cmml">E</mi><mi id="S3.Ex43.m1.2.2.1.4.2.3" xref="S3.Ex43.m1.2.2.1.4.2.3.cmml">l</mi></msub><mo id="S3.Ex43.m1.2.2.1.4.1" xref="S3.Ex43.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S3.Ex43.m1.2.2.1.4.3.2" xref="S3.Ex43.m1.2.2.1.4.cmml"><mo id="S3.Ex43.m1.2.2.1.4.3.2.1" stretchy="false" xref="S3.Ex43.m1.2.2.1.4.cmml">(</mo><mi id="S3.Ex43.m1.2.2.1.1" xref="S3.Ex43.m1.2.2.1.1.cmml">P</mi><mo id="S3.Ex43.m1.2.2.1.4.3.2.2" stretchy="false" xref="S3.Ex43.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><msub id="S3.Ex43.m1.6.6.1.1.3.2.5.2" xref="S3.Ex43.m1.6.6.1.1.3.2.5.2.cmml"><mn id="S3.Ex43.m1.6.6.1.1.3.2.5.2.2" xref="S3.Ex43.m1.6.6.1.1.3.2.5.2.2.cmml">𝟙</mn><msubsup id="S3.Ex43.m1.6.6.1.1.3.2.5.2.3" 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id="S3.Ex43.m1.6c">\mathds{1}_{P}-1=-n_{a}(P)-h+d(P)+\sum_{v\in V(P)}\mathds{1}_{Q^{v}_{P}}+\sum_% {e\in E_{l}(P)}\mathds{1}_{H^{e}_{P}}-\sum_{e\in E_{b}(P)}\mathds{1}_{H^{e}_{P% }}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex43.m1.6d">blackboard_1 start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - 1 = - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - italic_h + italic_d ( italic_P ) + ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUBSCRIPT - ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_P ) end_POSTSUBSCRIPT blackboard_1 start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT 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.SS1.SSS3.5.p1.37">Adding <math alttext="1" class="ltx_Math" display="inline" id="S3.SS1.SSS3.5.p1.36.m1.1"><semantics id="S3.SS1.SSS3.5.p1.36.m1.1a"><mn id="S3.SS1.SSS3.5.p1.36.m1.1.1" xref="S3.SS1.SSS3.5.p1.36.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.36.m1.1b"><cn id="S3.SS1.SSS3.5.p1.36.m1.1.1.cmml" type="integer" xref="S3.SS1.SSS3.5.p1.36.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.36.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.36.m1.1d">1</annotation></semantics></math> to both 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xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.1a" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.1.cmml">−</mo><mi id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.4" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.4.cmml">h</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS3.5.p1.37.m2.3b"><apply id="S3.SS1.SSS3.5.p1.37.m2.3.4.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4"><eq id="S3.SS1.SSS3.5.p1.37.m2.3.4.1.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.1"></eq><apply id="S3.SS1.SSS3.5.p1.37.m2.3.4.2.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.2"><times id="S3.SS1.SSS3.5.p1.37.m2.3.4.2.1.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.2.1"></times><ci id="S3.SS1.SSS3.5.p1.37.m2.3.4.2.2.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.2.2">𝑐</ci><ci id="S3.SS1.SSS3.5.p1.37.m2.1.1.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.1.1">𝑃</ci></apply><apply id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3"><minus id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.1.cmml" 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xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2.1.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2">subscript</csymbol><ci id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2.2.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2.2">𝑛</ci><ci id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2.3.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.3.2.3">𝑎</ci></apply><ci id="S3.SS1.SSS3.5.p1.37.m2.3.3.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.3">𝑃</ci></apply><ci id="S3.SS1.SSS3.5.p1.37.m2.3.4.3.4.cmml" xref="S3.SS1.SSS3.5.p1.37.m2.3.4.3.4">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS3.5.p1.37.m2.3c">c(P)=1+d(P)-n_{a}(P)-h</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS3.5.p1.37.m2.3d">italic_c ( italic_P ) = 1 + italic_d ( italic_P ) - italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_P ) - italic_h</annotation></semantics></math> yields the desired result. ∎</p> </div> </div> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>max-Representation of CPA Functions</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.12">The results of this section reduce the representation of <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">roman_CPA</annotation></semantics></math> functions, given by <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a>, to a sum of nested signed maxima. 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xref="S4.p1.2.m2.1.2.2">𝑣</ci><apply id="S4.p1.2.m2.1.2.3.cmml" xref="S4.p1.2.m2.1.2.3"><times id="S4.p1.2.m2.1.2.3.1.cmml" xref="S4.p1.2.m2.1.2.3.1"></times><ci id="S4.p1.2.m2.1.2.3.2.cmml" xref="S4.p1.2.m2.1.2.3.2">𝑉</ci><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">v\in V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_v ∈ italic_V ( italic_f )</annotation></semantics></math>, its <em class="ltx_emph ltx_font_italic" id="S4.p1.12.2">degree</em> <math alttext="\deg(v):=|\{e\in E(f):v\in e\}|" class="ltx_Math" display="inline" id="S4.p1.3.m3.4"><semantics id="S4.p1.3.m3.4a"><mrow id="S4.p1.3.m3.4.4" xref="S4.p1.3.m3.4.4.cmml"><mrow id="S4.p1.3.m3.4.4.3.2" xref="S4.p1.3.m3.4.4.3.1.cmml"><mi id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">deg</mi><mo id="S4.p1.3.m3.4.4.3.2a" xref="S4.p1.3.m3.4.4.3.1.cmml">⁡</mo><mrow id="S4.p1.3.m3.4.4.3.2.1" 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stretchy="false" xref="S4.p1.3.m3.4.4.1.1.1.3.1.cmml">}</mo></mrow><mo id="S4.p1.3.m3.4.4.1.1.3" stretchy="false" xref="S4.p1.3.m3.4.4.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.4b"><apply id="S4.p1.3.m3.4.4.cmml" xref="S4.p1.3.m3.4.4"><csymbol cd="latexml" id="S4.p1.3.m3.4.4.2.cmml" xref="S4.p1.3.m3.4.4.2">assign</csymbol><apply id="S4.p1.3.m3.4.4.3.1.cmml" xref="S4.p1.3.m3.4.4.3.2"><csymbol cd="latexml" id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1">degree</csymbol><ci id="S4.p1.3.m3.2.2.cmml" xref="S4.p1.3.m3.2.2">𝑣</ci></apply><apply id="S4.p1.3.m3.4.4.1.2.cmml" xref="S4.p1.3.m3.4.4.1.1"><abs id="S4.p1.3.m3.4.4.1.2.1.cmml" xref="S4.p1.3.m3.4.4.1.1.2"></abs><apply id="S4.p1.3.m3.4.4.1.1.1.3.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2"><csymbol cd="latexml" id="S4.p1.3.m3.4.4.1.1.1.3.1.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2.3">conditional-set</csymbol><apply id="S4.p1.3.m3.4.4.1.1.1.1.1.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1"><in id="S4.p1.3.m3.4.4.1.1.1.1.1.1.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1.1"></in><ci id="S4.p1.3.m3.4.4.1.1.1.1.1.2.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1.2">𝑒</ci><apply id="S4.p1.3.m3.4.4.1.1.1.1.1.3.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1.3"><times id="S4.p1.3.m3.4.4.1.1.1.1.1.3.1.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1.3.1"></times><ci id="S4.p1.3.m3.4.4.1.1.1.1.1.3.2.cmml" xref="S4.p1.3.m3.4.4.1.1.1.1.1.3.2">𝐸</ci><ci id="S4.p1.3.m3.3.3.cmml" xref="S4.p1.3.m3.3.3">𝑓</ci></apply></apply><apply id="S4.p1.3.m3.4.4.1.1.1.2.2.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2.2"><in id="S4.p1.3.m3.4.4.1.1.1.2.2.1.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2.2.1"></in><ci id="S4.p1.3.m3.4.4.1.1.1.2.2.2.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2.2.2">𝑣</ci><ci id="S4.p1.3.m3.4.4.1.1.1.2.2.3.cmml" xref="S4.p1.3.m3.4.4.1.1.1.2.2.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.4c">\deg(v):=|\{e\in E(f):v\in e\}|</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.4d">roman_deg ( italic_v ) := | { italic_e ∈ italic_E ( italic_f ) : italic_v ∈ italic_e } |</annotation></semantics></math> is defined as the number of edges that contain <math alttext="v" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><mi id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><ci id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">italic_v</annotation></semantics></math> as a vertex. A <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mi id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><ci id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">roman_CPA</annotation></semantics></math> function is called <em class="ltx_emph ltx_font_italic" id="S4.p1.6.1"><math alttext="v" class="ltx_Math" display="inline" id="S4.p1.6.1.m1.1"><semantics id="S4.p1.6.1.m1.1a"><mi id="S4.p1.6.1.m1.1.1" xref="S4.p1.6.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p1.6.1.m1.1b"><ci id="S4.p1.6.1.m1.1.1.cmml" xref="S4.p1.6.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.1.m1.1d">italic_v</annotation></semantics></math>-function</em> if <math alttext="v\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S4.p1.7.m6.1"><semantics id="S4.p1.7.m6.1a"><mrow id="S4.p1.7.m6.1.1" xref="S4.p1.7.m6.1.1.cmml"><mi id="S4.p1.7.m6.1.1.2" xref="S4.p1.7.m6.1.1.2.cmml">v</mi><mo id="S4.p1.7.m6.1.1.1" xref="S4.p1.7.m6.1.1.1.cmml">∈</mo><msup id="S4.p1.7.m6.1.1.3" xref="S4.p1.7.m6.1.1.3.cmml"><mi id="S4.p1.7.m6.1.1.3.2" xref="S4.p1.7.m6.1.1.3.2.cmml">ℝ</mi><mn id="S4.p1.7.m6.1.1.3.3" xref="S4.p1.7.m6.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.7.m6.1b"><apply id="S4.p1.7.m6.1.1.cmml" xref="S4.p1.7.m6.1.1"><in id="S4.p1.7.m6.1.1.1.cmml" xref="S4.p1.7.m6.1.1.1"></in><ci id="S4.p1.7.m6.1.1.2.cmml" xref="S4.p1.7.m6.1.1.2">𝑣</ci><apply id="S4.p1.7.m6.1.1.3.cmml" xref="S4.p1.7.m6.1.1.3"><csymbol cd="ambiguous" id="S4.p1.7.m6.1.1.3.1.cmml" xref="S4.p1.7.m6.1.1.3">superscript</csymbol><ci id="S4.p1.7.m6.1.1.3.2.cmml" xref="S4.p1.7.m6.1.1.3.2">ℝ</ci><cn id="S4.p1.7.m6.1.1.3.3.cmml" type="integer" xref="S4.p1.7.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m6.1c">v\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m6.1d">italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is its only vertex and all its edges are rays. Note, that <math alttext="f\in\operatorname{CPA}_{\deg(v)}" class="ltx_Math" display="inline" id="S4.p1.8.m7.2"><semantics id="S4.p1.8.m7.2a"><mrow id="S4.p1.8.m7.2.3" xref="S4.p1.8.m7.2.3.cmml"><mi id="S4.p1.8.m7.2.3.2" xref="S4.p1.8.m7.2.3.2.cmml">f</mi><mo id="S4.p1.8.m7.2.3.1" xref="S4.p1.8.m7.2.3.1.cmml">∈</mo><msub id="S4.p1.8.m7.2.3.3" xref="S4.p1.8.m7.2.3.3.cmml"><mi id="S4.p1.8.m7.2.3.3.2" xref="S4.p1.8.m7.2.3.3.2.cmml">CPA</mi><mrow id="S4.p1.8.m7.2.2.2.4" xref="S4.p1.8.m7.2.2.2.3.cmml"><mi id="S4.p1.8.m7.1.1.1.1" xref="S4.p1.8.m7.1.1.1.1.cmml">deg</mi><mo id="S4.p1.8.m7.2.2.2.4a" xref="S4.p1.8.m7.2.2.2.3.cmml">⁡</mo><mrow id="S4.p1.8.m7.2.2.2.4.1" xref="S4.p1.8.m7.2.2.2.3.cmml"><mo id="S4.p1.8.m7.2.2.2.4.1.1" stretchy="false" xref="S4.p1.8.m7.2.2.2.3.cmml">(</mo><mi id="S4.p1.8.m7.2.2.2.2" xref="S4.p1.8.m7.2.2.2.2.cmml">v</mi><mo id="S4.p1.8.m7.2.2.2.4.1.2" stretchy="false" xref="S4.p1.8.m7.2.2.2.3.cmml">)</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.8.m7.2b"><apply id="S4.p1.8.m7.2.3.cmml" xref="S4.p1.8.m7.2.3"><in id="S4.p1.8.m7.2.3.1.cmml" xref="S4.p1.8.m7.2.3.1"></in><ci id="S4.p1.8.m7.2.3.2.cmml" xref="S4.p1.8.m7.2.3.2">𝑓</ci><apply id="S4.p1.8.m7.2.3.3.cmml" xref="S4.p1.8.m7.2.3.3"><csymbol cd="ambiguous" id="S4.p1.8.m7.2.3.3.1.cmml" xref="S4.p1.8.m7.2.3.3">subscript</csymbol><ci id="S4.p1.8.m7.2.3.3.2.cmml" xref="S4.p1.8.m7.2.3.3.2">CPA</ci><apply id="S4.p1.8.m7.2.2.2.3.cmml" xref="S4.p1.8.m7.2.2.2.4"><csymbol cd="latexml" id="S4.p1.8.m7.1.1.1.1.cmml" xref="S4.p1.8.m7.1.1.1.1">degree</csymbol><ci id="S4.p1.8.m7.2.2.2.2.cmml" xref="S4.p1.8.m7.2.2.2.2">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m7.2c">f\in\operatorname{CPA}_{\deg(v)}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m7.2d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT roman_deg ( italic_v ) end_POSTSUBSCRIPT</annotation></semantics></math> for every <math alttext="v" class="ltx_Math" display="inline" id="S4.p1.9.m8.1"><semantics id="S4.p1.9.m8.1a"><mi id="S4.p1.9.m8.1.1" xref="S4.p1.9.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p1.9.m8.1b"><ci id="S4.p1.9.m8.1.1.cmml" xref="S4.p1.9.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m8.1d">italic_v</annotation></semantics></math>-function <math alttext="f" class="ltx_Math" display="inline" id="S4.p1.10.m9.1"><semantics id="S4.p1.10.m9.1a"><mi id="S4.p1.10.m9.1.1" xref="S4.p1.10.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.p1.10.m9.1b"><ci id="S4.p1.10.m9.1.1.cmml" xref="S4.p1.10.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.10.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.p1.10.m9.1d">italic_f</annotation></semantics></math>, and that any <math alttext="f\in\operatorname{CPL}" class="ltx_Math" display="inline" id="S4.p1.11.m10.1"><semantics id="S4.p1.11.m10.1a"><mrow id="S4.p1.11.m10.1.1" xref="S4.p1.11.m10.1.1.cmml"><mi id="S4.p1.11.m10.1.1.2" xref="S4.p1.11.m10.1.1.2.cmml">f</mi><mo id="S4.p1.11.m10.1.1.1" xref="S4.p1.11.m10.1.1.1.cmml">∈</mo><mi id="S4.p1.11.m10.1.1.3" xref="S4.p1.11.m10.1.1.3.cmml">CPL</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.11.m10.1b"><apply id="S4.p1.11.m10.1.1.cmml" xref="S4.p1.11.m10.1.1"><in id="S4.p1.11.m10.1.1.1.cmml" xref="S4.p1.11.m10.1.1.1"></in><ci id="S4.p1.11.m10.1.1.2.cmml" xref="S4.p1.11.m10.1.1.2">𝑓</ci><ci id="S4.p1.11.m10.1.1.3.cmml" xref="S4.p1.11.m10.1.1.3">CPL</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.11.m10.1c">f\in\operatorname{CPL}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.11.m10.1d">italic_f ∈ roman_CPL</annotation></semantics></math> is a <math alttext="0" class="ltx_Math" display="inline" id="S4.p1.12.m11.1"><semantics id="S4.p1.12.m11.1a"><mn id="S4.p1.12.m11.1.1" xref="S4.p1.12.m11.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.p1.12.m11.1b"><cn id="S4.p1.12.m11.1.1.cmml" type="integer" xref="S4.p1.12.m11.1.1">0</cn></annotation-xml></semantics></math>-function.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.2">In general, the number of vertices and edges of a <math alttext="\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><msub id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">CPA</mi><mi id="S4.p2.1.m1.1.1.3" xref="S4.p2.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1">subscript</csymbol><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">CPA</ci><ci id="S4.p2.1.m1.1.1.3.cmml" xref="S4.p2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> function is not bounded by <math alttext="p" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mi id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><ci id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_p</annotation></semantics></math>, since arbitrary edges can be split by adding vertices. Consequently, the number of summands in the representation from <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a> is not known. However, there always exists a sparse choice of vertices, as described in the following lemma.</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">For every <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem1.p1.1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.1b"><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1"><in id="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.2">CPA</ci><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, there exists an admissible set of pieces <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">caligraphic_P</annotation></semantics></math>, with corresponding sets of vertices <math alttext="V" 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">V</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">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.1d">italic_V</annotation></semantics></math> and edges <math alttext="E" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.4.m4.1a"><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.1b"><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.1d">italic_E</annotation></semantics></math>, such that <math alttext="|\mathcal{P}|\leq p" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.1"><semantics id="S4.Thmtheorem1.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem1.p1.5.5.m5.1.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml"><mrow id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">𝒫</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.1.cmml">≤</mo><mi id="S4.Thmtheorem1.p1.5.5.m5.1.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.1b"><apply id="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2"><leq id="S4.Thmtheorem1.p1.5.5.m5.1.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.1"></leq><apply id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2"><abs id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2.1"></abs><ci id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">𝒫</ci></apply><ci id="S4.Thmtheorem1.p1.5.5.m5.1.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.1c">|\mathcal{P}|\leq p</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.1d">| caligraphic_P | ≤ italic_p</annotation></semantics></math>, <math alttext="\deg(v)\geq 3" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.6.m6.2"><semantics id="S4.Thmtheorem1.p1.6.6.m6.2a"><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.cmml"><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml"><mi id="S4.Thmtheorem1.p1.6.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml">deg</mi><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2a" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml">⁡</mo><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2.1" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml"><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2.1.1" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml">(</mo><mi id="S4.Thmtheorem1.p1.6.6.m6.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.2.cmml">v</mi><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.1" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.1.cmml">≥</mo><mn id="S4.Thmtheorem1.p1.6.6.m6.2.3.3" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.6.m6.2b"><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3"><geq id="S4.Thmtheorem1.p1.6.6.m6.2.3.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.1"></geq><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2"><csymbol cd="latexml" id="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1">degree</csymbol><ci id="S4.Thmtheorem1.p1.6.6.m6.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.2">𝑣</ci></apply><cn id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.6.m6.2c">\deg(v)\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.6.m6.2d">roman_deg ( italic_v ) ≥ 3</annotation></semantics></math> for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.7.7.m7.1"><semantics id="S4.Thmtheorem1.p1.7.7.m7.1a"><mrow id="S4.Thmtheorem1.p1.7.7.m7.1.1" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem1.p1.7.7.m7.1.1.2" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem1.p1.7.7.m7.1.1.1" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem1.p1.7.7.m7.1.1.3" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.7.7.m7.1b"><apply id="S4.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem1.p1.7.7.m7.1.1"><in id="S4.Thmtheorem1.p1.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.1"></in><ci id="S4.Thmtheorem1.p1.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.2">𝑣</ci><ci id="S4.Thmtheorem1.p1.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem1.p1.7.7.m7.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.7.7.m7.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.7.7.m7.1d">italic_v ∈ italic_V</annotation></semantics></math>, and <math alttext="|E|\leq 3p" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.8.8.m8.1"><semantics id="S4.Thmtheorem1.p1.8.8.m8.1a"><mrow id="S4.Thmtheorem1.p1.8.8.m8.1.2" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.cmml"><mrow id="S4.Thmtheorem1.p1.8.8.m8.1.2.2.2" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.2.1.cmml"><mo id="S4.Thmtheorem1.p1.8.8.m8.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem1.p1.8.8.m8.1.1" xref="S4.Thmtheorem1.p1.8.8.m8.1.1.cmml">E</mi><mo id="S4.Thmtheorem1.p1.8.8.m8.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem1.p1.8.8.m8.1.2.1" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.1.cmml">≤</mo><mrow id="S4.Thmtheorem1.p1.8.8.m8.1.2.3" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.cmml"><mn id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.2" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.2.cmml">3</mn><mo id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.1" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.3" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.8.8.m8.1b"><apply id="S4.Thmtheorem1.p1.8.8.m8.1.2.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2"><leq id="S4.Thmtheorem1.p1.8.8.m8.1.2.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.1"></leq><apply id="S4.Thmtheorem1.p1.8.8.m8.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.2.2"><abs id="S4.Thmtheorem1.p1.8.8.m8.1.2.2.1.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.2.2.1"></abs><ci id="S4.Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.1">𝐸</ci></apply><apply id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3"><times id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.1.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.1"></times><cn id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.2">3</cn><ci id="S4.Thmtheorem1.p1.8.8.m8.1.2.3.3.cmml" xref="S4.Thmtheorem1.p1.8.8.m8.1.2.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.8.8.m8.1c">|E|\leq 3p</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.8.8.m8.1d">| italic_E | ≤ 3 italic_p</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.1.p1"> <p class="ltx_p" id="S4.1.p1.7">Let <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S4.1.p1.1.m1.1"><semantics id="S4.1.p1.1.m1.1a"><mrow id="S4.1.p1.1.m1.1.1" xref="S4.1.p1.1.m1.1.1.cmml"><mi id="S4.1.p1.1.m1.1.1.2" xref="S4.1.p1.1.m1.1.1.2.cmml">f</mi><mo id="S4.1.p1.1.m1.1.1.1" xref="S4.1.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.1.p1.1.m1.1.1.3" xref="S4.1.p1.1.m1.1.1.3.cmml"><mi id="S4.1.p1.1.m1.1.1.3.2" xref="S4.1.p1.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S4.1.p1.1.m1.1.1.3.3" xref="S4.1.p1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.1.m1.1b"><apply id="S4.1.p1.1.m1.1.1.cmml" xref="S4.1.p1.1.m1.1.1"><in id="S4.1.p1.1.m1.1.1.1.cmml" xref="S4.1.p1.1.m1.1.1.1"></in><ci id="S4.1.p1.1.m1.1.1.2.cmml" xref="S4.1.p1.1.m1.1.1.2">𝑓</ci><apply id="S4.1.p1.1.m1.1.1.3.cmml" xref="S4.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.1.p1.1.m1.1.1.3.1.cmml" xref="S4.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.1.p1.1.m1.1.1.3.2.cmml" xref="S4.1.p1.1.m1.1.1.3.2">CPA</ci><ci id="S4.1.p1.1.m1.1.1.3.3.cmml" xref="S4.1.p1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> be an arbitrary function. By definition, there is an admissible set of pieces <math alttext="\mathcal{P}(f)" class="ltx_Math" display="inline" id="S4.1.p1.2.m2.1"><semantics id="S4.1.p1.2.m2.1a"><mrow id="S4.1.p1.2.m2.1.2" xref="S4.1.p1.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.1.p1.2.m2.1.2.2" xref="S4.1.p1.2.m2.1.2.2.cmml">𝒫</mi><mo id="S4.1.p1.2.m2.1.2.1" xref="S4.1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.1.p1.2.m2.1.2.3.2" xref="S4.1.p1.2.m2.1.2.cmml"><mo id="S4.1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S4.1.p1.2.m2.1.2.cmml">(</mo><mi id="S4.1.p1.2.m2.1.1" xref="S4.1.p1.2.m2.1.1.cmml">f</mi><mo id="S4.1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S4.1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.2.m2.1b"><apply id="S4.1.p1.2.m2.1.2.cmml" xref="S4.1.p1.2.m2.1.2"><times id="S4.1.p1.2.m2.1.2.1.cmml" xref="S4.1.p1.2.m2.1.2.1"></times><ci id="S4.1.p1.2.m2.1.2.2.cmml" xref="S4.1.p1.2.m2.1.2.2">𝒫</ci><ci id="S4.1.p1.2.m2.1.1.cmml" xref="S4.1.p1.2.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.2.m2.1c">\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.2.m2.1d">caligraphic_P ( italic_f )</annotation></semantics></math> such that <math alttext="|\mathcal{P}(f)|=p" class="ltx_Math" display="inline" id="S4.1.p1.3.m3.2"><semantics id="S4.1.p1.3.m3.2a"><mrow id="S4.1.p1.3.m3.2.2" xref="S4.1.p1.3.m3.2.2.cmml"><mrow id="S4.1.p1.3.m3.2.2.1.1" xref="S4.1.p1.3.m3.2.2.1.2.cmml"><mo id="S4.1.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S4.1.p1.3.m3.2.2.1.2.1.cmml">|</mo><mrow id="S4.1.p1.3.m3.2.2.1.1.1" xref="S4.1.p1.3.m3.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.1.p1.3.m3.2.2.1.1.1.2" xref="S4.1.p1.3.m3.2.2.1.1.1.2.cmml">𝒫</mi><mo id="S4.1.p1.3.m3.2.2.1.1.1.1" xref="S4.1.p1.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.1.p1.3.m3.2.2.1.1.1.3.2" xref="S4.1.p1.3.m3.2.2.1.1.1.cmml"><mo id="S4.1.p1.3.m3.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.1.p1.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S4.1.p1.3.m3.1.1" xref="S4.1.p1.3.m3.1.1.cmml">f</mi><mo id="S4.1.p1.3.m3.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.1.p1.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.1.p1.3.m3.2.2.1.1.3" stretchy="false" xref="S4.1.p1.3.m3.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.1.p1.3.m3.2.2.2" xref="S4.1.p1.3.m3.2.2.2.cmml">=</mo><mi id="S4.1.p1.3.m3.2.2.3" xref="S4.1.p1.3.m3.2.2.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.3.m3.2b"><apply id="S4.1.p1.3.m3.2.2.cmml" xref="S4.1.p1.3.m3.2.2"><eq id="S4.1.p1.3.m3.2.2.2.cmml" xref="S4.1.p1.3.m3.2.2.2"></eq><apply id="S4.1.p1.3.m3.2.2.1.2.cmml" xref="S4.1.p1.3.m3.2.2.1.1"><abs id="S4.1.p1.3.m3.2.2.1.2.1.cmml" xref="S4.1.p1.3.m3.2.2.1.1.2"></abs><apply id="S4.1.p1.3.m3.2.2.1.1.1.cmml" xref="S4.1.p1.3.m3.2.2.1.1.1"><times id="S4.1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.1.p1.3.m3.2.2.1.1.1.1"></times><ci id="S4.1.p1.3.m3.2.2.1.1.1.2.cmml" xref="S4.1.p1.3.m3.2.2.1.1.1.2">𝒫</ci><ci id="S4.1.p1.3.m3.1.1.cmml" xref="S4.1.p1.3.m3.1.1">𝑓</ci></apply></apply><ci id="S4.1.p1.3.m3.2.2.3.cmml" xref="S4.1.p1.3.m3.2.2.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.3.m3.2c">|\mathcal{P}(f)|=p</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.3.m3.2d">| caligraphic_P ( italic_f ) | = italic_p</annotation></semantics></math>. Let <math alttext="V(f)" class="ltx_Math" display="inline" id="S4.1.p1.4.m4.1"><semantics id="S4.1.p1.4.m4.1a"><mrow id="S4.1.p1.4.m4.1.2" xref="S4.1.p1.4.m4.1.2.cmml"><mi id="S4.1.p1.4.m4.1.2.2" xref="S4.1.p1.4.m4.1.2.2.cmml">V</mi><mo id="S4.1.p1.4.m4.1.2.1" xref="S4.1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.1.p1.4.m4.1.2.3.2" xref="S4.1.p1.4.m4.1.2.cmml"><mo id="S4.1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.1.p1.4.m4.1.2.cmml">(</mo><mi id="S4.1.p1.4.m4.1.1" xref="S4.1.p1.4.m4.1.1.cmml">f</mi><mo id="S4.1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.4.m4.1b"><apply id="S4.1.p1.4.m4.1.2.cmml" xref="S4.1.p1.4.m4.1.2"><times id="S4.1.p1.4.m4.1.2.1.cmml" xref="S4.1.p1.4.m4.1.2.1"></times><ci id="S4.1.p1.4.m4.1.2.2.cmml" xref="S4.1.p1.4.m4.1.2.2">𝑉</ci><ci id="S4.1.p1.4.m4.1.1.cmml" xref="S4.1.p1.4.m4.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.4.m4.1c">V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.4.m4.1d">italic_V ( italic_f )</annotation></semantics></math> and <math alttext="E(f)" class="ltx_Math" display="inline" id="S4.1.p1.5.m5.1"><semantics id="S4.1.p1.5.m5.1a"><mrow id="S4.1.p1.5.m5.1.2" xref="S4.1.p1.5.m5.1.2.cmml"><mi id="S4.1.p1.5.m5.1.2.2" xref="S4.1.p1.5.m5.1.2.2.cmml">E</mi><mo id="S4.1.p1.5.m5.1.2.1" xref="S4.1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.1.p1.5.m5.1.2.3.2" xref="S4.1.p1.5.m5.1.2.cmml"><mo id="S4.1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S4.1.p1.5.m5.1.2.cmml">(</mo><mi id="S4.1.p1.5.m5.1.1" xref="S4.1.p1.5.m5.1.1.cmml">f</mi><mo id="S4.1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S4.1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.5.m5.1b"><apply id="S4.1.p1.5.m5.1.2.cmml" xref="S4.1.p1.5.m5.1.2"><times id="S4.1.p1.5.m5.1.2.1.cmml" xref="S4.1.p1.5.m5.1.2.1"></times><ci id="S4.1.p1.5.m5.1.2.2.cmml" xref="S4.1.p1.5.m5.1.2.2">𝐸</ci><ci id="S4.1.p1.5.m5.1.1.cmml" xref="S4.1.p1.5.m5.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.5.m5.1c">E(f)</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.5.m5.1d">italic_E ( italic_f )</annotation></semantics></math> be the corresponding sets of vertices and edges. The homeomorphisms of the boundary components of a polygon ensure that <math alttext="\deg(v)\geq 2" class="ltx_Math" display="inline" id="S4.1.p1.6.m6.2"><semantics id="S4.1.p1.6.m6.2a"><mrow id="S4.1.p1.6.m6.2.3" xref="S4.1.p1.6.m6.2.3.cmml"><mrow id="S4.1.p1.6.m6.2.3.2.2" xref="S4.1.p1.6.m6.2.3.2.1.cmml"><mi id="S4.1.p1.6.m6.1.1" xref="S4.1.p1.6.m6.1.1.cmml">deg</mi><mo id="S4.1.p1.6.m6.2.3.2.2a" xref="S4.1.p1.6.m6.2.3.2.1.cmml">⁡</mo><mrow id="S4.1.p1.6.m6.2.3.2.2.1" xref="S4.1.p1.6.m6.2.3.2.1.cmml"><mo id="S4.1.p1.6.m6.2.3.2.2.1.1" stretchy="false" xref="S4.1.p1.6.m6.2.3.2.1.cmml">(</mo><mi id="S4.1.p1.6.m6.2.2" xref="S4.1.p1.6.m6.2.2.cmml">v</mi><mo id="S4.1.p1.6.m6.2.3.2.2.1.2" stretchy="false" xref="S4.1.p1.6.m6.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.1.p1.6.m6.2.3.1" xref="S4.1.p1.6.m6.2.3.1.cmml">≥</mo><mn id="S4.1.p1.6.m6.2.3.3" xref="S4.1.p1.6.m6.2.3.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.6.m6.2b"><apply id="S4.1.p1.6.m6.2.3.cmml" xref="S4.1.p1.6.m6.2.3"><geq id="S4.1.p1.6.m6.2.3.1.cmml" xref="S4.1.p1.6.m6.2.3.1"></geq><apply id="S4.1.p1.6.m6.2.3.2.1.cmml" xref="S4.1.p1.6.m6.2.3.2.2"><csymbol cd="latexml" id="S4.1.p1.6.m6.1.1.cmml" xref="S4.1.p1.6.m6.1.1">degree</csymbol><ci id="S4.1.p1.6.m6.2.2.cmml" xref="S4.1.p1.6.m6.2.2">𝑣</ci></apply><cn id="S4.1.p1.6.m6.2.3.3.cmml" type="integer" xref="S4.1.p1.6.m6.2.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.6.m6.2c">\deg(v)\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.6.m6.2d">roman_deg ( italic_v ) ≥ 2</annotation></semantics></math> for all <math alttext="v\in V(f)" class="ltx_Math" display="inline" id="S4.1.p1.7.m7.1"><semantics id="S4.1.p1.7.m7.1a"><mrow id="S4.1.p1.7.m7.1.2" xref="S4.1.p1.7.m7.1.2.cmml"><mi id="S4.1.p1.7.m7.1.2.2" xref="S4.1.p1.7.m7.1.2.2.cmml">v</mi><mo id="S4.1.p1.7.m7.1.2.1" xref="S4.1.p1.7.m7.1.2.1.cmml">∈</mo><mrow id="S4.1.p1.7.m7.1.2.3" xref="S4.1.p1.7.m7.1.2.3.cmml"><mi id="S4.1.p1.7.m7.1.2.3.2" xref="S4.1.p1.7.m7.1.2.3.2.cmml">V</mi><mo id="S4.1.p1.7.m7.1.2.3.1" xref="S4.1.p1.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S4.1.p1.7.m7.1.2.3.3.2" xref="S4.1.p1.7.m7.1.2.3.cmml"><mo id="S4.1.p1.7.m7.1.2.3.3.2.1" stretchy="false" xref="S4.1.p1.7.m7.1.2.3.cmml">(</mo><mi id="S4.1.p1.7.m7.1.1" xref="S4.1.p1.7.m7.1.1.cmml">f</mi><mo id="S4.1.p1.7.m7.1.2.3.3.2.2" stretchy="false" xref="S4.1.p1.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.7.m7.1b"><apply id="S4.1.p1.7.m7.1.2.cmml" xref="S4.1.p1.7.m7.1.2"><in id="S4.1.p1.7.m7.1.2.1.cmml" xref="S4.1.p1.7.m7.1.2.1"></in><ci id="S4.1.p1.7.m7.1.2.2.cmml" xref="S4.1.p1.7.m7.1.2.2">𝑣</ci><apply id="S4.1.p1.7.m7.1.2.3.cmml" xref="S4.1.p1.7.m7.1.2.3"><times id="S4.1.p1.7.m7.1.2.3.1.cmml" xref="S4.1.p1.7.m7.1.2.3.1"></times><ci id="S4.1.p1.7.m7.1.2.3.2.cmml" xref="S4.1.p1.7.m7.1.2.3.2">𝑉</ci><ci id="S4.1.p1.7.m7.1.1.cmml" xref="S4.1.p1.7.m7.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.7.m7.1c">v\in V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.7.m7.1d">italic_v ∈ italic_V ( italic_f )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.2.p2"> <p class="ltx_p" id="S4.2.p2.16">If there exists some <math alttext="v\in V(f)" class="ltx_Math" display="inline" id="S4.2.p2.1.m1.1"><semantics id="S4.2.p2.1.m1.1a"><mrow id="S4.2.p2.1.m1.1.2" xref="S4.2.p2.1.m1.1.2.cmml"><mi id="S4.2.p2.1.m1.1.2.2" xref="S4.2.p2.1.m1.1.2.2.cmml">v</mi><mo id="S4.2.p2.1.m1.1.2.1" xref="S4.2.p2.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.2.p2.1.m1.1.2.3" xref="S4.2.p2.1.m1.1.2.3.cmml"><mi id="S4.2.p2.1.m1.1.2.3.2" xref="S4.2.p2.1.m1.1.2.3.2.cmml">V</mi><mo id="S4.2.p2.1.m1.1.2.3.1" xref="S4.2.p2.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.2.p2.1.m1.1.2.3.3.2" xref="S4.2.p2.1.m1.1.2.3.cmml"><mo id="S4.2.p2.1.m1.1.2.3.3.2.1" stretchy="false" xref="S4.2.p2.1.m1.1.2.3.cmml">(</mo><mi id="S4.2.p2.1.m1.1.1" xref="S4.2.p2.1.m1.1.1.cmml">f</mi><mo id="S4.2.p2.1.m1.1.2.3.3.2.2" stretchy="false" xref="S4.2.p2.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.1.m1.1b"><apply id="S4.2.p2.1.m1.1.2.cmml" xref="S4.2.p2.1.m1.1.2"><in id="S4.2.p2.1.m1.1.2.1.cmml" xref="S4.2.p2.1.m1.1.2.1"></in><ci id="S4.2.p2.1.m1.1.2.2.cmml" xref="S4.2.p2.1.m1.1.2.2">𝑣</ci><apply id="S4.2.p2.1.m1.1.2.3.cmml" xref="S4.2.p2.1.m1.1.2.3"><times id="S4.2.p2.1.m1.1.2.3.1.cmml" xref="S4.2.p2.1.m1.1.2.3.1"></times><ci id="S4.2.p2.1.m1.1.2.3.2.cmml" xref="S4.2.p2.1.m1.1.2.3.2">𝑉</ci><ci id="S4.2.p2.1.m1.1.1.cmml" xref="S4.2.p2.1.m1.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.1.m1.1c">v\in V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.1.m1.1d">italic_v ∈ italic_V ( italic_f )</annotation></semantics></math> with <math alttext="\deg(v)=2" class="ltx_Math" display="inline" id="S4.2.p2.2.m2.2"><semantics id="S4.2.p2.2.m2.2a"><mrow id="S4.2.p2.2.m2.2.3" xref="S4.2.p2.2.m2.2.3.cmml"><mrow id="S4.2.p2.2.m2.2.3.2.2" xref="S4.2.p2.2.m2.2.3.2.1.cmml"><mi id="S4.2.p2.2.m2.1.1" xref="S4.2.p2.2.m2.1.1.cmml">deg</mi><mo id="S4.2.p2.2.m2.2.3.2.2a" xref="S4.2.p2.2.m2.2.3.2.1.cmml">⁡</mo><mrow id="S4.2.p2.2.m2.2.3.2.2.1" xref="S4.2.p2.2.m2.2.3.2.1.cmml"><mo id="S4.2.p2.2.m2.2.3.2.2.1.1" stretchy="false" xref="S4.2.p2.2.m2.2.3.2.1.cmml">(</mo><mi id="S4.2.p2.2.m2.2.2" xref="S4.2.p2.2.m2.2.2.cmml">v</mi><mo id="S4.2.p2.2.m2.2.3.2.2.1.2" stretchy="false" xref="S4.2.p2.2.m2.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.2.p2.2.m2.2.3.1" xref="S4.2.p2.2.m2.2.3.1.cmml">=</mo><mn id="S4.2.p2.2.m2.2.3.3" xref="S4.2.p2.2.m2.2.3.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.2.m2.2b"><apply id="S4.2.p2.2.m2.2.3.cmml" xref="S4.2.p2.2.m2.2.3"><eq id="S4.2.p2.2.m2.2.3.1.cmml" xref="S4.2.p2.2.m2.2.3.1"></eq><apply id="S4.2.p2.2.m2.2.3.2.1.cmml" xref="S4.2.p2.2.m2.2.3.2.2"><csymbol cd="latexml" id="S4.2.p2.2.m2.1.1.cmml" xref="S4.2.p2.2.m2.1.1">degree</csymbol><ci id="S4.2.p2.2.m2.2.2.cmml" xref="S4.2.p2.2.m2.2.2">𝑣</ci></apply><cn id="S4.2.p2.2.m2.2.3.3.cmml" type="integer" xref="S4.2.p2.2.m2.2.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.2.m2.2c">\deg(v)=2</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.2.m2.2d">roman_deg ( italic_v ) = 2</annotation></semantics></math>, then there must be two pieces <math alttext="P,Q\in\mathcal{P}(f)" class="ltx_Math" display="inline" id="S4.2.p2.3.m3.3"><semantics id="S4.2.p2.3.m3.3a"><mrow id="S4.2.p2.3.m3.3.4" xref="S4.2.p2.3.m3.3.4.cmml"><mrow id="S4.2.p2.3.m3.3.4.2.2" xref="S4.2.p2.3.m3.3.4.2.1.cmml"><mi id="S4.2.p2.3.m3.2.2" xref="S4.2.p2.3.m3.2.2.cmml">P</mi><mo id="S4.2.p2.3.m3.3.4.2.2.1" xref="S4.2.p2.3.m3.3.4.2.1.cmml">,</mo><mi id="S4.2.p2.3.m3.3.3" xref="S4.2.p2.3.m3.3.3.cmml">Q</mi></mrow><mo id="S4.2.p2.3.m3.3.4.1" xref="S4.2.p2.3.m3.3.4.1.cmml">∈</mo><mrow id="S4.2.p2.3.m3.3.4.3" xref="S4.2.p2.3.m3.3.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.2.p2.3.m3.3.4.3.2" xref="S4.2.p2.3.m3.3.4.3.2.cmml">𝒫</mi><mo id="S4.2.p2.3.m3.3.4.3.1" xref="S4.2.p2.3.m3.3.4.3.1.cmml">⁢</mo><mrow id="S4.2.p2.3.m3.3.4.3.3.2" xref="S4.2.p2.3.m3.3.4.3.cmml"><mo id="S4.2.p2.3.m3.3.4.3.3.2.1" stretchy="false" xref="S4.2.p2.3.m3.3.4.3.cmml">(</mo><mi id="S4.2.p2.3.m3.1.1" xref="S4.2.p2.3.m3.1.1.cmml">f</mi><mo id="S4.2.p2.3.m3.3.4.3.3.2.2" stretchy="false" xref="S4.2.p2.3.m3.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.3.m3.3b"><apply id="S4.2.p2.3.m3.3.4.cmml" xref="S4.2.p2.3.m3.3.4"><in id="S4.2.p2.3.m3.3.4.1.cmml" xref="S4.2.p2.3.m3.3.4.1"></in><list id="S4.2.p2.3.m3.3.4.2.1.cmml" xref="S4.2.p2.3.m3.3.4.2.2"><ci id="S4.2.p2.3.m3.2.2.cmml" xref="S4.2.p2.3.m3.2.2">𝑃</ci><ci id="S4.2.p2.3.m3.3.3.cmml" xref="S4.2.p2.3.m3.3.3">𝑄</ci></list><apply id="S4.2.p2.3.m3.3.4.3.cmml" xref="S4.2.p2.3.m3.3.4.3"><times id="S4.2.p2.3.m3.3.4.3.1.cmml" xref="S4.2.p2.3.m3.3.4.3.1"></times><ci id="S4.2.p2.3.m3.3.4.3.2.cmml" xref="S4.2.p2.3.m3.3.4.3.2">𝒫</ci><ci id="S4.2.p2.3.m3.1.1.cmml" xref="S4.2.p2.3.m3.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.3.m3.3c">P,Q\in\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.3.m3.3d">italic_P , italic_Q ∈ caligraphic_P ( italic_f )</annotation></semantics></math> that share the two incident edges. Let <math alttext="f_{P}" class="ltx_Math" display="inline" id="S4.2.p2.4.m4.1"><semantics id="S4.2.p2.4.m4.1a"><msub id="S4.2.p2.4.m4.1.1" xref="S4.2.p2.4.m4.1.1.cmml"><mi id="S4.2.p2.4.m4.1.1.2" xref="S4.2.p2.4.m4.1.1.2.cmml">f</mi><mi id="S4.2.p2.4.m4.1.1.3" xref="S4.2.p2.4.m4.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S4.2.p2.4.m4.1b"><apply id="S4.2.p2.4.m4.1.1.cmml" xref="S4.2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.2.p2.4.m4.1.1.1.cmml" xref="S4.2.p2.4.m4.1.1">subscript</csymbol><ci id="S4.2.p2.4.m4.1.1.2.cmml" xref="S4.2.p2.4.m4.1.1.2">𝑓</ci><ci id="S4.2.p2.4.m4.1.1.3.cmml" xref="S4.2.p2.4.m4.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.4.m4.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.4.m4.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{Q}" class="ltx_Math" display="inline" id="S4.2.p2.5.m5.1"><semantics id="S4.2.p2.5.m5.1a"><msub id="S4.2.p2.5.m5.1.1" xref="S4.2.p2.5.m5.1.1.cmml"><mi id="S4.2.p2.5.m5.1.1.2" xref="S4.2.p2.5.m5.1.1.2.cmml">f</mi><mi id="S4.2.p2.5.m5.1.1.3" xref="S4.2.p2.5.m5.1.1.3.cmml">Q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.2.p2.5.m5.1b"><apply id="S4.2.p2.5.m5.1.1.cmml" xref="S4.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.2.p2.5.m5.1.1.1.cmml" xref="S4.2.p2.5.m5.1.1">subscript</csymbol><ci id="S4.2.p2.5.m5.1.1.2.cmml" xref="S4.2.p2.5.m5.1.1.2">𝑓</ci><ci id="S4.2.p2.5.m5.1.1.3.cmml" xref="S4.2.p2.5.m5.1.1.3">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.5.m5.1c">f_{Q}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT</annotation></semantics></math> be the respective affine components of <math alttext="P" class="ltx_Math" display="inline" id="S4.2.p2.6.m6.1"><semantics id="S4.2.p2.6.m6.1a"><mi id="S4.2.p2.6.m6.1.1" xref="S4.2.p2.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.6.m6.1b"><ci id="S4.2.p2.6.m6.1.1.cmml" xref="S4.2.p2.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.6.m6.1d">italic_P</annotation></semantics></math> and <math alttext="Q" class="ltx_Math" display="inline" id="S4.2.p2.7.m7.1"><semantics id="S4.2.p2.7.m7.1a"><mi id="S4.2.p2.7.m7.1.1" xref="S4.2.p2.7.m7.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.7.m7.1b"><ci id="S4.2.p2.7.m7.1.1.cmml" xref="S4.2.p2.7.m7.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.7.m7.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.7.m7.1d">italic_Q</annotation></semantics></math>. If <math alttext="f_{P}\neq f_{Q}" class="ltx_Math" display="inline" id="S4.2.p2.8.m8.1"><semantics id="S4.2.p2.8.m8.1a"><mrow id="S4.2.p2.8.m8.1.1" xref="S4.2.p2.8.m8.1.1.cmml"><msub id="S4.2.p2.8.m8.1.1.2" xref="S4.2.p2.8.m8.1.1.2.cmml"><mi id="S4.2.p2.8.m8.1.1.2.2" xref="S4.2.p2.8.m8.1.1.2.2.cmml">f</mi><mi id="S4.2.p2.8.m8.1.1.2.3" xref="S4.2.p2.8.m8.1.1.2.3.cmml">P</mi></msub><mo id="S4.2.p2.8.m8.1.1.1" xref="S4.2.p2.8.m8.1.1.1.cmml">≠</mo><msub id="S4.2.p2.8.m8.1.1.3" xref="S4.2.p2.8.m8.1.1.3.cmml"><mi id="S4.2.p2.8.m8.1.1.3.2" xref="S4.2.p2.8.m8.1.1.3.2.cmml">f</mi><mi id="S4.2.p2.8.m8.1.1.3.3" xref="S4.2.p2.8.m8.1.1.3.3.cmml">Q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.8.m8.1b"><apply id="S4.2.p2.8.m8.1.1.cmml" xref="S4.2.p2.8.m8.1.1"><neq id="S4.2.p2.8.m8.1.1.1.cmml" xref="S4.2.p2.8.m8.1.1.1"></neq><apply id="S4.2.p2.8.m8.1.1.2.cmml" xref="S4.2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.2.p2.8.m8.1.1.2.1.cmml" xref="S4.2.p2.8.m8.1.1.2">subscript</csymbol><ci id="S4.2.p2.8.m8.1.1.2.2.cmml" xref="S4.2.p2.8.m8.1.1.2.2">𝑓</ci><ci id="S4.2.p2.8.m8.1.1.2.3.cmml" xref="S4.2.p2.8.m8.1.1.2.3">𝑃</ci></apply><apply id="S4.2.p2.8.m8.1.1.3.cmml" xref="S4.2.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.2.p2.8.m8.1.1.3.1.cmml" xref="S4.2.p2.8.m8.1.1.3">subscript</csymbol><ci id="S4.2.p2.8.m8.1.1.3.2.cmml" xref="S4.2.p2.8.m8.1.1.3.2">𝑓</ci><ci id="S4.2.p2.8.m8.1.1.3.3.cmml" xref="S4.2.p2.8.m8.1.1.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.8.m8.1c">f_{P}\neq f_{Q}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.8.m8.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ≠ italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT</annotation></semantics></math>, then the incident edges lie on the same line, and the vertex can be removed from <math alttext="V(f)" class="ltx_Math" display="inline" id="S4.2.p2.9.m9.1"><semantics id="S4.2.p2.9.m9.1a"><mrow id="S4.2.p2.9.m9.1.2" xref="S4.2.p2.9.m9.1.2.cmml"><mi id="S4.2.p2.9.m9.1.2.2" xref="S4.2.p2.9.m9.1.2.2.cmml">V</mi><mo id="S4.2.p2.9.m9.1.2.1" xref="S4.2.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.2.p2.9.m9.1.2.3.2" xref="S4.2.p2.9.m9.1.2.cmml"><mo id="S4.2.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S4.2.p2.9.m9.1.2.cmml">(</mo><mi id="S4.2.p2.9.m9.1.1" xref="S4.2.p2.9.m9.1.1.cmml">f</mi><mo id="S4.2.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S4.2.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.9.m9.1b"><apply id="S4.2.p2.9.m9.1.2.cmml" xref="S4.2.p2.9.m9.1.2"><times id="S4.2.p2.9.m9.1.2.1.cmml" xref="S4.2.p2.9.m9.1.2.1"></times><ci id="S4.2.p2.9.m9.1.2.2.cmml" xref="S4.2.p2.9.m9.1.2.2">𝑉</ci><ci id="S4.2.p2.9.m9.1.1.cmml" xref="S4.2.p2.9.m9.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.9.m9.1c">V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.9.m9.1d">italic_V ( italic_f )</annotation></semantics></math> by merging the incident edges. If <math alttext="f_{P}=f_{Q}" class="ltx_Math" display="inline" id="S4.2.p2.10.m10.1"><semantics id="S4.2.p2.10.m10.1a"><mrow id="S4.2.p2.10.m10.1.1" xref="S4.2.p2.10.m10.1.1.cmml"><msub id="S4.2.p2.10.m10.1.1.2" xref="S4.2.p2.10.m10.1.1.2.cmml"><mi id="S4.2.p2.10.m10.1.1.2.2" xref="S4.2.p2.10.m10.1.1.2.2.cmml">f</mi><mi id="S4.2.p2.10.m10.1.1.2.3" xref="S4.2.p2.10.m10.1.1.2.3.cmml">P</mi></msub><mo id="S4.2.p2.10.m10.1.1.1" xref="S4.2.p2.10.m10.1.1.1.cmml">=</mo><msub id="S4.2.p2.10.m10.1.1.3" xref="S4.2.p2.10.m10.1.1.3.cmml"><mi id="S4.2.p2.10.m10.1.1.3.2" xref="S4.2.p2.10.m10.1.1.3.2.cmml">f</mi><mi id="S4.2.p2.10.m10.1.1.3.3" xref="S4.2.p2.10.m10.1.1.3.3.cmml">Q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.10.m10.1b"><apply id="S4.2.p2.10.m10.1.1.cmml" xref="S4.2.p2.10.m10.1.1"><eq id="S4.2.p2.10.m10.1.1.1.cmml" xref="S4.2.p2.10.m10.1.1.1"></eq><apply id="S4.2.p2.10.m10.1.1.2.cmml" xref="S4.2.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.2.p2.10.m10.1.1.2.1.cmml" xref="S4.2.p2.10.m10.1.1.2">subscript</csymbol><ci id="S4.2.p2.10.m10.1.1.2.2.cmml" xref="S4.2.p2.10.m10.1.1.2.2">𝑓</ci><ci id="S4.2.p2.10.m10.1.1.2.3.cmml" xref="S4.2.p2.10.m10.1.1.2.3">𝑃</ci></apply><apply id="S4.2.p2.10.m10.1.1.3.cmml" xref="S4.2.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.2.p2.10.m10.1.1.3.1.cmml" xref="S4.2.p2.10.m10.1.1.3">subscript</csymbol><ci id="S4.2.p2.10.m10.1.1.3.2.cmml" xref="S4.2.p2.10.m10.1.1.3.2">𝑓</ci><ci id="S4.2.p2.10.m10.1.1.3.3.cmml" xref="S4.2.p2.10.m10.1.1.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.10.m10.1c">f_{P}=f_{Q}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.10.m10.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT</annotation></semantics></math>, then the pieces <math alttext="P" class="ltx_Math" display="inline" id="S4.2.p2.11.m11.1"><semantics id="S4.2.p2.11.m11.1a"><mi id="S4.2.p2.11.m11.1.1" xref="S4.2.p2.11.m11.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.11.m11.1b"><ci id="S4.2.p2.11.m11.1.1.cmml" xref="S4.2.p2.11.m11.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.11.m11.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.11.m11.1d">italic_P</annotation></semantics></math> and <math alttext="Q" class="ltx_Math" display="inline" id="S4.2.p2.12.m12.1"><semantics id="S4.2.p2.12.m12.1a"><mi id="S4.2.p2.12.m12.1.1" xref="S4.2.p2.12.m12.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.12.m12.1b"><ci id="S4.2.p2.12.m12.1.1.cmml" xref="S4.2.p2.12.m12.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.12.m12.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.12.m12.1d">italic_Q</annotation></semantics></math> can be replaced by the single piece <math alttext="P\cup Q" class="ltx_Math" display="inline" id="S4.2.p2.13.m13.1"><semantics id="S4.2.p2.13.m13.1a"><mrow id="S4.2.p2.13.m13.1.1" xref="S4.2.p2.13.m13.1.1.cmml"><mi id="S4.2.p2.13.m13.1.1.2" xref="S4.2.p2.13.m13.1.1.2.cmml">P</mi><mo id="S4.2.p2.13.m13.1.1.1" xref="S4.2.p2.13.m13.1.1.1.cmml">∪</mo><mi id="S4.2.p2.13.m13.1.1.3" xref="S4.2.p2.13.m13.1.1.3.cmml">Q</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.13.m13.1b"><apply id="S4.2.p2.13.m13.1.1.cmml" xref="S4.2.p2.13.m13.1.1"><union id="S4.2.p2.13.m13.1.1.1.cmml" xref="S4.2.p2.13.m13.1.1.1"></union><ci id="S4.2.p2.13.m13.1.1.2.cmml" xref="S4.2.p2.13.m13.1.1.2">𝑃</ci><ci id="S4.2.p2.13.m13.1.1.3.cmml" xref="S4.2.p2.13.m13.1.1.3">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.13.m13.1c">P\cup Q</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.13.m13.1d">italic_P ∪ italic_Q</annotation></semantics></math>, effectively removing <math alttext="v" class="ltx_Math" display="inline" id="S4.2.p2.14.m14.1"><semantics id="S4.2.p2.14.m14.1a"><mi id="S4.2.p2.14.m14.1.1" xref="S4.2.p2.14.m14.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.14.m14.1b"><ci id="S4.2.p2.14.m14.1.1.cmml" xref="S4.2.p2.14.m14.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.14.m14.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.14.m14.1d">italic_v</annotation></semantics></math> from <math alttext="V(f)" class="ltx_Math" display="inline" id="S4.2.p2.15.m15.1"><semantics id="S4.2.p2.15.m15.1a"><mrow id="S4.2.p2.15.m15.1.2" xref="S4.2.p2.15.m15.1.2.cmml"><mi id="S4.2.p2.15.m15.1.2.2" xref="S4.2.p2.15.m15.1.2.2.cmml">V</mi><mo id="S4.2.p2.15.m15.1.2.1" xref="S4.2.p2.15.m15.1.2.1.cmml">⁢</mo><mrow id="S4.2.p2.15.m15.1.2.3.2" xref="S4.2.p2.15.m15.1.2.cmml"><mo id="S4.2.p2.15.m15.1.2.3.2.1" stretchy="false" xref="S4.2.p2.15.m15.1.2.cmml">(</mo><mi id="S4.2.p2.15.m15.1.1" xref="S4.2.p2.15.m15.1.1.cmml">f</mi><mo id="S4.2.p2.15.m15.1.2.3.2.2" stretchy="false" xref="S4.2.p2.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.15.m15.1b"><apply id="S4.2.p2.15.m15.1.2.cmml" xref="S4.2.p2.15.m15.1.2"><times id="S4.2.p2.15.m15.1.2.1.cmml" xref="S4.2.p2.15.m15.1.2.1"></times><ci id="S4.2.p2.15.m15.1.2.2.cmml" xref="S4.2.p2.15.m15.1.2.2">𝑉</ci><ci id="S4.2.p2.15.m15.1.1.cmml" xref="S4.2.p2.15.m15.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.15.m15.1c">V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.15.m15.1d">italic_V ( italic_f )</annotation></semantics></math>, removing the incident edges from <math alttext="E(f)" class="ltx_Math" display="inline" id="S4.2.p2.16.m16.1"><semantics id="S4.2.p2.16.m16.1a"><mrow id="S4.2.p2.16.m16.1.2" xref="S4.2.p2.16.m16.1.2.cmml"><mi id="S4.2.p2.16.m16.1.2.2" xref="S4.2.p2.16.m16.1.2.2.cmml">E</mi><mo id="S4.2.p2.16.m16.1.2.1" xref="S4.2.p2.16.m16.1.2.1.cmml">⁢</mo><mrow id="S4.2.p2.16.m16.1.2.3.2" xref="S4.2.p2.16.m16.1.2.cmml"><mo id="S4.2.p2.16.m16.1.2.3.2.1" stretchy="false" xref="S4.2.p2.16.m16.1.2.cmml">(</mo><mi id="S4.2.p2.16.m16.1.1" xref="S4.2.p2.16.m16.1.1.cmml">f</mi><mo id="S4.2.p2.16.m16.1.2.3.2.2" stretchy="false" xref="S4.2.p2.16.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.16.m16.1b"><apply id="S4.2.p2.16.m16.1.2.cmml" xref="S4.2.p2.16.m16.1.2"><times id="S4.2.p2.16.m16.1.2.1.cmml" xref="S4.2.p2.16.m16.1.2.1"></times><ci id="S4.2.p2.16.m16.1.2.2.cmml" xref="S4.2.p2.16.m16.1.2.2">𝐸</ci><ci id="S4.2.p2.16.m16.1.1.cmml" xref="S4.2.p2.16.m16.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.16.m16.1c">E(f)</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.16.m16.1d">italic_E ( italic_f )</annotation></semantics></math>, and reducing the number of pieces.</p> </div> <div class="ltx_para" id="S4.3.p3"> <p class="ltx_p" id="S4.3.p3.9">By iterating this process for all vertices <math alttext="v\in V(f)" class="ltx_Math" display="inline" id="S4.3.p3.1.m1.1"><semantics id="S4.3.p3.1.m1.1a"><mrow id="S4.3.p3.1.m1.1.2" xref="S4.3.p3.1.m1.1.2.cmml"><mi id="S4.3.p3.1.m1.1.2.2" xref="S4.3.p3.1.m1.1.2.2.cmml">v</mi><mo id="S4.3.p3.1.m1.1.2.1" xref="S4.3.p3.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.3.p3.1.m1.1.2.3" xref="S4.3.p3.1.m1.1.2.3.cmml"><mi id="S4.3.p3.1.m1.1.2.3.2" xref="S4.3.p3.1.m1.1.2.3.2.cmml">V</mi><mo id="S4.3.p3.1.m1.1.2.3.1" xref="S4.3.p3.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.3.p3.1.m1.1.2.3.3.2" xref="S4.3.p3.1.m1.1.2.3.cmml"><mo id="S4.3.p3.1.m1.1.2.3.3.2.1" stretchy="false" xref="S4.3.p3.1.m1.1.2.3.cmml">(</mo><mi id="S4.3.p3.1.m1.1.1" xref="S4.3.p3.1.m1.1.1.cmml">f</mi><mo id="S4.3.p3.1.m1.1.2.3.3.2.2" stretchy="false" xref="S4.3.p3.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p3.1.m1.1b"><apply id="S4.3.p3.1.m1.1.2.cmml" xref="S4.3.p3.1.m1.1.2"><in id="S4.3.p3.1.m1.1.2.1.cmml" xref="S4.3.p3.1.m1.1.2.1"></in><ci id="S4.3.p3.1.m1.1.2.2.cmml" xref="S4.3.p3.1.m1.1.2.2">𝑣</ci><apply id="S4.3.p3.1.m1.1.2.3.cmml" xref="S4.3.p3.1.m1.1.2.3"><times id="S4.3.p3.1.m1.1.2.3.1.cmml" xref="S4.3.p3.1.m1.1.2.3.1"></times><ci id="S4.3.p3.1.m1.1.2.3.2.cmml" xref="S4.3.p3.1.m1.1.2.3.2">𝑉</ci><ci id="S4.3.p3.1.m1.1.1.cmml" xref="S4.3.p3.1.m1.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.1.m1.1c">v\in V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.1.m1.1d">italic_v ∈ italic_V ( italic_f )</annotation></semantics></math> with <math alttext="\deg(v)=2" class="ltx_Math" display="inline" id="S4.3.p3.2.m2.2"><semantics id="S4.3.p3.2.m2.2a"><mrow id="S4.3.p3.2.m2.2.3" xref="S4.3.p3.2.m2.2.3.cmml"><mrow id="S4.3.p3.2.m2.2.3.2.2" xref="S4.3.p3.2.m2.2.3.2.1.cmml"><mi id="S4.3.p3.2.m2.1.1" xref="S4.3.p3.2.m2.1.1.cmml">deg</mi><mo id="S4.3.p3.2.m2.2.3.2.2a" xref="S4.3.p3.2.m2.2.3.2.1.cmml">⁡</mo><mrow id="S4.3.p3.2.m2.2.3.2.2.1" xref="S4.3.p3.2.m2.2.3.2.1.cmml"><mo id="S4.3.p3.2.m2.2.3.2.2.1.1" stretchy="false" xref="S4.3.p3.2.m2.2.3.2.1.cmml">(</mo><mi id="S4.3.p3.2.m2.2.2" xref="S4.3.p3.2.m2.2.2.cmml">v</mi><mo id="S4.3.p3.2.m2.2.3.2.2.1.2" stretchy="false" xref="S4.3.p3.2.m2.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p3.2.m2.2.3.1" xref="S4.3.p3.2.m2.2.3.1.cmml">=</mo><mn id="S4.3.p3.2.m2.2.3.3" xref="S4.3.p3.2.m2.2.3.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p3.2.m2.2b"><apply id="S4.3.p3.2.m2.2.3.cmml" xref="S4.3.p3.2.m2.2.3"><eq id="S4.3.p3.2.m2.2.3.1.cmml" xref="S4.3.p3.2.m2.2.3.1"></eq><apply id="S4.3.p3.2.m2.2.3.2.1.cmml" xref="S4.3.p3.2.m2.2.3.2.2"><csymbol cd="latexml" id="S4.3.p3.2.m2.1.1.cmml" xref="S4.3.p3.2.m2.1.1">degree</csymbol><ci id="S4.3.p3.2.m2.2.2.cmml" xref="S4.3.p3.2.m2.2.2">𝑣</ci></apply><cn id="S4.3.p3.2.m2.2.3.3.cmml" type="integer" xref="S4.3.p3.2.m2.2.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.2.m2.2c">\deg(v)=2</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.2.m2.2d">roman_deg ( italic_v ) = 2</annotation></semantics></math>, we obtain a new set <math alttext="\mathcal{P}^{\prime}" class="ltx_Math" display="inline" id="S4.3.p3.3.m3.1"><semantics id="S4.3.p3.3.m3.1a"><msup id="S4.3.p3.3.m3.1.1" xref="S4.3.p3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.3.p3.3.m3.1.1.2" xref="S4.3.p3.3.m3.1.1.2.cmml">𝒫</mi><mo id="S4.3.p3.3.m3.1.1.3" xref="S4.3.p3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.3.p3.3.m3.1b"><apply id="S4.3.p3.3.m3.1.1.cmml" xref="S4.3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.3.p3.3.m3.1.1.1.cmml" xref="S4.3.p3.3.m3.1.1">superscript</csymbol><ci id="S4.3.p3.3.m3.1.1.2.cmml" xref="S4.3.p3.3.m3.1.1.2">𝒫</ci><ci id="S4.3.p3.3.m3.1.1.3.cmml" xref="S4.3.p3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.3.m3.1c">\mathcal{P}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.3.m3.1d">caligraphic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="p^{\prime}\leq p" class="ltx_Math" display="inline" id="S4.3.p3.4.m4.1"><semantics id="S4.3.p3.4.m4.1a"><mrow id="S4.3.p3.4.m4.1.1" xref="S4.3.p3.4.m4.1.1.cmml"><msup id="S4.3.p3.4.m4.1.1.2" xref="S4.3.p3.4.m4.1.1.2.cmml"><mi id="S4.3.p3.4.m4.1.1.2.2" xref="S4.3.p3.4.m4.1.1.2.2.cmml">p</mi><mo id="S4.3.p3.4.m4.1.1.2.3" xref="S4.3.p3.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.3.p3.4.m4.1.1.1" xref="S4.3.p3.4.m4.1.1.1.cmml">≤</mo><mi id="S4.3.p3.4.m4.1.1.3" xref="S4.3.p3.4.m4.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p3.4.m4.1b"><apply id="S4.3.p3.4.m4.1.1.cmml" xref="S4.3.p3.4.m4.1.1"><leq id="S4.3.p3.4.m4.1.1.1.cmml" xref="S4.3.p3.4.m4.1.1.1"></leq><apply id="S4.3.p3.4.m4.1.1.2.cmml" xref="S4.3.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.3.p3.4.m4.1.1.2.1.cmml" xref="S4.3.p3.4.m4.1.1.2">superscript</csymbol><ci id="S4.3.p3.4.m4.1.1.2.2.cmml" xref="S4.3.p3.4.m4.1.1.2.2">𝑝</ci><ci id="S4.3.p3.4.m4.1.1.2.3.cmml" xref="S4.3.p3.4.m4.1.1.2.3">′</ci></apply><ci id="S4.3.p3.4.m4.1.1.3.cmml" xref="S4.3.p3.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.4.m4.1c">p^{\prime}\leq p</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.4.m4.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_p</annotation></semantics></math> admissible pieces for <math alttext="f" class="ltx_Math" display="inline" id="S4.3.p3.5.m5.1"><semantics id="S4.3.p3.5.m5.1a"><mi id="S4.3.p3.5.m5.1.1" xref="S4.3.p3.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.3.p3.5.m5.1b"><ci id="S4.3.p3.5.m5.1.1.cmml" xref="S4.3.p3.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.5.m5.1d">italic_f</annotation></semantics></math>, along with corresponding vertex and edge sets <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.3.p3.6.m6.1"><semantics id="S4.3.p3.6.m6.1a"><msup id="S4.3.p3.6.m6.1.1" xref="S4.3.p3.6.m6.1.1.cmml"><mi id="S4.3.p3.6.m6.1.1.2" xref="S4.3.p3.6.m6.1.1.2.cmml">V</mi><mo id="S4.3.p3.6.m6.1.1.3" xref="S4.3.p3.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.3.p3.6.m6.1b"><apply id="S4.3.p3.6.m6.1.1.cmml" xref="S4.3.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.3.p3.6.m6.1.1.1.cmml" xref="S4.3.p3.6.m6.1.1">superscript</csymbol><ci id="S4.3.p3.6.m6.1.1.2.cmml" xref="S4.3.p3.6.m6.1.1.2">𝑉</ci><ci id="S4.3.p3.6.m6.1.1.3.cmml" xref="S4.3.p3.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.6.m6.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.6.m6.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.3.p3.7.m7.1"><semantics id="S4.3.p3.7.m7.1a"><msup id="S4.3.p3.7.m7.1.1" xref="S4.3.p3.7.m7.1.1.cmml"><mi id="S4.3.p3.7.m7.1.1.2" xref="S4.3.p3.7.m7.1.1.2.cmml">E</mi><mo id="S4.3.p3.7.m7.1.1.3" xref="S4.3.p3.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.3.p3.7.m7.1b"><apply id="S4.3.p3.7.m7.1.1.cmml" xref="S4.3.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.3.p3.7.m7.1.1.1.cmml" xref="S4.3.p3.7.m7.1.1">superscript</csymbol><ci id="S4.3.p3.7.m7.1.1.2.cmml" xref="S4.3.p3.7.m7.1.1.2">𝐸</ci><ci id="S4.3.p3.7.m7.1.1.3.cmml" xref="S4.3.p3.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.7.m7.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.7.m7.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\deg(v)\geq 3" class="ltx_Math" display="inline" id="S4.3.p3.8.m8.2"><semantics id="S4.3.p3.8.m8.2a"><mrow id="S4.3.p3.8.m8.2.3" xref="S4.3.p3.8.m8.2.3.cmml"><mrow id="S4.3.p3.8.m8.2.3.2.2" xref="S4.3.p3.8.m8.2.3.2.1.cmml"><mi id="S4.3.p3.8.m8.1.1" xref="S4.3.p3.8.m8.1.1.cmml">deg</mi><mo id="S4.3.p3.8.m8.2.3.2.2a" xref="S4.3.p3.8.m8.2.3.2.1.cmml">⁡</mo><mrow id="S4.3.p3.8.m8.2.3.2.2.1" xref="S4.3.p3.8.m8.2.3.2.1.cmml"><mo id="S4.3.p3.8.m8.2.3.2.2.1.1" stretchy="false" xref="S4.3.p3.8.m8.2.3.2.1.cmml">(</mo><mi id="S4.3.p3.8.m8.2.2" xref="S4.3.p3.8.m8.2.2.cmml">v</mi><mo id="S4.3.p3.8.m8.2.3.2.2.1.2" stretchy="false" xref="S4.3.p3.8.m8.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p3.8.m8.2.3.1" xref="S4.3.p3.8.m8.2.3.1.cmml">≥</mo><mn id="S4.3.p3.8.m8.2.3.3" xref="S4.3.p3.8.m8.2.3.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p3.8.m8.2b"><apply id="S4.3.p3.8.m8.2.3.cmml" xref="S4.3.p3.8.m8.2.3"><geq id="S4.3.p3.8.m8.2.3.1.cmml" xref="S4.3.p3.8.m8.2.3.1"></geq><apply id="S4.3.p3.8.m8.2.3.2.1.cmml" xref="S4.3.p3.8.m8.2.3.2.2"><csymbol cd="latexml" id="S4.3.p3.8.m8.1.1.cmml" xref="S4.3.p3.8.m8.1.1">degree</csymbol><ci id="S4.3.p3.8.m8.2.2.cmml" xref="S4.3.p3.8.m8.2.2">𝑣</ci></apply><cn id="S4.3.p3.8.m8.2.3.3.cmml" type="integer" xref="S4.3.p3.8.m8.2.3.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.8.m8.2c">\deg(v)\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.8.m8.2d">roman_deg ( italic_v ) ≥ 3</annotation></semantics></math> for all <math alttext="v\in V^{\prime}" class="ltx_Math" display="inline" id="S4.3.p3.9.m9.1"><semantics id="S4.3.p3.9.m9.1a"><mrow id="S4.3.p3.9.m9.1.1" xref="S4.3.p3.9.m9.1.1.cmml"><mi id="S4.3.p3.9.m9.1.1.2" xref="S4.3.p3.9.m9.1.1.2.cmml">v</mi><mo id="S4.3.p3.9.m9.1.1.1" xref="S4.3.p3.9.m9.1.1.1.cmml">∈</mo><msup id="S4.3.p3.9.m9.1.1.3" xref="S4.3.p3.9.m9.1.1.3.cmml"><mi id="S4.3.p3.9.m9.1.1.3.2" xref="S4.3.p3.9.m9.1.1.3.2.cmml">V</mi><mo id="S4.3.p3.9.m9.1.1.3.3" xref="S4.3.p3.9.m9.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p3.9.m9.1b"><apply id="S4.3.p3.9.m9.1.1.cmml" xref="S4.3.p3.9.m9.1.1"><in id="S4.3.p3.9.m9.1.1.1.cmml" xref="S4.3.p3.9.m9.1.1.1"></in><ci id="S4.3.p3.9.m9.1.1.2.cmml" xref="S4.3.p3.9.m9.1.1.2">𝑣</ci><apply id="S4.3.p3.9.m9.1.1.3.cmml" xref="S4.3.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.3.p3.9.m9.1.1.3.1.cmml" xref="S4.3.p3.9.m9.1.1.3">superscript</csymbol><ci id="S4.3.p3.9.m9.1.1.3.2.cmml" xref="S4.3.p3.9.m9.1.1.3.2">𝑉</ci><ci id="S4.3.p3.9.m9.1.1.3.3.cmml" xref="S4.3.p3.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p3.9.m9.1c">v\in V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p3.9.m9.1d">italic_v ∈ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.4.p4"> <p class="ltx_p" id="S4.4.p4.4">The sets <math alttext="\mathcal{P}^{\prime}" class="ltx_Math" display="inline" id="S4.4.p4.1.m1.1"><semantics id="S4.4.p4.1.m1.1a"><msup id="S4.4.p4.1.m1.1.1" xref="S4.4.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.4.p4.1.m1.1.1.2" xref="S4.4.p4.1.m1.1.1.2.cmml">𝒫</mi><mo id="S4.4.p4.1.m1.1.1.3" xref="S4.4.p4.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p4.1.m1.1b"><apply id="S4.4.p4.1.m1.1.1.cmml" xref="S4.4.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.4.p4.1.m1.1.1.1.cmml" xref="S4.4.p4.1.m1.1.1">superscript</csymbol><ci id="S4.4.p4.1.m1.1.1.2.cmml" xref="S4.4.p4.1.m1.1.1.2">𝒫</ci><ci id="S4.4.p4.1.m1.1.1.3.cmml" xref="S4.4.p4.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p4.1.m1.1c">\mathcal{P}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p4.1.m1.1d">caligraphic_P start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.4.p4.2.m2.1"><semantics id="S4.4.p4.2.m2.1a"><msup id="S4.4.p4.2.m2.1.1" xref="S4.4.p4.2.m2.1.1.cmml"><mi id="S4.4.p4.2.m2.1.1.2" xref="S4.4.p4.2.m2.1.1.2.cmml">V</mi><mo id="S4.4.p4.2.m2.1.1.3" xref="S4.4.p4.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p4.2.m2.1b"><apply id="S4.4.p4.2.m2.1.1.cmml" xref="S4.4.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.4.p4.2.m2.1.1.1.cmml" xref="S4.4.p4.2.m2.1.1">superscript</csymbol><ci id="S4.4.p4.2.m2.1.1.2.cmml" xref="S4.4.p4.2.m2.1.1.2">𝑉</ci><ci id="S4.4.p4.2.m2.1.1.3.cmml" xref="S4.4.p4.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p4.2.m2.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p4.2.m2.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.4.p4.3.m3.1"><semantics id="S4.4.p4.3.m3.1a"><msup id="S4.4.p4.3.m3.1.1" xref="S4.4.p4.3.m3.1.1.cmml"><mi id="S4.4.p4.3.m3.1.1.2" xref="S4.4.p4.3.m3.1.1.2.cmml">E</mi><mo id="S4.4.p4.3.m3.1.1.3" xref="S4.4.p4.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p4.3.m3.1b"><apply id="S4.4.p4.3.m3.1.1.cmml" xref="S4.4.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.4.p4.3.m3.1.1.1.cmml" xref="S4.4.p4.3.m3.1.1">superscript</csymbol><ci id="S4.4.p4.3.m3.1.1.2.cmml" xref="S4.4.p4.3.m3.1.1.2">𝐸</ci><ci id="S4.4.p4.3.m3.1.1.3.cmml" xref="S4.4.p4.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p4.3.m3.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p4.3.m3.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are the desired sets. It is left to prove that <math alttext="|E^{\prime}|\leq 3p" class="ltx_Math" display="inline" id="S4.4.p4.4.m4.1"><semantics id="S4.4.p4.4.m4.1a"><mrow id="S4.4.p4.4.m4.1.1" xref="S4.4.p4.4.m4.1.1.cmml"><mrow id="S4.4.p4.4.m4.1.1.1.1" xref="S4.4.p4.4.m4.1.1.1.2.cmml"><mo id="S4.4.p4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.4.p4.4.m4.1.1.1.2.1.cmml">|</mo><msup id="S4.4.p4.4.m4.1.1.1.1.1" xref="S4.4.p4.4.m4.1.1.1.1.1.cmml"><mi id="S4.4.p4.4.m4.1.1.1.1.1.2" xref="S4.4.p4.4.m4.1.1.1.1.1.2.cmml">E</mi><mo id="S4.4.p4.4.m4.1.1.1.1.1.3" xref="S4.4.p4.4.m4.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.4.p4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.4.p4.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.4.p4.4.m4.1.1.2" xref="S4.4.p4.4.m4.1.1.2.cmml">≤</mo><mrow id="S4.4.p4.4.m4.1.1.3" xref="S4.4.p4.4.m4.1.1.3.cmml"><mn id="S4.4.p4.4.m4.1.1.3.2" xref="S4.4.p4.4.m4.1.1.3.2.cmml">3</mn><mo id="S4.4.p4.4.m4.1.1.3.1" xref="S4.4.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.4.p4.4.m4.1.1.3.3" xref="S4.4.p4.4.m4.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p4.4.m4.1b"><apply id="S4.4.p4.4.m4.1.1.cmml" xref="S4.4.p4.4.m4.1.1"><leq id="S4.4.p4.4.m4.1.1.2.cmml" xref="S4.4.p4.4.m4.1.1.2"></leq><apply id="S4.4.p4.4.m4.1.1.1.2.cmml" xref="S4.4.p4.4.m4.1.1.1.1"><abs id="S4.4.p4.4.m4.1.1.1.2.1.cmml" xref="S4.4.p4.4.m4.1.1.1.1.2"></abs><apply id="S4.4.p4.4.m4.1.1.1.1.1.cmml" xref="S4.4.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.4.p4.4.m4.1.1.1.1.1.1.cmml" xref="S4.4.p4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S4.4.p4.4.m4.1.1.1.1.1.2.cmml" xref="S4.4.p4.4.m4.1.1.1.1.1.2">𝐸</ci><ci id="S4.4.p4.4.m4.1.1.1.1.1.3.cmml" xref="S4.4.p4.4.m4.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.4.p4.4.m4.1.1.3.cmml" xref="S4.4.p4.4.m4.1.1.3"><times id="S4.4.p4.4.m4.1.1.3.1.cmml" xref="S4.4.p4.4.m4.1.1.3.1"></times><cn id="S4.4.p4.4.m4.1.1.3.2.cmml" type="integer" xref="S4.4.p4.4.m4.1.1.3.2">3</cn><ci id="S4.4.p4.4.m4.1.1.3.3.cmml" xref="S4.4.p4.4.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p4.4.m4.1c">|E^{\prime}|\leq 3p</annotation><annotation encoding="application/x-llamapun" id="S4.4.p4.4.m4.1d">| italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ 3 italic_p</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.5.p5"> <p class="ltx_p" id="S4.5.p5.1">Since all vertices in <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.5.p5.1.m1.1"><semantics id="S4.5.p5.1.m1.1a"><msup id="S4.5.p5.1.m1.1.1" xref="S4.5.p5.1.m1.1.1.cmml"><mi id="S4.5.p5.1.m1.1.1.2" xref="S4.5.p5.1.m1.1.1.2.cmml">V</mi><mo id="S4.5.p5.1.m1.1.1.3" xref="S4.5.p5.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.5.p5.1.m1.1b"><apply id="S4.5.p5.1.m1.1.1.cmml" xref="S4.5.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.5.p5.1.m1.1.1.1.cmml" xref="S4.5.p5.1.m1.1.1">superscript</csymbol><ci id="S4.5.p5.1.m1.1.1.2.cmml" xref="S4.5.p5.1.m1.1.1.2">𝑉</ci><ci id="S4.5.p5.1.m1.1.1.3.cmml" xref="S4.5.p5.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.1.m1.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.1.m1.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> have degree at least three, we have</p> <table class="ltx_equation ltx_eqn_table" id="S4.E34"> <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="3|V^{\prime}|\leq\sum_{v\in V^{\prime}}\deg(v)\leq 2|E^{\prime}|," class="ltx_Math" display="block" id="S4.E34.m1.3"><semantics id="S4.E34.m1.3a"><mrow id="S4.E34.m1.3.3.1" xref="S4.E34.m1.3.3.1.1.cmml"><mrow id="S4.E34.m1.3.3.1.1" xref="S4.E34.m1.3.3.1.1.cmml"><mrow id="S4.E34.m1.3.3.1.1.1" xref="S4.E34.m1.3.3.1.1.1.cmml"><mn id="S4.E34.m1.3.3.1.1.1.3" xref="S4.E34.m1.3.3.1.1.1.3.cmml">3</mn><mo id="S4.E34.m1.3.3.1.1.1.2" xref="S4.E34.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E34.m1.3.3.1.1.1.1.1" xref="S4.E34.m1.3.3.1.1.1.1.2.cmml"><mo id="S4.E34.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.E34.m1.3.3.1.1.1.1.2.1.cmml">|</mo><msup id="S4.E34.m1.3.3.1.1.1.1.1.1" xref="S4.E34.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.E34.m1.3.3.1.1.1.1.1.1.2" xref="S4.E34.m1.3.3.1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.E34.m1.3.3.1.1.1.1.1.1.3" xref="S4.E34.m1.3.3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E34.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.E34.m1.3.3.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.E34.m1.3.3.1.1.4" rspace="0.111em" xref="S4.E34.m1.3.3.1.1.4.cmml">≤</mo><mrow id="S4.E34.m1.3.3.1.1.5" xref="S4.E34.m1.3.3.1.1.5.cmml"><munder id="S4.E34.m1.3.3.1.1.5.1" xref="S4.E34.m1.3.3.1.1.5.1.cmml"><mo id="S4.E34.m1.3.3.1.1.5.1.2" movablelimits="false" xref="S4.E34.m1.3.3.1.1.5.1.2.cmml">∑</mo><mrow id="S4.E34.m1.3.3.1.1.5.1.3" xref="S4.E34.m1.3.3.1.1.5.1.3.cmml"><mi id="S4.E34.m1.3.3.1.1.5.1.3.2" xref="S4.E34.m1.3.3.1.1.5.1.3.2.cmml">v</mi><mo id="S4.E34.m1.3.3.1.1.5.1.3.1" xref="S4.E34.m1.3.3.1.1.5.1.3.1.cmml">∈</mo><msup id="S4.E34.m1.3.3.1.1.5.1.3.3" xref="S4.E34.m1.3.3.1.1.5.1.3.3.cmml"><mi id="S4.E34.m1.3.3.1.1.5.1.3.3.2" xref="S4.E34.m1.3.3.1.1.5.1.3.3.2.cmml">V</mi><mo id="S4.E34.m1.3.3.1.1.5.1.3.3.3" xref="S4.E34.m1.3.3.1.1.5.1.3.3.3.cmml">′</mo></msup></mrow></munder><mrow id="S4.E34.m1.3.3.1.1.5.2.2" xref="S4.E34.m1.3.3.1.1.5.2.1.cmml"><mi id="S4.E34.m1.1.1" xref="S4.E34.m1.1.1.cmml">deg</mi><mo id="S4.E34.m1.3.3.1.1.5.2.2a" xref="S4.E34.m1.3.3.1.1.5.2.1.cmml">⁡</mo><mrow id="S4.E34.m1.3.3.1.1.5.2.2.1" xref="S4.E34.m1.3.3.1.1.5.2.1.cmml"><mo id="S4.E34.m1.3.3.1.1.5.2.2.1.1" stretchy="false" xref="S4.E34.m1.3.3.1.1.5.2.1.cmml">(</mo><mi id="S4.E34.m1.2.2" xref="S4.E34.m1.2.2.cmml">v</mi><mo id="S4.E34.m1.3.3.1.1.5.2.2.1.2" stretchy="false" xref="S4.E34.m1.3.3.1.1.5.2.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E34.m1.3.3.1.1.6" xref="S4.E34.m1.3.3.1.1.6.cmml">≤</mo><mrow id="S4.E34.m1.3.3.1.1.2" xref="S4.E34.m1.3.3.1.1.2.cmml"><mn id="S4.E34.m1.3.3.1.1.2.3" xref="S4.E34.m1.3.3.1.1.2.3.cmml">2</mn><mo id="S4.E34.m1.3.3.1.1.2.2" xref="S4.E34.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="S4.E34.m1.3.3.1.1.2.1.1" xref="S4.E34.m1.3.3.1.1.2.1.2.cmml"><mo id="S4.E34.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="S4.E34.m1.3.3.1.1.2.1.2.1.cmml">|</mo><msup id="S4.E34.m1.3.3.1.1.2.1.1.1" xref="S4.E34.m1.3.3.1.1.2.1.1.1.cmml"><mi id="S4.E34.m1.3.3.1.1.2.1.1.1.2" xref="S4.E34.m1.3.3.1.1.2.1.1.1.2.cmml">E</mi><mo id="S4.E34.m1.3.3.1.1.2.1.1.1.3" xref="S4.E34.m1.3.3.1.1.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.E34.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="S4.E34.m1.3.3.1.1.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><mo id="S4.E34.m1.3.3.1.2" xref="S4.E34.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E34.m1.3b"><apply id="S4.E34.m1.3.3.1.1.cmml" xref="S4.E34.m1.3.3.1"><and id="S4.E34.m1.3.3.1.1a.cmml" xref="S4.E34.m1.3.3.1"></and><apply id="S4.E34.m1.3.3.1.1b.cmml" xref="S4.E34.m1.3.3.1"><leq id="S4.E34.m1.3.3.1.1.4.cmml" xref="S4.E34.m1.3.3.1.1.4"></leq><apply id="S4.E34.m1.3.3.1.1.1.cmml" xref="S4.E34.m1.3.3.1.1.1"><times id="S4.E34.m1.3.3.1.1.1.2.cmml" xref="S4.E34.m1.3.3.1.1.1.2"></times><cn id="S4.E34.m1.3.3.1.1.1.3.cmml" type="integer" xref="S4.E34.m1.3.3.1.1.1.3">3</cn><apply id="S4.E34.m1.3.3.1.1.1.1.2.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1"><abs id="S4.E34.m1.3.3.1.1.1.1.2.1.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1.2"></abs><apply id="S4.E34.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E34.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S4.E34.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1.1.2">𝑉</ci><ci id="S4.E34.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E34.m1.3.3.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S4.E34.m1.3.3.1.1.5.cmml" xref="S4.E34.m1.3.3.1.1.5"><apply id="S4.E34.m1.3.3.1.1.5.1.cmml" xref="S4.E34.m1.3.3.1.1.5.1"><csymbol cd="ambiguous" id="S4.E34.m1.3.3.1.1.5.1.1.cmml" xref="S4.E34.m1.3.3.1.1.5.1">subscript</csymbol><sum id="S4.E34.m1.3.3.1.1.5.1.2.cmml" xref="S4.E34.m1.3.3.1.1.5.1.2"></sum><apply id="S4.E34.m1.3.3.1.1.5.1.3.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3"><in id="S4.E34.m1.3.3.1.1.5.1.3.1.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.1"></in><ci id="S4.E34.m1.3.3.1.1.5.1.3.2.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.2">𝑣</ci><apply id="S4.E34.m1.3.3.1.1.5.1.3.3.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.3"><csymbol cd="ambiguous" id="S4.E34.m1.3.3.1.1.5.1.3.3.1.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.3">superscript</csymbol><ci id="S4.E34.m1.3.3.1.1.5.1.3.3.2.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.3.2">𝑉</ci><ci id="S4.E34.m1.3.3.1.1.5.1.3.3.3.cmml" xref="S4.E34.m1.3.3.1.1.5.1.3.3.3">′</ci></apply></apply></apply><apply id="S4.E34.m1.3.3.1.1.5.2.1.cmml" xref="S4.E34.m1.3.3.1.1.5.2.2"><csymbol cd="latexml" id="S4.E34.m1.1.1.cmml" xref="S4.E34.m1.1.1">degree</csymbol><ci id="S4.E34.m1.2.2.cmml" xref="S4.E34.m1.2.2">𝑣</ci></apply></apply></apply><apply id="S4.E34.m1.3.3.1.1c.cmml" xref="S4.E34.m1.3.3.1"><leq id="S4.E34.m1.3.3.1.1.6.cmml" xref="S4.E34.m1.3.3.1.1.6"></leq><share href="https://arxiv.org/html/2503.13001v1#S4.E34.m1.3.3.1.1.5.cmml" id="S4.E34.m1.3.3.1.1d.cmml" xref="S4.E34.m1.3.3.1"></share><apply id="S4.E34.m1.3.3.1.1.2.cmml" xref="S4.E34.m1.3.3.1.1.2"><times id="S4.E34.m1.3.3.1.1.2.2.cmml" xref="S4.E34.m1.3.3.1.1.2.2"></times><cn id="S4.E34.m1.3.3.1.1.2.3.cmml" type="integer" xref="S4.E34.m1.3.3.1.1.2.3">2</cn><apply id="S4.E34.m1.3.3.1.1.2.1.2.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1"><abs id="S4.E34.m1.3.3.1.1.2.1.2.1.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1.2"></abs><apply id="S4.E34.m1.3.3.1.1.2.1.1.1.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S4.E34.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1.1">superscript</csymbol><ci id="S4.E34.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1.1.2">𝐸</ci><ci id="S4.E34.m1.3.3.1.1.2.1.1.1.3.cmml" xref="S4.E34.m1.3.3.1.1.2.1.1.1.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E34.m1.3c">3|V^{\prime}|\leq\sum_{v\in V^{\prime}}\deg(v)\leq 2|E^{\prime}|,</annotation><annotation encoding="application/x-llamapun" id="S4.E34.m1.3d">3 | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_deg ( italic_v ) ≤ 2 | italic_E start_POSTSUPERSCRIPT ′ 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">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.5.p5.13">where the last inequality holds as an equality if all edges are line segments. Let <math alttext="D" class="ltx_Math" display="inline" id="S4.5.p5.2.m1.1"><semantics id="S4.5.p5.2.m1.1a"><mi id="S4.5.p5.2.m1.1.1" xref="S4.5.p5.2.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.5.p5.2.m1.1b"><ci id="S4.5.p5.2.m1.1.1.cmml" xref="S4.5.p5.2.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.2.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.2.m1.1d">italic_D</annotation></semantics></math> be a disk that contains all vertices in <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.5.p5.3.m2.1"><semantics id="S4.5.p5.3.m2.1a"><msup id="S4.5.p5.3.m2.1.1" xref="S4.5.p5.3.m2.1.1.cmml"><mi id="S4.5.p5.3.m2.1.1.2" xref="S4.5.p5.3.m2.1.1.2.cmml">V</mi><mo id="S4.5.p5.3.m2.1.1.3" xref="S4.5.p5.3.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.5.p5.3.m2.1b"><apply id="S4.5.p5.3.m2.1.1.cmml" xref="S4.5.p5.3.m2.1.1"><csymbol cd="ambiguous" id="S4.5.p5.3.m2.1.1.1.cmml" xref="S4.5.p5.3.m2.1.1">superscript</csymbol><ci id="S4.5.p5.3.m2.1.1.2.cmml" xref="S4.5.p5.3.m2.1.1.2">𝑉</ci><ci id="S4.5.p5.3.m2.1.1.3.cmml" xref="S4.5.p5.3.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.3.m2.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.3.m2.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and intersects every ray and line in <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.5.p5.4.m3.1"><semantics id="S4.5.p5.4.m3.1a"><msup id="S4.5.p5.4.m3.1.1" xref="S4.5.p5.4.m3.1.1.cmml"><mi id="S4.5.p5.4.m3.1.1.2" xref="S4.5.p5.4.m3.1.1.2.cmml">E</mi><mo id="S4.5.p5.4.m3.1.1.3" xref="S4.5.p5.4.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.5.p5.4.m3.1b"><apply id="S4.5.p5.4.m3.1.1.cmml" xref="S4.5.p5.4.m3.1.1"><csymbol cd="ambiguous" id="S4.5.p5.4.m3.1.1.1.cmml" xref="S4.5.p5.4.m3.1.1">superscript</csymbol><ci id="S4.5.p5.4.m3.1.1.2.cmml" xref="S4.5.p5.4.m3.1.1.2">𝐸</ci><ci id="S4.5.p5.4.m3.1.1.3.cmml" xref="S4.5.p5.4.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.4.m3.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.4.m3.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We form a plane graph <math alttext="\overline{G}:=(\overline{V},\overline{E})" class="ltx_Math" display="inline" id="S4.5.p5.5.m4.2"><semantics id="S4.5.p5.5.m4.2a"><mrow id="S4.5.p5.5.m4.2.3" xref="S4.5.p5.5.m4.2.3.cmml"><mover accent="true" id="S4.5.p5.5.m4.2.3.2" xref="S4.5.p5.5.m4.2.3.2.cmml"><mi id="S4.5.p5.5.m4.2.3.2.2" xref="S4.5.p5.5.m4.2.3.2.2.cmml">G</mi><mo id="S4.5.p5.5.m4.2.3.2.1" xref="S4.5.p5.5.m4.2.3.2.1.cmml">¯</mo></mover><mo id="S4.5.p5.5.m4.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.5.p5.5.m4.2.3.1.cmml">:=</mo><mrow id="S4.5.p5.5.m4.2.3.3.2" xref="S4.5.p5.5.m4.2.3.3.1.cmml"><mo id="S4.5.p5.5.m4.2.3.3.2.1" stretchy="false" xref="S4.5.p5.5.m4.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.5.p5.5.m4.1.1" xref="S4.5.p5.5.m4.1.1.cmml"><mi id="S4.5.p5.5.m4.1.1.2" xref="S4.5.p5.5.m4.1.1.2.cmml">V</mi><mo id="S4.5.p5.5.m4.1.1.1" xref="S4.5.p5.5.m4.1.1.1.cmml">¯</mo></mover><mo id="S4.5.p5.5.m4.2.3.3.2.2" xref="S4.5.p5.5.m4.2.3.3.1.cmml">,</mo><mover accent="true" id="S4.5.p5.5.m4.2.2" xref="S4.5.p5.5.m4.2.2.cmml"><mi id="S4.5.p5.5.m4.2.2.2" xref="S4.5.p5.5.m4.2.2.2.cmml">E</mi><mo id="S4.5.p5.5.m4.2.2.1" xref="S4.5.p5.5.m4.2.2.1.cmml">¯</mo></mover><mo id="S4.5.p5.5.m4.2.3.3.2.3" stretchy="false" xref="S4.5.p5.5.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p5.5.m4.2b"><apply id="S4.5.p5.5.m4.2.3.cmml" xref="S4.5.p5.5.m4.2.3"><csymbol cd="latexml" id="S4.5.p5.5.m4.2.3.1.cmml" xref="S4.5.p5.5.m4.2.3.1">assign</csymbol><apply id="S4.5.p5.5.m4.2.3.2.cmml" xref="S4.5.p5.5.m4.2.3.2"><ci id="S4.5.p5.5.m4.2.3.2.1.cmml" xref="S4.5.p5.5.m4.2.3.2.1">¯</ci><ci id="S4.5.p5.5.m4.2.3.2.2.cmml" xref="S4.5.p5.5.m4.2.3.2.2">𝐺</ci></apply><interval closure="open" id="S4.5.p5.5.m4.2.3.3.1.cmml" xref="S4.5.p5.5.m4.2.3.3.2"><apply id="S4.5.p5.5.m4.1.1.cmml" xref="S4.5.p5.5.m4.1.1"><ci id="S4.5.p5.5.m4.1.1.1.cmml" xref="S4.5.p5.5.m4.1.1.1">¯</ci><ci id="S4.5.p5.5.m4.1.1.2.cmml" xref="S4.5.p5.5.m4.1.1.2">𝑉</ci></apply><apply id="S4.5.p5.5.m4.2.2.cmml" xref="S4.5.p5.5.m4.2.2"><ci id="S4.5.p5.5.m4.2.2.1.cmml" xref="S4.5.p5.5.m4.2.2.1">¯</ci><ci id="S4.5.p5.5.m4.2.2.2.cmml" xref="S4.5.p5.5.m4.2.2.2">𝐸</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.5.m4.2c">\overline{G}:=(\overline{V},\overline{E})</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.5.m4.2d">over¯ start_ARG italic_G end_ARG := ( over¯ start_ARG italic_V end_ARG , over¯ start_ARG italic_E end_ARG )</annotation></semantics></math> from <math alttext="(V^{\prime},E^{\prime})" class="ltx_Math" display="inline" id="S4.5.p5.6.m5.2"><semantics id="S4.5.p5.6.m5.2a"><mrow id="S4.5.p5.6.m5.2.2.2" xref="S4.5.p5.6.m5.2.2.3.cmml"><mo id="S4.5.p5.6.m5.2.2.2.3" stretchy="false" xref="S4.5.p5.6.m5.2.2.3.cmml">(</mo><msup id="S4.5.p5.6.m5.1.1.1.1" xref="S4.5.p5.6.m5.1.1.1.1.cmml"><mi id="S4.5.p5.6.m5.1.1.1.1.2" xref="S4.5.p5.6.m5.1.1.1.1.2.cmml">V</mi><mo id="S4.5.p5.6.m5.1.1.1.1.3" xref="S4.5.p5.6.m5.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.5.p5.6.m5.2.2.2.4" xref="S4.5.p5.6.m5.2.2.3.cmml">,</mo><msup id="S4.5.p5.6.m5.2.2.2.2" xref="S4.5.p5.6.m5.2.2.2.2.cmml"><mi id="S4.5.p5.6.m5.2.2.2.2.2" xref="S4.5.p5.6.m5.2.2.2.2.2.cmml">E</mi><mo id="S4.5.p5.6.m5.2.2.2.2.3" xref="S4.5.p5.6.m5.2.2.2.2.3.cmml">′</mo></msup><mo id="S4.5.p5.6.m5.2.2.2.5" stretchy="false" xref="S4.5.p5.6.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p5.6.m5.2b"><interval closure="open" id="S4.5.p5.6.m5.2.2.3.cmml" xref="S4.5.p5.6.m5.2.2.2"><apply id="S4.5.p5.6.m5.1.1.1.1.cmml" xref="S4.5.p5.6.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p5.6.m5.1.1.1.1.1.cmml" xref="S4.5.p5.6.m5.1.1.1.1">superscript</csymbol><ci id="S4.5.p5.6.m5.1.1.1.1.2.cmml" xref="S4.5.p5.6.m5.1.1.1.1.2">𝑉</ci><ci id="S4.5.p5.6.m5.1.1.1.1.3.cmml" xref="S4.5.p5.6.m5.1.1.1.1.3">′</ci></apply><apply id="S4.5.p5.6.m5.2.2.2.2.cmml" xref="S4.5.p5.6.m5.2.2.2.2"><csymbol cd="ambiguous" id="S4.5.p5.6.m5.2.2.2.2.1.cmml" xref="S4.5.p5.6.m5.2.2.2.2">superscript</csymbol><ci id="S4.5.p5.6.m5.2.2.2.2.2.cmml" xref="S4.5.p5.6.m5.2.2.2.2.2">𝐸</ci><ci id="S4.5.p5.6.m5.2.2.2.2.3.cmml" xref="S4.5.p5.6.m5.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.6.m5.2c">(V^{\prime},E^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.6.m5.2d">( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> by adding an auxiliary vertex outside of <math alttext="D" class="ltx_Math" display="inline" id="S4.5.p5.7.m6.1"><semantics id="S4.5.p5.7.m6.1a"><mi id="S4.5.p5.7.m6.1.1" xref="S4.5.p5.7.m6.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.5.p5.7.m6.1b"><ci id="S4.5.p5.7.m6.1.1.cmml" xref="S4.5.p5.7.m6.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.7.m6.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.7.m6.1d">italic_D</annotation></semantics></math> and connecting all unbounded edges to this vertex. Hence, <math alttext="|\overline{V}|=|V^{\prime}|+1" class="ltx_Math" display="inline" id="S4.5.p5.8.m7.2"><semantics id="S4.5.p5.8.m7.2a"><mrow id="S4.5.p5.8.m7.2.2" xref="S4.5.p5.8.m7.2.2.cmml"><mrow id="S4.5.p5.8.m7.2.2.3.2" xref="S4.5.p5.8.m7.2.2.3.1.cmml"><mo id="S4.5.p5.8.m7.2.2.3.2.1" stretchy="false" xref="S4.5.p5.8.m7.2.2.3.1.1.cmml">|</mo><mover accent="true" id="S4.5.p5.8.m7.1.1" xref="S4.5.p5.8.m7.1.1.cmml"><mi id="S4.5.p5.8.m7.1.1.2" xref="S4.5.p5.8.m7.1.1.2.cmml">V</mi><mo id="S4.5.p5.8.m7.1.1.1" xref="S4.5.p5.8.m7.1.1.1.cmml">¯</mo></mover><mo id="S4.5.p5.8.m7.2.2.3.2.2" stretchy="false" xref="S4.5.p5.8.m7.2.2.3.1.1.cmml">|</mo></mrow><mo id="S4.5.p5.8.m7.2.2.2" xref="S4.5.p5.8.m7.2.2.2.cmml">=</mo><mrow id="S4.5.p5.8.m7.2.2.1" xref="S4.5.p5.8.m7.2.2.1.cmml"><mrow id="S4.5.p5.8.m7.2.2.1.1.1" xref="S4.5.p5.8.m7.2.2.1.1.2.cmml"><mo id="S4.5.p5.8.m7.2.2.1.1.1.2" stretchy="false" xref="S4.5.p5.8.m7.2.2.1.1.2.1.cmml">|</mo><msup id="S4.5.p5.8.m7.2.2.1.1.1.1" xref="S4.5.p5.8.m7.2.2.1.1.1.1.cmml"><mi id="S4.5.p5.8.m7.2.2.1.1.1.1.2" xref="S4.5.p5.8.m7.2.2.1.1.1.1.2.cmml">V</mi><mo id="S4.5.p5.8.m7.2.2.1.1.1.1.3" xref="S4.5.p5.8.m7.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.5.p5.8.m7.2.2.1.1.1.3" stretchy="false" xref="S4.5.p5.8.m7.2.2.1.1.2.1.cmml">|</mo></mrow><mo id="S4.5.p5.8.m7.2.2.1.2" xref="S4.5.p5.8.m7.2.2.1.2.cmml">+</mo><mn id="S4.5.p5.8.m7.2.2.1.3" xref="S4.5.p5.8.m7.2.2.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p5.8.m7.2b"><apply id="S4.5.p5.8.m7.2.2.cmml" xref="S4.5.p5.8.m7.2.2"><eq id="S4.5.p5.8.m7.2.2.2.cmml" xref="S4.5.p5.8.m7.2.2.2"></eq><apply id="S4.5.p5.8.m7.2.2.3.1.cmml" xref="S4.5.p5.8.m7.2.2.3.2"><abs id="S4.5.p5.8.m7.2.2.3.1.1.cmml" xref="S4.5.p5.8.m7.2.2.3.2.1"></abs><apply id="S4.5.p5.8.m7.1.1.cmml" xref="S4.5.p5.8.m7.1.1"><ci id="S4.5.p5.8.m7.1.1.1.cmml" xref="S4.5.p5.8.m7.1.1.1">¯</ci><ci id="S4.5.p5.8.m7.1.1.2.cmml" xref="S4.5.p5.8.m7.1.1.2">𝑉</ci></apply></apply><apply id="S4.5.p5.8.m7.2.2.1.cmml" xref="S4.5.p5.8.m7.2.2.1"><plus id="S4.5.p5.8.m7.2.2.1.2.cmml" xref="S4.5.p5.8.m7.2.2.1.2"></plus><apply id="S4.5.p5.8.m7.2.2.1.1.2.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1"><abs id="S4.5.p5.8.m7.2.2.1.1.2.1.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1.2"></abs><apply id="S4.5.p5.8.m7.2.2.1.1.1.1.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p5.8.m7.2.2.1.1.1.1.1.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1.1">superscript</csymbol><ci id="S4.5.p5.8.m7.2.2.1.1.1.1.2.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1.1.2">𝑉</ci><ci id="S4.5.p5.8.m7.2.2.1.1.1.1.3.cmml" xref="S4.5.p5.8.m7.2.2.1.1.1.1.3">′</ci></apply></apply><cn id="S4.5.p5.8.m7.2.2.1.3.cmml" type="integer" xref="S4.5.p5.8.m7.2.2.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.8.m7.2c">|\overline{V}|=|V^{\prime}|+1</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.8.m7.2d">| over¯ start_ARG italic_V end_ARG | = | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | + 1</annotation></semantics></math>, <math alttext="|\overline{E}|=|E^{\prime}|" class="ltx_Math" display="inline" id="S4.5.p5.9.m8.2"><semantics id="S4.5.p5.9.m8.2a"><mrow id="S4.5.p5.9.m8.2.2" xref="S4.5.p5.9.m8.2.2.cmml"><mrow id="S4.5.p5.9.m8.2.2.3.2" xref="S4.5.p5.9.m8.2.2.3.1.cmml"><mo id="S4.5.p5.9.m8.2.2.3.2.1" stretchy="false" xref="S4.5.p5.9.m8.2.2.3.1.1.cmml">|</mo><mover accent="true" id="S4.5.p5.9.m8.1.1" xref="S4.5.p5.9.m8.1.1.cmml"><mi id="S4.5.p5.9.m8.1.1.2" xref="S4.5.p5.9.m8.1.1.2.cmml">E</mi><mo id="S4.5.p5.9.m8.1.1.1" xref="S4.5.p5.9.m8.1.1.1.cmml">¯</mo></mover><mo id="S4.5.p5.9.m8.2.2.3.2.2" stretchy="false" xref="S4.5.p5.9.m8.2.2.3.1.1.cmml">|</mo></mrow><mo id="S4.5.p5.9.m8.2.2.2" xref="S4.5.p5.9.m8.2.2.2.cmml">=</mo><mrow id="S4.5.p5.9.m8.2.2.1.1" xref="S4.5.p5.9.m8.2.2.1.2.cmml"><mo id="S4.5.p5.9.m8.2.2.1.1.2" stretchy="false" xref="S4.5.p5.9.m8.2.2.1.2.1.cmml">|</mo><msup id="S4.5.p5.9.m8.2.2.1.1.1" xref="S4.5.p5.9.m8.2.2.1.1.1.cmml"><mi id="S4.5.p5.9.m8.2.2.1.1.1.2" xref="S4.5.p5.9.m8.2.2.1.1.1.2.cmml">E</mi><mo id="S4.5.p5.9.m8.2.2.1.1.1.3" xref="S4.5.p5.9.m8.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.5.p5.9.m8.2.2.1.1.3" stretchy="false" xref="S4.5.p5.9.m8.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p5.9.m8.2b"><apply id="S4.5.p5.9.m8.2.2.cmml" xref="S4.5.p5.9.m8.2.2"><eq id="S4.5.p5.9.m8.2.2.2.cmml" xref="S4.5.p5.9.m8.2.2.2"></eq><apply id="S4.5.p5.9.m8.2.2.3.1.cmml" xref="S4.5.p5.9.m8.2.2.3.2"><abs id="S4.5.p5.9.m8.2.2.3.1.1.cmml" xref="S4.5.p5.9.m8.2.2.3.2.1"></abs><apply id="S4.5.p5.9.m8.1.1.cmml" xref="S4.5.p5.9.m8.1.1"><ci id="S4.5.p5.9.m8.1.1.1.cmml" xref="S4.5.p5.9.m8.1.1.1">¯</ci><ci id="S4.5.p5.9.m8.1.1.2.cmml" xref="S4.5.p5.9.m8.1.1.2">𝐸</ci></apply></apply><apply id="S4.5.p5.9.m8.2.2.1.2.cmml" xref="S4.5.p5.9.m8.2.2.1.1"><abs id="S4.5.p5.9.m8.2.2.1.2.1.cmml" xref="S4.5.p5.9.m8.2.2.1.1.2"></abs><apply id="S4.5.p5.9.m8.2.2.1.1.1.cmml" xref="S4.5.p5.9.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.5.p5.9.m8.2.2.1.1.1.1.cmml" xref="S4.5.p5.9.m8.2.2.1.1.1">superscript</csymbol><ci id="S4.5.p5.9.m8.2.2.1.1.1.2.cmml" xref="S4.5.p5.9.m8.2.2.1.1.1.2">𝐸</ci><ci id="S4.5.p5.9.m8.2.2.1.1.1.3.cmml" xref="S4.5.p5.9.m8.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.9.m8.2c">|\overline{E}|=|E^{\prime}|</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.9.m8.2d">| over¯ start_ARG italic_E end_ARG | = | italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT |</annotation></semantics></math>, and <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.5.p5.10.m9.1"><semantics id="S4.5.p5.10.m9.1a"><mover accent="true" id="S4.5.p5.10.m9.1.1" xref="S4.5.p5.10.m9.1.1.cmml"><mi id="S4.5.p5.10.m9.1.1.2" xref="S4.5.p5.10.m9.1.1.2.cmml">G</mi><mo id="S4.5.p5.10.m9.1.1.1" xref="S4.5.p5.10.m9.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.5.p5.10.m9.1b"><apply id="S4.5.p5.10.m9.1.1.cmml" xref="S4.5.p5.10.m9.1.1"><ci id="S4.5.p5.10.m9.1.1.1.cmml" xref="S4.5.p5.10.m9.1.1.1">¯</ci><ci id="S4.5.p5.10.m9.1.1.2.cmml" xref="S4.5.p5.10.m9.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.10.m9.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.10.m9.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math> has <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="S4.5.p5.11.m10.1"><semantics id="S4.5.p5.11.m10.1a"><msup id="S4.5.p5.11.m10.1.1" xref="S4.5.p5.11.m10.1.1.cmml"><mi id="S4.5.p5.11.m10.1.1.2" xref="S4.5.p5.11.m10.1.1.2.cmml">p</mi><mo id="S4.5.p5.11.m10.1.1.3" xref="S4.5.p5.11.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.5.p5.11.m10.1b"><apply id="S4.5.p5.11.m10.1.1.cmml" xref="S4.5.p5.11.m10.1.1"><csymbol cd="ambiguous" id="S4.5.p5.11.m10.1.1.1.cmml" xref="S4.5.p5.11.m10.1.1">superscript</csymbol><ci id="S4.5.p5.11.m10.1.1.2.cmml" xref="S4.5.p5.11.m10.1.1.2">𝑝</ci><ci id="S4.5.p5.11.m10.1.1.3.cmml" xref="S4.5.p5.11.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.11.m10.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.11.m10.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> faces. Let <math alttext="k" class="ltx_Math" display="inline" id="S4.5.p5.12.m11.1"><semantics id="S4.5.p5.12.m11.1a"><mi id="S4.5.p5.12.m11.1.1" xref="S4.5.p5.12.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.5.p5.12.m11.1b"><ci id="S4.5.p5.12.m11.1.1.cmml" xref="S4.5.p5.12.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.12.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.12.m11.1d">italic_k</annotation></semantics></math> be the number of connected components of <math alttext="\overline{G}" class="ltx_Math" display="inline" id="S4.5.p5.13.m12.1"><semantics id="S4.5.p5.13.m12.1a"><mover accent="true" id="S4.5.p5.13.m12.1.1" xref="S4.5.p5.13.m12.1.1.cmml"><mi id="S4.5.p5.13.m12.1.1.2" xref="S4.5.p5.13.m12.1.1.2.cmml">G</mi><mo id="S4.5.p5.13.m12.1.1.1" xref="S4.5.p5.13.m12.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.5.p5.13.m12.1b"><apply id="S4.5.p5.13.m12.1.1.cmml" xref="S4.5.p5.13.m12.1.1"><ci id="S4.5.p5.13.m12.1.1.1.cmml" xref="S4.5.p5.13.m12.1.1.1">¯</ci><ci id="S4.5.p5.13.m12.1.1.2.cmml" xref="S4.5.p5.13.m12.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.13.m12.1c">\overline{G}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.13.m12.1d">over¯ start_ARG italic_G end_ARG</annotation></semantics></math>. A straightforward extension of Euler’s formula for disconnected graphs (see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Diestel2017</span>, Thm. 4.2.9]</cite> for the standard version) is given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.E35"> <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="\absolutevalue{\overline{V}}-\absolutevalue{\overline{E}}+p^{\prime}=k+1." class="ltx_Math" display="block" id="S4.E35.m1.3"><semantics id="S4.E35.m1.3a"><mrow id="S4.E35.m1.3.3.1" xref="S4.E35.m1.3.3.1.1.cmml"><mrow id="S4.E35.m1.3.3.1.1" xref="S4.E35.m1.3.3.1.1.cmml"><mrow id="S4.E35.m1.3.3.1.1.2" xref="S4.E35.m1.3.3.1.1.2.cmml"><mrow id="S4.E35.m1.3.3.1.1.2.2" xref="S4.E35.m1.3.3.1.1.2.2.cmml"><mrow id="S4.E35.m1.1.1.3" xref="S4.E35.m1.1.1.2.cmml"><mo id="S4.E35.m1.1.1.3.1" xref="S4.E35.m1.1.1.2.1.cmml">|</mo><mover accent="true" id="S4.E35.m1.1.1.1.1.1" xref="S4.E35.m1.1.1.1.1.1.cmml"><mi id="S4.E35.m1.1.1.1.1.1.2" xref="S4.E35.m1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.E35.m1.1.1.1.1.1.1" xref="S4.E35.m1.1.1.1.1.1.1.cmml">¯</mo></mover><mo id="S4.E35.m1.1.1.3.2" xref="S4.E35.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.E35.m1.3.3.1.1.2.2.1" xref="S4.E35.m1.3.3.1.1.2.2.1.cmml">−</mo><mrow id="S4.E35.m1.2.2.3" xref="S4.E35.m1.2.2.2.cmml"><mo id="S4.E35.m1.2.2.3.1" xref="S4.E35.m1.2.2.2.1.cmml">|</mo><mover accent="true" id="S4.E35.m1.2.2.1.1.1" xref="S4.E35.m1.2.2.1.1.1.cmml"><mi id="S4.E35.m1.2.2.1.1.1.2" xref="S4.E35.m1.2.2.1.1.1.2.cmml">E</mi><mo id="S4.E35.m1.2.2.1.1.1.1" xref="S4.E35.m1.2.2.1.1.1.1.cmml">¯</mo></mover><mo id="S4.E35.m1.2.2.3.2" xref="S4.E35.m1.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S4.E35.m1.3.3.1.1.2.1" xref="S4.E35.m1.3.3.1.1.2.1.cmml">+</mo><msup id="S4.E35.m1.3.3.1.1.2.3" xref="S4.E35.m1.3.3.1.1.2.3.cmml"><mi id="S4.E35.m1.3.3.1.1.2.3.2" xref="S4.E35.m1.3.3.1.1.2.3.2.cmml">p</mi><mo id="S4.E35.m1.3.3.1.1.2.3.3" xref="S4.E35.m1.3.3.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S4.E35.m1.3.3.1.1.1" xref="S4.E35.m1.3.3.1.1.1.cmml">=</mo><mrow id="S4.E35.m1.3.3.1.1.3" xref="S4.E35.m1.3.3.1.1.3.cmml"><mi id="S4.E35.m1.3.3.1.1.3.2" xref="S4.E35.m1.3.3.1.1.3.2.cmml">k</mi><mo id="S4.E35.m1.3.3.1.1.3.1" xref="S4.E35.m1.3.3.1.1.3.1.cmml">+</mo><mn id="S4.E35.m1.3.3.1.1.3.3" xref="S4.E35.m1.3.3.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.E35.m1.3.3.1.2" lspace="0em" xref="S4.E35.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E35.m1.3b"><apply id="S4.E35.m1.3.3.1.1.cmml" xref="S4.E35.m1.3.3.1"><eq id="S4.E35.m1.3.3.1.1.1.cmml" xref="S4.E35.m1.3.3.1.1.1"></eq><apply id="S4.E35.m1.3.3.1.1.2.cmml" xref="S4.E35.m1.3.3.1.1.2"><plus id="S4.E35.m1.3.3.1.1.2.1.cmml" xref="S4.E35.m1.3.3.1.1.2.1"></plus><apply id="S4.E35.m1.3.3.1.1.2.2.cmml" xref="S4.E35.m1.3.3.1.1.2.2"><minus id="S4.E35.m1.3.3.1.1.2.2.1.cmml" xref="S4.E35.m1.3.3.1.1.2.2.1"></minus><apply id="S4.E35.m1.1.1.2.cmml" xref="S4.E35.m1.1.1.3"><abs id="S4.E35.m1.1.1.2.1.cmml" xref="S4.E35.m1.1.1.3.1"></abs><apply id="S4.E35.m1.1.1.1.1.1.cmml" xref="S4.E35.m1.1.1.1.1.1"><ci id="S4.E35.m1.1.1.1.1.1.1.cmml" xref="S4.E35.m1.1.1.1.1.1.1">¯</ci><ci id="S4.E35.m1.1.1.1.1.1.2.cmml" xref="S4.E35.m1.1.1.1.1.1.2">𝑉</ci></apply></apply><apply id="S4.E35.m1.2.2.2.cmml" xref="S4.E35.m1.2.2.3"><abs id="S4.E35.m1.2.2.2.1.cmml" xref="S4.E35.m1.2.2.3.1"></abs><apply id="S4.E35.m1.2.2.1.1.1.cmml" xref="S4.E35.m1.2.2.1.1.1"><ci id="S4.E35.m1.2.2.1.1.1.1.cmml" xref="S4.E35.m1.2.2.1.1.1.1">¯</ci><ci id="S4.E35.m1.2.2.1.1.1.2.cmml" xref="S4.E35.m1.2.2.1.1.1.2">𝐸</ci></apply></apply></apply><apply id="S4.E35.m1.3.3.1.1.2.3.cmml" xref="S4.E35.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S4.E35.m1.3.3.1.1.2.3.1.cmml" xref="S4.E35.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S4.E35.m1.3.3.1.1.2.3.2.cmml" xref="S4.E35.m1.3.3.1.1.2.3.2">𝑝</ci><ci id="S4.E35.m1.3.3.1.1.2.3.3.cmml" xref="S4.E35.m1.3.3.1.1.2.3.3">′</ci></apply></apply><apply id="S4.E35.m1.3.3.1.1.3.cmml" xref="S4.E35.m1.3.3.1.1.3"><plus id="S4.E35.m1.3.3.1.1.3.1.cmml" xref="S4.E35.m1.3.3.1.1.3.1"></plus><ci id="S4.E35.m1.3.3.1.1.3.2.cmml" xref="S4.E35.m1.3.3.1.1.3.2">𝑘</ci><cn id="S4.E35.m1.3.3.1.1.3.3.cmml" type="integer" xref="S4.E35.m1.3.3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E35.m1.3c">\absolutevalue{\overline{V}}-\absolutevalue{\overline{E}}+p^{\prime}=k+1.</annotation><annotation encoding="application/x-llamapun" id="S4.E35.m1.3d">| start_ARG over¯ start_ARG italic_V end_ARG end_ARG | - | start_ARG over¯ start_ARG italic_E end_ARG end_ARG | + italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_k + 1 .</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">(35)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.5.p5.14">For <math alttext="(V^{\prime},E^{\prime})" class="ltx_Math" display="inline" id="S4.5.p5.14.m1.2"><semantics id="S4.5.p5.14.m1.2a"><mrow id="S4.5.p5.14.m1.2.2.2" xref="S4.5.p5.14.m1.2.2.3.cmml"><mo id="S4.5.p5.14.m1.2.2.2.3" stretchy="false" xref="S4.5.p5.14.m1.2.2.3.cmml">(</mo><msup id="S4.5.p5.14.m1.1.1.1.1" xref="S4.5.p5.14.m1.1.1.1.1.cmml"><mi id="S4.5.p5.14.m1.1.1.1.1.2" xref="S4.5.p5.14.m1.1.1.1.1.2.cmml">V</mi><mo id="S4.5.p5.14.m1.1.1.1.1.3" xref="S4.5.p5.14.m1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.5.p5.14.m1.2.2.2.4" xref="S4.5.p5.14.m1.2.2.3.cmml">,</mo><msup id="S4.5.p5.14.m1.2.2.2.2" xref="S4.5.p5.14.m1.2.2.2.2.cmml"><mi id="S4.5.p5.14.m1.2.2.2.2.2" xref="S4.5.p5.14.m1.2.2.2.2.2.cmml">E</mi><mo id="S4.5.p5.14.m1.2.2.2.2.3" xref="S4.5.p5.14.m1.2.2.2.2.3.cmml">′</mo></msup><mo id="S4.5.p5.14.m1.2.2.2.5" stretchy="false" xref="S4.5.p5.14.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p5.14.m1.2b"><interval closure="open" id="S4.5.p5.14.m1.2.2.3.cmml" xref="S4.5.p5.14.m1.2.2.2"><apply id="S4.5.p5.14.m1.1.1.1.1.cmml" xref="S4.5.p5.14.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p5.14.m1.1.1.1.1.1.cmml" xref="S4.5.p5.14.m1.1.1.1.1">superscript</csymbol><ci id="S4.5.p5.14.m1.1.1.1.1.2.cmml" xref="S4.5.p5.14.m1.1.1.1.1.2">𝑉</ci><ci id="S4.5.p5.14.m1.1.1.1.1.3.cmml" xref="S4.5.p5.14.m1.1.1.1.1.3">′</ci></apply><apply id="S4.5.p5.14.m1.2.2.2.2.cmml" xref="S4.5.p5.14.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.5.p5.14.m1.2.2.2.2.1.cmml" xref="S4.5.p5.14.m1.2.2.2.2">superscript</csymbol><ci id="S4.5.p5.14.m1.2.2.2.2.2.cmml" xref="S4.5.p5.14.m1.2.2.2.2.2">𝐸</ci><ci id="S4.5.p5.14.m1.2.2.2.2.3.cmml" xref="S4.5.p5.14.m1.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p5.14.m1.2c">(V^{\prime},E^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.5.p5.14.m1.2d">( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E35" title="Equation 35 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">35</span></a>) gives</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex44"> <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="\absolutevalue{V^{\prime}}-\absolutevalue{E^{\prime}}+p^{\prime}=k\geq 0." class="ltx_Math" display="block" id="S4.Ex44.m1.3"><semantics id="S4.Ex44.m1.3a"><mrow id="S4.Ex44.m1.3.3.1" xref="S4.Ex44.m1.3.3.1.1.cmml"><mrow id="S4.Ex44.m1.3.3.1.1" xref="S4.Ex44.m1.3.3.1.1.cmml"><mrow id="S4.Ex44.m1.3.3.1.1.2" xref="S4.Ex44.m1.3.3.1.1.2.cmml"><mrow id="S4.Ex44.m1.3.3.1.1.2.2" xref="S4.Ex44.m1.3.3.1.1.2.2.cmml"><mrow id="S4.Ex44.m1.1.1.3" xref="S4.Ex44.m1.1.1.2.cmml"><mo id="S4.Ex44.m1.1.1.3.1" xref="S4.Ex44.m1.1.1.2.1.cmml">|</mo><msup id="S4.Ex44.m1.1.1.1.1.1" xref="S4.Ex44.m1.1.1.1.1.1.cmml"><mi id="S4.Ex44.m1.1.1.1.1.1.2" xref="S4.Ex44.m1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.Ex44.m1.1.1.1.1.1.3" xref="S4.Ex44.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex44.m1.1.1.3.2" xref="S4.Ex44.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex44.m1.3.3.1.1.2.2.1" xref="S4.Ex44.m1.3.3.1.1.2.2.1.cmml">−</mo><mrow id="S4.Ex44.m1.2.2.3" xref="S4.Ex44.m1.2.2.2.cmml"><mo id="S4.Ex44.m1.2.2.3.1" xref="S4.Ex44.m1.2.2.2.1.cmml">|</mo><msup id="S4.Ex44.m1.2.2.1.1.1" xref="S4.Ex44.m1.2.2.1.1.1.cmml"><mi id="S4.Ex44.m1.2.2.1.1.1.2" xref="S4.Ex44.m1.2.2.1.1.1.2.cmml">E</mi><mo id="S4.Ex44.m1.2.2.1.1.1.3" xref="S4.Ex44.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex44.m1.2.2.3.2" xref="S4.Ex44.m1.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex44.m1.3.3.1.1.2.1" xref="S4.Ex44.m1.3.3.1.1.2.1.cmml">+</mo><msup id="S4.Ex44.m1.3.3.1.1.2.3" xref="S4.Ex44.m1.3.3.1.1.2.3.cmml"><mi id="S4.Ex44.m1.3.3.1.1.2.3.2" xref="S4.Ex44.m1.3.3.1.1.2.3.2.cmml">p</mi><mo id="S4.Ex44.m1.3.3.1.1.2.3.3" xref="S4.Ex44.m1.3.3.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S4.Ex44.m1.3.3.1.1.3" xref="S4.Ex44.m1.3.3.1.1.3.cmml">=</mo><mi id="S4.Ex44.m1.3.3.1.1.4" xref="S4.Ex44.m1.3.3.1.1.4.cmml">k</mi><mo id="S4.Ex44.m1.3.3.1.1.5" xref="S4.Ex44.m1.3.3.1.1.5.cmml">≥</mo><mn id="S4.Ex44.m1.3.3.1.1.6" xref="S4.Ex44.m1.3.3.1.1.6.cmml">0</mn></mrow><mo id="S4.Ex44.m1.3.3.1.2" lspace="0em" xref="S4.Ex44.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex44.m1.3b"><apply id="S4.Ex44.m1.3.3.1.1.cmml" xref="S4.Ex44.m1.3.3.1"><and id="S4.Ex44.m1.3.3.1.1a.cmml" 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id="S4.Ex44.m1.2.2.2.1.cmml" xref="S4.Ex44.m1.2.2.3.1"></abs><apply id="S4.Ex44.m1.2.2.1.1.1.cmml" xref="S4.Ex44.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Ex44.m1.2.2.1.1.1.1.cmml" xref="S4.Ex44.m1.2.2.1.1.1">superscript</csymbol><ci id="S4.Ex44.m1.2.2.1.1.1.2.cmml" xref="S4.Ex44.m1.2.2.1.1.1.2">𝐸</ci><ci id="S4.Ex44.m1.2.2.1.1.1.3.cmml" xref="S4.Ex44.m1.2.2.1.1.1.3">′</ci></apply></apply></apply><apply id="S4.Ex44.m1.3.3.1.1.2.3.cmml" xref="S4.Ex44.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S4.Ex44.m1.3.3.1.1.2.3.1.cmml" xref="S4.Ex44.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S4.Ex44.m1.3.3.1.1.2.3.2.cmml" xref="S4.Ex44.m1.3.3.1.1.2.3.2">𝑝</ci><ci id="S4.Ex44.m1.3.3.1.1.2.3.3.cmml" xref="S4.Ex44.m1.3.3.1.1.2.3.3">′</ci></apply></apply><ci id="S4.Ex44.m1.3.3.1.1.4.cmml" xref="S4.Ex44.m1.3.3.1.1.4">𝑘</ci></apply><apply id="S4.Ex44.m1.3.3.1.1c.cmml" xref="S4.Ex44.m1.3.3.1"><geq id="S4.Ex44.m1.3.3.1.1.5.cmml" xref="S4.Ex44.m1.3.3.1.1.5"></geq><share href="https://arxiv.org/html/2503.13001v1#S4.Ex44.m1.3.3.1.1.4.cmml" id="S4.Ex44.m1.3.3.1.1d.cmml" xref="S4.Ex44.m1.3.3.1"></share><cn id="S4.Ex44.m1.3.3.1.1.6.cmml" type="integer" xref="S4.Ex44.m1.3.3.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex44.m1.3c">\absolutevalue{V^{\prime}}-\absolutevalue{E^{\prime}}+p^{\prime}=k\geq 0.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex44.m1.3d">| start_ARG italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG | - | start_ARG italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG | + italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_k ≥ 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.5.p5.15">Combining this with (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E34" title="Equation 34 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">34</span></a>) yields</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex45"> <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="\absolutevalue{E^{\prime}}\leq 3p^{\prime}\leq 3p." class="ltx_Math" display="block" id="S4.Ex45.m1.2"><semantics id="S4.Ex45.m1.2a"><mrow id="S4.Ex45.m1.2.2.1" xref="S4.Ex45.m1.2.2.1.1.cmml"><mrow id="S4.Ex45.m1.2.2.1.1" xref="S4.Ex45.m1.2.2.1.1.cmml"><mrow id="S4.Ex45.m1.1.1.3" xref="S4.Ex45.m1.1.1.2.cmml"><mo id="S4.Ex45.m1.1.1.3.1" xref="S4.Ex45.m1.1.1.2.1.cmml">|</mo><msup id="S4.Ex45.m1.1.1.1.1.1" xref="S4.Ex45.m1.1.1.1.1.1.cmml"><mi id="S4.Ex45.m1.1.1.1.1.1.2" xref="S4.Ex45.m1.1.1.1.1.1.2.cmml">E</mi><mo id="S4.Ex45.m1.1.1.1.1.1.3" xref="S4.Ex45.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex45.m1.1.1.3.2" xref="S4.Ex45.m1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex45.m1.2.2.1.1.2" xref="S4.Ex45.m1.2.2.1.1.2.cmml">≤</mo><mrow id="S4.Ex45.m1.2.2.1.1.3" xref="S4.Ex45.m1.2.2.1.1.3.cmml"><mn id="S4.Ex45.m1.2.2.1.1.3.2" xref="S4.Ex45.m1.2.2.1.1.3.2.cmml">3</mn><mo id="S4.Ex45.m1.2.2.1.1.3.1" xref="S4.Ex45.m1.2.2.1.1.3.1.cmml">⁢</mo><msup id="S4.Ex45.m1.2.2.1.1.3.3" xref="S4.Ex45.m1.2.2.1.1.3.3.cmml"><mi id="S4.Ex45.m1.2.2.1.1.3.3.2" xref="S4.Ex45.m1.2.2.1.1.3.3.2.cmml">p</mi><mo id="S4.Ex45.m1.2.2.1.1.3.3.3" xref="S4.Ex45.m1.2.2.1.1.3.3.3.cmml">′</mo></msup></mrow><mo id="S4.Ex45.m1.2.2.1.1.4" xref="S4.Ex45.m1.2.2.1.1.4.cmml">≤</mo><mrow id="S4.Ex45.m1.2.2.1.1.5" xref="S4.Ex45.m1.2.2.1.1.5.cmml"><mn id="S4.Ex45.m1.2.2.1.1.5.2" xref="S4.Ex45.m1.2.2.1.1.5.2.cmml">3</mn><mo id="S4.Ex45.m1.2.2.1.1.5.1" xref="S4.Ex45.m1.2.2.1.1.5.1.cmml">⁢</mo><mi id="S4.Ex45.m1.2.2.1.1.5.3" xref="S4.Ex45.m1.2.2.1.1.5.3.cmml">p</mi></mrow></mrow><mo id="S4.Ex45.m1.2.2.1.2" lspace="0em" xref="S4.Ex45.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex45.m1.2b"><apply id="S4.Ex45.m1.2.2.1.1.cmml" xref="S4.Ex45.m1.2.2.1"><and id="S4.Ex45.m1.2.2.1.1a.cmml" xref="S4.Ex45.m1.2.2.1"></and><apply id="S4.Ex45.m1.2.2.1.1b.cmml" xref="S4.Ex45.m1.2.2.1"><leq id="S4.Ex45.m1.2.2.1.1.2.cmml" xref="S4.Ex45.m1.2.2.1.1.2"></leq><apply id="S4.Ex45.m1.1.1.2.cmml" xref="S4.Ex45.m1.1.1.3"><abs id="S4.Ex45.m1.1.1.2.1.cmml" xref="S4.Ex45.m1.1.1.3.1"></abs><apply id="S4.Ex45.m1.1.1.1.1.1.cmml" xref="S4.Ex45.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex45.m1.1.1.1.1.1.1.cmml" xref="S4.Ex45.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex45.m1.1.1.1.1.1.2.cmml" xref="S4.Ex45.m1.1.1.1.1.1.2">𝐸</ci><ci id="S4.Ex45.m1.1.1.1.1.1.3.cmml" xref="S4.Ex45.m1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.Ex45.m1.2.2.1.1.3.cmml" xref="S4.Ex45.m1.2.2.1.1.3"><times id="S4.Ex45.m1.2.2.1.1.3.1.cmml" xref="S4.Ex45.m1.2.2.1.1.3.1"></times><cn id="S4.Ex45.m1.2.2.1.1.3.2.cmml" type="integer" xref="S4.Ex45.m1.2.2.1.1.3.2">3</cn><apply id="S4.Ex45.m1.2.2.1.1.3.3.cmml" xref="S4.Ex45.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S4.Ex45.m1.2.2.1.1.3.3.1.cmml" xref="S4.Ex45.m1.2.2.1.1.3.3">superscript</csymbol><ci id="S4.Ex45.m1.2.2.1.1.3.3.2.cmml" xref="S4.Ex45.m1.2.2.1.1.3.3.2">𝑝</ci><ci id="S4.Ex45.m1.2.2.1.1.3.3.3.cmml" xref="S4.Ex45.m1.2.2.1.1.3.3.3">′</ci></apply></apply></apply><apply id="S4.Ex45.m1.2.2.1.1c.cmml" xref="S4.Ex45.m1.2.2.1"><leq id="S4.Ex45.m1.2.2.1.1.4.cmml" xref="S4.Ex45.m1.2.2.1.1.4"></leq><share href="https://arxiv.org/html/2503.13001v1#S4.Ex45.m1.2.2.1.1.3.cmml" id="S4.Ex45.m1.2.2.1.1d.cmml" xref="S4.Ex45.m1.2.2.1"></share><apply id="S4.Ex45.m1.2.2.1.1.5.cmml" xref="S4.Ex45.m1.2.2.1.1.5"><times id="S4.Ex45.m1.2.2.1.1.5.1.cmml" xref="S4.Ex45.m1.2.2.1.1.5.1"></times><cn id="S4.Ex45.m1.2.2.1.1.5.2.cmml" type="integer" xref="S4.Ex45.m1.2.2.1.1.5.2">3</cn><ci id="S4.Ex45.m1.2.2.1.1.5.3.cmml" xref="S4.Ex45.m1.2.2.1.1.5.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex45.m1.2c">\absolutevalue{E^{\prime}}\leq 3p^{\prime}\leq 3p.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex45.m1.2d">| start_ARG italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG | ≤ 3 italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ 3 italic_p .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.5.p5.16">∎</p> </div> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.5">The first step to derive the <math alttext="\max" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">max</mi><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><max id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"></max></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">\max</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">roman_max</annotation></semantics></math>-representation of <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mi id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">roman_CPA</annotation></semantics></math> functions is <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>, which can be applied to the functions <math alttext="f^{v}" class="ltx_Math" display="inline" id="S4.p3.3.m3.1"><semantics id="S4.p3.3.m3.1a"><msup id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml"><mi id="S4.p3.3.m3.1.1.2" xref="S4.p3.3.m3.1.1.2.cmml">f</mi><mi id="S4.p3.3.m3.1.1.3" xref="S4.p3.3.m3.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.1b"><apply id="S4.p3.3.m3.1.1.cmml" xref="S4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p3.3.m3.1.1.1.cmml" xref="S4.p3.3.m3.1.1">superscript</csymbol><ci id="S4.p3.3.m3.1.1.2.cmml" xref="S4.p3.3.m3.1.1.2">𝑓</ci><ci id="S4.p3.3.m3.1.1.3.cmml" xref="S4.p3.3.m3.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math>. It states that every <math alttext="v" class="ltx_Math" display="inline" id="S4.p3.4.m4.1"><semantics id="S4.p3.4.m4.1a"><mi id="S4.p3.4.m4.1.1" xref="S4.p3.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p3.4.m4.1b"><ci id="S4.p3.4.m4.1.1.cmml" xref="S4.p3.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m4.1d">italic_v</annotation></semantics></math>-function can be expressed as a sum of <math alttext="v" class="ltx_Math" display="inline" id="S4.p3.5.m5.1"><semantics id="S4.p3.5.m5.1a"><mi id="S4.p3.5.m5.1.1" xref="S4.p3.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p3.5.m5.1b"><ci id="S4.p3.5.m5.1.1.cmml" xref="S4.p3.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m5.1d">italic_v</annotation></semantics></math>-functions with three pieces each.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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 class="ltx_p" id="S4.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.5.5">For <math alttext="v\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S4.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.1b"><apply id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1"><in id="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2">𝑣</ci><apply id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.2">ℝ</ci><cn id="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.1c">v\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.1d">italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">CPA</mi><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1"><in id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.2">𝑓</ci><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.2">CPA</ci><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> be an arbitrary <math alttext="v" 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">v</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">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.1d">italic_v</annotation></semantics></math>-function with <math alttext="p\geq 3" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.4.m4.1a"><mrow 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" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">p</mi><mo id="S4.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">3</mn></mrow><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"><geq id="S4.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.1"></geq><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.2">𝑝</ci><cn id="S4.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.4.m4.1c">p\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.4.m4.1d">italic_p ≥ 3</annotation></semantics></math> pieces. Then, <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.5.m5.1a"><mi id="S4.Thmtheorem2.p1.5.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.5.m5.1b"><ci id="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.5.m5.1d">italic_f</annotation></semantics></math> can be written as</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex46"> <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=\sum_{n=1}^{p-2}f^{(n)}," class="ltx_Math" display="block" id="S4.Ex46.m1.2"><semantics id="S4.Ex46.m1.2a"><mrow id="S4.Ex46.m1.2.2.1" xref="S4.Ex46.m1.2.2.1.1.cmml"><mrow id="S4.Ex46.m1.2.2.1.1" xref="S4.Ex46.m1.2.2.1.1.cmml"><mi id="S4.Ex46.m1.2.2.1.1.2" xref="S4.Ex46.m1.2.2.1.1.2.cmml">f</mi><mo id="S4.Ex46.m1.2.2.1.1.1" rspace="0.111em" xref="S4.Ex46.m1.2.2.1.1.1.cmml">=</mo><mrow id="S4.Ex46.m1.2.2.1.1.3" xref="S4.Ex46.m1.2.2.1.1.3.cmml"><munderover id="S4.Ex46.m1.2.2.1.1.3.1" xref="S4.Ex46.m1.2.2.1.1.3.1.cmml"><mo id="S4.Ex46.m1.2.2.1.1.3.1.2.2" movablelimits="false" xref="S4.Ex46.m1.2.2.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.Ex46.m1.2.2.1.1.3.1.2.3" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.cmml"><mi id="S4.Ex46.m1.2.2.1.1.3.1.2.3.2" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.2.cmml">n</mi><mo id="S4.Ex46.m1.2.2.1.1.3.1.2.3.1" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.Ex46.m1.2.2.1.1.3.1.2.3.3" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.3.cmml">1</mn></mrow><mrow id="S4.Ex46.m1.2.2.1.1.3.1.3" xref="S4.Ex46.m1.2.2.1.1.3.1.3.cmml"><mi id="S4.Ex46.m1.2.2.1.1.3.1.3.2" xref="S4.Ex46.m1.2.2.1.1.3.1.3.2.cmml">p</mi><mo id="S4.Ex46.m1.2.2.1.1.3.1.3.1" xref="S4.Ex46.m1.2.2.1.1.3.1.3.1.cmml">−</mo><mn id="S4.Ex46.m1.2.2.1.1.3.1.3.3" xref="S4.Ex46.m1.2.2.1.1.3.1.3.3.cmml">2</mn></mrow></munderover><msup id="S4.Ex46.m1.2.2.1.1.3.2" xref="S4.Ex46.m1.2.2.1.1.3.2.cmml"><mi id="S4.Ex46.m1.2.2.1.1.3.2.2" xref="S4.Ex46.m1.2.2.1.1.3.2.2.cmml">f</mi><mrow id="S4.Ex46.m1.1.1.1.3" xref="S4.Ex46.m1.2.2.1.1.3.2.cmml"><mo id="S4.Ex46.m1.1.1.1.3.1" stretchy="false" xref="S4.Ex46.m1.2.2.1.1.3.2.cmml">(</mo><mi id="S4.Ex46.m1.1.1.1.1" xref="S4.Ex46.m1.1.1.1.1.cmml">n</mi><mo id="S4.Ex46.m1.1.1.1.3.2" stretchy="false" xref="S4.Ex46.m1.2.2.1.1.3.2.cmml">)</mo></mrow></msup></mrow></mrow><mo id="S4.Ex46.m1.2.2.1.2" xref="S4.Ex46.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex46.m1.2b"><apply id="S4.Ex46.m1.2.2.1.1.cmml" xref="S4.Ex46.m1.2.2.1"><eq id="S4.Ex46.m1.2.2.1.1.1.cmml" xref="S4.Ex46.m1.2.2.1.1.1"></eq><ci id="S4.Ex46.m1.2.2.1.1.2.cmml" xref="S4.Ex46.m1.2.2.1.1.2">𝑓</ci><apply id="S4.Ex46.m1.2.2.1.1.3.cmml" xref="S4.Ex46.m1.2.2.1.1.3"><apply id="S4.Ex46.m1.2.2.1.1.3.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex46.m1.2.2.1.1.3.1.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1">superscript</csymbol><apply id="S4.Ex46.m1.2.2.1.1.3.1.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex46.m1.2.2.1.1.3.1.2.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1">subscript</csymbol><sum id="S4.Ex46.m1.2.2.1.1.3.1.2.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.2.2"></sum><apply id="S4.Ex46.m1.2.2.1.1.3.1.2.3.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3"><eq id="S4.Ex46.m1.2.2.1.1.3.1.2.3.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.1"></eq><ci id="S4.Ex46.m1.2.2.1.1.3.1.2.3.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.2">𝑛</ci><cn id="S4.Ex46.m1.2.2.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.Ex46.m1.2.2.1.1.3.1.2.3.3">1</cn></apply></apply><apply id="S4.Ex46.m1.2.2.1.1.3.1.3.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.3"><minus id="S4.Ex46.m1.2.2.1.1.3.1.3.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.3.1"></minus><ci id="S4.Ex46.m1.2.2.1.1.3.1.3.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.1.3.2">𝑝</ci><cn id="S4.Ex46.m1.2.2.1.1.3.1.3.3.cmml" type="integer" xref="S4.Ex46.m1.2.2.1.1.3.1.3.3">2</cn></apply></apply><apply id="S4.Ex46.m1.2.2.1.1.3.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex46.m1.2.2.1.1.3.2.1.cmml" xref="S4.Ex46.m1.2.2.1.1.3.2">superscript</csymbol><ci id="S4.Ex46.m1.2.2.1.1.3.2.2.cmml" xref="S4.Ex46.m1.2.2.1.1.3.2.2">𝑓</ci><ci id="S4.Ex46.m1.1.1.1.1.cmml" xref="S4.Ex46.m1.1.1.1.1">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex46.m1.2c">f=\sum_{n=1}^{p-2}f^{(n)},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex46.m1.2d">italic_f = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p - 2 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT ( italic_n ) 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.Thmtheorem2.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.8.3">where <math alttext="f^{(n)}\in\operatorname{CPA}_{3}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.1.m1.1"><semantics id="S4.Thmtheorem2.p1.6.1.m1.1a"><mrow id="S4.Thmtheorem2.p1.6.1.m1.1.2" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.cmml"><msup id="S4.Thmtheorem2.p1.6.1.m1.1.2.2" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.cmml"><mi id="S4.Thmtheorem2.p1.6.1.m1.1.2.2.2" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.2.cmml">f</mi><mrow id="S4.Thmtheorem2.p1.6.1.m1.1.1.1.3" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.cmml"><mo id="S4.Thmtheorem2.p1.6.1.m1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem2.p1.6.1.m1.1.1.1.1" xref="S4.Thmtheorem2.p1.6.1.m1.1.1.1.1.cmml">n</mi><mo id="S4.Thmtheorem2.p1.6.1.m1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.cmml">)</mo></mrow></msup><mo id="S4.Thmtheorem2.p1.6.1.m1.1.2.1" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.1.cmml">∈</mo><msub id="S4.Thmtheorem2.p1.6.1.m1.1.2.3" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3.cmml"><mi id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.2" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3.2.cmml">CPA</mi><mn id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.3" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.1.m1.1b"><apply id="S4.Thmtheorem2.p1.6.1.m1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2"><in id="S4.Thmtheorem2.p1.6.1.m1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.1"></in><apply id="S4.Thmtheorem2.p1.6.1.m1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.1.m1.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem2.p1.6.1.m1.1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.2.2">𝑓</ci><ci id="S4.Thmtheorem2.p1.6.1.m1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.1.1.1">𝑛</ci></apply><apply id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.1.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3">subscript</csymbol><ci id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.2.cmml" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3.2">CPA</ci><cn id="S4.Thmtheorem2.p1.6.1.m1.1.2.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.6.1.m1.1.2.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.1.m1.1c">f^{(n)}\in\operatorname{CPA}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.1.m1.1d">italic_f start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ∈ roman_CPA start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> is a <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.7.2.m2.1"><semantics id="S4.Thmtheorem2.p1.7.2.m2.1a"><mi id="S4.Thmtheorem2.p1.7.2.m2.1.1" xref="S4.Thmtheorem2.p1.7.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.7.2.m2.1b"><ci id="S4.Thmtheorem2.p1.7.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.7.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.7.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.7.2.m2.1d">italic_v</annotation></semantics></math>-function for every <math alttext="n\in[p-2]" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.8.3.m3.1"><semantics id="S4.Thmtheorem2.p1.8.3.m3.1a"><mrow id="S4.Thmtheorem2.p1.8.3.m3.1.1" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.cmml"><mi id="S4.Thmtheorem2.p1.8.3.m3.1.1.3" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.3.cmml">n</mi><mo id="S4.Thmtheorem2.p1.8.3.m3.1.1.2" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.2.cmml">∈</mo><mrow id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.2.cmml"><mo id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.2.1.cmml">[</mo><mrow id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.2.cmml">p</mi><mo id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.8.3.m3.1b"><apply id="S4.Thmtheorem2.p1.8.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1"><in id="S4.Thmtheorem2.p1.8.3.m3.1.1.2.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.2"></in><ci id="S4.Thmtheorem2.p1.8.3.m3.1.1.3.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.3">𝑛</ci><apply id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1"><minus id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.1"></minus><ci id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.2">𝑝</ci><cn id="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.8.3.m3.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.8.3.m3.1c">n\in[p-2]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.8.3.m3.1d">italic_n ∈ [ italic_p - 2 ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.2">In the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>, I will use an auxiliary lemma that describes how to merge two adjacent pieces of a <math alttext="\operatorname{CPL}" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mi id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml">CPL</mi><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><ci id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1">CPL</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">\operatorname{CPL}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">roman_CPL</annotation></semantics></math> function by subtracting a function <math alttext="f^{(n)}" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><msup id="S4.p4.2.m2.1.2" xref="S4.p4.2.m2.1.2.cmml"><mi id="S4.p4.2.m2.1.2.2" xref="S4.p4.2.m2.1.2.2.cmml">f</mi><mrow id="S4.p4.2.m2.1.1.1.3" xref="S4.p4.2.m2.1.2.cmml"><mo id="S4.p4.2.m2.1.1.1.3.1" stretchy="false" xref="S4.p4.2.m2.1.2.cmml">(</mo><mi id="S4.p4.2.m2.1.1.1.1" xref="S4.p4.2.m2.1.1.1.1.cmml">n</mi><mo id="S4.p4.2.m2.1.1.1.3.2" stretchy="false" xref="S4.p4.2.m2.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><apply id="S4.p4.2.m2.1.2.cmml" xref="S4.p4.2.m2.1.2"><csymbol cd="ambiguous" id="S4.p4.2.m2.1.2.1.cmml" xref="S4.p4.2.m2.1.2">superscript</csymbol><ci id="S4.p4.2.m2.1.2.2.cmml" xref="S4.p4.2.m2.1.2.2">𝑓</ci><ci id="S4.p4.2.m2.1.1.1.1.cmml" xref="S4.p4.2.m2.1.1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">f^{(n)}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">italic_f start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math> with three pieces. Two pieces are called adjacent if their intersection contains a line segment.</p> </div> <figure class="ltx_figure" id="S4.F10"> <p class="ltx_p ltx_align_center" id="S4.F10.1"><span class="ltx_text" id="S4.F10.1.1"><foreignobject height="60.0pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="79.5pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="115" id="S4.F10.1.1.1.g1" src="x26.png" width="152"/></foreignobject></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10: </span> Pieces (enclosed by solid lines) of a <math alttext="\operatorname{CPL}" class="ltx_Math" display="inline" id="S4.F10.12.m1.1"><semantics id="S4.F10.12.m1.1b"><mi id="S4.F10.12.m1.1.1" xref="S4.F10.12.m1.1.1.cmml">CPL</mi><annotation-xml encoding="MathML-Content" id="S4.F10.12.m1.1c"><ci id="S4.F10.12.m1.1.1.cmml" xref="S4.F10.12.m1.1.1">CPL</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.12.m1.1d">\operatorname{CPL}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.12.m1.1e">roman_CPL</annotation></semantics></math> function <math alttext="f" class="ltx_Math" display="inline" id="S4.F10.13.m2.1"><semantics id="S4.F10.13.m2.1b"><mi id="S4.F10.13.m2.1.1" xref="S4.F10.13.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.F10.13.m2.1c"><ci id="S4.F10.13.m2.1.1.cmml" xref="S4.F10.13.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.13.m2.1d">f</annotation><annotation encoding="application/x-llamapun" id="S4.F10.13.m2.1e">italic_f</annotation></semantics></math> with adjacent pieces <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.F10.14.m3.1"><semantics id="S4.F10.14.m3.1b"><msub id="S4.F10.14.m3.1.1" xref="S4.F10.14.m3.1.1.cmml"><mi id="S4.F10.14.m3.1.1.2" xref="S4.F10.14.m3.1.1.2.cmml">P</mi><mn id="S4.F10.14.m3.1.1.3" xref="S4.F10.14.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.14.m3.1c"><apply id="S4.F10.14.m3.1.1.cmml" xref="S4.F10.14.m3.1.1"><csymbol cd="ambiguous" id="S4.F10.14.m3.1.1.1.cmml" xref="S4.F10.14.m3.1.1">subscript</csymbol><ci id="S4.F10.14.m3.1.1.2.cmml" xref="S4.F10.14.m3.1.1.2">𝑃</ci><cn id="S4.F10.14.m3.1.1.3.cmml" type="integer" xref="S4.F10.14.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.14.m3.1d">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.14.m3.1e">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.F10.15.m4.1"><semantics id="S4.F10.15.m4.1b"><msub id="S4.F10.15.m4.1.1" xref="S4.F10.15.m4.1.1.cmml"><mi id="S4.F10.15.m4.1.1.2" xref="S4.F10.15.m4.1.1.2.cmml">P</mi><mn id="S4.F10.15.m4.1.1.3" xref="S4.F10.15.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.15.m4.1c"><apply id="S4.F10.15.m4.1.1.cmml" xref="S4.F10.15.m4.1.1"><csymbol cd="ambiguous" id="S4.F10.15.m4.1.1.1.cmml" xref="S4.F10.15.m4.1.1">subscript</csymbol><ci id="S4.F10.15.m4.1.1.2.cmml" xref="S4.F10.15.m4.1.1.2">𝑃</ci><cn id="S4.F10.15.m4.1.1.3.cmml" type="integer" xref="S4.F10.15.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.15.m4.1d">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.15.m4.1e">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. One can define a <math alttext="3" class="ltx_Math" display="inline" id="S4.F10.16.m5.1"><semantics id="S4.F10.16.m5.1b"><mn id="S4.F10.16.m5.1.1" xref="S4.F10.16.m5.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S4.F10.16.m5.1c"><cn id="S4.F10.16.m5.1.1.cmml" type="integer" xref="S4.F10.16.m5.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.16.m5.1d">3</annotation><annotation encoding="application/x-llamapun" id="S4.F10.16.m5.1e">3</annotation></semantics></math>-piece function that equals <math alttext="f" class="ltx_Math" display="inline" id="S4.F10.17.m6.1"><semantics id="S4.F10.17.m6.1b"><mi id="S4.F10.17.m6.1.1" xref="S4.F10.17.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.F10.17.m6.1c"><ci id="S4.F10.17.m6.1.1.cmml" xref="S4.F10.17.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.17.m6.1d">f</annotation><annotation encoding="application/x-llamapun" id="S4.F10.17.m6.1e">italic_f</annotation></semantics></math> on <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.F10.18.m7.1"><semantics id="S4.F10.18.m7.1b"><msub id="S4.F10.18.m7.1.1" xref="S4.F10.18.m7.1.1.cmml"><mi id="S4.F10.18.m7.1.1.2" xref="S4.F10.18.m7.1.1.2.cmml">P</mi><mn id="S4.F10.18.m7.1.1.3" xref="S4.F10.18.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.18.m7.1c"><apply id="S4.F10.18.m7.1.1.cmml" xref="S4.F10.18.m7.1.1"><csymbol cd="ambiguous" id="S4.F10.18.m7.1.1.1.cmml" xref="S4.F10.18.m7.1.1">subscript</csymbol><ci id="S4.F10.18.m7.1.1.2.cmml" xref="S4.F10.18.m7.1.1.2">𝑃</ci><cn id="S4.F10.18.m7.1.1.3.cmml" type="integer" xref="S4.F10.18.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.18.m7.1d">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.18.m7.1e">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.F10.19.m8.1"><semantics id="S4.F10.19.m8.1b"><msub id="S4.F10.19.m8.1.1" xref="S4.F10.19.m8.1.1.cmml"><mi id="S4.F10.19.m8.1.1.2" xref="S4.F10.19.m8.1.1.2.cmml">P</mi><mn id="S4.F10.19.m8.1.1.3" xref="S4.F10.19.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.19.m8.1c"><apply id="S4.F10.19.m8.1.1.cmml" xref="S4.F10.19.m8.1.1"><csymbol cd="ambiguous" id="S4.F10.19.m8.1.1.1.cmml" xref="S4.F10.19.m8.1.1">subscript</csymbol><ci id="S4.F10.19.m8.1.1.2.cmml" xref="S4.F10.19.m8.1.1.2">𝑃</ci><cn id="S4.F10.19.m8.1.1.3.cmml" type="integer" xref="S4.F10.19.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.19.m8.1d">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.19.m8.1e">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> by interpolating <math alttext="f_{1}" class="ltx_Math" display="inline" id="S4.F10.20.m9.1"><semantics id="S4.F10.20.m9.1b"><msub id="S4.F10.20.m9.1.1" xref="S4.F10.20.m9.1.1.cmml"><mi id="S4.F10.20.m9.1.1.2" xref="S4.F10.20.m9.1.1.2.cmml">f</mi><mn id="S4.F10.20.m9.1.1.3" xref="S4.F10.20.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.20.m9.1c"><apply id="S4.F10.20.m9.1.1.cmml" xref="S4.F10.20.m9.1.1"><csymbol cd="ambiguous" id="S4.F10.20.m9.1.1.1.cmml" xref="S4.F10.20.m9.1.1">subscript</csymbol><ci id="S4.F10.20.m9.1.1.2.cmml" xref="S4.F10.20.m9.1.1.2">𝑓</ci><cn id="S4.F10.20.m9.1.1.3.cmml" type="integer" xref="S4.F10.20.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.20.m9.1d">f_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.20.m9.1e">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{2}" class="ltx_Math" display="inline" id="S4.F10.21.m10.1"><semantics id="S4.F10.21.m10.1b"><msub id="S4.F10.21.m10.1.1" xref="S4.F10.21.m10.1.1.cmml"><mi id="S4.F10.21.m10.1.1.2" xref="S4.F10.21.m10.1.1.2.cmml">f</mi><mn id="S4.F10.21.m10.1.1.3" xref="S4.F10.21.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F10.21.m10.1c"><apply id="S4.F10.21.m10.1.1.cmml" xref="S4.F10.21.m10.1.1"><csymbol cd="ambiguous" id="S4.F10.21.m10.1.1.1.cmml" xref="S4.F10.21.m10.1.1">subscript</csymbol><ci id="S4.F10.21.m10.1.1.2.cmml" xref="S4.F10.21.m10.1.1.2">𝑓</ci><cn id="S4.F10.21.m10.1.1.3.cmml" type="integer" xref="S4.F10.21.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.21.m10.1d">f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.21.m10.1e">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> on their outer rays. </figcaption> </figure> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.5.5">Let <math alttext="f\in\operatorname{CPL}_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml">CPL</mi><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1"><in id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.2">CPL</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">f\in\operatorname{CPL}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">italic_f ∈ roman_CPL start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> be a piecewise linear function with <math alttext="p\geq 3" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">p</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><apply id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1"><geq id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1"></geq><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2">𝑝</ci><cn id="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">p\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_p ≥ 3</annotation></semantics></math> pieces, such that there are two adjacent pieces that enclose an angle less than <math alttext="\pi" 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">π</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">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.1d">italic_π</annotation></semantics></math>. Then, there exists a function <math alttext="f^{\prime}\in\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.4.4.m4.1"><semantics id="S4.Thmtheorem3.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem3.p1.4.4.m4.1.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.cmml"><msup id="S4.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.cmml"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.2.cmml">f</mi><mo id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem3.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.2" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.2.cmml">CPL</mi><mn id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.3" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.3.cmml">3</mn></msub></mrow><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"><in id="S4.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.1"></in><apply id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.2">𝑓</ci><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.2.3.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.2">CPL</ci><cn id="S4.Thmtheorem3.p1.4.4.m4.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.4.4.m4.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.4.4.m4.1c">f^{\prime}\in\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.4.4.m4.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f-f^{\prime}\in\operatorname{CPL}_{p-1}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.5.5.m5.1"><semantics id="S4.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml"><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.2.cmml">f</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.1.cmml">−</mo><msup id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.2.cmml">f</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S4.Thmtheorem3.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.2.cmml">CPL</mi><mrow id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml"><mi id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.2" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.2.cmml">p</mi><mo id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.1" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.1.cmml">−</mo><mn id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.3" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.5.5.m5.1b"><apply id="S4.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1"><in id="S4.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.1"></in><apply id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2"><minus id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.1"></minus><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.2">𝑓</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3">superscript</csymbol><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.2">𝑓</ci><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.2.3.3">′</ci></apply></apply><apply id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.2">CPL</ci><apply id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3"><minus id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.1"></minus><ci id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.2.cmml" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.2">𝑝</ci><cn id="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.5.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.5.5.m5.1c">f-f^{\prime}\in\operatorname{CPL}_{p-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.5.5.m5.1d">italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT italic_p - 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.6.p1"> <p class="ltx_p" id="S4.6.p1.26">Let <math alttext="\{P_{i}\}_{i\in[p]}" class="ltx_Math" display="inline" id="S4.6.p1.1.m1.2"><semantics id="S4.6.p1.1.m1.2a"><msub id="S4.6.p1.1.m1.2.2" xref="S4.6.p1.1.m1.2.2.cmml"><mrow id="S4.6.p1.1.m1.2.2.1.1" xref="S4.6.p1.1.m1.2.2.1.2.cmml"><mo id="S4.6.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S4.6.p1.1.m1.2.2.1.2.cmml">{</mo><msub id="S4.6.p1.1.m1.2.2.1.1.1" xref="S4.6.p1.1.m1.2.2.1.1.1.cmml"><mi id="S4.6.p1.1.m1.2.2.1.1.1.2" xref="S4.6.p1.1.m1.2.2.1.1.1.2.cmml">P</mi><mi id="S4.6.p1.1.m1.2.2.1.1.1.3" xref="S4.6.p1.1.m1.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S4.6.p1.1.m1.2.2.1.1.3" stretchy="false" xref="S4.6.p1.1.m1.2.2.1.2.cmml">}</mo></mrow><mrow id="S4.6.p1.1.m1.1.1.1" xref="S4.6.p1.1.m1.1.1.1.cmml"><mi id="S4.6.p1.1.m1.1.1.1.3" xref="S4.6.p1.1.m1.1.1.1.3.cmml">i</mi><mo id="S4.6.p1.1.m1.1.1.1.2" xref="S4.6.p1.1.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.6.p1.1.m1.1.1.1.4.2" xref="S4.6.p1.1.m1.1.1.1.4.1.cmml"><mo id="S4.6.p1.1.m1.1.1.1.4.2.1" stretchy="false" xref="S4.6.p1.1.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S4.6.p1.1.m1.1.1.1.1" xref="S4.6.p1.1.m1.1.1.1.1.cmml">p</mi><mo id="S4.6.p1.1.m1.1.1.1.4.2.2" stretchy="false" xref="S4.6.p1.1.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.1.m1.2b"><apply id="S4.6.p1.1.m1.2.2.cmml" xref="S4.6.p1.1.m1.2.2"><csymbol cd="ambiguous" id="S4.6.p1.1.m1.2.2.2.cmml" xref="S4.6.p1.1.m1.2.2">subscript</csymbol><set id="S4.6.p1.1.m1.2.2.1.2.cmml" xref="S4.6.p1.1.m1.2.2.1.1"><apply id="S4.6.p1.1.m1.2.2.1.1.1.cmml" xref="S4.6.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.1.m1.2.2.1.1.1.1.cmml" xref="S4.6.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S4.6.p1.1.m1.2.2.1.1.1.2.cmml" xref="S4.6.p1.1.m1.2.2.1.1.1.2">𝑃</ci><ci id="S4.6.p1.1.m1.2.2.1.1.1.3.cmml" xref="S4.6.p1.1.m1.2.2.1.1.1.3">𝑖</ci></apply></set><apply id="S4.6.p1.1.m1.1.1.1.cmml" xref="S4.6.p1.1.m1.1.1.1"><in id="S4.6.p1.1.m1.1.1.1.2.cmml" xref="S4.6.p1.1.m1.1.1.1.2"></in><ci id="S4.6.p1.1.m1.1.1.1.3.cmml" xref="S4.6.p1.1.m1.1.1.1.3">𝑖</ci><apply id="S4.6.p1.1.m1.1.1.1.4.1.cmml" xref="S4.6.p1.1.m1.1.1.1.4.2"><csymbol cd="latexml" id="S4.6.p1.1.m1.1.1.1.4.1.1.cmml" xref="S4.6.p1.1.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.6.p1.1.m1.1.1.1.1.cmml" xref="S4.6.p1.1.m1.1.1.1.1">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.1.m1.2c">\{P_{i}\}_{i\in[p]}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.1.m1.2d">{ italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_p ] end_POSTSUBSCRIPT</annotation></semantics></math> be the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S4.6.p1.2.m2.1"><semantics id="S4.6.p1.2.m2.1a"><mi id="S4.6.p1.2.m2.1.1" xref="S4.6.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.2.m2.1b"><ci id="S4.6.p1.2.m2.1.1.cmml" xref="S4.6.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.2.m2.1d">italic_f</annotation></semantics></math>, such that <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.6.p1.3.m3.1"><semantics id="S4.6.p1.3.m3.1a"><msub id="S4.6.p1.3.m3.1.1" xref="S4.6.p1.3.m3.1.1.cmml"><mi id="S4.6.p1.3.m3.1.1.2" xref="S4.6.p1.3.m3.1.1.2.cmml">P</mi><mn id="S4.6.p1.3.m3.1.1.3" xref="S4.6.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.3.m3.1b"><apply id="S4.6.p1.3.m3.1.1.cmml" xref="S4.6.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.6.p1.3.m3.1.1.1.cmml" xref="S4.6.p1.3.m3.1.1">subscript</csymbol><ci id="S4.6.p1.3.m3.1.1.2.cmml" xref="S4.6.p1.3.m3.1.1.2">𝑃</ci><cn id="S4.6.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.6.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.3.m3.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.3.m3.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.6.p1.4.m4.1"><semantics id="S4.6.p1.4.m4.1a"><msub id="S4.6.p1.4.m4.1.1" xref="S4.6.p1.4.m4.1.1.cmml"><mi id="S4.6.p1.4.m4.1.1.2" xref="S4.6.p1.4.m4.1.1.2.cmml">P</mi><mn id="S4.6.p1.4.m4.1.1.3" xref="S4.6.p1.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.4.m4.1b"><apply id="S4.6.p1.4.m4.1.1.cmml" xref="S4.6.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.6.p1.4.m4.1.1.1.cmml" xref="S4.6.p1.4.m4.1.1">subscript</csymbol><ci id="S4.6.p1.4.m4.1.1.2.cmml" xref="S4.6.p1.4.m4.1.1.2">𝑃</ci><cn id="S4.6.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.6.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.4.m4.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.4.m4.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are adjacent pieces that enclose an angle less than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.6.p1.5.m5.1"><semantics id="S4.6.p1.5.m5.1a"><mi id="S4.6.p1.5.m5.1.1" xref="S4.6.p1.5.m5.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.5.m5.1b"><ci id="S4.6.p1.5.m5.1.1.cmml" xref="S4.6.p1.5.m5.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.5.m5.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.5.m5.1d">italic_π</annotation></semantics></math>, and let <math alttext="\{f_{i}\}_{i\in[p]}" class="ltx_Math" display="inline" id="S4.6.p1.6.m6.2"><semantics id="S4.6.p1.6.m6.2a"><msub id="S4.6.p1.6.m6.2.2" xref="S4.6.p1.6.m6.2.2.cmml"><mrow id="S4.6.p1.6.m6.2.2.1.1" xref="S4.6.p1.6.m6.2.2.1.2.cmml"><mo id="S4.6.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S4.6.p1.6.m6.2.2.1.2.cmml">{</mo><msub id="S4.6.p1.6.m6.2.2.1.1.1" xref="S4.6.p1.6.m6.2.2.1.1.1.cmml"><mi id="S4.6.p1.6.m6.2.2.1.1.1.2" xref="S4.6.p1.6.m6.2.2.1.1.1.2.cmml">f</mi><mi id="S4.6.p1.6.m6.2.2.1.1.1.3" xref="S4.6.p1.6.m6.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S4.6.p1.6.m6.2.2.1.1.3" stretchy="false" xref="S4.6.p1.6.m6.2.2.1.2.cmml">}</mo></mrow><mrow id="S4.6.p1.6.m6.1.1.1" xref="S4.6.p1.6.m6.1.1.1.cmml"><mi id="S4.6.p1.6.m6.1.1.1.3" xref="S4.6.p1.6.m6.1.1.1.3.cmml">i</mi><mo id="S4.6.p1.6.m6.1.1.1.2" xref="S4.6.p1.6.m6.1.1.1.2.cmml">∈</mo><mrow id="S4.6.p1.6.m6.1.1.1.4.2" xref="S4.6.p1.6.m6.1.1.1.4.1.cmml"><mo id="S4.6.p1.6.m6.1.1.1.4.2.1" stretchy="false" xref="S4.6.p1.6.m6.1.1.1.4.1.1.cmml">[</mo><mi id="S4.6.p1.6.m6.1.1.1.1" xref="S4.6.p1.6.m6.1.1.1.1.cmml">p</mi><mo id="S4.6.p1.6.m6.1.1.1.4.2.2" stretchy="false" xref="S4.6.p1.6.m6.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.6.m6.2b"><apply id="S4.6.p1.6.m6.2.2.cmml" xref="S4.6.p1.6.m6.2.2"><csymbol cd="ambiguous" id="S4.6.p1.6.m6.2.2.2.cmml" xref="S4.6.p1.6.m6.2.2">subscript</csymbol><set id="S4.6.p1.6.m6.2.2.1.2.cmml" xref="S4.6.p1.6.m6.2.2.1.1"><apply id="S4.6.p1.6.m6.2.2.1.1.1.cmml" xref="S4.6.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.6.m6.2.2.1.1.1.1.cmml" xref="S4.6.p1.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S4.6.p1.6.m6.2.2.1.1.1.2.cmml" xref="S4.6.p1.6.m6.2.2.1.1.1.2">𝑓</ci><ci id="S4.6.p1.6.m6.2.2.1.1.1.3.cmml" xref="S4.6.p1.6.m6.2.2.1.1.1.3">𝑖</ci></apply></set><apply id="S4.6.p1.6.m6.1.1.1.cmml" xref="S4.6.p1.6.m6.1.1.1"><in id="S4.6.p1.6.m6.1.1.1.2.cmml" xref="S4.6.p1.6.m6.1.1.1.2"></in><ci id="S4.6.p1.6.m6.1.1.1.3.cmml" xref="S4.6.p1.6.m6.1.1.1.3">𝑖</ci><apply id="S4.6.p1.6.m6.1.1.1.4.1.cmml" xref="S4.6.p1.6.m6.1.1.1.4.2"><csymbol cd="latexml" id="S4.6.p1.6.m6.1.1.1.4.1.1.cmml" xref="S4.6.p1.6.m6.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.6.p1.6.m6.1.1.1.1.cmml" xref="S4.6.p1.6.m6.1.1.1.1">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.6.m6.2c">\{f_{i}\}_{i\in[p]}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.6.m6.2d">{ italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_p ] end_POSTSUBSCRIPT</annotation></semantics></math> be the corresponding linear components. The boundaries of the pieces are each the union of two rays with vertex <math alttext="0" class="ltx_Math" display="inline" id="S4.6.p1.7.m7.1"><semantics id="S4.6.p1.7.m7.1a"><mn id="S4.6.p1.7.m7.1.1" xref="S4.6.p1.7.m7.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.6.p1.7.m7.1b"><cn id="S4.6.p1.7.m7.1.1.cmml" type="integer" xref="S4.6.p1.7.m7.1.1">0</cn></annotation-xml></semantics></math>. As <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.6.p1.8.m8.1"><semantics id="S4.6.p1.8.m8.1a"><msub id="S4.6.p1.8.m8.1.1" xref="S4.6.p1.8.m8.1.1.cmml"><mi id="S4.6.p1.8.m8.1.1.2" xref="S4.6.p1.8.m8.1.1.2.cmml">P</mi><mn id="S4.6.p1.8.m8.1.1.3" xref="S4.6.p1.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.8.m8.1b"><apply id="S4.6.p1.8.m8.1.1.cmml" xref="S4.6.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.6.p1.8.m8.1.1.1.cmml" xref="S4.6.p1.8.m8.1.1">subscript</csymbol><ci id="S4.6.p1.8.m8.1.1.2.cmml" xref="S4.6.p1.8.m8.1.1.2">𝑃</ci><cn id="S4.6.p1.8.m8.1.1.3.cmml" type="integer" xref="S4.6.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.8.m8.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.8.m8.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.6.p1.9.m9.1"><semantics id="S4.6.p1.9.m9.1a"><msub id="S4.6.p1.9.m9.1.1" xref="S4.6.p1.9.m9.1.1.cmml"><mi id="S4.6.p1.9.m9.1.1.2" xref="S4.6.p1.9.m9.1.1.2.cmml">P</mi><mn id="S4.6.p1.9.m9.1.1.3" xref="S4.6.p1.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.9.m9.1b"><apply id="S4.6.p1.9.m9.1.1.cmml" xref="S4.6.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.6.p1.9.m9.1.1.1.cmml" xref="S4.6.p1.9.m9.1.1">subscript</csymbol><ci id="S4.6.p1.9.m9.1.1.2.cmml" xref="S4.6.p1.9.m9.1.1.2">𝑃</ci><cn id="S4.6.p1.9.m9.1.1.3.cmml" type="integer" xref="S4.6.p1.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.9.m9.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.9.m9.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are adjacent, they share one of these rays. Let <math alttext="p_{1},p_{2}\in\mathds{R}^{2}\setminus\{0\}" class="ltx_Math" display="inline" id="S4.6.p1.10.m10.3"><semantics id="S4.6.p1.10.m10.3a"><mrow id="S4.6.p1.10.m10.3.3" xref="S4.6.p1.10.m10.3.3.cmml"><mrow id="S4.6.p1.10.m10.3.3.2.2" xref="S4.6.p1.10.m10.3.3.2.3.cmml"><msub id="S4.6.p1.10.m10.2.2.1.1.1" xref="S4.6.p1.10.m10.2.2.1.1.1.cmml"><mi id="S4.6.p1.10.m10.2.2.1.1.1.2" xref="S4.6.p1.10.m10.2.2.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.10.m10.2.2.1.1.1.3" xref="S4.6.p1.10.m10.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.6.p1.10.m10.3.3.2.2.3" xref="S4.6.p1.10.m10.3.3.2.3.cmml">,</mo><msub id="S4.6.p1.10.m10.3.3.2.2.2" xref="S4.6.p1.10.m10.3.3.2.2.2.cmml"><mi id="S4.6.p1.10.m10.3.3.2.2.2.2" xref="S4.6.p1.10.m10.3.3.2.2.2.2.cmml">p</mi><mn id="S4.6.p1.10.m10.3.3.2.2.2.3" xref="S4.6.p1.10.m10.3.3.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.6.p1.10.m10.3.3.3" xref="S4.6.p1.10.m10.3.3.3.cmml">∈</mo><mrow id="S4.6.p1.10.m10.3.3.4" xref="S4.6.p1.10.m10.3.3.4.cmml"><msup id="S4.6.p1.10.m10.3.3.4.2" xref="S4.6.p1.10.m10.3.3.4.2.cmml"><mi id="S4.6.p1.10.m10.3.3.4.2.2" xref="S4.6.p1.10.m10.3.3.4.2.2.cmml">ℝ</mi><mn id="S4.6.p1.10.m10.3.3.4.2.3" xref="S4.6.p1.10.m10.3.3.4.2.3.cmml">2</mn></msup><mo id="S4.6.p1.10.m10.3.3.4.1" xref="S4.6.p1.10.m10.3.3.4.1.cmml">∖</mo><mrow id="S4.6.p1.10.m10.3.3.4.3.2" xref="S4.6.p1.10.m10.3.3.4.3.1.cmml"><mo id="S4.6.p1.10.m10.3.3.4.3.2.1" stretchy="false" xref="S4.6.p1.10.m10.3.3.4.3.1.cmml">{</mo><mn id="S4.6.p1.10.m10.1.1" xref="S4.6.p1.10.m10.1.1.cmml">0</mn><mo id="S4.6.p1.10.m10.3.3.4.3.2.2" stretchy="false" xref="S4.6.p1.10.m10.3.3.4.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.10.m10.3b"><apply id="S4.6.p1.10.m10.3.3.cmml" xref="S4.6.p1.10.m10.3.3"><in id="S4.6.p1.10.m10.3.3.3.cmml" xref="S4.6.p1.10.m10.3.3.3"></in><list id="S4.6.p1.10.m10.3.3.2.3.cmml" xref="S4.6.p1.10.m10.3.3.2.2"><apply id="S4.6.p1.10.m10.2.2.1.1.1.cmml" xref="S4.6.p1.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.10.m10.2.2.1.1.1.1.cmml" xref="S4.6.p1.10.m10.2.2.1.1.1">subscript</csymbol><ci id="S4.6.p1.10.m10.2.2.1.1.1.2.cmml" xref="S4.6.p1.10.m10.2.2.1.1.1.2">𝑝</ci><cn id="S4.6.p1.10.m10.2.2.1.1.1.3.cmml" type="integer" xref="S4.6.p1.10.m10.2.2.1.1.1.3">1</cn></apply><apply id="S4.6.p1.10.m10.3.3.2.2.2.cmml" xref="S4.6.p1.10.m10.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.6.p1.10.m10.3.3.2.2.2.1.cmml" xref="S4.6.p1.10.m10.3.3.2.2.2">subscript</csymbol><ci id="S4.6.p1.10.m10.3.3.2.2.2.2.cmml" xref="S4.6.p1.10.m10.3.3.2.2.2.2">𝑝</ci><cn id="S4.6.p1.10.m10.3.3.2.2.2.3.cmml" type="integer" xref="S4.6.p1.10.m10.3.3.2.2.2.3">2</cn></apply></list><apply id="S4.6.p1.10.m10.3.3.4.cmml" xref="S4.6.p1.10.m10.3.3.4"><setdiff id="S4.6.p1.10.m10.3.3.4.1.cmml" xref="S4.6.p1.10.m10.3.3.4.1"></setdiff><apply id="S4.6.p1.10.m10.3.3.4.2.cmml" xref="S4.6.p1.10.m10.3.3.4.2"><csymbol cd="ambiguous" id="S4.6.p1.10.m10.3.3.4.2.1.cmml" xref="S4.6.p1.10.m10.3.3.4.2">superscript</csymbol><ci id="S4.6.p1.10.m10.3.3.4.2.2.cmml" xref="S4.6.p1.10.m10.3.3.4.2.2">ℝ</ci><cn id="S4.6.p1.10.m10.3.3.4.2.3.cmml" type="integer" xref="S4.6.p1.10.m10.3.3.4.2.3">2</cn></apply><set id="S4.6.p1.10.m10.3.3.4.3.1.cmml" xref="S4.6.p1.10.m10.3.3.4.3.2"><cn id="S4.6.p1.10.m10.1.1.cmml" type="integer" xref="S4.6.p1.10.m10.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.10.m10.3c">p_{1},p_{2}\in\mathds{R}^{2}\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.10.m10.3d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ { 0 }</annotation></semantics></math> be points on the non-shared ray of <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.6.p1.11.m11.1"><semantics id="S4.6.p1.11.m11.1a"><msub id="S4.6.p1.11.m11.1.1" xref="S4.6.p1.11.m11.1.1.cmml"><mi id="S4.6.p1.11.m11.1.1.2" xref="S4.6.p1.11.m11.1.1.2.cmml">P</mi><mn id="S4.6.p1.11.m11.1.1.3" xref="S4.6.p1.11.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.11.m11.1b"><apply id="S4.6.p1.11.m11.1.1.cmml" xref="S4.6.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.6.p1.11.m11.1.1.1.cmml" xref="S4.6.p1.11.m11.1.1">subscript</csymbol><ci id="S4.6.p1.11.m11.1.1.2.cmml" xref="S4.6.p1.11.m11.1.1.2">𝑃</ci><cn id="S4.6.p1.11.m11.1.1.3.cmml" type="integer" xref="S4.6.p1.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.11.m11.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.11.m11.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.6.p1.12.m12.1"><semantics id="S4.6.p1.12.m12.1a"><msub id="S4.6.p1.12.m12.1.1" xref="S4.6.p1.12.m12.1.1.cmml"><mi id="S4.6.p1.12.m12.1.1.2" xref="S4.6.p1.12.m12.1.1.2.cmml">P</mi><mn id="S4.6.p1.12.m12.1.1.3" xref="S4.6.p1.12.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.12.m12.1b"><apply id="S4.6.p1.12.m12.1.1.cmml" xref="S4.6.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.6.p1.12.m12.1.1.1.cmml" xref="S4.6.p1.12.m12.1.1">subscript</csymbol><ci id="S4.6.p1.12.m12.1.1.2.cmml" xref="S4.6.p1.12.m12.1.1.2">𝑃</ci><cn id="S4.6.p1.12.m12.1.1.3.cmml" type="integer" xref="S4.6.p1.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.12.m12.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.12.m12.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively, as shown in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.F10" title="Figure 10 ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 10</span></a>. As the angle at <math alttext="0" class="ltx_Math" display="inline" id="S4.6.p1.13.m13.1"><semantics id="S4.6.p1.13.m13.1a"><mn id="S4.6.p1.13.m13.1.1" xref="S4.6.p1.13.m13.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.6.p1.13.m13.1b"><cn id="S4.6.p1.13.m13.1.1.cmml" type="integer" xref="S4.6.p1.13.m13.1.1">0</cn></annotation-xml></semantics></math> enclosed by <math alttext="P_{1}\cup P_{2}" class="ltx_Math" display="inline" id="S4.6.p1.14.m14.1"><semantics id="S4.6.p1.14.m14.1a"><mrow id="S4.6.p1.14.m14.1.1" xref="S4.6.p1.14.m14.1.1.cmml"><msub id="S4.6.p1.14.m14.1.1.2" xref="S4.6.p1.14.m14.1.1.2.cmml"><mi id="S4.6.p1.14.m14.1.1.2.2" xref="S4.6.p1.14.m14.1.1.2.2.cmml">P</mi><mn id="S4.6.p1.14.m14.1.1.2.3" xref="S4.6.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S4.6.p1.14.m14.1.1.1" xref="S4.6.p1.14.m14.1.1.1.cmml">∪</mo><msub id="S4.6.p1.14.m14.1.1.3" xref="S4.6.p1.14.m14.1.1.3.cmml"><mi id="S4.6.p1.14.m14.1.1.3.2" xref="S4.6.p1.14.m14.1.1.3.2.cmml">P</mi><mn id="S4.6.p1.14.m14.1.1.3.3" xref="S4.6.p1.14.m14.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.14.m14.1b"><apply id="S4.6.p1.14.m14.1.1.cmml" xref="S4.6.p1.14.m14.1.1"><union id="S4.6.p1.14.m14.1.1.1.cmml" xref="S4.6.p1.14.m14.1.1.1"></union><apply id="S4.6.p1.14.m14.1.1.2.cmml" xref="S4.6.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.6.p1.14.m14.1.1.2.1.cmml" xref="S4.6.p1.14.m14.1.1.2">subscript</csymbol><ci id="S4.6.p1.14.m14.1.1.2.2.cmml" xref="S4.6.p1.14.m14.1.1.2.2">𝑃</ci><cn id="S4.6.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S4.6.p1.14.m14.1.1.2.3">1</cn></apply><apply id="S4.6.p1.14.m14.1.1.3.cmml" xref="S4.6.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.14.m14.1.1.3.1.cmml" xref="S4.6.p1.14.m14.1.1.3">subscript</csymbol><ci id="S4.6.p1.14.m14.1.1.3.2.cmml" xref="S4.6.p1.14.m14.1.1.3.2">𝑃</ci><cn id="S4.6.p1.14.m14.1.1.3.3.cmml" type="integer" xref="S4.6.p1.14.m14.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.14.m14.1c">P_{1}\cup P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.14.m14.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is smaller than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.6.p1.15.m15.1"><semantics id="S4.6.p1.15.m15.1a"><mi id="S4.6.p1.15.m15.1.1" xref="S4.6.p1.15.m15.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.15.m15.1b"><ci id="S4.6.p1.15.m15.1.1.cmml" xref="S4.6.p1.15.m15.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.15.m15.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.15.m15.1d">italic_π</annotation></semantics></math>, <math alttext="p_{1}" class="ltx_Math" display="inline" id="S4.6.p1.16.m16.1"><semantics id="S4.6.p1.16.m16.1a"><msub id="S4.6.p1.16.m16.1.1" xref="S4.6.p1.16.m16.1.1.cmml"><mi id="S4.6.p1.16.m16.1.1.2" xref="S4.6.p1.16.m16.1.1.2.cmml">p</mi><mn id="S4.6.p1.16.m16.1.1.3" xref="S4.6.p1.16.m16.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.16.m16.1b"><apply id="S4.6.p1.16.m16.1.1.cmml" xref="S4.6.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S4.6.p1.16.m16.1.1.1.cmml" xref="S4.6.p1.16.m16.1.1">subscript</csymbol><ci id="S4.6.p1.16.m16.1.1.2.cmml" xref="S4.6.p1.16.m16.1.1.2">𝑝</ci><cn id="S4.6.p1.16.m16.1.1.3.cmml" type="integer" xref="S4.6.p1.16.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.16.m16.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.16.m16.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{2}" class="ltx_Math" display="inline" id="S4.6.p1.17.m17.1"><semantics id="S4.6.p1.17.m17.1a"><msub id="S4.6.p1.17.m17.1.1" xref="S4.6.p1.17.m17.1.1.cmml"><mi id="S4.6.p1.17.m17.1.1.2" xref="S4.6.p1.17.m17.1.1.2.cmml">p</mi><mn id="S4.6.p1.17.m17.1.1.3" xref="S4.6.p1.17.m17.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.17.m17.1b"><apply id="S4.6.p1.17.m17.1.1.cmml" xref="S4.6.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S4.6.p1.17.m17.1.1.1.cmml" xref="S4.6.p1.17.m17.1.1">subscript</csymbol><ci id="S4.6.p1.17.m17.1.1.2.cmml" xref="S4.6.p1.17.m17.1.1.2">𝑝</ci><cn id="S4.6.p1.17.m17.1.1.3.cmml" type="integer" xref="S4.6.p1.17.m17.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.17.m17.1c">p_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.17.m17.1d">italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are linearly independent. Therefore, there exists a linear function <math alttext="f_{P}" class="ltx_Math" display="inline" id="S4.6.p1.18.m18.1"><semantics id="S4.6.p1.18.m18.1a"><msub id="S4.6.p1.18.m18.1.1" xref="S4.6.p1.18.m18.1.1.cmml"><mi id="S4.6.p1.18.m18.1.1.2" xref="S4.6.p1.18.m18.1.1.2.cmml">f</mi><mi id="S4.6.p1.18.m18.1.1.3" xref="S4.6.p1.18.m18.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.18.m18.1b"><apply id="S4.6.p1.18.m18.1.1.cmml" xref="S4.6.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.6.p1.18.m18.1.1.1.cmml" xref="S4.6.p1.18.m18.1.1">subscript</csymbol><ci id="S4.6.p1.18.m18.1.1.2.cmml" xref="S4.6.p1.18.m18.1.1.2">𝑓</ci><ci id="S4.6.p1.18.m18.1.1.3.cmml" xref="S4.6.p1.18.m18.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.18.m18.1c">f_{P}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.18.m18.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f_{P}(p_{1})=f_{1}(p_{1})" class="ltx_Math" display="inline" id="S4.6.p1.19.m19.2"><semantics id="S4.6.p1.19.m19.2a"><mrow id="S4.6.p1.19.m19.2.2" xref="S4.6.p1.19.m19.2.2.cmml"><mrow id="S4.6.p1.19.m19.1.1.1" xref="S4.6.p1.19.m19.1.1.1.cmml"><msub id="S4.6.p1.19.m19.1.1.1.3" xref="S4.6.p1.19.m19.1.1.1.3.cmml"><mi id="S4.6.p1.19.m19.1.1.1.3.2" xref="S4.6.p1.19.m19.1.1.1.3.2.cmml">f</mi><mi id="S4.6.p1.19.m19.1.1.1.3.3" xref="S4.6.p1.19.m19.1.1.1.3.3.cmml">P</mi></msub><mo id="S4.6.p1.19.m19.1.1.1.2" xref="S4.6.p1.19.m19.1.1.1.2.cmml">⁢</mo><mrow id="S4.6.p1.19.m19.1.1.1.1.1" xref="S4.6.p1.19.m19.1.1.1.1.1.1.cmml"><mo id="S4.6.p1.19.m19.1.1.1.1.1.2" stretchy="false" xref="S4.6.p1.19.m19.1.1.1.1.1.1.cmml">(</mo><msub id="S4.6.p1.19.m19.1.1.1.1.1.1" xref="S4.6.p1.19.m19.1.1.1.1.1.1.cmml"><mi id="S4.6.p1.19.m19.1.1.1.1.1.1.2" xref="S4.6.p1.19.m19.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.19.m19.1.1.1.1.1.1.3" xref="S4.6.p1.19.m19.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.6.p1.19.m19.1.1.1.1.1.3" stretchy="false" xref="S4.6.p1.19.m19.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.6.p1.19.m19.2.2.3" xref="S4.6.p1.19.m19.2.2.3.cmml">=</mo><mrow id="S4.6.p1.19.m19.2.2.2" xref="S4.6.p1.19.m19.2.2.2.cmml"><msub id="S4.6.p1.19.m19.2.2.2.3" xref="S4.6.p1.19.m19.2.2.2.3.cmml"><mi id="S4.6.p1.19.m19.2.2.2.3.2" xref="S4.6.p1.19.m19.2.2.2.3.2.cmml">f</mi><mn id="S4.6.p1.19.m19.2.2.2.3.3" xref="S4.6.p1.19.m19.2.2.2.3.3.cmml">1</mn></msub><mo id="S4.6.p1.19.m19.2.2.2.2" xref="S4.6.p1.19.m19.2.2.2.2.cmml">⁢</mo><mrow id="S4.6.p1.19.m19.2.2.2.1.1" xref="S4.6.p1.19.m19.2.2.2.1.1.1.cmml"><mo id="S4.6.p1.19.m19.2.2.2.1.1.2" stretchy="false" xref="S4.6.p1.19.m19.2.2.2.1.1.1.cmml">(</mo><msub id="S4.6.p1.19.m19.2.2.2.1.1.1" xref="S4.6.p1.19.m19.2.2.2.1.1.1.cmml"><mi id="S4.6.p1.19.m19.2.2.2.1.1.1.2" xref="S4.6.p1.19.m19.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.19.m19.2.2.2.1.1.1.3" xref="S4.6.p1.19.m19.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.6.p1.19.m19.2.2.2.1.1.3" stretchy="false" xref="S4.6.p1.19.m19.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.19.m19.2b"><apply id="S4.6.p1.19.m19.2.2.cmml" xref="S4.6.p1.19.m19.2.2"><eq id="S4.6.p1.19.m19.2.2.3.cmml" xref="S4.6.p1.19.m19.2.2.3"></eq><apply id="S4.6.p1.19.m19.1.1.1.cmml" xref="S4.6.p1.19.m19.1.1.1"><times id="S4.6.p1.19.m19.1.1.1.2.cmml" xref="S4.6.p1.19.m19.1.1.1.2"></times><apply id="S4.6.p1.19.m19.1.1.1.3.cmml" xref="S4.6.p1.19.m19.1.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.19.m19.1.1.1.3.1.cmml" xref="S4.6.p1.19.m19.1.1.1.3">subscript</csymbol><ci id="S4.6.p1.19.m19.1.1.1.3.2.cmml" xref="S4.6.p1.19.m19.1.1.1.3.2">𝑓</ci><ci id="S4.6.p1.19.m19.1.1.1.3.3.cmml" xref="S4.6.p1.19.m19.1.1.1.3.3">𝑃</ci></apply><apply id="S4.6.p1.19.m19.1.1.1.1.1.1.cmml" xref="S4.6.p1.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.19.m19.1.1.1.1.1.1.1.cmml" xref="S4.6.p1.19.m19.1.1.1.1.1">subscript</csymbol><ci id="S4.6.p1.19.m19.1.1.1.1.1.1.2.cmml" xref="S4.6.p1.19.m19.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.6.p1.19.m19.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.6.p1.19.m19.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.6.p1.19.m19.2.2.2.cmml" xref="S4.6.p1.19.m19.2.2.2"><times id="S4.6.p1.19.m19.2.2.2.2.cmml" xref="S4.6.p1.19.m19.2.2.2.2"></times><apply id="S4.6.p1.19.m19.2.2.2.3.cmml" xref="S4.6.p1.19.m19.2.2.2.3"><csymbol cd="ambiguous" id="S4.6.p1.19.m19.2.2.2.3.1.cmml" xref="S4.6.p1.19.m19.2.2.2.3">subscript</csymbol><ci id="S4.6.p1.19.m19.2.2.2.3.2.cmml" xref="S4.6.p1.19.m19.2.2.2.3.2">𝑓</ci><cn id="S4.6.p1.19.m19.2.2.2.3.3.cmml" type="integer" xref="S4.6.p1.19.m19.2.2.2.3.3">1</cn></apply><apply id="S4.6.p1.19.m19.2.2.2.1.1.1.cmml" xref="S4.6.p1.19.m19.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.6.p1.19.m19.2.2.2.1.1.1.1.cmml" xref="S4.6.p1.19.m19.2.2.2.1.1">subscript</csymbol><ci id="S4.6.p1.19.m19.2.2.2.1.1.1.2.cmml" xref="S4.6.p1.19.m19.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.6.p1.19.m19.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.6.p1.19.m19.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.19.m19.2c">f_{P}(p_{1})=f_{1}(p_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.19.m19.2d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="f_{P}(p_{2})=f_{2}(p_{2})" class="ltx_Math" display="inline" id="S4.6.p1.20.m20.2"><semantics id="S4.6.p1.20.m20.2a"><mrow id="S4.6.p1.20.m20.2.2" xref="S4.6.p1.20.m20.2.2.cmml"><mrow id="S4.6.p1.20.m20.1.1.1" xref="S4.6.p1.20.m20.1.1.1.cmml"><msub id="S4.6.p1.20.m20.1.1.1.3" xref="S4.6.p1.20.m20.1.1.1.3.cmml"><mi id="S4.6.p1.20.m20.1.1.1.3.2" xref="S4.6.p1.20.m20.1.1.1.3.2.cmml">f</mi><mi id="S4.6.p1.20.m20.1.1.1.3.3" xref="S4.6.p1.20.m20.1.1.1.3.3.cmml">P</mi></msub><mo id="S4.6.p1.20.m20.1.1.1.2" xref="S4.6.p1.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S4.6.p1.20.m20.1.1.1.1.1" xref="S4.6.p1.20.m20.1.1.1.1.1.1.cmml"><mo id="S4.6.p1.20.m20.1.1.1.1.1.2" stretchy="false" xref="S4.6.p1.20.m20.1.1.1.1.1.1.cmml">(</mo><msub id="S4.6.p1.20.m20.1.1.1.1.1.1" xref="S4.6.p1.20.m20.1.1.1.1.1.1.cmml"><mi id="S4.6.p1.20.m20.1.1.1.1.1.1.2" xref="S4.6.p1.20.m20.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.20.m20.1.1.1.1.1.1.3" xref="S4.6.p1.20.m20.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.6.p1.20.m20.1.1.1.1.1.3" stretchy="false" xref="S4.6.p1.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.6.p1.20.m20.2.2.3" xref="S4.6.p1.20.m20.2.2.3.cmml">=</mo><mrow id="S4.6.p1.20.m20.2.2.2" xref="S4.6.p1.20.m20.2.2.2.cmml"><msub id="S4.6.p1.20.m20.2.2.2.3" xref="S4.6.p1.20.m20.2.2.2.3.cmml"><mi id="S4.6.p1.20.m20.2.2.2.3.2" xref="S4.6.p1.20.m20.2.2.2.3.2.cmml">f</mi><mn id="S4.6.p1.20.m20.2.2.2.3.3" xref="S4.6.p1.20.m20.2.2.2.3.3.cmml">2</mn></msub><mo id="S4.6.p1.20.m20.2.2.2.2" xref="S4.6.p1.20.m20.2.2.2.2.cmml">⁢</mo><mrow id="S4.6.p1.20.m20.2.2.2.1.1" xref="S4.6.p1.20.m20.2.2.2.1.1.1.cmml"><mo id="S4.6.p1.20.m20.2.2.2.1.1.2" stretchy="false" xref="S4.6.p1.20.m20.2.2.2.1.1.1.cmml">(</mo><msub id="S4.6.p1.20.m20.2.2.2.1.1.1" xref="S4.6.p1.20.m20.2.2.2.1.1.1.cmml"><mi id="S4.6.p1.20.m20.2.2.2.1.1.1.2" xref="S4.6.p1.20.m20.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.20.m20.2.2.2.1.1.1.3" xref="S4.6.p1.20.m20.2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.6.p1.20.m20.2.2.2.1.1.3" stretchy="false" xref="S4.6.p1.20.m20.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.20.m20.2b"><apply id="S4.6.p1.20.m20.2.2.cmml" xref="S4.6.p1.20.m20.2.2"><eq id="S4.6.p1.20.m20.2.2.3.cmml" xref="S4.6.p1.20.m20.2.2.3"></eq><apply id="S4.6.p1.20.m20.1.1.1.cmml" xref="S4.6.p1.20.m20.1.1.1"><times id="S4.6.p1.20.m20.1.1.1.2.cmml" xref="S4.6.p1.20.m20.1.1.1.2"></times><apply id="S4.6.p1.20.m20.1.1.1.3.cmml" xref="S4.6.p1.20.m20.1.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.20.m20.1.1.1.3.1.cmml" xref="S4.6.p1.20.m20.1.1.1.3">subscript</csymbol><ci id="S4.6.p1.20.m20.1.1.1.3.2.cmml" xref="S4.6.p1.20.m20.1.1.1.3.2">𝑓</ci><ci id="S4.6.p1.20.m20.1.1.1.3.3.cmml" xref="S4.6.p1.20.m20.1.1.1.3.3">𝑃</ci></apply><apply id="S4.6.p1.20.m20.1.1.1.1.1.1.cmml" xref="S4.6.p1.20.m20.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.20.m20.1.1.1.1.1.1.1.cmml" xref="S4.6.p1.20.m20.1.1.1.1.1">subscript</csymbol><ci id="S4.6.p1.20.m20.1.1.1.1.1.1.2.cmml" xref="S4.6.p1.20.m20.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.6.p1.20.m20.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.6.p1.20.m20.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S4.6.p1.20.m20.2.2.2.cmml" xref="S4.6.p1.20.m20.2.2.2"><times id="S4.6.p1.20.m20.2.2.2.2.cmml" xref="S4.6.p1.20.m20.2.2.2.2"></times><apply id="S4.6.p1.20.m20.2.2.2.3.cmml" xref="S4.6.p1.20.m20.2.2.2.3"><csymbol cd="ambiguous" id="S4.6.p1.20.m20.2.2.2.3.1.cmml" xref="S4.6.p1.20.m20.2.2.2.3">subscript</csymbol><ci id="S4.6.p1.20.m20.2.2.2.3.2.cmml" xref="S4.6.p1.20.m20.2.2.2.3.2">𝑓</ci><cn id="S4.6.p1.20.m20.2.2.2.3.3.cmml" type="integer" xref="S4.6.p1.20.m20.2.2.2.3.3">2</cn></apply><apply id="S4.6.p1.20.m20.2.2.2.1.1.1.cmml" xref="S4.6.p1.20.m20.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.6.p1.20.m20.2.2.2.1.1.1.1.cmml" xref="S4.6.p1.20.m20.2.2.2.1.1">subscript</csymbol><ci id="S4.6.p1.20.m20.2.2.2.1.1.1.2.cmml" xref="S4.6.p1.20.m20.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.6.p1.20.m20.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.6.p1.20.m20.2.2.2.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.20.m20.2c">f_{P}(p_{2})=f_{2}(p_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.20.m20.2d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. By linearity of <math alttext="f_{i}" class="ltx_Math" display="inline" id="S4.6.p1.21.m21.1"><semantics id="S4.6.p1.21.m21.1a"><msub id="S4.6.p1.21.m21.1.1" xref="S4.6.p1.21.m21.1.1.cmml"><mi id="S4.6.p1.21.m21.1.1.2" xref="S4.6.p1.21.m21.1.1.2.cmml">f</mi><mi id="S4.6.p1.21.m21.1.1.3" xref="S4.6.p1.21.m21.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.21.m21.1b"><apply id="S4.6.p1.21.m21.1.1.cmml" xref="S4.6.p1.21.m21.1.1"><csymbol cd="ambiguous" id="S4.6.p1.21.m21.1.1.1.cmml" xref="S4.6.p1.21.m21.1.1">subscript</csymbol><ci id="S4.6.p1.21.m21.1.1.2.cmml" xref="S4.6.p1.21.m21.1.1.2">𝑓</ci><ci id="S4.6.p1.21.m21.1.1.3.cmml" xref="S4.6.p1.21.m21.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.21.m21.1c">f_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.21.m21.1d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we have <math alttext="f_{P}=f_{1}" class="ltx_Math" display="inline" id="S4.6.p1.22.m22.1"><semantics id="S4.6.p1.22.m22.1a"><mrow id="S4.6.p1.22.m22.1.1" xref="S4.6.p1.22.m22.1.1.cmml"><msub id="S4.6.p1.22.m22.1.1.2" xref="S4.6.p1.22.m22.1.1.2.cmml"><mi id="S4.6.p1.22.m22.1.1.2.2" xref="S4.6.p1.22.m22.1.1.2.2.cmml">f</mi><mi id="S4.6.p1.22.m22.1.1.2.3" xref="S4.6.p1.22.m22.1.1.2.3.cmml">P</mi></msub><mo id="S4.6.p1.22.m22.1.1.1" xref="S4.6.p1.22.m22.1.1.1.cmml">=</mo><msub id="S4.6.p1.22.m22.1.1.3" xref="S4.6.p1.22.m22.1.1.3.cmml"><mi id="S4.6.p1.22.m22.1.1.3.2" xref="S4.6.p1.22.m22.1.1.3.2.cmml">f</mi><mn id="S4.6.p1.22.m22.1.1.3.3" xref="S4.6.p1.22.m22.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.22.m22.1b"><apply id="S4.6.p1.22.m22.1.1.cmml" xref="S4.6.p1.22.m22.1.1"><eq id="S4.6.p1.22.m22.1.1.1.cmml" xref="S4.6.p1.22.m22.1.1.1"></eq><apply id="S4.6.p1.22.m22.1.1.2.cmml" xref="S4.6.p1.22.m22.1.1.2"><csymbol cd="ambiguous" id="S4.6.p1.22.m22.1.1.2.1.cmml" xref="S4.6.p1.22.m22.1.1.2">subscript</csymbol><ci id="S4.6.p1.22.m22.1.1.2.2.cmml" xref="S4.6.p1.22.m22.1.1.2.2">𝑓</ci><ci id="S4.6.p1.22.m22.1.1.2.3.cmml" xref="S4.6.p1.22.m22.1.1.2.3">𝑃</ci></apply><apply id="S4.6.p1.22.m22.1.1.3.cmml" xref="S4.6.p1.22.m22.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.22.m22.1.1.3.1.cmml" xref="S4.6.p1.22.m22.1.1.3">subscript</csymbol><ci id="S4.6.p1.22.m22.1.1.3.2.cmml" xref="S4.6.p1.22.m22.1.1.3.2">𝑓</ci><cn id="S4.6.p1.22.m22.1.1.3.3.cmml" type="integer" xref="S4.6.p1.22.m22.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.22.m22.1c">f_{P}=f_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.22.m22.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\operatorname*{span}(p_{1})" class="ltx_Math" display="inline" id="S4.6.p1.23.m23.2"><semantics id="S4.6.p1.23.m23.2a"><mrow id="S4.6.p1.23.m23.2.2.1" xref="S4.6.p1.23.m23.2.2.2.cmml"><mo id="S4.6.p1.23.m23.1.1" rspace="0em" xref="S4.6.p1.23.m23.1.1.cmml">span</mo><mrow id="S4.6.p1.23.m23.2.2.1.1" xref="S4.6.p1.23.m23.2.2.2.cmml"><mo id="S4.6.p1.23.m23.2.2.1.1.2" stretchy="false" xref="S4.6.p1.23.m23.2.2.2.cmml">(</mo><msub id="S4.6.p1.23.m23.2.2.1.1.1" xref="S4.6.p1.23.m23.2.2.1.1.1.cmml"><mi id="S4.6.p1.23.m23.2.2.1.1.1.2" xref="S4.6.p1.23.m23.2.2.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.23.m23.2.2.1.1.1.3" xref="S4.6.p1.23.m23.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.6.p1.23.m23.2.2.1.1.3" stretchy="false" xref="S4.6.p1.23.m23.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.23.m23.2b"><apply id="S4.6.p1.23.m23.2.2.2.cmml" xref="S4.6.p1.23.m23.2.2.1"><ci id="S4.6.p1.23.m23.1.1.cmml" xref="S4.6.p1.23.m23.1.1">span</ci><apply id="S4.6.p1.23.m23.2.2.1.1.1.cmml" xref="S4.6.p1.23.m23.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.23.m23.2.2.1.1.1.1.cmml" xref="S4.6.p1.23.m23.2.2.1.1.1">subscript</csymbol><ci id="S4.6.p1.23.m23.2.2.1.1.1.2.cmml" xref="S4.6.p1.23.m23.2.2.1.1.1.2">𝑝</ci><cn id="S4.6.p1.23.m23.2.2.1.1.1.3.cmml" type="integer" xref="S4.6.p1.23.m23.2.2.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.23.m23.2c">\operatorname*{span}(p_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.23.m23.2d">roman_span ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="f_{P}=f_{2}" class="ltx_Math" display="inline" id="S4.6.p1.24.m24.1"><semantics id="S4.6.p1.24.m24.1a"><mrow id="S4.6.p1.24.m24.1.1" xref="S4.6.p1.24.m24.1.1.cmml"><msub id="S4.6.p1.24.m24.1.1.2" xref="S4.6.p1.24.m24.1.1.2.cmml"><mi id="S4.6.p1.24.m24.1.1.2.2" xref="S4.6.p1.24.m24.1.1.2.2.cmml">f</mi><mi id="S4.6.p1.24.m24.1.1.2.3" xref="S4.6.p1.24.m24.1.1.2.3.cmml">P</mi></msub><mo id="S4.6.p1.24.m24.1.1.1" xref="S4.6.p1.24.m24.1.1.1.cmml">=</mo><msub id="S4.6.p1.24.m24.1.1.3" xref="S4.6.p1.24.m24.1.1.3.cmml"><mi id="S4.6.p1.24.m24.1.1.3.2" xref="S4.6.p1.24.m24.1.1.3.2.cmml">f</mi><mn id="S4.6.p1.24.m24.1.1.3.3" xref="S4.6.p1.24.m24.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.24.m24.1b"><apply id="S4.6.p1.24.m24.1.1.cmml" xref="S4.6.p1.24.m24.1.1"><eq id="S4.6.p1.24.m24.1.1.1.cmml" xref="S4.6.p1.24.m24.1.1.1"></eq><apply id="S4.6.p1.24.m24.1.1.2.cmml" xref="S4.6.p1.24.m24.1.1.2"><csymbol cd="ambiguous" id="S4.6.p1.24.m24.1.1.2.1.cmml" xref="S4.6.p1.24.m24.1.1.2">subscript</csymbol><ci id="S4.6.p1.24.m24.1.1.2.2.cmml" xref="S4.6.p1.24.m24.1.1.2.2">𝑓</ci><ci id="S4.6.p1.24.m24.1.1.2.3.cmml" xref="S4.6.p1.24.m24.1.1.2.3">𝑃</ci></apply><apply id="S4.6.p1.24.m24.1.1.3.cmml" xref="S4.6.p1.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.24.m24.1.1.3.1.cmml" xref="S4.6.p1.24.m24.1.1.3">subscript</csymbol><ci id="S4.6.p1.24.m24.1.1.3.2.cmml" xref="S4.6.p1.24.m24.1.1.3.2">𝑓</ci><cn id="S4.6.p1.24.m24.1.1.3.3.cmml" type="integer" xref="S4.6.p1.24.m24.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.24.m24.1c">f_{P}=f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.24.m24.1d">italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\operatorname*{span}(p_{2})" class="ltx_Math" display="inline" id="S4.6.p1.25.m25.2"><semantics id="S4.6.p1.25.m25.2a"><mrow id="S4.6.p1.25.m25.2.2.1" xref="S4.6.p1.25.m25.2.2.2.cmml"><mo id="S4.6.p1.25.m25.1.1" rspace="0em" xref="S4.6.p1.25.m25.1.1.cmml">span</mo><mrow id="S4.6.p1.25.m25.2.2.1.1" xref="S4.6.p1.25.m25.2.2.2.cmml"><mo id="S4.6.p1.25.m25.2.2.1.1.2" stretchy="false" xref="S4.6.p1.25.m25.2.2.2.cmml">(</mo><msub id="S4.6.p1.25.m25.2.2.1.1.1" xref="S4.6.p1.25.m25.2.2.1.1.1.cmml"><mi id="S4.6.p1.25.m25.2.2.1.1.1.2" xref="S4.6.p1.25.m25.2.2.1.1.1.2.cmml">p</mi><mn id="S4.6.p1.25.m25.2.2.1.1.1.3" xref="S4.6.p1.25.m25.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.6.p1.25.m25.2.2.1.1.3" stretchy="false" xref="S4.6.p1.25.m25.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.25.m25.2b"><apply id="S4.6.p1.25.m25.2.2.2.cmml" xref="S4.6.p1.25.m25.2.2.1"><ci id="S4.6.p1.25.m25.1.1.cmml" xref="S4.6.p1.25.m25.1.1">span</ci><apply id="S4.6.p1.25.m25.2.2.1.1.1.cmml" xref="S4.6.p1.25.m25.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.25.m25.2.2.1.1.1.1.cmml" xref="S4.6.p1.25.m25.2.2.1.1.1">subscript</csymbol><ci id="S4.6.p1.25.m25.2.2.1.1.1.2.cmml" xref="S4.6.p1.25.m25.2.2.1.1.1.2">𝑝</ci><cn id="S4.6.p1.25.m25.2.2.1.1.1.3.cmml" type="integer" xref="S4.6.p1.25.m25.2.2.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.25.m25.2c">\operatorname*{span}(p_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.25.m25.2d">roman_span ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. Therefore, with <math alttext="P:=\bigcup_{i=3}^{p}P_{i}" class="ltx_Math" display="inline" id="S4.6.p1.26.m26.1"><semantics id="S4.6.p1.26.m26.1a"><mrow id="S4.6.p1.26.m26.1.1" xref="S4.6.p1.26.m26.1.1.cmml"><mi id="S4.6.p1.26.m26.1.1.2" xref="S4.6.p1.26.m26.1.1.2.cmml">P</mi><mo id="S4.6.p1.26.m26.1.1.1" lspace="0.278em" rspace="0.111em" xref="S4.6.p1.26.m26.1.1.1.cmml">:=</mo><mrow id="S4.6.p1.26.m26.1.1.3" xref="S4.6.p1.26.m26.1.1.3.cmml"><msubsup id="S4.6.p1.26.m26.1.1.3.1" xref="S4.6.p1.26.m26.1.1.3.1.cmml"><mo id="S4.6.p1.26.m26.1.1.3.1.2.2" xref="S4.6.p1.26.m26.1.1.3.1.2.2.cmml">⋃</mo><mrow id="S4.6.p1.26.m26.1.1.3.1.2.3" xref="S4.6.p1.26.m26.1.1.3.1.2.3.cmml"><mi id="S4.6.p1.26.m26.1.1.3.1.2.3.2" xref="S4.6.p1.26.m26.1.1.3.1.2.3.2.cmml">i</mi><mo id="S4.6.p1.26.m26.1.1.3.1.2.3.1" xref="S4.6.p1.26.m26.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.6.p1.26.m26.1.1.3.1.2.3.3" xref="S4.6.p1.26.m26.1.1.3.1.2.3.3.cmml">3</mn></mrow><mi id="S4.6.p1.26.m26.1.1.3.1.3" xref="S4.6.p1.26.m26.1.1.3.1.3.cmml">p</mi></msubsup><msub id="S4.6.p1.26.m26.1.1.3.2" xref="S4.6.p1.26.m26.1.1.3.2.cmml"><mi id="S4.6.p1.26.m26.1.1.3.2.2" xref="S4.6.p1.26.m26.1.1.3.2.2.cmml">P</mi><mi id="S4.6.p1.26.m26.1.1.3.2.3" xref="S4.6.p1.26.m26.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.26.m26.1b"><apply id="S4.6.p1.26.m26.1.1.cmml" xref="S4.6.p1.26.m26.1.1"><csymbol cd="latexml" id="S4.6.p1.26.m26.1.1.1.cmml" xref="S4.6.p1.26.m26.1.1.1">assign</csymbol><ci id="S4.6.p1.26.m26.1.1.2.cmml" xref="S4.6.p1.26.m26.1.1.2">𝑃</ci><apply id="S4.6.p1.26.m26.1.1.3.cmml" xref="S4.6.p1.26.m26.1.1.3"><apply id="S4.6.p1.26.m26.1.1.3.1.cmml" xref="S4.6.p1.26.m26.1.1.3.1"><csymbol cd="ambiguous" id="S4.6.p1.26.m26.1.1.3.1.1.cmml" xref="S4.6.p1.26.m26.1.1.3.1">superscript</csymbol><apply id="S4.6.p1.26.m26.1.1.3.1.2.cmml" xref="S4.6.p1.26.m26.1.1.3.1"><csymbol cd="ambiguous" id="S4.6.p1.26.m26.1.1.3.1.2.1.cmml" xref="S4.6.p1.26.m26.1.1.3.1">subscript</csymbol><union id="S4.6.p1.26.m26.1.1.3.1.2.2.cmml" xref="S4.6.p1.26.m26.1.1.3.1.2.2"></union><apply id="S4.6.p1.26.m26.1.1.3.1.2.3.cmml" xref="S4.6.p1.26.m26.1.1.3.1.2.3"><eq id="S4.6.p1.26.m26.1.1.3.1.2.3.1.cmml" xref="S4.6.p1.26.m26.1.1.3.1.2.3.1"></eq><ci id="S4.6.p1.26.m26.1.1.3.1.2.3.2.cmml" xref="S4.6.p1.26.m26.1.1.3.1.2.3.2">𝑖</ci><cn id="S4.6.p1.26.m26.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.6.p1.26.m26.1.1.3.1.2.3.3">3</cn></apply></apply><ci id="S4.6.p1.26.m26.1.1.3.1.3.cmml" xref="S4.6.p1.26.m26.1.1.3.1.3">𝑝</ci></apply><apply id="S4.6.p1.26.m26.1.1.3.2.cmml" xref="S4.6.p1.26.m26.1.1.3.2"><csymbol cd="ambiguous" id="S4.6.p1.26.m26.1.1.3.2.1.cmml" xref="S4.6.p1.26.m26.1.1.3.2">subscript</csymbol><ci id="S4.6.p1.26.m26.1.1.3.2.2.cmml" xref="S4.6.p1.26.m26.1.1.3.2.2">𝑃</ci><ci id="S4.6.p1.26.m26.1.1.3.2.3.cmml" xref="S4.6.p1.26.m26.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.26.m26.1c">P:=\bigcup_{i=3}^{p}P_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.26.m26.1d">italic_P := ⋃ start_POSTSUBSCRIPT italic_i = 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the function</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex47"> <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^{\prime}(x):=\begin{cases}f_{1},\quad&\text{in }P_{1}\\ f_{2},\quad&\text{in }P_{2}\\ f_{P},\quad&\text{in }P\\ \end{cases}" class="ltx_Math" display="block" id="S4.Ex47.m1.7"><semantics id="S4.Ex47.m1.7a"><mrow id="S4.Ex47.m1.7.8" xref="S4.Ex47.m1.7.8.cmml"><mrow id="S4.Ex47.m1.7.8.2" xref="S4.Ex47.m1.7.8.2.cmml"><msup id="S4.Ex47.m1.7.8.2.2" xref="S4.Ex47.m1.7.8.2.2.cmml"><mi id="S4.Ex47.m1.7.8.2.2.2" xref="S4.Ex47.m1.7.8.2.2.2.cmml">f</mi><mo id="S4.Ex47.m1.7.8.2.2.3" xref="S4.Ex47.m1.7.8.2.2.3.cmml">′</mo></msup><mo id="S4.Ex47.m1.7.8.2.1" xref="S4.Ex47.m1.7.8.2.1.cmml">⁢</mo><mrow id="S4.Ex47.m1.7.8.2.3.2" xref="S4.Ex47.m1.7.8.2.cmml"><mo id="S4.Ex47.m1.7.8.2.3.2.1" stretchy="false" xref="S4.Ex47.m1.7.8.2.cmml">(</mo><mi id="S4.Ex47.m1.7.7" xref="S4.Ex47.m1.7.7.cmml">x</mi><mo id="S4.Ex47.m1.7.8.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.Ex47.m1.7.8.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex47.m1.7.8.1" rspace="0.278em" xref="S4.Ex47.m1.7.8.1.cmml">:=</mo><mrow id="S4.Ex47.m1.6.6" xref="S4.Ex47.m1.7.8.3.1.cmml"><mo id="S4.Ex47.m1.6.6.7" xref="S4.Ex47.m1.7.8.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex47.m1.6.6.6" rowspacing="0pt" xref="S4.Ex47.m1.7.8.3.1.cmml"><mtr id="S4.Ex47.m1.6.6.6a" xref="S4.Ex47.m1.7.8.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6b" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.1.1.1.1.1.1.1" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex47.m1.1.1.1.1.1.1.1.1" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex47.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.2.cmml">f</mi><mn id="S4.Ex47.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex47.m1.1.1.1.1.1.1.1.2" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6c" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.2.2.2.2.2.1" xref="S4.Ex47.m1.2.2.2.2.2.1.cmml"><mtext id="S4.Ex47.m1.2.2.2.2.2.1.2" xref="S4.Ex47.m1.2.2.2.2.2.1.2a.cmml">in </mtext><mo id="S4.Ex47.m1.2.2.2.2.2.1.1" xref="S4.Ex47.m1.2.2.2.2.2.1.1.cmml">⁢</mo><msub id="S4.Ex47.m1.2.2.2.2.2.1.3" xref="S4.Ex47.m1.2.2.2.2.2.1.3.cmml"><mi id="S4.Ex47.m1.2.2.2.2.2.1.3.2" xref="S4.Ex47.m1.2.2.2.2.2.1.3.2.cmml">P</mi><mn id="S4.Ex47.m1.2.2.2.2.2.1.3.3" xref="S4.Ex47.m1.2.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow></mtd></mtr><mtr id="S4.Ex47.m1.6.6.6d" xref="S4.Ex47.m1.7.8.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6e" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.3.3.3.3.1.1.1" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.cmml"><msub id="S4.Ex47.m1.3.3.3.3.1.1.1.1" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.cmml"><mi id="S4.Ex47.m1.3.3.3.3.1.1.1.1.2" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.2.cmml">f</mi><mn id="S4.Ex47.m1.3.3.3.3.1.1.1.1.3" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.Ex47.m1.3.3.3.3.1.1.1.2" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6f" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.4.4.4.4.2.1" xref="S4.Ex47.m1.4.4.4.4.2.1.cmml"><mtext id="S4.Ex47.m1.4.4.4.4.2.1.2" xref="S4.Ex47.m1.4.4.4.4.2.1.2a.cmml">in </mtext><mo id="S4.Ex47.m1.4.4.4.4.2.1.1" xref="S4.Ex47.m1.4.4.4.4.2.1.1.cmml">⁢</mo><msub id="S4.Ex47.m1.4.4.4.4.2.1.3" xref="S4.Ex47.m1.4.4.4.4.2.1.3.cmml"><mi id="S4.Ex47.m1.4.4.4.4.2.1.3.2" xref="S4.Ex47.m1.4.4.4.4.2.1.3.2.cmml">P</mi><mn id="S4.Ex47.m1.4.4.4.4.2.1.3.3" xref="S4.Ex47.m1.4.4.4.4.2.1.3.3.cmml">2</mn></msub></mrow></mtd></mtr><mtr id="S4.Ex47.m1.6.6.6g" xref="S4.Ex47.m1.7.8.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6h" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.5.5.5.5.1.1.1" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.cmml"><msub id="S4.Ex47.m1.5.5.5.5.1.1.1.1" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.cmml"><mi id="S4.Ex47.m1.5.5.5.5.1.1.1.1.2" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.2.cmml">f</mi><mi id="S4.Ex47.m1.5.5.5.5.1.1.1.1.3" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.3.cmml">P</mi></msub><mo id="S4.Ex47.m1.5.5.5.5.1.1.1.2" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex47.m1.6.6.6i" xref="S4.Ex47.m1.7.8.3.1.cmml"><mrow id="S4.Ex47.m1.6.6.6.6.2.1" xref="S4.Ex47.m1.6.6.6.6.2.1.cmml"><mtext id="S4.Ex47.m1.6.6.6.6.2.1.2" xref="S4.Ex47.m1.6.6.6.6.2.1.2a.cmml">in </mtext><mo id="S4.Ex47.m1.6.6.6.6.2.1.1" xref="S4.Ex47.m1.6.6.6.6.2.1.1.cmml">⁢</mo><mi id="S4.Ex47.m1.6.6.6.6.2.1.3" xref="S4.Ex47.m1.6.6.6.6.2.1.3.cmml">P</mi></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex47.m1.7b"><apply id="S4.Ex47.m1.7.8.cmml" xref="S4.Ex47.m1.7.8"><csymbol cd="latexml" id="S4.Ex47.m1.7.8.1.cmml" xref="S4.Ex47.m1.7.8.1">assign</csymbol><apply id="S4.Ex47.m1.7.8.2.cmml" xref="S4.Ex47.m1.7.8.2"><times id="S4.Ex47.m1.7.8.2.1.cmml" xref="S4.Ex47.m1.7.8.2.1"></times><apply id="S4.Ex47.m1.7.8.2.2.cmml" xref="S4.Ex47.m1.7.8.2.2"><csymbol cd="ambiguous" id="S4.Ex47.m1.7.8.2.2.1.cmml" xref="S4.Ex47.m1.7.8.2.2">superscript</csymbol><ci id="S4.Ex47.m1.7.8.2.2.2.cmml" xref="S4.Ex47.m1.7.8.2.2.2">𝑓</ci><ci id="S4.Ex47.m1.7.8.2.2.3.cmml" xref="S4.Ex47.m1.7.8.2.2.3">′</ci></apply><ci id="S4.Ex47.m1.7.7.cmml" xref="S4.Ex47.m1.7.7">𝑥</ci></apply><apply id="S4.Ex47.m1.7.8.3.1.cmml" xref="S4.Ex47.m1.6.6"><csymbol cd="latexml" id="S4.Ex47.m1.7.8.3.1.1.cmml" xref="S4.Ex47.m1.6.6.7">cases</csymbol><apply id="S4.Ex47.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex47.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex47.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex47.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex47.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.2">𝑓</ci><cn id="S4.Ex47.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex47.m1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.Ex47.m1.2.2.2.2.2.1.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1"><times id="S4.Ex47.m1.2.2.2.2.2.1.1.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.1"></times><ci id="S4.Ex47.m1.2.2.2.2.2.1.2a.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.2"><mtext id="S4.Ex47.m1.2.2.2.2.2.1.2.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.2">in </mtext></ci><apply id="S4.Ex47.m1.2.2.2.2.2.1.3.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.Ex47.m1.2.2.2.2.2.1.3.1.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.3">subscript</csymbol><ci id="S4.Ex47.m1.2.2.2.2.2.1.3.2.cmml" xref="S4.Ex47.m1.2.2.2.2.2.1.3.2">𝑃</ci><cn id="S4.Ex47.m1.2.2.2.2.2.1.3.3.cmml" type="integer" xref="S4.Ex47.m1.2.2.2.2.2.1.3.3">1</cn></apply></apply><apply id="S4.Ex47.m1.3.3.3.3.1.1.1.1.cmml" xref="S4.Ex47.m1.3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex47.m1.3.3.3.3.1.1.1.1.1.cmml" xref="S4.Ex47.m1.3.3.3.3.1.1.1">subscript</csymbol><ci id="S4.Ex47.m1.3.3.3.3.1.1.1.1.2.cmml" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.2">𝑓</ci><cn id="S4.Ex47.m1.3.3.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.Ex47.m1.3.3.3.3.1.1.1.1.3">2</cn></apply><apply id="S4.Ex47.m1.4.4.4.4.2.1.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1"><times id="S4.Ex47.m1.4.4.4.4.2.1.1.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.1"></times><ci id="S4.Ex47.m1.4.4.4.4.2.1.2a.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.2"><mtext id="S4.Ex47.m1.4.4.4.4.2.1.2.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.2">in </mtext></ci><apply id="S4.Ex47.m1.4.4.4.4.2.1.3.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.3"><csymbol cd="ambiguous" id="S4.Ex47.m1.4.4.4.4.2.1.3.1.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.3">subscript</csymbol><ci id="S4.Ex47.m1.4.4.4.4.2.1.3.2.cmml" xref="S4.Ex47.m1.4.4.4.4.2.1.3.2">𝑃</ci><cn id="S4.Ex47.m1.4.4.4.4.2.1.3.3.cmml" type="integer" xref="S4.Ex47.m1.4.4.4.4.2.1.3.3">2</cn></apply></apply><apply id="S4.Ex47.m1.5.5.5.5.1.1.1.1.cmml" xref="S4.Ex47.m1.5.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.Ex47.m1.5.5.5.5.1.1.1.1.1.cmml" xref="S4.Ex47.m1.5.5.5.5.1.1.1">subscript</csymbol><ci id="S4.Ex47.m1.5.5.5.5.1.1.1.1.2.cmml" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.2">𝑓</ci><ci id="S4.Ex47.m1.5.5.5.5.1.1.1.1.3.cmml" xref="S4.Ex47.m1.5.5.5.5.1.1.1.1.3">𝑃</ci></apply><apply id="S4.Ex47.m1.6.6.6.6.2.1.cmml" xref="S4.Ex47.m1.6.6.6.6.2.1"><times id="S4.Ex47.m1.6.6.6.6.2.1.1.cmml" xref="S4.Ex47.m1.6.6.6.6.2.1.1"></times><ci id="S4.Ex47.m1.6.6.6.6.2.1.2a.cmml" xref="S4.Ex47.m1.6.6.6.6.2.1.2"><mtext id="S4.Ex47.m1.6.6.6.6.2.1.2.cmml" xref="S4.Ex47.m1.6.6.6.6.2.1.2">in </mtext></ci><ci id="S4.Ex47.m1.6.6.6.6.2.1.3.cmml" xref="S4.Ex47.m1.6.6.6.6.2.1.3">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex47.m1.7c">f^{\prime}(x):=\begin{cases}f_{1},\quad&\text{in }P_{1}\\ f_{2},\quad&\text{in }P_{2}\\ f_{P},\quad&\text{in }P\\ \end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex47.m1.7d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) := { start_ROW start_CELL italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL start_CELL in italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , end_CELL start_CELL in italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT , end_CELL start_CELL in italic_P 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="S4.6.p1.31">is continuous and piecewise linear. Since the interior of <math alttext="P" class="ltx_Math" display="inline" id="S4.6.p1.27.m1.1"><semantics id="S4.6.p1.27.m1.1a"><mi id="S4.6.p1.27.m1.1.1" xref="S4.6.p1.27.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.27.m1.1b"><ci id="S4.6.p1.27.m1.1.1.cmml" xref="S4.6.p1.27.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.27.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.27.m1.1d">italic_P</annotation></semantics></math> is connected, <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.6.p1.28.m2.1"><semantics id="S4.6.p1.28.m2.1a"><msub id="S4.6.p1.28.m2.1.1" xref="S4.6.p1.28.m2.1.1.cmml"><mi id="S4.6.p1.28.m2.1.1.2" xref="S4.6.p1.28.m2.1.1.2.cmml">P</mi><mn id="S4.6.p1.28.m2.1.1.3" xref="S4.6.p1.28.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.28.m2.1b"><apply id="S4.6.p1.28.m2.1.1.cmml" xref="S4.6.p1.28.m2.1.1"><csymbol cd="ambiguous" id="S4.6.p1.28.m2.1.1.1.cmml" xref="S4.6.p1.28.m2.1.1">subscript</csymbol><ci id="S4.6.p1.28.m2.1.1.2.cmml" xref="S4.6.p1.28.m2.1.1.2">𝑃</ci><cn id="S4.6.p1.28.m2.1.1.3.cmml" type="integer" xref="S4.6.p1.28.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.28.m2.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.28.m2.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.6.p1.29.m3.1"><semantics id="S4.6.p1.29.m3.1a"><msub id="S4.6.p1.29.m3.1.1" xref="S4.6.p1.29.m3.1.1.cmml"><mi id="S4.6.p1.29.m3.1.1.2" xref="S4.6.p1.29.m3.1.1.2.cmml">P</mi><mn id="S4.6.p1.29.m3.1.1.3" xref="S4.6.p1.29.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p1.29.m3.1b"><apply id="S4.6.p1.29.m3.1.1.cmml" xref="S4.6.p1.29.m3.1.1"><csymbol cd="ambiguous" id="S4.6.p1.29.m3.1.1.1.cmml" xref="S4.6.p1.29.m3.1.1">subscript</csymbol><ci id="S4.6.p1.29.m3.1.1.2.cmml" xref="S4.6.p1.29.m3.1.1.2">𝑃</ci><cn id="S4.6.p1.29.m3.1.1.3.cmml" type="integer" xref="S4.6.p1.29.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.29.m3.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.29.m3.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="P" class="ltx_Math" display="inline" id="S4.6.p1.30.m4.1"><semantics id="S4.6.p1.30.m4.1a"><mi id="S4.6.p1.30.m4.1.1" xref="S4.6.p1.30.m4.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.30.m4.1b"><ci id="S4.6.p1.30.m4.1.1.cmml" xref="S4.6.p1.30.m4.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.30.m4.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.30.m4.1d">italic_P</annotation></semantics></math> are admissible pieces, thus <math alttext="f^{\prime}\in\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.6.p1.31.m5.1"><semantics id="S4.6.p1.31.m5.1a"><mrow id="S4.6.p1.31.m5.1.1" xref="S4.6.p1.31.m5.1.1.cmml"><msup id="S4.6.p1.31.m5.1.1.2" xref="S4.6.p1.31.m5.1.1.2.cmml"><mi id="S4.6.p1.31.m5.1.1.2.2" xref="S4.6.p1.31.m5.1.1.2.2.cmml">f</mi><mo id="S4.6.p1.31.m5.1.1.2.3" xref="S4.6.p1.31.m5.1.1.2.3.cmml">′</mo></msup><mo id="S4.6.p1.31.m5.1.1.1" xref="S4.6.p1.31.m5.1.1.1.cmml">∈</mo><msub id="S4.6.p1.31.m5.1.1.3" xref="S4.6.p1.31.m5.1.1.3.cmml"><mi id="S4.6.p1.31.m5.1.1.3.2" xref="S4.6.p1.31.m5.1.1.3.2.cmml">CPL</mi><mn id="S4.6.p1.31.m5.1.1.3.3" xref="S4.6.p1.31.m5.1.1.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.31.m5.1b"><apply id="S4.6.p1.31.m5.1.1.cmml" xref="S4.6.p1.31.m5.1.1"><in id="S4.6.p1.31.m5.1.1.1.cmml" xref="S4.6.p1.31.m5.1.1.1"></in><apply id="S4.6.p1.31.m5.1.1.2.cmml" xref="S4.6.p1.31.m5.1.1.2"><csymbol cd="ambiguous" id="S4.6.p1.31.m5.1.1.2.1.cmml" xref="S4.6.p1.31.m5.1.1.2">superscript</csymbol><ci id="S4.6.p1.31.m5.1.1.2.2.cmml" xref="S4.6.p1.31.m5.1.1.2.2">𝑓</ci><ci id="S4.6.p1.31.m5.1.1.2.3.cmml" xref="S4.6.p1.31.m5.1.1.2.3">′</ci></apply><apply id="S4.6.p1.31.m5.1.1.3.cmml" xref="S4.6.p1.31.m5.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.31.m5.1.1.3.1.cmml" xref="S4.6.p1.31.m5.1.1.3">subscript</csymbol><ci id="S4.6.p1.31.m5.1.1.3.2.cmml" xref="S4.6.p1.31.m5.1.1.3.2">CPL</ci><cn id="S4.6.p1.31.m5.1.1.3.3.cmml" type="integer" xref="S4.6.p1.31.m5.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.31.m5.1c">f^{\prime}\in\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.31.m5.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.7.p2"> <p class="ltx_p" id="S4.7.p2.9">For <math alttext="i>2" class="ltx_Math" display="inline" id="S4.7.p2.1.m1.1"><semantics id="S4.7.p2.1.m1.1a"><mrow id="S4.7.p2.1.m1.1.1" xref="S4.7.p2.1.m1.1.1.cmml"><mi id="S4.7.p2.1.m1.1.1.2" xref="S4.7.p2.1.m1.1.1.2.cmml">i</mi><mo id="S4.7.p2.1.m1.1.1.1" xref="S4.7.p2.1.m1.1.1.1.cmml">></mo><mn id="S4.7.p2.1.m1.1.1.3" xref="S4.7.p2.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.1.m1.1b"><apply id="S4.7.p2.1.m1.1.1.cmml" xref="S4.7.p2.1.m1.1.1"><gt id="S4.7.p2.1.m1.1.1.1.cmml" xref="S4.7.p2.1.m1.1.1.1"></gt><ci id="S4.7.p2.1.m1.1.1.2.cmml" xref="S4.7.p2.1.m1.1.1.2">𝑖</ci><cn id="S4.7.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.7.p2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.1.m1.1c">i>2</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.1.m1.1d">italic_i > 2</annotation></semantics></math>, we have <math alttext="(f-f^{\prime})|_{P_{i}}=f|_{P_{i}}-f_{P}" class="ltx_Math" display="inline" id="S4.7.p2.2.m2.4"><semantics id="S4.7.p2.2.m2.4a"><mrow id="S4.7.p2.2.m2.4.4" xref="S4.7.p2.2.m2.4.4.cmml"><msub id="S4.7.p2.2.m2.4.4.1.1" xref="S4.7.p2.2.m2.4.4.1.2.cmml"><mrow id="S4.7.p2.2.m2.4.4.1.1.1" xref="S4.7.p2.2.m2.4.4.1.2.cmml"><mrow id="S4.7.p2.2.m2.4.4.1.1.1.1.1" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.cmml"><mo id="S4.7.p2.2.m2.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.cmml"><mi id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.2" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.2.cmml">f</mi><mo id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.1" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.cmml"><mi id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.2" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.2.cmml">f</mi><mo id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.3" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.7.p2.2.m2.4.4.1.1.1.1.1.3" stretchy="false" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.7.p2.2.m2.4.4.1.1.1.2" stretchy="false" xref="S4.7.p2.2.m2.4.4.1.2.1.cmml">|</mo></mrow><msub id="S4.7.p2.2.m2.1.1.1" xref="S4.7.p2.2.m2.1.1.1.cmml"><mi id="S4.7.p2.2.m2.1.1.1.2" xref="S4.7.p2.2.m2.1.1.1.2.cmml">P</mi><mi id="S4.7.p2.2.m2.1.1.1.3" xref="S4.7.p2.2.m2.1.1.1.3.cmml">i</mi></msub></msub><mo id="S4.7.p2.2.m2.4.4.2" xref="S4.7.p2.2.m2.4.4.2.cmml">=</mo><mrow id="S4.7.p2.2.m2.4.4.3" xref="S4.7.p2.2.m2.4.4.3.cmml"><msub id="S4.7.p2.2.m2.4.4.3.2.2" xref="S4.7.p2.2.m2.4.4.3.2.1.cmml"><mrow id="S4.7.p2.2.m2.4.4.3.2.2.2" xref="S4.7.p2.2.m2.4.4.3.2.1.cmml"><mi id="S4.7.p2.2.m2.2.2" xref="S4.7.p2.2.m2.2.2.cmml">f</mi><mo id="S4.7.p2.2.m2.4.4.3.2.2.2.1" stretchy="false" xref="S4.7.p2.2.m2.4.4.3.2.1.1.cmml">|</mo></mrow><msub id="S4.7.p2.2.m2.3.3.1" xref="S4.7.p2.2.m2.3.3.1.cmml"><mi id="S4.7.p2.2.m2.3.3.1.2" xref="S4.7.p2.2.m2.3.3.1.2.cmml">P</mi><mi id="S4.7.p2.2.m2.3.3.1.3" xref="S4.7.p2.2.m2.3.3.1.3.cmml">i</mi></msub></msub><mo id="S4.7.p2.2.m2.4.4.3.1" xref="S4.7.p2.2.m2.4.4.3.1.cmml">−</mo><msub id="S4.7.p2.2.m2.4.4.3.3" xref="S4.7.p2.2.m2.4.4.3.3.cmml"><mi id="S4.7.p2.2.m2.4.4.3.3.2" xref="S4.7.p2.2.m2.4.4.3.3.2.cmml">f</mi><mi id="S4.7.p2.2.m2.4.4.3.3.3" xref="S4.7.p2.2.m2.4.4.3.3.3.cmml">P</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.2.m2.4b"><apply id="S4.7.p2.2.m2.4.4.cmml" xref="S4.7.p2.2.m2.4.4"><eq id="S4.7.p2.2.m2.4.4.2.cmml" xref="S4.7.p2.2.m2.4.4.2"></eq><apply id="S4.7.p2.2.m2.4.4.1.2.cmml" xref="S4.7.p2.2.m2.4.4.1.1"><csymbol cd="latexml" id="S4.7.p2.2.m2.4.4.1.2.1.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.2">evaluated-at</csymbol><apply id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1"><minus id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.1.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.1"></minus><ci id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.2.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.2">𝑓</ci><apply id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.1.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.2.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.2">𝑓</ci><ci id="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.3.cmml" xref="S4.7.p2.2.m2.4.4.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S4.7.p2.2.m2.1.1.1.cmml" xref="S4.7.p2.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.7.p2.2.m2.1.1.1.1.cmml" xref="S4.7.p2.2.m2.1.1.1">subscript</csymbol><ci id="S4.7.p2.2.m2.1.1.1.2.cmml" xref="S4.7.p2.2.m2.1.1.1.2">𝑃</ci><ci id="S4.7.p2.2.m2.1.1.1.3.cmml" xref="S4.7.p2.2.m2.1.1.1.3">𝑖</ci></apply></apply><apply id="S4.7.p2.2.m2.4.4.3.cmml" xref="S4.7.p2.2.m2.4.4.3"><minus id="S4.7.p2.2.m2.4.4.3.1.cmml" xref="S4.7.p2.2.m2.4.4.3.1"></minus><apply id="S4.7.p2.2.m2.4.4.3.2.1.cmml" xref="S4.7.p2.2.m2.4.4.3.2.2"><csymbol cd="latexml" id="S4.7.p2.2.m2.4.4.3.2.1.1.cmml" xref="S4.7.p2.2.m2.4.4.3.2.2.2.1">evaluated-at</csymbol><ci id="S4.7.p2.2.m2.2.2.cmml" xref="S4.7.p2.2.m2.2.2">𝑓</ci><apply id="S4.7.p2.2.m2.3.3.1.cmml" xref="S4.7.p2.2.m2.3.3.1"><csymbol cd="ambiguous" id="S4.7.p2.2.m2.3.3.1.1.cmml" xref="S4.7.p2.2.m2.3.3.1">subscript</csymbol><ci id="S4.7.p2.2.m2.3.3.1.2.cmml" xref="S4.7.p2.2.m2.3.3.1.2">𝑃</ci><ci id="S4.7.p2.2.m2.3.3.1.3.cmml" xref="S4.7.p2.2.m2.3.3.1.3">𝑖</ci></apply></apply><apply id="S4.7.p2.2.m2.4.4.3.3.cmml" xref="S4.7.p2.2.m2.4.4.3.3"><csymbol cd="ambiguous" id="S4.7.p2.2.m2.4.4.3.3.1.cmml" xref="S4.7.p2.2.m2.4.4.3.3">subscript</csymbol><ci id="S4.7.p2.2.m2.4.4.3.3.2.cmml" xref="S4.7.p2.2.m2.4.4.3.3.2">𝑓</ci><ci id="S4.7.p2.2.m2.4.4.3.3.3.cmml" xref="S4.7.p2.2.m2.4.4.3.3.3">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.2.m2.4c">(f-f^{\prime})|_{P_{i}}=f|_{P_{i}}-f_{P}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.2.m2.4d">( italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_f | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>, which is a linear function. Moreover, <math alttext="(f-f^{\prime})|_{P_{1}}=(f-f^{\prime})|_{P_{2}}=0" class="ltx_Math" display="inline" id="S4.7.p2.3.m3.4"><semantics id="S4.7.p2.3.m3.4a"><mrow id="S4.7.p2.3.m3.4.4" xref="S4.7.p2.3.m3.4.4.cmml"><msub id="S4.7.p2.3.m3.3.3.1.1" xref="S4.7.p2.3.m3.3.3.1.2.cmml"><mrow id="S4.7.p2.3.m3.3.3.1.1.1" xref="S4.7.p2.3.m3.3.3.1.2.cmml"><mrow id="S4.7.p2.3.m3.3.3.1.1.1.1.1" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.cmml"><mo id="S4.7.p2.3.m3.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.cmml"><mi id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.2" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.2.cmml">f</mi><mo id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.1" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.cmml"><mi id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.2" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.2.cmml">f</mi><mo id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.3" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.7.p2.3.m3.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.7.p2.3.m3.3.3.1.1.1.2" stretchy="false" xref="S4.7.p2.3.m3.3.3.1.2.1.cmml">|</mo></mrow><msub id="S4.7.p2.3.m3.1.1.1" xref="S4.7.p2.3.m3.1.1.1.cmml"><mi id="S4.7.p2.3.m3.1.1.1.2" xref="S4.7.p2.3.m3.1.1.1.2.cmml">P</mi><mn id="S4.7.p2.3.m3.1.1.1.3" xref="S4.7.p2.3.m3.1.1.1.3.cmml">1</mn></msub></msub><mo id="S4.7.p2.3.m3.4.4.4" xref="S4.7.p2.3.m3.4.4.4.cmml">=</mo><msub id="S4.7.p2.3.m3.4.4.2.1" xref="S4.7.p2.3.m3.4.4.2.2.cmml"><mrow id="S4.7.p2.3.m3.4.4.2.1.1" xref="S4.7.p2.3.m3.4.4.2.2.cmml"><mrow id="S4.7.p2.3.m3.4.4.2.1.1.1.1" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.cmml"><mo id="S4.7.p2.3.m3.4.4.2.1.1.1.1.2" stretchy="false" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.cmml"><mi id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.2" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.2.cmml">f</mi><mo id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.1" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.1.cmml">−</mo><msup id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.cmml"><mi id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.2" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.2.cmml">f</mi><mo id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.3" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.7.p2.3.m3.4.4.2.1.1.1.1.3" stretchy="false" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.7.p2.3.m3.4.4.2.1.1.2" stretchy="false" xref="S4.7.p2.3.m3.4.4.2.2.1.cmml">|</mo></mrow><msub id="S4.7.p2.3.m3.2.2.1" xref="S4.7.p2.3.m3.2.2.1.cmml"><mi id="S4.7.p2.3.m3.2.2.1.2" xref="S4.7.p2.3.m3.2.2.1.2.cmml">P</mi><mn id="S4.7.p2.3.m3.2.2.1.3" xref="S4.7.p2.3.m3.2.2.1.3.cmml">2</mn></msub></msub><mo id="S4.7.p2.3.m3.4.4.5" xref="S4.7.p2.3.m3.4.4.5.cmml">=</mo><mn id="S4.7.p2.3.m3.4.4.6" xref="S4.7.p2.3.m3.4.4.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.3.m3.4b"><apply id="S4.7.p2.3.m3.4.4.cmml" xref="S4.7.p2.3.m3.4.4"><and id="S4.7.p2.3.m3.4.4a.cmml" xref="S4.7.p2.3.m3.4.4"></and><apply id="S4.7.p2.3.m3.4.4b.cmml" xref="S4.7.p2.3.m3.4.4"><eq id="S4.7.p2.3.m3.4.4.4.cmml" xref="S4.7.p2.3.m3.4.4.4"></eq><apply id="S4.7.p2.3.m3.3.3.1.2.cmml" xref="S4.7.p2.3.m3.3.3.1.1"><csymbol cd="latexml" id="S4.7.p2.3.m3.3.3.1.2.1.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.2">evaluated-at</csymbol><apply id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1"><minus id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.1"></minus><ci id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.2.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.2">𝑓</ci><apply id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.1.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.2.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.2">𝑓</ci><ci id="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.3.cmml" xref="S4.7.p2.3.m3.3.3.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S4.7.p2.3.m3.1.1.1.cmml" xref="S4.7.p2.3.m3.1.1.1"><csymbol cd="ambiguous" id="S4.7.p2.3.m3.1.1.1.1.cmml" xref="S4.7.p2.3.m3.1.1.1">subscript</csymbol><ci id="S4.7.p2.3.m3.1.1.1.2.cmml" xref="S4.7.p2.3.m3.1.1.1.2">𝑃</ci><cn id="S4.7.p2.3.m3.1.1.1.3.cmml" type="integer" xref="S4.7.p2.3.m3.1.1.1.3">1</cn></apply></apply><apply id="S4.7.p2.3.m3.4.4.2.2.cmml" xref="S4.7.p2.3.m3.4.4.2.1"><csymbol cd="latexml" id="S4.7.p2.3.m3.4.4.2.2.1.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.2">evaluated-at</csymbol><apply id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1"><minus id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.1.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.1"></minus><ci id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.2.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.2">𝑓</ci><apply id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.1.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3">superscript</csymbol><ci id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.2.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.2">𝑓</ci><ci id="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.3.cmml" xref="S4.7.p2.3.m3.4.4.2.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S4.7.p2.3.m3.2.2.1.cmml" xref="S4.7.p2.3.m3.2.2.1"><csymbol cd="ambiguous" id="S4.7.p2.3.m3.2.2.1.1.cmml" xref="S4.7.p2.3.m3.2.2.1">subscript</csymbol><ci id="S4.7.p2.3.m3.2.2.1.2.cmml" xref="S4.7.p2.3.m3.2.2.1.2">𝑃</ci><cn id="S4.7.p2.3.m3.2.2.1.3.cmml" type="integer" xref="S4.7.p2.3.m3.2.2.1.3">2</cn></apply></apply></apply><apply id="S4.7.p2.3.m3.4.4c.cmml" xref="S4.7.p2.3.m3.4.4"><eq id="S4.7.p2.3.m3.4.4.5.cmml" xref="S4.7.p2.3.m3.4.4.5"></eq><share href="https://arxiv.org/html/2503.13001v1#S4.7.p2.3.m3.4.4.2.cmml" id="S4.7.p2.3.m3.4.4d.cmml" xref="S4.7.p2.3.m3.4.4"></share><cn id="S4.7.p2.3.m3.4.4.6.cmml" type="integer" xref="S4.7.p2.3.m3.4.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.3.m3.4c">(f-f^{\prime})|_{P_{1}}=(f-f^{\prime})|_{P_{2}}=0</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.3.m3.4d">( italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ( italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>, i.e. <math alttext="f-f^{\prime}" class="ltx_Math" display="inline" id="S4.7.p2.4.m4.1"><semantics id="S4.7.p2.4.m4.1a"><mrow id="S4.7.p2.4.m4.1.1" xref="S4.7.p2.4.m4.1.1.cmml"><mi id="S4.7.p2.4.m4.1.1.2" xref="S4.7.p2.4.m4.1.1.2.cmml">f</mi><mo id="S4.7.p2.4.m4.1.1.1" xref="S4.7.p2.4.m4.1.1.1.cmml">−</mo><msup id="S4.7.p2.4.m4.1.1.3" xref="S4.7.p2.4.m4.1.1.3.cmml"><mi id="S4.7.p2.4.m4.1.1.3.2" xref="S4.7.p2.4.m4.1.1.3.2.cmml">f</mi><mo id="S4.7.p2.4.m4.1.1.3.3" xref="S4.7.p2.4.m4.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.4.m4.1b"><apply id="S4.7.p2.4.m4.1.1.cmml" xref="S4.7.p2.4.m4.1.1"><minus id="S4.7.p2.4.m4.1.1.1.cmml" xref="S4.7.p2.4.m4.1.1.1"></minus><ci id="S4.7.p2.4.m4.1.1.2.cmml" xref="S4.7.p2.4.m4.1.1.2">𝑓</ci><apply id="S4.7.p2.4.m4.1.1.3.cmml" xref="S4.7.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.4.m4.1.1.3.1.cmml" xref="S4.7.p2.4.m4.1.1.3">superscript</csymbol><ci id="S4.7.p2.4.m4.1.1.3.2.cmml" xref="S4.7.p2.4.m4.1.1.3.2">𝑓</ci><ci id="S4.7.p2.4.m4.1.1.3.3.cmml" xref="S4.7.p2.4.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.4.m4.1c">f-f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.4.m4.1d">italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is linear on <math alttext="P_{1}\cup P_{2}" class="ltx_Math" display="inline" id="S4.7.p2.5.m5.1"><semantics id="S4.7.p2.5.m5.1a"><mrow id="S4.7.p2.5.m5.1.1" xref="S4.7.p2.5.m5.1.1.cmml"><msub id="S4.7.p2.5.m5.1.1.2" xref="S4.7.p2.5.m5.1.1.2.cmml"><mi id="S4.7.p2.5.m5.1.1.2.2" xref="S4.7.p2.5.m5.1.1.2.2.cmml">P</mi><mn id="S4.7.p2.5.m5.1.1.2.3" xref="S4.7.p2.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S4.7.p2.5.m5.1.1.1" xref="S4.7.p2.5.m5.1.1.1.cmml">∪</mo><msub id="S4.7.p2.5.m5.1.1.3" xref="S4.7.p2.5.m5.1.1.3.cmml"><mi id="S4.7.p2.5.m5.1.1.3.2" xref="S4.7.p2.5.m5.1.1.3.2.cmml">P</mi><mn id="S4.7.p2.5.m5.1.1.3.3" xref="S4.7.p2.5.m5.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.5.m5.1b"><apply id="S4.7.p2.5.m5.1.1.cmml" xref="S4.7.p2.5.m5.1.1"><union id="S4.7.p2.5.m5.1.1.1.cmml" xref="S4.7.p2.5.m5.1.1.1"></union><apply id="S4.7.p2.5.m5.1.1.2.cmml" xref="S4.7.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.7.p2.5.m5.1.1.2.1.cmml" xref="S4.7.p2.5.m5.1.1.2">subscript</csymbol><ci id="S4.7.p2.5.m5.1.1.2.2.cmml" xref="S4.7.p2.5.m5.1.1.2.2">𝑃</ci><cn id="S4.7.p2.5.m5.1.1.2.3.cmml" type="integer" xref="S4.7.p2.5.m5.1.1.2.3">1</cn></apply><apply id="S4.7.p2.5.m5.1.1.3.cmml" xref="S4.7.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.5.m5.1.1.3.1.cmml" xref="S4.7.p2.5.m5.1.1.3">subscript</csymbol><ci id="S4.7.p2.5.m5.1.1.3.2.cmml" xref="S4.7.p2.5.m5.1.1.3.2">𝑃</ci><cn id="S4.7.p2.5.m5.1.1.3.3.cmml" type="integer" xref="S4.7.p2.5.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.5.m5.1c">P_{1}\cup P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.5.m5.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Continuity of <math alttext="f-f^{\prime}" class="ltx_Math" display="inline" id="S4.7.p2.6.m6.1"><semantics id="S4.7.p2.6.m6.1a"><mrow id="S4.7.p2.6.m6.1.1" xref="S4.7.p2.6.m6.1.1.cmml"><mi id="S4.7.p2.6.m6.1.1.2" xref="S4.7.p2.6.m6.1.1.2.cmml">f</mi><mo id="S4.7.p2.6.m6.1.1.1" xref="S4.7.p2.6.m6.1.1.1.cmml">−</mo><msup id="S4.7.p2.6.m6.1.1.3" xref="S4.7.p2.6.m6.1.1.3.cmml"><mi id="S4.7.p2.6.m6.1.1.3.2" xref="S4.7.p2.6.m6.1.1.3.2.cmml">f</mi><mo id="S4.7.p2.6.m6.1.1.3.3" xref="S4.7.p2.6.m6.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.6.m6.1b"><apply id="S4.7.p2.6.m6.1.1.cmml" xref="S4.7.p2.6.m6.1.1"><minus id="S4.7.p2.6.m6.1.1.1.cmml" xref="S4.7.p2.6.m6.1.1.1"></minus><ci id="S4.7.p2.6.m6.1.1.2.cmml" xref="S4.7.p2.6.m6.1.1.2">𝑓</ci><apply id="S4.7.p2.6.m6.1.1.3.cmml" xref="S4.7.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.6.m6.1.1.3.1.cmml" xref="S4.7.p2.6.m6.1.1.3">superscript</csymbol><ci id="S4.7.p2.6.m6.1.1.3.2.cmml" xref="S4.7.p2.6.m6.1.1.3.2">𝑓</ci><ci id="S4.7.p2.6.m6.1.1.3.3.cmml" xref="S4.7.p2.6.m6.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.6.m6.1c">f-f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.6.m6.1d">italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is implied by the continuity of <math alttext="f" class="ltx_Math" display="inline" id="S4.7.p2.7.m7.1"><semantics id="S4.7.p2.7.m7.1a"><mi id="S4.7.p2.7.m7.1.1" xref="S4.7.p2.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.7.p2.7.m7.1b"><ci id="S4.7.p2.7.m7.1.1.cmml" xref="S4.7.p2.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.7.m7.1d">italic_f</annotation></semantics></math> and <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S4.7.p2.8.m8.1"><semantics id="S4.7.p2.8.m8.1a"><msup id="S4.7.p2.8.m8.1.1" xref="S4.7.p2.8.m8.1.1.cmml"><mi id="S4.7.p2.8.m8.1.1.2" xref="S4.7.p2.8.m8.1.1.2.cmml">f</mi><mo id="S4.7.p2.8.m8.1.1.3" xref="S4.7.p2.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.7.p2.8.m8.1b"><apply id="S4.7.p2.8.m8.1.1.cmml" xref="S4.7.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.7.p2.8.m8.1.1.1.cmml" xref="S4.7.p2.8.m8.1.1">superscript</csymbol><ci id="S4.7.p2.8.m8.1.1.2.cmml" xref="S4.7.p2.8.m8.1.1.2">𝑓</ci><ci id="S4.7.p2.8.m8.1.1.3.cmml" xref="S4.7.p2.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.8.m8.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.8.m8.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and we get <math alttext="f-f^{\prime}\in\operatorname{CPA}_{p-1}" class="ltx_Math" display="inline" id="S4.7.p2.9.m9.1"><semantics id="S4.7.p2.9.m9.1a"><mrow id="S4.7.p2.9.m9.1.1" xref="S4.7.p2.9.m9.1.1.cmml"><mrow id="S4.7.p2.9.m9.1.1.2" xref="S4.7.p2.9.m9.1.1.2.cmml"><mi id="S4.7.p2.9.m9.1.1.2.2" xref="S4.7.p2.9.m9.1.1.2.2.cmml">f</mi><mo id="S4.7.p2.9.m9.1.1.2.1" xref="S4.7.p2.9.m9.1.1.2.1.cmml">−</mo><msup id="S4.7.p2.9.m9.1.1.2.3" xref="S4.7.p2.9.m9.1.1.2.3.cmml"><mi id="S4.7.p2.9.m9.1.1.2.3.2" xref="S4.7.p2.9.m9.1.1.2.3.2.cmml">f</mi><mo id="S4.7.p2.9.m9.1.1.2.3.3" xref="S4.7.p2.9.m9.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S4.7.p2.9.m9.1.1.1" xref="S4.7.p2.9.m9.1.1.1.cmml">∈</mo><msub id="S4.7.p2.9.m9.1.1.3" xref="S4.7.p2.9.m9.1.1.3.cmml"><mi id="S4.7.p2.9.m9.1.1.3.2" xref="S4.7.p2.9.m9.1.1.3.2.cmml">CPA</mi><mrow id="S4.7.p2.9.m9.1.1.3.3" xref="S4.7.p2.9.m9.1.1.3.3.cmml"><mi id="S4.7.p2.9.m9.1.1.3.3.2" xref="S4.7.p2.9.m9.1.1.3.3.2.cmml">p</mi><mo id="S4.7.p2.9.m9.1.1.3.3.1" xref="S4.7.p2.9.m9.1.1.3.3.1.cmml">−</mo><mn id="S4.7.p2.9.m9.1.1.3.3.3" xref="S4.7.p2.9.m9.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p2.9.m9.1b"><apply id="S4.7.p2.9.m9.1.1.cmml" xref="S4.7.p2.9.m9.1.1"><in id="S4.7.p2.9.m9.1.1.1.cmml" xref="S4.7.p2.9.m9.1.1.1"></in><apply id="S4.7.p2.9.m9.1.1.2.cmml" xref="S4.7.p2.9.m9.1.1.2"><minus id="S4.7.p2.9.m9.1.1.2.1.cmml" xref="S4.7.p2.9.m9.1.1.2.1"></minus><ci id="S4.7.p2.9.m9.1.1.2.2.cmml" xref="S4.7.p2.9.m9.1.1.2.2">𝑓</ci><apply id="S4.7.p2.9.m9.1.1.2.3.cmml" xref="S4.7.p2.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S4.7.p2.9.m9.1.1.2.3.1.cmml" xref="S4.7.p2.9.m9.1.1.2.3">superscript</csymbol><ci id="S4.7.p2.9.m9.1.1.2.3.2.cmml" xref="S4.7.p2.9.m9.1.1.2.3.2">𝑓</ci><ci id="S4.7.p2.9.m9.1.1.2.3.3.cmml" xref="S4.7.p2.9.m9.1.1.2.3.3">′</ci></apply></apply><apply id="S4.7.p2.9.m9.1.1.3.cmml" xref="S4.7.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.7.p2.9.m9.1.1.3.1.cmml" xref="S4.7.p2.9.m9.1.1.3">subscript</csymbol><ci id="S4.7.p2.9.m9.1.1.3.2.cmml" xref="S4.7.p2.9.m9.1.1.3.2">CPA</ci><apply id="S4.7.p2.9.m9.1.1.3.3.cmml" xref="S4.7.p2.9.m9.1.1.3.3"><minus id="S4.7.p2.9.m9.1.1.3.3.1.cmml" xref="S4.7.p2.9.m9.1.1.3.3.1"></minus><ci id="S4.7.p2.9.m9.1.1.3.3.2.cmml" xref="S4.7.p2.9.m9.1.1.3.3.2">𝑝</ci><cn id="S4.7.p2.9.m9.1.1.3.3.3.cmml" type="integer" xref="S4.7.p2.9.m9.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p2.9.m9.1c">f-f^{\prime}\in\operatorname{CPA}_{p-1}</annotation><annotation encoding="application/x-llamapun" id="S4.7.p2.9.m9.1d">italic_f - italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_CPA start_POSTSUBSCRIPT italic_p - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">With this preparation, we now come to the proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <div class="ltx_proof" id="S4.11"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</h6> <div class="ltx_para" id="S4.8.p1"> <p class="ltx_p" id="S4.8.p1.6">We may assume that <math alttext="f\in\operatorname{CPL}_{p}" class="ltx_Math" display="inline" id="S4.8.p1.1.m1.1"><semantics id="S4.8.p1.1.m1.1a"><mrow id="S4.8.p1.1.m1.1.1" xref="S4.8.p1.1.m1.1.1.cmml"><mi id="S4.8.p1.1.m1.1.1.2" xref="S4.8.p1.1.m1.1.1.2.cmml">f</mi><mo id="S4.8.p1.1.m1.1.1.1" xref="S4.8.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.8.p1.1.m1.1.1.3" xref="S4.8.p1.1.m1.1.1.3.cmml"><mi id="S4.8.p1.1.m1.1.1.3.2" xref="S4.8.p1.1.m1.1.1.3.2.cmml">CPL</mi><mi id="S4.8.p1.1.m1.1.1.3.3" xref="S4.8.p1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.1.m1.1b"><apply id="S4.8.p1.1.m1.1.1.cmml" xref="S4.8.p1.1.m1.1.1"><in id="S4.8.p1.1.m1.1.1.1.cmml" xref="S4.8.p1.1.m1.1.1.1"></in><ci id="S4.8.p1.1.m1.1.1.2.cmml" xref="S4.8.p1.1.m1.1.1.2">𝑓</ci><apply id="S4.8.p1.1.m1.1.1.3.cmml" xref="S4.8.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.8.p1.1.m1.1.1.3.1.cmml" xref="S4.8.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.8.p1.1.m1.1.1.3.2.cmml" xref="S4.8.p1.1.m1.1.1.3.2">CPL</ci><ci id="S4.8.p1.1.m1.1.1.3.3.cmml" xref="S4.8.p1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.1.m1.1c">f\in\operatorname{CPL}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.1.m1.1d">italic_f ∈ roman_CPL start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. Otherwise, represent <math alttext="\tilde{f}(x):=f(v-x)-f(v)" class="ltx_Math" display="inline" id="S4.8.p1.2.m2.3"><semantics id="S4.8.p1.2.m2.3a"><mrow id="S4.8.p1.2.m2.3.3" xref="S4.8.p1.2.m2.3.3.cmml"><mrow id="S4.8.p1.2.m2.3.3.3" xref="S4.8.p1.2.m2.3.3.3.cmml"><mover accent="true" id="S4.8.p1.2.m2.3.3.3.2" xref="S4.8.p1.2.m2.3.3.3.2.cmml"><mi id="S4.8.p1.2.m2.3.3.3.2.2" xref="S4.8.p1.2.m2.3.3.3.2.2.cmml">f</mi><mo id="S4.8.p1.2.m2.3.3.3.2.1" xref="S4.8.p1.2.m2.3.3.3.2.1.cmml">~</mo></mover><mo id="S4.8.p1.2.m2.3.3.3.1" xref="S4.8.p1.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S4.8.p1.2.m2.3.3.3.3.2" xref="S4.8.p1.2.m2.3.3.3.cmml"><mo id="S4.8.p1.2.m2.3.3.3.3.2.1" stretchy="false" xref="S4.8.p1.2.m2.3.3.3.cmml">(</mo><mi id="S4.8.p1.2.m2.1.1" xref="S4.8.p1.2.m2.1.1.cmml">x</mi><mo id="S4.8.p1.2.m2.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.8.p1.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.2.m2.3.3.2" rspace="0.278em" xref="S4.8.p1.2.m2.3.3.2.cmml">:=</mo><mrow id="S4.8.p1.2.m2.3.3.1" xref="S4.8.p1.2.m2.3.3.1.cmml"><mrow id="S4.8.p1.2.m2.3.3.1.1" xref="S4.8.p1.2.m2.3.3.1.1.cmml"><mi id="S4.8.p1.2.m2.3.3.1.1.3" xref="S4.8.p1.2.m2.3.3.1.1.3.cmml">f</mi><mo id="S4.8.p1.2.m2.3.3.1.1.2" xref="S4.8.p1.2.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.8.p1.2.m2.3.3.1.1.1.1" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.cmml"><mo id="S4.8.p1.2.m2.3.3.1.1.1.1.2" stretchy="false" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S4.8.p1.2.m2.3.3.1.1.1.1.1" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.cmml"><mi id="S4.8.p1.2.m2.3.3.1.1.1.1.1.2" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.2.cmml">v</mi><mo id="S4.8.p1.2.m2.3.3.1.1.1.1.1.1" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="S4.8.p1.2.m2.3.3.1.1.1.1.1.3" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.8.p1.2.m2.3.3.1.1.1.1.3" stretchy="false" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.2.m2.3.3.1.2" xref="S4.8.p1.2.m2.3.3.1.2.cmml">−</mo><mrow id="S4.8.p1.2.m2.3.3.1.3" xref="S4.8.p1.2.m2.3.3.1.3.cmml"><mi id="S4.8.p1.2.m2.3.3.1.3.2" xref="S4.8.p1.2.m2.3.3.1.3.2.cmml">f</mi><mo id="S4.8.p1.2.m2.3.3.1.3.1" xref="S4.8.p1.2.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="S4.8.p1.2.m2.3.3.1.3.3.2" xref="S4.8.p1.2.m2.3.3.1.3.cmml"><mo id="S4.8.p1.2.m2.3.3.1.3.3.2.1" stretchy="false" xref="S4.8.p1.2.m2.3.3.1.3.cmml">(</mo><mi id="S4.8.p1.2.m2.2.2" xref="S4.8.p1.2.m2.2.2.cmml">v</mi><mo id="S4.8.p1.2.m2.3.3.1.3.3.2.2" stretchy="false" xref="S4.8.p1.2.m2.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.2.m2.3b"><apply id="S4.8.p1.2.m2.3.3.cmml" xref="S4.8.p1.2.m2.3.3"><csymbol cd="latexml" id="S4.8.p1.2.m2.3.3.2.cmml" xref="S4.8.p1.2.m2.3.3.2">assign</csymbol><apply id="S4.8.p1.2.m2.3.3.3.cmml" xref="S4.8.p1.2.m2.3.3.3"><times id="S4.8.p1.2.m2.3.3.3.1.cmml" xref="S4.8.p1.2.m2.3.3.3.1"></times><apply id="S4.8.p1.2.m2.3.3.3.2.cmml" xref="S4.8.p1.2.m2.3.3.3.2"><ci id="S4.8.p1.2.m2.3.3.3.2.1.cmml" xref="S4.8.p1.2.m2.3.3.3.2.1">~</ci><ci id="S4.8.p1.2.m2.3.3.3.2.2.cmml" xref="S4.8.p1.2.m2.3.3.3.2.2">𝑓</ci></apply><ci id="S4.8.p1.2.m2.1.1.cmml" xref="S4.8.p1.2.m2.1.1">𝑥</ci></apply><apply id="S4.8.p1.2.m2.3.3.1.cmml" xref="S4.8.p1.2.m2.3.3.1"><minus id="S4.8.p1.2.m2.3.3.1.2.cmml" xref="S4.8.p1.2.m2.3.3.1.2"></minus><apply id="S4.8.p1.2.m2.3.3.1.1.cmml" xref="S4.8.p1.2.m2.3.3.1.1"><times id="S4.8.p1.2.m2.3.3.1.1.2.cmml" xref="S4.8.p1.2.m2.3.3.1.1.2"></times><ci id="S4.8.p1.2.m2.3.3.1.1.3.cmml" xref="S4.8.p1.2.m2.3.3.1.1.3">𝑓</ci><apply id="S4.8.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S4.8.p1.2.m2.3.3.1.1.1.1"><minus id="S4.8.p1.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.1"></minus><ci id="S4.8.p1.2.m2.3.3.1.1.1.1.1.2.cmml" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.2">𝑣</ci><ci id="S4.8.p1.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S4.8.p1.2.m2.3.3.1.1.1.1.1.3">𝑥</ci></apply></apply><apply id="S4.8.p1.2.m2.3.3.1.3.cmml" xref="S4.8.p1.2.m2.3.3.1.3"><times id="S4.8.p1.2.m2.3.3.1.3.1.cmml" xref="S4.8.p1.2.m2.3.3.1.3.1"></times><ci id="S4.8.p1.2.m2.3.3.1.3.2.cmml" xref="S4.8.p1.2.m2.3.3.1.3.2">𝑓</ci><ci id="S4.8.p1.2.m2.2.2.cmml" xref="S4.8.p1.2.m2.2.2">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.2.m2.3c">\tilde{f}(x):=f(v-x)-f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.2.m2.3d">over~ start_ARG italic_f end_ARG ( italic_x ) := italic_f ( italic_v - italic_x ) - italic_f ( italic_v )</annotation></semantics></math>, which is contained in <math alttext="\operatorname{CPL}_{p}" class="ltx_Math" display="inline" id="S4.8.p1.3.m3.1"><semantics id="S4.8.p1.3.m3.1a"><msub id="S4.8.p1.3.m3.1.1" xref="S4.8.p1.3.m3.1.1.cmml"><mi id="S4.8.p1.3.m3.1.1.2" xref="S4.8.p1.3.m3.1.1.2.cmml">CPL</mi><mi id="S4.8.p1.3.m3.1.1.3" xref="S4.8.p1.3.m3.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.8.p1.3.m3.1b"><apply id="S4.8.p1.3.m3.1.1.cmml" xref="S4.8.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.8.p1.3.m3.1.1.1.cmml" xref="S4.8.p1.3.m3.1.1">subscript</csymbol><ci id="S4.8.p1.3.m3.1.1.2.cmml" xref="S4.8.p1.3.m3.1.1.2">CPL</ci><ci id="S4.8.p1.3.m3.1.1.3.cmml" xref="S4.8.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.3.m3.1c">\operatorname{CPL}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.3.m3.1d">roman_CPL start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, as <math alttext="\tilde{f}=\sum_{n\in[p-2]}\tilde{f}^{(n)}" class="ltx_Math" display="inline" id="S4.8.p1.4.m4.2"><semantics id="S4.8.p1.4.m4.2a"><mrow id="S4.8.p1.4.m4.2.3" xref="S4.8.p1.4.m4.2.3.cmml"><mover accent="true" id="S4.8.p1.4.m4.2.3.2" xref="S4.8.p1.4.m4.2.3.2.cmml"><mi id="S4.8.p1.4.m4.2.3.2.2" xref="S4.8.p1.4.m4.2.3.2.2.cmml">f</mi><mo id="S4.8.p1.4.m4.2.3.2.1" xref="S4.8.p1.4.m4.2.3.2.1.cmml">~</mo></mover><mo id="S4.8.p1.4.m4.2.3.1" rspace="0.111em" xref="S4.8.p1.4.m4.2.3.1.cmml">=</mo><mrow id="S4.8.p1.4.m4.2.3.3" xref="S4.8.p1.4.m4.2.3.3.cmml"><msub id="S4.8.p1.4.m4.2.3.3.1" xref="S4.8.p1.4.m4.2.3.3.1.cmml"><mo id="S4.8.p1.4.m4.2.3.3.1.2" xref="S4.8.p1.4.m4.2.3.3.1.2.cmml">∑</mo><mrow id="S4.8.p1.4.m4.1.1.1" xref="S4.8.p1.4.m4.1.1.1.cmml"><mi id="S4.8.p1.4.m4.1.1.1.3" xref="S4.8.p1.4.m4.1.1.1.3.cmml">n</mi><mo id="S4.8.p1.4.m4.1.1.1.2" xref="S4.8.p1.4.m4.1.1.1.2.cmml">∈</mo><mrow id="S4.8.p1.4.m4.1.1.1.1.1" xref="S4.8.p1.4.m4.1.1.1.1.2.cmml"><mo id="S4.8.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.8.p1.4.m4.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.8.p1.4.m4.1.1.1.1.1.1" xref="S4.8.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.8.p1.4.m4.1.1.1.1.1.1.2" xref="S4.8.p1.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mo id="S4.8.p1.4.m4.1.1.1.1.1.1.1" xref="S4.8.p1.4.m4.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.8.p1.4.m4.1.1.1.1.1.1.3" xref="S4.8.p1.4.m4.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S4.8.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.8.p1.4.m4.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></msub><msup id="S4.8.p1.4.m4.2.3.3.2" xref="S4.8.p1.4.m4.2.3.3.2.cmml"><mover accent="true" id="S4.8.p1.4.m4.2.3.3.2.2" xref="S4.8.p1.4.m4.2.3.3.2.2.cmml"><mi id="S4.8.p1.4.m4.2.3.3.2.2.2" xref="S4.8.p1.4.m4.2.3.3.2.2.2.cmml">f</mi><mo id="S4.8.p1.4.m4.2.3.3.2.2.1" xref="S4.8.p1.4.m4.2.3.3.2.2.1.cmml">~</mo></mover><mrow id="S4.8.p1.4.m4.2.2.1.3" xref="S4.8.p1.4.m4.2.3.3.2.cmml"><mo id="S4.8.p1.4.m4.2.2.1.3.1" stretchy="false" xref="S4.8.p1.4.m4.2.3.3.2.cmml">(</mo><mi id="S4.8.p1.4.m4.2.2.1.1" xref="S4.8.p1.4.m4.2.2.1.1.cmml">n</mi><mo id="S4.8.p1.4.m4.2.2.1.3.2" stretchy="false" xref="S4.8.p1.4.m4.2.3.3.2.cmml">)</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.4.m4.2b"><apply id="S4.8.p1.4.m4.2.3.cmml" xref="S4.8.p1.4.m4.2.3"><eq id="S4.8.p1.4.m4.2.3.1.cmml" xref="S4.8.p1.4.m4.2.3.1"></eq><apply id="S4.8.p1.4.m4.2.3.2.cmml" xref="S4.8.p1.4.m4.2.3.2"><ci id="S4.8.p1.4.m4.2.3.2.1.cmml" xref="S4.8.p1.4.m4.2.3.2.1">~</ci><ci id="S4.8.p1.4.m4.2.3.2.2.cmml" xref="S4.8.p1.4.m4.2.3.2.2">𝑓</ci></apply><apply id="S4.8.p1.4.m4.2.3.3.cmml" xref="S4.8.p1.4.m4.2.3.3"><apply id="S4.8.p1.4.m4.2.3.3.1.cmml" xref="S4.8.p1.4.m4.2.3.3.1"><csymbol cd="ambiguous" id="S4.8.p1.4.m4.2.3.3.1.1.cmml" xref="S4.8.p1.4.m4.2.3.3.1">subscript</csymbol><sum id="S4.8.p1.4.m4.2.3.3.1.2.cmml" xref="S4.8.p1.4.m4.2.3.3.1.2"></sum><apply id="S4.8.p1.4.m4.1.1.1.cmml" xref="S4.8.p1.4.m4.1.1.1"><in id="S4.8.p1.4.m4.1.1.1.2.cmml" xref="S4.8.p1.4.m4.1.1.1.2"></in><ci id="S4.8.p1.4.m4.1.1.1.3.cmml" xref="S4.8.p1.4.m4.1.1.1.3">𝑛</ci><apply id="S4.8.p1.4.m4.1.1.1.1.2.cmml" xref="S4.8.p1.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="S4.8.p1.4.m4.1.1.1.1.2.1.cmml" xref="S4.8.p1.4.m4.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.8.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.8.p1.4.m4.1.1.1.1.1.1"><minus id="S4.8.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.8.p1.4.m4.1.1.1.1.1.1.1"></minus><ci id="S4.8.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.8.p1.4.m4.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.8.p1.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.8.p1.4.m4.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply><apply id="S4.8.p1.4.m4.2.3.3.2.cmml" xref="S4.8.p1.4.m4.2.3.3.2"><csymbol cd="ambiguous" id="S4.8.p1.4.m4.2.3.3.2.1.cmml" xref="S4.8.p1.4.m4.2.3.3.2">superscript</csymbol><apply id="S4.8.p1.4.m4.2.3.3.2.2.cmml" xref="S4.8.p1.4.m4.2.3.3.2.2"><ci id="S4.8.p1.4.m4.2.3.3.2.2.1.cmml" xref="S4.8.p1.4.m4.2.3.3.2.2.1">~</ci><ci id="S4.8.p1.4.m4.2.3.3.2.2.2.cmml" xref="S4.8.p1.4.m4.2.3.3.2.2.2">𝑓</ci></apply><ci id="S4.8.p1.4.m4.2.2.1.1.cmml" xref="S4.8.p1.4.m4.2.2.1.1">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.4.m4.2c">\tilde{f}=\sum_{n\in[p-2]}\tilde{f}^{(n)}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.4.m4.2d">over~ start_ARG italic_f end_ARG = ∑ start_POSTSUBSCRIPT italic_n ∈ [ italic_p - 2 ] end_POSTSUBSCRIPT over~ start_ARG italic_f end_ARG start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\tilde{f}^{(n)}\in\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.8.p1.5.m5.1"><semantics id="S4.8.p1.5.m5.1a"><mrow id="S4.8.p1.5.m5.1.2" xref="S4.8.p1.5.m5.1.2.cmml"><msup id="S4.8.p1.5.m5.1.2.2" xref="S4.8.p1.5.m5.1.2.2.cmml"><mover accent="true" id="S4.8.p1.5.m5.1.2.2.2" xref="S4.8.p1.5.m5.1.2.2.2.cmml"><mi id="S4.8.p1.5.m5.1.2.2.2.2" xref="S4.8.p1.5.m5.1.2.2.2.2.cmml">f</mi><mo id="S4.8.p1.5.m5.1.2.2.2.1" xref="S4.8.p1.5.m5.1.2.2.2.1.cmml">~</mo></mover><mrow id="S4.8.p1.5.m5.1.1.1.3" xref="S4.8.p1.5.m5.1.2.2.cmml"><mo id="S4.8.p1.5.m5.1.1.1.3.1" stretchy="false" xref="S4.8.p1.5.m5.1.2.2.cmml">(</mo><mi id="S4.8.p1.5.m5.1.1.1.1" xref="S4.8.p1.5.m5.1.1.1.1.cmml">n</mi><mo id="S4.8.p1.5.m5.1.1.1.3.2" stretchy="false" xref="S4.8.p1.5.m5.1.2.2.cmml">)</mo></mrow></msup><mo id="S4.8.p1.5.m5.1.2.1" xref="S4.8.p1.5.m5.1.2.1.cmml">∈</mo><msub id="S4.8.p1.5.m5.1.2.3" xref="S4.8.p1.5.m5.1.2.3.cmml"><mi id="S4.8.p1.5.m5.1.2.3.2" xref="S4.8.p1.5.m5.1.2.3.2.cmml">CPL</mi><mn id="S4.8.p1.5.m5.1.2.3.3" xref="S4.8.p1.5.m5.1.2.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.5.m5.1b"><apply id="S4.8.p1.5.m5.1.2.cmml" xref="S4.8.p1.5.m5.1.2"><in id="S4.8.p1.5.m5.1.2.1.cmml" xref="S4.8.p1.5.m5.1.2.1"></in><apply id="S4.8.p1.5.m5.1.2.2.cmml" xref="S4.8.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.8.p1.5.m5.1.2.2.1.cmml" xref="S4.8.p1.5.m5.1.2.2">superscript</csymbol><apply id="S4.8.p1.5.m5.1.2.2.2.cmml" xref="S4.8.p1.5.m5.1.2.2.2"><ci id="S4.8.p1.5.m5.1.2.2.2.1.cmml" xref="S4.8.p1.5.m5.1.2.2.2.1">~</ci><ci id="S4.8.p1.5.m5.1.2.2.2.2.cmml" xref="S4.8.p1.5.m5.1.2.2.2.2">𝑓</ci></apply><ci id="S4.8.p1.5.m5.1.1.1.1.cmml" xref="S4.8.p1.5.m5.1.1.1.1">𝑛</ci></apply><apply id="S4.8.p1.5.m5.1.2.3.cmml" xref="S4.8.p1.5.m5.1.2.3"><csymbol cd="ambiguous" id="S4.8.p1.5.m5.1.2.3.1.cmml" xref="S4.8.p1.5.m5.1.2.3">subscript</csymbol><ci id="S4.8.p1.5.m5.1.2.3.2.cmml" xref="S4.8.p1.5.m5.1.2.3.2">CPL</ci><cn id="S4.8.p1.5.m5.1.2.3.3.cmml" type="integer" xref="S4.8.p1.5.m5.1.2.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.5.m5.1c">\tilde{f}^{(n)}\in\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.5.m5.1d">over~ start_ARG italic_f end_ARG start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. This gives <math alttext="f(x)=\sum_{n\in[p-2]}\tilde{f}^{(n)}(v-x)+f(v)" class="ltx_Math" display="inline" id="S4.8.p1.6.m6.5"><semantics id="S4.8.p1.6.m6.5a"><mrow id="S4.8.p1.6.m6.5.5" xref="S4.8.p1.6.m6.5.5.cmml"><mrow id="S4.8.p1.6.m6.5.5.3" xref="S4.8.p1.6.m6.5.5.3.cmml"><mi id="S4.8.p1.6.m6.5.5.3.2" xref="S4.8.p1.6.m6.5.5.3.2.cmml">f</mi><mo id="S4.8.p1.6.m6.5.5.3.1" xref="S4.8.p1.6.m6.5.5.3.1.cmml">⁢</mo><mrow id="S4.8.p1.6.m6.5.5.3.3.2" xref="S4.8.p1.6.m6.5.5.3.cmml"><mo id="S4.8.p1.6.m6.5.5.3.3.2.1" stretchy="false" xref="S4.8.p1.6.m6.5.5.3.cmml">(</mo><mi id="S4.8.p1.6.m6.3.3" xref="S4.8.p1.6.m6.3.3.cmml">x</mi><mo id="S4.8.p1.6.m6.5.5.3.3.2.2" stretchy="false" xref="S4.8.p1.6.m6.5.5.3.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.6.m6.5.5.2" rspace="0.111em" xref="S4.8.p1.6.m6.5.5.2.cmml">=</mo><mrow id="S4.8.p1.6.m6.5.5.1" xref="S4.8.p1.6.m6.5.5.1.cmml"><mrow id="S4.8.p1.6.m6.5.5.1.1" xref="S4.8.p1.6.m6.5.5.1.1.cmml"><msub id="S4.8.p1.6.m6.5.5.1.1.2" xref="S4.8.p1.6.m6.5.5.1.1.2.cmml"><mo id="S4.8.p1.6.m6.5.5.1.1.2.2" xref="S4.8.p1.6.m6.5.5.1.1.2.2.cmml">∑</mo><mrow id="S4.8.p1.6.m6.1.1.1" xref="S4.8.p1.6.m6.1.1.1.cmml"><mi id="S4.8.p1.6.m6.1.1.1.3" xref="S4.8.p1.6.m6.1.1.1.3.cmml">n</mi><mo id="S4.8.p1.6.m6.1.1.1.2" xref="S4.8.p1.6.m6.1.1.1.2.cmml">∈</mo><mrow id="S4.8.p1.6.m6.1.1.1.1.1" xref="S4.8.p1.6.m6.1.1.1.1.2.cmml"><mo id="S4.8.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S4.8.p1.6.m6.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.8.p1.6.m6.1.1.1.1.1.1" xref="S4.8.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S4.8.p1.6.m6.1.1.1.1.1.1.2" xref="S4.8.p1.6.m6.1.1.1.1.1.1.2.cmml">p</mi><mo id="S4.8.p1.6.m6.1.1.1.1.1.1.1" xref="S4.8.p1.6.m6.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.8.p1.6.m6.1.1.1.1.1.1.3" xref="S4.8.p1.6.m6.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S4.8.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S4.8.p1.6.m6.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></msub><mrow id="S4.8.p1.6.m6.5.5.1.1.1" xref="S4.8.p1.6.m6.5.5.1.1.1.cmml"><msup 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xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.cmml"><mi id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.2" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.2.cmml">v</mi><mo id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.1" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.3" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.8.p1.6.m6.5.5.1.1.1.1.1.3" stretchy="false" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.8.p1.6.m6.5.5.1.2" xref="S4.8.p1.6.m6.5.5.1.2.cmml">+</mo><mrow id="S4.8.p1.6.m6.5.5.1.3" xref="S4.8.p1.6.m6.5.5.1.3.cmml"><mi id="S4.8.p1.6.m6.5.5.1.3.2" xref="S4.8.p1.6.m6.5.5.1.3.2.cmml">f</mi><mo id="S4.8.p1.6.m6.5.5.1.3.1" xref="S4.8.p1.6.m6.5.5.1.3.1.cmml">⁢</mo><mrow id="S4.8.p1.6.m6.5.5.1.3.3.2" xref="S4.8.p1.6.m6.5.5.1.3.cmml"><mo id="S4.8.p1.6.m6.5.5.1.3.3.2.1" stretchy="false" xref="S4.8.p1.6.m6.5.5.1.3.cmml">(</mo><mi id="S4.8.p1.6.m6.4.4" xref="S4.8.p1.6.m6.4.4.cmml">v</mi><mo id="S4.8.p1.6.m6.5.5.1.3.3.2.2" stretchy="false" xref="S4.8.p1.6.m6.5.5.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.6.m6.5b"><apply id="S4.8.p1.6.m6.5.5.cmml" xref="S4.8.p1.6.m6.5.5"><eq id="S4.8.p1.6.m6.5.5.2.cmml" xref="S4.8.p1.6.m6.5.5.2"></eq><apply id="S4.8.p1.6.m6.5.5.3.cmml" xref="S4.8.p1.6.m6.5.5.3"><times id="S4.8.p1.6.m6.5.5.3.1.cmml" xref="S4.8.p1.6.m6.5.5.3.1"></times><ci id="S4.8.p1.6.m6.5.5.3.2.cmml" xref="S4.8.p1.6.m6.5.5.3.2">𝑓</ci><ci id="S4.8.p1.6.m6.3.3.cmml" xref="S4.8.p1.6.m6.3.3">𝑥</ci></apply><apply id="S4.8.p1.6.m6.5.5.1.cmml" xref="S4.8.p1.6.m6.5.5.1"><plus id="S4.8.p1.6.m6.5.5.1.2.cmml" xref="S4.8.p1.6.m6.5.5.1.2"></plus><apply id="S4.8.p1.6.m6.5.5.1.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1"><apply id="S4.8.p1.6.m6.5.5.1.1.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.2"><csymbol cd="ambiguous" id="S4.8.p1.6.m6.5.5.1.1.2.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.2">subscript</csymbol><sum id="S4.8.p1.6.m6.5.5.1.1.2.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.2.2"></sum><apply id="S4.8.p1.6.m6.1.1.1.cmml" xref="S4.8.p1.6.m6.1.1.1"><in id="S4.8.p1.6.m6.1.1.1.2.cmml" xref="S4.8.p1.6.m6.1.1.1.2"></in><ci id="S4.8.p1.6.m6.1.1.1.3.cmml" xref="S4.8.p1.6.m6.1.1.1.3">𝑛</ci><apply id="S4.8.p1.6.m6.1.1.1.1.2.cmml" xref="S4.8.p1.6.m6.1.1.1.1.1"><csymbol cd="latexml" id="S4.8.p1.6.m6.1.1.1.1.2.1.cmml" xref="S4.8.p1.6.m6.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.8.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.8.p1.6.m6.1.1.1.1.1.1"><minus id="S4.8.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S4.8.p1.6.m6.1.1.1.1.1.1.1"></minus><ci id="S4.8.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S4.8.p1.6.m6.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.8.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.8.p1.6.m6.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply><apply id="S4.8.p1.6.m6.5.5.1.1.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1"><times id="S4.8.p1.6.m6.5.5.1.1.1.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.2"></times><apply id="S4.8.p1.6.m6.5.5.1.1.1.3.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.3"><csymbol cd="ambiguous" id="S4.8.p1.6.m6.5.5.1.1.1.3.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.3">superscript</csymbol><apply id="S4.8.p1.6.m6.5.5.1.1.1.3.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.3.2"><ci id="S4.8.p1.6.m6.5.5.1.1.1.3.2.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.3.2.1">~</ci><ci id="S4.8.p1.6.m6.5.5.1.1.1.3.2.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.3.2.2">𝑓</ci></apply><ci id="S4.8.p1.6.m6.2.2.1.1.cmml" xref="S4.8.p1.6.m6.2.2.1.1">𝑛</ci></apply><apply id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1"><minus id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.1.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.1"></minus><ci id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.2.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.2">𝑣</ci><ci id="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.3.cmml" xref="S4.8.p1.6.m6.5.5.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply><apply id="S4.8.p1.6.m6.5.5.1.3.cmml" xref="S4.8.p1.6.m6.5.5.1.3"><times id="S4.8.p1.6.m6.5.5.1.3.1.cmml" xref="S4.8.p1.6.m6.5.5.1.3.1"></times><ci id="S4.8.p1.6.m6.5.5.1.3.2.cmml" xref="S4.8.p1.6.m6.5.5.1.3.2">𝑓</ci><ci id="S4.8.p1.6.m6.4.4.cmml" xref="S4.8.p1.6.m6.4.4">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.6.m6.5c">f(x)=\sum_{n\in[p-2]}\tilde{f}^{(n)}(v-x)+f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.6.m6.5d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n ∈ [ italic_p - 2 ] end_POSTSUBSCRIPT over~ start_ARG italic_f end_ARG start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_v - italic_x ) + italic_f ( italic_v )</annotation></semantics></math> as desired.</p> </div> <div class="ltx_para" id="S4.9.p2"> <p class="ltx_p" id="S4.9.p2.2">We will proceed by induction. For <math alttext="p=3" class="ltx_Math" display="inline" id="S4.9.p2.1.m1.1"><semantics id="S4.9.p2.1.m1.1a"><mrow id="S4.9.p2.1.m1.1.1" xref="S4.9.p2.1.m1.1.1.cmml"><mi id="S4.9.p2.1.m1.1.1.2" xref="S4.9.p2.1.m1.1.1.2.cmml">p</mi><mo id="S4.9.p2.1.m1.1.1.1" xref="S4.9.p2.1.m1.1.1.1.cmml">=</mo><mn id="S4.9.p2.1.m1.1.1.3" xref="S4.9.p2.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.1.m1.1b"><apply id="S4.9.p2.1.m1.1.1.cmml" xref="S4.9.p2.1.m1.1.1"><eq id="S4.9.p2.1.m1.1.1.1.cmml" xref="S4.9.p2.1.m1.1.1.1"></eq><ci id="S4.9.p2.1.m1.1.1.2.cmml" xref="S4.9.p2.1.m1.1.1.2">𝑝</ci><cn id="S4.9.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.9.p2.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.1.m1.1c">p=3</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.1.m1.1d">italic_p = 3</annotation></semantics></math>, the statement is trivial with <math alttext="f^{(1)}:=f" class="ltx_Math" display="inline" id="S4.9.p2.2.m2.1"><semantics id="S4.9.p2.2.m2.1a"><mrow id="S4.9.p2.2.m2.1.2" xref="S4.9.p2.2.m2.1.2.cmml"><msup id="S4.9.p2.2.m2.1.2.2" xref="S4.9.p2.2.m2.1.2.2.cmml"><mi id="S4.9.p2.2.m2.1.2.2.2" xref="S4.9.p2.2.m2.1.2.2.2.cmml">f</mi><mrow id="S4.9.p2.2.m2.1.1.1.3" xref="S4.9.p2.2.m2.1.2.2.cmml"><mo id="S4.9.p2.2.m2.1.1.1.3.1" stretchy="false" xref="S4.9.p2.2.m2.1.2.2.cmml">(</mo><mn id="S4.9.p2.2.m2.1.1.1.1" xref="S4.9.p2.2.m2.1.1.1.1.cmml">1</mn><mo id="S4.9.p2.2.m2.1.1.1.3.2" stretchy="false" xref="S4.9.p2.2.m2.1.2.2.cmml">)</mo></mrow></msup><mo id="S4.9.p2.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S4.9.p2.2.m2.1.2.1.cmml">:=</mo><mi id="S4.9.p2.2.m2.1.2.3" xref="S4.9.p2.2.m2.1.2.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.2.m2.1b"><apply id="S4.9.p2.2.m2.1.2.cmml" xref="S4.9.p2.2.m2.1.2"><csymbol cd="latexml" id="S4.9.p2.2.m2.1.2.1.cmml" xref="S4.9.p2.2.m2.1.2.1">assign</csymbol><apply id="S4.9.p2.2.m2.1.2.2.cmml" xref="S4.9.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.9.p2.2.m2.1.2.2.1.cmml" xref="S4.9.p2.2.m2.1.2.2">superscript</csymbol><ci id="S4.9.p2.2.m2.1.2.2.2.cmml" xref="S4.9.p2.2.m2.1.2.2.2">𝑓</ci><cn id="S4.9.p2.2.m2.1.1.1.1.cmml" type="integer" xref="S4.9.p2.2.m2.1.1.1.1">1</cn></apply><ci id="S4.9.p2.2.m2.1.2.3.cmml" xref="S4.9.p2.2.m2.1.2.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.2.m2.1c">f^{(1)}:=f</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.2.m2.1d">italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT := italic_f</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.10.p3"> <p class="ltx_p" id="S4.10.p3.4">Let <math alttext="p\geq 4" class="ltx_Math" display="inline" id="S4.10.p3.1.m1.1"><semantics id="S4.10.p3.1.m1.1a"><mrow id="S4.10.p3.1.m1.1.1" xref="S4.10.p3.1.m1.1.1.cmml"><mi id="S4.10.p3.1.m1.1.1.2" xref="S4.10.p3.1.m1.1.1.2.cmml">p</mi><mo id="S4.10.p3.1.m1.1.1.1" xref="S4.10.p3.1.m1.1.1.1.cmml">≥</mo><mn id="S4.10.p3.1.m1.1.1.3" xref="S4.10.p3.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.1.m1.1b"><apply id="S4.10.p3.1.m1.1.1.cmml" xref="S4.10.p3.1.m1.1.1"><geq id="S4.10.p3.1.m1.1.1.1.cmml" xref="S4.10.p3.1.m1.1.1.1"></geq><ci id="S4.10.p3.1.m1.1.1.2.cmml" xref="S4.10.p3.1.m1.1.1.2">𝑝</ci><cn id="S4.10.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.10.p3.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.1.m1.1c">p\geq 4</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.1.m1.1d">italic_p ≥ 4</annotation></semantics></math>, and assume that the statement is proven for <math alttext="p-1" class="ltx_Math" display="inline" id="S4.10.p3.2.m2.1"><semantics id="S4.10.p3.2.m2.1a"><mrow id="S4.10.p3.2.m2.1.1" xref="S4.10.p3.2.m2.1.1.cmml"><mi id="S4.10.p3.2.m2.1.1.2" xref="S4.10.p3.2.m2.1.1.2.cmml">p</mi><mo id="S4.10.p3.2.m2.1.1.1" xref="S4.10.p3.2.m2.1.1.1.cmml">−</mo><mn id="S4.10.p3.2.m2.1.1.3" xref="S4.10.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.2.m2.1b"><apply id="S4.10.p3.2.m2.1.1.cmml" xref="S4.10.p3.2.m2.1.1"><minus id="S4.10.p3.2.m2.1.1.1.cmml" xref="S4.10.p3.2.m2.1.1.1"></minus><ci id="S4.10.p3.2.m2.1.1.2.cmml" xref="S4.10.p3.2.m2.1.1.2">𝑝</ci><cn id="S4.10.p3.2.m2.1.1.3.cmml" type="integer" xref="S4.10.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.2.m2.1c">p-1</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.2.m2.1d">italic_p - 1</annotation></semantics></math>. First, assume that there are two adjacent pieces that enclose an angle smaller than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.10.p3.3.m3.1"><semantics id="S4.10.p3.3.m3.1a"><mi id="S4.10.p3.3.m3.1.1" xref="S4.10.p3.3.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.10.p3.3.m3.1b"><ci id="S4.10.p3.3.m3.1.1.cmml" xref="S4.10.p3.3.m3.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.3.m3.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.3.m3.1d">italic_π</annotation></semantics></math>. This is always the case for <math alttext="p>4" class="ltx_Math" display="inline" id="S4.10.p3.4.m4.1"><semantics id="S4.10.p3.4.m4.1a"><mrow id="S4.10.p3.4.m4.1.1" xref="S4.10.p3.4.m4.1.1.cmml"><mi id="S4.10.p3.4.m4.1.1.2" xref="S4.10.p3.4.m4.1.1.2.cmml">p</mi><mo id="S4.10.p3.4.m4.1.1.1" xref="S4.10.p3.4.m4.1.1.1.cmml">></mo><mn id="S4.10.p3.4.m4.1.1.3" xref="S4.10.p3.4.m4.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.4.m4.1b"><apply id="S4.10.p3.4.m4.1.1.cmml" xref="S4.10.p3.4.m4.1.1"><gt id="S4.10.p3.4.m4.1.1.1.cmml" xref="S4.10.p3.4.m4.1.1.1"></gt><ci id="S4.10.p3.4.m4.1.1.2.cmml" xref="S4.10.p3.4.m4.1.1.2">𝑝</ci><cn id="S4.10.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.10.p3.4.m4.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.4.m4.1c">p>4</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.4.m4.1d">italic_p > 4</annotation></semantics></math>. Then, we can apply <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem3" title="Lemma 4.3. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.3</span></a> and are done by induction.</p> </div> <div class="ltx_para" id="S4.11.p4"> <p class="ltx_p" id="S4.11.p4.10">Now, consider the remaining case where <math alttext="p=4" class="ltx_Math" display="inline" id="S4.11.p4.1.m1.1"><semantics id="S4.11.p4.1.m1.1a"><mrow id="S4.11.p4.1.m1.1.1" xref="S4.11.p4.1.m1.1.1.cmml"><mi id="S4.11.p4.1.m1.1.1.2" xref="S4.11.p4.1.m1.1.1.2.cmml">p</mi><mo id="S4.11.p4.1.m1.1.1.1" xref="S4.11.p4.1.m1.1.1.1.cmml">=</mo><mn id="S4.11.p4.1.m1.1.1.3" xref="S4.11.p4.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.1.m1.1b"><apply id="S4.11.p4.1.m1.1.1.cmml" xref="S4.11.p4.1.m1.1.1"><eq id="S4.11.p4.1.m1.1.1.1.cmml" xref="S4.11.p4.1.m1.1.1.1"></eq><ci id="S4.11.p4.1.m1.1.1.2.cmml" xref="S4.11.p4.1.m1.1.1.2">𝑝</ci><cn id="S4.11.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.11.p4.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.1.m1.1c">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.1.m1.1d">italic_p = 4</annotation></semantics></math>, and <math alttext="f" class="ltx_Math" display="inline" id="S4.11.p4.2.m2.1"><semantics id="S4.11.p4.2.m2.1a"><mi id="S4.11.p4.2.m2.1.1" xref="S4.11.p4.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.11.p4.2.m2.1b"><ci id="S4.11.p4.2.m2.1.1.cmml" xref="S4.11.p4.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.2.m2.1d">italic_f</annotation></semantics></math> is such that no two pieces form an angle smaller than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.11.p4.3.m3.1"><semantics id="S4.11.p4.3.m3.1a"><mi id="S4.11.p4.3.m3.1.1" xref="S4.11.p4.3.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.11.p4.3.m3.1b"><ci id="S4.11.p4.3.m3.1.1.cmml" xref="S4.11.p4.3.m3.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.3.m3.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.3.m3.1d">italic_π</annotation></semantics></math>. Then, the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S4.11.p4.4.m4.1"><semantics id="S4.11.p4.4.m4.1a"><mi id="S4.11.p4.4.m4.1.1" xref="S4.11.p4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.11.p4.4.m4.1b"><ci id="S4.11.p4.4.m4.1.1.cmml" xref="S4.11.p4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.4.m4.1d">italic_f</annotation></semantics></math> are defined by two intersecting lines. Let <math alttext="P_{1},P_{2},P_{3},P_{4}" class="ltx_Math" display="inline" id="S4.11.p4.5.m5.4"><semantics id="S4.11.p4.5.m5.4a"><mrow id="S4.11.p4.5.m5.4.4.4" xref="S4.11.p4.5.m5.4.4.5.cmml"><msub id="S4.11.p4.5.m5.1.1.1.1" xref="S4.11.p4.5.m5.1.1.1.1.cmml"><mi id="S4.11.p4.5.m5.1.1.1.1.2" xref="S4.11.p4.5.m5.1.1.1.1.2.cmml">P</mi><mn id="S4.11.p4.5.m5.1.1.1.1.3" xref="S4.11.p4.5.m5.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.11.p4.5.m5.4.4.4.5" xref="S4.11.p4.5.m5.4.4.5.cmml">,</mo><msub id="S4.11.p4.5.m5.2.2.2.2" xref="S4.11.p4.5.m5.2.2.2.2.cmml"><mi id="S4.11.p4.5.m5.2.2.2.2.2" xref="S4.11.p4.5.m5.2.2.2.2.2.cmml">P</mi><mn id="S4.11.p4.5.m5.2.2.2.2.3" xref="S4.11.p4.5.m5.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.11.p4.5.m5.4.4.4.6" xref="S4.11.p4.5.m5.4.4.5.cmml">,</mo><msub id="S4.11.p4.5.m5.3.3.3.3" xref="S4.11.p4.5.m5.3.3.3.3.cmml"><mi id="S4.11.p4.5.m5.3.3.3.3.2" xref="S4.11.p4.5.m5.3.3.3.3.2.cmml">P</mi><mn id="S4.11.p4.5.m5.3.3.3.3.3" xref="S4.11.p4.5.m5.3.3.3.3.3.cmml">3</mn></msub><mo id="S4.11.p4.5.m5.4.4.4.7" xref="S4.11.p4.5.m5.4.4.5.cmml">,</mo><msub id="S4.11.p4.5.m5.4.4.4.4" xref="S4.11.p4.5.m5.4.4.4.4.cmml"><mi id="S4.11.p4.5.m5.4.4.4.4.2" xref="S4.11.p4.5.m5.4.4.4.4.2.cmml">P</mi><mn id="S4.11.p4.5.m5.4.4.4.4.3" xref="S4.11.p4.5.m5.4.4.4.4.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.5.m5.4b"><list id="S4.11.p4.5.m5.4.4.5.cmml" xref="S4.11.p4.5.m5.4.4.4"><apply id="S4.11.p4.5.m5.1.1.1.1.cmml" xref="S4.11.p4.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.1.1.1.1.1.cmml" xref="S4.11.p4.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.11.p4.5.m5.1.1.1.1.2.cmml" xref="S4.11.p4.5.m5.1.1.1.1.2">𝑃</ci><cn id="S4.11.p4.5.m5.1.1.1.1.3.cmml" type="integer" xref="S4.11.p4.5.m5.1.1.1.1.3">1</cn></apply><apply id="S4.11.p4.5.m5.2.2.2.2.cmml" xref="S4.11.p4.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.2.2.2.2.1.cmml" xref="S4.11.p4.5.m5.2.2.2.2">subscript</csymbol><ci id="S4.11.p4.5.m5.2.2.2.2.2.cmml" xref="S4.11.p4.5.m5.2.2.2.2.2">𝑃</ci><cn id="S4.11.p4.5.m5.2.2.2.2.3.cmml" type="integer" xref="S4.11.p4.5.m5.2.2.2.2.3">2</cn></apply><apply id="S4.11.p4.5.m5.3.3.3.3.cmml" xref="S4.11.p4.5.m5.3.3.3.3"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.3.3.3.3.1.cmml" xref="S4.11.p4.5.m5.3.3.3.3">subscript</csymbol><ci id="S4.11.p4.5.m5.3.3.3.3.2.cmml" xref="S4.11.p4.5.m5.3.3.3.3.2">𝑃</ci><cn id="S4.11.p4.5.m5.3.3.3.3.3.cmml" type="integer" xref="S4.11.p4.5.m5.3.3.3.3.3">3</cn></apply><apply id="S4.11.p4.5.m5.4.4.4.4.cmml" xref="S4.11.p4.5.m5.4.4.4.4"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.4.4.4.4.1.cmml" xref="S4.11.p4.5.m5.4.4.4.4">subscript</csymbol><ci id="S4.11.p4.5.m5.4.4.4.4.2.cmml" xref="S4.11.p4.5.m5.4.4.4.4.2">𝑃</ci><cn id="S4.11.p4.5.m5.4.4.4.4.3.cmml" type="integer" xref="S4.11.p4.5.m5.4.4.4.4.3">4</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.5.m5.4c">P_{1},P_{2},P_{3},P_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.5.m5.4d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> be the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S4.11.p4.6.m6.1"><semantics id="S4.11.p4.6.m6.1a"><mi id="S4.11.p4.6.m6.1.1" xref="S4.11.p4.6.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.11.p4.6.m6.1b"><ci id="S4.11.p4.6.m6.1.1.cmml" xref="S4.11.p4.6.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.6.m6.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.6.m6.1d">italic_f</annotation></semantics></math> in the order depicted in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.F11" title="Figure 11 ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 11</span></a>, and <math alttext="a\in(P_{1}\cap P_{2})\setminus\{0\}" class="ltx_Math" display="inline" id="S4.11.p4.7.m7.2"><semantics id="S4.11.p4.7.m7.2a"><mrow id="S4.11.p4.7.m7.2.2" xref="S4.11.p4.7.m7.2.2.cmml"><mi id="S4.11.p4.7.m7.2.2.3" xref="S4.11.p4.7.m7.2.2.3.cmml">a</mi><mo id="S4.11.p4.7.m7.2.2.2" xref="S4.11.p4.7.m7.2.2.2.cmml">∈</mo><mrow id="S4.11.p4.7.m7.2.2.1" xref="S4.11.p4.7.m7.2.2.1.cmml"><mrow id="S4.11.p4.7.m7.2.2.1.1.1" xref="S4.11.p4.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.11.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.11.p4.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.11.p4.7.m7.2.2.1.1.1.1" xref="S4.11.p4.7.m7.2.2.1.1.1.1.cmml"><msub id="S4.11.p4.7.m7.2.2.1.1.1.1.2" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2.cmml"><mi id="S4.11.p4.7.m7.2.2.1.1.1.1.2.2" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2.2.cmml">P</mi><mn id="S4.11.p4.7.m7.2.2.1.1.1.1.2.3" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.11.p4.7.m7.2.2.1.1.1.1.1" xref="S4.11.p4.7.m7.2.2.1.1.1.1.1.cmml">∩</mo><msub id="S4.11.p4.7.m7.2.2.1.1.1.1.3" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3.cmml"><mi id="S4.11.p4.7.m7.2.2.1.1.1.1.3.2" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3.2.cmml">P</mi><mn id="S4.11.p4.7.m7.2.2.1.1.1.1.3.3" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S4.11.p4.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.11.p4.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S4.11.p4.7.m7.2.2.1.2" xref="S4.11.p4.7.m7.2.2.1.2.cmml">∖</mo><mrow id="S4.11.p4.7.m7.2.2.1.3.2" xref="S4.11.p4.7.m7.2.2.1.3.1.cmml"><mo id="S4.11.p4.7.m7.2.2.1.3.2.1" stretchy="false" xref="S4.11.p4.7.m7.2.2.1.3.1.cmml">{</mo><mn id="S4.11.p4.7.m7.1.1" xref="S4.11.p4.7.m7.1.1.cmml">0</mn><mo id="S4.11.p4.7.m7.2.2.1.3.2.2" stretchy="false" xref="S4.11.p4.7.m7.2.2.1.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.7.m7.2b"><apply id="S4.11.p4.7.m7.2.2.cmml" xref="S4.11.p4.7.m7.2.2"><in id="S4.11.p4.7.m7.2.2.2.cmml" xref="S4.11.p4.7.m7.2.2.2"></in><ci id="S4.11.p4.7.m7.2.2.3.cmml" xref="S4.11.p4.7.m7.2.2.3">𝑎</ci><apply id="S4.11.p4.7.m7.2.2.1.cmml" xref="S4.11.p4.7.m7.2.2.1"><setdiff id="S4.11.p4.7.m7.2.2.1.2.cmml" xref="S4.11.p4.7.m7.2.2.1.2"></setdiff><apply id="S4.11.p4.7.m7.2.2.1.1.1.1.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1"><intersect id="S4.11.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.1"></intersect><apply id="S4.11.p4.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.11.p4.7.m7.2.2.1.1.1.1.2.1.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2">subscript</csymbol><ci id="S4.11.p4.7.m7.2.2.1.1.1.1.2.2.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2.2">𝑃</ci><cn id="S4.11.p4.7.m7.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S4.11.p4.7.m7.2.2.1.1.1.1.2.3">1</cn></apply><apply id="S4.11.p4.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.7.m7.2.2.1.1.1.1.3.1.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.11.p4.7.m7.2.2.1.1.1.1.3.2.cmml" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3.2">𝑃</ci><cn id="S4.11.p4.7.m7.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S4.11.p4.7.m7.2.2.1.1.1.1.3.3">2</cn></apply></apply><set id="S4.11.p4.7.m7.2.2.1.3.1.cmml" xref="S4.11.p4.7.m7.2.2.1.3.2"><cn id="S4.11.p4.7.m7.1.1.cmml" type="integer" xref="S4.11.p4.7.m7.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.7.m7.2c">a\in(P_{1}\cap P_{2})\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.7.m7.2d">italic_a ∈ ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∖ { 0 }</annotation></semantics></math>. Since <math alttext="f_{1}=f_{2}" class="ltx_Math" display="inline" id="S4.11.p4.8.m8.1"><semantics id="S4.11.p4.8.m8.1a"><mrow id="S4.11.p4.8.m8.1.1" xref="S4.11.p4.8.m8.1.1.cmml"><msub id="S4.11.p4.8.m8.1.1.2" xref="S4.11.p4.8.m8.1.1.2.cmml"><mi id="S4.11.p4.8.m8.1.1.2.2" xref="S4.11.p4.8.m8.1.1.2.2.cmml">f</mi><mn id="S4.11.p4.8.m8.1.1.2.3" xref="S4.11.p4.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S4.11.p4.8.m8.1.1.1" xref="S4.11.p4.8.m8.1.1.1.cmml">=</mo><msub id="S4.11.p4.8.m8.1.1.3" xref="S4.11.p4.8.m8.1.1.3.cmml"><mi id="S4.11.p4.8.m8.1.1.3.2" xref="S4.11.p4.8.m8.1.1.3.2.cmml">f</mi><mn id="S4.11.p4.8.m8.1.1.3.3" xref="S4.11.p4.8.m8.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.8.m8.1b"><apply id="S4.11.p4.8.m8.1.1.cmml" xref="S4.11.p4.8.m8.1.1"><eq id="S4.11.p4.8.m8.1.1.1.cmml" xref="S4.11.p4.8.m8.1.1.1"></eq><apply id="S4.11.p4.8.m8.1.1.2.cmml" xref="S4.11.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.11.p4.8.m8.1.1.2.1.cmml" xref="S4.11.p4.8.m8.1.1.2">subscript</csymbol><ci id="S4.11.p4.8.m8.1.1.2.2.cmml" xref="S4.11.p4.8.m8.1.1.2.2">𝑓</ci><cn id="S4.11.p4.8.m8.1.1.2.3.cmml" type="integer" xref="S4.11.p4.8.m8.1.1.2.3">1</cn></apply><apply id="S4.11.p4.8.m8.1.1.3.cmml" xref="S4.11.p4.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.8.m8.1.1.3.1.cmml" xref="S4.11.p4.8.m8.1.1.3">subscript</csymbol><ci id="S4.11.p4.8.m8.1.1.3.2.cmml" xref="S4.11.p4.8.m8.1.1.3.2">𝑓</ci><cn id="S4.11.p4.8.m8.1.1.3.3.cmml" type="integer" xref="S4.11.p4.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.8.m8.1c">f_{1}=f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.8.m8.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\operatorname*{span}(a)" class="ltx_Math" display="inline" id="S4.11.p4.9.m9.2"><semantics id="S4.11.p4.9.m9.2a"><mrow id="S4.11.p4.9.m9.2.3.2" xref="S4.11.p4.9.m9.2.3.1.cmml"><mo id="S4.11.p4.9.m9.1.1" rspace="0em" xref="S4.11.p4.9.m9.1.1.cmml">span</mo><mrow id="S4.11.p4.9.m9.2.3.2.1" xref="S4.11.p4.9.m9.2.3.1.cmml"><mo id="S4.11.p4.9.m9.2.3.2.1.1" stretchy="false" xref="S4.11.p4.9.m9.2.3.1.cmml">(</mo><mi id="S4.11.p4.9.m9.2.2" xref="S4.11.p4.9.m9.2.2.cmml">a</mi><mo id="S4.11.p4.9.m9.2.3.2.1.2" stretchy="false" xref="S4.11.p4.9.m9.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.9.m9.2b"><apply id="S4.11.p4.9.m9.2.3.1.cmml" xref="S4.11.p4.9.m9.2.3.2"><ci id="S4.11.p4.9.m9.1.1.cmml" xref="S4.11.p4.9.m9.1.1">span</ci><ci id="S4.11.p4.9.m9.2.2.cmml" xref="S4.11.p4.9.m9.2.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.9.m9.2c">\operatorname*{span}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.9.m9.2d">roman_span ( italic_a )</annotation></semantics></math> and <math alttext="(P_{1}\cup P_{4})\cap(P_{2}\cup P_{3})=\operatorname*{span}(a)" class="ltx_Math" display="inline" id="S4.11.p4.10.m10.4"><semantics id="S4.11.p4.10.m10.4a"><mrow id="S4.11.p4.10.m10.4.4" xref="S4.11.p4.10.m10.4.4.cmml"><mrow id="S4.11.p4.10.m10.4.4.2" xref="S4.11.p4.10.m10.4.4.2.cmml"><mrow id="S4.11.p4.10.m10.3.3.1.1.1" xref="S4.11.p4.10.m10.3.3.1.1.1.1.cmml"><mo id="S4.11.p4.10.m10.3.3.1.1.1.2" stretchy="false" xref="S4.11.p4.10.m10.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.11.p4.10.m10.3.3.1.1.1.1" xref="S4.11.p4.10.m10.3.3.1.1.1.1.cmml"><msub id="S4.11.p4.10.m10.3.3.1.1.1.1.2" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2.cmml"><mi id="S4.11.p4.10.m10.3.3.1.1.1.1.2.2" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2.2.cmml">P</mi><mn id="S4.11.p4.10.m10.3.3.1.1.1.1.2.3" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.11.p4.10.m10.3.3.1.1.1.1.1" xref="S4.11.p4.10.m10.3.3.1.1.1.1.1.cmml">∪</mo><msub id="S4.11.p4.10.m10.3.3.1.1.1.1.3" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3.cmml"><mi id="S4.11.p4.10.m10.3.3.1.1.1.1.3.2" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3.2.cmml">P</mi><mn id="S4.11.p4.10.m10.3.3.1.1.1.1.3.3" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3.3.cmml">4</mn></msub></mrow><mo id="S4.11.p4.10.m10.3.3.1.1.1.3" stretchy="false" xref="S4.11.p4.10.m10.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.11.p4.10.m10.4.4.2.3" xref="S4.11.p4.10.m10.4.4.2.3.cmml">∩</mo><mrow id="S4.11.p4.10.m10.4.4.2.2.1" xref="S4.11.p4.10.m10.4.4.2.2.1.1.cmml"><mo id="S4.11.p4.10.m10.4.4.2.2.1.2" stretchy="false" xref="S4.11.p4.10.m10.4.4.2.2.1.1.cmml">(</mo><mrow id="S4.11.p4.10.m10.4.4.2.2.1.1" xref="S4.11.p4.10.m10.4.4.2.2.1.1.cmml"><msub id="S4.11.p4.10.m10.4.4.2.2.1.1.2" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2.cmml"><mi id="S4.11.p4.10.m10.4.4.2.2.1.1.2.2" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2.2.cmml">P</mi><mn id="S4.11.p4.10.m10.4.4.2.2.1.1.2.3" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2.3.cmml">2</mn></msub><mo id="S4.11.p4.10.m10.4.4.2.2.1.1.1" xref="S4.11.p4.10.m10.4.4.2.2.1.1.1.cmml">∪</mo><msub id="S4.11.p4.10.m10.4.4.2.2.1.1.3" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3.cmml"><mi id="S4.11.p4.10.m10.4.4.2.2.1.1.3.2" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3.2.cmml">P</mi><mn id="S4.11.p4.10.m10.4.4.2.2.1.1.3.3" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3.3.cmml">3</mn></msub></mrow><mo id="S4.11.p4.10.m10.4.4.2.2.1.3" stretchy="false" xref="S4.11.p4.10.m10.4.4.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.11.p4.10.m10.4.4.3" rspace="0.1389em" xref="S4.11.p4.10.m10.4.4.3.cmml">=</mo><mrow id="S4.11.p4.10.m10.4.4.4.2" xref="S4.11.p4.10.m10.4.4.4.1.cmml"><mo id="S4.11.p4.10.m10.1.1" lspace="0.1389em" rspace="0em" xref="S4.11.p4.10.m10.1.1.cmml">span</mo><mrow id="S4.11.p4.10.m10.4.4.4.2.1" xref="S4.11.p4.10.m10.4.4.4.1.cmml"><mo id="S4.11.p4.10.m10.4.4.4.2.1.1" stretchy="false" xref="S4.11.p4.10.m10.4.4.4.1.cmml">(</mo><mi id="S4.11.p4.10.m10.2.2" xref="S4.11.p4.10.m10.2.2.cmml">a</mi><mo id="S4.11.p4.10.m10.4.4.4.2.1.2" stretchy="false" xref="S4.11.p4.10.m10.4.4.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.10.m10.4b"><apply id="S4.11.p4.10.m10.4.4.cmml" xref="S4.11.p4.10.m10.4.4"><eq id="S4.11.p4.10.m10.4.4.3.cmml" xref="S4.11.p4.10.m10.4.4.3"></eq><apply id="S4.11.p4.10.m10.4.4.2.cmml" xref="S4.11.p4.10.m10.4.4.2"><intersect id="S4.11.p4.10.m10.4.4.2.3.cmml" xref="S4.11.p4.10.m10.4.4.2.3"></intersect><apply id="S4.11.p4.10.m10.3.3.1.1.1.1.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1"><union id="S4.11.p4.10.m10.3.3.1.1.1.1.1.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.1"></union><apply id="S4.11.p4.10.m10.3.3.1.1.1.1.2.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.11.p4.10.m10.3.3.1.1.1.1.2.1.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2">subscript</csymbol><ci id="S4.11.p4.10.m10.3.3.1.1.1.1.2.2.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2.2">𝑃</ci><cn id="S4.11.p4.10.m10.3.3.1.1.1.1.2.3.cmml" type="integer" xref="S4.11.p4.10.m10.3.3.1.1.1.1.2.3">1</cn></apply><apply id="S4.11.p4.10.m10.3.3.1.1.1.1.3.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.10.m10.3.3.1.1.1.1.3.1.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3">subscript</csymbol><ci id="S4.11.p4.10.m10.3.3.1.1.1.1.3.2.cmml" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3.2">𝑃</ci><cn id="S4.11.p4.10.m10.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S4.11.p4.10.m10.3.3.1.1.1.1.3.3">4</cn></apply></apply><apply id="S4.11.p4.10.m10.4.4.2.2.1.1.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1"><union id="S4.11.p4.10.m10.4.4.2.2.1.1.1.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.1"></union><apply id="S4.11.p4.10.m10.4.4.2.2.1.1.2.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.11.p4.10.m10.4.4.2.2.1.1.2.1.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2">subscript</csymbol><ci id="S4.11.p4.10.m10.4.4.2.2.1.1.2.2.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2.2">𝑃</ci><cn id="S4.11.p4.10.m10.4.4.2.2.1.1.2.3.cmml" type="integer" xref="S4.11.p4.10.m10.4.4.2.2.1.1.2.3">2</cn></apply><apply id="S4.11.p4.10.m10.4.4.2.2.1.1.3.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.10.m10.4.4.2.2.1.1.3.1.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3">subscript</csymbol><ci id="S4.11.p4.10.m10.4.4.2.2.1.1.3.2.cmml" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3.2">𝑃</ci><cn id="S4.11.p4.10.m10.4.4.2.2.1.1.3.3.cmml" type="integer" xref="S4.11.p4.10.m10.4.4.2.2.1.1.3.3">3</cn></apply></apply></apply><apply id="S4.11.p4.10.m10.4.4.4.1.cmml" xref="S4.11.p4.10.m10.4.4.4.2"><ci id="S4.11.p4.10.m10.1.1.cmml" xref="S4.11.p4.10.m10.1.1">span</ci><ci id="S4.11.p4.10.m10.2.2.cmml" xref="S4.11.p4.10.m10.2.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.10.m10.4c">(P_{1}\cup P_{4})\cap(P_{2}\cup P_{3})=\operatorname*{span}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.10.m10.4d">( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∩ ( italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = roman_span ( italic_a )</annotation></semantics></math>, the function defined by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex48"> <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^{(1)}(x):=\begin{cases}f_{1},\quad&\text{in }P_{1}\cup P_{4}\\ f_{2},\quad&\text{in }P_{2}\cup P_{3}\end{cases}" class="ltx_Math" display="block" id="S4.Ex48.m1.6"><semantics id="S4.Ex48.m1.6a"><mrow id="S4.Ex48.m1.6.7" xref="S4.Ex48.m1.6.7.cmml"><mrow id="S4.Ex48.m1.6.7.2" xref="S4.Ex48.m1.6.7.2.cmml"><msup id="S4.Ex48.m1.6.7.2.2" xref="S4.Ex48.m1.6.7.2.2.cmml"><mi id="S4.Ex48.m1.6.7.2.2.2" xref="S4.Ex48.m1.6.7.2.2.2.cmml">f</mi><mrow id="S4.Ex48.m1.5.5.1.3" xref="S4.Ex48.m1.6.7.2.2.cmml"><mo id="S4.Ex48.m1.5.5.1.3.1" stretchy="false" xref="S4.Ex48.m1.6.7.2.2.cmml">(</mo><mn id="S4.Ex48.m1.5.5.1.1" xref="S4.Ex48.m1.5.5.1.1.cmml">1</mn><mo id="S4.Ex48.m1.5.5.1.3.2" stretchy="false" xref="S4.Ex48.m1.6.7.2.2.cmml">)</mo></mrow></msup><mo id="S4.Ex48.m1.6.7.2.1" xref="S4.Ex48.m1.6.7.2.1.cmml">⁢</mo><mrow id="S4.Ex48.m1.6.7.2.3.2" xref="S4.Ex48.m1.6.7.2.cmml"><mo id="S4.Ex48.m1.6.7.2.3.2.1" stretchy="false" xref="S4.Ex48.m1.6.7.2.cmml">(</mo><mi id="S4.Ex48.m1.6.6" xref="S4.Ex48.m1.6.6.cmml">x</mi><mo id="S4.Ex48.m1.6.7.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.Ex48.m1.6.7.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex48.m1.6.7.1" rspace="0.278em" xref="S4.Ex48.m1.6.7.1.cmml">:=</mo><mrow id="S4.Ex48.m1.4.4" xref="S4.Ex48.m1.6.7.3.1.cmml"><mo id="S4.Ex48.m1.4.4.5" xref="S4.Ex48.m1.6.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex48.m1.4.4.4" rowspacing="0pt" xref="S4.Ex48.m1.6.7.3.1.cmml"><mtr id="S4.Ex48.m1.4.4.4a" xref="S4.Ex48.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex48.m1.4.4.4b" xref="S4.Ex48.m1.6.7.3.1.cmml"><mrow id="S4.Ex48.m1.1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex48.m1.1.1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex48.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.2.cmml">f</mi><mn id="S4.Ex48.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex48.m1.1.1.1.1.1.1.1.2" xref="S4.Ex48.m1.1.1.1.1.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex48.m1.4.4.4c" xref="S4.Ex48.m1.6.7.3.1.cmml"><mrow id="S4.Ex48.m1.2.2.2.2.2.1" xref="S4.Ex48.m1.2.2.2.2.2.1.cmml"><mrow id="S4.Ex48.m1.2.2.2.2.2.1.2" xref="S4.Ex48.m1.2.2.2.2.2.1.2.cmml"><mtext id="S4.Ex48.m1.2.2.2.2.2.1.2.2" xref="S4.Ex48.m1.2.2.2.2.2.1.2.2a.cmml">in </mtext><mo id="S4.Ex48.m1.2.2.2.2.2.1.2.1" xref="S4.Ex48.m1.2.2.2.2.2.1.2.1.cmml">⁢</mo><msub id="S4.Ex48.m1.2.2.2.2.2.1.2.3" 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xref="S4.Ex48.m1.4.4.4.4.2.1.2.3.3">2</cn></apply></apply><apply id="S4.Ex48.m1.4.4.4.4.2.1.3.cmml" xref="S4.Ex48.m1.4.4.4.4.2.1.3"><csymbol cd="ambiguous" id="S4.Ex48.m1.4.4.4.4.2.1.3.1.cmml" xref="S4.Ex48.m1.4.4.4.4.2.1.3">subscript</csymbol><ci id="S4.Ex48.m1.4.4.4.4.2.1.3.2.cmml" xref="S4.Ex48.m1.4.4.4.4.2.1.3.2">𝑃</ci><cn id="S4.Ex48.m1.4.4.4.4.2.1.3.3.cmml" type="integer" xref="S4.Ex48.m1.4.4.4.4.2.1.3.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex48.m1.6c">f^{(1)}(x):=\begin{cases}f_{1},\quad&\text{in }P_{1}\cup P_{4}\\ f_{2},\quad&\text{in }P_{2}\cup P_{3}\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex48.m1.6d">italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x ) := { start_ROW start_CELL italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL start_CELL in italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , end_CELL start_CELL in italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 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="S4.11.p4.11">is a continuous function defined on two pieces that intersect in a line. In particular <math alttext="f^{(1)}\in\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.11.p4.11.m1.1"><semantics id="S4.11.p4.11.m1.1a"><mrow id="S4.11.p4.11.m1.1.2" xref="S4.11.p4.11.m1.1.2.cmml"><msup id="S4.11.p4.11.m1.1.2.2" xref="S4.11.p4.11.m1.1.2.2.cmml"><mi id="S4.11.p4.11.m1.1.2.2.2" xref="S4.11.p4.11.m1.1.2.2.2.cmml">f</mi><mrow id="S4.11.p4.11.m1.1.1.1.3" xref="S4.11.p4.11.m1.1.2.2.cmml"><mo id="S4.11.p4.11.m1.1.1.1.3.1" stretchy="false" xref="S4.11.p4.11.m1.1.2.2.cmml">(</mo><mn id="S4.11.p4.11.m1.1.1.1.1" xref="S4.11.p4.11.m1.1.1.1.1.cmml">1</mn><mo id="S4.11.p4.11.m1.1.1.1.3.2" stretchy="false" xref="S4.11.p4.11.m1.1.2.2.cmml">)</mo></mrow></msup><mo id="S4.11.p4.11.m1.1.2.1" xref="S4.11.p4.11.m1.1.2.1.cmml">∈</mo><msub id="S4.11.p4.11.m1.1.2.3" xref="S4.11.p4.11.m1.1.2.3.cmml"><mi id="S4.11.p4.11.m1.1.2.3.2" xref="S4.11.p4.11.m1.1.2.3.2.cmml">CPL</mi><mn id="S4.11.p4.11.m1.1.2.3.3" xref="S4.11.p4.11.m1.1.2.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.11.m1.1b"><apply id="S4.11.p4.11.m1.1.2.cmml" xref="S4.11.p4.11.m1.1.2"><in id="S4.11.p4.11.m1.1.2.1.cmml" xref="S4.11.p4.11.m1.1.2.1"></in><apply id="S4.11.p4.11.m1.1.2.2.cmml" xref="S4.11.p4.11.m1.1.2.2"><csymbol cd="ambiguous" id="S4.11.p4.11.m1.1.2.2.1.cmml" xref="S4.11.p4.11.m1.1.2.2">superscript</csymbol><ci id="S4.11.p4.11.m1.1.2.2.2.cmml" xref="S4.11.p4.11.m1.1.2.2.2">𝑓</ci><cn id="S4.11.p4.11.m1.1.1.1.1.cmml" type="integer" xref="S4.11.p4.11.m1.1.1.1.1">1</cn></apply><apply id="S4.11.p4.11.m1.1.2.3.cmml" xref="S4.11.p4.11.m1.1.2.3"><csymbol cd="ambiguous" id="S4.11.p4.11.m1.1.2.3.1.cmml" xref="S4.11.p4.11.m1.1.2.3">subscript</csymbol><ci id="S4.11.p4.11.m1.1.2.3.2.cmml" xref="S4.11.p4.11.m1.1.2.3.2">CPL</ci><cn id="S4.11.p4.11.m1.1.2.3.3.cmml" type="integer" xref="S4.11.p4.11.m1.1.2.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.11.m1.1c">f^{(1)}\in\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.11.m1.1d">italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover,</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex49"> <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^{(2)}(x):=f(x)-f^{(1)}(x)=\begin{cases}0,&x\in P_{1}\cup P_{2}\\ f_{3}(x)-f_{2}(x),\quad&x\in P_{3}\\ f_{4}(x)-f_{1}(x),\quad&x\in P_{4}\\ \end{cases}," class="ltx_Math" display="block" id="S4.Ex49.m1.12"><semantics id="S4.Ex49.m1.12a"><mrow id="S4.Ex49.m1.12.12.1" xref="S4.Ex49.m1.12.12.1.1.cmml"><mrow id="S4.Ex49.m1.12.12.1.1" xref="S4.Ex49.m1.12.12.1.1.cmml"><mrow id="S4.Ex49.m1.12.12.1.1.2" xref="S4.Ex49.m1.12.12.1.1.2.cmml"><msup id="S4.Ex49.m1.12.12.1.1.2.2" xref="S4.Ex49.m1.12.12.1.1.2.2.cmml"><mi id="S4.Ex49.m1.12.12.1.1.2.2.2" xref="S4.Ex49.m1.12.12.1.1.2.2.2.cmml">f</mi><mrow id="S4.Ex49.m1.7.7.1.3" xref="S4.Ex49.m1.12.12.1.1.2.2.cmml"><mo id="S4.Ex49.m1.7.7.1.3.1" stretchy="false" xref="S4.Ex49.m1.12.12.1.1.2.2.cmml">(</mo><mn 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italic_x ) - italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) , end_CELL start_CELL italic_x ∈ italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_f start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_x ) - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) , end_CELL start_CELL italic_x ∈ italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT 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="S4.11.p4.15">is a <math alttext="\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.11.p4.12.m1.1"><semantics id="S4.11.p4.12.m1.1a"><msub id="S4.11.p4.12.m1.1.1" xref="S4.11.p4.12.m1.1.1.cmml"><mi id="S4.11.p4.12.m1.1.1.2" xref="S4.11.p4.12.m1.1.1.2.cmml">CPL</mi><mn id="S4.11.p4.12.m1.1.1.3" xref="S4.11.p4.12.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.11.p4.12.m1.1b"><apply id="S4.11.p4.12.m1.1.1.cmml" xref="S4.11.p4.12.m1.1.1"><csymbol cd="ambiguous" id="S4.11.p4.12.m1.1.1.1.cmml" xref="S4.11.p4.12.m1.1.1">subscript</csymbol><ci id="S4.11.p4.12.m1.1.1.2.cmml" xref="S4.11.p4.12.m1.1.1.2">CPL</ci><cn id="S4.11.p4.12.m1.1.1.3.cmml" type="integer" xref="S4.11.p4.12.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.12.m1.1c">\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.12.m1.1d">roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> function (actually even <math alttext="\operatorname{CPL}_{2}" class="ltx_Math" display="inline" id="S4.11.p4.13.m2.1"><semantics id="S4.11.p4.13.m2.1a"><msub id="S4.11.p4.13.m2.1.1" xref="S4.11.p4.13.m2.1.1.cmml"><mi id="S4.11.p4.13.m2.1.1.2" xref="S4.11.p4.13.m2.1.1.2.cmml">CPL</mi><mn id="S4.11.p4.13.m2.1.1.3" xref="S4.11.p4.13.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.11.p4.13.m2.1b"><apply id="S4.11.p4.13.m2.1.1.cmml" xref="S4.11.p4.13.m2.1.1"><csymbol cd="ambiguous" id="S4.11.p4.13.m2.1.1.1.cmml" xref="S4.11.p4.13.m2.1.1">subscript</csymbol><ci id="S4.11.p4.13.m2.1.1.2.cmml" xref="S4.11.p4.13.m2.1.1.2">CPL</ci><cn id="S4.11.p4.13.m2.1.1.3.cmml" type="integer" xref="S4.11.p4.13.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.13.m2.1c">\operatorname{CPL}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.13.m2.1d">roman_CPL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>). Thus, <math alttext="f=f^{(1)}+f^{(2)}" class="ltx_Math" display="inline" id="S4.11.p4.14.m3.2"><semantics id="S4.11.p4.14.m3.2a"><mrow id="S4.11.p4.14.m3.2.3" xref="S4.11.p4.14.m3.2.3.cmml"><mi id="S4.11.p4.14.m3.2.3.2" xref="S4.11.p4.14.m3.2.3.2.cmml">f</mi><mo id="S4.11.p4.14.m3.2.3.1" xref="S4.11.p4.14.m3.2.3.1.cmml">=</mo><mrow id="S4.11.p4.14.m3.2.3.3" xref="S4.11.p4.14.m3.2.3.3.cmml"><msup id="S4.11.p4.14.m3.2.3.3.2" xref="S4.11.p4.14.m3.2.3.3.2.cmml"><mi id="S4.11.p4.14.m3.2.3.3.2.2" xref="S4.11.p4.14.m3.2.3.3.2.2.cmml">f</mi><mrow id="S4.11.p4.14.m3.1.1.1.3" xref="S4.11.p4.14.m3.2.3.3.2.cmml"><mo id="S4.11.p4.14.m3.1.1.1.3.1" stretchy="false" xref="S4.11.p4.14.m3.2.3.3.2.cmml">(</mo><mn id="S4.11.p4.14.m3.1.1.1.1" xref="S4.11.p4.14.m3.1.1.1.1.cmml">1</mn><mo id="S4.11.p4.14.m3.1.1.1.3.2" stretchy="false" xref="S4.11.p4.14.m3.2.3.3.2.cmml">)</mo></mrow></msup><mo id="S4.11.p4.14.m3.2.3.3.1" xref="S4.11.p4.14.m3.2.3.3.1.cmml">+</mo><msup id="S4.11.p4.14.m3.2.3.3.3" xref="S4.11.p4.14.m3.2.3.3.3.cmml"><mi id="S4.11.p4.14.m3.2.3.3.3.2" xref="S4.11.p4.14.m3.2.3.3.3.2.cmml">f</mi><mrow id="S4.11.p4.14.m3.2.2.1.3" xref="S4.11.p4.14.m3.2.3.3.3.cmml"><mo id="S4.11.p4.14.m3.2.2.1.3.1" stretchy="false" xref="S4.11.p4.14.m3.2.3.3.3.cmml">(</mo><mn id="S4.11.p4.14.m3.2.2.1.1" xref="S4.11.p4.14.m3.2.2.1.1.cmml">2</mn><mo id="S4.11.p4.14.m3.2.2.1.3.2" stretchy="false" xref="S4.11.p4.14.m3.2.3.3.3.cmml">)</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.14.m3.2b"><apply id="S4.11.p4.14.m3.2.3.cmml" xref="S4.11.p4.14.m3.2.3"><eq id="S4.11.p4.14.m3.2.3.1.cmml" xref="S4.11.p4.14.m3.2.3.1"></eq><ci id="S4.11.p4.14.m3.2.3.2.cmml" xref="S4.11.p4.14.m3.2.3.2">𝑓</ci><apply id="S4.11.p4.14.m3.2.3.3.cmml" xref="S4.11.p4.14.m3.2.3.3"><plus id="S4.11.p4.14.m3.2.3.3.1.cmml" xref="S4.11.p4.14.m3.2.3.3.1"></plus><apply id="S4.11.p4.14.m3.2.3.3.2.cmml" xref="S4.11.p4.14.m3.2.3.3.2"><csymbol cd="ambiguous" id="S4.11.p4.14.m3.2.3.3.2.1.cmml" xref="S4.11.p4.14.m3.2.3.3.2">superscript</csymbol><ci id="S4.11.p4.14.m3.2.3.3.2.2.cmml" xref="S4.11.p4.14.m3.2.3.3.2.2">𝑓</ci><cn id="S4.11.p4.14.m3.1.1.1.1.cmml" type="integer" xref="S4.11.p4.14.m3.1.1.1.1">1</cn></apply><apply id="S4.11.p4.14.m3.2.3.3.3.cmml" xref="S4.11.p4.14.m3.2.3.3.3"><csymbol cd="ambiguous" id="S4.11.p4.14.m3.2.3.3.3.1.cmml" xref="S4.11.p4.14.m3.2.3.3.3">superscript</csymbol><ci id="S4.11.p4.14.m3.2.3.3.3.2.cmml" xref="S4.11.p4.14.m3.2.3.3.3.2">𝑓</ci><cn id="S4.11.p4.14.m3.2.2.1.1.cmml" type="integer" xref="S4.11.p4.14.m3.2.2.1.1">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.14.m3.2c">f=f^{(1)}+f^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.14.m3.2d">italic_f = italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT + italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="f^{(1)},f^{(2)}\in\operatorname{CPL}_{3}" class="ltx_Math" display="inline" id="S4.11.p4.15.m4.4"><semantics id="S4.11.p4.15.m4.4a"><mrow id="S4.11.p4.15.m4.4.4" xref="S4.11.p4.15.m4.4.4.cmml"><mrow id="S4.11.p4.15.m4.4.4.2.2" xref="S4.11.p4.15.m4.4.4.2.3.cmml"><msup id="S4.11.p4.15.m4.3.3.1.1.1" xref="S4.11.p4.15.m4.3.3.1.1.1.cmml"><mi id="S4.11.p4.15.m4.3.3.1.1.1.2" xref="S4.11.p4.15.m4.3.3.1.1.1.2.cmml">f</mi><mrow id="S4.11.p4.15.m4.1.1.1.3" xref="S4.11.p4.15.m4.3.3.1.1.1.cmml"><mo id="S4.11.p4.15.m4.1.1.1.3.1" stretchy="false" xref="S4.11.p4.15.m4.3.3.1.1.1.cmml">(</mo><mn id="S4.11.p4.15.m4.1.1.1.1" xref="S4.11.p4.15.m4.1.1.1.1.cmml">1</mn><mo id="S4.11.p4.15.m4.1.1.1.3.2" stretchy="false" xref="S4.11.p4.15.m4.3.3.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.11.p4.15.m4.4.4.2.2.3" xref="S4.11.p4.15.m4.4.4.2.3.cmml">,</mo><msup id="S4.11.p4.15.m4.4.4.2.2.2" xref="S4.11.p4.15.m4.4.4.2.2.2.cmml"><mi id="S4.11.p4.15.m4.4.4.2.2.2.2" xref="S4.11.p4.15.m4.4.4.2.2.2.2.cmml">f</mi><mrow id="S4.11.p4.15.m4.2.2.1.3" xref="S4.11.p4.15.m4.4.4.2.2.2.cmml"><mo id="S4.11.p4.15.m4.2.2.1.3.1" stretchy="false" xref="S4.11.p4.15.m4.4.4.2.2.2.cmml">(</mo><mn id="S4.11.p4.15.m4.2.2.1.1" xref="S4.11.p4.15.m4.2.2.1.1.cmml">2</mn><mo id="S4.11.p4.15.m4.2.2.1.3.2" stretchy="false" xref="S4.11.p4.15.m4.4.4.2.2.2.cmml">)</mo></mrow></msup></mrow><mo id="S4.11.p4.15.m4.4.4.3" xref="S4.11.p4.15.m4.4.4.3.cmml">∈</mo><msub id="S4.11.p4.15.m4.4.4.4" xref="S4.11.p4.15.m4.4.4.4.cmml"><mi id="S4.11.p4.15.m4.4.4.4.2" xref="S4.11.p4.15.m4.4.4.4.2.cmml">CPL</mi><mn id="S4.11.p4.15.m4.4.4.4.3" xref="S4.11.p4.15.m4.4.4.4.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.15.m4.4b"><apply id="S4.11.p4.15.m4.4.4.cmml" xref="S4.11.p4.15.m4.4.4"><in id="S4.11.p4.15.m4.4.4.3.cmml" xref="S4.11.p4.15.m4.4.4.3"></in><list id="S4.11.p4.15.m4.4.4.2.3.cmml" xref="S4.11.p4.15.m4.4.4.2.2"><apply id="S4.11.p4.15.m4.3.3.1.1.1.cmml" xref="S4.11.p4.15.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.11.p4.15.m4.3.3.1.1.1.1.cmml" xref="S4.11.p4.15.m4.3.3.1.1.1">superscript</csymbol><ci id="S4.11.p4.15.m4.3.3.1.1.1.2.cmml" xref="S4.11.p4.15.m4.3.3.1.1.1.2">𝑓</ci><cn id="S4.11.p4.15.m4.1.1.1.1.cmml" type="integer" xref="S4.11.p4.15.m4.1.1.1.1">1</cn></apply><apply id="S4.11.p4.15.m4.4.4.2.2.2.cmml" xref="S4.11.p4.15.m4.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.11.p4.15.m4.4.4.2.2.2.1.cmml" xref="S4.11.p4.15.m4.4.4.2.2.2">superscript</csymbol><ci id="S4.11.p4.15.m4.4.4.2.2.2.2.cmml" xref="S4.11.p4.15.m4.4.4.2.2.2.2">𝑓</ci><cn id="S4.11.p4.15.m4.2.2.1.1.cmml" type="integer" xref="S4.11.p4.15.m4.2.2.1.1">2</cn></apply></list><apply id="S4.11.p4.15.m4.4.4.4.cmml" xref="S4.11.p4.15.m4.4.4.4"><csymbol cd="ambiguous" id="S4.11.p4.15.m4.4.4.4.1.cmml" xref="S4.11.p4.15.m4.4.4.4">subscript</csymbol><ci id="S4.11.p4.15.m4.4.4.4.2.cmml" xref="S4.11.p4.15.m4.4.4.4.2">CPL</ci><cn id="S4.11.p4.15.m4.4.4.4.3.cmml" type="integer" xref="S4.11.p4.15.m4.4.4.4.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.15.m4.4c">f^{(1)},f^{(2)}\in\operatorname{CPL}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.15.m4.4d">italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ∈ roman_CPL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <figure class="ltx_figure" id="S4.F11"> <p class="ltx_p ltx_align_center" id="S4.F11.1"><span class="ltx_text" id="S4.F11.1.1"><foreignobject height="52.8pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="65.8pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="101" id="S4.F11.1.1.1.g1" src="x27.png" width="126"/></foreignobject></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 11: </span>A four-piece <math alttext="\operatorname{CPL}" class="ltx_Math" display="inline" id="S4.F11.4.m1.1"><semantics id="S4.F11.4.m1.1b"><mi id="S4.F11.4.m1.1.1" xref="S4.F11.4.m1.1.1.cmml">CPL</mi><annotation-xml encoding="MathML-Content" id="S4.F11.4.m1.1c"><ci id="S4.F11.4.m1.1.1.cmml" xref="S4.F11.4.m1.1.1">CPL</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F11.4.m1.1d">\operatorname{CPL}</annotation><annotation encoding="application/x-llamapun" id="S4.F11.4.m1.1e">roman_CPL</annotation></semantics></math> function with no two adjacent pieces forming an angle smaller than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.F11.5.m2.1"><semantics id="S4.F11.5.m2.1b"><mi id="S4.F11.5.m2.1.1" xref="S4.F11.5.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.F11.5.m2.1c"><ci id="S4.F11.5.m2.1.1.cmml" xref="S4.F11.5.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F11.5.m2.1d">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.F11.5.m2.1e">italic_π</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">I will now give an explicit expression for <math alttext="v" class="ltx_Math" display="inline" id="S4.p6.1.m1.1"><semantics id="S4.p6.1.m1.1a"><mi id="S4.p6.1.m1.1.1" xref="S4.p6.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.p6.1.m1.1b"><ci id="S4.p6.1.m1.1.1.cmml" xref="S4.p6.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.p6.1.m1.1d">italic_v</annotation></semantics></math>-functions with three pieces.</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">Let <math alttext="f\in\operatorname{CPA}_{3}" 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">f</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml">∈</mo><msub 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" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml">CPA</mi><mn id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml">3</mn></msub></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"><in id="S4.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.2">𝑓</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><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.2">CPA</ci><cn id="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.1c">f\in\operatorname{CPA}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> be a <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_v</annotation></semantics></math>-function with three pieces. Then, there are signs <math alttext="\sigma_{1},\sigma_{2}\in\{-1,1\}" 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.m3.3.3.2.2" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.3.cmml"><msub id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.3.cmml">,</mo><msub id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.2" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.2.cmml">σ</mi><mn id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.3" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.4" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.4.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.2.cmml"><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.2.cmml">{</mo><mrow id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.cmml"><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1a" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.2.cmml">,</mo><mn id="S4.Thmtheorem4.p1.3.3.m3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml">1</mn><mo id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.4" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.2.cmml">}</mo></mrow></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"><in id="S4.Thmtheorem4.p1.3.3.m3.4.4.4.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.4"></in><list id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.3.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2"><apply id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.2">𝜎</ci><cn id="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.3.3.2.2.2.3">2</cn></apply></list><set id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1"><apply id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1"><minus id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1"></minus><cn id="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.4.4.3.1.1.2">1</cn></apply><cn id="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.1.1">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.3.m3.4c">\sigma_{1},\sigma_{2}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.3.m3.4d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math>, such that <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.4.m4.1"><semantics id="S4.Thmtheorem4.p1.4.4.m4.1a"><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.4.m4.1b"><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.4.m4.1d">italic_f</annotation></semantics></math> can be expressed as</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.E36"> <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=\sigma_{1}\max(g_{1},\sigma_{2}\max(g_{2},g_{3}))," class="ltx_Math" display="block" id="S4.E36.m1.3"><semantics id="S4.E36.m1.3a"><mrow id="S4.E36.m1.3.3.1" xref="S4.E36.m1.3.3.1.1.cmml"><mrow id="S4.E36.m1.3.3.1.1" xref="S4.E36.m1.3.3.1.1.cmml"><mi id="S4.E36.m1.3.3.1.1.4" xref="S4.E36.m1.3.3.1.1.4.cmml">f</mi><mo id="S4.E36.m1.3.3.1.1.3" xref="S4.E36.m1.3.3.1.1.3.cmml">=</mo><mrow id="S4.E36.m1.3.3.1.1.2" xref="S4.E36.m1.3.3.1.1.2.cmml"><msub id="S4.E36.m1.3.3.1.1.2.4" xref="S4.E36.m1.3.3.1.1.2.4.cmml"><mi id="S4.E36.m1.3.3.1.1.2.4.2" xref="S4.E36.m1.3.3.1.1.2.4.2.cmml">σ</mi><mn id="S4.E36.m1.3.3.1.1.2.4.3" xref="S4.E36.m1.3.3.1.1.2.4.3.cmml">1</mn></msub><mo id="S4.E36.m1.3.3.1.1.2.3" lspace="0.167em" xref="S4.E36.m1.3.3.1.1.2.3.cmml">⁢</mo><mrow id="S4.E36.m1.3.3.1.1.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml"><mi id="S4.E36.m1.2.2" xref="S4.E36.m1.2.2.cmml">max</mi><mo id="S4.E36.m1.3.3.1.1.2.2.2a" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml">⁡</mo><mrow id="S4.E36.m1.3.3.1.1.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml"><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.3" stretchy="false" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml">(</mo><msub id="S4.E36.m1.3.3.1.1.1.1.1.1.1" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S4.E36.m1.3.3.1.1.1.1.1.1.1.2" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1.2.cmml">g</mi><mn id="S4.E36.m1.3.3.1.1.1.1.1.1.1.3" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.4" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml">,</mo><mrow id="S4.E36.m1.3.3.1.1.2.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.cmml"><msub id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.cmml"><mi id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.2.cmml">σ</mi><mn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.3" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.3.cmml">2</mn></msub><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.2.3" lspace="0.167em" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml"><mi id="S4.E36.m1.1.1" xref="S4.E36.m1.1.1.cmml">max</mi><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2a" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml">⁡</mo><mrow id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml"><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.3" stretchy="false" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml">(</mo><msub id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.cmml"><mi id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.2.cmml">g</mi><mn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.3" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.4" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml">,</mo><msub id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.cmml"><mi id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.cmml">g</mi><mn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.3" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.5" stretchy="false" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E36.m1.3.3.1.1.2.2.2.2.5" stretchy="false" xref="S4.E36.m1.3.3.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E36.m1.3.3.1.2" xref="S4.E36.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E36.m1.3b"><apply id="S4.E36.m1.3.3.1.1.cmml" xref="S4.E36.m1.3.3.1"><eq id="S4.E36.m1.3.3.1.1.3.cmml" xref="S4.E36.m1.3.3.1.1.3"></eq><ci id="S4.E36.m1.3.3.1.1.4.cmml" xref="S4.E36.m1.3.3.1.1.4">𝑓</ci><apply id="S4.E36.m1.3.3.1.1.2.cmml" xref="S4.E36.m1.3.3.1.1.2"><times id="S4.E36.m1.3.3.1.1.2.3.cmml" xref="S4.E36.m1.3.3.1.1.2.3"></times><apply id="S4.E36.m1.3.3.1.1.2.4.cmml" xref="S4.E36.m1.3.3.1.1.2.4"><csymbol cd="ambiguous" id="S4.E36.m1.3.3.1.1.2.4.1.cmml" xref="S4.E36.m1.3.3.1.1.2.4">subscript</csymbol><ci id="S4.E36.m1.3.3.1.1.2.4.2.cmml" xref="S4.E36.m1.3.3.1.1.2.4.2">𝜎</ci><cn id="S4.E36.m1.3.3.1.1.2.4.3.cmml" type="integer" xref="S4.E36.m1.3.3.1.1.2.4.3">1</cn></apply><apply id="S4.E36.m1.3.3.1.1.2.2.3.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2"><max id="S4.E36.m1.2.2.cmml" xref="S4.E36.m1.2.2"></max><apply id="S4.E36.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E36.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E36.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1.2">𝑔</ci><cn id="S4.E36.m1.3.3.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E36.m1.3.3.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.E36.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2"><times id="S4.E36.m1.3.3.1.1.2.2.2.2.2.3.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.3"></times><apply id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4"><csymbol cd="ambiguous" id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.1.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4">subscript</csymbol><ci id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.2.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.2">𝜎</ci><cn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.3.cmml" type="integer" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.4.3">2</cn></apply><apply id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.3.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2"><max id="S4.E36.m1.1.1.cmml" xref="S4.E36.m1.1.1"></max><apply id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.1.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1">subscript</csymbol><ci id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.2.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.2">𝑔</ci><cn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.3">2</cn></apply><apply id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.1.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2">𝑔</ci><cn id="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.E36.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.3">3</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E36.m1.3c">f=\sigma_{1}\max(g_{1},\sigma_{2}\max(g_{2},g_{3})),</annotation><annotation encoding="application/x-llamapun" id="S4.E36.m1.3d">italic_f = italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_max ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_max ( italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 3 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">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem4.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.8.4">where <math alttext="g_{i}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.1.m1.1"><semantics id="S4.Thmtheorem4.p1.5.1.m1.1a"><msub id="S4.Thmtheorem4.p1.5.1.m1.1.1" xref="S4.Thmtheorem4.p1.5.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.5.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.5.1.m1.1.1.2.cmml">g</mi><mi id="S4.Thmtheorem4.p1.5.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.5.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.1.m1.1b"><apply id="S4.Thmtheorem4.p1.5.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.5.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.5.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.5.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.5.1.m1.1.1.2">𝑔</ci><ci id="S4.Thmtheorem4.p1.5.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.5.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.1.m1.1c">g_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.1.m1.1d">italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are affine components of <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.6.2.m2.1"><semantics id="S4.Thmtheorem4.p1.6.2.m2.1a"><mi id="S4.Thmtheorem4.p1.6.2.m2.1.1" xref="S4.Thmtheorem4.p1.6.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.6.2.m2.1b"><ci id="S4.Thmtheorem4.p1.6.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.6.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.6.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.6.2.m2.1d">italic_f</annotation></semantics></math> multiplied by <math alttext="-1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.7.3.m3.1"><semantics id="S4.Thmtheorem4.p1.7.3.m3.1a"><mrow id="S4.Thmtheorem4.p1.7.3.m3.1.1" xref="S4.Thmtheorem4.p1.7.3.m3.1.1.cmml"><mo id="S4.Thmtheorem4.p1.7.3.m3.1.1a" xref="S4.Thmtheorem4.p1.7.3.m3.1.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.7.3.m3.1.1.2" xref="S4.Thmtheorem4.p1.7.3.m3.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.7.3.m3.1b"><apply id="S4.Thmtheorem4.p1.7.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.7.3.m3.1.1"><minus id="S4.Thmtheorem4.p1.7.3.m3.1.1.1.cmml" xref="S4.Thmtheorem4.p1.7.3.m3.1.1"></minus><cn id="S4.Thmtheorem4.p1.7.3.m3.1.1.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.7.3.m3.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.7.3.m3.1c">-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.7.3.m3.1d">- 1</annotation></semantics></math> or <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.8.4.m4.1"><semantics id="S4.Thmtheorem4.p1.8.4.m4.1a"><mn id="S4.Thmtheorem4.p1.8.4.m4.1.1" xref="S4.Thmtheorem4.p1.8.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.8.4.m4.1b"><cn id="S4.Thmtheorem4.p1.8.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem4.p1.8.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.8.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.8.4.m4.1d">1</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.15"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.12.p1"> <p class="ltx_p" id="S4.12.p1.20">First, I note a trivial observation. Let <math alttext="g" class="ltx_Math" display="inline" id="S4.12.p1.1.m1.1"><semantics id="S4.12.p1.1.m1.1a"><mi id="S4.12.p1.1.m1.1.1" xref="S4.12.p1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.1.m1.1b"><ci id="S4.12.p1.1.m1.1.1.cmml" xref="S4.12.p1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.1.m1.1d">italic_g</annotation></semantics></math> and <math alttext="h" class="ltx_Math" display="inline" id="S4.12.p1.2.m2.1"><semantics id="S4.12.p1.2.m2.1a"><mi id="S4.12.p1.2.m2.1.1" xref="S4.12.p1.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.2.m2.1b"><ci id="S4.12.p1.2.m2.1.1.cmml" xref="S4.12.p1.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.2.m2.1d">italic_h</annotation></semantics></math> be affine functions such that <math alttext="g-h" class="ltx_Math" display="inline" id="S4.12.p1.3.m3.1"><semantics id="S4.12.p1.3.m3.1a"><mrow id="S4.12.p1.3.m3.1.1" xref="S4.12.p1.3.m3.1.1.cmml"><mi id="S4.12.p1.3.m3.1.1.2" xref="S4.12.p1.3.m3.1.1.2.cmml">g</mi><mo id="S4.12.p1.3.m3.1.1.1" xref="S4.12.p1.3.m3.1.1.1.cmml">−</mo><mi id="S4.12.p1.3.m3.1.1.3" xref="S4.12.p1.3.m3.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.3.m3.1b"><apply id="S4.12.p1.3.m3.1.1.cmml" xref="S4.12.p1.3.m3.1.1"><minus id="S4.12.p1.3.m3.1.1.1.cmml" xref="S4.12.p1.3.m3.1.1.1"></minus><ci id="S4.12.p1.3.m3.1.1.2.cmml" xref="S4.12.p1.3.m3.1.1.2">𝑔</ci><ci id="S4.12.p1.3.m3.1.1.3.cmml" xref="S4.12.p1.3.m3.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.3.m3.1c">g-h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.3.m3.1d">italic_g - italic_h</annotation></semantics></math> is not constant. Then <math alttext="g-h" class="ltx_Math" display="inline" id="S4.12.p1.4.m4.1"><semantics id="S4.12.p1.4.m4.1a"><mrow id="S4.12.p1.4.m4.1.1" xref="S4.12.p1.4.m4.1.1.cmml"><mi id="S4.12.p1.4.m4.1.1.2" xref="S4.12.p1.4.m4.1.1.2.cmml">g</mi><mo id="S4.12.p1.4.m4.1.1.1" xref="S4.12.p1.4.m4.1.1.1.cmml">−</mo><mi id="S4.12.p1.4.m4.1.1.3" xref="S4.12.p1.4.m4.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.4.m4.1b"><apply id="S4.12.p1.4.m4.1.1.cmml" xref="S4.12.p1.4.m4.1.1"><minus id="S4.12.p1.4.m4.1.1.1.cmml" xref="S4.12.p1.4.m4.1.1.1"></minus><ci id="S4.12.p1.4.m4.1.1.2.cmml" xref="S4.12.p1.4.m4.1.1.2">𝑔</ci><ci id="S4.12.p1.4.m4.1.1.3.cmml" xref="S4.12.p1.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.4.m4.1c">g-h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.4.m4.1d">italic_g - italic_h</annotation></semantics></math> vanishes precisely on a line <math alttext="l" class="ltx_Math" display="inline" id="S4.12.p1.5.m5.1"><semantics id="S4.12.p1.5.m5.1a"><mi id="S4.12.p1.5.m5.1.1" xref="S4.12.p1.5.m5.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.5.m5.1b"><ci id="S4.12.p1.5.m5.1.1.cmml" xref="S4.12.p1.5.m5.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.5.m5.1c">l</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.5.m5.1d">italic_l</annotation></semantics></math> that divides <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S4.12.p1.6.m6.1"><semantics id="S4.12.p1.6.m6.1a"><msup id="S4.12.p1.6.m6.1.1" xref="S4.12.p1.6.m6.1.1.cmml"><mi id="S4.12.p1.6.m6.1.1.2" xref="S4.12.p1.6.m6.1.1.2.cmml">ℝ</mi><mn id="S4.12.p1.6.m6.1.1.3" xref="S4.12.p1.6.m6.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.12.p1.6.m6.1b"><apply id="S4.12.p1.6.m6.1.1.cmml" xref="S4.12.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.12.p1.6.m6.1.1.1.cmml" xref="S4.12.p1.6.m6.1.1">superscript</csymbol><ci id="S4.12.p1.6.m6.1.1.2.cmml" xref="S4.12.p1.6.m6.1.1.2">ℝ</ci><cn id="S4.12.p1.6.m6.1.1.3.cmml" type="integer" xref="S4.12.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.6.m6.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> into two open half-planes. If <math alttext="x_{1},x_{2}\in\mathds{R}^{2}" class="ltx_Math" display="inline" id="S4.12.p1.7.m7.2"><semantics id="S4.12.p1.7.m7.2a"><mrow id="S4.12.p1.7.m7.2.2" xref="S4.12.p1.7.m7.2.2.cmml"><mrow id="S4.12.p1.7.m7.2.2.2.2" xref="S4.12.p1.7.m7.2.2.2.3.cmml"><msub id="S4.12.p1.7.m7.1.1.1.1.1" xref="S4.12.p1.7.m7.1.1.1.1.1.cmml"><mi id="S4.12.p1.7.m7.1.1.1.1.1.2" xref="S4.12.p1.7.m7.1.1.1.1.1.2.cmml">x</mi><mn id="S4.12.p1.7.m7.1.1.1.1.1.3" xref="S4.12.p1.7.m7.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.12.p1.7.m7.2.2.2.2.3" xref="S4.12.p1.7.m7.2.2.2.3.cmml">,</mo><msub id="S4.12.p1.7.m7.2.2.2.2.2" xref="S4.12.p1.7.m7.2.2.2.2.2.cmml"><mi id="S4.12.p1.7.m7.2.2.2.2.2.2" xref="S4.12.p1.7.m7.2.2.2.2.2.2.cmml">x</mi><mn id="S4.12.p1.7.m7.2.2.2.2.2.3" xref="S4.12.p1.7.m7.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.12.p1.7.m7.2.2.3" xref="S4.12.p1.7.m7.2.2.3.cmml">∈</mo><msup id="S4.12.p1.7.m7.2.2.4" xref="S4.12.p1.7.m7.2.2.4.cmml"><mi id="S4.12.p1.7.m7.2.2.4.2" xref="S4.12.p1.7.m7.2.2.4.2.cmml">ℝ</mi><mn id="S4.12.p1.7.m7.2.2.4.3" xref="S4.12.p1.7.m7.2.2.4.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.7.m7.2b"><apply id="S4.12.p1.7.m7.2.2.cmml" xref="S4.12.p1.7.m7.2.2"><in id="S4.12.p1.7.m7.2.2.3.cmml" xref="S4.12.p1.7.m7.2.2.3"></in><list id="S4.12.p1.7.m7.2.2.2.3.cmml" xref="S4.12.p1.7.m7.2.2.2.2"><apply id="S4.12.p1.7.m7.1.1.1.1.1.cmml" xref="S4.12.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.12.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.12.p1.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S4.12.p1.7.m7.1.1.1.1.1.2.cmml" xref="S4.12.p1.7.m7.1.1.1.1.1.2">𝑥</ci><cn id="S4.12.p1.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S4.12.p1.7.m7.1.1.1.1.1.3">1</cn></apply><apply id="S4.12.p1.7.m7.2.2.2.2.2.cmml" xref="S4.12.p1.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.12.p1.7.m7.2.2.2.2.2.1.cmml" xref="S4.12.p1.7.m7.2.2.2.2.2">subscript</csymbol><ci id="S4.12.p1.7.m7.2.2.2.2.2.2.cmml" xref="S4.12.p1.7.m7.2.2.2.2.2.2">𝑥</ci><cn id="S4.12.p1.7.m7.2.2.2.2.2.3.cmml" type="integer" xref="S4.12.p1.7.m7.2.2.2.2.2.3">2</cn></apply></list><apply id="S4.12.p1.7.m7.2.2.4.cmml" xref="S4.12.p1.7.m7.2.2.4"><csymbol cd="ambiguous" id="S4.12.p1.7.m7.2.2.4.1.cmml" xref="S4.12.p1.7.m7.2.2.4">superscript</csymbol><ci id="S4.12.p1.7.m7.2.2.4.2.cmml" xref="S4.12.p1.7.m7.2.2.4.2">ℝ</ci><cn id="S4.12.p1.7.m7.2.2.4.3.cmml" type="integer" xref="S4.12.p1.7.m7.2.2.4.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.7.m7.2c">x_{1},x_{2}\in\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.7.m7.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> are contained in the same half-plane, then the line segment <math alttext="\overline{x_{1}x_{2}}" class="ltx_Math" display="inline" id="S4.12.p1.8.m8.1"><semantics id="S4.12.p1.8.m8.1a"><mover accent="true" id="S4.12.p1.8.m8.1.1" xref="S4.12.p1.8.m8.1.1.cmml"><mrow id="S4.12.p1.8.m8.1.1.2" xref="S4.12.p1.8.m8.1.1.2.cmml"><msub id="S4.12.p1.8.m8.1.1.2.2" xref="S4.12.p1.8.m8.1.1.2.2.cmml"><mi id="S4.12.p1.8.m8.1.1.2.2.2" xref="S4.12.p1.8.m8.1.1.2.2.2.cmml">x</mi><mn id="S4.12.p1.8.m8.1.1.2.2.3" xref="S4.12.p1.8.m8.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.12.p1.8.m8.1.1.2.1" xref="S4.12.p1.8.m8.1.1.2.1.cmml">⁢</mo><msub id="S4.12.p1.8.m8.1.1.2.3" xref="S4.12.p1.8.m8.1.1.2.3.cmml"><mi id="S4.12.p1.8.m8.1.1.2.3.2" xref="S4.12.p1.8.m8.1.1.2.3.2.cmml">x</mi><mn id="S4.12.p1.8.m8.1.1.2.3.3" xref="S4.12.p1.8.m8.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S4.12.p1.8.m8.1.1.1" xref="S4.12.p1.8.m8.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.12.p1.8.m8.1b"><apply id="S4.12.p1.8.m8.1.1.cmml" xref="S4.12.p1.8.m8.1.1"><ci id="S4.12.p1.8.m8.1.1.1.cmml" xref="S4.12.p1.8.m8.1.1.1">¯</ci><apply id="S4.12.p1.8.m8.1.1.2.cmml" xref="S4.12.p1.8.m8.1.1.2"><times id="S4.12.p1.8.m8.1.1.2.1.cmml" xref="S4.12.p1.8.m8.1.1.2.1"></times><apply id="S4.12.p1.8.m8.1.1.2.2.cmml" xref="S4.12.p1.8.m8.1.1.2.2"><csymbol cd="ambiguous" id="S4.12.p1.8.m8.1.1.2.2.1.cmml" xref="S4.12.p1.8.m8.1.1.2.2">subscript</csymbol><ci id="S4.12.p1.8.m8.1.1.2.2.2.cmml" xref="S4.12.p1.8.m8.1.1.2.2.2">𝑥</ci><cn id="S4.12.p1.8.m8.1.1.2.2.3.cmml" type="integer" xref="S4.12.p1.8.m8.1.1.2.2.3">1</cn></apply><apply id="S4.12.p1.8.m8.1.1.2.3.cmml" xref="S4.12.p1.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S4.12.p1.8.m8.1.1.2.3.1.cmml" xref="S4.12.p1.8.m8.1.1.2.3">subscript</csymbol><ci id="S4.12.p1.8.m8.1.1.2.3.2.cmml" xref="S4.12.p1.8.m8.1.1.2.3.2">𝑥</ci><cn id="S4.12.p1.8.m8.1.1.2.3.3.cmml" type="integer" xref="S4.12.p1.8.m8.1.1.2.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.8.m8.1c">\overline{x_{1}x_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.8.m8.1d">over¯ start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> does not intersect <math alttext="l" class="ltx_Math" display="inline" id="S4.12.p1.9.m9.1"><semantics id="S4.12.p1.9.m9.1a"><mi id="S4.12.p1.9.m9.1.1" xref="S4.12.p1.9.m9.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.9.m9.1b"><ci id="S4.12.p1.9.m9.1.1.cmml" xref="S4.12.p1.9.m9.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.9.m9.1c">l</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.9.m9.1d">italic_l</annotation></semantics></math> due to the convexity of half-planes. Consequently, the continuous function <math alttext="g-h" class="ltx_Math" display="inline" id="S4.12.p1.10.m10.1"><semantics id="S4.12.p1.10.m10.1a"><mrow id="S4.12.p1.10.m10.1.1" xref="S4.12.p1.10.m10.1.1.cmml"><mi id="S4.12.p1.10.m10.1.1.2" xref="S4.12.p1.10.m10.1.1.2.cmml">g</mi><mo id="S4.12.p1.10.m10.1.1.1" xref="S4.12.p1.10.m10.1.1.1.cmml">−</mo><mi id="S4.12.p1.10.m10.1.1.3" xref="S4.12.p1.10.m10.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.10.m10.1b"><apply id="S4.12.p1.10.m10.1.1.cmml" xref="S4.12.p1.10.m10.1.1"><minus id="S4.12.p1.10.m10.1.1.1.cmml" xref="S4.12.p1.10.m10.1.1.1"></minus><ci id="S4.12.p1.10.m10.1.1.2.cmml" xref="S4.12.p1.10.m10.1.1.2">𝑔</ci><ci id="S4.12.p1.10.m10.1.1.3.cmml" xref="S4.12.p1.10.m10.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.10.m10.1c">g-h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.10.m10.1d">italic_g - italic_h</annotation></semantics></math> does not change sign along <math alttext="\overline{x_{1}x_{2}}" class="ltx_Math" display="inline" id="S4.12.p1.11.m11.1"><semantics id="S4.12.p1.11.m11.1a"><mover accent="true" id="S4.12.p1.11.m11.1.1" xref="S4.12.p1.11.m11.1.1.cmml"><mrow id="S4.12.p1.11.m11.1.1.2" xref="S4.12.p1.11.m11.1.1.2.cmml"><msub id="S4.12.p1.11.m11.1.1.2.2" xref="S4.12.p1.11.m11.1.1.2.2.cmml"><mi id="S4.12.p1.11.m11.1.1.2.2.2" xref="S4.12.p1.11.m11.1.1.2.2.2.cmml">x</mi><mn id="S4.12.p1.11.m11.1.1.2.2.3" xref="S4.12.p1.11.m11.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.12.p1.11.m11.1.1.2.1" xref="S4.12.p1.11.m11.1.1.2.1.cmml">⁢</mo><msub id="S4.12.p1.11.m11.1.1.2.3" xref="S4.12.p1.11.m11.1.1.2.3.cmml"><mi id="S4.12.p1.11.m11.1.1.2.3.2" xref="S4.12.p1.11.m11.1.1.2.3.2.cmml">x</mi><mn id="S4.12.p1.11.m11.1.1.2.3.3" xref="S4.12.p1.11.m11.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S4.12.p1.11.m11.1.1.1" xref="S4.12.p1.11.m11.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.12.p1.11.m11.1b"><apply id="S4.12.p1.11.m11.1.1.cmml" xref="S4.12.p1.11.m11.1.1"><ci id="S4.12.p1.11.m11.1.1.1.cmml" xref="S4.12.p1.11.m11.1.1.1">¯</ci><apply id="S4.12.p1.11.m11.1.1.2.cmml" xref="S4.12.p1.11.m11.1.1.2"><times id="S4.12.p1.11.m11.1.1.2.1.cmml" xref="S4.12.p1.11.m11.1.1.2.1"></times><apply id="S4.12.p1.11.m11.1.1.2.2.cmml" xref="S4.12.p1.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S4.12.p1.11.m11.1.1.2.2.1.cmml" xref="S4.12.p1.11.m11.1.1.2.2">subscript</csymbol><ci id="S4.12.p1.11.m11.1.1.2.2.2.cmml" xref="S4.12.p1.11.m11.1.1.2.2.2">𝑥</ci><cn id="S4.12.p1.11.m11.1.1.2.2.3.cmml" type="integer" xref="S4.12.p1.11.m11.1.1.2.2.3">1</cn></apply><apply id="S4.12.p1.11.m11.1.1.2.3.cmml" xref="S4.12.p1.11.m11.1.1.2.3"><csymbol cd="ambiguous" id="S4.12.p1.11.m11.1.1.2.3.1.cmml" xref="S4.12.p1.11.m11.1.1.2.3">subscript</csymbol><ci id="S4.12.p1.11.m11.1.1.2.3.2.cmml" xref="S4.12.p1.11.m11.1.1.2.3.2">𝑥</ci><cn id="S4.12.p1.11.m11.1.1.2.3.3.cmml" type="integer" xref="S4.12.p1.11.m11.1.1.2.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.11.m11.1c">\overline{x_{1}x_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.11.m11.1d">over¯ start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. However, if <math alttext="x_{1}" class="ltx_Math" display="inline" id="S4.12.p1.12.m12.1"><semantics id="S4.12.p1.12.m12.1a"><msub id="S4.12.p1.12.m12.1.1" xref="S4.12.p1.12.m12.1.1.cmml"><mi id="S4.12.p1.12.m12.1.1.2" xref="S4.12.p1.12.m12.1.1.2.cmml">x</mi><mn id="S4.12.p1.12.m12.1.1.3" xref="S4.12.p1.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.12.p1.12.m12.1b"><apply id="S4.12.p1.12.m12.1.1.cmml" xref="S4.12.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.12.p1.12.m12.1.1.1.cmml" xref="S4.12.p1.12.m12.1.1">subscript</csymbol><ci id="S4.12.p1.12.m12.1.1.2.cmml" xref="S4.12.p1.12.m12.1.1.2">𝑥</ci><cn id="S4.12.p1.12.m12.1.1.3.cmml" type="integer" xref="S4.12.p1.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.12.m12.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.12.m12.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{2}" class="ltx_Math" display="inline" id="S4.12.p1.13.m13.1"><semantics id="S4.12.p1.13.m13.1a"><msub id="S4.12.p1.13.m13.1.1" xref="S4.12.p1.13.m13.1.1.cmml"><mi id="S4.12.p1.13.m13.1.1.2" xref="S4.12.p1.13.m13.1.1.2.cmml">x</mi><mn id="S4.12.p1.13.m13.1.1.3" xref="S4.12.p1.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.12.p1.13.m13.1b"><apply id="S4.12.p1.13.m13.1.1.cmml" xref="S4.12.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S4.12.p1.13.m13.1.1.1.cmml" xref="S4.12.p1.13.m13.1.1">subscript</csymbol><ci id="S4.12.p1.13.m13.1.1.2.cmml" xref="S4.12.p1.13.m13.1.1.2">𝑥</ci><cn id="S4.12.p1.13.m13.1.1.3.cmml" type="integer" xref="S4.12.p1.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.13.m13.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.13.m13.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> lie in opposite half-planes, the segment <math alttext="\overline{x_{1}x_{2}}" class="ltx_Math" display="inline" id="S4.12.p1.14.m14.1"><semantics id="S4.12.p1.14.m14.1a"><mover accent="true" id="S4.12.p1.14.m14.1.1" xref="S4.12.p1.14.m14.1.1.cmml"><mrow id="S4.12.p1.14.m14.1.1.2" xref="S4.12.p1.14.m14.1.1.2.cmml"><msub id="S4.12.p1.14.m14.1.1.2.2" xref="S4.12.p1.14.m14.1.1.2.2.cmml"><mi id="S4.12.p1.14.m14.1.1.2.2.2" xref="S4.12.p1.14.m14.1.1.2.2.2.cmml">x</mi><mn id="S4.12.p1.14.m14.1.1.2.2.3" xref="S4.12.p1.14.m14.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.12.p1.14.m14.1.1.2.1" xref="S4.12.p1.14.m14.1.1.2.1.cmml">⁢</mo><msub id="S4.12.p1.14.m14.1.1.2.3" xref="S4.12.p1.14.m14.1.1.2.3.cmml"><mi id="S4.12.p1.14.m14.1.1.2.3.2" xref="S4.12.p1.14.m14.1.1.2.3.2.cmml">x</mi><mn id="S4.12.p1.14.m14.1.1.2.3.3" xref="S4.12.p1.14.m14.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S4.12.p1.14.m14.1.1.1" xref="S4.12.p1.14.m14.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.12.p1.14.m14.1b"><apply id="S4.12.p1.14.m14.1.1.cmml" xref="S4.12.p1.14.m14.1.1"><ci id="S4.12.p1.14.m14.1.1.1.cmml" xref="S4.12.p1.14.m14.1.1.1">¯</ci><apply id="S4.12.p1.14.m14.1.1.2.cmml" xref="S4.12.p1.14.m14.1.1.2"><times id="S4.12.p1.14.m14.1.1.2.1.cmml" xref="S4.12.p1.14.m14.1.1.2.1"></times><apply id="S4.12.p1.14.m14.1.1.2.2.cmml" xref="S4.12.p1.14.m14.1.1.2.2"><csymbol cd="ambiguous" id="S4.12.p1.14.m14.1.1.2.2.1.cmml" xref="S4.12.p1.14.m14.1.1.2.2">subscript</csymbol><ci id="S4.12.p1.14.m14.1.1.2.2.2.cmml" xref="S4.12.p1.14.m14.1.1.2.2.2">𝑥</ci><cn id="S4.12.p1.14.m14.1.1.2.2.3.cmml" type="integer" xref="S4.12.p1.14.m14.1.1.2.2.3">1</cn></apply><apply id="S4.12.p1.14.m14.1.1.2.3.cmml" xref="S4.12.p1.14.m14.1.1.2.3"><csymbol cd="ambiguous" id="S4.12.p1.14.m14.1.1.2.3.1.cmml" xref="S4.12.p1.14.m14.1.1.2.3">subscript</csymbol><ci id="S4.12.p1.14.m14.1.1.2.3.2.cmml" xref="S4.12.p1.14.m14.1.1.2.3.2">𝑥</ci><cn id="S4.12.p1.14.m14.1.1.2.3.3.cmml" type="integer" xref="S4.12.p1.14.m14.1.1.2.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.14.m14.1c">\overline{x_{1}x_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.14.m14.1d">over¯ start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> intersects <math alttext="l" class="ltx_Math" display="inline" id="S4.12.p1.15.m15.1"><semantics id="S4.12.p1.15.m15.1a"><mi id="S4.12.p1.15.m15.1.1" xref="S4.12.p1.15.m15.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.15.m15.1b"><ci id="S4.12.p1.15.m15.1.1.cmml" xref="S4.12.p1.15.m15.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.15.m15.1c">l</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.15.m15.1d">italic_l</annotation></semantics></math> exactly once, causing the function to change sign. Therefore, <math alttext="g>h" class="ltx_Math" display="inline" id="S4.12.p1.16.m16.1"><semantics id="S4.12.p1.16.m16.1a"><mrow id="S4.12.p1.16.m16.1.1" xref="S4.12.p1.16.m16.1.1.cmml"><mi id="S4.12.p1.16.m16.1.1.2" xref="S4.12.p1.16.m16.1.1.2.cmml">g</mi><mo id="S4.12.p1.16.m16.1.1.1" xref="S4.12.p1.16.m16.1.1.1.cmml">></mo><mi id="S4.12.p1.16.m16.1.1.3" xref="S4.12.p1.16.m16.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.16.m16.1b"><apply id="S4.12.p1.16.m16.1.1.cmml" xref="S4.12.p1.16.m16.1.1"><gt id="S4.12.p1.16.m16.1.1.1.cmml" xref="S4.12.p1.16.m16.1.1.1"></gt><ci id="S4.12.p1.16.m16.1.1.2.cmml" xref="S4.12.p1.16.m16.1.1.2">𝑔</ci><ci id="S4.12.p1.16.m16.1.1.3.cmml" xref="S4.12.p1.16.m16.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.16.m16.1c">g>h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.16.m16.1d">italic_g > italic_h</annotation></semantics></math> in one half-plane, and <math alttext="h>g" class="ltx_Math" display="inline" id="S4.12.p1.17.m17.1"><semantics id="S4.12.p1.17.m17.1a"><mrow id="S4.12.p1.17.m17.1.1" xref="S4.12.p1.17.m17.1.1.cmml"><mi id="S4.12.p1.17.m17.1.1.2" xref="S4.12.p1.17.m17.1.1.2.cmml">h</mi><mo id="S4.12.p1.17.m17.1.1.1" xref="S4.12.p1.17.m17.1.1.1.cmml">></mo><mi id="S4.12.p1.17.m17.1.1.3" xref="S4.12.p1.17.m17.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.17.m17.1b"><apply id="S4.12.p1.17.m17.1.1.cmml" xref="S4.12.p1.17.m17.1.1"><gt id="S4.12.p1.17.m17.1.1.1.cmml" xref="S4.12.p1.17.m17.1.1.1"></gt><ci id="S4.12.p1.17.m17.1.1.2.cmml" xref="S4.12.p1.17.m17.1.1.2">ℎ</ci><ci id="S4.12.p1.17.m17.1.1.3.cmml" xref="S4.12.p1.17.m17.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.17.m17.1c">h>g</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.17.m17.1d">italic_h > italic_g</annotation></semantics></math> in the other. More generally, it holds that <math alttext="g\geq h" class="ltx_Math" display="inline" id="S4.12.p1.18.m18.1"><semantics id="S4.12.p1.18.m18.1a"><mrow id="S4.12.p1.18.m18.1.1" xref="S4.12.p1.18.m18.1.1.cmml"><mi id="S4.12.p1.18.m18.1.1.2" xref="S4.12.p1.18.m18.1.1.2.cmml">g</mi><mo id="S4.12.p1.18.m18.1.1.1" xref="S4.12.p1.18.m18.1.1.1.cmml">≥</mo><mi id="S4.12.p1.18.m18.1.1.3" xref="S4.12.p1.18.m18.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.18.m18.1b"><apply id="S4.12.p1.18.m18.1.1.cmml" xref="S4.12.p1.18.m18.1.1"><geq id="S4.12.p1.18.m18.1.1.1.cmml" xref="S4.12.p1.18.m18.1.1.1"></geq><ci id="S4.12.p1.18.m18.1.1.2.cmml" xref="S4.12.p1.18.m18.1.1.2">𝑔</ci><ci id="S4.12.p1.18.m18.1.1.3.cmml" xref="S4.12.p1.18.m18.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.18.m18.1c">g\geq h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.18.m18.1d">italic_g ≥ italic_h</annotation></semantics></math>, respectively <math alttext="h\geq g" class="ltx_Math" display="inline" id="S4.12.p1.19.m19.1"><semantics id="S4.12.p1.19.m19.1a"><mrow id="S4.12.p1.19.m19.1.1" xref="S4.12.p1.19.m19.1.1.cmml"><mi id="S4.12.p1.19.m19.1.1.2" xref="S4.12.p1.19.m19.1.1.2.cmml">h</mi><mo id="S4.12.p1.19.m19.1.1.1" xref="S4.12.p1.19.m19.1.1.1.cmml">≥</mo><mi id="S4.12.p1.19.m19.1.1.3" xref="S4.12.p1.19.m19.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.19.m19.1b"><apply id="S4.12.p1.19.m19.1.1.cmml" xref="S4.12.p1.19.m19.1.1"><geq id="S4.12.p1.19.m19.1.1.1.cmml" xref="S4.12.p1.19.m19.1.1.1"></geq><ci id="S4.12.p1.19.m19.1.1.2.cmml" xref="S4.12.p1.19.m19.1.1.2">ℎ</ci><ci id="S4.12.p1.19.m19.1.1.3.cmml" xref="S4.12.p1.19.m19.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.19.m19.1c">h\geq g</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.19.m19.1d">italic_h ≥ italic_g</annotation></semantics></math>, in the closed half-planes, which also allows for the case that <math alttext="g=h" class="ltx_Math" display="inline" id="S4.12.p1.20.m20.1"><semantics id="S4.12.p1.20.m20.1a"><mrow id="S4.12.p1.20.m20.1.1" xref="S4.12.p1.20.m20.1.1.cmml"><mi id="S4.12.p1.20.m20.1.1.2" xref="S4.12.p1.20.m20.1.1.2.cmml">g</mi><mo id="S4.12.p1.20.m20.1.1.1" xref="S4.12.p1.20.m20.1.1.1.cmml">=</mo><mi id="S4.12.p1.20.m20.1.1.3" xref="S4.12.p1.20.m20.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.20.m20.1b"><apply id="S4.12.p1.20.m20.1.1.cmml" xref="S4.12.p1.20.m20.1.1"><eq id="S4.12.p1.20.m20.1.1.1.cmml" xref="S4.12.p1.20.m20.1.1.1"></eq><ci id="S4.12.p1.20.m20.1.1.2.cmml" xref="S4.12.p1.20.m20.1.1.2">𝑔</ci><ci id="S4.12.p1.20.m20.1.1.3.cmml" xref="S4.12.p1.20.m20.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.20.m20.1c">g=h</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.20.m20.1d">italic_g = italic_h</annotation></semantics></math> (with any line).</p> </div> <figure class="ltx_figure" id="S4.F12"> <table class="ltx_tabular ltx_align_middle" id="S4.F12.2"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.F12.2.2"> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S4.F12.1.1.1" style="width:142.3pt;"> <span class="ltx_inline-block ltx_align_top" id="S4.F12.1.1.1.1"> <span class="ltx_p" id="S4.F12.1.1.1.1.1"><foreignobject height="62.9pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="80.2pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="121" id="S4.F12.1.1.1.1.1.1.g1" src="x28.png" width="154"/></foreignobject></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_middle" id="S4.F12.2.2.2" style="width:142.3pt;"> <span class="ltx_inline-block ltx_align_top" id="S4.F12.2.2.2.1"> <span class="ltx_p" id="S4.F12.2.2.2.1.1"><foreignobject height="80.2pt" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="80.2pt"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="154" id="S4.F12.2.2.2.1.1.1.g1" src="x29.png" width="154"/></foreignobject></span> </span> </td> </tr> </tbody> </table> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 12: </span><math alttext="v" class="ltx_Math" display="inline" id="S4.F12.9.m1.1"><semantics id="S4.F12.9.m1.1b"><mi id="S4.F12.9.m1.1.1" xref="S4.F12.9.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.F12.9.m1.1c"><ci id="S4.F12.9.m1.1.1.cmml" xref="S4.F12.9.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.9.m1.1d">v</annotation><annotation encoding="application/x-llamapun" id="S4.F12.9.m1.1e">italic_v</annotation></semantics></math>-functions with three pieces that are enclosed by rays (solid lines). The sign of <math alttext="f_{i}-f_{j}" class="ltx_Math" display="inline" id="S4.F12.10.m2.1"><semantics id="S4.F12.10.m2.1b"><mrow id="S4.F12.10.m2.1.1" xref="S4.F12.10.m2.1.1.cmml"><msub id="S4.F12.10.m2.1.1.2" xref="S4.F12.10.m2.1.1.2.cmml"><mi id="S4.F12.10.m2.1.1.2.2" xref="S4.F12.10.m2.1.1.2.2.cmml">f</mi><mi id="S4.F12.10.m2.1.1.2.3" xref="S4.F12.10.m2.1.1.2.3.cmml">i</mi></msub><mo id="S4.F12.10.m2.1.1.1" xref="S4.F12.10.m2.1.1.1.cmml">−</mo><msub id="S4.F12.10.m2.1.1.3" xref="S4.F12.10.m2.1.1.3.cmml"><mi id="S4.F12.10.m2.1.1.3.2" xref="S4.F12.10.m2.1.1.3.2.cmml">f</mi><mi id="S4.F12.10.m2.1.1.3.3" xref="S4.F12.10.m2.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F12.10.m2.1c"><apply id="S4.F12.10.m2.1.1.cmml" xref="S4.F12.10.m2.1.1"><minus id="S4.F12.10.m2.1.1.1.cmml" xref="S4.F12.10.m2.1.1.1"></minus><apply id="S4.F12.10.m2.1.1.2.cmml" xref="S4.F12.10.m2.1.1.2"><csymbol cd="ambiguous" id="S4.F12.10.m2.1.1.2.1.cmml" xref="S4.F12.10.m2.1.1.2">subscript</csymbol><ci id="S4.F12.10.m2.1.1.2.2.cmml" xref="S4.F12.10.m2.1.1.2.2">𝑓</ci><ci id="S4.F12.10.m2.1.1.2.3.cmml" xref="S4.F12.10.m2.1.1.2.3">𝑖</ci></apply><apply id="S4.F12.10.m2.1.1.3.cmml" xref="S4.F12.10.m2.1.1.3"><csymbol cd="ambiguous" id="S4.F12.10.m2.1.1.3.1.cmml" xref="S4.F12.10.m2.1.1.3">subscript</csymbol><ci id="S4.F12.10.m2.1.1.3.2.cmml" xref="S4.F12.10.m2.1.1.3.2">𝑓</ci><ci id="S4.F12.10.m2.1.1.3.3.cmml" xref="S4.F12.10.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.10.m2.1d">f_{i}-f_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.F12.10.m2.1e">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> only changes on the affine hull of the ray between <math alttext="P_{i}" class="ltx_Math" display="inline" id="S4.F12.11.m3.1"><semantics id="S4.F12.11.m3.1b"><msub id="S4.F12.11.m3.1.1" xref="S4.F12.11.m3.1.1.cmml"><mi id="S4.F12.11.m3.1.1.2" xref="S4.F12.11.m3.1.1.2.cmml">P</mi><mi id="S4.F12.11.m3.1.1.3" xref="S4.F12.11.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F12.11.m3.1c"><apply id="S4.F12.11.m3.1.1.cmml" xref="S4.F12.11.m3.1.1"><csymbol cd="ambiguous" id="S4.F12.11.m3.1.1.1.cmml" xref="S4.F12.11.m3.1.1">subscript</csymbol><ci id="S4.F12.11.m3.1.1.2.cmml" xref="S4.F12.11.m3.1.1.2">𝑃</ci><ci id="S4.F12.11.m3.1.1.3.cmml" xref="S4.F12.11.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.11.m3.1d">P_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.F12.11.m3.1e">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{j}" class="ltx_Math" display="inline" id="S4.F12.12.m4.1"><semantics id="S4.F12.12.m4.1b"><msub id="S4.F12.12.m4.1.1" xref="S4.F12.12.m4.1.1.cmml"><mi id="S4.F12.12.m4.1.1.2" xref="S4.F12.12.m4.1.1.2.cmml">P</mi><mi id="S4.F12.12.m4.1.1.3" xref="S4.F12.12.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F12.12.m4.1c"><apply id="S4.F12.12.m4.1.1.cmml" xref="S4.F12.12.m4.1.1"><csymbol cd="ambiguous" id="S4.F12.12.m4.1.1.1.cmml" xref="S4.F12.12.m4.1.1">subscript</csymbol><ci id="S4.F12.12.m4.1.1.2.cmml" xref="S4.F12.12.m4.1.1.2">𝑃</ci><ci id="S4.F12.12.m4.1.1.3.cmml" xref="S4.F12.12.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.12.m4.1d">P_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.F12.12.m4.1e">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> (dashed lines). Left: all angles smaller than <math alttext="\pi" class="ltx_Math" display="inline" id="S4.F12.13.m5.1"><semantics id="S4.F12.13.m5.1b"><mi id="S4.F12.13.m5.1.1" xref="S4.F12.13.m5.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.F12.13.m5.1c"><ci id="S4.F12.13.m5.1.1.cmml" xref="S4.F12.13.m5.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.13.m5.1d">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.F12.13.m5.1e">italic_π</annotation></semantics></math>. Right: <math alttext="\alpha_{0}>\pi" class="ltx_Math" display="inline" id="S4.F12.14.m6.1"><semantics id="S4.F12.14.m6.1b"><mrow id="S4.F12.14.m6.1.1" xref="S4.F12.14.m6.1.1.cmml"><msub id="S4.F12.14.m6.1.1.2" xref="S4.F12.14.m6.1.1.2.cmml"><mi id="S4.F12.14.m6.1.1.2.2" xref="S4.F12.14.m6.1.1.2.2.cmml">α</mi><mn id="S4.F12.14.m6.1.1.2.3" xref="S4.F12.14.m6.1.1.2.3.cmml">0</mn></msub><mo id="S4.F12.14.m6.1.1.1" xref="S4.F12.14.m6.1.1.1.cmml">></mo><mi id="S4.F12.14.m6.1.1.3" xref="S4.F12.14.m6.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F12.14.m6.1c"><apply id="S4.F12.14.m6.1.1.cmml" xref="S4.F12.14.m6.1.1"><gt id="S4.F12.14.m6.1.1.1.cmml" xref="S4.F12.14.m6.1.1.1"></gt><apply id="S4.F12.14.m6.1.1.2.cmml" xref="S4.F12.14.m6.1.1.2"><csymbol cd="ambiguous" id="S4.F12.14.m6.1.1.2.1.cmml" xref="S4.F12.14.m6.1.1.2">subscript</csymbol><ci id="S4.F12.14.m6.1.1.2.2.cmml" xref="S4.F12.14.m6.1.1.2.2">𝛼</ci><cn id="S4.F12.14.m6.1.1.2.3.cmml" type="integer" xref="S4.F12.14.m6.1.1.2.3">0</cn></apply><ci id="S4.F12.14.m6.1.1.3.cmml" xref="S4.F12.14.m6.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F12.14.m6.1d">\alpha_{0}>\pi</annotation><annotation encoding="application/x-llamapun" id="S4.F12.14.m6.1e">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_π</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S4.13.p2"> <p class="ltx_p" id="S4.13.p2.16">Let <math alttext="P_{0},P_{1},P_{2}" class="ltx_Math" display="inline" id="S4.13.p2.1.m1.3"><semantics id="S4.13.p2.1.m1.3a"><mrow id="S4.13.p2.1.m1.3.3.3" xref="S4.13.p2.1.m1.3.3.4.cmml"><msub id="S4.13.p2.1.m1.1.1.1.1" xref="S4.13.p2.1.m1.1.1.1.1.cmml"><mi id="S4.13.p2.1.m1.1.1.1.1.2" xref="S4.13.p2.1.m1.1.1.1.1.2.cmml">P</mi><mn id="S4.13.p2.1.m1.1.1.1.1.3" xref="S4.13.p2.1.m1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.13.p2.1.m1.3.3.3.4" xref="S4.13.p2.1.m1.3.3.4.cmml">,</mo><msub id="S4.13.p2.1.m1.2.2.2.2" xref="S4.13.p2.1.m1.2.2.2.2.cmml"><mi id="S4.13.p2.1.m1.2.2.2.2.2" xref="S4.13.p2.1.m1.2.2.2.2.2.cmml">P</mi><mn id="S4.13.p2.1.m1.2.2.2.2.3" xref="S4.13.p2.1.m1.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.13.p2.1.m1.3.3.3.5" xref="S4.13.p2.1.m1.3.3.4.cmml">,</mo><msub id="S4.13.p2.1.m1.3.3.3.3" xref="S4.13.p2.1.m1.3.3.3.3.cmml"><mi id="S4.13.p2.1.m1.3.3.3.3.2" xref="S4.13.p2.1.m1.3.3.3.3.2.cmml">P</mi><mn id="S4.13.p2.1.m1.3.3.3.3.3" xref="S4.13.p2.1.m1.3.3.3.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.1.m1.3b"><list id="S4.13.p2.1.m1.3.3.4.cmml" xref="S4.13.p2.1.m1.3.3.3"><apply id="S4.13.p2.1.m1.1.1.1.1.cmml" xref="S4.13.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.13.p2.1.m1.1.1.1.1.1.cmml" xref="S4.13.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.13.p2.1.m1.1.1.1.1.2.cmml" xref="S4.13.p2.1.m1.1.1.1.1.2">𝑃</ci><cn id="S4.13.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.13.p2.1.m1.1.1.1.1.3">0</cn></apply><apply id="S4.13.p2.1.m1.2.2.2.2.cmml" xref="S4.13.p2.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.13.p2.1.m1.2.2.2.2.1.cmml" xref="S4.13.p2.1.m1.2.2.2.2">subscript</csymbol><ci id="S4.13.p2.1.m1.2.2.2.2.2.cmml" xref="S4.13.p2.1.m1.2.2.2.2.2">𝑃</ci><cn id="S4.13.p2.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.13.p2.1.m1.2.2.2.2.3">1</cn></apply><apply id="S4.13.p2.1.m1.3.3.3.3.cmml" xref="S4.13.p2.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S4.13.p2.1.m1.3.3.3.3.1.cmml" xref="S4.13.p2.1.m1.3.3.3.3">subscript</csymbol><ci id="S4.13.p2.1.m1.3.3.3.3.2.cmml" xref="S4.13.p2.1.m1.3.3.3.3.2">𝑃</ci><cn id="S4.13.p2.1.m1.3.3.3.3.3.cmml" type="integer" xref="S4.13.p2.1.m1.3.3.3.3.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.1.m1.3c">P_{0},P_{1},P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.1.m1.3d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S4.13.p2.2.m2.1"><semantics id="S4.13.p2.2.m2.1a"><mi id="S4.13.p2.2.m2.1.1" xref="S4.13.p2.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.13.p2.2.m2.1b"><ci id="S4.13.p2.2.m2.1.1.cmml" xref="S4.13.p2.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.2.m2.1d">italic_f</annotation></semantics></math>, and denote their interior angles by <math alttext="\alpha_{0},\alpha_{1},\alpha_{2}" class="ltx_Math" display="inline" id="S4.13.p2.3.m3.3"><semantics id="S4.13.p2.3.m3.3a"><mrow id="S4.13.p2.3.m3.3.3.3" xref="S4.13.p2.3.m3.3.3.4.cmml"><msub id="S4.13.p2.3.m3.1.1.1.1" xref="S4.13.p2.3.m3.1.1.1.1.cmml"><mi id="S4.13.p2.3.m3.1.1.1.1.2" xref="S4.13.p2.3.m3.1.1.1.1.2.cmml">α</mi><mn id="S4.13.p2.3.m3.1.1.1.1.3" xref="S4.13.p2.3.m3.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.13.p2.3.m3.3.3.3.4" xref="S4.13.p2.3.m3.3.3.4.cmml">,</mo><msub id="S4.13.p2.3.m3.2.2.2.2" xref="S4.13.p2.3.m3.2.2.2.2.cmml"><mi id="S4.13.p2.3.m3.2.2.2.2.2" xref="S4.13.p2.3.m3.2.2.2.2.2.cmml">α</mi><mn id="S4.13.p2.3.m3.2.2.2.2.3" xref="S4.13.p2.3.m3.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.13.p2.3.m3.3.3.3.5" xref="S4.13.p2.3.m3.3.3.4.cmml">,</mo><msub id="S4.13.p2.3.m3.3.3.3.3" xref="S4.13.p2.3.m3.3.3.3.3.cmml"><mi id="S4.13.p2.3.m3.3.3.3.3.2" xref="S4.13.p2.3.m3.3.3.3.3.2.cmml">α</mi><mn id="S4.13.p2.3.m3.3.3.3.3.3" xref="S4.13.p2.3.m3.3.3.3.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.3.m3.3b"><list id="S4.13.p2.3.m3.3.3.4.cmml" xref="S4.13.p2.3.m3.3.3.3"><apply id="S4.13.p2.3.m3.1.1.1.1.cmml" xref="S4.13.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.13.p2.3.m3.1.1.1.1.1.cmml" xref="S4.13.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="S4.13.p2.3.m3.1.1.1.1.2.cmml" xref="S4.13.p2.3.m3.1.1.1.1.2">𝛼</ci><cn id="S4.13.p2.3.m3.1.1.1.1.3.cmml" type="integer" xref="S4.13.p2.3.m3.1.1.1.1.3">0</cn></apply><apply id="S4.13.p2.3.m3.2.2.2.2.cmml" xref="S4.13.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.13.p2.3.m3.2.2.2.2.1.cmml" xref="S4.13.p2.3.m3.2.2.2.2">subscript</csymbol><ci id="S4.13.p2.3.m3.2.2.2.2.2.cmml" xref="S4.13.p2.3.m3.2.2.2.2.2">𝛼</ci><cn id="S4.13.p2.3.m3.2.2.2.2.3.cmml" type="integer" xref="S4.13.p2.3.m3.2.2.2.2.3">1</cn></apply><apply id="S4.13.p2.3.m3.3.3.3.3.cmml" xref="S4.13.p2.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="S4.13.p2.3.m3.3.3.3.3.1.cmml" xref="S4.13.p2.3.m3.3.3.3.3">subscript</csymbol><ci id="S4.13.p2.3.m3.3.3.3.3.2.cmml" xref="S4.13.p2.3.m3.3.3.3.3.2">𝛼</ci><cn id="S4.13.p2.3.m3.3.3.3.3.3.cmml" type="integer" xref="S4.13.p2.3.m3.3.3.3.3.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.3.m3.3c">\alpha_{0},\alpha_{1},\alpha_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.3.m3.3d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, cf. <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.F12" title="Figure 12 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 12</span></a>. For <math alttext="i\in\{0,1,2\}" class="ltx_Math" display="inline" id="S4.13.p2.4.m4.3"><semantics id="S4.13.p2.4.m4.3a"><mrow id="S4.13.p2.4.m4.3.4" xref="S4.13.p2.4.m4.3.4.cmml"><mi id="S4.13.p2.4.m4.3.4.2" xref="S4.13.p2.4.m4.3.4.2.cmml">i</mi><mo id="S4.13.p2.4.m4.3.4.1" xref="S4.13.p2.4.m4.3.4.1.cmml">∈</mo><mrow id="S4.13.p2.4.m4.3.4.3.2" xref="S4.13.p2.4.m4.3.4.3.1.cmml"><mo id="S4.13.p2.4.m4.3.4.3.2.1" stretchy="false" xref="S4.13.p2.4.m4.3.4.3.1.cmml">{</mo><mn id="S4.13.p2.4.m4.1.1" xref="S4.13.p2.4.m4.1.1.cmml">0</mn><mo id="S4.13.p2.4.m4.3.4.3.2.2" xref="S4.13.p2.4.m4.3.4.3.1.cmml">,</mo><mn id="S4.13.p2.4.m4.2.2" xref="S4.13.p2.4.m4.2.2.cmml">1</mn><mo id="S4.13.p2.4.m4.3.4.3.2.3" xref="S4.13.p2.4.m4.3.4.3.1.cmml">,</mo><mn id="S4.13.p2.4.m4.3.3" xref="S4.13.p2.4.m4.3.3.cmml">2</mn><mo id="S4.13.p2.4.m4.3.4.3.2.4" stretchy="false" xref="S4.13.p2.4.m4.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.4.m4.3b"><apply id="S4.13.p2.4.m4.3.4.cmml" xref="S4.13.p2.4.m4.3.4"><in id="S4.13.p2.4.m4.3.4.1.cmml" xref="S4.13.p2.4.m4.3.4.1"></in><ci id="S4.13.p2.4.m4.3.4.2.cmml" xref="S4.13.p2.4.m4.3.4.2">𝑖</ci><set id="S4.13.p2.4.m4.3.4.3.1.cmml" xref="S4.13.p2.4.m4.3.4.3.2"><cn id="S4.13.p2.4.m4.1.1.cmml" type="integer" xref="S4.13.p2.4.m4.1.1">0</cn><cn id="S4.13.p2.4.m4.2.2.cmml" type="integer" xref="S4.13.p2.4.m4.2.2">1</cn><cn id="S4.13.p2.4.m4.3.3.cmml" type="integer" xref="S4.13.p2.4.m4.3.3">2</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.4.m4.3c">i\in\{0,1,2\}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.4.m4.3d">italic_i ∈ { 0 , 1 , 2 }</annotation></semantics></math>, let <math alttext="f_{i}" class="ltx_Math" display="inline" id="S4.13.p2.5.m5.1"><semantics id="S4.13.p2.5.m5.1a"><msub id="S4.13.p2.5.m5.1.1" xref="S4.13.p2.5.m5.1.1.cmml"><mi id="S4.13.p2.5.m5.1.1.2" xref="S4.13.p2.5.m5.1.1.2.cmml">f</mi><mi id="S4.13.p2.5.m5.1.1.3" xref="S4.13.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.5.m5.1b"><apply id="S4.13.p2.5.m5.1.1.cmml" xref="S4.13.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.13.p2.5.m5.1.1.1.cmml" xref="S4.13.p2.5.m5.1.1">subscript</csymbol><ci id="S4.13.p2.5.m5.1.1.2.cmml" xref="S4.13.p2.5.m5.1.1.2">𝑓</ci><ci id="S4.13.p2.5.m5.1.1.3.cmml" xref="S4.13.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.5.m5.1c">f_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the affine component of <math alttext="P_{i}" class="ltx_Math" display="inline" id="S4.13.p2.6.m6.1"><semantics id="S4.13.p2.6.m6.1a"><msub id="S4.13.p2.6.m6.1.1" xref="S4.13.p2.6.m6.1.1.cmml"><mi id="S4.13.p2.6.m6.1.1.2" xref="S4.13.p2.6.m6.1.1.2.cmml">P</mi><mi id="S4.13.p2.6.m6.1.1.3" xref="S4.13.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.6.m6.1b"><apply id="S4.13.p2.6.m6.1.1.cmml" xref="S4.13.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.13.p2.6.m6.1.1.1.cmml" xref="S4.13.p2.6.m6.1.1">subscript</csymbol><ci id="S4.13.p2.6.m6.1.1.2.cmml" xref="S4.13.p2.6.m6.1.1.2">𝑃</ci><ci id="S4.13.p2.6.m6.1.1.3.cmml" xref="S4.13.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.6.m6.1c">P_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.6.m6.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="p_{0},p_{1},p_{2}\neq 0" class="ltx_Math" display="inline" id="S4.13.p2.7.m7.3"><semantics id="S4.13.p2.7.m7.3a"><mrow id="S4.13.p2.7.m7.3.3" xref="S4.13.p2.7.m7.3.3.cmml"><mrow id="S4.13.p2.7.m7.3.3.3.3" xref="S4.13.p2.7.m7.3.3.3.4.cmml"><msub id="S4.13.p2.7.m7.1.1.1.1.1" xref="S4.13.p2.7.m7.1.1.1.1.1.cmml"><mi id="S4.13.p2.7.m7.1.1.1.1.1.2" xref="S4.13.p2.7.m7.1.1.1.1.1.2.cmml">p</mi><mn id="S4.13.p2.7.m7.1.1.1.1.1.3" xref="S4.13.p2.7.m7.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.13.p2.7.m7.3.3.3.3.4" xref="S4.13.p2.7.m7.3.3.3.4.cmml">,</mo><msub id="S4.13.p2.7.m7.2.2.2.2.2" xref="S4.13.p2.7.m7.2.2.2.2.2.cmml"><mi id="S4.13.p2.7.m7.2.2.2.2.2.2" xref="S4.13.p2.7.m7.2.2.2.2.2.2.cmml">p</mi><mn id="S4.13.p2.7.m7.2.2.2.2.2.3" xref="S4.13.p2.7.m7.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.13.p2.7.m7.3.3.3.3.5" xref="S4.13.p2.7.m7.3.3.3.4.cmml">,</mo><msub id="S4.13.p2.7.m7.3.3.3.3.3" xref="S4.13.p2.7.m7.3.3.3.3.3.cmml"><mi id="S4.13.p2.7.m7.3.3.3.3.3.2" xref="S4.13.p2.7.m7.3.3.3.3.3.2.cmml">p</mi><mn id="S4.13.p2.7.m7.3.3.3.3.3.3" xref="S4.13.p2.7.m7.3.3.3.3.3.3.cmml">2</mn></msub></mrow><mo id="S4.13.p2.7.m7.3.3.4" xref="S4.13.p2.7.m7.3.3.4.cmml">≠</mo><mn id="S4.13.p2.7.m7.3.3.5" xref="S4.13.p2.7.m7.3.3.5.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.7.m7.3b"><apply id="S4.13.p2.7.m7.3.3.cmml" xref="S4.13.p2.7.m7.3.3"><neq id="S4.13.p2.7.m7.3.3.4.cmml" xref="S4.13.p2.7.m7.3.3.4"></neq><list id="S4.13.p2.7.m7.3.3.3.4.cmml" xref="S4.13.p2.7.m7.3.3.3.3"><apply id="S4.13.p2.7.m7.1.1.1.1.1.cmml" xref="S4.13.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.13.p2.7.m7.1.1.1.1.1.1.cmml" xref="S4.13.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S4.13.p2.7.m7.1.1.1.1.1.2.cmml" xref="S4.13.p2.7.m7.1.1.1.1.1.2">𝑝</ci><cn id="S4.13.p2.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S4.13.p2.7.m7.1.1.1.1.1.3">0</cn></apply><apply id="S4.13.p2.7.m7.2.2.2.2.2.cmml" xref="S4.13.p2.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.13.p2.7.m7.2.2.2.2.2.1.cmml" xref="S4.13.p2.7.m7.2.2.2.2.2">subscript</csymbol><ci id="S4.13.p2.7.m7.2.2.2.2.2.2.cmml" xref="S4.13.p2.7.m7.2.2.2.2.2.2">𝑝</ci><cn id="S4.13.p2.7.m7.2.2.2.2.2.3.cmml" type="integer" xref="S4.13.p2.7.m7.2.2.2.2.2.3">1</cn></apply><apply id="S4.13.p2.7.m7.3.3.3.3.3.cmml" xref="S4.13.p2.7.m7.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.13.p2.7.m7.3.3.3.3.3.1.cmml" xref="S4.13.p2.7.m7.3.3.3.3.3">subscript</csymbol><ci id="S4.13.p2.7.m7.3.3.3.3.3.2.cmml" xref="S4.13.p2.7.m7.3.3.3.3.3.2">𝑝</ci><cn id="S4.13.p2.7.m7.3.3.3.3.3.3.cmml" type="integer" xref="S4.13.p2.7.m7.3.3.3.3.3.3">2</cn></apply></list><cn id="S4.13.p2.7.m7.3.3.5.cmml" type="integer" xref="S4.13.p2.7.m7.3.3.5">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.7.m7.3c">p_{0},p_{1},p_{2}\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.7.m7.3d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ 0</annotation></semantics></math> be points on the ray between <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.13.p2.8.m8.1"><semantics id="S4.13.p2.8.m8.1a"><msub id="S4.13.p2.8.m8.1.1" xref="S4.13.p2.8.m8.1.1.cmml"><mi id="S4.13.p2.8.m8.1.1.2" xref="S4.13.p2.8.m8.1.1.2.cmml">P</mi><mn id="S4.13.p2.8.m8.1.1.3" xref="S4.13.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.8.m8.1b"><apply id="S4.13.p2.8.m8.1.1.cmml" xref="S4.13.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.13.p2.8.m8.1.1.1.cmml" xref="S4.13.p2.8.m8.1.1">subscript</csymbol><ci id="S4.13.p2.8.m8.1.1.2.cmml" xref="S4.13.p2.8.m8.1.1.2">𝑃</ci><cn id="S4.13.p2.8.m8.1.1.3.cmml" type="integer" xref="S4.13.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.8.m8.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.8.m8.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.13.p2.9.m9.1"><semantics id="S4.13.p2.9.m9.1a"><msub id="S4.13.p2.9.m9.1.1" xref="S4.13.p2.9.m9.1.1.cmml"><mi id="S4.13.p2.9.m9.1.1.2" xref="S4.13.p2.9.m9.1.1.2.cmml">P</mi><mn id="S4.13.p2.9.m9.1.1.3" xref="S4.13.p2.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.9.m9.1b"><apply id="S4.13.p2.9.m9.1.1.cmml" xref="S4.13.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.13.p2.9.m9.1.1.1.cmml" xref="S4.13.p2.9.m9.1.1">subscript</csymbol><ci id="S4.13.p2.9.m9.1.1.2.cmml" xref="S4.13.p2.9.m9.1.1.2">𝑃</ci><cn id="S4.13.p2.9.m9.1.1.3.cmml" type="integer" xref="S4.13.p2.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.9.m9.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.9.m9.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.13.p2.10.m10.1"><semantics id="S4.13.p2.10.m10.1a"><msub id="S4.13.p2.10.m10.1.1" xref="S4.13.p2.10.m10.1.1.cmml"><mi id="S4.13.p2.10.m10.1.1.2" xref="S4.13.p2.10.m10.1.1.2.cmml">P</mi><mn id="S4.13.p2.10.m10.1.1.3" xref="S4.13.p2.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.10.m10.1b"><apply id="S4.13.p2.10.m10.1.1.cmml" xref="S4.13.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.13.p2.10.m10.1.1.1.cmml" xref="S4.13.p2.10.m10.1.1">subscript</csymbol><ci id="S4.13.p2.10.m10.1.1.2.cmml" xref="S4.13.p2.10.m10.1.1.2">𝑃</ci><cn id="S4.13.p2.10.m10.1.1.3.cmml" type="integer" xref="S4.13.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.10.m10.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.10.m10.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.13.p2.11.m11.1"><semantics id="S4.13.p2.11.m11.1a"><msub id="S4.13.p2.11.m11.1.1" xref="S4.13.p2.11.m11.1.1.cmml"><mi id="S4.13.p2.11.m11.1.1.2" xref="S4.13.p2.11.m11.1.1.2.cmml">P</mi><mn id="S4.13.p2.11.m11.1.1.3" xref="S4.13.p2.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.11.m11.1b"><apply id="S4.13.p2.11.m11.1.1.cmml" xref="S4.13.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S4.13.p2.11.m11.1.1.1.cmml" xref="S4.13.p2.11.m11.1.1">subscript</csymbol><ci id="S4.13.p2.11.m11.1.1.2.cmml" xref="S4.13.p2.11.m11.1.1.2">𝑃</ci><cn id="S4.13.p2.11.m11.1.1.3.cmml" type="integer" xref="S4.13.p2.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.11.m11.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.11.m11.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.13.p2.12.m12.1"><semantics id="S4.13.p2.12.m12.1a"><msub id="S4.13.p2.12.m12.1.1" xref="S4.13.p2.12.m12.1.1.cmml"><mi id="S4.13.p2.12.m12.1.1.2" xref="S4.13.p2.12.m12.1.1.2.cmml">P</mi><mn id="S4.13.p2.12.m12.1.1.3" xref="S4.13.p2.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.12.m12.1b"><apply id="S4.13.p2.12.m12.1.1.cmml" xref="S4.13.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S4.13.p2.12.m12.1.1.1.cmml" xref="S4.13.p2.12.m12.1.1">subscript</csymbol><ci id="S4.13.p2.12.m12.1.1.2.cmml" xref="S4.13.p2.12.m12.1.1.2">𝑃</ci><cn id="S4.13.p2.12.m12.1.1.3.cmml" type="integer" xref="S4.13.p2.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.12.m12.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.12.m12.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.13.p2.13.m13.1"><semantics id="S4.13.p2.13.m13.1a"><msub id="S4.13.p2.13.m13.1.1" xref="S4.13.p2.13.m13.1.1.cmml"><mi id="S4.13.p2.13.m13.1.1.2" xref="S4.13.p2.13.m13.1.1.2.cmml">P</mi><mn id="S4.13.p2.13.m13.1.1.3" xref="S4.13.p2.13.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.13.p2.13.m13.1b"><apply id="S4.13.p2.13.m13.1.1.cmml" xref="S4.13.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.13.p2.13.m13.1.1.1.cmml" xref="S4.13.p2.13.m13.1.1">subscript</csymbol><ci id="S4.13.p2.13.m13.1.1.2.cmml" xref="S4.13.p2.13.m13.1.1.2">𝑃</ci><cn id="S4.13.p2.13.m13.1.1.3.cmml" type="integer" xref="S4.13.p2.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.13.m13.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.13.m13.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> respectively. Further, define <math alttext="l_{0}:=\operatorname*{aff}(\overline{vp_{0}})" class="ltx_Math" display="inline" id="S4.13.p2.14.m14.2"><semantics id="S4.13.p2.14.m14.2a"><mrow id="S4.13.p2.14.m14.2.3" xref="S4.13.p2.14.m14.2.3.cmml"><msub id="S4.13.p2.14.m14.2.3.2" xref="S4.13.p2.14.m14.2.3.2.cmml"><mi id="S4.13.p2.14.m14.2.3.2.2" xref="S4.13.p2.14.m14.2.3.2.2.cmml">l</mi><mn id="S4.13.p2.14.m14.2.3.2.3" xref="S4.13.p2.14.m14.2.3.2.3.cmml">0</mn></msub><mo id="S4.13.p2.14.m14.2.3.1" lspace="0.278em" xref="S4.13.p2.14.m14.2.3.1.cmml">:=</mo><mrow id="S4.13.p2.14.m14.2.3.3.2" xref="S4.13.p2.14.m14.2.3.3.1.cmml"><mo id="S4.13.p2.14.m14.1.1" lspace="0.111em" rspace="0em" xref="S4.13.p2.14.m14.1.1.cmml">aff</mo><mrow id="S4.13.p2.14.m14.2.3.3.2.1" xref="S4.13.p2.14.m14.2.3.3.1.cmml"><mo id="S4.13.p2.14.m14.2.3.3.2.1.1" stretchy="false" xref="S4.13.p2.14.m14.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.13.p2.14.m14.2.2" xref="S4.13.p2.14.m14.2.2.cmml"><mrow id="S4.13.p2.14.m14.2.2.2" xref="S4.13.p2.14.m14.2.2.2.cmml"><mi id="S4.13.p2.14.m14.2.2.2.2" xref="S4.13.p2.14.m14.2.2.2.2.cmml">v</mi><mo id="S4.13.p2.14.m14.2.2.2.1" xref="S4.13.p2.14.m14.2.2.2.1.cmml">⁢</mo><msub id="S4.13.p2.14.m14.2.2.2.3" xref="S4.13.p2.14.m14.2.2.2.3.cmml"><mi id="S4.13.p2.14.m14.2.2.2.3.2" xref="S4.13.p2.14.m14.2.2.2.3.2.cmml">p</mi><mn id="S4.13.p2.14.m14.2.2.2.3.3" xref="S4.13.p2.14.m14.2.2.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.13.p2.14.m14.2.2.1" xref="S4.13.p2.14.m14.2.2.1.cmml">¯</mo></mover><mo id="S4.13.p2.14.m14.2.3.3.2.1.2" stretchy="false" xref="S4.13.p2.14.m14.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.14.m14.2b"><apply id="S4.13.p2.14.m14.2.3.cmml" xref="S4.13.p2.14.m14.2.3"><csymbol cd="latexml" id="S4.13.p2.14.m14.2.3.1.cmml" xref="S4.13.p2.14.m14.2.3.1">assign</csymbol><apply id="S4.13.p2.14.m14.2.3.2.cmml" xref="S4.13.p2.14.m14.2.3.2"><csymbol cd="ambiguous" id="S4.13.p2.14.m14.2.3.2.1.cmml" xref="S4.13.p2.14.m14.2.3.2">subscript</csymbol><ci id="S4.13.p2.14.m14.2.3.2.2.cmml" xref="S4.13.p2.14.m14.2.3.2.2">𝑙</ci><cn id="S4.13.p2.14.m14.2.3.2.3.cmml" type="integer" xref="S4.13.p2.14.m14.2.3.2.3">0</cn></apply><apply id="S4.13.p2.14.m14.2.3.3.1.cmml" xref="S4.13.p2.14.m14.2.3.3.2"><ci id="S4.13.p2.14.m14.1.1.cmml" xref="S4.13.p2.14.m14.1.1">aff</ci><apply id="S4.13.p2.14.m14.2.2.cmml" xref="S4.13.p2.14.m14.2.2"><ci id="S4.13.p2.14.m14.2.2.1.cmml" xref="S4.13.p2.14.m14.2.2.1">¯</ci><apply id="S4.13.p2.14.m14.2.2.2.cmml" xref="S4.13.p2.14.m14.2.2.2"><times id="S4.13.p2.14.m14.2.2.2.1.cmml" xref="S4.13.p2.14.m14.2.2.2.1"></times><ci id="S4.13.p2.14.m14.2.2.2.2.cmml" xref="S4.13.p2.14.m14.2.2.2.2">𝑣</ci><apply id="S4.13.p2.14.m14.2.2.2.3.cmml" xref="S4.13.p2.14.m14.2.2.2.3"><csymbol cd="ambiguous" id="S4.13.p2.14.m14.2.2.2.3.1.cmml" xref="S4.13.p2.14.m14.2.2.2.3">subscript</csymbol><ci id="S4.13.p2.14.m14.2.2.2.3.2.cmml" xref="S4.13.p2.14.m14.2.2.2.3.2">𝑝</ci><cn id="S4.13.p2.14.m14.2.2.2.3.3.cmml" type="integer" xref="S4.13.p2.14.m14.2.2.2.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.14.m14.2c">l_{0}:=\operatorname*{aff}(\overline{vp_{0}})</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.14.m14.2d">italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := roman_aff ( over¯ start_ARG italic_v italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>, <math alttext="l_{1}:=\operatorname*{aff}(\overline{vp_{1}})" class="ltx_Math" display="inline" id="S4.13.p2.15.m15.2"><semantics id="S4.13.p2.15.m15.2a"><mrow id="S4.13.p2.15.m15.2.3" xref="S4.13.p2.15.m15.2.3.cmml"><msub id="S4.13.p2.15.m15.2.3.2" xref="S4.13.p2.15.m15.2.3.2.cmml"><mi id="S4.13.p2.15.m15.2.3.2.2" xref="S4.13.p2.15.m15.2.3.2.2.cmml">l</mi><mn id="S4.13.p2.15.m15.2.3.2.3" xref="S4.13.p2.15.m15.2.3.2.3.cmml">1</mn></msub><mo id="S4.13.p2.15.m15.2.3.1" lspace="0.278em" xref="S4.13.p2.15.m15.2.3.1.cmml">:=</mo><mrow id="S4.13.p2.15.m15.2.3.3.2" xref="S4.13.p2.15.m15.2.3.3.1.cmml"><mo id="S4.13.p2.15.m15.1.1" lspace="0.111em" rspace="0em" xref="S4.13.p2.15.m15.1.1.cmml">aff</mo><mrow id="S4.13.p2.15.m15.2.3.3.2.1" xref="S4.13.p2.15.m15.2.3.3.1.cmml"><mo id="S4.13.p2.15.m15.2.3.3.2.1.1" stretchy="false" xref="S4.13.p2.15.m15.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.13.p2.15.m15.2.2" xref="S4.13.p2.15.m15.2.2.cmml"><mrow id="S4.13.p2.15.m15.2.2.2" xref="S4.13.p2.15.m15.2.2.2.cmml"><mi id="S4.13.p2.15.m15.2.2.2.2" xref="S4.13.p2.15.m15.2.2.2.2.cmml">v</mi><mo id="S4.13.p2.15.m15.2.2.2.1" xref="S4.13.p2.15.m15.2.2.2.1.cmml">⁢</mo><msub id="S4.13.p2.15.m15.2.2.2.3" xref="S4.13.p2.15.m15.2.2.2.3.cmml"><mi id="S4.13.p2.15.m15.2.2.2.3.2" xref="S4.13.p2.15.m15.2.2.2.3.2.cmml">p</mi><mn id="S4.13.p2.15.m15.2.2.2.3.3" xref="S4.13.p2.15.m15.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.13.p2.15.m15.2.2.1" xref="S4.13.p2.15.m15.2.2.1.cmml">¯</mo></mover><mo id="S4.13.p2.15.m15.2.3.3.2.1.2" stretchy="false" xref="S4.13.p2.15.m15.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.15.m15.2b"><apply id="S4.13.p2.15.m15.2.3.cmml" xref="S4.13.p2.15.m15.2.3"><csymbol cd="latexml" id="S4.13.p2.15.m15.2.3.1.cmml" xref="S4.13.p2.15.m15.2.3.1">assign</csymbol><apply id="S4.13.p2.15.m15.2.3.2.cmml" xref="S4.13.p2.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.13.p2.15.m15.2.3.2.1.cmml" xref="S4.13.p2.15.m15.2.3.2">subscript</csymbol><ci id="S4.13.p2.15.m15.2.3.2.2.cmml" xref="S4.13.p2.15.m15.2.3.2.2">𝑙</ci><cn id="S4.13.p2.15.m15.2.3.2.3.cmml" type="integer" xref="S4.13.p2.15.m15.2.3.2.3">1</cn></apply><apply id="S4.13.p2.15.m15.2.3.3.1.cmml" xref="S4.13.p2.15.m15.2.3.3.2"><ci id="S4.13.p2.15.m15.1.1.cmml" xref="S4.13.p2.15.m15.1.1">aff</ci><apply id="S4.13.p2.15.m15.2.2.cmml" xref="S4.13.p2.15.m15.2.2"><ci id="S4.13.p2.15.m15.2.2.1.cmml" xref="S4.13.p2.15.m15.2.2.1">¯</ci><apply id="S4.13.p2.15.m15.2.2.2.cmml" xref="S4.13.p2.15.m15.2.2.2"><times id="S4.13.p2.15.m15.2.2.2.1.cmml" xref="S4.13.p2.15.m15.2.2.2.1"></times><ci id="S4.13.p2.15.m15.2.2.2.2.cmml" xref="S4.13.p2.15.m15.2.2.2.2">𝑣</ci><apply id="S4.13.p2.15.m15.2.2.2.3.cmml" xref="S4.13.p2.15.m15.2.2.2.3"><csymbol cd="ambiguous" id="S4.13.p2.15.m15.2.2.2.3.1.cmml" xref="S4.13.p2.15.m15.2.2.2.3">subscript</csymbol><ci id="S4.13.p2.15.m15.2.2.2.3.2.cmml" xref="S4.13.p2.15.m15.2.2.2.3.2">𝑝</ci><cn id="S4.13.p2.15.m15.2.2.2.3.3.cmml" type="integer" xref="S4.13.p2.15.m15.2.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.15.m15.2c">l_{1}:=\operatorname*{aff}(\overline{vp_{1}})</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.15.m15.2d">italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := roman_aff ( over¯ start_ARG italic_v italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>, and <math alttext="l_{2}:=\operatorname*{aff}(\overline{vp_{2}})" class="ltx_Math" display="inline" id="S4.13.p2.16.m16.2"><semantics id="S4.13.p2.16.m16.2a"><mrow id="S4.13.p2.16.m16.2.3" xref="S4.13.p2.16.m16.2.3.cmml"><msub id="S4.13.p2.16.m16.2.3.2" xref="S4.13.p2.16.m16.2.3.2.cmml"><mi id="S4.13.p2.16.m16.2.3.2.2" xref="S4.13.p2.16.m16.2.3.2.2.cmml">l</mi><mn id="S4.13.p2.16.m16.2.3.2.3" xref="S4.13.p2.16.m16.2.3.2.3.cmml">2</mn></msub><mo id="S4.13.p2.16.m16.2.3.1" lspace="0.278em" xref="S4.13.p2.16.m16.2.3.1.cmml">:=</mo><mrow id="S4.13.p2.16.m16.2.3.3.2" xref="S4.13.p2.16.m16.2.3.3.1.cmml"><mo id="S4.13.p2.16.m16.1.1" lspace="0.111em" rspace="0em" xref="S4.13.p2.16.m16.1.1.cmml">aff</mo><mrow id="S4.13.p2.16.m16.2.3.3.2.1" xref="S4.13.p2.16.m16.2.3.3.1.cmml"><mo id="S4.13.p2.16.m16.2.3.3.2.1.1" stretchy="false" xref="S4.13.p2.16.m16.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.13.p2.16.m16.2.2" xref="S4.13.p2.16.m16.2.2.cmml"><mrow id="S4.13.p2.16.m16.2.2.2" xref="S4.13.p2.16.m16.2.2.2.cmml"><mi id="S4.13.p2.16.m16.2.2.2.2" xref="S4.13.p2.16.m16.2.2.2.2.cmml">v</mi><mo id="S4.13.p2.16.m16.2.2.2.1" xref="S4.13.p2.16.m16.2.2.2.1.cmml">⁢</mo><msub id="S4.13.p2.16.m16.2.2.2.3" xref="S4.13.p2.16.m16.2.2.2.3.cmml"><mi id="S4.13.p2.16.m16.2.2.2.3.2" xref="S4.13.p2.16.m16.2.2.2.3.2.cmml">p</mi><mn id="S4.13.p2.16.m16.2.2.2.3.3" xref="S4.13.p2.16.m16.2.2.2.3.3.cmml">2</mn></msub></mrow><mo id="S4.13.p2.16.m16.2.2.1" xref="S4.13.p2.16.m16.2.2.1.cmml">¯</mo></mover><mo id="S4.13.p2.16.m16.2.3.3.2.1.2" stretchy="false" xref="S4.13.p2.16.m16.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.13.p2.16.m16.2b"><apply id="S4.13.p2.16.m16.2.3.cmml" xref="S4.13.p2.16.m16.2.3"><csymbol cd="latexml" id="S4.13.p2.16.m16.2.3.1.cmml" xref="S4.13.p2.16.m16.2.3.1">assign</csymbol><apply id="S4.13.p2.16.m16.2.3.2.cmml" xref="S4.13.p2.16.m16.2.3.2"><csymbol cd="ambiguous" id="S4.13.p2.16.m16.2.3.2.1.cmml" xref="S4.13.p2.16.m16.2.3.2">subscript</csymbol><ci id="S4.13.p2.16.m16.2.3.2.2.cmml" xref="S4.13.p2.16.m16.2.3.2.2">𝑙</ci><cn id="S4.13.p2.16.m16.2.3.2.3.cmml" type="integer" xref="S4.13.p2.16.m16.2.3.2.3">2</cn></apply><apply id="S4.13.p2.16.m16.2.3.3.1.cmml" xref="S4.13.p2.16.m16.2.3.3.2"><ci id="S4.13.p2.16.m16.1.1.cmml" xref="S4.13.p2.16.m16.1.1">aff</ci><apply id="S4.13.p2.16.m16.2.2.cmml" xref="S4.13.p2.16.m16.2.2"><ci id="S4.13.p2.16.m16.2.2.1.cmml" xref="S4.13.p2.16.m16.2.2.1">¯</ci><apply id="S4.13.p2.16.m16.2.2.2.cmml" xref="S4.13.p2.16.m16.2.2.2"><times id="S4.13.p2.16.m16.2.2.2.1.cmml" xref="S4.13.p2.16.m16.2.2.2.1"></times><ci id="S4.13.p2.16.m16.2.2.2.2.cmml" xref="S4.13.p2.16.m16.2.2.2.2">𝑣</ci><apply id="S4.13.p2.16.m16.2.2.2.3.cmml" xref="S4.13.p2.16.m16.2.2.2.3"><csymbol cd="ambiguous" id="S4.13.p2.16.m16.2.2.2.3.1.cmml" xref="S4.13.p2.16.m16.2.2.2.3">subscript</csymbol><ci id="S4.13.p2.16.m16.2.2.2.3.2.cmml" xref="S4.13.p2.16.m16.2.2.2.3.2">𝑝</ci><cn id="S4.13.p2.16.m16.2.2.2.3.3.cmml" type="integer" xref="S4.13.p2.16.m16.2.2.2.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.13.p2.16.m16.2c">l_{2}:=\operatorname*{aff}(\overline{vp_{2}})</annotation><annotation encoding="application/x-llamapun" id="S4.13.p2.16.m16.2d">italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := roman_aff ( over¯ start_ARG italic_v italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.14.p3"> <p class="ltx_p" id="S4.14.p3.37">First, we consider the case that <math alttext="\alpha_{i}<\pi" class="ltx_Math" display="inline" id="S4.14.p3.1.m1.1"><semantics id="S4.14.p3.1.m1.1a"><mrow id="S4.14.p3.1.m1.1.1" xref="S4.14.p3.1.m1.1.1.cmml"><msub id="S4.14.p3.1.m1.1.1.2" xref="S4.14.p3.1.m1.1.1.2.cmml"><mi id="S4.14.p3.1.m1.1.1.2.2" xref="S4.14.p3.1.m1.1.1.2.2.cmml">α</mi><mi id="S4.14.p3.1.m1.1.1.2.3" xref="S4.14.p3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.14.p3.1.m1.1.1.1" xref="S4.14.p3.1.m1.1.1.1.cmml"><</mo><mi id="S4.14.p3.1.m1.1.1.3" xref="S4.14.p3.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.1.m1.1b"><apply id="S4.14.p3.1.m1.1.1.cmml" xref="S4.14.p3.1.m1.1.1"><lt id="S4.14.p3.1.m1.1.1.1.cmml" xref="S4.14.p3.1.m1.1.1.1"></lt><apply id="S4.14.p3.1.m1.1.1.2.cmml" xref="S4.14.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.1.m1.1.1.2.1.cmml" xref="S4.14.p3.1.m1.1.1.2">subscript</csymbol><ci id="S4.14.p3.1.m1.1.1.2.2.cmml" xref="S4.14.p3.1.m1.1.1.2.2">𝛼</ci><ci id="S4.14.p3.1.m1.1.1.2.3.cmml" xref="S4.14.p3.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S4.14.p3.1.m1.1.1.3.cmml" xref="S4.14.p3.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.1.m1.1c">\alpha_{i}<\pi</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < italic_π</annotation></semantics></math> for all <math alttext="i\in\{0,1,2\}" class="ltx_Math" display="inline" id="S4.14.p3.2.m2.3"><semantics id="S4.14.p3.2.m2.3a"><mrow id="S4.14.p3.2.m2.3.4" xref="S4.14.p3.2.m2.3.4.cmml"><mi id="S4.14.p3.2.m2.3.4.2" xref="S4.14.p3.2.m2.3.4.2.cmml">i</mi><mo id="S4.14.p3.2.m2.3.4.1" xref="S4.14.p3.2.m2.3.4.1.cmml">∈</mo><mrow id="S4.14.p3.2.m2.3.4.3.2" xref="S4.14.p3.2.m2.3.4.3.1.cmml"><mo id="S4.14.p3.2.m2.3.4.3.2.1" stretchy="false" xref="S4.14.p3.2.m2.3.4.3.1.cmml">{</mo><mn id="S4.14.p3.2.m2.1.1" xref="S4.14.p3.2.m2.1.1.cmml">0</mn><mo id="S4.14.p3.2.m2.3.4.3.2.2" xref="S4.14.p3.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.14.p3.2.m2.2.2" xref="S4.14.p3.2.m2.2.2.cmml">1</mn><mo id="S4.14.p3.2.m2.3.4.3.2.3" xref="S4.14.p3.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.14.p3.2.m2.3.3" xref="S4.14.p3.2.m2.3.3.cmml">2</mn><mo id="S4.14.p3.2.m2.3.4.3.2.4" stretchy="false" xref="S4.14.p3.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.2.m2.3b"><apply id="S4.14.p3.2.m2.3.4.cmml" xref="S4.14.p3.2.m2.3.4"><in id="S4.14.p3.2.m2.3.4.1.cmml" xref="S4.14.p3.2.m2.3.4.1"></in><ci id="S4.14.p3.2.m2.3.4.2.cmml" xref="S4.14.p3.2.m2.3.4.2">𝑖</ci><set id="S4.14.p3.2.m2.3.4.3.1.cmml" xref="S4.14.p3.2.m2.3.4.3.2"><cn id="S4.14.p3.2.m2.1.1.cmml" type="integer" xref="S4.14.p3.2.m2.1.1">0</cn><cn id="S4.14.p3.2.m2.2.2.cmml" type="integer" xref="S4.14.p3.2.m2.2.2">1</cn><cn id="S4.14.p3.2.m2.3.3.cmml" type="integer" xref="S4.14.p3.2.m2.3.3">2</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.2.m2.3c">i\in\{0,1,2\}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.2.m2.3d">italic_i ∈ { 0 , 1 , 2 }</annotation></semantics></math>. Consider the case that <math alttext="f(p_{0})=f_{1}(p_{0})=f_{2}(p_{0})\geq f_{0}(p_{0})" class="ltx_Math" display="inline" id="S4.14.p3.3.m3.4"><semantics id="S4.14.p3.3.m3.4a"><mrow id="S4.14.p3.3.m3.4.4" xref="S4.14.p3.3.m3.4.4.cmml"><mrow id="S4.14.p3.3.m3.1.1.1" xref="S4.14.p3.3.m3.1.1.1.cmml"><mi id="S4.14.p3.3.m3.1.1.1.3" xref="S4.14.p3.3.m3.1.1.1.3.cmml">f</mi><mo id="S4.14.p3.3.m3.1.1.1.2" xref="S4.14.p3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.14.p3.3.m3.1.1.1.1.1" xref="S4.14.p3.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.14.p3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.14.p3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S4.14.p3.3.m3.1.1.1.1.1.1" xref="S4.14.p3.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.14.p3.3.m3.1.1.1.1.1.1.2" xref="S4.14.p3.3.m3.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.3.m3.1.1.1.1.1.1.3" xref="S4.14.p3.3.m3.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.14.p3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.3.m3.4.4.6" xref="S4.14.p3.3.m3.4.4.6.cmml">=</mo><mrow id="S4.14.p3.3.m3.2.2.2" xref="S4.14.p3.3.m3.2.2.2.cmml"><msub id="S4.14.p3.3.m3.2.2.2.3" xref="S4.14.p3.3.m3.2.2.2.3.cmml"><mi id="S4.14.p3.3.m3.2.2.2.3.2" xref="S4.14.p3.3.m3.2.2.2.3.2.cmml">f</mi><mn id="S4.14.p3.3.m3.2.2.2.3.3" xref="S4.14.p3.3.m3.2.2.2.3.3.cmml">1</mn></msub><mo id="S4.14.p3.3.m3.2.2.2.2" xref="S4.14.p3.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S4.14.p3.3.m3.2.2.2.1.1" xref="S4.14.p3.3.m3.2.2.2.1.1.1.cmml"><mo id="S4.14.p3.3.m3.2.2.2.1.1.2" stretchy="false" xref="S4.14.p3.3.m3.2.2.2.1.1.1.cmml">(</mo><msub id="S4.14.p3.3.m3.2.2.2.1.1.1" xref="S4.14.p3.3.m3.2.2.2.1.1.1.cmml"><mi id="S4.14.p3.3.m3.2.2.2.1.1.1.2" xref="S4.14.p3.3.m3.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.3.m3.2.2.2.1.1.1.3" xref="S4.14.p3.3.m3.2.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.3.m3.2.2.2.1.1.3" stretchy="false" xref="S4.14.p3.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.3.m3.4.4.7" xref="S4.14.p3.3.m3.4.4.7.cmml">=</mo><mrow id="S4.14.p3.3.m3.3.3.3" xref="S4.14.p3.3.m3.3.3.3.cmml"><msub id="S4.14.p3.3.m3.3.3.3.3" xref="S4.14.p3.3.m3.3.3.3.3.cmml"><mi id="S4.14.p3.3.m3.3.3.3.3.2" xref="S4.14.p3.3.m3.3.3.3.3.2.cmml">f</mi><mn id="S4.14.p3.3.m3.3.3.3.3.3" xref="S4.14.p3.3.m3.3.3.3.3.3.cmml">2</mn></msub><mo id="S4.14.p3.3.m3.3.3.3.2" xref="S4.14.p3.3.m3.3.3.3.2.cmml">⁢</mo><mrow id="S4.14.p3.3.m3.3.3.3.1.1" xref="S4.14.p3.3.m3.3.3.3.1.1.1.cmml"><mo id="S4.14.p3.3.m3.3.3.3.1.1.2" stretchy="false" xref="S4.14.p3.3.m3.3.3.3.1.1.1.cmml">(</mo><msub id="S4.14.p3.3.m3.3.3.3.1.1.1" xref="S4.14.p3.3.m3.3.3.3.1.1.1.cmml"><mi id="S4.14.p3.3.m3.3.3.3.1.1.1.2" xref="S4.14.p3.3.m3.3.3.3.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.3.m3.3.3.3.1.1.1.3" xref="S4.14.p3.3.m3.3.3.3.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.3.m3.3.3.3.1.1.3" stretchy="false" xref="S4.14.p3.3.m3.3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.3.m3.4.4.8" xref="S4.14.p3.3.m3.4.4.8.cmml">≥</mo><mrow id="S4.14.p3.3.m3.4.4.4" xref="S4.14.p3.3.m3.4.4.4.cmml"><msub id="S4.14.p3.3.m3.4.4.4.3" xref="S4.14.p3.3.m3.4.4.4.3.cmml"><mi id="S4.14.p3.3.m3.4.4.4.3.2" xref="S4.14.p3.3.m3.4.4.4.3.2.cmml">f</mi><mn id="S4.14.p3.3.m3.4.4.4.3.3" xref="S4.14.p3.3.m3.4.4.4.3.3.cmml">0</mn></msub><mo id="S4.14.p3.3.m3.4.4.4.2" xref="S4.14.p3.3.m3.4.4.4.2.cmml">⁢</mo><mrow id="S4.14.p3.3.m3.4.4.4.1.1" xref="S4.14.p3.3.m3.4.4.4.1.1.1.cmml"><mo id="S4.14.p3.3.m3.4.4.4.1.1.2" stretchy="false" xref="S4.14.p3.3.m3.4.4.4.1.1.1.cmml">(</mo><msub id="S4.14.p3.3.m3.4.4.4.1.1.1" xref="S4.14.p3.3.m3.4.4.4.1.1.1.cmml"><mi id="S4.14.p3.3.m3.4.4.4.1.1.1.2" xref="S4.14.p3.3.m3.4.4.4.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.3.m3.4.4.4.1.1.1.3" xref="S4.14.p3.3.m3.4.4.4.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.3.m3.4.4.4.1.1.3" stretchy="false" xref="S4.14.p3.3.m3.4.4.4.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.3.m3.4b"><apply id="S4.14.p3.3.m3.4.4.cmml" xref="S4.14.p3.3.m3.4.4"><and id="S4.14.p3.3.m3.4.4a.cmml" xref="S4.14.p3.3.m3.4.4"></and><apply id="S4.14.p3.3.m3.4.4b.cmml" xref="S4.14.p3.3.m3.4.4"><eq id="S4.14.p3.3.m3.4.4.6.cmml" xref="S4.14.p3.3.m3.4.4.6"></eq><apply id="S4.14.p3.3.m3.1.1.1.cmml" xref="S4.14.p3.3.m3.1.1.1"><times id="S4.14.p3.3.m3.1.1.1.2.cmml" xref="S4.14.p3.3.m3.1.1.1.2"></times><ci id="S4.14.p3.3.m3.1.1.1.3.cmml" xref="S4.14.p3.3.m3.1.1.1.3">𝑓</ci><apply id="S4.14.p3.3.m3.1.1.1.1.1.1.cmml" xref="S4.14.p3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.14.p3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.14.p3.3.m3.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.14.p3.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.3.m3.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S4.14.p3.3.m3.2.2.2.cmml" xref="S4.14.p3.3.m3.2.2.2"><times id="S4.14.p3.3.m3.2.2.2.2.cmml" xref="S4.14.p3.3.m3.2.2.2.2"></times><apply id="S4.14.p3.3.m3.2.2.2.3.cmml" xref="S4.14.p3.3.m3.2.2.2.3"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.2.2.2.3.1.cmml" xref="S4.14.p3.3.m3.2.2.2.3">subscript</csymbol><ci id="S4.14.p3.3.m3.2.2.2.3.2.cmml" xref="S4.14.p3.3.m3.2.2.2.3.2">𝑓</ci><cn id="S4.14.p3.3.m3.2.2.2.3.3.cmml" type="integer" xref="S4.14.p3.3.m3.2.2.2.3.3">1</cn></apply><apply id="S4.14.p3.3.m3.2.2.2.1.1.1.cmml" xref="S4.14.p3.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.2.2.2.1.1.1.1.cmml" xref="S4.14.p3.3.m3.2.2.2.1.1">subscript</csymbol><ci id="S4.14.p3.3.m3.2.2.2.1.1.1.2.cmml" xref="S4.14.p3.3.m3.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.14.p3.3.m3.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.14.p3.3.m3.2.2.2.1.1.1.3">0</cn></apply></apply></apply><apply id="S4.14.p3.3.m3.4.4c.cmml" xref="S4.14.p3.3.m3.4.4"><eq id="S4.14.p3.3.m3.4.4.7.cmml" xref="S4.14.p3.3.m3.4.4.7"></eq><share href="https://arxiv.org/html/2503.13001v1#S4.14.p3.3.m3.2.2.2.cmml" id="S4.14.p3.3.m3.4.4d.cmml" xref="S4.14.p3.3.m3.4.4"></share><apply id="S4.14.p3.3.m3.3.3.3.cmml" xref="S4.14.p3.3.m3.3.3.3"><times id="S4.14.p3.3.m3.3.3.3.2.cmml" xref="S4.14.p3.3.m3.3.3.3.2"></times><apply id="S4.14.p3.3.m3.3.3.3.3.cmml" xref="S4.14.p3.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.3.3.3.3.1.cmml" xref="S4.14.p3.3.m3.3.3.3.3">subscript</csymbol><ci id="S4.14.p3.3.m3.3.3.3.3.2.cmml" xref="S4.14.p3.3.m3.3.3.3.3.2">𝑓</ci><cn id="S4.14.p3.3.m3.3.3.3.3.3.cmml" type="integer" xref="S4.14.p3.3.m3.3.3.3.3.3">2</cn></apply><apply id="S4.14.p3.3.m3.3.3.3.1.1.1.cmml" xref="S4.14.p3.3.m3.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.3.3.3.1.1.1.1.cmml" xref="S4.14.p3.3.m3.3.3.3.1.1">subscript</csymbol><ci id="S4.14.p3.3.m3.3.3.3.1.1.1.2.cmml" xref="S4.14.p3.3.m3.3.3.3.1.1.1.2">𝑝</ci><cn id="S4.14.p3.3.m3.3.3.3.1.1.1.3.cmml" type="integer" xref="S4.14.p3.3.m3.3.3.3.1.1.1.3">0</cn></apply></apply></apply><apply id="S4.14.p3.3.m3.4.4e.cmml" xref="S4.14.p3.3.m3.4.4"><geq id="S4.14.p3.3.m3.4.4.8.cmml" xref="S4.14.p3.3.m3.4.4.8"></geq><share href="https://arxiv.org/html/2503.13001v1#S4.14.p3.3.m3.3.3.3.cmml" id="S4.14.p3.3.m3.4.4f.cmml" xref="S4.14.p3.3.m3.4.4"></share><apply id="S4.14.p3.3.m3.4.4.4.cmml" xref="S4.14.p3.3.m3.4.4.4"><times id="S4.14.p3.3.m3.4.4.4.2.cmml" xref="S4.14.p3.3.m3.4.4.4.2"></times><apply id="S4.14.p3.3.m3.4.4.4.3.cmml" xref="S4.14.p3.3.m3.4.4.4.3"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.4.4.4.3.1.cmml" xref="S4.14.p3.3.m3.4.4.4.3">subscript</csymbol><ci id="S4.14.p3.3.m3.4.4.4.3.2.cmml" xref="S4.14.p3.3.m3.4.4.4.3.2">𝑓</ci><cn id="S4.14.p3.3.m3.4.4.4.3.3.cmml" type="integer" xref="S4.14.p3.3.m3.4.4.4.3.3">0</cn></apply><apply id="S4.14.p3.3.m3.4.4.4.1.1.1.cmml" xref="S4.14.p3.3.m3.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.14.p3.3.m3.4.4.4.1.1.1.1.cmml" xref="S4.14.p3.3.m3.4.4.4.1.1">subscript</csymbol><ci id="S4.14.p3.3.m3.4.4.4.1.1.1.2.cmml" xref="S4.14.p3.3.m3.4.4.4.1.1.1.2">𝑝</ci><cn id="S4.14.p3.3.m3.4.4.4.1.1.1.3.cmml" type="integer" xref="S4.14.p3.3.m3.4.4.4.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.3.m3.4c">f(p_{0})=f_{1}(p_{0})=f_{2}(p_{0})\geq f_{0}(p_{0})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.3.m3.4d">italic_f ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. Then, as <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.14.p3.4.m4.1"><semantics id="S4.14.p3.4.m4.1a"><msub id="S4.14.p3.4.m4.1.1" xref="S4.14.p3.4.m4.1.1.cmml"><mi id="S4.14.p3.4.m4.1.1.2" xref="S4.14.p3.4.m4.1.1.2.cmml">P</mi><mn id="S4.14.p3.4.m4.1.1.3" xref="S4.14.p3.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.4.m4.1b"><apply id="S4.14.p3.4.m4.1.1.cmml" xref="S4.14.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.14.p3.4.m4.1.1.1.cmml" xref="S4.14.p3.4.m4.1.1">subscript</csymbol><ci id="S4.14.p3.4.m4.1.1.2.cmml" xref="S4.14.p3.4.m4.1.1.2">𝑃</ci><cn id="S4.14.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.14.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.4.m4.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.4.m4.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{0}" class="ltx_Math" display="inline" id="S4.14.p3.5.m5.1"><semantics id="S4.14.p3.5.m5.1a"><msub id="S4.14.p3.5.m5.1.1" xref="S4.14.p3.5.m5.1.1.cmml"><mi id="S4.14.p3.5.m5.1.1.2" xref="S4.14.p3.5.m5.1.1.2.cmml">p</mi><mn id="S4.14.p3.5.m5.1.1.3" xref="S4.14.p3.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.5.m5.1b"><apply id="S4.14.p3.5.m5.1.1.cmml" xref="S4.14.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.14.p3.5.m5.1.1.1.cmml" xref="S4.14.p3.5.m5.1.1">subscript</csymbol><ci id="S4.14.p3.5.m5.1.1.2.cmml" xref="S4.14.p3.5.m5.1.1.2">𝑝</ci><cn id="S4.14.p3.5.m5.1.1.3.cmml" type="integer" xref="S4.14.p3.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.5.m5.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.5.m5.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> lie on the same side of <math alttext="l_{2}" class="ltx_Math" display="inline" id="S4.14.p3.6.m6.1"><semantics id="S4.14.p3.6.m6.1a"><msub id="S4.14.p3.6.m6.1.1" xref="S4.14.p3.6.m6.1.1.cmml"><mi id="S4.14.p3.6.m6.1.1.2" xref="S4.14.p3.6.m6.1.1.2.cmml">l</mi><mn id="S4.14.p3.6.m6.1.1.3" xref="S4.14.p3.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.6.m6.1b"><apply id="S4.14.p3.6.m6.1.1.cmml" xref="S4.14.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.14.p3.6.m6.1.1.1.cmml" xref="S4.14.p3.6.m6.1.1">subscript</csymbol><ci id="S4.14.p3.6.m6.1.1.2.cmml" xref="S4.14.p3.6.m6.1.1.2">𝑙</ci><cn id="S4.14.p3.6.m6.1.1.3.cmml" type="integer" xref="S4.14.p3.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.6.m6.1c">l_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.6.m6.1d">italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which is the line with <math alttext="f_{0}=f_{1}" class="ltx_Math" display="inline" id="S4.14.p3.7.m7.1"><semantics id="S4.14.p3.7.m7.1a"><mrow id="S4.14.p3.7.m7.1.1" xref="S4.14.p3.7.m7.1.1.cmml"><msub id="S4.14.p3.7.m7.1.1.2" xref="S4.14.p3.7.m7.1.1.2.cmml"><mi id="S4.14.p3.7.m7.1.1.2.2" xref="S4.14.p3.7.m7.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.7.m7.1.1.2.3" xref="S4.14.p3.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S4.14.p3.7.m7.1.1.1" xref="S4.14.p3.7.m7.1.1.1.cmml">=</mo><msub id="S4.14.p3.7.m7.1.1.3" xref="S4.14.p3.7.m7.1.1.3.cmml"><mi id="S4.14.p3.7.m7.1.1.3.2" xref="S4.14.p3.7.m7.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.7.m7.1.1.3.3" xref="S4.14.p3.7.m7.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.7.m7.1b"><apply id="S4.14.p3.7.m7.1.1.cmml" xref="S4.14.p3.7.m7.1.1"><eq id="S4.14.p3.7.m7.1.1.1.cmml" xref="S4.14.p3.7.m7.1.1.1"></eq><apply id="S4.14.p3.7.m7.1.1.2.cmml" xref="S4.14.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.7.m7.1.1.2.1.cmml" xref="S4.14.p3.7.m7.1.1.2">subscript</csymbol><ci id="S4.14.p3.7.m7.1.1.2.2.cmml" xref="S4.14.p3.7.m7.1.1.2.2">𝑓</ci><cn id="S4.14.p3.7.m7.1.1.2.3.cmml" type="integer" xref="S4.14.p3.7.m7.1.1.2.3">0</cn></apply><apply id="S4.14.p3.7.m7.1.1.3.cmml" xref="S4.14.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.7.m7.1.1.3.1.cmml" xref="S4.14.p3.7.m7.1.1.3">subscript</csymbol><ci id="S4.14.p3.7.m7.1.1.3.2.cmml" xref="S4.14.p3.7.m7.1.1.3.2">𝑓</ci><cn id="S4.14.p3.7.m7.1.1.3.3.cmml" type="integer" xref="S4.14.p3.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.7.m7.1c">f_{0}=f_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.7.m7.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, this extends to <math alttext="f_{1}\geq f_{0}" class="ltx_Math" display="inline" id="S4.14.p3.8.m8.1"><semantics id="S4.14.p3.8.m8.1a"><mrow id="S4.14.p3.8.m8.1.1" xref="S4.14.p3.8.m8.1.1.cmml"><msub id="S4.14.p3.8.m8.1.1.2" xref="S4.14.p3.8.m8.1.1.2.cmml"><mi id="S4.14.p3.8.m8.1.1.2.2" xref="S4.14.p3.8.m8.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.8.m8.1.1.2.3" xref="S4.14.p3.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S4.14.p3.8.m8.1.1.1" xref="S4.14.p3.8.m8.1.1.1.cmml">≥</mo><msub id="S4.14.p3.8.m8.1.1.3" xref="S4.14.p3.8.m8.1.1.3.cmml"><mi id="S4.14.p3.8.m8.1.1.3.2" xref="S4.14.p3.8.m8.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.8.m8.1.1.3.3" xref="S4.14.p3.8.m8.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.8.m8.1b"><apply id="S4.14.p3.8.m8.1.1.cmml" xref="S4.14.p3.8.m8.1.1"><geq id="S4.14.p3.8.m8.1.1.1.cmml" xref="S4.14.p3.8.m8.1.1.1"></geq><apply id="S4.14.p3.8.m8.1.1.2.cmml" xref="S4.14.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.8.m8.1.1.2.1.cmml" xref="S4.14.p3.8.m8.1.1.2">subscript</csymbol><ci id="S4.14.p3.8.m8.1.1.2.2.cmml" xref="S4.14.p3.8.m8.1.1.2.2">𝑓</ci><cn id="S4.14.p3.8.m8.1.1.2.3.cmml" type="integer" xref="S4.14.p3.8.m8.1.1.2.3">1</cn></apply><apply id="S4.14.p3.8.m8.1.1.3.cmml" xref="S4.14.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.8.m8.1.1.3.1.cmml" xref="S4.14.p3.8.m8.1.1.3">subscript</csymbol><ci id="S4.14.p3.8.m8.1.1.3.2.cmml" xref="S4.14.p3.8.m8.1.1.3.2">𝑓</ci><cn id="S4.14.p3.8.m8.1.1.3.3.cmml" type="integer" xref="S4.14.p3.8.m8.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.8.m8.1c">f_{1}\geq f_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.8.m8.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.14.p3.9.m9.1"><semantics id="S4.14.p3.9.m9.1a"><msub id="S4.14.p3.9.m9.1.1" xref="S4.14.p3.9.m9.1.1.cmml"><mi id="S4.14.p3.9.m9.1.1.2" xref="S4.14.p3.9.m9.1.1.2.cmml">P</mi><mn id="S4.14.p3.9.m9.1.1.3" xref="S4.14.p3.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.9.m9.1b"><apply id="S4.14.p3.9.m9.1.1.cmml" xref="S4.14.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S4.14.p3.9.m9.1.1.1.cmml" xref="S4.14.p3.9.m9.1.1">subscript</csymbol><ci id="S4.14.p3.9.m9.1.1.2.cmml" xref="S4.14.p3.9.m9.1.1.2">𝑃</ci><cn id="S4.14.p3.9.m9.1.1.3.cmml" type="integer" xref="S4.14.p3.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.9.m9.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.9.m9.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, <math alttext="f_{2}\geq f_{0}" class="ltx_Math" display="inline" id="S4.14.p3.10.m10.1"><semantics id="S4.14.p3.10.m10.1a"><mrow id="S4.14.p3.10.m10.1.1" xref="S4.14.p3.10.m10.1.1.cmml"><msub id="S4.14.p3.10.m10.1.1.2" xref="S4.14.p3.10.m10.1.1.2.cmml"><mi id="S4.14.p3.10.m10.1.1.2.2" xref="S4.14.p3.10.m10.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.10.m10.1.1.2.3" xref="S4.14.p3.10.m10.1.1.2.3.cmml">2</mn></msub><mo id="S4.14.p3.10.m10.1.1.1" xref="S4.14.p3.10.m10.1.1.1.cmml">≥</mo><msub id="S4.14.p3.10.m10.1.1.3" xref="S4.14.p3.10.m10.1.1.3.cmml"><mi id="S4.14.p3.10.m10.1.1.3.2" xref="S4.14.p3.10.m10.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.10.m10.1.1.3.3" xref="S4.14.p3.10.m10.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.10.m10.1b"><apply id="S4.14.p3.10.m10.1.1.cmml" xref="S4.14.p3.10.m10.1.1"><geq id="S4.14.p3.10.m10.1.1.1.cmml" xref="S4.14.p3.10.m10.1.1.1"></geq><apply id="S4.14.p3.10.m10.1.1.2.cmml" xref="S4.14.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.10.m10.1.1.2.1.cmml" xref="S4.14.p3.10.m10.1.1.2">subscript</csymbol><ci id="S4.14.p3.10.m10.1.1.2.2.cmml" xref="S4.14.p3.10.m10.1.1.2.2">𝑓</ci><cn id="S4.14.p3.10.m10.1.1.2.3.cmml" type="integer" xref="S4.14.p3.10.m10.1.1.2.3">2</cn></apply><apply id="S4.14.p3.10.m10.1.1.3.cmml" xref="S4.14.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.10.m10.1.1.3.1.cmml" xref="S4.14.p3.10.m10.1.1.3">subscript</csymbol><ci id="S4.14.p3.10.m10.1.1.3.2.cmml" xref="S4.14.p3.10.m10.1.1.3.2">𝑓</ci><cn id="S4.14.p3.10.m10.1.1.3.3.cmml" type="integer" xref="S4.14.p3.10.m10.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.10.m10.1c">f_{2}\geq f_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.10.m10.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.14.p3.11.m11.1"><semantics id="S4.14.p3.11.m11.1a"><msub id="S4.14.p3.11.m11.1.1" xref="S4.14.p3.11.m11.1.1.cmml"><mi id="S4.14.p3.11.m11.1.1.2" xref="S4.14.p3.11.m11.1.1.2.cmml">P</mi><mn id="S4.14.p3.11.m11.1.1.3" xref="S4.14.p3.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.11.m11.1b"><apply id="S4.14.p3.11.m11.1.1.cmml" xref="S4.14.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S4.14.p3.11.m11.1.1.1.cmml" xref="S4.14.p3.11.m11.1.1">subscript</csymbol><ci id="S4.14.p3.11.m11.1.1.2.cmml" xref="S4.14.p3.11.m11.1.1.2">𝑃</ci><cn id="S4.14.p3.11.m11.1.1.3.cmml" type="integer" xref="S4.14.p3.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.11.m11.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.11.m11.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.14.p3.12.m12.1"><semantics id="S4.14.p3.12.m12.1a"><msub id="S4.14.p3.12.m12.1.1" xref="S4.14.p3.12.m12.1.1.cmml"><mi id="S4.14.p3.12.m12.1.1.2" xref="S4.14.p3.12.m12.1.1.2.cmml">P</mi><mn id="S4.14.p3.12.m12.1.1.3" xref="S4.14.p3.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.12.m12.1b"><apply id="S4.14.p3.12.m12.1.1.cmml" xref="S4.14.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S4.14.p3.12.m12.1.1.1.cmml" xref="S4.14.p3.12.m12.1.1">subscript</csymbol><ci id="S4.14.p3.12.m12.1.1.2.cmml" xref="S4.14.p3.12.m12.1.1.2">𝑃</ci><cn id="S4.14.p3.12.m12.1.1.3.cmml" type="integer" xref="S4.14.p3.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.12.m12.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.12.m12.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{0}" class="ltx_Math" display="inline" id="S4.14.p3.13.m13.1"><semantics id="S4.14.p3.13.m13.1a"><msub id="S4.14.p3.13.m13.1.1" xref="S4.14.p3.13.m13.1.1.cmml"><mi id="S4.14.p3.13.m13.1.1.2" xref="S4.14.p3.13.m13.1.1.2.cmml">p</mi><mn id="S4.14.p3.13.m13.1.1.3" xref="S4.14.p3.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.13.m13.1b"><apply id="S4.14.p3.13.m13.1.1.cmml" xref="S4.14.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S4.14.p3.13.m13.1.1.1.cmml" xref="S4.14.p3.13.m13.1.1">subscript</csymbol><ci id="S4.14.p3.13.m13.1.1.2.cmml" xref="S4.14.p3.13.m13.1.1.2">𝑝</ci><cn id="S4.14.p3.13.m13.1.1.3.cmml" type="integer" xref="S4.14.p3.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.13.m13.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.13.m13.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> lie on opposite sides of <math alttext="l_{1}" class="ltx_Math" display="inline" id="S4.14.p3.14.m14.1"><semantics id="S4.14.p3.14.m14.1a"><msub id="S4.14.p3.14.m14.1.1" xref="S4.14.p3.14.m14.1.1.cmml"><mi id="S4.14.p3.14.m14.1.1.2" xref="S4.14.p3.14.m14.1.1.2.cmml">l</mi><mn id="S4.14.p3.14.m14.1.1.3" xref="S4.14.p3.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.14.m14.1b"><apply id="S4.14.p3.14.m14.1.1.cmml" xref="S4.14.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S4.14.p3.14.m14.1.1.1.cmml" xref="S4.14.p3.14.m14.1.1">subscript</csymbol><ci id="S4.14.p3.14.m14.1.1.2.cmml" xref="S4.14.p3.14.m14.1.1.2">𝑙</ci><cn id="S4.14.p3.14.m14.1.1.3.cmml" type="integer" xref="S4.14.p3.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.14.m14.1c">l_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.14.m14.1d">italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="l_{2}" class="ltx_Math" display="inline" id="S4.14.p3.15.m15.1"><semantics id="S4.14.p3.15.m15.1a"><msub id="S4.14.p3.15.m15.1.1" xref="S4.14.p3.15.m15.1.1.cmml"><mi id="S4.14.p3.15.m15.1.1.2" xref="S4.14.p3.15.m15.1.1.2.cmml">l</mi><mn id="S4.14.p3.15.m15.1.1.3" xref="S4.14.p3.15.m15.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.15.m15.1b"><apply id="S4.14.p3.15.m15.1.1.cmml" xref="S4.14.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S4.14.p3.15.m15.1.1.1.cmml" xref="S4.14.p3.15.m15.1.1">subscript</csymbol><ci id="S4.14.p3.15.m15.1.1.2.cmml" xref="S4.14.p3.15.m15.1.1.2">𝑙</ci><cn id="S4.14.p3.15.m15.1.1.3.cmml" type="integer" xref="S4.14.p3.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.15.m15.1c">l_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.15.m15.1d">italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which implies that <math alttext="f_{0}\geq f_{1}" class="ltx_Math" display="inline" id="S4.14.p3.16.m16.1"><semantics id="S4.14.p3.16.m16.1a"><mrow id="S4.14.p3.16.m16.1.1" xref="S4.14.p3.16.m16.1.1.cmml"><msub id="S4.14.p3.16.m16.1.1.2" xref="S4.14.p3.16.m16.1.1.2.cmml"><mi id="S4.14.p3.16.m16.1.1.2.2" xref="S4.14.p3.16.m16.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.16.m16.1.1.2.3" xref="S4.14.p3.16.m16.1.1.2.3.cmml">0</mn></msub><mo id="S4.14.p3.16.m16.1.1.1" xref="S4.14.p3.16.m16.1.1.1.cmml">≥</mo><msub id="S4.14.p3.16.m16.1.1.3" xref="S4.14.p3.16.m16.1.1.3.cmml"><mi id="S4.14.p3.16.m16.1.1.3.2" xref="S4.14.p3.16.m16.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.16.m16.1.1.3.3" xref="S4.14.p3.16.m16.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.16.m16.1b"><apply id="S4.14.p3.16.m16.1.1.cmml" xref="S4.14.p3.16.m16.1.1"><geq id="S4.14.p3.16.m16.1.1.1.cmml" xref="S4.14.p3.16.m16.1.1.1"></geq><apply id="S4.14.p3.16.m16.1.1.2.cmml" xref="S4.14.p3.16.m16.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.16.m16.1.1.2.1.cmml" xref="S4.14.p3.16.m16.1.1.2">subscript</csymbol><ci id="S4.14.p3.16.m16.1.1.2.2.cmml" xref="S4.14.p3.16.m16.1.1.2.2">𝑓</ci><cn id="S4.14.p3.16.m16.1.1.2.3.cmml" type="integer" xref="S4.14.p3.16.m16.1.1.2.3">0</cn></apply><apply id="S4.14.p3.16.m16.1.1.3.cmml" xref="S4.14.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.16.m16.1.1.3.1.cmml" xref="S4.14.p3.16.m16.1.1.3">subscript</csymbol><ci id="S4.14.p3.16.m16.1.1.3.2.cmml" xref="S4.14.p3.16.m16.1.1.3.2">𝑓</ci><cn id="S4.14.p3.16.m16.1.1.3.3.cmml" type="integer" xref="S4.14.p3.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.16.m16.1c">f_{0}\geq f_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.16.m16.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{0}\geq f_{2}" class="ltx_Math" display="inline" id="S4.14.p3.17.m17.1"><semantics id="S4.14.p3.17.m17.1a"><mrow id="S4.14.p3.17.m17.1.1" xref="S4.14.p3.17.m17.1.1.cmml"><msub id="S4.14.p3.17.m17.1.1.2" xref="S4.14.p3.17.m17.1.1.2.cmml"><mi id="S4.14.p3.17.m17.1.1.2.2" xref="S4.14.p3.17.m17.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.17.m17.1.1.2.3" xref="S4.14.p3.17.m17.1.1.2.3.cmml">0</mn></msub><mo id="S4.14.p3.17.m17.1.1.1" xref="S4.14.p3.17.m17.1.1.1.cmml">≥</mo><msub id="S4.14.p3.17.m17.1.1.3" xref="S4.14.p3.17.m17.1.1.3.cmml"><mi id="S4.14.p3.17.m17.1.1.3.2" xref="S4.14.p3.17.m17.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.17.m17.1.1.3.3" xref="S4.14.p3.17.m17.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.17.m17.1b"><apply id="S4.14.p3.17.m17.1.1.cmml" xref="S4.14.p3.17.m17.1.1"><geq id="S4.14.p3.17.m17.1.1.1.cmml" xref="S4.14.p3.17.m17.1.1.1"></geq><apply id="S4.14.p3.17.m17.1.1.2.cmml" xref="S4.14.p3.17.m17.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.17.m17.1.1.2.1.cmml" xref="S4.14.p3.17.m17.1.1.2">subscript</csymbol><ci id="S4.14.p3.17.m17.1.1.2.2.cmml" xref="S4.14.p3.17.m17.1.1.2.2">𝑓</ci><cn id="S4.14.p3.17.m17.1.1.2.3.cmml" type="integer" xref="S4.14.p3.17.m17.1.1.2.3">0</cn></apply><apply id="S4.14.p3.17.m17.1.1.3.cmml" xref="S4.14.p3.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.17.m17.1.1.3.1.cmml" xref="S4.14.p3.17.m17.1.1.3">subscript</csymbol><ci id="S4.14.p3.17.m17.1.1.3.2.cmml" xref="S4.14.p3.17.m17.1.1.3.2">𝑓</ci><cn id="S4.14.p3.17.m17.1.1.3.3.cmml" type="integer" xref="S4.14.p3.17.m17.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.17.m17.1c">f_{0}\geq f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.17.m17.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.14.p3.18.m18.1"><semantics id="S4.14.p3.18.m18.1a"><msub id="S4.14.p3.18.m18.1.1" xref="S4.14.p3.18.m18.1.1.cmml"><mi id="S4.14.p3.18.m18.1.1.2" xref="S4.14.p3.18.m18.1.1.2.cmml">P</mi><mn id="S4.14.p3.18.m18.1.1.3" xref="S4.14.p3.18.m18.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.18.m18.1b"><apply id="S4.14.p3.18.m18.1.1.cmml" xref="S4.14.p3.18.m18.1.1"><csymbol cd="ambiguous" id="S4.14.p3.18.m18.1.1.1.cmml" xref="S4.14.p3.18.m18.1.1">subscript</csymbol><ci id="S4.14.p3.18.m18.1.1.2.cmml" xref="S4.14.p3.18.m18.1.1.2">𝑃</ci><cn id="S4.14.p3.18.m18.1.1.3.cmml" type="integer" xref="S4.14.p3.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.18.m18.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.18.m18.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, <math alttext="f_{2}(p_{1})=f_{0}(p_{1})\geq f_{1}(p_{1})" class="ltx_Math" display="inline" id="S4.14.p3.19.m19.3"><semantics id="S4.14.p3.19.m19.3a"><mrow id="S4.14.p3.19.m19.3.3" xref="S4.14.p3.19.m19.3.3.cmml"><mrow id="S4.14.p3.19.m19.1.1.1" xref="S4.14.p3.19.m19.1.1.1.cmml"><msub id="S4.14.p3.19.m19.1.1.1.3" xref="S4.14.p3.19.m19.1.1.1.3.cmml"><mi id="S4.14.p3.19.m19.1.1.1.3.2" xref="S4.14.p3.19.m19.1.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.19.m19.1.1.1.3.3" xref="S4.14.p3.19.m19.1.1.1.3.3.cmml">2</mn></msub><mo id="S4.14.p3.19.m19.1.1.1.2" xref="S4.14.p3.19.m19.1.1.1.2.cmml">⁢</mo><mrow id="S4.14.p3.19.m19.1.1.1.1.1" xref="S4.14.p3.19.m19.1.1.1.1.1.1.cmml"><mo id="S4.14.p3.19.m19.1.1.1.1.1.2" stretchy="false" xref="S4.14.p3.19.m19.1.1.1.1.1.1.cmml">(</mo><msub id="S4.14.p3.19.m19.1.1.1.1.1.1" xref="S4.14.p3.19.m19.1.1.1.1.1.1.cmml"><mi id="S4.14.p3.19.m19.1.1.1.1.1.1.2" xref="S4.14.p3.19.m19.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.19.m19.1.1.1.1.1.1.3" xref="S4.14.p3.19.m19.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.14.p3.19.m19.1.1.1.1.1.3" stretchy="false" xref="S4.14.p3.19.m19.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.19.m19.3.3.5" xref="S4.14.p3.19.m19.3.3.5.cmml">=</mo><mrow id="S4.14.p3.19.m19.2.2.2" xref="S4.14.p3.19.m19.2.2.2.cmml"><msub id="S4.14.p3.19.m19.2.2.2.3" xref="S4.14.p3.19.m19.2.2.2.3.cmml"><mi id="S4.14.p3.19.m19.2.2.2.3.2" xref="S4.14.p3.19.m19.2.2.2.3.2.cmml">f</mi><mn id="S4.14.p3.19.m19.2.2.2.3.3" xref="S4.14.p3.19.m19.2.2.2.3.3.cmml">0</mn></msub><mo id="S4.14.p3.19.m19.2.2.2.2" xref="S4.14.p3.19.m19.2.2.2.2.cmml">⁢</mo><mrow id="S4.14.p3.19.m19.2.2.2.1.1" xref="S4.14.p3.19.m19.2.2.2.1.1.1.cmml"><mo id="S4.14.p3.19.m19.2.2.2.1.1.2" stretchy="false" xref="S4.14.p3.19.m19.2.2.2.1.1.1.cmml">(</mo><msub id="S4.14.p3.19.m19.2.2.2.1.1.1" xref="S4.14.p3.19.m19.2.2.2.1.1.1.cmml"><mi id="S4.14.p3.19.m19.2.2.2.1.1.1.2" xref="S4.14.p3.19.m19.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.19.m19.2.2.2.1.1.1.3" xref="S4.14.p3.19.m19.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.14.p3.19.m19.2.2.2.1.1.3" stretchy="false" xref="S4.14.p3.19.m19.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.19.m19.3.3.6" xref="S4.14.p3.19.m19.3.3.6.cmml">≥</mo><mrow id="S4.14.p3.19.m19.3.3.3" xref="S4.14.p3.19.m19.3.3.3.cmml"><msub id="S4.14.p3.19.m19.3.3.3.3" xref="S4.14.p3.19.m19.3.3.3.3.cmml"><mi id="S4.14.p3.19.m19.3.3.3.3.2" xref="S4.14.p3.19.m19.3.3.3.3.2.cmml">f</mi><mn id="S4.14.p3.19.m19.3.3.3.3.3" xref="S4.14.p3.19.m19.3.3.3.3.3.cmml">1</mn></msub><mo id="S4.14.p3.19.m19.3.3.3.2" xref="S4.14.p3.19.m19.3.3.3.2.cmml">⁢</mo><mrow id="S4.14.p3.19.m19.3.3.3.1.1" xref="S4.14.p3.19.m19.3.3.3.1.1.1.cmml"><mo id="S4.14.p3.19.m19.3.3.3.1.1.2" stretchy="false" xref="S4.14.p3.19.m19.3.3.3.1.1.1.cmml">(</mo><msub id="S4.14.p3.19.m19.3.3.3.1.1.1" xref="S4.14.p3.19.m19.3.3.3.1.1.1.cmml"><mi id="S4.14.p3.19.m19.3.3.3.1.1.1.2" xref="S4.14.p3.19.m19.3.3.3.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.19.m19.3.3.3.1.1.1.3" xref="S4.14.p3.19.m19.3.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S4.14.p3.19.m19.3.3.3.1.1.3" stretchy="false" xref="S4.14.p3.19.m19.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.19.m19.3b"><apply id="S4.14.p3.19.m19.3.3.cmml" xref="S4.14.p3.19.m19.3.3"><and id="S4.14.p3.19.m19.3.3a.cmml" xref="S4.14.p3.19.m19.3.3"></and><apply id="S4.14.p3.19.m19.3.3b.cmml" xref="S4.14.p3.19.m19.3.3"><eq id="S4.14.p3.19.m19.3.3.5.cmml" xref="S4.14.p3.19.m19.3.3.5"></eq><apply id="S4.14.p3.19.m19.1.1.1.cmml" xref="S4.14.p3.19.m19.1.1.1"><times id="S4.14.p3.19.m19.1.1.1.2.cmml" xref="S4.14.p3.19.m19.1.1.1.2"></times><apply id="S4.14.p3.19.m19.1.1.1.3.cmml" xref="S4.14.p3.19.m19.1.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.1.1.1.3.1.cmml" xref="S4.14.p3.19.m19.1.1.1.3">subscript</csymbol><ci id="S4.14.p3.19.m19.1.1.1.3.2.cmml" xref="S4.14.p3.19.m19.1.1.1.3.2">𝑓</ci><cn id="S4.14.p3.19.m19.1.1.1.3.3.cmml" type="integer" xref="S4.14.p3.19.m19.1.1.1.3.3">2</cn></apply><apply id="S4.14.p3.19.m19.1.1.1.1.1.1.cmml" xref="S4.14.p3.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.1.1.1.1.1.1.1.cmml" xref="S4.14.p3.19.m19.1.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.19.m19.1.1.1.1.1.1.2.cmml" xref="S4.14.p3.19.m19.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.14.p3.19.m19.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.19.m19.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.14.p3.19.m19.2.2.2.cmml" xref="S4.14.p3.19.m19.2.2.2"><times id="S4.14.p3.19.m19.2.2.2.2.cmml" xref="S4.14.p3.19.m19.2.2.2.2"></times><apply id="S4.14.p3.19.m19.2.2.2.3.cmml" xref="S4.14.p3.19.m19.2.2.2.3"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.2.2.2.3.1.cmml" xref="S4.14.p3.19.m19.2.2.2.3">subscript</csymbol><ci id="S4.14.p3.19.m19.2.2.2.3.2.cmml" xref="S4.14.p3.19.m19.2.2.2.3.2">𝑓</ci><cn id="S4.14.p3.19.m19.2.2.2.3.3.cmml" type="integer" xref="S4.14.p3.19.m19.2.2.2.3.3">0</cn></apply><apply id="S4.14.p3.19.m19.2.2.2.1.1.1.cmml" xref="S4.14.p3.19.m19.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.2.2.2.1.1.1.1.cmml" xref="S4.14.p3.19.m19.2.2.2.1.1">subscript</csymbol><ci id="S4.14.p3.19.m19.2.2.2.1.1.1.2.cmml" xref="S4.14.p3.19.m19.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.14.p3.19.m19.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.14.p3.19.m19.2.2.2.1.1.1.3">1</cn></apply></apply></apply><apply id="S4.14.p3.19.m19.3.3c.cmml" xref="S4.14.p3.19.m19.3.3"><geq id="S4.14.p3.19.m19.3.3.6.cmml" xref="S4.14.p3.19.m19.3.3.6"></geq><share href="https://arxiv.org/html/2503.13001v1#S4.14.p3.19.m19.2.2.2.cmml" id="S4.14.p3.19.m19.3.3d.cmml" xref="S4.14.p3.19.m19.3.3"></share><apply id="S4.14.p3.19.m19.3.3.3.cmml" xref="S4.14.p3.19.m19.3.3.3"><times id="S4.14.p3.19.m19.3.3.3.2.cmml" xref="S4.14.p3.19.m19.3.3.3.2"></times><apply id="S4.14.p3.19.m19.3.3.3.3.cmml" xref="S4.14.p3.19.m19.3.3.3.3"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.3.3.3.3.1.cmml" xref="S4.14.p3.19.m19.3.3.3.3">subscript</csymbol><ci id="S4.14.p3.19.m19.3.3.3.3.2.cmml" xref="S4.14.p3.19.m19.3.3.3.3.2">𝑓</ci><cn id="S4.14.p3.19.m19.3.3.3.3.3.cmml" type="integer" xref="S4.14.p3.19.m19.3.3.3.3.3">1</cn></apply><apply id="S4.14.p3.19.m19.3.3.3.1.1.1.cmml" xref="S4.14.p3.19.m19.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.14.p3.19.m19.3.3.3.1.1.1.1.cmml" xref="S4.14.p3.19.m19.3.3.3.1.1">subscript</csymbol><ci id="S4.14.p3.19.m19.3.3.3.1.1.1.2.cmml" xref="S4.14.p3.19.m19.3.3.3.1.1.1.2">𝑝</ci><cn id="S4.14.p3.19.m19.3.3.3.1.1.1.3.cmml" type="integer" xref="S4.14.p3.19.m19.3.3.3.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.19.m19.3c">f_{2}(p_{1})=f_{0}(p_{1})\geq f_{1}(p_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.19.m19.3d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≥ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Since <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.14.p3.20.m20.1"><semantics id="S4.14.p3.20.m20.1a"><msub id="S4.14.p3.20.m20.1.1" xref="S4.14.p3.20.m20.1.1.cmml"><mi id="S4.14.p3.20.m20.1.1.2" xref="S4.14.p3.20.m20.1.1.2.cmml">P</mi><mn id="S4.14.p3.20.m20.1.1.3" xref="S4.14.p3.20.m20.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.20.m20.1b"><apply id="S4.14.p3.20.m20.1.1.cmml" xref="S4.14.p3.20.m20.1.1"><csymbol cd="ambiguous" id="S4.14.p3.20.m20.1.1.1.cmml" xref="S4.14.p3.20.m20.1.1">subscript</csymbol><ci id="S4.14.p3.20.m20.1.1.2.cmml" xref="S4.14.p3.20.m20.1.1.2">𝑃</ci><cn id="S4.14.p3.20.m20.1.1.3.cmml" type="integer" xref="S4.14.p3.20.m20.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.20.m20.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.20.m20.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{1}" class="ltx_Math" display="inline" id="S4.14.p3.21.m21.1"><semantics id="S4.14.p3.21.m21.1a"><msub id="S4.14.p3.21.m21.1.1" xref="S4.14.p3.21.m21.1.1.cmml"><mi id="S4.14.p3.21.m21.1.1.2" xref="S4.14.p3.21.m21.1.1.2.cmml">p</mi><mn id="S4.14.p3.21.m21.1.1.3" xref="S4.14.p3.21.m21.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.21.m21.1b"><apply id="S4.14.p3.21.m21.1.1.cmml" xref="S4.14.p3.21.m21.1.1"><csymbol cd="ambiguous" id="S4.14.p3.21.m21.1.1.1.cmml" xref="S4.14.p3.21.m21.1.1">subscript</csymbol><ci id="S4.14.p3.21.m21.1.1.2.cmml" xref="S4.14.p3.21.m21.1.1.2">𝑝</ci><cn id="S4.14.p3.21.m21.1.1.3.cmml" type="integer" xref="S4.14.p3.21.m21.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.21.m21.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.21.m21.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> lie in the same half-plane w.r.t <math alttext="l_{0}" class="ltx_Math" display="inline" id="S4.14.p3.22.m22.1"><semantics id="S4.14.p3.22.m22.1a"><msub id="S4.14.p3.22.m22.1.1" xref="S4.14.p3.22.m22.1.1.cmml"><mi id="S4.14.p3.22.m22.1.1.2" xref="S4.14.p3.22.m22.1.1.2.cmml">l</mi><mn id="S4.14.p3.22.m22.1.1.3" xref="S4.14.p3.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.22.m22.1b"><apply id="S4.14.p3.22.m22.1.1.cmml" xref="S4.14.p3.22.m22.1.1"><csymbol cd="ambiguous" id="S4.14.p3.22.m22.1.1.1.cmml" xref="S4.14.p3.22.m22.1.1">subscript</csymbol><ci id="S4.14.p3.22.m22.1.1.2.cmml" xref="S4.14.p3.22.m22.1.1.2">𝑙</ci><cn id="S4.14.p3.22.m22.1.1.3.cmml" type="integer" xref="S4.14.p3.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.22.m22.1c">l_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.22.m22.1d">italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, opposite to that of <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.14.p3.23.m23.1"><semantics id="S4.14.p3.23.m23.1a"><msub id="S4.14.p3.23.m23.1.1" xref="S4.14.p3.23.m23.1.1.cmml"><mi id="S4.14.p3.23.m23.1.1.2" xref="S4.14.p3.23.m23.1.1.2.cmml">P</mi><mn id="S4.14.p3.23.m23.1.1.3" xref="S4.14.p3.23.m23.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.23.m23.1b"><apply id="S4.14.p3.23.m23.1.1.cmml" xref="S4.14.p3.23.m23.1.1"><csymbol cd="ambiguous" id="S4.14.p3.23.m23.1.1.1.cmml" xref="S4.14.p3.23.m23.1.1">subscript</csymbol><ci id="S4.14.p3.23.m23.1.1.2.cmml" xref="S4.14.p3.23.m23.1.1.2">𝑃</ci><cn id="S4.14.p3.23.m23.1.1.3.cmml" type="integer" xref="S4.14.p3.23.m23.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.23.m23.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.23.m23.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, this means that <math alttext="f_{2}\geq f_{1}" class="ltx_Math" display="inline" id="S4.14.p3.24.m24.1"><semantics id="S4.14.p3.24.m24.1a"><mrow id="S4.14.p3.24.m24.1.1" xref="S4.14.p3.24.m24.1.1.cmml"><msub id="S4.14.p3.24.m24.1.1.2" xref="S4.14.p3.24.m24.1.1.2.cmml"><mi id="S4.14.p3.24.m24.1.1.2.2" xref="S4.14.p3.24.m24.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.24.m24.1.1.2.3" xref="S4.14.p3.24.m24.1.1.2.3.cmml">2</mn></msub><mo id="S4.14.p3.24.m24.1.1.1" xref="S4.14.p3.24.m24.1.1.1.cmml">≥</mo><msub id="S4.14.p3.24.m24.1.1.3" xref="S4.14.p3.24.m24.1.1.3.cmml"><mi id="S4.14.p3.24.m24.1.1.3.2" xref="S4.14.p3.24.m24.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.24.m24.1.1.3.3" xref="S4.14.p3.24.m24.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.24.m24.1b"><apply id="S4.14.p3.24.m24.1.1.cmml" xref="S4.14.p3.24.m24.1.1"><geq id="S4.14.p3.24.m24.1.1.1.cmml" xref="S4.14.p3.24.m24.1.1.1"></geq><apply id="S4.14.p3.24.m24.1.1.2.cmml" xref="S4.14.p3.24.m24.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.24.m24.1.1.2.1.cmml" xref="S4.14.p3.24.m24.1.1.2">subscript</csymbol><ci id="S4.14.p3.24.m24.1.1.2.2.cmml" xref="S4.14.p3.24.m24.1.1.2.2">𝑓</ci><cn id="S4.14.p3.24.m24.1.1.2.3.cmml" type="integer" xref="S4.14.p3.24.m24.1.1.2.3">2</cn></apply><apply id="S4.14.p3.24.m24.1.1.3.cmml" xref="S4.14.p3.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.24.m24.1.1.3.1.cmml" xref="S4.14.p3.24.m24.1.1.3">subscript</csymbol><ci id="S4.14.p3.24.m24.1.1.3.2.cmml" xref="S4.14.p3.24.m24.1.1.3.2">𝑓</ci><cn id="S4.14.p3.24.m24.1.1.3.3.cmml" type="integer" xref="S4.14.p3.24.m24.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.24.m24.1c">f_{2}\geq f_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.24.m24.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.14.p3.25.m25.1"><semantics id="S4.14.p3.25.m25.1a"><msub id="S4.14.p3.25.m25.1.1" xref="S4.14.p3.25.m25.1.1.cmml"><mi id="S4.14.p3.25.m25.1.1.2" xref="S4.14.p3.25.m25.1.1.2.cmml">P</mi><mn id="S4.14.p3.25.m25.1.1.3" xref="S4.14.p3.25.m25.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.25.m25.1b"><apply id="S4.14.p3.25.m25.1.1.cmml" xref="S4.14.p3.25.m25.1.1"><csymbol cd="ambiguous" id="S4.14.p3.25.m25.1.1.1.cmml" xref="S4.14.p3.25.m25.1.1">subscript</csymbol><ci id="S4.14.p3.25.m25.1.1.2.cmml" xref="S4.14.p3.25.m25.1.1.2">𝑃</ci><cn id="S4.14.p3.25.m25.1.1.3.cmml" type="integer" xref="S4.14.p3.25.m25.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.25.m25.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.25.m25.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{1}\geq f_{2}" class="ltx_Math" display="inline" id="S4.14.p3.26.m26.1"><semantics id="S4.14.p3.26.m26.1a"><mrow id="S4.14.p3.26.m26.1.1" xref="S4.14.p3.26.m26.1.1.cmml"><msub id="S4.14.p3.26.m26.1.1.2" xref="S4.14.p3.26.m26.1.1.2.cmml"><mi id="S4.14.p3.26.m26.1.1.2.2" xref="S4.14.p3.26.m26.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.26.m26.1.1.2.3" xref="S4.14.p3.26.m26.1.1.2.3.cmml">1</mn></msub><mo id="S4.14.p3.26.m26.1.1.1" xref="S4.14.p3.26.m26.1.1.1.cmml">≥</mo><msub id="S4.14.p3.26.m26.1.1.3" xref="S4.14.p3.26.m26.1.1.3.cmml"><mi id="S4.14.p3.26.m26.1.1.3.2" xref="S4.14.p3.26.m26.1.1.3.2.cmml">f</mi><mn id="S4.14.p3.26.m26.1.1.3.3" xref="S4.14.p3.26.m26.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.26.m26.1b"><apply id="S4.14.p3.26.m26.1.1.cmml" xref="S4.14.p3.26.m26.1.1"><geq id="S4.14.p3.26.m26.1.1.1.cmml" xref="S4.14.p3.26.m26.1.1.1"></geq><apply id="S4.14.p3.26.m26.1.1.2.cmml" xref="S4.14.p3.26.m26.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.26.m26.1.1.2.1.cmml" xref="S4.14.p3.26.m26.1.1.2">subscript</csymbol><ci id="S4.14.p3.26.m26.1.1.2.2.cmml" xref="S4.14.p3.26.m26.1.1.2.2">𝑓</ci><cn id="S4.14.p3.26.m26.1.1.2.3.cmml" type="integer" xref="S4.14.p3.26.m26.1.1.2.3">1</cn></apply><apply id="S4.14.p3.26.m26.1.1.3.cmml" xref="S4.14.p3.26.m26.1.1.3"><csymbol cd="ambiguous" id="S4.14.p3.26.m26.1.1.3.1.cmml" xref="S4.14.p3.26.m26.1.1.3">subscript</csymbol><ci id="S4.14.p3.26.m26.1.1.3.2.cmml" xref="S4.14.p3.26.m26.1.1.3.2">𝑓</ci><cn id="S4.14.p3.26.m26.1.1.3.3.cmml" type="integer" xref="S4.14.p3.26.m26.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.26.m26.1c">f_{1}\geq f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.26.m26.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.14.p3.27.m27.1"><semantics id="S4.14.p3.27.m27.1a"><msub id="S4.14.p3.27.m27.1.1" xref="S4.14.p3.27.m27.1.1.cmml"><mi id="S4.14.p3.27.m27.1.1.2" xref="S4.14.p3.27.m27.1.1.2.cmml">P</mi><mn id="S4.14.p3.27.m27.1.1.3" xref="S4.14.p3.27.m27.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.27.m27.1b"><apply id="S4.14.p3.27.m27.1.1.cmml" xref="S4.14.p3.27.m27.1.1"><csymbol cd="ambiguous" id="S4.14.p3.27.m27.1.1.1.cmml" xref="S4.14.p3.27.m27.1.1">subscript</csymbol><ci id="S4.14.p3.27.m27.1.1.2.cmml" xref="S4.14.p3.27.m27.1.1.2">𝑃</ci><cn id="S4.14.p3.27.m27.1.1.3.cmml" type="integer" xref="S4.14.p3.27.m27.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.27.m27.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.27.m27.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. In total, we have <math alttext="f_{1}\geq\max(f_{0},f_{2})" class="ltx_Math" display="inline" id="S4.14.p3.28.m28.3"><semantics id="S4.14.p3.28.m28.3a"><mrow id="S4.14.p3.28.m28.3.3" xref="S4.14.p3.28.m28.3.3.cmml"><msub id="S4.14.p3.28.m28.3.3.4" xref="S4.14.p3.28.m28.3.3.4.cmml"><mi id="S4.14.p3.28.m28.3.3.4.2" xref="S4.14.p3.28.m28.3.3.4.2.cmml">f</mi><mn id="S4.14.p3.28.m28.3.3.4.3" xref="S4.14.p3.28.m28.3.3.4.3.cmml">1</mn></msub><mo id="S4.14.p3.28.m28.3.3.3" xref="S4.14.p3.28.m28.3.3.3.cmml">≥</mo><mrow id="S4.14.p3.28.m28.3.3.2.2" xref="S4.14.p3.28.m28.3.3.2.3.cmml"><mi id="S4.14.p3.28.m28.1.1" xref="S4.14.p3.28.m28.1.1.cmml">max</mi><mo id="S4.14.p3.28.m28.3.3.2.2a" xref="S4.14.p3.28.m28.3.3.2.3.cmml">⁡</mo><mrow id="S4.14.p3.28.m28.3.3.2.2.2" xref="S4.14.p3.28.m28.3.3.2.3.cmml"><mo id="S4.14.p3.28.m28.3.3.2.2.2.3" stretchy="false" xref="S4.14.p3.28.m28.3.3.2.3.cmml">(</mo><msub id="S4.14.p3.28.m28.2.2.1.1.1.1" xref="S4.14.p3.28.m28.2.2.1.1.1.1.cmml"><mi id="S4.14.p3.28.m28.2.2.1.1.1.1.2" xref="S4.14.p3.28.m28.2.2.1.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.28.m28.2.2.1.1.1.1.3" xref="S4.14.p3.28.m28.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.28.m28.3.3.2.2.2.4" xref="S4.14.p3.28.m28.3.3.2.3.cmml">,</mo><msub id="S4.14.p3.28.m28.3.3.2.2.2.2" xref="S4.14.p3.28.m28.3.3.2.2.2.2.cmml"><mi id="S4.14.p3.28.m28.3.3.2.2.2.2.2" xref="S4.14.p3.28.m28.3.3.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.28.m28.3.3.2.2.2.2.3" xref="S4.14.p3.28.m28.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.14.p3.28.m28.3.3.2.2.2.5" stretchy="false" xref="S4.14.p3.28.m28.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.28.m28.3b"><apply id="S4.14.p3.28.m28.3.3.cmml" xref="S4.14.p3.28.m28.3.3"><geq id="S4.14.p3.28.m28.3.3.3.cmml" xref="S4.14.p3.28.m28.3.3.3"></geq><apply id="S4.14.p3.28.m28.3.3.4.cmml" xref="S4.14.p3.28.m28.3.3.4"><csymbol cd="ambiguous" id="S4.14.p3.28.m28.3.3.4.1.cmml" xref="S4.14.p3.28.m28.3.3.4">subscript</csymbol><ci id="S4.14.p3.28.m28.3.3.4.2.cmml" xref="S4.14.p3.28.m28.3.3.4.2">𝑓</ci><cn id="S4.14.p3.28.m28.3.3.4.3.cmml" type="integer" xref="S4.14.p3.28.m28.3.3.4.3">1</cn></apply><apply id="S4.14.p3.28.m28.3.3.2.3.cmml" xref="S4.14.p3.28.m28.3.3.2.2"><max id="S4.14.p3.28.m28.1.1.cmml" xref="S4.14.p3.28.m28.1.1"></max><apply id="S4.14.p3.28.m28.2.2.1.1.1.1.cmml" xref="S4.14.p3.28.m28.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.28.m28.2.2.1.1.1.1.1.cmml" xref="S4.14.p3.28.m28.2.2.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.28.m28.2.2.1.1.1.1.2.cmml" xref="S4.14.p3.28.m28.2.2.1.1.1.1.2">𝑓</ci><cn id="S4.14.p3.28.m28.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.28.m28.2.2.1.1.1.1.3">0</cn></apply><apply id="S4.14.p3.28.m28.3.3.2.2.2.2.cmml" xref="S4.14.p3.28.m28.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.28.m28.3.3.2.2.2.2.1.cmml" xref="S4.14.p3.28.m28.3.3.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.28.m28.3.3.2.2.2.2.2.cmml" xref="S4.14.p3.28.m28.3.3.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.28.m28.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.28.m28.3.3.2.2.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.28.m28.3c">f_{1}\geq\max(f_{0},f_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.28.m28.3d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ roman_max ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.14.p3.29.m29.1"><semantics id="S4.14.p3.29.m29.1a"><msub id="S4.14.p3.29.m29.1.1" xref="S4.14.p3.29.m29.1.1.cmml"><mi id="S4.14.p3.29.m29.1.1.2" xref="S4.14.p3.29.m29.1.1.2.cmml">P</mi><mn id="S4.14.p3.29.m29.1.1.3" xref="S4.14.p3.29.m29.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.29.m29.1b"><apply id="S4.14.p3.29.m29.1.1.cmml" xref="S4.14.p3.29.m29.1.1"><csymbol cd="ambiguous" id="S4.14.p3.29.m29.1.1.1.cmml" xref="S4.14.p3.29.m29.1.1">subscript</csymbol><ci id="S4.14.p3.29.m29.1.1.2.cmml" xref="S4.14.p3.29.m29.1.1.2">𝑃</ci><cn id="S4.14.p3.29.m29.1.1.3.cmml" type="integer" xref="S4.14.p3.29.m29.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.29.m29.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.29.m29.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="f_{2}\geq\max(f_{0},f_{1})" class="ltx_Math" display="inline" id="S4.14.p3.30.m30.3"><semantics id="S4.14.p3.30.m30.3a"><mrow id="S4.14.p3.30.m30.3.3" xref="S4.14.p3.30.m30.3.3.cmml"><msub id="S4.14.p3.30.m30.3.3.4" xref="S4.14.p3.30.m30.3.3.4.cmml"><mi id="S4.14.p3.30.m30.3.3.4.2" xref="S4.14.p3.30.m30.3.3.4.2.cmml">f</mi><mn id="S4.14.p3.30.m30.3.3.4.3" xref="S4.14.p3.30.m30.3.3.4.3.cmml">2</mn></msub><mo id="S4.14.p3.30.m30.3.3.3" xref="S4.14.p3.30.m30.3.3.3.cmml">≥</mo><mrow id="S4.14.p3.30.m30.3.3.2.2" xref="S4.14.p3.30.m30.3.3.2.3.cmml"><mi id="S4.14.p3.30.m30.1.1" xref="S4.14.p3.30.m30.1.1.cmml">max</mi><mo id="S4.14.p3.30.m30.3.3.2.2a" xref="S4.14.p3.30.m30.3.3.2.3.cmml">⁡</mo><mrow id="S4.14.p3.30.m30.3.3.2.2.2" xref="S4.14.p3.30.m30.3.3.2.3.cmml"><mo id="S4.14.p3.30.m30.3.3.2.2.2.3" stretchy="false" xref="S4.14.p3.30.m30.3.3.2.3.cmml">(</mo><msub id="S4.14.p3.30.m30.2.2.1.1.1.1" xref="S4.14.p3.30.m30.2.2.1.1.1.1.cmml"><mi id="S4.14.p3.30.m30.2.2.1.1.1.1.2" xref="S4.14.p3.30.m30.2.2.1.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.30.m30.2.2.1.1.1.1.3" xref="S4.14.p3.30.m30.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.30.m30.3.3.2.2.2.4" xref="S4.14.p3.30.m30.3.3.2.3.cmml">,</mo><msub id="S4.14.p3.30.m30.3.3.2.2.2.2" xref="S4.14.p3.30.m30.3.3.2.2.2.2.cmml"><mi id="S4.14.p3.30.m30.3.3.2.2.2.2.2" xref="S4.14.p3.30.m30.3.3.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.30.m30.3.3.2.2.2.2.3" xref="S4.14.p3.30.m30.3.3.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.14.p3.30.m30.3.3.2.2.2.5" stretchy="false" xref="S4.14.p3.30.m30.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.30.m30.3b"><apply id="S4.14.p3.30.m30.3.3.cmml" xref="S4.14.p3.30.m30.3.3"><geq id="S4.14.p3.30.m30.3.3.3.cmml" xref="S4.14.p3.30.m30.3.3.3"></geq><apply id="S4.14.p3.30.m30.3.3.4.cmml" xref="S4.14.p3.30.m30.3.3.4"><csymbol cd="ambiguous" id="S4.14.p3.30.m30.3.3.4.1.cmml" xref="S4.14.p3.30.m30.3.3.4">subscript</csymbol><ci id="S4.14.p3.30.m30.3.3.4.2.cmml" xref="S4.14.p3.30.m30.3.3.4.2">𝑓</ci><cn id="S4.14.p3.30.m30.3.3.4.3.cmml" type="integer" xref="S4.14.p3.30.m30.3.3.4.3">2</cn></apply><apply id="S4.14.p3.30.m30.3.3.2.3.cmml" xref="S4.14.p3.30.m30.3.3.2.2"><max id="S4.14.p3.30.m30.1.1.cmml" xref="S4.14.p3.30.m30.1.1"></max><apply id="S4.14.p3.30.m30.2.2.1.1.1.1.cmml" xref="S4.14.p3.30.m30.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.30.m30.2.2.1.1.1.1.1.cmml" xref="S4.14.p3.30.m30.2.2.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.30.m30.2.2.1.1.1.1.2.cmml" xref="S4.14.p3.30.m30.2.2.1.1.1.1.2">𝑓</ci><cn id="S4.14.p3.30.m30.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.30.m30.2.2.1.1.1.1.3">0</cn></apply><apply id="S4.14.p3.30.m30.3.3.2.2.2.2.cmml" xref="S4.14.p3.30.m30.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.30.m30.3.3.2.2.2.2.1.cmml" xref="S4.14.p3.30.m30.3.3.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.30.m30.3.3.2.2.2.2.2.cmml" xref="S4.14.p3.30.m30.3.3.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.30.m30.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.30.m30.3.3.2.2.2.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.30.m30.3c">f_{2}\geq\max(f_{0},f_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.30.m30.3d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ roman_max ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.14.p3.31.m31.1"><semantics id="S4.14.p3.31.m31.1a"><msub id="S4.14.p3.31.m31.1.1" xref="S4.14.p3.31.m31.1.1.cmml"><mi id="S4.14.p3.31.m31.1.1.2" xref="S4.14.p3.31.m31.1.1.2.cmml">P</mi><mn id="S4.14.p3.31.m31.1.1.3" xref="S4.14.p3.31.m31.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.31.m31.1b"><apply id="S4.14.p3.31.m31.1.1.cmml" xref="S4.14.p3.31.m31.1.1"><csymbol cd="ambiguous" id="S4.14.p3.31.m31.1.1.1.cmml" xref="S4.14.p3.31.m31.1.1">subscript</csymbol><ci id="S4.14.p3.31.m31.1.1.2.cmml" xref="S4.14.p3.31.m31.1.1.2">𝑃</ci><cn id="S4.14.p3.31.m31.1.1.3.cmml" type="integer" xref="S4.14.p3.31.m31.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.31.m31.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.31.m31.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="f_{0}\geq\max(f_{1},f_{2})" class="ltx_Math" display="inline" id="S4.14.p3.32.m32.3"><semantics id="S4.14.p3.32.m32.3a"><mrow id="S4.14.p3.32.m32.3.3" xref="S4.14.p3.32.m32.3.3.cmml"><msub id="S4.14.p3.32.m32.3.3.4" xref="S4.14.p3.32.m32.3.3.4.cmml"><mi id="S4.14.p3.32.m32.3.3.4.2" xref="S4.14.p3.32.m32.3.3.4.2.cmml">f</mi><mn id="S4.14.p3.32.m32.3.3.4.3" xref="S4.14.p3.32.m32.3.3.4.3.cmml">0</mn></msub><mo id="S4.14.p3.32.m32.3.3.3" xref="S4.14.p3.32.m32.3.3.3.cmml">≥</mo><mrow id="S4.14.p3.32.m32.3.3.2.2" xref="S4.14.p3.32.m32.3.3.2.3.cmml"><mi id="S4.14.p3.32.m32.1.1" xref="S4.14.p3.32.m32.1.1.cmml">max</mi><mo id="S4.14.p3.32.m32.3.3.2.2a" xref="S4.14.p3.32.m32.3.3.2.3.cmml">⁡</mo><mrow id="S4.14.p3.32.m32.3.3.2.2.2" xref="S4.14.p3.32.m32.3.3.2.3.cmml"><mo id="S4.14.p3.32.m32.3.3.2.2.2.3" stretchy="false" xref="S4.14.p3.32.m32.3.3.2.3.cmml">(</mo><msub id="S4.14.p3.32.m32.2.2.1.1.1.1" xref="S4.14.p3.32.m32.2.2.1.1.1.1.cmml"><mi id="S4.14.p3.32.m32.2.2.1.1.1.1.2" xref="S4.14.p3.32.m32.2.2.1.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.32.m32.2.2.1.1.1.1.3" xref="S4.14.p3.32.m32.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.14.p3.32.m32.3.3.2.2.2.4" xref="S4.14.p3.32.m32.3.3.2.3.cmml">,</mo><msub id="S4.14.p3.32.m32.3.3.2.2.2.2" xref="S4.14.p3.32.m32.3.3.2.2.2.2.cmml"><mi id="S4.14.p3.32.m32.3.3.2.2.2.2.2" xref="S4.14.p3.32.m32.3.3.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.32.m32.3.3.2.2.2.2.3" xref="S4.14.p3.32.m32.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.14.p3.32.m32.3.3.2.2.2.5" stretchy="false" xref="S4.14.p3.32.m32.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.32.m32.3b"><apply id="S4.14.p3.32.m32.3.3.cmml" xref="S4.14.p3.32.m32.3.3"><geq id="S4.14.p3.32.m32.3.3.3.cmml" xref="S4.14.p3.32.m32.3.3.3"></geq><apply id="S4.14.p3.32.m32.3.3.4.cmml" xref="S4.14.p3.32.m32.3.3.4"><csymbol cd="ambiguous" id="S4.14.p3.32.m32.3.3.4.1.cmml" xref="S4.14.p3.32.m32.3.3.4">subscript</csymbol><ci id="S4.14.p3.32.m32.3.3.4.2.cmml" xref="S4.14.p3.32.m32.3.3.4.2">𝑓</ci><cn id="S4.14.p3.32.m32.3.3.4.3.cmml" type="integer" xref="S4.14.p3.32.m32.3.3.4.3">0</cn></apply><apply id="S4.14.p3.32.m32.3.3.2.3.cmml" xref="S4.14.p3.32.m32.3.3.2.2"><max id="S4.14.p3.32.m32.1.1.cmml" xref="S4.14.p3.32.m32.1.1"></max><apply id="S4.14.p3.32.m32.2.2.1.1.1.1.cmml" xref="S4.14.p3.32.m32.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.32.m32.2.2.1.1.1.1.1.cmml" xref="S4.14.p3.32.m32.2.2.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.32.m32.2.2.1.1.1.1.2.cmml" xref="S4.14.p3.32.m32.2.2.1.1.1.1.2">𝑓</ci><cn id="S4.14.p3.32.m32.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.32.m32.2.2.1.1.1.1.3">1</cn></apply><apply id="S4.14.p3.32.m32.3.3.2.2.2.2.cmml" xref="S4.14.p3.32.m32.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.32.m32.3.3.2.2.2.2.1.cmml" xref="S4.14.p3.32.m32.3.3.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.32.m32.3.3.2.2.2.2.2.cmml" xref="S4.14.p3.32.m32.3.3.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.32.m32.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.32.m32.3.3.2.2.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.32.m32.3c">f_{0}\geq\max(f_{1},f_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.32.m32.3d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.14.p3.33.m33.1"><semantics id="S4.14.p3.33.m33.1a"><msub id="S4.14.p3.33.m33.1.1" xref="S4.14.p3.33.m33.1.1.cmml"><mi id="S4.14.p3.33.m33.1.1.2" xref="S4.14.p3.33.m33.1.1.2.cmml">P</mi><mn id="S4.14.p3.33.m33.1.1.3" xref="S4.14.p3.33.m33.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.14.p3.33.m33.1b"><apply id="S4.14.p3.33.m33.1.1.cmml" xref="S4.14.p3.33.m33.1.1"><csymbol cd="ambiguous" id="S4.14.p3.33.m33.1.1.1.cmml" xref="S4.14.p3.33.m33.1.1">subscript</csymbol><ci id="S4.14.p3.33.m33.1.1.2.cmml" xref="S4.14.p3.33.m33.1.1.2">𝑃</ci><cn id="S4.14.p3.33.m33.1.1.3.cmml" type="integer" xref="S4.14.p3.33.m33.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.33.m33.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.33.m33.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="f=\max(f_{0},f_{1},f_{2})=\max(f_{0},\max(f_{1},f_{2}))" class="ltx_Math" display="inline" id="S4.14.p3.34.m34.8"><semantics id="S4.14.p3.34.m34.8a"><mrow id="S4.14.p3.34.m34.8.8" xref="S4.14.p3.34.m34.8.8.cmml"><mi id="S4.14.p3.34.m34.8.8.7" xref="S4.14.p3.34.m34.8.8.7.cmml">f</mi><mo id="S4.14.p3.34.m34.8.8.8" xref="S4.14.p3.34.m34.8.8.8.cmml">=</mo><mrow id="S4.14.p3.34.m34.6.6.3.3" xref="S4.14.p3.34.m34.6.6.3.4.cmml"><mi id="S4.14.p3.34.m34.1.1" xref="S4.14.p3.34.m34.1.1.cmml">max</mi><mo id="S4.14.p3.34.m34.6.6.3.3a" xref="S4.14.p3.34.m34.6.6.3.4.cmml">⁡</mo><mrow id="S4.14.p3.34.m34.6.6.3.3.3" xref="S4.14.p3.34.m34.6.6.3.4.cmml"><mo id="S4.14.p3.34.m34.6.6.3.3.3.4" stretchy="false" xref="S4.14.p3.34.m34.6.6.3.4.cmml">(</mo><msub id="S4.14.p3.34.m34.4.4.1.1.1.1" xref="S4.14.p3.34.m34.4.4.1.1.1.1.cmml"><mi id="S4.14.p3.34.m34.4.4.1.1.1.1.2" xref="S4.14.p3.34.m34.4.4.1.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.34.m34.4.4.1.1.1.1.3" xref="S4.14.p3.34.m34.4.4.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.34.m34.6.6.3.3.3.5" xref="S4.14.p3.34.m34.6.6.3.4.cmml">,</mo><msub id="S4.14.p3.34.m34.5.5.2.2.2.2" xref="S4.14.p3.34.m34.5.5.2.2.2.2.cmml"><mi id="S4.14.p3.34.m34.5.5.2.2.2.2.2" xref="S4.14.p3.34.m34.5.5.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.34.m34.5.5.2.2.2.2.3" xref="S4.14.p3.34.m34.5.5.2.2.2.2.3.cmml">1</mn></msub><mo id="S4.14.p3.34.m34.6.6.3.3.3.6" xref="S4.14.p3.34.m34.6.6.3.4.cmml">,</mo><msub id="S4.14.p3.34.m34.6.6.3.3.3.3" xref="S4.14.p3.34.m34.6.6.3.3.3.3.cmml"><mi id="S4.14.p3.34.m34.6.6.3.3.3.3.2" xref="S4.14.p3.34.m34.6.6.3.3.3.3.2.cmml">f</mi><mn id="S4.14.p3.34.m34.6.6.3.3.3.3.3" xref="S4.14.p3.34.m34.6.6.3.3.3.3.3.cmml">2</mn></msub><mo id="S4.14.p3.34.m34.6.6.3.3.3.7" stretchy="false" xref="S4.14.p3.34.m34.6.6.3.4.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.34.m34.8.8.9" xref="S4.14.p3.34.m34.8.8.9.cmml">=</mo><mrow id="S4.14.p3.34.m34.8.8.5.2" xref="S4.14.p3.34.m34.8.8.5.3.cmml"><mi id="S4.14.p3.34.m34.3.3" xref="S4.14.p3.34.m34.3.3.cmml">max</mi><mo id="S4.14.p3.34.m34.8.8.5.2a" xref="S4.14.p3.34.m34.8.8.5.3.cmml">⁡</mo><mrow id="S4.14.p3.34.m34.8.8.5.2.2" xref="S4.14.p3.34.m34.8.8.5.3.cmml"><mo id="S4.14.p3.34.m34.8.8.5.2.2.3" stretchy="false" xref="S4.14.p3.34.m34.8.8.5.3.cmml">(</mo><msub id="S4.14.p3.34.m34.7.7.4.1.1.1" xref="S4.14.p3.34.m34.7.7.4.1.1.1.cmml"><mi id="S4.14.p3.34.m34.7.7.4.1.1.1.2" xref="S4.14.p3.34.m34.7.7.4.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.34.m34.7.7.4.1.1.1.3" xref="S4.14.p3.34.m34.7.7.4.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.34.m34.8.8.5.2.2.4" xref="S4.14.p3.34.m34.8.8.5.3.cmml">,</mo><mrow id="S4.14.p3.34.m34.8.8.5.2.2.2.2" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml"><mi id="S4.14.p3.34.m34.2.2" xref="S4.14.p3.34.m34.2.2.cmml">max</mi><mo id="S4.14.p3.34.m34.8.8.5.2.2.2.2a" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml">⁡</mo><mrow id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml"><mo id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.3" stretchy="false" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml">(</mo><msub id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.cmml"><mi id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.2" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.2.cmml">f</mi><mn id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.3" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.4" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml">,</mo><msub id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.cmml"><mi id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.2" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.3" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.5" stretchy="false" xref="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.34.m34.8.8.5.2.2.5" stretchy="false" xref="S4.14.p3.34.m34.8.8.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.34.m34.8b"><apply id="S4.14.p3.34.m34.8.8.cmml" xref="S4.14.p3.34.m34.8.8"><and id="S4.14.p3.34.m34.8.8a.cmml" xref="S4.14.p3.34.m34.8.8"></and><apply id="S4.14.p3.34.m34.8.8b.cmml" xref="S4.14.p3.34.m34.8.8"><eq id="S4.14.p3.34.m34.8.8.8.cmml" xref="S4.14.p3.34.m34.8.8.8"></eq><ci id="S4.14.p3.34.m34.8.8.7.cmml" xref="S4.14.p3.34.m34.8.8.7">𝑓</ci><apply id="S4.14.p3.34.m34.6.6.3.4.cmml" xref="S4.14.p3.34.m34.6.6.3.3"><max id="S4.14.p3.34.m34.1.1.cmml" xref="S4.14.p3.34.m34.1.1"></max><apply id="S4.14.p3.34.m34.4.4.1.1.1.1.cmml" xref="S4.14.p3.34.m34.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.4.4.1.1.1.1.1.cmml" xref="S4.14.p3.34.m34.4.4.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.34.m34.4.4.1.1.1.1.2.cmml" xref="S4.14.p3.34.m34.4.4.1.1.1.1.2">𝑓</ci><cn id="S4.14.p3.34.m34.4.4.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.34.m34.4.4.1.1.1.1.3">0</cn></apply><apply id="S4.14.p3.34.m34.5.5.2.2.2.2.cmml" xref="S4.14.p3.34.m34.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.5.5.2.2.2.2.1.cmml" xref="S4.14.p3.34.m34.5.5.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.34.m34.5.5.2.2.2.2.2.cmml" xref="S4.14.p3.34.m34.5.5.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.34.m34.5.5.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.34.m34.5.5.2.2.2.2.3">1</cn></apply><apply id="S4.14.p3.34.m34.6.6.3.3.3.3.cmml" xref="S4.14.p3.34.m34.6.6.3.3.3.3"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.6.6.3.3.3.3.1.cmml" xref="S4.14.p3.34.m34.6.6.3.3.3.3">subscript</csymbol><ci id="S4.14.p3.34.m34.6.6.3.3.3.3.2.cmml" xref="S4.14.p3.34.m34.6.6.3.3.3.3.2">𝑓</ci><cn id="S4.14.p3.34.m34.6.6.3.3.3.3.3.cmml" type="integer" xref="S4.14.p3.34.m34.6.6.3.3.3.3.3">2</cn></apply></apply></apply><apply id="S4.14.p3.34.m34.8.8c.cmml" xref="S4.14.p3.34.m34.8.8"><eq id="S4.14.p3.34.m34.8.8.9.cmml" xref="S4.14.p3.34.m34.8.8.9"></eq><share href="https://arxiv.org/html/2503.13001v1#S4.14.p3.34.m34.6.6.3.cmml" id="S4.14.p3.34.m34.8.8d.cmml" xref="S4.14.p3.34.m34.8.8"></share><apply id="S4.14.p3.34.m34.8.8.5.3.cmml" xref="S4.14.p3.34.m34.8.8.5.2"><max id="S4.14.p3.34.m34.3.3.cmml" xref="S4.14.p3.34.m34.3.3"></max><apply id="S4.14.p3.34.m34.7.7.4.1.1.1.cmml" xref="S4.14.p3.34.m34.7.7.4.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.7.7.4.1.1.1.1.cmml" xref="S4.14.p3.34.m34.7.7.4.1.1.1">subscript</csymbol><ci id="S4.14.p3.34.m34.7.7.4.1.1.1.2.cmml" xref="S4.14.p3.34.m34.7.7.4.1.1.1.2">𝑓</ci><cn id="S4.14.p3.34.m34.7.7.4.1.1.1.3.cmml" type="integer" xref="S4.14.p3.34.m34.7.7.4.1.1.1.3">0</cn></apply><apply id="S4.14.p3.34.m34.8.8.5.2.2.2.3.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2"><max id="S4.14.p3.34.m34.2.2.cmml" xref="S4.14.p3.34.m34.2.2"></max><apply id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.1.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1">subscript</csymbol><ci id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.2.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.2">𝑓</ci><cn id="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.14.p3.34.m34.8.8.5.2.2.2.1.1.1.3">1</cn></apply><apply id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.1.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.2.cmml" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.34.m34.8.8.5.2.2.2.2.2.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.34.m34.8c">f=\max(f_{0},f_{1},f_{2})=\max(f_{0},\max(f_{1},f_{2}))</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.34.m34.8d">italic_f = roman_max ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = roman_max ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math>, which is of the form (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E36" title="Equation 36 ‣ Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">36</span></a>). For the case that <math alttext="f(p_{0})<f_{0}(p_{0})" class="ltx_Math" display="inline" id="S4.14.p3.35.m35.2"><semantics id="S4.14.p3.35.m35.2a"><mrow id="S4.14.p3.35.m35.2.2" xref="S4.14.p3.35.m35.2.2.cmml"><mrow id="S4.14.p3.35.m35.1.1.1" xref="S4.14.p3.35.m35.1.1.1.cmml"><mi id="S4.14.p3.35.m35.1.1.1.3" xref="S4.14.p3.35.m35.1.1.1.3.cmml">f</mi><mo id="S4.14.p3.35.m35.1.1.1.2" xref="S4.14.p3.35.m35.1.1.1.2.cmml">⁢</mo><mrow id="S4.14.p3.35.m35.1.1.1.1.1" xref="S4.14.p3.35.m35.1.1.1.1.1.1.cmml"><mo id="S4.14.p3.35.m35.1.1.1.1.1.2" stretchy="false" xref="S4.14.p3.35.m35.1.1.1.1.1.1.cmml">(</mo><msub id="S4.14.p3.35.m35.1.1.1.1.1.1" xref="S4.14.p3.35.m35.1.1.1.1.1.1.cmml"><mi id="S4.14.p3.35.m35.1.1.1.1.1.1.2" xref="S4.14.p3.35.m35.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.35.m35.1.1.1.1.1.1.3" xref="S4.14.p3.35.m35.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.35.m35.1.1.1.1.1.3" stretchy="false" xref="S4.14.p3.35.m35.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.35.m35.2.2.3" xref="S4.14.p3.35.m35.2.2.3.cmml"><</mo><mrow id="S4.14.p3.35.m35.2.2.2" xref="S4.14.p3.35.m35.2.2.2.cmml"><msub id="S4.14.p3.35.m35.2.2.2.3" xref="S4.14.p3.35.m35.2.2.2.3.cmml"><mi id="S4.14.p3.35.m35.2.2.2.3.2" xref="S4.14.p3.35.m35.2.2.2.3.2.cmml">f</mi><mn id="S4.14.p3.35.m35.2.2.2.3.3" xref="S4.14.p3.35.m35.2.2.2.3.3.cmml">0</mn></msub><mo id="S4.14.p3.35.m35.2.2.2.2" xref="S4.14.p3.35.m35.2.2.2.2.cmml">⁢</mo><mrow id="S4.14.p3.35.m35.2.2.2.1.1" xref="S4.14.p3.35.m35.2.2.2.1.1.1.cmml"><mo id="S4.14.p3.35.m35.2.2.2.1.1.2" stretchy="false" xref="S4.14.p3.35.m35.2.2.2.1.1.1.cmml">(</mo><msub id="S4.14.p3.35.m35.2.2.2.1.1.1" xref="S4.14.p3.35.m35.2.2.2.1.1.1.cmml"><mi id="S4.14.p3.35.m35.2.2.2.1.1.1.2" xref="S4.14.p3.35.m35.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.14.p3.35.m35.2.2.2.1.1.1.3" xref="S4.14.p3.35.m35.2.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S4.14.p3.35.m35.2.2.2.1.1.3" stretchy="false" xref="S4.14.p3.35.m35.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.35.m35.2b"><apply id="S4.14.p3.35.m35.2.2.cmml" xref="S4.14.p3.35.m35.2.2"><lt id="S4.14.p3.35.m35.2.2.3.cmml" xref="S4.14.p3.35.m35.2.2.3"></lt><apply id="S4.14.p3.35.m35.1.1.1.cmml" xref="S4.14.p3.35.m35.1.1.1"><times id="S4.14.p3.35.m35.1.1.1.2.cmml" xref="S4.14.p3.35.m35.1.1.1.2"></times><ci id="S4.14.p3.35.m35.1.1.1.3.cmml" xref="S4.14.p3.35.m35.1.1.1.3">𝑓</ci><apply id="S4.14.p3.35.m35.1.1.1.1.1.1.cmml" xref="S4.14.p3.35.m35.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.14.p3.35.m35.1.1.1.1.1.1.1.cmml" xref="S4.14.p3.35.m35.1.1.1.1.1">subscript</csymbol><ci id="S4.14.p3.35.m35.1.1.1.1.1.1.2.cmml" xref="S4.14.p3.35.m35.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.14.p3.35.m35.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.14.p3.35.m35.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S4.14.p3.35.m35.2.2.2.cmml" xref="S4.14.p3.35.m35.2.2.2"><times id="S4.14.p3.35.m35.2.2.2.2.cmml" xref="S4.14.p3.35.m35.2.2.2.2"></times><apply id="S4.14.p3.35.m35.2.2.2.3.cmml" xref="S4.14.p3.35.m35.2.2.2.3"><csymbol cd="ambiguous" id="S4.14.p3.35.m35.2.2.2.3.1.cmml" xref="S4.14.p3.35.m35.2.2.2.3">subscript</csymbol><ci id="S4.14.p3.35.m35.2.2.2.3.2.cmml" xref="S4.14.p3.35.m35.2.2.2.3.2">𝑓</ci><cn id="S4.14.p3.35.m35.2.2.2.3.3.cmml" type="integer" xref="S4.14.p3.35.m35.2.2.2.3.3">0</cn></apply><apply id="S4.14.p3.35.m35.2.2.2.1.1.1.cmml" xref="S4.14.p3.35.m35.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.14.p3.35.m35.2.2.2.1.1.1.1.cmml" xref="S4.14.p3.35.m35.2.2.2.1.1">subscript</csymbol><ci id="S4.14.p3.35.m35.2.2.2.1.1.1.2.cmml" xref="S4.14.p3.35.m35.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.14.p3.35.m35.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.14.p3.35.m35.2.2.2.1.1.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.35.m35.2c">f(p_{0})<f_{0}(p_{0})</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.35.m35.2d">italic_f ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>, one can apply the same arguments to <math alttext="-f" class="ltx_Math" display="inline" id="S4.14.p3.36.m36.1"><semantics id="S4.14.p3.36.m36.1a"><mrow id="S4.14.p3.36.m36.1.1" xref="S4.14.p3.36.m36.1.1.cmml"><mo id="S4.14.p3.36.m36.1.1a" xref="S4.14.p3.36.m36.1.1.cmml">−</mo><mi id="S4.14.p3.36.m36.1.1.2" xref="S4.14.p3.36.m36.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.36.m36.1b"><apply id="S4.14.p3.36.m36.1.1.cmml" xref="S4.14.p3.36.m36.1.1"><minus id="S4.14.p3.36.m36.1.1.1.cmml" xref="S4.14.p3.36.m36.1.1"></minus><ci id="S4.14.p3.36.m36.1.1.2.cmml" xref="S4.14.p3.36.m36.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.36.m36.1c">-f</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.36.m36.1d">- italic_f</annotation></semantics></math> and get <math alttext="f=-\max(-f_{0},\max(-f_{1},-f_{2}))" class="ltx_Math" display="inline" id="S4.14.p3.37.m37.4"><semantics id="S4.14.p3.37.m37.4a"><mrow id="S4.14.p3.37.m37.4.4" xref="S4.14.p3.37.m37.4.4.cmml"><mi id="S4.14.p3.37.m37.4.4.4" xref="S4.14.p3.37.m37.4.4.4.cmml">f</mi><mo id="S4.14.p3.37.m37.4.4.3" xref="S4.14.p3.37.m37.4.4.3.cmml">=</mo><mrow id="S4.14.p3.37.m37.4.4.2" xref="S4.14.p3.37.m37.4.4.2.cmml"><mo id="S4.14.p3.37.m37.4.4.2a" rspace="0.167em" xref="S4.14.p3.37.m37.4.4.2.cmml">−</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml"><mi id="S4.14.p3.37.m37.2.2" xref="S4.14.p3.37.m37.2.2.cmml">max</mi><mo id="S4.14.p3.37.m37.4.4.2.2.2a" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml">⁡</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml"><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.3" stretchy="false" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml">(</mo><mrow id="S4.14.p3.37.m37.3.3.1.1.1.1.1" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.cmml"><mo id="S4.14.p3.37.m37.3.3.1.1.1.1.1a" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.cmml">−</mo><msub id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.cmml"><mi id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.2" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.3" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.3.cmml">0</mn></msub></mrow><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.4" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml">,</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml"><mi id="S4.14.p3.37.m37.1.1" xref="S4.14.p3.37.m37.1.1.cmml">max</mi><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2a" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml">⁡</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml"><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.3" stretchy="false" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml">(</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.cmml"><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1a" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.cmml">−</mo><msub id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.cmml"><mi id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.2.cmml">f</mi><mn id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.3" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.4" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml">,</mo><mrow id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.cmml"><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2a" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.cmml">−</mo><msub id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.cmml"><mi id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.2" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.2.cmml">f</mi><mn id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.3" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.5" stretchy="false" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.14.p3.37.m37.4.4.2.2.2.2.5" stretchy="false" xref="S4.14.p3.37.m37.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.14.p3.37.m37.4b"><apply id="S4.14.p3.37.m37.4.4.cmml" xref="S4.14.p3.37.m37.4.4"><eq id="S4.14.p3.37.m37.4.4.3.cmml" xref="S4.14.p3.37.m37.4.4.3"></eq><ci id="S4.14.p3.37.m37.4.4.4.cmml" xref="S4.14.p3.37.m37.4.4.4">𝑓</ci><apply id="S4.14.p3.37.m37.4.4.2.cmml" xref="S4.14.p3.37.m37.4.4.2"><minus id="S4.14.p3.37.m37.4.4.2.3.cmml" xref="S4.14.p3.37.m37.4.4.2"></minus><apply id="S4.14.p3.37.m37.4.4.2.2.3.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2"><max id="S4.14.p3.37.m37.2.2.cmml" xref="S4.14.p3.37.m37.2.2"></max><apply id="S4.14.p3.37.m37.3.3.1.1.1.1.1.cmml" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1"><minus id="S4.14.p3.37.m37.3.3.1.1.1.1.1.1.cmml" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1"></minus><apply id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.cmml" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.1.cmml" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2">subscript</csymbol><ci id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.2.cmml" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.2">𝑓</ci><cn id="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.14.p3.37.m37.3.3.1.1.1.1.1.2.3">0</cn></apply></apply><apply id="S4.14.p3.37.m37.4.4.2.2.2.2.2.3.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2"><max id="S4.14.p3.37.m37.1.1.cmml" xref="S4.14.p3.37.m37.1.1"></max><apply id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1"><minus id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.1.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1"></minus><apply id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.1.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2">subscript</csymbol><ci id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.2.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.2">𝑓</ci><cn id="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.1.1.1.2.3">1</cn></apply></apply><apply id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2"><minus id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.1.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2"></minus><apply id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.1.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.2">𝑓</ci><cn id="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.14.p3.37.m37.4.4.2.2.2.2.2.2.2.2.2.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.14.p3.37.m37.4c">f=-\max(-f_{0},\max(-f_{1},-f_{2}))</annotation><annotation encoding="application/x-llamapun" id="S4.14.p3.37.m37.4d">italic_f = - roman_max ( - italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_max ( - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math>, which also satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E36" title="Equation 36 ‣ Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">36</span></a>).</p> </div> <div class="ltx_para" id="S4.15.p4"> <p class="ltx_p" id="S4.15.p4.25">Now, consider the case that <math alttext="\alpha_{i}\geq\pi" class="ltx_Math" display="inline" id="S4.15.p4.1.m1.1"><semantics id="S4.15.p4.1.m1.1a"><mrow id="S4.15.p4.1.m1.1.1" xref="S4.15.p4.1.m1.1.1.cmml"><msub id="S4.15.p4.1.m1.1.1.2" xref="S4.15.p4.1.m1.1.1.2.cmml"><mi id="S4.15.p4.1.m1.1.1.2.2" xref="S4.15.p4.1.m1.1.1.2.2.cmml">α</mi><mi id="S4.15.p4.1.m1.1.1.2.3" xref="S4.15.p4.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.15.p4.1.m1.1.1.1" xref="S4.15.p4.1.m1.1.1.1.cmml">≥</mo><mi id="S4.15.p4.1.m1.1.1.3" xref="S4.15.p4.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.1.m1.1b"><apply id="S4.15.p4.1.m1.1.1.cmml" xref="S4.15.p4.1.m1.1.1"><geq id="S4.15.p4.1.m1.1.1.1.cmml" xref="S4.15.p4.1.m1.1.1.1"></geq><apply id="S4.15.p4.1.m1.1.1.2.cmml" xref="S4.15.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.15.p4.1.m1.1.1.2.1.cmml" xref="S4.15.p4.1.m1.1.1.2">subscript</csymbol><ci id="S4.15.p4.1.m1.1.1.2.2.cmml" xref="S4.15.p4.1.m1.1.1.2.2">𝛼</ci><ci id="S4.15.p4.1.m1.1.1.2.3.cmml" xref="S4.15.p4.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S4.15.p4.1.m1.1.1.3.cmml" xref="S4.15.p4.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.1.m1.1c">\alpha_{i}\geq\pi</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ italic_π</annotation></semantics></math> for some <math alttext="i\in\{0,1,2\}" class="ltx_Math" display="inline" id="S4.15.p4.2.m2.3"><semantics id="S4.15.p4.2.m2.3a"><mrow id="S4.15.p4.2.m2.3.4" xref="S4.15.p4.2.m2.3.4.cmml"><mi id="S4.15.p4.2.m2.3.4.2" xref="S4.15.p4.2.m2.3.4.2.cmml">i</mi><mo id="S4.15.p4.2.m2.3.4.1" xref="S4.15.p4.2.m2.3.4.1.cmml">∈</mo><mrow id="S4.15.p4.2.m2.3.4.3.2" xref="S4.15.p4.2.m2.3.4.3.1.cmml"><mo id="S4.15.p4.2.m2.3.4.3.2.1" stretchy="false" xref="S4.15.p4.2.m2.3.4.3.1.cmml">{</mo><mn id="S4.15.p4.2.m2.1.1" xref="S4.15.p4.2.m2.1.1.cmml">0</mn><mo id="S4.15.p4.2.m2.3.4.3.2.2" xref="S4.15.p4.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.15.p4.2.m2.2.2" xref="S4.15.p4.2.m2.2.2.cmml">1</mn><mo id="S4.15.p4.2.m2.3.4.3.2.3" xref="S4.15.p4.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.15.p4.2.m2.3.3" xref="S4.15.p4.2.m2.3.3.cmml">2</mn><mo id="S4.15.p4.2.m2.3.4.3.2.4" stretchy="false" xref="S4.15.p4.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.2.m2.3b"><apply id="S4.15.p4.2.m2.3.4.cmml" xref="S4.15.p4.2.m2.3.4"><in id="S4.15.p4.2.m2.3.4.1.cmml" xref="S4.15.p4.2.m2.3.4.1"></in><ci id="S4.15.p4.2.m2.3.4.2.cmml" xref="S4.15.p4.2.m2.3.4.2">𝑖</ci><set id="S4.15.p4.2.m2.3.4.3.1.cmml" xref="S4.15.p4.2.m2.3.4.3.2"><cn id="S4.15.p4.2.m2.1.1.cmml" type="integer" xref="S4.15.p4.2.m2.1.1">0</cn><cn id="S4.15.p4.2.m2.2.2.cmml" type="integer" xref="S4.15.p4.2.m2.2.2">1</cn><cn id="S4.15.p4.2.m2.3.3.cmml" type="integer" xref="S4.15.p4.2.m2.3.3">2</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.2.m2.3c">i\in\{0,1,2\}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.2.m2.3d">italic_i ∈ { 0 , 1 , 2 }</annotation></semantics></math>, say <math alttext="i=0" class="ltx_Math" display="inline" id="S4.15.p4.3.m3.1"><semantics id="S4.15.p4.3.m3.1a"><mrow id="S4.15.p4.3.m3.1.1" xref="S4.15.p4.3.m3.1.1.cmml"><mi id="S4.15.p4.3.m3.1.1.2" xref="S4.15.p4.3.m3.1.1.2.cmml">i</mi><mo id="S4.15.p4.3.m3.1.1.1" xref="S4.15.p4.3.m3.1.1.1.cmml">=</mo><mn id="S4.15.p4.3.m3.1.1.3" xref="S4.15.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.3.m3.1b"><apply id="S4.15.p4.3.m3.1.1.cmml" xref="S4.15.p4.3.m3.1.1"><eq id="S4.15.p4.3.m3.1.1.1.cmml" xref="S4.15.p4.3.m3.1.1.1"></eq><ci id="S4.15.p4.3.m3.1.1.2.cmml" xref="S4.15.p4.3.m3.1.1.2">𝑖</ci><cn id="S4.15.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.15.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.3.m3.1c">i=0</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.3.m3.1d">italic_i = 0</annotation></semantics></math>. First, assume that <math alttext="f(p_{0})=f_{1}(p_{0})=f_{2}(p_{0})\geq f_{0}(p_{0})" class="ltx_Math" display="inline" id="S4.15.p4.4.m4.4"><semantics id="S4.15.p4.4.m4.4a"><mrow id="S4.15.p4.4.m4.4.4" xref="S4.15.p4.4.m4.4.4.cmml"><mrow id="S4.15.p4.4.m4.1.1.1" xref="S4.15.p4.4.m4.1.1.1.cmml"><mi id="S4.15.p4.4.m4.1.1.1.3" xref="S4.15.p4.4.m4.1.1.1.3.cmml">f</mi><mo id="S4.15.p4.4.m4.1.1.1.2" xref="S4.15.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.15.p4.4.m4.1.1.1.1.1" xref="S4.15.p4.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.15.p4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.15.p4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S4.15.p4.4.m4.1.1.1.1.1.1" xref="S4.15.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.15.p4.4.m4.1.1.1.1.1.1.2" xref="S4.15.p4.4.m4.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.4.m4.1.1.1.1.1.1.3" xref="S4.15.p4.4.m4.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.15.p4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.4.m4.4.4.6" xref="S4.15.p4.4.m4.4.4.6.cmml">=</mo><mrow id="S4.15.p4.4.m4.2.2.2" xref="S4.15.p4.4.m4.2.2.2.cmml"><msub id="S4.15.p4.4.m4.2.2.2.3" xref="S4.15.p4.4.m4.2.2.2.3.cmml"><mi id="S4.15.p4.4.m4.2.2.2.3.2" xref="S4.15.p4.4.m4.2.2.2.3.2.cmml">f</mi><mn id="S4.15.p4.4.m4.2.2.2.3.3" xref="S4.15.p4.4.m4.2.2.2.3.3.cmml">1</mn></msub><mo id="S4.15.p4.4.m4.2.2.2.2" xref="S4.15.p4.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="S4.15.p4.4.m4.2.2.2.1.1" xref="S4.15.p4.4.m4.2.2.2.1.1.1.cmml"><mo id="S4.15.p4.4.m4.2.2.2.1.1.2" stretchy="false" xref="S4.15.p4.4.m4.2.2.2.1.1.1.cmml">(</mo><msub id="S4.15.p4.4.m4.2.2.2.1.1.1" xref="S4.15.p4.4.m4.2.2.2.1.1.1.cmml"><mi id="S4.15.p4.4.m4.2.2.2.1.1.1.2" xref="S4.15.p4.4.m4.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.4.m4.2.2.2.1.1.1.3" xref="S4.15.p4.4.m4.2.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.4.m4.2.2.2.1.1.3" stretchy="false" xref="S4.15.p4.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.4.m4.4.4.7" xref="S4.15.p4.4.m4.4.4.7.cmml">=</mo><mrow id="S4.15.p4.4.m4.3.3.3" xref="S4.15.p4.4.m4.3.3.3.cmml"><msub id="S4.15.p4.4.m4.3.3.3.3" xref="S4.15.p4.4.m4.3.3.3.3.cmml"><mi id="S4.15.p4.4.m4.3.3.3.3.2" xref="S4.15.p4.4.m4.3.3.3.3.2.cmml">f</mi><mn id="S4.15.p4.4.m4.3.3.3.3.3" xref="S4.15.p4.4.m4.3.3.3.3.3.cmml">2</mn></msub><mo id="S4.15.p4.4.m4.3.3.3.2" xref="S4.15.p4.4.m4.3.3.3.2.cmml">⁢</mo><mrow id="S4.15.p4.4.m4.3.3.3.1.1" xref="S4.15.p4.4.m4.3.3.3.1.1.1.cmml"><mo id="S4.15.p4.4.m4.3.3.3.1.1.2" stretchy="false" xref="S4.15.p4.4.m4.3.3.3.1.1.1.cmml">(</mo><msub id="S4.15.p4.4.m4.3.3.3.1.1.1" xref="S4.15.p4.4.m4.3.3.3.1.1.1.cmml"><mi id="S4.15.p4.4.m4.3.3.3.1.1.1.2" xref="S4.15.p4.4.m4.3.3.3.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.4.m4.3.3.3.1.1.1.3" xref="S4.15.p4.4.m4.3.3.3.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.4.m4.3.3.3.1.1.3" stretchy="false" xref="S4.15.p4.4.m4.3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.4.m4.4.4.8" xref="S4.15.p4.4.m4.4.4.8.cmml">≥</mo><mrow id="S4.15.p4.4.m4.4.4.4" xref="S4.15.p4.4.m4.4.4.4.cmml"><msub id="S4.15.p4.4.m4.4.4.4.3" xref="S4.15.p4.4.m4.4.4.4.3.cmml"><mi id="S4.15.p4.4.m4.4.4.4.3.2" xref="S4.15.p4.4.m4.4.4.4.3.2.cmml">f</mi><mn id="S4.15.p4.4.m4.4.4.4.3.3" xref="S4.15.p4.4.m4.4.4.4.3.3.cmml">0</mn></msub><mo id="S4.15.p4.4.m4.4.4.4.2" xref="S4.15.p4.4.m4.4.4.4.2.cmml">⁢</mo><mrow id="S4.15.p4.4.m4.4.4.4.1.1" xref="S4.15.p4.4.m4.4.4.4.1.1.1.cmml"><mo id="S4.15.p4.4.m4.4.4.4.1.1.2" stretchy="false" xref="S4.15.p4.4.m4.4.4.4.1.1.1.cmml">(</mo><msub id="S4.15.p4.4.m4.4.4.4.1.1.1" xref="S4.15.p4.4.m4.4.4.4.1.1.1.cmml"><mi id="S4.15.p4.4.m4.4.4.4.1.1.1.2" xref="S4.15.p4.4.m4.4.4.4.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.4.m4.4.4.4.1.1.1.3" xref="S4.15.p4.4.m4.4.4.4.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.4.m4.4.4.4.1.1.3" stretchy="false" xref="S4.15.p4.4.m4.4.4.4.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.4.m4.4b"><apply id="S4.15.p4.4.m4.4.4.cmml" xref="S4.15.p4.4.m4.4.4"><and id="S4.15.p4.4.m4.4.4a.cmml" xref="S4.15.p4.4.m4.4.4"></and><apply id="S4.15.p4.4.m4.4.4b.cmml" xref="S4.15.p4.4.m4.4.4"><eq id="S4.15.p4.4.m4.4.4.6.cmml" xref="S4.15.p4.4.m4.4.4.6"></eq><apply id="S4.15.p4.4.m4.1.1.1.cmml" xref="S4.15.p4.4.m4.1.1.1"><times id="S4.15.p4.4.m4.1.1.1.2.cmml" xref="S4.15.p4.4.m4.1.1.1.2"></times><ci id="S4.15.p4.4.m4.1.1.1.3.cmml" xref="S4.15.p4.4.m4.1.1.1.3">𝑓</ci><apply id="S4.15.p4.4.m4.1.1.1.1.1.1.cmml" xref="S4.15.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.15.p4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.15.p4.4.m4.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.15.p4.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.4.m4.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S4.15.p4.4.m4.2.2.2.cmml" xref="S4.15.p4.4.m4.2.2.2"><times id="S4.15.p4.4.m4.2.2.2.2.cmml" xref="S4.15.p4.4.m4.2.2.2.2"></times><apply id="S4.15.p4.4.m4.2.2.2.3.cmml" xref="S4.15.p4.4.m4.2.2.2.3"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.2.2.2.3.1.cmml" xref="S4.15.p4.4.m4.2.2.2.3">subscript</csymbol><ci id="S4.15.p4.4.m4.2.2.2.3.2.cmml" xref="S4.15.p4.4.m4.2.2.2.3.2">𝑓</ci><cn id="S4.15.p4.4.m4.2.2.2.3.3.cmml" type="integer" xref="S4.15.p4.4.m4.2.2.2.3.3">1</cn></apply><apply id="S4.15.p4.4.m4.2.2.2.1.1.1.cmml" xref="S4.15.p4.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.2.2.2.1.1.1.1.cmml" xref="S4.15.p4.4.m4.2.2.2.1.1">subscript</csymbol><ci id="S4.15.p4.4.m4.2.2.2.1.1.1.2.cmml" xref="S4.15.p4.4.m4.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.15.p4.4.m4.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.15.p4.4.m4.2.2.2.1.1.1.3">0</cn></apply></apply></apply><apply id="S4.15.p4.4.m4.4.4c.cmml" xref="S4.15.p4.4.m4.4.4"><eq id="S4.15.p4.4.m4.4.4.7.cmml" xref="S4.15.p4.4.m4.4.4.7"></eq><share href="https://arxiv.org/html/2503.13001v1#S4.15.p4.4.m4.2.2.2.cmml" id="S4.15.p4.4.m4.4.4d.cmml" xref="S4.15.p4.4.m4.4.4"></share><apply id="S4.15.p4.4.m4.3.3.3.cmml" xref="S4.15.p4.4.m4.3.3.3"><times id="S4.15.p4.4.m4.3.3.3.2.cmml" xref="S4.15.p4.4.m4.3.3.3.2"></times><apply id="S4.15.p4.4.m4.3.3.3.3.cmml" xref="S4.15.p4.4.m4.3.3.3.3"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.3.3.3.3.1.cmml" xref="S4.15.p4.4.m4.3.3.3.3">subscript</csymbol><ci id="S4.15.p4.4.m4.3.3.3.3.2.cmml" xref="S4.15.p4.4.m4.3.3.3.3.2">𝑓</ci><cn id="S4.15.p4.4.m4.3.3.3.3.3.cmml" type="integer" xref="S4.15.p4.4.m4.3.3.3.3.3">2</cn></apply><apply id="S4.15.p4.4.m4.3.3.3.1.1.1.cmml" xref="S4.15.p4.4.m4.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.3.3.3.1.1.1.1.cmml" xref="S4.15.p4.4.m4.3.3.3.1.1">subscript</csymbol><ci id="S4.15.p4.4.m4.3.3.3.1.1.1.2.cmml" xref="S4.15.p4.4.m4.3.3.3.1.1.1.2">𝑝</ci><cn id="S4.15.p4.4.m4.3.3.3.1.1.1.3.cmml" type="integer" xref="S4.15.p4.4.m4.3.3.3.1.1.1.3">0</cn></apply></apply></apply><apply id="S4.15.p4.4.m4.4.4e.cmml" xref="S4.15.p4.4.m4.4.4"><geq id="S4.15.p4.4.m4.4.4.8.cmml" xref="S4.15.p4.4.m4.4.4.8"></geq><share href="https://arxiv.org/html/2503.13001v1#S4.15.p4.4.m4.3.3.3.cmml" id="S4.15.p4.4.m4.4.4f.cmml" xref="S4.15.p4.4.m4.4.4"></share><apply id="S4.15.p4.4.m4.4.4.4.cmml" xref="S4.15.p4.4.m4.4.4.4"><times id="S4.15.p4.4.m4.4.4.4.2.cmml" xref="S4.15.p4.4.m4.4.4.4.2"></times><apply id="S4.15.p4.4.m4.4.4.4.3.cmml" xref="S4.15.p4.4.m4.4.4.4.3"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.4.4.4.3.1.cmml" xref="S4.15.p4.4.m4.4.4.4.3">subscript</csymbol><ci id="S4.15.p4.4.m4.4.4.4.3.2.cmml" xref="S4.15.p4.4.m4.4.4.4.3.2">𝑓</ci><cn id="S4.15.p4.4.m4.4.4.4.3.3.cmml" type="integer" xref="S4.15.p4.4.m4.4.4.4.3.3">0</cn></apply><apply id="S4.15.p4.4.m4.4.4.4.1.1.1.cmml" xref="S4.15.p4.4.m4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.15.p4.4.m4.4.4.4.1.1.1.1.cmml" xref="S4.15.p4.4.m4.4.4.4.1.1">subscript</csymbol><ci id="S4.15.p4.4.m4.4.4.4.1.1.1.2.cmml" xref="S4.15.p4.4.m4.4.4.4.1.1.1.2">𝑝</ci><cn id="S4.15.p4.4.m4.4.4.4.1.1.1.3.cmml" type="integer" xref="S4.15.p4.4.m4.4.4.4.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.4.m4.4c">f(p_{0})=f_{1}(p_{0})=f_{2}(p_{0})\geq f_{0}(p_{0})</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.4.m4.4d">italic_f ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. Since <math alttext="\alpha_{1}+\alpha_{2}<\pi" class="ltx_Math" display="inline" id="S4.15.p4.5.m5.1"><semantics id="S4.15.p4.5.m5.1a"><mrow id="S4.15.p4.5.m5.1.1" xref="S4.15.p4.5.m5.1.1.cmml"><mrow id="S4.15.p4.5.m5.1.1.2" xref="S4.15.p4.5.m5.1.1.2.cmml"><msub id="S4.15.p4.5.m5.1.1.2.2" xref="S4.15.p4.5.m5.1.1.2.2.cmml"><mi id="S4.15.p4.5.m5.1.1.2.2.2" xref="S4.15.p4.5.m5.1.1.2.2.2.cmml">α</mi><mn id="S4.15.p4.5.m5.1.1.2.2.3" xref="S4.15.p4.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.15.p4.5.m5.1.1.2.1" xref="S4.15.p4.5.m5.1.1.2.1.cmml">+</mo><msub id="S4.15.p4.5.m5.1.1.2.3" xref="S4.15.p4.5.m5.1.1.2.3.cmml"><mi id="S4.15.p4.5.m5.1.1.2.3.2" xref="S4.15.p4.5.m5.1.1.2.3.2.cmml">α</mi><mn id="S4.15.p4.5.m5.1.1.2.3.3" xref="S4.15.p4.5.m5.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S4.15.p4.5.m5.1.1.1" xref="S4.15.p4.5.m5.1.1.1.cmml"><</mo><mi id="S4.15.p4.5.m5.1.1.3" xref="S4.15.p4.5.m5.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.5.m5.1b"><apply id="S4.15.p4.5.m5.1.1.cmml" xref="S4.15.p4.5.m5.1.1"><lt id="S4.15.p4.5.m5.1.1.1.cmml" xref="S4.15.p4.5.m5.1.1.1"></lt><apply id="S4.15.p4.5.m5.1.1.2.cmml" xref="S4.15.p4.5.m5.1.1.2"><plus id="S4.15.p4.5.m5.1.1.2.1.cmml" xref="S4.15.p4.5.m5.1.1.2.1"></plus><apply id="S4.15.p4.5.m5.1.1.2.2.cmml" xref="S4.15.p4.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S4.15.p4.5.m5.1.1.2.2.1.cmml" xref="S4.15.p4.5.m5.1.1.2.2">subscript</csymbol><ci id="S4.15.p4.5.m5.1.1.2.2.2.cmml" xref="S4.15.p4.5.m5.1.1.2.2.2">𝛼</ci><cn id="S4.15.p4.5.m5.1.1.2.2.3.cmml" type="integer" xref="S4.15.p4.5.m5.1.1.2.2.3">1</cn></apply><apply id="S4.15.p4.5.m5.1.1.2.3.cmml" xref="S4.15.p4.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S4.15.p4.5.m5.1.1.2.3.1.cmml" xref="S4.15.p4.5.m5.1.1.2.3">subscript</csymbol><ci id="S4.15.p4.5.m5.1.1.2.3.2.cmml" xref="S4.15.p4.5.m5.1.1.2.3.2">𝛼</ci><cn id="S4.15.p4.5.m5.1.1.2.3.3.cmml" type="integer" xref="S4.15.p4.5.m5.1.1.2.3.3">2</cn></apply></apply><ci id="S4.15.p4.5.m5.1.1.3.cmml" xref="S4.15.p4.5.m5.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.5.m5.1c">\alpha_{1}+\alpha_{2}<\pi</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.5.m5.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_π</annotation></semantics></math>, all of <math alttext="P_{1}\cup P_{2}" class="ltx_Math" display="inline" id="S4.15.p4.6.m6.1"><semantics id="S4.15.p4.6.m6.1a"><mrow id="S4.15.p4.6.m6.1.1" xref="S4.15.p4.6.m6.1.1.cmml"><msub id="S4.15.p4.6.m6.1.1.2" xref="S4.15.p4.6.m6.1.1.2.cmml"><mi id="S4.15.p4.6.m6.1.1.2.2" xref="S4.15.p4.6.m6.1.1.2.2.cmml">P</mi><mn id="S4.15.p4.6.m6.1.1.2.3" xref="S4.15.p4.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S4.15.p4.6.m6.1.1.1" xref="S4.15.p4.6.m6.1.1.1.cmml">∪</mo><msub id="S4.15.p4.6.m6.1.1.3" xref="S4.15.p4.6.m6.1.1.3.cmml"><mi id="S4.15.p4.6.m6.1.1.3.2" xref="S4.15.p4.6.m6.1.1.3.2.cmml">P</mi><mn id="S4.15.p4.6.m6.1.1.3.3" xref="S4.15.p4.6.m6.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.6.m6.1b"><apply id="S4.15.p4.6.m6.1.1.cmml" xref="S4.15.p4.6.m6.1.1"><union id="S4.15.p4.6.m6.1.1.1.cmml" xref="S4.15.p4.6.m6.1.1.1"></union><apply id="S4.15.p4.6.m6.1.1.2.cmml" xref="S4.15.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.15.p4.6.m6.1.1.2.1.cmml" xref="S4.15.p4.6.m6.1.1.2">subscript</csymbol><ci id="S4.15.p4.6.m6.1.1.2.2.cmml" xref="S4.15.p4.6.m6.1.1.2.2">𝑃</ci><cn id="S4.15.p4.6.m6.1.1.2.3.cmml" type="integer" xref="S4.15.p4.6.m6.1.1.2.3">1</cn></apply><apply id="S4.15.p4.6.m6.1.1.3.cmml" xref="S4.15.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.15.p4.6.m6.1.1.3.1.cmml" xref="S4.15.p4.6.m6.1.1.3">subscript</csymbol><ci id="S4.15.p4.6.m6.1.1.3.2.cmml" xref="S4.15.p4.6.m6.1.1.3.2">𝑃</ci><cn id="S4.15.p4.6.m6.1.1.3.3.cmml" type="integer" xref="S4.15.p4.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.6.m6.1c">P_{1}\cup P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.6.m6.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> lies on the same side as <math alttext="p_{0}" class="ltx_Math" display="inline" id="S4.15.p4.7.m7.1"><semantics id="S4.15.p4.7.m7.1a"><msub id="S4.15.p4.7.m7.1.1" xref="S4.15.p4.7.m7.1.1.cmml"><mi id="S4.15.p4.7.m7.1.1.2" xref="S4.15.p4.7.m7.1.1.2.cmml">p</mi><mn id="S4.15.p4.7.m7.1.1.3" xref="S4.15.p4.7.m7.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.7.m7.1b"><apply id="S4.15.p4.7.m7.1.1.cmml" xref="S4.15.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.15.p4.7.m7.1.1.1.cmml" xref="S4.15.p4.7.m7.1.1">subscript</csymbol><ci id="S4.15.p4.7.m7.1.1.2.cmml" xref="S4.15.p4.7.m7.1.1.2">𝑝</ci><cn id="S4.15.p4.7.m7.1.1.3.cmml" type="integer" xref="S4.15.p4.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.7.m7.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.7.m7.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of both <math alttext="l_{1}" class="ltx_Math" display="inline" id="S4.15.p4.8.m8.1"><semantics id="S4.15.p4.8.m8.1a"><msub id="S4.15.p4.8.m8.1.1" xref="S4.15.p4.8.m8.1.1.cmml"><mi id="S4.15.p4.8.m8.1.1.2" xref="S4.15.p4.8.m8.1.1.2.cmml">l</mi><mn id="S4.15.p4.8.m8.1.1.3" xref="S4.15.p4.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.8.m8.1b"><apply id="S4.15.p4.8.m8.1.1.cmml" xref="S4.15.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S4.15.p4.8.m8.1.1.1.cmml" xref="S4.15.p4.8.m8.1.1">subscript</csymbol><ci id="S4.15.p4.8.m8.1.1.2.cmml" xref="S4.15.p4.8.m8.1.1.2">𝑙</ci><cn id="S4.15.p4.8.m8.1.1.3.cmml" type="integer" xref="S4.15.p4.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.8.m8.1c">l_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.8.m8.1d">italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="l_{2}" class="ltx_Math" display="inline" id="S4.15.p4.9.m9.1"><semantics id="S4.15.p4.9.m9.1a"><msub id="S4.15.p4.9.m9.1.1" xref="S4.15.p4.9.m9.1.1.cmml"><mi id="S4.15.p4.9.m9.1.1.2" xref="S4.15.p4.9.m9.1.1.2.cmml">l</mi><mn id="S4.15.p4.9.m9.1.1.3" xref="S4.15.p4.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.9.m9.1b"><apply id="S4.15.p4.9.m9.1.1.cmml" xref="S4.15.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.15.p4.9.m9.1.1.1.cmml" xref="S4.15.p4.9.m9.1.1">subscript</csymbol><ci id="S4.15.p4.9.m9.1.1.2.cmml" xref="S4.15.p4.9.m9.1.1.2">𝑙</ci><cn id="S4.15.p4.9.m9.1.1.3.cmml" type="integer" xref="S4.15.p4.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.9.m9.1c">l_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.9.m9.1d">italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, <math alttext="\min(f_{1},f_{2})\geq f_{0}" class="ltx_Math" display="inline" id="S4.15.p4.10.m10.3"><semantics id="S4.15.p4.10.m10.3a"><mrow id="S4.15.p4.10.m10.3.3" xref="S4.15.p4.10.m10.3.3.cmml"><mrow id="S4.15.p4.10.m10.3.3.2.2" xref="S4.15.p4.10.m10.3.3.2.3.cmml"><mi id="S4.15.p4.10.m10.1.1" xref="S4.15.p4.10.m10.1.1.cmml">min</mi><mo id="S4.15.p4.10.m10.3.3.2.2a" xref="S4.15.p4.10.m10.3.3.2.3.cmml">⁡</mo><mrow id="S4.15.p4.10.m10.3.3.2.2.2" xref="S4.15.p4.10.m10.3.3.2.3.cmml"><mo id="S4.15.p4.10.m10.3.3.2.2.2.3" stretchy="false" xref="S4.15.p4.10.m10.3.3.2.3.cmml">(</mo><msub id="S4.15.p4.10.m10.2.2.1.1.1.1" xref="S4.15.p4.10.m10.2.2.1.1.1.1.cmml"><mi id="S4.15.p4.10.m10.2.2.1.1.1.1.2" xref="S4.15.p4.10.m10.2.2.1.1.1.1.2.cmml">f</mi><mn id="S4.15.p4.10.m10.2.2.1.1.1.1.3" xref="S4.15.p4.10.m10.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.15.p4.10.m10.3.3.2.2.2.4" xref="S4.15.p4.10.m10.3.3.2.3.cmml">,</mo><msub id="S4.15.p4.10.m10.3.3.2.2.2.2" xref="S4.15.p4.10.m10.3.3.2.2.2.2.cmml"><mi id="S4.15.p4.10.m10.3.3.2.2.2.2.2" xref="S4.15.p4.10.m10.3.3.2.2.2.2.2.cmml">f</mi><mn id="S4.15.p4.10.m10.3.3.2.2.2.2.3" xref="S4.15.p4.10.m10.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.15.p4.10.m10.3.3.2.2.2.5" stretchy="false" xref="S4.15.p4.10.m10.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.10.m10.3.3.3" xref="S4.15.p4.10.m10.3.3.3.cmml">≥</mo><msub id="S4.15.p4.10.m10.3.3.4" xref="S4.15.p4.10.m10.3.3.4.cmml"><mi id="S4.15.p4.10.m10.3.3.4.2" xref="S4.15.p4.10.m10.3.3.4.2.cmml">f</mi><mn id="S4.15.p4.10.m10.3.3.4.3" xref="S4.15.p4.10.m10.3.3.4.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.10.m10.3b"><apply id="S4.15.p4.10.m10.3.3.cmml" xref="S4.15.p4.10.m10.3.3"><geq id="S4.15.p4.10.m10.3.3.3.cmml" xref="S4.15.p4.10.m10.3.3.3"></geq><apply id="S4.15.p4.10.m10.3.3.2.3.cmml" xref="S4.15.p4.10.m10.3.3.2.2"><min id="S4.15.p4.10.m10.1.1.cmml" xref="S4.15.p4.10.m10.1.1"></min><apply id="S4.15.p4.10.m10.2.2.1.1.1.1.cmml" xref="S4.15.p4.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.10.m10.2.2.1.1.1.1.1.cmml" xref="S4.15.p4.10.m10.2.2.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.10.m10.2.2.1.1.1.1.2.cmml" xref="S4.15.p4.10.m10.2.2.1.1.1.1.2">𝑓</ci><cn id="S4.15.p4.10.m10.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.10.m10.2.2.1.1.1.1.3">1</cn></apply><apply id="S4.15.p4.10.m10.3.3.2.2.2.2.cmml" xref="S4.15.p4.10.m10.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.15.p4.10.m10.3.3.2.2.2.2.1.cmml" xref="S4.15.p4.10.m10.3.3.2.2.2.2">subscript</csymbol><ci id="S4.15.p4.10.m10.3.3.2.2.2.2.2.cmml" xref="S4.15.p4.10.m10.3.3.2.2.2.2.2">𝑓</ci><cn id="S4.15.p4.10.m10.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.15.p4.10.m10.3.3.2.2.2.2.3">2</cn></apply></apply><apply id="S4.15.p4.10.m10.3.3.4.cmml" xref="S4.15.p4.10.m10.3.3.4"><csymbol cd="ambiguous" id="S4.15.p4.10.m10.3.3.4.1.cmml" xref="S4.15.p4.10.m10.3.3.4">subscript</csymbol><ci id="S4.15.p4.10.m10.3.3.4.2.cmml" xref="S4.15.p4.10.m10.3.3.4.2">𝑓</ci><cn id="S4.15.p4.10.m10.3.3.4.3.cmml" type="integer" xref="S4.15.p4.10.m10.3.3.4.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.10.m10.3c">\min(f_{1},f_{2})\geq f_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.10.m10.3d">roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="P_{1}\cup P_{2}" class="ltx_Math" display="inline" id="S4.15.p4.11.m11.1"><semantics id="S4.15.p4.11.m11.1a"><mrow id="S4.15.p4.11.m11.1.1" xref="S4.15.p4.11.m11.1.1.cmml"><msub id="S4.15.p4.11.m11.1.1.2" xref="S4.15.p4.11.m11.1.1.2.cmml"><mi id="S4.15.p4.11.m11.1.1.2.2" xref="S4.15.p4.11.m11.1.1.2.2.cmml">P</mi><mn id="S4.15.p4.11.m11.1.1.2.3" xref="S4.15.p4.11.m11.1.1.2.3.cmml">1</mn></msub><mo id="S4.15.p4.11.m11.1.1.1" xref="S4.15.p4.11.m11.1.1.1.cmml">∪</mo><msub id="S4.15.p4.11.m11.1.1.3" xref="S4.15.p4.11.m11.1.1.3.cmml"><mi id="S4.15.p4.11.m11.1.1.3.2" xref="S4.15.p4.11.m11.1.1.3.2.cmml">P</mi><mn id="S4.15.p4.11.m11.1.1.3.3" xref="S4.15.p4.11.m11.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.11.m11.1b"><apply id="S4.15.p4.11.m11.1.1.cmml" xref="S4.15.p4.11.m11.1.1"><union id="S4.15.p4.11.m11.1.1.1.cmml" xref="S4.15.p4.11.m11.1.1.1"></union><apply id="S4.15.p4.11.m11.1.1.2.cmml" xref="S4.15.p4.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.15.p4.11.m11.1.1.2.1.cmml" xref="S4.15.p4.11.m11.1.1.2">subscript</csymbol><ci id="S4.15.p4.11.m11.1.1.2.2.cmml" xref="S4.15.p4.11.m11.1.1.2.2">𝑃</ci><cn id="S4.15.p4.11.m11.1.1.2.3.cmml" type="integer" xref="S4.15.p4.11.m11.1.1.2.3">1</cn></apply><apply id="S4.15.p4.11.m11.1.1.3.cmml" xref="S4.15.p4.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.15.p4.11.m11.1.1.3.1.cmml" xref="S4.15.p4.11.m11.1.1.3">subscript</csymbol><ci id="S4.15.p4.11.m11.1.1.3.2.cmml" xref="S4.15.p4.11.m11.1.1.3.2">𝑃</ci><cn id="S4.15.p4.11.m11.1.1.3.3.cmml" type="integer" xref="S4.15.p4.11.m11.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.11.m11.1c">P_{1}\cup P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.11.m11.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, <math alttext="f_{2}(p_{2})\geq f_{0}(p_{2})=f_{1}(p_{2})" class="ltx_Math" display="inline" id="S4.15.p4.12.m12.3"><semantics id="S4.15.p4.12.m12.3a"><mrow id="S4.15.p4.12.m12.3.3" xref="S4.15.p4.12.m12.3.3.cmml"><mrow id="S4.15.p4.12.m12.1.1.1" xref="S4.15.p4.12.m12.1.1.1.cmml"><msub id="S4.15.p4.12.m12.1.1.1.3" xref="S4.15.p4.12.m12.1.1.1.3.cmml"><mi id="S4.15.p4.12.m12.1.1.1.3.2" xref="S4.15.p4.12.m12.1.1.1.3.2.cmml">f</mi><mn id="S4.15.p4.12.m12.1.1.1.3.3" xref="S4.15.p4.12.m12.1.1.1.3.3.cmml">2</mn></msub><mo id="S4.15.p4.12.m12.1.1.1.2" xref="S4.15.p4.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S4.15.p4.12.m12.1.1.1.1.1" xref="S4.15.p4.12.m12.1.1.1.1.1.1.cmml"><mo id="S4.15.p4.12.m12.1.1.1.1.1.2" stretchy="false" xref="S4.15.p4.12.m12.1.1.1.1.1.1.cmml">(</mo><msub id="S4.15.p4.12.m12.1.1.1.1.1.1" xref="S4.15.p4.12.m12.1.1.1.1.1.1.cmml"><mi id="S4.15.p4.12.m12.1.1.1.1.1.1.2" xref="S4.15.p4.12.m12.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.12.m12.1.1.1.1.1.1.3" xref="S4.15.p4.12.m12.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.15.p4.12.m12.1.1.1.1.1.3" stretchy="false" xref="S4.15.p4.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.12.m12.3.3.5" xref="S4.15.p4.12.m12.3.3.5.cmml">≥</mo><mrow id="S4.15.p4.12.m12.2.2.2" xref="S4.15.p4.12.m12.2.2.2.cmml"><msub id="S4.15.p4.12.m12.2.2.2.3" xref="S4.15.p4.12.m12.2.2.2.3.cmml"><mi id="S4.15.p4.12.m12.2.2.2.3.2" xref="S4.15.p4.12.m12.2.2.2.3.2.cmml">f</mi><mn id="S4.15.p4.12.m12.2.2.2.3.3" xref="S4.15.p4.12.m12.2.2.2.3.3.cmml">0</mn></msub><mo id="S4.15.p4.12.m12.2.2.2.2" xref="S4.15.p4.12.m12.2.2.2.2.cmml">⁢</mo><mrow id="S4.15.p4.12.m12.2.2.2.1.1" xref="S4.15.p4.12.m12.2.2.2.1.1.1.cmml"><mo id="S4.15.p4.12.m12.2.2.2.1.1.2" stretchy="false" xref="S4.15.p4.12.m12.2.2.2.1.1.1.cmml">(</mo><msub id="S4.15.p4.12.m12.2.2.2.1.1.1" xref="S4.15.p4.12.m12.2.2.2.1.1.1.cmml"><mi id="S4.15.p4.12.m12.2.2.2.1.1.1.2" xref="S4.15.p4.12.m12.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.12.m12.2.2.2.1.1.1.3" xref="S4.15.p4.12.m12.2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.15.p4.12.m12.2.2.2.1.1.3" stretchy="false" xref="S4.15.p4.12.m12.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.12.m12.3.3.6" xref="S4.15.p4.12.m12.3.3.6.cmml">=</mo><mrow id="S4.15.p4.12.m12.3.3.3" xref="S4.15.p4.12.m12.3.3.3.cmml"><msub id="S4.15.p4.12.m12.3.3.3.3" xref="S4.15.p4.12.m12.3.3.3.3.cmml"><mi id="S4.15.p4.12.m12.3.3.3.3.2" xref="S4.15.p4.12.m12.3.3.3.3.2.cmml">f</mi><mn id="S4.15.p4.12.m12.3.3.3.3.3" xref="S4.15.p4.12.m12.3.3.3.3.3.cmml">1</mn></msub><mo id="S4.15.p4.12.m12.3.3.3.2" xref="S4.15.p4.12.m12.3.3.3.2.cmml">⁢</mo><mrow id="S4.15.p4.12.m12.3.3.3.1.1" xref="S4.15.p4.12.m12.3.3.3.1.1.1.cmml"><mo id="S4.15.p4.12.m12.3.3.3.1.1.2" stretchy="false" xref="S4.15.p4.12.m12.3.3.3.1.1.1.cmml">(</mo><msub id="S4.15.p4.12.m12.3.3.3.1.1.1" xref="S4.15.p4.12.m12.3.3.3.1.1.1.cmml"><mi id="S4.15.p4.12.m12.3.3.3.1.1.1.2" xref="S4.15.p4.12.m12.3.3.3.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.12.m12.3.3.3.1.1.1.3" xref="S4.15.p4.12.m12.3.3.3.1.1.1.3.cmml">2</mn></msub><mo id="S4.15.p4.12.m12.3.3.3.1.1.3" stretchy="false" xref="S4.15.p4.12.m12.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.12.m12.3b"><apply id="S4.15.p4.12.m12.3.3.cmml" xref="S4.15.p4.12.m12.3.3"><and id="S4.15.p4.12.m12.3.3a.cmml" xref="S4.15.p4.12.m12.3.3"></and><apply id="S4.15.p4.12.m12.3.3b.cmml" xref="S4.15.p4.12.m12.3.3"><geq id="S4.15.p4.12.m12.3.3.5.cmml" xref="S4.15.p4.12.m12.3.3.5"></geq><apply id="S4.15.p4.12.m12.1.1.1.cmml" xref="S4.15.p4.12.m12.1.1.1"><times id="S4.15.p4.12.m12.1.1.1.2.cmml" xref="S4.15.p4.12.m12.1.1.1.2"></times><apply id="S4.15.p4.12.m12.1.1.1.3.cmml" xref="S4.15.p4.12.m12.1.1.1.3"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.1.1.1.3.1.cmml" xref="S4.15.p4.12.m12.1.1.1.3">subscript</csymbol><ci id="S4.15.p4.12.m12.1.1.1.3.2.cmml" xref="S4.15.p4.12.m12.1.1.1.3.2">𝑓</ci><cn id="S4.15.p4.12.m12.1.1.1.3.3.cmml" type="integer" xref="S4.15.p4.12.m12.1.1.1.3.3">2</cn></apply><apply id="S4.15.p4.12.m12.1.1.1.1.1.1.cmml" xref="S4.15.p4.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.1.1.1.1.1.1.1.cmml" xref="S4.15.p4.12.m12.1.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.12.m12.1.1.1.1.1.1.2.cmml" xref="S4.15.p4.12.m12.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.15.p4.12.m12.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.12.m12.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S4.15.p4.12.m12.2.2.2.cmml" xref="S4.15.p4.12.m12.2.2.2"><times id="S4.15.p4.12.m12.2.2.2.2.cmml" xref="S4.15.p4.12.m12.2.2.2.2"></times><apply id="S4.15.p4.12.m12.2.2.2.3.cmml" xref="S4.15.p4.12.m12.2.2.2.3"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.2.2.2.3.1.cmml" xref="S4.15.p4.12.m12.2.2.2.3">subscript</csymbol><ci id="S4.15.p4.12.m12.2.2.2.3.2.cmml" xref="S4.15.p4.12.m12.2.2.2.3.2">𝑓</ci><cn id="S4.15.p4.12.m12.2.2.2.3.3.cmml" type="integer" xref="S4.15.p4.12.m12.2.2.2.3.3">0</cn></apply><apply id="S4.15.p4.12.m12.2.2.2.1.1.1.cmml" xref="S4.15.p4.12.m12.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.2.2.2.1.1.1.1.cmml" xref="S4.15.p4.12.m12.2.2.2.1.1">subscript</csymbol><ci id="S4.15.p4.12.m12.2.2.2.1.1.1.2.cmml" xref="S4.15.p4.12.m12.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.15.p4.12.m12.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.15.p4.12.m12.2.2.2.1.1.1.3">2</cn></apply></apply></apply><apply id="S4.15.p4.12.m12.3.3c.cmml" xref="S4.15.p4.12.m12.3.3"><eq id="S4.15.p4.12.m12.3.3.6.cmml" xref="S4.15.p4.12.m12.3.3.6"></eq><share href="https://arxiv.org/html/2503.13001v1#S4.15.p4.12.m12.2.2.2.cmml" id="S4.15.p4.12.m12.3.3d.cmml" xref="S4.15.p4.12.m12.3.3"></share><apply id="S4.15.p4.12.m12.3.3.3.cmml" xref="S4.15.p4.12.m12.3.3.3"><times id="S4.15.p4.12.m12.3.3.3.2.cmml" xref="S4.15.p4.12.m12.3.3.3.2"></times><apply id="S4.15.p4.12.m12.3.3.3.3.cmml" xref="S4.15.p4.12.m12.3.3.3.3"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.3.3.3.3.1.cmml" xref="S4.15.p4.12.m12.3.3.3.3">subscript</csymbol><ci id="S4.15.p4.12.m12.3.3.3.3.2.cmml" xref="S4.15.p4.12.m12.3.3.3.3.2">𝑓</ci><cn id="S4.15.p4.12.m12.3.3.3.3.3.cmml" type="integer" xref="S4.15.p4.12.m12.3.3.3.3.3">1</cn></apply><apply id="S4.15.p4.12.m12.3.3.3.1.1.1.cmml" xref="S4.15.p4.12.m12.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.15.p4.12.m12.3.3.3.1.1.1.1.cmml" xref="S4.15.p4.12.m12.3.3.3.1.1">subscript</csymbol><ci id="S4.15.p4.12.m12.3.3.3.1.1.1.2.cmml" xref="S4.15.p4.12.m12.3.3.3.1.1.1.2">𝑝</ci><cn id="S4.15.p4.12.m12.3.3.3.1.1.1.3.cmml" type="integer" xref="S4.15.p4.12.m12.3.3.3.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.12.m12.3c">f_{2}(p_{2})\geq f_{0}(p_{2})=f_{1}(p_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.12.m12.3d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, and since <math alttext="l_{0}" class="ltx_Math" display="inline" id="S4.15.p4.13.m13.1"><semantics id="S4.15.p4.13.m13.1a"><msub id="S4.15.p4.13.m13.1.1" xref="S4.15.p4.13.m13.1.1.cmml"><mi id="S4.15.p4.13.m13.1.1.2" xref="S4.15.p4.13.m13.1.1.2.cmml">l</mi><mn id="S4.15.p4.13.m13.1.1.3" xref="S4.15.p4.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.13.m13.1b"><apply id="S4.15.p4.13.m13.1.1.cmml" xref="S4.15.p4.13.m13.1.1"><csymbol cd="ambiguous" id="S4.15.p4.13.m13.1.1.1.cmml" xref="S4.15.p4.13.m13.1.1">subscript</csymbol><ci id="S4.15.p4.13.m13.1.1.2.cmml" xref="S4.15.p4.13.m13.1.1.2">𝑙</ci><cn id="S4.15.p4.13.m13.1.1.3.cmml" type="integer" xref="S4.15.p4.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.13.m13.1c">l_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.13.m13.1d">italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> separates <math alttext="P_{1}" class="ltx_Math" display="inline" id="S4.15.p4.14.m14.1"><semantics id="S4.15.p4.14.m14.1a"><msub id="S4.15.p4.14.m14.1.1" xref="S4.15.p4.14.m14.1.1.cmml"><mi id="S4.15.p4.14.m14.1.1.2" xref="S4.15.p4.14.m14.1.1.2.cmml">P</mi><mn id="S4.15.p4.14.m14.1.1.3" xref="S4.15.p4.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.14.m14.1b"><apply id="S4.15.p4.14.m14.1.1.cmml" xref="S4.15.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S4.15.p4.14.m14.1.1.1.cmml" xref="S4.15.p4.14.m14.1.1">subscript</csymbol><ci id="S4.15.p4.14.m14.1.1.2.cmml" xref="S4.15.p4.14.m14.1.1.2">𝑃</ci><cn id="S4.15.p4.14.m14.1.1.3.cmml" type="integer" xref="S4.15.p4.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.14.m14.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.14.m14.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="P_{2}" class="ltx_Math" display="inline" id="S4.15.p4.15.m15.1"><semantics id="S4.15.p4.15.m15.1a"><msub id="S4.15.p4.15.m15.1.1" xref="S4.15.p4.15.m15.1.1.cmml"><mi id="S4.15.p4.15.m15.1.1.2" xref="S4.15.p4.15.m15.1.1.2.cmml">P</mi><mn id="S4.15.p4.15.m15.1.1.3" xref="S4.15.p4.15.m15.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.15.m15.1b"><apply id="S4.15.p4.15.m15.1.1.cmml" xref="S4.15.p4.15.m15.1.1"><csymbol cd="ambiguous" id="S4.15.p4.15.m15.1.1.1.cmml" xref="S4.15.p4.15.m15.1.1">subscript</csymbol><ci id="S4.15.p4.15.m15.1.1.2.cmml" xref="S4.15.p4.15.m15.1.1.2">𝑃</ci><cn id="S4.15.p4.15.m15.1.1.3.cmml" type="integer" xref="S4.15.p4.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.15.m15.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.15.m15.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, we have <math alttext="f|_{P_{1}\cup P_{2}}=\min(f_{1},f_{2})|_{P_{1}\cup P_{2}}" class="ltx_Math" display="inline" id="S4.15.p4.16.m16.5"><semantics id="S4.15.p4.16.m16.5a"><mrow id="S4.15.p4.16.m16.5.5" xref="S4.15.p4.16.m16.5.5.cmml"><msub id="S4.15.p4.16.m16.5.5.3.2" xref="S4.15.p4.16.m16.5.5.3.1.cmml"><mrow id="S4.15.p4.16.m16.5.5.3.2.2" xref="S4.15.p4.16.m16.5.5.3.1.cmml"><mi id="S4.15.p4.16.m16.1.1" xref="S4.15.p4.16.m16.1.1.cmml">f</mi><mo id="S4.15.p4.16.m16.5.5.3.2.2.1" stretchy="false" xref="S4.15.p4.16.m16.5.5.3.1.1.cmml">|</mo></mrow><mrow id="S4.15.p4.16.m16.2.2.1" xref="S4.15.p4.16.m16.2.2.1.cmml"><msub id="S4.15.p4.16.m16.2.2.1.2" xref="S4.15.p4.16.m16.2.2.1.2.cmml"><mi id="S4.15.p4.16.m16.2.2.1.2.2" xref="S4.15.p4.16.m16.2.2.1.2.2.cmml">P</mi><mn id="S4.15.p4.16.m16.2.2.1.2.3" xref="S4.15.p4.16.m16.2.2.1.2.3.cmml">1</mn></msub><mo id="S4.15.p4.16.m16.2.2.1.1" xref="S4.15.p4.16.m16.2.2.1.1.cmml">∪</mo><msub id="S4.15.p4.16.m16.2.2.1.3" xref="S4.15.p4.16.m16.2.2.1.3.cmml"><mi id="S4.15.p4.16.m16.2.2.1.3.2" xref="S4.15.p4.16.m16.2.2.1.3.2.cmml">P</mi><mn id="S4.15.p4.16.m16.2.2.1.3.3" xref="S4.15.p4.16.m16.2.2.1.3.3.cmml">2</mn></msub></mrow></msub><mo id="S4.15.p4.16.m16.5.5.2" xref="S4.15.p4.16.m16.5.5.2.cmml">=</mo><msub id="S4.15.p4.16.m16.5.5.1.1" xref="S4.15.p4.16.m16.5.5.1.2.cmml"><mrow id="S4.15.p4.16.m16.5.5.1.1.1" xref="S4.15.p4.16.m16.5.5.1.2.cmml"><mrow id="S4.15.p4.16.m16.5.5.1.1.1.1.2" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml"><mi id="S4.15.p4.16.m16.3.3" xref="S4.15.p4.16.m16.3.3.cmml">min</mi><mo id="S4.15.p4.16.m16.5.5.1.1.1.1.2a" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml">⁡</mo><mrow id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml"><mo id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.3" stretchy="false" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml">(</mo><msub id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.cmml"><mi id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.2" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.2.cmml">f</mi><mn id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.3" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.4" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml">,</mo><msub id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.cmml"><mi id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.2" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.2.cmml">f</mi><mn id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.3" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.3.cmml">2</mn></msub><mo id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.5" stretchy="false" xref="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.16.m16.5.5.1.1.1.2" stretchy="false" xref="S4.15.p4.16.m16.5.5.1.2.1.cmml">|</mo></mrow><mrow id="S4.15.p4.16.m16.4.4.1" xref="S4.15.p4.16.m16.4.4.1.cmml"><msub id="S4.15.p4.16.m16.4.4.1.2" xref="S4.15.p4.16.m16.4.4.1.2.cmml"><mi id="S4.15.p4.16.m16.4.4.1.2.2" xref="S4.15.p4.16.m16.4.4.1.2.2.cmml">P</mi><mn id="S4.15.p4.16.m16.4.4.1.2.3" xref="S4.15.p4.16.m16.4.4.1.2.3.cmml">1</mn></msub><mo id="S4.15.p4.16.m16.4.4.1.1" xref="S4.15.p4.16.m16.4.4.1.1.cmml">∪</mo><msub id="S4.15.p4.16.m16.4.4.1.3" xref="S4.15.p4.16.m16.4.4.1.3.cmml"><mi id="S4.15.p4.16.m16.4.4.1.3.2" xref="S4.15.p4.16.m16.4.4.1.3.2.cmml">P</mi><mn id="S4.15.p4.16.m16.4.4.1.3.3" xref="S4.15.p4.16.m16.4.4.1.3.3.cmml">2</mn></msub></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.16.m16.5b"><apply id="S4.15.p4.16.m16.5.5.cmml" xref="S4.15.p4.16.m16.5.5"><eq id="S4.15.p4.16.m16.5.5.2.cmml" xref="S4.15.p4.16.m16.5.5.2"></eq><apply id="S4.15.p4.16.m16.5.5.3.1.cmml" xref="S4.15.p4.16.m16.5.5.3.2"><csymbol cd="latexml" id="S4.15.p4.16.m16.5.5.3.1.1.cmml" xref="S4.15.p4.16.m16.5.5.3.2.2.1">evaluated-at</csymbol><ci id="S4.15.p4.16.m16.1.1.cmml" xref="S4.15.p4.16.m16.1.1">𝑓</ci><apply id="S4.15.p4.16.m16.2.2.1.cmml" xref="S4.15.p4.16.m16.2.2.1"><union id="S4.15.p4.16.m16.2.2.1.1.cmml" xref="S4.15.p4.16.m16.2.2.1.1"></union><apply id="S4.15.p4.16.m16.2.2.1.2.cmml" xref="S4.15.p4.16.m16.2.2.1.2"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.2.2.1.2.1.cmml" xref="S4.15.p4.16.m16.2.2.1.2">subscript</csymbol><ci id="S4.15.p4.16.m16.2.2.1.2.2.cmml" xref="S4.15.p4.16.m16.2.2.1.2.2">𝑃</ci><cn id="S4.15.p4.16.m16.2.2.1.2.3.cmml" type="integer" xref="S4.15.p4.16.m16.2.2.1.2.3">1</cn></apply><apply id="S4.15.p4.16.m16.2.2.1.3.cmml" xref="S4.15.p4.16.m16.2.2.1.3"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.2.2.1.3.1.cmml" xref="S4.15.p4.16.m16.2.2.1.3">subscript</csymbol><ci id="S4.15.p4.16.m16.2.2.1.3.2.cmml" xref="S4.15.p4.16.m16.2.2.1.3.2">𝑃</ci><cn id="S4.15.p4.16.m16.2.2.1.3.3.cmml" type="integer" xref="S4.15.p4.16.m16.2.2.1.3.3">2</cn></apply></apply></apply><apply id="S4.15.p4.16.m16.5.5.1.2.cmml" xref="S4.15.p4.16.m16.5.5.1.1"><csymbol cd="latexml" id="S4.15.p4.16.m16.5.5.1.2.1.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.2">evaluated-at</csymbol><apply id="S4.15.p4.16.m16.5.5.1.1.1.1.3.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2"><min id="S4.15.p4.16.m16.3.3.cmml" xref="S4.15.p4.16.m16.3.3"></min><apply id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.1.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.2.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.2">𝑓</ci><cn id="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.16.m16.5.5.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.1.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.2.cmml" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.2">𝑓</ci><cn id="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.3.cmml" type="integer" xref="S4.15.p4.16.m16.5.5.1.1.1.1.2.2.2.3">2</cn></apply></apply><apply id="S4.15.p4.16.m16.4.4.1.cmml" xref="S4.15.p4.16.m16.4.4.1"><union id="S4.15.p4.16.m16.4.4.1.1.cmml" xref="S4.15.p4.16.m16.4.4.1.1"></union><apply id="S4.15.p4.16.m16.4.4.1.2.cmml" xref="S4.15.p4.16.m16.4.4.1.2"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.4.4.1.2.1.cmml" xref="S4.15.p4.16.m16.4.4.1.2">subscript</csymbol><ci id="S4.15.p4.16.m16.4.4.1.2.2.cmml" xref="S4.15.p4.16.m16.4.4.1.2.2">𝑃</ci><cn id="S4.15.p4.16.m16.4.4.1.2.3.cmml" type="integer" xref="S4.15.p4.16.m16.4.4.1.2.3">1</cn></apply><apply id="S4.15.p4.16.m16.4.4.1.3.cmml" xref="S4.15.p4.16.m16.4.4.1.3"><csymbol cd="ambiguous" id="S4.15.p4.16.m16.4.4.1.3.1.cmml" xref="S4.15.p4.16.m16.4.4.1.3">subscript</csymbol><ci id="S4.15.p4.16.m16.4.4.1.3.2.cmml" xref="S4.15.p4.16.m16.4.4.1.3.2">𝑃</ci><cn id="S4.15.p4.16.m16.4.4.1.3.3.cmml" type="integer" xref="S4.15.p4.16.m16.4.4.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.16.m16.5c">f|_{P_{1}\cup P_{2}}=\min(f_{1},f_{2})|_{P_{1}\cup P_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.16.m16.5d">italic_f | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. In contrast, any <math alttext="x\in P_{0}" class="ltx_Math" display="inline" id="S4.15.p4.17.m17.1"><semantics id="S4.15.p4.17.m17.1a"><mrow id="S4.15.p4.17.m17.1.1" xref="S4.15.p4.17.m17.1.1.cmml"><mi id="S4.15.p4.17.m17.1.1.2" xref="S4.15.p4.17.m17.1.1.2.cmml">x</mi><mo id="S4.15.p4.17.m17.1.1.1" xref="S4.15.p4.17.m17.1.1.1.cmml">∈</mo><msub id="S4.15.p4.17.m17.1.1.3" xref="S4.15.p4.17.m17.1.1.3.cmml"><mi id="S4.15.p4.17.m17.1.1.3.2" xref="S4.15.p4.17.m17.1.1.3.2.cmml">P</mi><mn id="S4.15.p4.17.m17.1.1.3.3" xref="S4.15.p4.17.m17.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.17.m17.1b"><apply id="S4.15.p4.17.m17.1.1.cmml" xref="S4.15.p4.17.m17.1.1"><in id="S4.15.p4.17.m17.1.1.1.cmml" xref="S4.15.p4.17.m17.1.1.1"></in><ci id="S4.15.p4.17.m17.1.1.2.cmml" xref="S4.15.p4.17.m17.1.1.2">𝑥</ci><apply id="S4.15.p4.17.m17.1.1.3.cmml" xref="S4.15.p4.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.15.p4.17.m17.1.1.3.1.cmml" xref="S4.15.p4.17.m17.1.1.3">subscript</csymbol><ci id="S4.15.p4.17.m17.1.1.3.2.cmml" xref="S4.15.p4.17.m17.1.1.3.2">𝑃</ci><cn id="S4.15.p4.17.m17.1.1.3.3.cmml" type="integer" xref="S4.15.p4.17.m17.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.17.m17.1c">x\in P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.17.m17.1d">italic_x ∈ italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{0}" class="ltx_Math" display="inline" id="S4.15.p4.18.m18.1"><semantics id="S4.15.p4.18.m18.1a"><msub id="S4.15.p4.18.m18.1.1" xref="S4.15.p4.18.m18.1.1.cmml"><mi id="S4.15.p4.18.m18.1.1.2" xref="S4.15.p4.18.m18.1.1.2.cmml">p</mi><mn id="S4.15.p4.18.m18.1.1.3" xref="S4.15.p4.18.m18.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.18.m18.1b"><apply id="S4.15.p4.18.m18.1.1.cmml" xref="S4.15.p4.18.m18.1.1"><csymbol cd="ambiguous" id="S4.15.p4.18.m18.1.1.1.cmml" xref="S4.15.p4.18.m18.1.1">subscript</csymbol><ci id="S4.15.p4.18.m18.1.1.2.cmml" xref="S4.15.p4.18.m18.1.1.2">𝑝</ci><cn id="S4.15.p4.18.m18.1.1.3.cmml" type="integer" xref="S4.15.p4.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.18.m18.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.18.m18.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> lie on opposite sides of either <math alttext="l_{1}" class="ltx_Math" display="inline" id="S4.15.p4.19.m19.1"><semantics id="S4.15.p4.19.m19.1a"><msub id="S4.15.p4.19.m19.1.1" xref="S4.15.p4.19.m19.1.1.cmml"><mi id="S4.15.p4.19.m19.1.1.2" xref="S4.15.p4.19.m19.1.1.2.cmml">l</mi><mn id="S4.15.p4.19.m19.1.1.3" xref="S4.15.p4.19.m19.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.19.m19.1b"><apply id="S4.15.p4.19.m19.1.1.cmml" xref="S4.15.p4.19.m19.1.1"><csymbol cd="ambiguous" id="S4.15.p4.19.m19.1.1.1.cmml" xref="S4.15.p4.19.m19.1.1">subscript</csymbol><ci id="S4.15.p4.19.m19.1.1.2.cmml" xref="S4.15.p4.19.m19.1.1.2">𝑙</ci><cn id="S4.15.p4.19.m19.1.1.3.cmml" type="integer" xref="S4.15.p4.19.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.19.m19.1c">l_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.19.m19.1d">italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="l_{2}" class="ltx_Math" display="inline" id="S4.15.p4.20.m20.1"><semantics id="S4.15.p4.20.m20.1a"><msub id="S4.15.p4.20.m20.1.1" xref="S4.15.p4.20.m20.1.1.cmml"><mi id="S4.15.p4.20.m20.1.1.2" xref="S4.15.p4.20.m20.1.1.2.cmml">l</mi><mn id="S4.15.p4.20.m20.1.1.3" xref="S4.15.p4.20.m20.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.20.m20.1b"><apply id="S4.15.p4.20.m20.1.1.cmml" xref="S4.15.p4.20.m20.1.1"><csymbol cd="ambiguous" id="S4.15.p4.20.m20.1.1.1.cmml" xref="S4.15.p4.20.m20.1.1">subscript</csymbol><ci id="S4.15.p4.20.m20.1.1.2.cmml" xref="S4.15.p4.20.m20.1.1.2">𝑙</ci><cn id="S4.15.p4.20.m20.1.1.3.cmml" type="integer" xref="S4.15.p4.20.m20.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.20.m20.1c">l_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.20.m20.1d">italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and therefore <math alttext="f_{0}\geq\min(f_{1},f_{2})" class="ltx_Math" display="inline" id="S4.15.p4.21.m21.3"><semantics id="S4.15.p4.21.m21.3a"><mrow id="S4.15.p4.21.m21.3.3" xref="S4.15.p4.21.m21.3.3.cmml"><msub id="S4.15.p4.21.m21.3.3.4" xref="S4.15.p4.21.m21.3.3.4.cmml"><mi id="S4.15.p4.21.m21.3.3.4.2" xref="S4.15.p4.21.m21.3.3.4.2.cmml">f</mi><mn id="S4.15.p4.21.m21.3.3.4.3" xref="S4.15.p4.21.m21.3.3.4.3.cmml">0</mn></msub><mo id="S4.15.p4.21.m21.3.3.3" xref="S4.15.p4.21.m21.3.3.3.cmml">≥</mo><mrow id="S4.15.p4.21.m21.3.3.2.2" xref="S4.15.p4.21.m21.3.3.2.3.cmml"><mi id="S4.15.p4.21.m21.1.1" xref="S4.15.p4.21.m21.1.1.cmml">min</mi><mo id="S4.15.p4.21.m21.3.3.2.2a" xref="S4.15.p4.21.m21.3.3.2.3.cmml">⁡</mo><mrow id="S4.15.p4.21.m21.3.3.2.2.2" xref="S4.15.p4.21.m21.3.3.2.3.cmml"><mo id="S4.15.p4.21.m21.3.3.2.2.2.3" stretchy="false" xref="S4.15.p4.21.m21.3.3.2.3.cmml">(</mo><msub id="S4.15.p4.21.m21.2.2.1.1.1.1" xref="S4.15.p4.21.m21.2.2.1.1.1.1.cmml"><mi id="S4.15.p4.21.m21.2.2.1.1.1.1.2" xref="S4.15.p4.21.m21.2.2.1.1.1.1.2.cmml">f</mi><mn id="S4.15.p4.21.m21.2.2.1.1.1.1.3" xref="S4.15.p4.21.m21.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.15.p4.21.m21.3.3.2.2.2.4" xref="S4.15.p4.21.m21.3.3.2.3.cmml">,</mo><msub id="S4.15.p4.21.m21.3.3.2.2.2.2" xref="S4.15.p4.21.m21.3.3.2.2.2.2.cmml"><mi id="S4.15.p4.21.m21.3.3.2.2.2.2.2" xref="S4.15.p4.21.m21.3.3.2.2.2.2.2.cmml">f</mi><mn id="S4.15.p4.21.m21.3.3.2.2.2.2.3" xref="S4.15.p4.21.m21.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.15.p4.21.m21.3.3.2.2.2.5" stretchy="false" xref="S4.15.p4.21.m21.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.21.m21.3b"><apply id="S4.15.p4.21.m21.3.3.cmml" xref="S4.15.p4.21.m21.3.3"><geq id="S4.15.p4.21.m21.3.3.3.cmml" xref="S4.15.p4.21.m21.3.3.3"></geq><apply id="S4.15.p4.21.m21.3.3.4.cmml" xref="S4.15.p4.21.m21.3.3.4"><csymbol cd="ambiguous" id="S4.15.p4.21.m21.3.3.4.1.cmml" xref="S4.15.p4.21.m21.3.3.4">subscript</csymbol><ci id="S4.15.p4.21.m21.3.3.4.2.cmml" xref="S4.15.p4.21.m21.3.3.4.2">𝑓</ci><cn id="S4.15.p4.21.m21.3.3.4.3.cmml" type="integer" xref="S4.15.p4.21.m21.3.3.4.3">0</cn></apply><apply id="S4.15.p4.21.m21.3.3.2.3.cmml" xref="S4.15.p4.21.m21.3.3.2.2"><min id="S4.15.p4.21.m21.1.1.cmml" xref="S4.15.p4.21.m21.1.1"></min><apply id="S4.15.p4.21.m21.2.2.1.1.1.1.cmml" xref="S4.15.p4.21.m21.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.21.m21.2.2.1.1.1.1.1.cmml" xref="S4.15.p4.21.m21.2.2.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.21.m21.2.2.1.1.1.1.2.cmml" xref="S4.15.p4.21.m21.2.2.1.1.1.1.2">𝑓</ci><cn id="S4.15.p4.21.m21.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.21.m21.2.2.1.1.1.1.3">1</cn></apply><apply id="S4.15.p4.21.m21.3.3.2.2.2.2.cmml" xref="S4.15.p4.21.m21.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.15.p4.21.m21.3.3.2.2.2.2.1.cmml" xref="S4.15.p4.21.m21.3.3.2.2.2.2">subscript</csymbol><ci id="S4.15.p4.21.m21.3.3.2.2.2.2.2.cmml" xref="S4.15.p4.21.m21.3.3.2.2.2.2.2">𝑓</ci><cn id="S4.15.p4.21.m21.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.15.p4.21.m21.3.3.2.2.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.21.m21.3c">f_{0}\geq\min(f_{1},f_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.21.m21.3d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="P_{0}" class="ltx_Math" display="inline" id="S4.15.p4.22.m22.1"><semantics id="S4.15.p4.22.m22.1a"><msub id="S4.15.p4.22.m22.1.1" xref="S4.15.p4.22.m22.1.1.cmml"><mi id="S4.15.p4.22.m22.1.1.2" xref="S4.15.p4.22.m22.1.1.2.cmml">P</mi><mn id="S4.15.p4.22.m22.1.1.3" xref="S4.15.p4.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.15.p4.22.m22.1b"><apply id="S4.15.p4.22.m22.1.1.cmml" xref="S4.15.p4.22.m22.1.1"><csymbol cd="ambiguous" id="S4.15.p4.22.m22.1.1.1.cmml" xref="S4.15.p4.22.m22.1.1">subscript</csymbol><ci id="S4.15.p4.22.m22.1.1.2.cmml" xref="S4.15.p4.22.m22.1.1.2">𝑃</ci><cn id="S4.15.p4.22.m22.1.1.3.cmml" type="integer" xref="S4.15.p4.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.22.m22.1c">P_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.22.m22.1d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. In total, we get <math alttext="f=\max(f_{0},\min(f_{1},f_{2}))" class="ltx_Math" display="inline" id="S4.15.p4.23.m23.4"><semantics id="S4.15.p4.23.m23.4a"><mrow id="S4.15.p4.23.m23.4.4" xref="S4.15.p4.23.m23.4.4.cmml"><mi id="S4.15.p4.23.m23.4.4.4" xref="S4.15.p4.23.m23.4.4.4.cmml">f</mi><mo id="S4.15.p4.23.m23.4.4.3" xref="S4.15.p4.23.m23.4.4.3.cmml">=</mo><mrow id="S4.15.p4.23.m23.4.4.2.2" xref="S4.15.p4.23.m23.4.4.2.3.cmml"><mi id="S4.15.p4.23.m23.2.2" xref="S4.15.p4.23.m23.2.2.cmml">max</mi><mo id="S4.15.p4.23.m23.4.4.2.2a" xref="S4.15.p4.23.m23.4.4.2.3.cmml">⁡</mo><mrow id="S4.15.p4.23.m23.4.4.2.2.2" xref="S4.15.p4.23.m23.4.4.2.3.cmml"><mo id="S4.15.p4.23.m23.4.4.2.2.2.3" stretchy="false" xref="S4.15.p4.23.m23.4.4.2.3.cmml">(</mo><msub id="S4.15.p4.23.m23.3.3.1.1.1.1" xref="S4.15.p4.23.m23.3.3.1.1.1.1.cmml"><mi id="S4.15.p4.23.m23.3.3.1.1.1.1.2" xref="S4.15.p4.23.m23.3.3.1.1.1.1.2.cmml">f</mi><mn id="S4.15.p4.23.m23.3.3.1.1.1.1.3" xref="S4.15.p4.23.m23.3.3.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.23.m23.4.4.2.2.2.4" xref="S4.15.p4.23.m23.4.4.2.3.cmml">,</mo><mrow id="S4.15.p4.23.m23.4.4.2.2.2.2.2" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml"><mi id="S4.15.p4.23.m23.1.1" xref="S4.15.p4.23.m23.1.1.cmml">min</mi><mo id="S4.15.p4.23.m23.4.4.2.2.2.2.2a" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml">⁡</mo><mrow id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml"><mo id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.3" stretchy="false" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml">(</mo><msub id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.cmml"><mi id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.2" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.2.cmml">f</mi><mn id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.3" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.4" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml">,</mo><msub id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.cmml"><mi id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.2" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.2.cmml">f</mi><mn id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.3" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.5" stretchy="false" xref="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.23.m23.4.4.2.2.2.5" stretchy="false" xref="S4.15.p4.23.m23.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.23.m23.4b"><apply id="S4.15.p4.23.m23.4.4.cmml" xref="S4.15.p4.23.m23.4.4"><eq id="S4.15.p4.23.m23.4.4.3.cmml" xref="S4.15.p4.23.m23.4.4.3"></eq><ci id="S4.15.p4.23.m23.4.4.4.cmml" xref="S4.15.p4.23.m23.4.4.4">𝑓</ci><apply id="S4.15.p4.23.m23.4.4.2.3.cmml" xref="S4.15.p4.23.m23.4.4.2.2"><max id="S4.15.p4.23.m23.2.2.cmml" xref="S4.15.p4.23.m23.2.2"></max><apply id="S4.15.p4.23.m23.3.3.1.1.1.1.cmml" xref="S4.15.p4.23.m23.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.23.m23.3.3.1.1.1.1.1.cmml" xref="S4.15.p4.23.m23.3.3.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.23.m23.3.3.1.1.1.1.2.cmml" xref="S4.15.p4.23.m23.3.3.1.1.1.1.2">𝑓</ci><cn id="S4.15.p4.23.m23.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.23.m23.3.3.1.1.1.1.3">0</cn></apply><apply id="S4.15.p4.23.m23.4.4.2.2.2.2.3.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2"><min id="S4.15.p4.23.m23.1.1.cmml" xref="S4.15.p4.23.m23.1.1"></min><apply id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.1.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1">subscript</csymbol><ci id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.2.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.2">𝑓</ci><cn id="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.15.p4.23.m23.4.4.2.2.2.2.1.1.1.3">1</cn></apply><apply id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.1.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.2.cmml" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.2">𝑓</ci><cn id="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.15.p4.23.m23.4.4.2.2.2.2.2.2.2.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.23.m23.4c">f=\max(f_{0},\min(f_{1},f_{2}))</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.23.m23.4d">italic_f = roman_max ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math>, which is of the form (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E36" title="Equation 36 ‣ Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">36</span></a>). Again, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E36" title="Equation 36 ‣ Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">36</span></a>) can also be achieved for the case that <math alttext="f(p_{0})<f_{0}(p_{0})" class="ltx_Math" display="inline" id="S4.15.p4.24.m24.2"><semantics id="S4.15.p4.24.m24.2a"><mrow id="S4.15.p4.24.m24.2.2" xref="S4.15.p4.24.m24.2.2.cmml"><mrow id="S4.15.p4.24.m24.1.1.1" xref="S4.15.p4.24.m24.1.1.1.cmml"><mi id="S4.15.p4.24.m24.1.1.1.3" xref="S4.15.p4.24.m24.1.1.1.3.cmml">f</mi><mo id="S4.15.p4.24.m24.1.1.1.2" xref="S4.15.p4.24.m24.1.1.1.2.cmml">⁢</mo><mrow id="S4.15.p4.24.m24.1.1.1.1.1" xref="S4.15.p4.24.m24.1.1.1.1.1.1.cmml"><mo id="S4.15.p4.24.m24.1.1.1.1.1.2" stretchy="false" xref="S4.15.p4.24.m24.1.1.1.1.1.1.cmml">(</mo><msub id="S4.15.p4.24.m24.1.1.1.1.1.1" xref="S4.15.p4.24.m24.1.1.1.1.1.1.cmml"><mi id="S4.15.p4.24.m24.1.1.1.1.1.1.2" xref="S4.15.p4.24.m24.1.1.1.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.24.m24.1.1.1.1.1.1.3" xref="S4.15.p4.24.m24.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.24.m24.1.1.1.1.1.3" stretchy="false" xref="S4.15.p4.24.m24.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.15.p4.24.m24.2.2.3" xref="S4.15.p4.24.m24.2.2.3.cmml"><</mo><mrow id="S4.15.p4.24.m24.2.2.2" xref="S4.15.p4.24.m24.2.2.2.cmml"><msub id="S4.15.p4.24.m24.2.2.2.3" xref="S4.15.p4.24.m24.2.2.2.3.cmml"><mi id="S4.15.p4.24.m24.2.2.2.3.2" xref="S4.15.p4.24.m24.2.2.2.3.2.cmml">f</mi><mn id="S4.15.p4.24.m24.2.2.2.3.3" xref="S4.15.p4.24.m24.2.2.2.3.3.cmml">0</mn></msub><mo id="S4.15.p4.24.m24.2.2.2.2" xref="S4.15.p4.24.m24.2.2.2.2.cmml">⁢</mo><mrow id="S4.15.p4.24.m24.2.2.2.1.1" xref="S4.15.p4.24.m24.2.2.2.1.1.1.cmml"><mo id="S4.15.p4.24.m24.2.2.2.1.1.2" stretchy="false" xref="S4.15.p4.24.m24.2.2.2.1.1.1.cmml">(</mo><msub id="S4.15.p4.24.m24.2.2.2.1.1.1" xref="S4.15.p4.24.m24.2.2.2.1.1.1.cmml"><mi id="S4.15.p4.24.m24.2.2.2.1.1.1.2" xref="S4.15.p4.24.m24.2.2.2.1.1.1.2.cmml">p</mi><mn id="S4.15.p4.24.m24.2.2.2.1.1.1.3" xref="S4.15.p4.24.m24.2.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S4.15.p4.24.m24.2.2.2.1.1.3" stretchy="false" xref="S4.15.p4.24.m24.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.24.m24.2b"><apply id="S4.15.p4.24.m24.2.2.cmml" xref="S4.15.p4.24.m24.2.2"><lt id="S4.15.p4.24.m24.2.2.3.cmml" xref="S4.15.p4.24.m24.2.2.3"></lt><apply id="S4.15.p4.24.m24.1.1.1.cmml" xref="S4.15.p4.24.m24.1.1.1"><times id="S4.15.p4.24.m24.1.1.1.2.cmml" xref="S4.15.p4.24.m24.1.1.1.2"></times><ci id="S4.15.p4.24.m24.1.1.1.3.cmml" xref="S4.15.p4.24.m24.1.1.1.3">𝑓</ci><apply id="S4.15.p4.24.m24.1.1.1.1.1.1.cmml" xref="S4.15.p4.24.m24.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.15.p4.24.m24.1.1.1.1.1.1.1.cmml" xref="S4.15.p4.24.m24.1.1.1.1.1">subscript</csymbol><ci id="S4.15.p4.24.m24.1.1.1.1.1.1.2.cmml" xref="S4.15.p4.24.m24.1.1.1.1.1.1.2">𝑝</ci><cn id="S4.15.p4.24.m24.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.15.p4.24.m24.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S4.15.p4.24.m24.2.2.2.cmml" xref="S4.15.p4.24.m24.2.2.2"><times id="S4.15.p4.24.m24.2.2.2.2.cmml" xref="S4.15.p4.24.m24.2.2.2.2"></times><apply id="S4.15.p4.24.m24.2.2.2.3.cmml" xref="S4.15.p4.24.m24.2.2.2.3"><csymbol cd="ambiguous" id="S4.15.p4.24.m24.2.2.2.3.1.cmml" xref="S4.15.p4.24.m24.2.2.2.3">subscript</csymbol><ci id="S4.15.p4.24.m24.2.2.2.3.2.cmml" xref="S4.15.p4.24.m24.2.2.2.3.2">𝑓</ci><cn id="S4.15.p4.24.m24.2.2.2.3.3.cmml" type="integer" xref="S4.15.p4.24.m24.2.2.2.3.3">0</cn></apply><apply id="S4.15.p4.24.m24.2.2.2.1.1.1.cmml" xref="S4.15.p4.24.m24.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.15.p4.24.m24.2.2.2.1.1.1.1.cmml" xref="S4.15.p4.24.m24.2.2.2.1.1">subscript</csymbol><ci id="S4.15.p4.24.m24.2.2.2.1.1.1.2.cmml" xref="S4.15.p4.24.m24.2.2.2.1.1.1.2">𝑝</ci><cn id="S4.15.p4.24.m24.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.15.p4.24.m24.2.2.2.1.1.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.24.m24.2c">f(p_{0})<f_{0}(p_{0})</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.24.m24.2d">italic_f ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math> by considering <math alttext="-f" class="ltx_Math" display="inline" id="S4.15.p4.25.m25.1"><semantics id="S4.15.p4.25.m25.1a"><mrow id="S4.15.p4.25.m25.1.1" xref="S4.15.p4.25.m25.1.1.cmml"><mo id="S4.15.p4.25.m25.1.1a" xref="S4.15.p4.25.m25.1.1.cmml">−</mo><mi id="S4.15.p4.25.m25.1.1.2" xref="S4.15.p4.25.m25.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.15.p4.25.m25.1b"><apply id="S4.15.p4.25.m25.1.1.cmml" xref="S4.15.p4.25.m25.1.1"><minus id="S4.15.p4.25.m25.1.1.1.cmml" xref="S4.15.p4.25.m25.1.1"></minus><ci id="S4.15.p4.25.m25.1.1.2.cmml" xref="S4.15.p4.25.m25.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.15.p4.25.m25.1c">-f</annotation><annotation encoding="application/x-llamapun" id="S4.15.p4.25.m25.1d">- italic_f</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.1">Together with <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.4</span></a> allows for a reformulation of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a> in a standardised form.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 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.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.3.3">Let <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.1"><semantics id="S4.Thmtheorem5.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.1b"><apply id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1"><in id="S4.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3.2">CPA</ci><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> be a continuous piecewise affine function. Then, there are affine functions <math alttext="f^{(k)}_{n}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.1"><semantics id="S4.Thmtheorem5.p1.2.2.m2.1a"><msubsup id="S4.Thmtheorem5.p1.2.2.m2.1.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.2.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.2.2.cmml">f</mi><mi id="S4.Thmtheorem5.p1.2.2.m2.1.2.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.3.cmml">n</mi><mrow id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.cmml"><mo id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.cmml">(</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.1b"><apply id="S4.Thmtheorem5.p1.2.2.m2.1.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.1.2.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2">subscript</csymbol><apply id="S4.Thmtheorem5.p1.2.2.m2.1.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.1.2.2.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2">superscript</csymbol><ci id="S4.Thmtheorem5.p1.2.2.m2.1.2.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.2.2">𝑓</ci><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1.1">𝑘</ci></apply><ci id="S4.Thmtheorem5.p1.2.2.m2.1.2.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.1c">f^{(k)}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.1d">italic_f start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, and signs <math alttext="\sigma_{n}^{(k)}\in\{-1,1\}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.3"><semantics id="S4.Thmtheorem5.p1.3.3.m3.3a"><mrow id="S4.Thmtheorem5.p1.3.3.m3.3.3" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.cmml"><msubsup id="S4.Thmtheorem5.p1.3.3.m3.3.3.3" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.2" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.2.cmml">σ</mi><mi id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.3" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.3.cmml">n</mi><mrow id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.cmml"><mo id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.cmml">(</mo><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1.1.cmml">k</mi><mo id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S4.Thmtheorem5.p1.3.3.m3.3.3.2" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.2.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.2.cmml"><mo id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.2.cmml">{</mo><mrow id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1a" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.2" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.2.cmml">,</mo><mn id="S4.Thmtheorem5.p1.3.3.m3.2.2" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.cmml">1</mn><mo id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.4" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.3b"><apply id="S4.Thmtheorem5.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3"><in id="S4.Thmtheorem5.p1.3.3.m3.3.3.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.2"></in><apply id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3">superscript</csymbol><apply id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.2">𝜎</ci><ci id="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.3.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.3.2.3">𝑛</ci></apply><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1.1">𝑘</ci></apply><set id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1"><apply id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1"><minus id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1"></minus><cn id="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem5.p1.3.3.m3.3.3.1.1.1.2">1</cn></apply><cn id="S4.Thmtheorem5.p1.3.3.m3.2.2.cmml" type="integer" xref="S4.Thmtheorem5.p1.3.3.m3.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.3c">\sigma_{n}^{(k)}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.3d">italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math>, such that</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex50"> <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 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id="S4.Ex50.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.3.cmml" xref="S4.Ex50.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.3">𝑛</ci></apply><cn id="S4.Ex50.m1.5.5.1.1.cmml" type="integer" xref="S4.Ex50.m1.5.5.1.1">3</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex50.m1.9c">f(x)=\sum_{n=1}^{9p}\sigma_{n}^{(1)}\max(f_{n}^{(1)},\sigma_{n}^{(2)}\max(f_{n% }^{(2)},f_{n}^{(3)})).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex50.m1.9d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 9 italic_p end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.19"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.16.p1"> <p class="ltx_p" id="S4.16.p1.5">Let <math alttext="\mathcal{P}(f)" class="ltx_Math" display="inline" id="S4.16.p1.1.m1.1"><semantics id="S4.16.p1.1.m1.1a"><mrow id="S4.16.p1.1.m1.1.2" xref="S4.16.p1.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.16.p1.1.m1.1.2.2" xref="S4.16.p1.1.m1.1.2.2.cmml">𝒫</mi><mo id="S4.16.p1.1.m1.1.2.1" xref="S4.16.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.16.p1.1.m1.1.2.3.2" xref="S4.16.p1.1.m1.1.2.cmml"><mo id="S4.16.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.16.p1.1.m1.1.2.cmml">(</mo><mi id="S4.16.p1.1.m1.1.1" xref="S4.16.p1.1.m1.1.1.cmml">f</mi><mo id="S4.16.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.16.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.1.m1.1b"><apply id="S4.16.p1.1.m1.1.2.cmml" xref="S4.16.p1.1.m1.1.2"><times id="S4.16.p1.1.m1.1.2.1.cmml" xref="S4.16.p1.1.m1.1.2.1"></times><ci id="S4.16.p1.1.m1.1.2.2.cmml" xref="S4.16.p1.1.m1.1.2.2">𝒫</ci><ci id="S4.16.p1.1.m1.1.1.cmml" xref="S4.16.p1.1.m1.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.1.m1.1c">\mathcal{P}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.1.m1.1d">caligraphic_P ( italic_f )</annotation></semantics></math>, <math alttext="V(f)" class="ltx_Math" display="inline" id="S4.16.p1.2.m2.1"><semantics id="S4.16.p1.2.m2.1a"><mrow id="S4.16.p1.2.m2.1.2" xref="S4.16.p1.2.m2.1.2.cmml"><mi id="S4.16.p1.2.m2.1.2.2" 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class="ltx_Math" display="inline" id="S4.16.p1.3.m3.1"><semantics id="S4.16.p1.3.m3.1a"><mrow id="S4.16.p1.3.m3.1.2" xref="S4.16.p1.3.m3.1.2.cmml"><mi id="S4.16.p1.3.m3.1.2.2" xref="S4.16.p1.3.m3.1.2.2.cmml">E</mi><mo id="S4.16.p1.3.m3.1.2.1" xref="S4.16.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.16.p1.3.m3.1.2.3.2" xref="S4.16.p1.3.m3.1.2.cmml"><mo id="S4.16.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S4.16.p1.3.m3.1.2.cmml">(</mo><mi id="S4.16.p1.3.m3.1.1" xref="S4.16.p1.3.m3.1.1.cmml">f</mi><mo id="S4.16.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S4.16.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.3.m3.1b"><apply id="S4.16.p1.3.m3.1.2.cmml" xref="S4.16.p1.3.m3.1.2"><times id="S4.16.p1.3.m3.1.2.1.cmml" xref="S4.16.p1.3.m3.1.2.1"></times><ci id="S4.16.p1.3.m3.1.2.2.cmml" xref="S4.16.p1.3.m3.1.2.2">𝐸</ci><ci id="S4.16.p1.3.m3.1.1.cmml" xref="S4.16.p1.3.m3.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.3.m3.1c">E(f)</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.3.m3.1d">italic_E ( italic_f )</annotation></semantics></math> be the sets given by <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.1</span></a>, and let <math alttext="E_{b}(f),E_{l}(f)\subseteq E(f)" class="ltx_Math" display="inline" id="S4.16.p1.4.m4.5"><semantics id="S4.16.p1.4.m4.5a"><mrow id="S4.16.p1.4.m4.5.5" xref="S4.16.p1.4.m4.5.5.cmml"><mrow id="S4.16.p1.4.m4.5.5.2.2" xref="S4.16.p1.4.m4.5.5.2.3.cmml"><mrow id="S4.16.p1.4.m4.4.4.1.1.1" xref="S4.16.p1.4.m4.4.4.1.1.1.cmml"><msub id="S4.16.p1.4.m4.4.4.1.1.1.2" xref="S4.16.p1.4.m4.4.4.1.1.1.2.cmml"><mi id="S4.16.p1.4.m4.4.4.1.1.1.2.2" xref="S4.16.p1.4.m4.4.4.1.1.1.2.2.cmml">E</mi><mi id="S4.16.p1.4.m4.4.4.1.1.1.2.3" xref="S4.16.p1.4.m4.4.4.1.1.1.2.3.cmml">b</mi></msub><mo id="S4.16.p1.4.m4.4.4.1.1.1.1" xref="S4.16.p1.4.m4.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S4.16.p1.4.m4.4.4.1.1.1.3.2" xref="S4.16.p1.4.m4.4.4.1.1.1.cmml"><mo id="S4.16.p1.4.m4.4.4.1.1.1.3.2.1" stretchy="false" xref="S4.16.p1.4.m4.4.4.1.1.1.cmml">(</mo><mi id="S4.16.p1.4.m4.1.1" xref="S4.16.p1.4.m4.1.1.cmml">f</mi><mo id="S4.16.p1.4.m4.4.4.1.1.1.3.2.2" stretchy="false" xref="S4.16.p1.4.m4.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.16.p1.4.m4.5.5.2.2.3" xref="S4.16.p1.4.m4.5.5.2.3.cmml">,</mo><mrow id="S4.16.p1.4.m4.5.5.2.2.2" xref="S4.16.p1.4.m4.5.5.2.2.2.cmml"><msub id="S4.16.p1.4.m4.5.5.2.2.2.2" xref="S4.16.p1.4.m4.5.5.2.2.2.2.cmml"><mi id="S4.16.p1.4.m4.5.5.2.2.2.2.2" xref="S4.16.p1.4.m4.5.5.2.2.2.2.2.cmml">E</mi><mi id="S4.16.p1.4.m4.5.5.2.2.2.2.3" xref="S4.16.p1.4.m4.5.5.2.2.2.2.3.cmml">l</mi></msub><mo id="S4.16.p1.4.m4.5.5.2.2.2.1" xref="S4.16.p1.4.m4.5.5.2.2.2.1.cmml">⁢</mo><mrow id="S4.16.p1.4.m4.5.5.2.2.2.3.2" xref="S4.16.p1.4.m4.5.5.2.2.2.cmml"><mo id="S4.16.p1.4.m4.5.5.2.2.2.3.2.1" stretchy="false" xref="S4.16.p1.4.m4.5.5.2.2.2.cmml">(</mo><mi id="S4.16.p1.4.m4.2.2" xref="S4.16.p1.4.m4.2.2.cmml">f</mi><mo id="S4.16.p1.4.m4.5.5.2.2.2.3.2.2" stretchy="false" xref="S4.16.p1.4.m4.5.5.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.16.p1.4.m4.5.5.3" xref="S4.16.p1.4.m4.5.5.3.cmml">⊆</mo><mrow id="S4.16.p1.4.m4.5.5.4" xref="S4.16.p1.4.m4.5.5.4.cmml"><mi id="S4.16.p1.4.m4.5.5.4.2" xref="S4.16.p1.4.m4.5.5.4.2.cmml">E</mi><mo id="S4.16.p1.4.m4.5.5.4.1" xref="S4.16.p1.4.m4.5.5.4.1.cmml">⁢</mo><mrow id="S4.16.p1.4.m4.5.5.4.3.2" xref="S4.16.p1.4.m4.5.5.4.cmml"><mo id="S4.16.p1.4.m4.5.5.4.3.2.1" stretchy="false" xref="S4.16.p1.4.m4.5.5.4.cmml">(</mo><mi id="S4.16.p1.4.m4.3.3" xref="S4.16.p1.4.m4.3.3.cmml">f</mi><mo id="S4.16.p1.4.m4.5.5.4.3.2.2" stretchy="false" xref="S4.16.p1.4.m4.5.5.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.4.m4.5b"><apply id="S4.16.p1.4.m4.5.5.cmml" xref="S4.16.p1.4.m4.5.5"><subset id="S4.16.p1.4.m4.5.5.3.cmml" xref="S4.16.p1.4.m4.5.5.3"></subset><list id="S4.16.p1.4.m4.5.5.2.3.cmml" xref="S4.16.p1.4.m4.5.5.2.2"><apply id="S4.16.p1.4.m4.4.4.1.1.1.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1"><times id="S4.16.p1.4.m4.4.4.1.1.1.1.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1.1"></times><apply id="S4.16.p1.4.m4.4.4.1.1.1.2.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S4.16.p1.4.m4.4.4.1.1.1.2.1.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1.2">subscript</csymbol><ci id="S4.16.p1.4.m4.4.4.1.1.1.2.2.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1.2.2">𝐸</ci><ci id="S4.16.p1.4.m4.4.4.1.1.1.2.3.cmml" xref="S4.16.p1.4.m4.4.4.1.1.1.2.3">𝑏</ci></apply><ci id="S4.16.p1.4.m4.1.1.cmml" xref="S4.16.p1.4.m4.1.1">𝑓</ci></apply><apply id="S4.16.p1.4.m4.5.5.2.2.2.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2"><times id="S4.16.p1.4.m4.5.5.2.2.2.1.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2.1"></times><apply id="S4.16.p1.4.m4.5.5.2.2.2.2.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.16.p1.4.m4.5.5.2.2.2.2.1.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S4.16.p1.4.m4.5.5.2.2.2.2.2.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2.2.2">𝐸</ci><ci id="S4.16.p1.4.m4.5.5.2.2.2.2.3.cmml" xref="S4.16.p1.4.m4.5.5.2.2.2.2.3">𝑙</ci></apply><ci id="S4.16.p1.4.m4.2.2.cmml" xref="S4.16.p1.4.m4.2.2">𝑓</ci></apply></list><apply id="S4.16.p1.4.m4.5.5.4.cmml" xref="S4.16.p1.4.m4.5.5.4"><times id="S4.16.p1.4.m4.5.5.4.1.cmml" xref="S4.16.p1.4.m4.5.5.4.1"></times><ci id="S4.16.p1.4.m4.5.5.4.2.cmml" xref="S4.16.p1.4.m4.5.5.4.2">𝐸</ci><ci id="S4.16.p1.4.m4.3.3.cmml" xref="S4.16.p1.4.m4.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.4.m4.5c">E_{b}(f),E_{l}(f)\subseteq E(f)</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.4.m4.5d">italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) , italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) ⊆ italic_E ( italic_f )</annotation></semantics></math> be the subsets of line segments and lines. By <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a>, there exists an affine function <math alttext="h" class="ltx_Math" display="inline" id="S4.16.p1.5.m5.1"><semantics id="S4.16.p1.5.m5.1a"><mi id="S4.16.p1.5.m5.1.1" xref="S4.16.p1.5.m5.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.16.p1.5.m5.1b"><ci id="S4.16.p1.5.m5.1.1.cmml" xref="S4.16.p1.5.m5.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.5.m5.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.5.m5.1d">italic_h</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E37"> <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)=\sum_{v\in V(f)}f^{{v}}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{% subarray}}f^{{e}}(x)-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end{subarray}}f^{{% e}}(x)+h." class="ltx_Math" display="block" id="S4.E37.m1.8"><semantics id="S4.E37.m1.8a"><mrow id="S4.E37.m1.8.8.1" xref="S4.E37.m1.8.8.1.1.cmml"><mrow id="S4.E37.m1.8.8.1.1" xref="S4.E37.m1.8.8.1.1.cmml"><mrow id="S4.E37.m1.8.8.1.1.2" xref="S4.E37.m1.8.8.1.1.2.cmml"><mi id="S4.E37.m1.8.8.1.1.2.2" xref="S4.E37.m1.8.8.1.1.2.2.cmml">f</mi><mo id="S4.E37.m1.8.8.1.1.2.1" xref="S4.E37.m1.8.8.1.1.2.1.cmml">⁢</mo><mrow id="S4.E37.m1.8.8.1.1.2.3.2" xref="S4.E37.m1.8.8.1.1.2.cmml"><mo id="S4.E37.m1.8.8.1.1.2.3.2.1" stretchy="false" xref="S4.E37.m1.8.8.1.1.2.cmml">(</mo><mi id="S4.E37.m1.4.4" xref="S4.E37.m1.4.4.cmml">x</mi><mo id="S4.E37.m1.8.8.1.1.2.3.2.2" stretchy="false" xref="S4.E37.m1.8.8.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E37.m1.8.8.1.1.1" rspace="0.111em" xref="S4.E37.m1.8.8.1.1.1.cmml">=</mo><mrow id="S4.E37.m1.8.8.1.1.3" xref="S4.E37.m1.8.8.1.1.3.cmml"><mrow id="S4.E37.m1.8.8.1.1.3.2" xref="S4.E37.m1.8.8.1.1.3.2.cmml"><mrow id="S4.E37.m1.8.8.1.1.3.2.2" xref="S4.E37.m1.8.8.1.1.3.2.2.cmml"><mrow id="S4.E37.m1.8.8.1.1.3.2.2.2" xref="S4.E37.m1.8.8.1.1.3.2.2.2.cmml"><munder id="S4.E37.m1.8.8.1.1.3.2.2.2.1" xref="S4.E37.m1.8.8.1.1.3.2.2.2.1.cmml"><mo id="S4.E37.m1.8.8.1.1.3.2.2.2.1.2" movablelimits="false" xref="S4.E37.m1.8.8.1.1.3.2.2.2.1.2.cmml">∑</mo><mrow id="S4.E37.m1.3.3.1" xref="S4.E37.m1.3.3.1.cmml"><mi id="S4.E37.m1.3.3.1.3" xref="S4.E37.m1.3.3.1.3.cmml">v</mi><mo id="S4.E37.m1.3.3.1.2" xref="S4.E37.m1.3.3.1.2.cmml">∈</mo><mrow id="S4.E37.m1.3.3.1.4" xref="S4.E37.m1.3.3.1.4.cmml"><mi id="S4.E37.m1.3.3.1.4.2" xref="S4.E37.m1.3.3.1.4.2.cmml">V</mi><mo id="S4.E37.m1.3.3.1.4.1" xref="S4.E37.m1.3.3.1.4.1.cmml">⁢</mo><mrow id="S4.E37.m1.3.3.1.4.3.2" xref="S4.E37.m1.3.3.1.4.cmml"><mo id="S4.E37.m1.3.3.1.4.3.2.1" 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xref="S4.E37.m1.8.8.1.1.3.2.3.2.2"><csymbol cd="ambiguous" id="S4.E37.m1.8.8.1.1.3.2.3.2.2.1.cmml" xref="S4.E37.m1.8.8.1.1.3.2.3.2.2">superscript</csymbol><ci id="S4.E37.m1.8.8.1.1.3.2.3.2.2.2.cmml" xref="S4.E37.m1.8.8.1.1.3.2.3.2.2.2">𝑓</ci><ci id="S4.E37.m1.8.8.1.1.3.2.3.2.2.3.cmml" xref="S4.E37.m1.8.8.1.1.3.2.3.2.2.3">𝑒</ci></apply><ci id="S4.E37.m1.7.7.cmml" xref="S4.E37.m1.7.7">𝑥</ci></apply></apply></apply><ci id="S4.E37.m1.8.8.1.1.3.3.cmml" xref="S4.E37.m1.8.8.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E37.m1.8c">f(x)=\sum_{v\in V(f)}f^{{v}}(x)+\sum_{\begin{subarray}{c}e\in E_{l}(f)\end{% subarray}}f^{{e}}(x)-\sum_{\begin{subarray}{c}e\in E_{b}(f)\end{subarray}}f^{{% e}}(x)+h.</annotation><annotation encoding="application/x-llamapun" id="S4.E37.m1.8d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_f ) end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT ( italic_x ) + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) + italic_h .</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">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.16.p1.14">For an edge <math alttext="e\in E(f)" class="ltx_Math" display="inline" id="S4.16.p1.6.m1.1"><semantics id="S4.16.p1.6.m1.1a"><mrow id="S4.16.p1.6.m1.1.2" xref="S4.16.p1.6.m1.1.2.cmml"><mi id="S4.16.p1.6.m1.1.2.2" xref="S4.16.p1.6.m1.1.2.2.cmml">e</mi><mo id="S4.16.p1.6.m1.1.2.1" xref="S4.16.p1.6.m1.1.2.1.cmml">∈</mo><mrow id="S4.16.p1.6.m1.1.2.3" xref="S4.16.p1.6.m1.1.2.3.cmml"><mi id="S4.16.p1.6.m1.1.2.3.2" xref="S4.16.p1.6.m1.1.2.3.2.cmml">E</mi><mo id="S4.16.p1.6.m1.1.2.3.1" xref="S4.16.p1.6.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.16.p1.6.m1.1.2.3.3.2" xref="S4.16.p1.6.m1.1.2.3.cmml"><mo id="S4.16.p1.6.m1.1.2.3.3.2.1" stretchy="false" xref="S4.16.p1.6.m1.1.2.3.cmml">(</mo><mi id="S4.16.p1.6.m1.1.1" xref="S4.16.p1.6.m1.1.1.cmml">f</mi><mo id="S4.16.p1.6.m1.1.2.3.3.2.2" stretchy="false" xref="S4.16.p1.6.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.6.m1.1b"><apply id="S4.16.p1.6.m1.1.2.cmml" xref="S4.16.p1.6.m1.1.2"><in id="S4.16.p1.6.m1.1.2.1.cmml" xref="S4.16.p1.6.m1.1.2.1"></in><ci id="S4.16.p1.6.m1.1.2.2.cmml" xref="S4.16.p1.6.m1.1.2.2">𝑒</ci><apply id="S4.16.p1.6.m1.1.2.3.cmml" xref="S4.16.p1.6.m1.1.2.3"><times id="S4.16.p1.6.m1.1.2.3.1.cmml" xref="S4.16.p1.6.m1.1.2.3.1"></times><ci id="S4.16.p1.6.m1.1.2.3.2.cmml" xref="S4.16.p1.6.m1.1.2.3.2">𝐸</ci><ci id="S4.16.p1.6.m1.1.1.cmml" xref="S4.16.p1.6.m1.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.6.m1.1c">e\in E(f)</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.6.m1.1d">italic_e ∈ italic_E ( italic_f )</annotation></semantics></math>, the function <math alttext="f^{e}" class="ltx_Math" display="inline" id="S4.16.p1.7.m2.1"><semantics id="S4.16.p1.7.m2.1a"><msup id="S4.16.p1.7.m2.1.1" xref="S4.16.p1.7.m2.1.1.cmml"><mi id="S4.16.p1.7.m2.1.1.2" xref="S4.16.p1.7.m2.1.1.2.cmml">f</mi><mi id="S4.16.p1.7.m2.1.1.3" xref="S4.16.p1.7.m2.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S4.16.p1.7.m2.1b"><apply id="S4.16.p1.7.m2.1.1.cmml" xref="S4.16.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S4.16.p1.7.m2.1.1.1.cmml" 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xref="S4.16.p1.8.m3.1.1">subscript</csymbol><ci id="S4.16.p1.8.m3.1.1.2.cmml" xref="S4.16.p1.8.m3.1.1.2">CPA</ci><cn id="S4.16.p1.8.m3.1.1.3.cmml" type="integer" xref="S4.16.p1.8.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.8.m3.1c">\operatorname{CPA}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.8.m3.1d">roman_CPA start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> function with two pieces that are half-planes separated by a line. If <math alttext="f_{1}^{e}" class="ltx_Math" display="inline" id="S4.16.p1.9.m4.1"><semantics id="S4.16.p1.9.m4.1a"><msubsup id="S4.16.p1.9.m4.1.1" xref="S4.16.p1.9.m4.1.1.cmml"><mi id="S4.16.p1.9.m4.1.1.2.2" xref="S4.16.p1.9.m4.1.1.2.2.cmml">f</mi><mn id="S4.16.p1.9.m4.1.1.2.3" xref="S4.16.p1.9.m4.1.1.2.3.cmml">1</mn><mi id="S4.16.p1.9.m4.1.1.3" xref="S4.16.p1.9.m4.1.1.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.16.p1.9.m4.1b"><apply id="S4.16.p1.9.m4.1.1.cmml" xref="S4.16.p1.9.m4.1.1"><csymbol cd="ambiguous" id="S4.16.p1.9.m4.1.1.1.cmml" xref="S4.16.p1.9.m4.1.1">superscript</csymbol><apply id="S4.16.p1.9.m4.1.1.2.cmml" xref="S4.16.p1.9.m4.1.1"><csymbol cd="ambiguous" id="S4.16.p1.9.m4.1.1.2.1.cmml" xref="S4.16.p1.9.m4.1.1">subscript</csymbol><ci id="S4.16.p1.9.m4.1.1.2.2.cmml" xref="S4.16.p1.9.m4.1.1.2.2">𝑓</ci><cn id="S4.16.p1.9.m4.1.1.2.3.cmml" type="integer" xref="S4.16.p1.9.m4.1.1.2.3">1</cn></apply><ci id="S4.16.p1.9.m4.1.1.3.cmml" xref="S4.16.p1.9.m4.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.9.m4.1c">f_{1}^{e}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.9.m4.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f_{2}^{e}" class="ltx_Math" display="inline" id="S4.16.p1.10.m5.1"><semantics id="S4.16.p1.10.m5.1a"><msubsup id="S4.16.p1.10.m5.1.1" xref="S4.16.p1.10.m5.1.1.cmml"><mi id="S4.16.p1.10.m5.1.1.2.2" xref="S4.16.p1.10.m5.1.1.2.2.cmml">f</mi><mn id="S4.16.p1.10.m5.1.1.2.3" xref="S4.16.p1.10.m5.1.1.2.3.cmml">2</mn><mi id="S4.16.p1.10.m5.1.1.3" xref="S4.16.p1.10.m5.1.1.3.cmml">e</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.16.p1.10.m5.1b"><apply id="S4.16.p1.10.m5.1.1.cmml" xref="S4.16.p1.10.m5.1.1"><csymbol cd="ambiguous" id="S4.16.p1.10.m5.1.1.1.cmml" xref="S4.16.p1.10.m5.1.1">superscript</csymbol><apply id="S4.16.p1.10.m5.1.1.2.cmml" xref="S4.16.p1.10.m5.1.1"><csymbol cd="ambiguous" id="S4.16.p1.10.m5.1.1.2.1.cmml" xref="S4.16.p1.10.m5.1.1">subscript</csymbol><ci id="S4.16.p1.10.m5.1.1.2.2.cmml" xref="S4.16.p1.10.m5.1.1.2.2">𝑓</ci><cn id="S4.16.p1.10.m5.1.1.2.3.cmml" type="integer" xref="S4.16.p1.10.m5.1.1.2.3">2</cn></apply><ci id="S4.16.p1.10.m5.1.1.3.cmml" xref="S4.16.p1.10.m5.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.10.m5.1c">f_{2}^{e}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.10.m5.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> are the affine components of <math alttext="f^{e}" class="ltx_Math" display="inline" id="S4.16.p1.11.m6.1"><semantics id="S4.16.p1.11.m6.1a"><msup id="S4.16.p1.11.m6.1.1" xref="S4.16.p1.11.m6.1.1.cmml"><mi id="S4.16.p1.11.m6.1.1.2" xref="S4.16.p1.11.m6.1.1.2.cmml">f</mi><mi id="S4.16.p1.11.m6.1.1.3" xref="S4.16.p1.11.m6.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S4.16.p1.11.m6.1b"><apply id="S4.16.p1.11.m6.1.1.cmml" xref="S4.16.p1.11.m6.1.1"><csymbol cd="ambiguous" id="S4.16.p1.11.m6.1.1.1.cmml" xref="S4.16.p1.11.m6.1.1">superscript</csymbol><ci id="S4.16.p1.11.m6.1.1.2.cmml" xref="S4.16.p1.11.m6.1.1.2">𝑓</ci><ci id="S4.16.p1.11.m6.1.1.3.cmml" xref="S4.16.p1.11.m6.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.11.m6.1c">f^{e}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.11.m6.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="f^{e}=\max(f_{1}^{e},f_{2}^{e})" class="ltx_Math" display="inline" id="S4.16.p1.12.m7.3"><semantics id="S4.16.p1.12.m7.3a"><mrow id="S4.16.p1.12.m7.3.3" xref="S4.16.p1.12.m7.3.3.cmml"><msup id="S4.16.p1.12.m7.3.3.4" xref="S4.16.p1.12.m7.3.3.4.cmml"><mi id="S4.16.p1.12.m7.3.3.4.2" xref="S4.16.p1.12.m7.3.3.4.2.cmml">f</mi><mi id="S4.16.p1.12.m7.3.3.4.3" xref="S4.16.p1.12.m7.3.3.4.3.cmml">e</mi></msup><mo id="S4.16.p1.12.m7.3.3.3" xref="S4.16.p1.12.m7.3.3.3.cmml">=</mo><mrow id="S4.16.p1.12.m7.3.3.2.2" xref="S4.16.p1.12.m7.3.3.2.3.cmml"><mi id="S4.16.p1.12.m7.1.1" xref="S4.16.p1.12.m7.1.1.cmml">max</mi><mo id="S4.16.p1.12.m7.3.3.2.2a" xref="S4.16.p1.12.m7.3.3.2.3.cmml">⁡</mo><mrow id="S4.16.p1.12.m7.3.3.2.2.2" xref="S4.16.p1.12.m7.3.3.2.3.cmml"><mo id="S4.16.p1.12.m7.3.3.2.2.2.3" stretchy="false" xref="S4.16.p1.12.m7.3.3.2.3.cmml">(</mo><msubsup id="S4.16.p1.12.m7.2.2.1.1.1.1" xref="S4.16.p1.12.m7.2.2.1.1.1.1.cmml"><mi id="S4.16.p1.12.m7.2.2.1.1.1.1.2.2" xref="S4.16.p1.12.m7.2.2.1.1.1.1.2.2.cmml">f</mi><mn id="S4.16.p1.12.m7.2.2.1.1.1.1.2.3" xref="S4.16.p1.12.m7.2.2.1.1.1.1.2.3.cmml">1</mn><mi id="S4.16.p1.12.m7.2.2.1.1.1.1.3" xref="S4.16.p1.12.m7.2.2.1.1.1.1.3.cmml">e</mi></msubsup><mo id="S4.16.p1.12.m7.3.3.2.2.2.4" xref="S4.16.p1.12.m7.3.3.2.3.cmml">,</mo><msubsup id="S4.16.p1.12.m7.3.3.2.2.2.2" xref="S4.16.p1.12.m7.3.3.2.2.2.2.cmml"><mi id="S4.16.p1.12.m7.3.3.2.2.2.2.2.2" xref="S4.16.p1.12.m7.3.3.2.2.2.2.2.2.cmml">f</mi><mn id="S4.16.p1.12.m7.3.3.2.2.2.2.2.3" xref="S4.16.p1.12.m7.3.3.2.2.2.2.2.3.cmml">2</mn><mi id="S4.16.p1.12.m7.3.3.2.2.2.2.3" xref="S4.16.p1.12.m7.3.3.2.2.2.2.3.cmml">e</mi></msubsup><mo id="S4.16.p1.12.m7.3.3.2.2.2.5" stretchy="false" xref="S4.16.p1.12.m7.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.12.m7.3b"><apply id="S4.16.p1.12.m7.3.3.cmml" xref="S4.16.p1.12.m7.3.3"><eq id="S4.16.p1.12.m7.3.3.3.cmml" xref="S4.16.p1.12.m7.3.3.3"></eq><apply id="S4.16.p1.12.m7.3.3.4.cmml" xref="S4.16.p1.12.m7.3.3.4"><csymbol cd="ambiguous" id="S4.16.p1.12.m7.3.3.4.1.cmml" xref="S4.16.p1.12.m7.3.3.4">superscript</csymbol><ci id="S4.16.p1.12.m7.3.3.4.2.cmml" xref="S4.16.p1.12.m7.3.3.4.2">𝑓</ci><ci id="S4.16.p1.12.m7.3.3.4.3.cmml" xref="S4.16.p1.12.m7.3.3.4.3">𝑒</ci></apply><apply id="S4.16.p1.12.m7.3.3.2.3.cmml" xref="S4.16.p1.12.m7.3.3.2.2"><max id="S4.16.p1.12.m7.1.1.cmml" xref="S4.16.p1.12.m7.1.1"></max><apply id="S4.16.p1.12.m7.2.2.1.1.1.1.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.16.p1.12.m7.2.2.1.1.1.1.1.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1">superscript</csymbol><apply id="S4.16.p1.12.m7.2.2.1.1.1.1.2.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.16.p1.12.m7.2.2.1.1.1.1.2.1.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1">subscript</csymbol><ci id="S4.16.p1.12.m7.2.2.1.1.1.1.2.2.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1.2.2">𝑓</ci><cn id="S4.16.p1.12.m7.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S4.16.p1.12.m7.2.2.1.1.1.1.2.3">1</cn></apply><ci id="S4.16.p1.12.m7.2.2.1.1.1.1.3.cmml" xref="S4.16.p1.12.m7.2.2.1.1.1.1.3">𝑒</ci></apply><apply id="S4.16.p1.12.m7.3.3.2.2.2.2.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.16.p1.12.m7.3.3.2.2.2.2.1.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2">superscript</csymbol><apply id="S4.16.p1.12.m7.3.3.2.2.2.2.2.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.16.p1.12.m7.3.3.2.2.2.2.2.1.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2">subscript</csymbol><ci id="S4.16.p1.12.m7.3.3.2.2.2.2.2.2.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2.2.2">𝑓</ci><cn id="S4.16.p1.12.m7.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S4.16.p1.12.m7.3.3.2.2.2.2.2.3">2</cn></apply><ci id="S4.16.p1.12.m7.3.3.2.2.2.2.3.cmml" xref="S4.16.p1.12.m7.3.3.2.2.2.2.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.12.m7.3c">f^{e}=\max(f_{1}^{e},f_{2}^{e})</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.12.m7.3d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT = roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT )</annotation></semantics></math> or <math alttext="f^{e}=\min(f_{1}^{e},f_{2}^{e})=-\max(-f_{1}^{e},-f_{2}^{e})" class="ltx_Math" display="inline" id="S4.16.p1.13.m8.6"><semantics id="S4.16.p1.13.m8.6a"><mrow id="S4.16.p1.13.m8.6.6" xref="S4.16.p1.13.m8.6.6.cmml"><msup id="S4.16.p1.13.m8.6.6.6" xref="S4.16.p1.13.m8.6.6.6.cmml"><mi id="S4.16.p1.13.m8.6.6.6.2" xref="S4.16.p1.13.m8.6.6.6.2.cmml">f</mi><mi id="S4.16.p1.13.m8.6.6.6.3" xref="S4.16.p1.13.m8.6.6.6.3.cmml">e</mi></msup><mo id="S4.16.p1.13.m8.6.6.7" xref="S4.16.p1.13.m8.6.6.7.cmml">=</mo><mrow id="S4.16.p1.13.m8.4.4.2.2" xref="S4.16.p1.13.m8.4.4.2.3.cmml"><mi id="S4.16.p1.13.m8.1.1" xref="S4.16.p1.13.m8.1.1.cmml">min</mi><mo id="S4.16.p1.13.m8.4.4.2.2a" xref="S4.16.p1.13.m8.4.4.2.3.cmml">⁡</mo><mrow id="S4.16.p1.13.m8.4.4.2.2.2" xref="S4.16.p1.13.m8.4.4.2.3.cmml"><mo id="S4.16.p1.13.m8.4.4.2.2.2.3" stretchy="false" xref="S4.16.p1.13.m8.4.4.2.3.cmml">(</mo><msubsup id="S4.16.p1.13.m8.3.3.1.1.1.1" xref="S4.16.p1.13.m8.3.3.1.1.1.1.cmml"><mi id="S4.16.p1.13.m8.3.3.1.1.1.1.2.2" xref="S4.16.p1.13.m8.3.3.1.1.1.1.2.2.cmml">f</mi><mn id="S4.16.p1.13.m8.3.3.1.1.1.1.2.3" xref="S4.16.p1.13.m8.3.3.1.1.1.1.2.3.cmml">1</mn><mi id="S4.16.p1.13.m8.3.3.1.1.1.1.3" xref="S4.16.p1.13.m8.3.3.1.1.1.1.3.cmml">e</mi></msubsup><mo id="S4.16.p1.13.m8.4.4.2.2.2.4" xref="S4.16.p1.13.m8.4.4.2.3.cmml">,</mo><msubsup id="S4.16.p1.13.m8.4.4.2.2.2.2" xref="S4.16.p1.13.m8.4.4.2.2.2.2.cmml"><mi id="S4.16.p1.13.m8.4.4.2.2.2.2.2.2" xref="S4.16.p1.13.m8.4.4.2.2.2.2.2.2.cmml">f</mi><mn id="S4.16.p1.13.m8.4.4.2.2.2.2.2.3" xref="S4.16.p1.13.m8.4.4.2.2.2.2.2.3.cmml">2</mn><mi id="S4.16.p1.13.m8.4.4.2.2.2.2.3" xref="S4.16.p1.13.m8.4.4.2.2.2.2.3.cmml">e</mi></msubsup><mo id="S4.16.p1.13.m8.4.4.2.2.2.5" stretchy="false" xref="S4.16.p1.13.m8.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S4.16.p1.13.m8.6.6.8" xref="S4.16.p1.13.m8.6.6.8.cmml">=</mo><mrow id="S4.16.p1.13.m8.6.6.4" xref="S4.16.p1.13.m8.6.6.4.cmml"><mo id="S4.16.p1.13.m8.6.6.4a" rspace="0.167em" xref="S4.16.p1.13.m8.6.6.4.cmml">−</mo><mrow id="S4.16.p1.13.m8.6.6.4.2.2" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml"><mi id="S4.16.p1.13.m8.2.2" xref="S4.16.p1.13.m8.2.2.cmml">max</mi><mo id="S4.16.p1.13.m8.6.6.4.2.2a" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml">⁡</mo><mrow id="S4.16.p1.13.m8.6.6.4.2.2.2" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml"><mo id="S4.16.p1.13.m8.6.6.4.2.2.2.3" stretchy="false" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml">(</mo><mrow id="S4.16.p1.13.m8.5.5.3.1.1.1.1" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.cmml"><mo id="S4.16.p1.13.m8.5.5.3.1.1.1.1a" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.cmml">−</mo><msubsup id="S4.16.p1.13.m8.5.5.3.1.1.1.1.2" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.cmml"><mi id="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.2.2" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.2.2.cmml">f</mi><mn id="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.2.3" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.2.3.cmml">1</mn><mi id="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.3" xref="S4.16.p1.13.m8.5.5.3.1.1.1.1.2.3.cmml">e</mi></msubsup></mrow><mo id="S4.16.p1.13.m8.6.6.4.2.2.2.4" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml">,</mo><mrow id="S4.16.p1.13.m8.6.6.4.2.2.2.2" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.cmml"><mo id="S4.16.p1.13.m8.6.6.4.2.2.2.2a" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.cmml">−</mo><msubsup id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.cmml"><mi id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.2" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.2.cmml">f</mi><mn id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.3" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.3.cmml">2</mn><mi id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.3" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.3.cmml">e</mi></msubsup></mrow><mo id="S4.16.p1.13.m8.6.6.4.2.2.2.5" stretchy="false" xref="S4.16.p1.13.m8.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.13.m8.6b"><apply 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xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2">superscript</csymbol><apply id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.cmml" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.1.cmml" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2">subscript</csymbol><ci id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.2.cmml" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.2">𝑓</ci><cn id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.2.3">2</cn></apply><ci id="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.3.cmml" xref="S4.16.p1.13.m8.6.6.4.2.2.2.2.2.3">𝑒</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.13.m8.6c">f^{e}=\min(f_{1}^{e},f_{2}^{e})=-\max(-f_{1}^{e},-f_{2}^{e})</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.13.m8.6d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT = roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ) = - roman_max ( - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT , - italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT )</annotation></semantics></math>. Since <math alttext="-f_{i}^{e}" class="ltx_Math" display="inline" id="S4.16.p1.14.m9.1"><semantics id="S4.16.p1.14.m9.1a"><mrow id="S4.16.p1.14.m9.1.1" xref="S4.16.p1.14.m9.1.1.cmml"><mo id="S4.16.p1.14.m9.1.1a" xref="S4.16.p1.14.m9.1.1.cmml">−</mo><msubsup id="S4.16.p1.14.m9.1.1.2" xref="S4.16.p1.14.m9.1.1.2.cmml"><mi id="S4.16.p1.14.m9.1.1.2.2.2" xref="S4.16.p1.14.m9.1.1.2.2.2.cmml">f</mi><mi id="S4.16.p1.14.m9.1.1.2.2.3" xref="S4.16.p1.14.m9.1.1.2.2.3.cmml">i</mi><mi id="S4.16.p1.14.m9.1.1.2.3" xref="S4.16.p1.14.m9.1.1.2.3.cmml">e</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.14.m9.1b"><apply id="S4.16.p1.14.m9.1.1.cmml" xref="S4.16.p1.14.m9.1.1"><minus id="S4.16.p1.14.m9.1.1.1.cmml" xref="S4.16.p1.14.m9.1.1"></minus><apply id="S4.16.p1.14.m9.1.1.2.cmml" xref="S4.16.p1.14.m9.1.1.2"><csymbol cd="ambiguous" id="S4.16.p1.14.m9.1.1.2.1.cmml" xref="S4.16.p1.14.m9.1.1.2">superscript</csymbol><apply id="S4.16.p1.14.m9.1.1.2.2.cmml" xref="S4.16.p1.14.m9.1.1.2"><csymbol cd="ambiguous" id="S4.16.p1.14.m9.1.1.2.2.1.cmml" xref="S4.16.p1.14.m9.1.1.2">subscript</csymbol><ci id="S4.16.p1.14.m9.1.1.2.2.2.cmml" xref="S4.16.p1.14.m9.1.1.2.2.2">𝑓</ci><ci id="S4.16.p1.14.m9.1.1.2.2.3.cmml" xref="S4.16.p1.14.m9.1.1.2.2.3">𝑖</ci></apply><ci id="S4.16.p1.14.m9.1.1.2.3.cmml" xref="S4.16.p1.14.m9.1.1.2.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.14.m9.1c">-f_{i}^{e}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.14.m9.1d">- italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> is also affine, we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx12"> <tbody id="S4.Ex51"><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 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encoding="application/x-llamapun" id="S4.Ex51.m1.4d">∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x ) - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_e ∈ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_f ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{n=1}^{\absolutevalue{E_{l}}+\absolutevalue{E_{b}}}\sigma_{% n}\max(f_{n}^{(1)},f_{n}^{(2)})" class="ltx_Math" display="inline" id="S4.Ex51.m2.7"><semantics id="S4.Ex51.m2.7a"><mrow id="S4.Ex51.m2.7.7" xref="S4.Ex51.m2.7.7.cmml"><mi id="S4.Ex51.m2.7.7.4" xref="S4.Ex51.m2.7.7.4.cmml"></mi><mo id="S4.Ex51.m2.7.7.3" 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id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.1.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2">superscript</csymbol><apply id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.1.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.2">𝑓</ci><ci id="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.3.cmml" xref="S4.E38.m1.8.8.1.1.2.2.2.2.2.2.2.2.2.2.3">𝑛</ci></apply><cn id="S4.E38.m1.5.5.1.1.cmml" type="integer" xref="S4.E38.m1.5.5.1.1">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E38.m1.8c">\displaystyle=\sum_{n=1}^{\absolutevalue{E_{l}}+\absolutevalue{E_{b}}}\sigma_{% n}\max(f_{n}^{(1)},\max(f_{n}^{(2)},f_{n}^{(2)})),</annotation><annotation encoding="application/x-llamapun" id="S4.E38.m1.8d">= ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT | start_ARG italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG | + | start_ARG italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_ARG | end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) 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">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.16.p1.17">for some signs <math alttext="\sigma_{n}\in\{-1,1\}" class="ltx_Math" display="inline" id="S4.16.p1.15.m1.2"><semantics id="S4.16.p1.15.m1.2a"><mrow id="S4.16.p1.15.m1.2.2" xref="S4.16.p1.15.m1.2.2.cmml"><msub id="S4.16.p1.15.m1.2.2.3" xref="S4.16.p1.15.m1.2.2.3.cmml"><mi id="S4.16.p1.15.m1.2.2.3.2" xref="S4.16.p1.15.m1.2.2.3.2.cmml">σ</mi><mi id="S4.16.p1.15.m1.2.2.3.3" xref="S4.16.p1.15.m1.2.2.3.3.cmml">n</mi></msub><mo id="S4.16.p1.15.m1.2.2.2" xref="S4.16.p1.15.m1.2.2.2.cmml">∈</mo><mrow id="S4.16.p1.15.m1.2.2.1.1" xref="S4.16.p1.15.m1.2.2.1.2.cmml"><mo id="S4.16.p1.15.m1.2.2.1.1.2" stretchy="false" xref="S4.16.p1.15.m1.2.2.1.2.cmml">{</mo><mrow id="S4.16.p1.15.m1.2.2.1.1.1" xref="S4.16.p1.15.m1.2.2.1.1.1.cmml"><mo id="S4.16.p1.15.m1.2.2.1.1.1a" xref="S4.16.p1.15.m1.2.2.1.1.1.cmml">−</mo><mn id="S4.16.p1.15.m1.2.2.1.1.1.2" xref="S4.16.p1.15.m1.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S4.16.p1.15.m1.2.2.1.1.3" xref="S4.16.p1.15.m1.2.2.1.2.cmml">,</mo><mn id="S4.16.p1.15.m1.1.1" xref="S4.16.p1.15.m1.1.1.cmml">1</mn><mo id="S4.16.p1.15.m1.2.2.1.1.4" stretchy="false" xref="S4.16.p1.15.m1.2.2.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.16.p1.15.m1.2b"><apply id="S4.16.p1.15.m1.2.2.cmml" xref="S4.16.p1.15.m1.2.2"><in id="S4.16.p1.15.m1.2.2.2.cmml" xref="S4.16.p1.15.m1.2.2.2"></in><apply id="S4.16.p1.15.m1.2.2.3.cmml" xref="S4.16.p1.15.m1.2.2.3"><csymbol cd="ambiguous" id="S4.16.p1.15.m1.2.2.3.1.cmml" xref="S4.16.p1.15.m1.2.2.3">subscript</csymbol><ci id="S4.16.p1.15.m1.2.2.3.2.cmml" xref="S4.16.p1.15.m1.2.2.3.2">𝜎</ci><ci id="S4.16.p1.15.m1.2.2.3.3.cmml" xref="S4.16.p1.15.m1.2.2.3.3">𝑛</ci></apply><set id="S4.16.p1.15.m1.2.2.1.2.cmml" xref="S4.16.p1.15.m1.2.2.1.1"><apply id="S4.16.p1.15.m1.2.2.1.1.1.cmml" xref="S4.16.p1.15.m1.2.2.1.1.1"><minus id="S4.16.p1.15.m1.2.2.1.1.1.1.cmml" xref="S4.16.p1.15.m1.2.2.1.1.1"></minus><cn id="S4.16.p1.15.m1.2.2.1.1.1.2.cmml" type="integer" xref="S4.16.p1.15.m1.2.2.1.1.1.2">1</cn></apply><cn id="S4.16.p1.15.m1.1.1.cmml" type="integer" xref="S4.16.p1.15.m1.1.1">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.15.m1.2c">\sigma_{n}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.15.m1.2d">italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math> and affine functions <math alttext="f_{n}^{(1)}" class="ltx_Math" display="inline" id="S4.16.p1.16.m2.1"><semantics id="S4.16.p1.16.m2.1a"><msubsup id="S4.16.p1.16.m2.1.2" xref="S4.16.p1.16.m2.1.2.cmml"><mi id="S4.16.p1.16.m2.1.2.2.2" xref="S4.16.p1.16.m2.1.2.2.2.cmml">f</mi><mi id="S4.16.p1.16.m2.1.2.2.3" xref="S4.16.p1.16.m2.1.2.2.3.cmml">n</mi><mrow id="S4.16.p1.16.m2.1.1.1.3" xref="S4.16.p1.16.m2.1.2.cmml"><mo id="S4.16.p1.16.m2.1.1.1.3.1" stretchy="false" xref="S4.16.p1.16.m2.1.2.cmml">(</mo><mn id="S4.16.p1.16.m2.1.1.1.1" xref="S4.16.p1.16.m2.1.1.1.1.cmml">1</mn><mo id="S4.16.p1.16.m2.1.1.1.3.2" stretchy="false" xref="S4.16.p1.16.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.16.p1.16.m2.1b"><apply id="S4.16.p1.16.m2.1.2.cmml" xref="S4.16.p1.16.m2.1.2"><csymbol cd="ambiguous" id="S4.16.p1.16.m2.1.2.1.cmml" xref="S4.16.p1.16.m2.1.2">superscript</csymbol><apply id="S4.16.p1.16.m2.1.2.2.cmml" xref="S4.16.p1.16.m2.1.2"><csymbol cd="ambiguous" id="S4.16.p1.16.m2.1.2.2.1.cmml" xref="S4.16.p1.16.m2.1.2">subscript</csymbol><ci id="S4.16.p1.16.m2.1.2.2.2.cmml" xref="S4.16.p1.16.m2.1.2.2.2">𝑓</ci><ci id="S4.16.p1.16.m2.1.2.2.3.cmml" xref="S4.16.p1.16.m2.1.2.2.3">𝑛</ci></apply><cn id="S4.16.p1.16.m2.1.1.1.1.cmml" type="integer" xref="S4.16.p1.16.m2.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.16.m2.1c">f_{n}^{(1)}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.16.m2.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="f_{n}^{(2)}" class="ltx_Math" display="inline" id="S4.16.p1.17.m3.1"><semantics id="S4.16.p1.17.m3.1a"><msubsup id="S4.16.p1.17.m3.1.2" xref="S4.16.p1.17.m3.1.2.cmml"><mi id="S4.16.p1.17.m3.1.2.2.2" xref="S4.16.p1.17.m3.1.2.2.2.cmml">f</mi><mi id="S4.16.p1.17.m3.1.2.2.3" xref="S4.16.p1.17.m3.1.2.2.3.cmml">n</mi><mrow id="S4.16.p1.17.m3.1.1.1.3" xref="S4.16.p1.17.m3.1.2.cmml"><mo id="S4.16.p1.17.m3.1.1.1.3.1" stretchy="false" xref="S4.16.p1.17.m3.1.2.cmml">(</mo><mn id="S4.16.p1.17.m3.1.1.1.1" xref="S4.16.p1.17.m3.1.1.1.1.cmml">2</mn><mo id="S4.16.p1.17.m3.1.1.1.3.2" stretchy="false" xref="S4.16.p1.17.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.16.p1.17.m3.1b"><apply id="S4.16.p1.17.m3.1.2.cmml" xref="S4.16.p1.17.m3.1.2"><csymbol cd="ambiguous" id="S4.16.p1.17.m3.1.2.1.cmml" xref="S4.16.p1.17.m3.1.2">superscript</csymbol><apply id="S4.16.p1.17.m3.1.2.2.cmml" xref="S4.16.p1.17.m3.1.2"><csymbol cd="ambiguous" id="S4.16.p1.17.m3.1.2.2.1.cmml" xref="S4.16.p1.17.m3.1.2">subscript</csymbol><ci id="S4.16.p1.17.m3.1.2.2.2.cmml" xref="S4.16.p1.17.m3.1.2.2.2">𝑓</ci><ci id="S4.16.p1.17.m3.1.2.2.3.cmml" xref="S4.16.p1.17.m3.1.2.2.3">𝑛</ci></apply><cn id="S4.16.p1.17.m3.1.1.1.1.cmml" type="integer" xref="S4.16.p1.17.m3.1.1.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.16.p1.17.m3.1c">f_{n}^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.16.p1.17.m3.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.17.p2"> <p class="ltx_p" id="S4.17.p2.5">For any <math alttext="v\in V(f)" class="ltx_Math" display="inline" id="S4.17.p2.1.m1.1"><semantics id="S4.17.p2.1.m1.1a"><mrow id="S4.17.p2.1.m1.1.2" xref="S4.17.p2.1.m1.1.2.cmml"><mi id="S4.17.p2.1.m1.1.2.2" xref="S4.17.p2.1.m1.1.2.2.cmml">v</mi><mo id="S4.17.p2.1.m1.1.2.1" xref="S4.17.p2.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.17.p2.1.m1.1.2.3" xref="S4.17.p2.1.m1.1.2.3.cmml"><mi id="S4.17.p2.1.m1.1.2.3.2" xref="S4.17.p2.1.m1.1.2.3.2.cmml">V</mi><mo id="S4.17.p2.1.m1.1.2.3.1" xref="S4.17.p2.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.17.p2.1.m1.1.2.3.3.2" xref="S4.17.p2.1.m1.1.2.3.cmml"><mo id="S4.17.p2.1.m1.1.2.3.3.2.1" stretchy="false" xref="S4.17.p2.1.m1.1.2.3.cmml">(</mo><mi id="S4.17.p2.1.m1.1.1" xref="S4.17.p2.1.m1.1.1.cmml">f</mi><mo id="S4.17.p2.1.m1.1.2.3.3.2.2" stretchy="false" xref="S4.17.p2.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.1.m1.1b"><apply id="S4.17.p2.1.m1.1.2.cmml" xref="S4.17.p2.1.m1.1.2"><in id="S4.17.p2.1.m1.1.2.1.cmml" xref="S4.17.p2.1.m1.1.2.1"></in><ci id="S4.17.p2.1.m1.1.2.2.cmml" xref="S4.17.p2.1.m1.1.2.2">𝑣</ci><apply id="S4.17.p2.1.m1.1.2.3.cmml" xref="S4.17.p2.1.m1.1.2.3"><times id="S4.17.p2.1.m1.1.2.3.1.cmml" xref="S4.17.p2.1.m1.1.2.3.1"></times><ci id="S4.17.p2.1.m1.1.2.3.2.cmml" xref="S4.17.p2.1.m1.1.2.3.2">𝑉</ci><ci id="S4.17.p2.1.m1.1.1.cmml" xref="S4.17.p2.1.m1.1.1">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.1.m1.1c">v\in V(f)</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.1.m1.1d">italic_v ∈ italic_V ( italic_f )</annotation></semantics></math>, <math alttext="f^{v}" class="ltx_Math" display="inline" id="S4.17.p2.2.m2.1"><semantics id="S4.17.p2.2.m2.1a"><msup id="S4.17.p2.2.m2.1.1" xref="S4.17.p2.2.m2.1.1.cmml"><mi id="S4.17.p2.2.m2.1.1.2" xref="S4.17.p2.2.m2.1.1.2.cmml">f</mi><mi id="S4.17.p2.2.m2.1.1.3" xref="S4.17.p2.2.m2.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S4.17.p2.2.m2.1b"><apply id="S4.17.p2.2.m2.1.1.cmml" xref="S4.17.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.17.p2.2.m2.1.1.1.cmml" xref="S4.17.p2.2.m2.1.1">superscript</csymbol><ci id="S4.17.p2.2.m2.1.1.2.cmml" xref="S4.17.p2.2.m2.1.1.2">𝑓</ci><ci id="S4.17.p2.2.m2.1.1.3.cmml" xref="S4.17.p2.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.2.m2.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.2.m2.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> is a <math alttext="v" class="ltx_Math" display="inline" id="S4.17.p2.3.m3.1"><semantics id="S4.17.p2.3.m3.1a"><mi id="S4.17.p2.3.m3.1.1" xref="S4.17.p2.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.17.p2.3.m3.1b"><ci id="S4.17.p2.3.m3.1.1.cmml" xref="S4.17.p2.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.3.m3.1d">italic_v</annotation></semantics></math>-function with <math alttext="\deg(v)\geq 3" class="ltx_Math" display="inline" id="S4.17.p2.4.m4.2"><semantics id="S4.17.p2.4.m4.2a"><mrow id="S4.17.p2.4.m4.2.3" xref="S4.17.p2.4.m4.2.3.cmml"><mrow id="S4.17.p2.4.m4.2.3.2.2" xref="S4.17.p2.4.m4.2.3.2.1.cmml"><mi id="S4.17.p2.4.m4.1.1" xref="S4.17.p2.4.m4.1.1.cmml">deg</mi><mo id="S4.17.p2.4.m4.2.3.2.2a" xref="S4.17.p2.4.m4.2.3.2.1.cmml">⁡</mo><mrow id="S4.17.p2.4.m4.2.3.2.2.1" xref="S4.17.p2.4.m4.2.3.2.1.cmml"><mo id="S4.17.p2.4.m4.2.3.2.2.1.1" stretchy="false" xref="S4.17.p2.4.m4.2.3.2.1.cmml">(</mo><mi id="S4.17.p2.4.m4.2.2" xref="S4.17.p2.4.m4.2.2.cmml">v</mi><mo id="S4.17.p2.4.m4.2.3.2.2.1.2" stretchy="false" xref="S4.17.p2.4.m4.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.17.p2.4.m4.2.3.1" xref="S4.17.p2.4.m4.2.3.1.cmml">≥</mo><mn id="S4.17.p2.4.m4.2.3.3" xref="S4.17.p2.4.m4.2.3.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.4.m4.2b"><apply id="S4.17.p2.4.m4.2.3.cmml" xref="S4.17.p2.4.m4.2.3"><geq id="S4.17.p2.4.m4.2.3.1.cmml" xref="S4.17.p2.4.m4.2.3.1"></geq><apply id="S4.17.p2.4.m4.2.3.2.1.cmml" xref="S4.17.p2.4.m4.2.3.2.2"><csymbol cd="latexml" id="S4.17.p2.4.m4.1.1.cmml" xref="S4.17.p2.4.m4.1.1">degree</csymbol><ci id="S4.17.p2.4.m4.2.2.cmml" xref="S4.17.p2.4.m4.2.2">𝑣</ci></apply><cn id="S4.17.p2.4.m4.2.3.3.cmml" type="integer" xref="S4.17.p2.4.m4.2.3.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.4.m4.2c">\deg(v)\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.4.m4.2d">roman_deg ( italic_v ) ≥ 3</annotation></semantics></math> pieces. Therefore, by <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.2</span></a>, we can express <math alttext="f^{v}" class="ltx_Math" display="inline" id="S4.17.p2.5.m5.1"><semantics id="S4.17.p2.5.m5.1a"><msup id="S4.17.p2.5.m5.1.1" xref="S4.17.p2.5.m5.1.1.cmml"><mi id="S4.17.p2.5.m5.1.1.2" xref="S4.17.p2.5.m5.1.1.2.cmml">f</mi><mi id="S4.17.p2.5.m5.1.1.3" xref="S4.17.p2.5.m5.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S4.17.p2.5.m5.1b"><apply id="S4.17.p2.5.m5.1.1.cmml" xref="S4.17.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.17.p2.5.m5.1.1.1.cmml" xref="S4.17.p2.5.m5.1.1">superscript</csymbol><ci id="S4.17.p2.5.m5.1.1.2.cmml" xref="S4.17.p2.5.m5.1.1.2">𝑓</ci><ci id="S4.17.p2.5.m5.1.1.3.cmml" xref="S4.17.p2.5.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.5.m5.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.5.m5.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex52"> <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^{v}=\sum_{n=1}^{d_{v}}f_{v,n}," class="ltx_Math" display="block" id="S4.Ex52.m1.3"><semantics id="S4.Ex52.m1.3a"><mrow id="S4.Ex52.m1.3.3.1" xref="S4.Ex52.m1.3.3.1.1.cmml"><mrow id="S4.Ex52.m1.3.3.1.1" xref="S4.Ex52.m1.3.3.1.1.cmml"><msup id="S4.Ex52.m1.3.3.1.1.2" xref="S4.Ex52.m1.3.3.1.1.2.cmml"><mi id="S4.Ex52.m1.3.3.1.1.2.2" xref="S4.Ex52.m1.3.3.1.1.2.2.cmml">f</mi><mi id="S4.Ex52.m1.3.3.1.1.2.3" xref="S4.Ex52.m1.3.3.1.1.2.3.cmml">v</mi></msup><mo id="S4.Ex52.m1.3.3.1.1.1" rspace="0.111em" xref="S4.Ex52.m1.3.3.1.1.1.cmml">=</mo><mrow id="S4.Ex52.m1.3.3.1.1.3" xref="S4.Ex52.m1.3.3.1.1.3.cmml"><munderover id="S4.Ex52.m1.3.3.1.1.3.1" xref="S4.Ex52.m1.3.3.1.1.3.1.cmml"><mo id="S4.Ex52.m1.3.3.1.1.3.1.2.2" movablelimits="false" xref="S4.Ex52.m1.3.3.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.Ex52.m1.3.3.1.1.3.1.2.3" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.cmml"><mi id="S4.Ex52.m1.3.3.1.1.3.1.2.3.2" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.2.cmml">n</mi><mo id="S4.Ex52.m1.3.3.1.1.3.1.2.3.1" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.Ex52.m1.3.3.1.1.3.1.2.3.3" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.3.cmml">1</mn></mrow><msub id="S4.Ex52.m1.3.3.1.1.3.1.3" xref="S4.Ex52.m1.3.3.1.1.3.1.3.cmml"><mi id="S4.Ex52.m1.3.3.1.1.3.1.3.2" xref="S4.Ex52.m1.3.3.1.1.3.1.3.2.cmml">d</mi><mi id="S4.Ex52.m1.3.3.1.1.3.1.3.3" xref="S4.Ex52.m1.3.3.1.1.3.1.3.3.cmml">v</mi></msub></munderover><msub id="S4.Ex52.m1.3.3.1.1.3.2" xref="S4.Ex52.m1.3.3.1.1.3.2.cmml"><mi id="S4.Ex52.m1.3.3.1.1.3.2.2" xref="S4.Ex52.m1.3.3.1.1.3.2.2.cmml">f</mi><mrow id="S4.Ex52.m1.2.2.2.4" xref="S4.Ex52.m1.2.2.2.3.cmml"><mi id="S4.Ex52.m1.1.1.1.1" xref="S4.Ex52.m1.1.1.1.1.cmml">v</mi><mo id="S4.Ex52.m1.2.2.2.4.1" xref="S4.Ex52.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex52.m1.2.2.2.2" xref="S4.Ex52.m1.2.2.2.2.cmml">n</mi></mrow></msub></mrow></mrow><mo id="S4.Ex52.m1.3.3.1.2" xref="S4.Ex52.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex52.m1.3b"><apply id="S4.Ex52.m1.3.3.1.1.cmml" xref="S4.Ex52.m1.3.3.1"><eq id="S4.Ex52.m1.3.3.1.1.1.cmml" xref="S4.Ex52.m1.3.3.1.1.1"></eq><apply id="S4.Ex52.m1.3.3.1.1.2.cmml" xref="S4.Ex52.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.Ex52.m1.3.3.1.1.2.1.cmml" xref="S4.Ex52.m1.3.3.1.1.2">superscript</csymbol><ci id="S4.Ex52.m1.3.3.1.1.2.2.cmml" xref="S4.Ex52.m1.3.3.1.1.2.2">𝑓</ci><ci id="S4.Ex52.m1.3.3.1.1.2.3.cmml" xref="S4.Ex52.m1.3.3.1.1.2.3">𝑣</ci></apply><apply id="S4.Ex52.m1.3.3.1.1.3.cmml" xref="S4.Ex52.m1.3.3.1.1.3"><apply id="S4.Ex52.m1.3.3.1.1.3.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex52.m1.3.3.1.1.3.1.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1">superscript</csymbol><apply id="S4.Ex52.m1.3.3.1.1.3.1.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex52.m1.3.3.1.1.3.1.2.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1">subscript</csymbol><sum id="S4.Ex52.m1.3.3.1.1.3.1.2.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.2.2"></sum><apply id="S4.Ex52.m1.3.3.1.1.3.1.2.3.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3"><eq id="S4.Ex52.m1.3.3.1.1.3.1.2.3.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.1"></eq><ci id="S4.Ex52.m1.3.3.1.1.3.1.2.3.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.2">𝑛</ci><cn id="S4.Ex52.m1.3.3.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.Ex52.m1.3.3.1.1.3.1.2.3.3">1</cn></apply></apply><apply id="S4.Ex52.m1.3.3.1.1.3.1.3.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.3"><csymbol cd="ambiguous" id="S4.Ex52.m1.3.3.1.1.3.1.3.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.3">subscript</csymbol><ci id="S4.Ex52.m1.3.3.1.1.3.1.3.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.3.2">𝑑</ci><ci id="S4.Ex52.m1.3.3.1.1.3.1.3.3.cmml" xref="S4.Ex52.m1.3.3.1.1.3.1.3.3">𝑣</ci></apply></apply><apply id="S4.Ex52.m1.3.3.1.1.3.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex52.m1.3.3.1.1.3.2.1.cmml" xref="S4.Ex52.m1.3.3.1.1.3.2">subscript</csymbol><ci id="S4.Ex52.m1.3.3.1.1.3.2.2.cmml" xref="S4.Ex52.m1.3.3.1.1.3.2.2">𝑓</ci><list id="S4.Ex52.m1.2.2.2.3.cmml" xref="S4.Ex52.m1.2.2.2.4"><ci id="S4.Ex52.m1.1.1.1.1.cmml" xref="S4.Ex52.m1.1.1.1.1">𝑣</ci><ci id="S4.Ex52.m1.2.2.2.2.cmml" xref="S4.Ex52.m1.2.2.2.2">𝑛</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex52.m1.3c">f^{v}=\sum_{n=1}^{d_{v}}f_{v,n},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex52.m1.3d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.17.p2.12">where <math alttext="f_{v,n}\in\operatorname{CPA}_{3}" class="ltx_Math" display="inline" id="S4.17.p2.6.m1.2"><semantics id="S4.17.p2.6.m1.2a"><mrow id="S4.17.p2.6.m1.2.3" xref="S4.17.p2.6.m1.2.3.cmml"><msub id="S4.17.p2.6.m1.2.3.2" xref="S4.17.p2.6.m1.2.3.2.cmml"><mi id="S4.17.p2.6.m1.2.3.2.2" xref="S4.17.p2.6.m1.2.3.2.2.cmml">f</mi><mrow id="S4.17.p2.6.m1.2.2.2.4" xref="S4.17.p2.6.m1.2.2.2.3.cmml"><mi id="S4.17.p2.6.m1.1.1.1.1" xref="S4.17.p2.6.m1.1.1.1.1.cmml">v</mi><mo id="S4.17.p2.6.m1.2.2.2.4.1" xref="S4.17.p2.6.m1.2.2.2.3.cmml">,</mo><mi id="S4.17.p2.6.m1.2.2.2.2" xref="S4.17.p2.6.m1.2.2.2.2.cmml">n</mi></mrow></msub><mo id="S4.17.p2.6.m1.2.3.1" xref="S4.17.p2.6.m1.2.3.1.cmml">∈</mo><msub id="S4.17.p2.6.m1.2.3.3" xref="S4.17.p2.6.m1.2.3.3.cmml"><mi id="S4.17.p2.6.m1.2.3.3.2" xref="S4.17.p2.6.m1.2.3.3.2.cmml">CPA</mi><mn id="S4.17.p2.6.m1.2.3.3.3" xref="S4.17.p2.6.m1.2.3.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.6.m1.2b"><apply id="S4.17.p2.6.m1.2.3.cmml" xref="S4.17.p2.6.m1.2.3"><in id="S4.17.p2.6.m1.2.3.1.cmml" xref="S4.17.p2.6.m1.2.3.1"></in><apply id="S4.17.p2.6.m1.2.3.2.cmml" xref="S4.17.p2.6.m1.2.3.2"><csymbol cd="ambiguous" id="S4.17.p2.6.m1.2.3.2.1.cmml" xref="S4.17.p2.6.m1.2.3.2">subscript</csymbol><ci id="S4.17.p2.6.m1.2.3.2.2.cmml" xref="S4.17.p2.6.m1.2.3.2.2">𝑓</ci><list id="S4.17.p2.6.m1.2.2.2.3.cmml" xref="S4.17.p2.6.m1.2.2.2.4"><ci id="S4.17.p2.6.m1.1.1.1.1.cmml" xref="S4.17.p2.6.m1.1.1.1.1">𝑣</ci><ci id="S4.17.p2.6.m1.2.2.2.2.cmml" xref="S4.17.p2.6.m1.2.2.2.2">𝑛</ci></list></apply><apply id="S4.17.p2.6.m1.2.3.3.cmml" xref="S4.17.p2.6.m1.2.3.3"><csymbol cd="ambiguous" id="S4.17.p2.6.m1.2.3.3.1.cmml" xref="S4.17.p2.6.m1.2.3.3">subscript</csymbol><ci id="S4.17.p2.6.m1.2.3.3.2.cmml" xref="S4.17.p2.6.m1.2.3.3.2">CPA</ci><cn id="S4.17.p2.6.m1.2.3.3.3.cmml" type="integer" xref="S4.17.p2.6.m1.2.3.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.6.m1.2c">f_{v,n}\in\operatorname{CPA}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.6.m1.2d">italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT ∈ roman_CPA start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="n\in[d_{v}]" class="ltx_Math" display="inline" id="S4.17.p2.7.m2.1"><semantics id="S4.17.p2.7.m2.1a"><mrow id="S4.17.p2.7.m2.1.1" xref="S4.17.p2.7.m2.1.1.cmml"><mi id="S4.17.p2.7.m2.1.1.3" xref="S4.17.p2.7.m2.1.1.3.cmml">n</mi><mo id="S4.17.p2.7.m2.1.1.2" xref="S4.17.p2.7.m2.1.1.2.cmml">∈</mo><mrow id="S4.17.p2.7.m2.1.1.1.1" xref="S4.17.p2.7.m2.1.1.1.2.cmml"><mo id="S4.17.p2.7.m2.1.1.1.1.2" stretchy="false" xref="S4.17.p2.7.m2.1.1.1.2.1.cmml">[</mo><msub id="S4.17.p2.7.m2.1.1.1.1.1" xref="S4.17.p2.7.m2.1.1.1.1.1.cmml"><mi id="S4.17.p2.7.m2.1.1.1.1.1.2" xref="S4.17.p2.7.m2.1.1.1.1.1.2.cmml">d</mi><mi id="S4.17.p2.7.m2.1.1.1.1.1.3" xref="S4.17.p2.7.m2.1.1.1.1.1.3.cmml">v</mi></msub><mo id="S4.17.p2.7.m2.1.1.1.1.3" stretchy="false" xref="S4.17.p2.7.m2.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.7.m2.1b"><apply id="S4.17.p2.7.m2.1.1.cmml" xref="S4.17.p2.7.m2.1.1"><in id="S4.17.p2.7.m2.1.1.2.cmml" xref="S4.17.p2.7.m2.1.1.2"></in><ci id="S4.17.p2.7.m2.1.1.3.cmml" xref="S4.17.p2.7.m2.1.1.3">𝑛</ci><apply id="S4.17.p2.7.m2.1.1.1.2.cmml" xref="S4.17.p2.7.m2.1.1.1.1"><csymbol cd="latexml" id="S4.17.p2.7.m2.1.1.1.2.1.cmml" xref="S4.17.p2.7.m2.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.17.p2.7.m2.1.1.1.1.1.cmml" xref="S4.17.p2.7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.17.p2.7.m2.1.1.1.1.1.1.cmml" xref="S4.17.p2.7.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.17.p2.7.m2.1.1.1.1.1.2.cmml" xref="S4.17.p2.7.m2.1.1.1.1.1.2">𝑑</ci><ci id="S4.17.p2.7.m2.1.1.1.1.1.3.cmml" xref="S4.17.p2.7.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.7.m2.1c">n\in[d_{v}]</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.7.m2.1d">italic_n ∈ [ italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ]</annotation></semantics></math> are <math alttext="v" class="ltx_Math" display="inline" id="S4.17.p2.8.m3.1"><semantics id="S4.17.p2.8.m3.1a"><mi id="S4.17.p2.8.m3.1.1" xref="S4.17.p2.8.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.17.p2.8.m3.1b"><ci id="S4.17.p2.8.m3.1.1.cmml" xref="S4.17.p2.8.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.8.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.8.m3.1d">italic_v</annotation></semantics></math>-functions, and <math alttext="d_{v}:=\deg(v)-2" class="ltx_Math" display="inline" id="S4.17.p2.9.m4.2"><semantics id="S4.17.p2.9.m4.2a"><mrow id="S4.17.p2.9.m4.2.3" xref="S4.17.p2.9.m4.2.3.cmml"><msub id="S4.17.p2.9.m4.2.3.2" xref="S4.17.p2.9.m4.2.3.2.cmml"><mi id="S4.17.p2.9.m4.2.3.2.2" xref="S4.17.p2.9.m4.2.3.2.2.cmml">d</mi><mi id="S4.17.p2.9.m4.2.3.2.3" xref="S4.17.p2.9.m4.2.3.2.3.cmml">v</mi></msub><mo id="S4.17.p2.9.m4.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.17.p2.9.m4.2.3.1.cmml">:=</mo><mrow id="S4.17.p2.9.m4.2.3.3" xref="S4.17.p2.9.m4.2.3.3.cmml"><mrow id="S4.17.p2.9.m4.2.3.3.2.2" xref="S4.17.p2.9.m4.2.3.3.2.1.cmml"><mi id="S4.17.p2.9.m4.1.1" xref="S4.17.p2.9.m4.1.1.cmml">deg</mi><mo id="S4.17.p2.9.m4.2.3.3.2.2a" xref="S4.17.p2.9.m4.2.3.3.2.1.cmml">⁡</mo><mrow id="S4.17.p2.9.m4.2.3.3.2.2.1" xref="S4.17.p2.9.m4.2.3.3.2.1.cmml"><mo id="S4.17.p2.9.m4.2.3.3.2.2.1.1" stretchy="false" xref="S4.17.p2.9.m4.2.3.3.2.1.cmml">(</mo><mi id="S4.17.p2.9.m4.2.2" xref="S4.17.p2.9.m4.2.2.cmml">v</mi><mo id="S4.17.p2.9.m4.2.3.3.2.2.1.2" stretchy="false" xref="S4.17.p2.9.m4.2.3.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.17.p2.9.m4.2.3.3.1" xref="S4.17.p2.9.m4.2.3.3.1.cmml">−</mo><mn id="S4.17.p2.9.m4.2.3.3.3" xref="S4.17.p2.9.m4.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.9.m4.2b"><apply id="S4.17.p2.9.m4.2.3.cmml" xref="S4.17.p2.9.m4.2.3"><csymbol cd="latexml" id="S4.17.p2.9.m4.2.3.1.cmml" xref="S4.17.p2.9.m4.2.3.1">assign</csymbol><apply id="S4.17.p2.9.m4.2.3.2.cmml" xref="S4.17.p2.9.m4.2.3.2"><csymbol cd="ambiguous" id="S4.17.p2.9.m4.2.3.2.1.cmml" xref="S4.17.p2.9.m4.2.3.2">subscript</csymbol><ci id="S4.17.p2.9.m4.2.3.2.2.cmml" xref="S4.17.p2.9.m4.2.3.2.2">𝑑</ci><ci id="S4.17.p2.9.m4.2.3.2.3.cmml" xref="S4.17.p2.9.m4.2.3.2.3">𝑣</ci></apply><apply id="S4.17.p2.9.m4.2.3.3.cmml" xref="S4.17.p2.9.m4.2.3.3"><minus id="S4.17.p2.9.m4.2.3.3.1.cmml" xref="S4.17.p2.9.m4.2.3.3.1"></minus><apply id="S4.17.p2.9.m4.2.3.3.2.1.cmml" xref="S4.17.p2.9.m4.2.3.3.2.2"><csymbol cd="latexml" id="S4.17.p2.9.m4.1.1.cmml" xref="S4.17.p2.9.m4.1.1">degree</csymbol><ci id="S4.17.p2.9.m4.2.2.cmml" xref="S4.17.p2.9.m4.2.2">𝑣</ci></apply><cn id="S4.17.p2.9.m4.2.3.3.3.cmml" type="integer" xref="S4.17.p2.9.m4.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.9.m4.2c">d_{v}:=\deg(v)-2</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.9.m4.2d">italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT := roman_deg ( italic_v ) - 2</annotation></semantics></math>. Applying <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.4</span></a> to the three piece functions <math alttext="f_{v,n}" class="ltx_Math" display="inline" id="S4.17.p2.10.m5.2"><semantics id="S4.17.p2.10.m5.2a"><msub id="S4.17.p2.10.m5.2.3" xref="S4.17.p2.10.m5.2.3.cmml"><mi id="S4.17.p2.10.m5.2.3.2" xref="S4.17.p2.10.m5.2.3.2.cmml">f</mi><mrow id="S4.17.p2.10.m5.2.2.2.4" xref="S4.17.p2.10.m5.2.2.2.3.cmml"><mi id="S4.17.p2.10.m5.1.1.1.1" xref="S4.17.p2.10.m5.1.1.1.1.cmml">v</mi><mo id="S4.17.p2.10.m5.2.2.2.4.1" xref="S4.17.p2.10.m5.2.2.2.3.cmml">,</mo><mi id="S4.17.p2.10.m5.2.2.2.2" xref="S4.17.p2.10.m5.2.2.2.2.cmml">n</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.17.p2.10.m5.2b"><apply id="S4.17.p2.10.m5.2.3.cmml" xref="S4.17.p2.10.m5.2.3"><csymbol cd="ambiguous" id="S4.17.p2.10.m5.2.3.1.cmml" xref="S4.17.p2.10.m5.2.3">subscript</csymbol><ci id="S4.17.p2.10.m5.2.3.2.cmml" xref="S4.17.p2.10.m5.2.3.2">𝑓</ci><list id="S4.17.p2.10.m5.2.2.2.3.cmml" xref="S4.17.p2.10.m5.2.2.2.4"><ci id="S4.17.p2.10.m5.1.1.1.1.cmml" xref="S4.17.p2.10.m5.1.1.1.1">𝑣</ci><ci id="S4.17.p2.10.m5.2.2.2.2.cmml" xref="S4.17.p2.10.m5.2.2.2.2">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.10.m5.2c">f_{v,n}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.10.m5.2d">italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT</annotation></semantics></math> gives <math alttext="\sigma_{v,n}^{(1)},\sigma_{v,n}^{(2)}\in\{-1,1\}" class="ltx_Math" display="inline" id="S4.17.p2.11.m6.10"><semantics id="S4.17.p2.11.m6.10a"><mrow id="S4.17.p2.11.m6.10.10" xref="S4.17.p2.11.m6.10.10.cmml"><mrow id="S4.17.p2.11.m6.9.9.2.2" xref="S4.17.p2.11.m6.9.9.2.3.cmml"><msubsup id="S4.17.p2.11.m6.8.8.1.1.1" xref="S4.17.p2.11.m6.8.8.1.1.1.cmml"><mi id="S4.17.p2.11.m6.8.8.1.1.1.2.2" xref="S4.17.p2.11.m6.8.8.1.1.1.2.2.cmml">σ</mi><mrow id="S4.17.p2.11.m6.2.2.2.4" xref="S4.17.p2.11.m6.2.2.2.3.cmml"><mi id="S4.17.p2.11.m6.1.1.1.1" xref="S4.17.p2.11.m6.1.1.1.1.cmml">v</mi><mo id="S4.17.p2.11.m6.2.2.2.4.1" xref="S4.17.p2.11.m6.2.2.2.3.cmml">,</mo><mi id="S4.17.p2.11.m6.2.2.2.2" xref="S4.17.p2.11.m6.2.2.2.2.cmml">n</mi></mrow><mrow id="S4.17.p2.11.m6.3.3.1.3" xref="S4.17.p2.11.m6.8.8.1.1.1.cmml"><mo id="S4.17.p2.11.m6.3.3.1.3.1" stretchy="false" xref="S4.17.p2.11.m6.8.8.1.1.1.cmml">(</mo><mn id="S4.17.p2.11.m6.3.3.1.1" xref="S4.17.p2.11.m6.3.3.1.1.cmml">1</mn><mo id="S4.17.p2.11.m6.3.3.1.3.2" stretchy="false" xref="S4.17.p2.11.m6.8.8.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S4.17.p2.11.m6.9.9.2.2.3" xref="S4.17.p2.11.m6.9.9.2.3.cmml">,</mo><msubsup id="S4.17.p2.11.m6.9.9.2.2.2" xref="S4.17.p2.11.m6.9.9.2.2.2.cmml"><mi id="S4.17.p2.11.m6.9.9.2.2.2.2.2" xref="S4.17.p2.11.m6.9.9.2.2.2.2.2.cmml">σ</mi><mrow id="S4.17.p2.11.m6.5.5.2.4" xref="S4.17.p2.11.m6.5.5.2.3.cmml"><mi id="S4.17.p2.11.m6.4.4.1.1" xref="S4.17.p2.11.m6.4.4.1.1.cmml">v</mi><mo id="S4.17.p2.11.m6.5.5.2.4.1" xref="S4.17.p2.11.m6.5.5.2.3.cmml">,</mo><mi id="S4.17.p2.11.m6.5.5.2.2" xref="S4.17.p2.11.m6.5.5.2.2.cmml">n</mi></mrow><mrow id="S4.17.p2.11.m6.6.6.1.3" xref="S4.17.p2.11.m6.9.9.2.2.2.cmml"><mo id="S4.17.p2.11.m6.6.6.1.3.1" stretchy="false" xref="S4.17.p2.11.m6.9.9.2.2.2.cmml">(</mo><mn id="S4.17.p2.11.m6.6.6.1.1" xref="S4.17.p2.11.m6.6.6.1.1.cmml">2</mn><mo id="S4.17.p2.11.m6.6.6.1.3.2" stretchy="false" xref="S4.17.p2.11.m6.9.9.2.2.2.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.17.p2.11.m6.10.10.4" xref="S4.17.p2.11.m6.10.10.4.cmml">∈</mo><mrow id="S4.17.p2.11.m6.10.10.3.1" xref="S4.17.p2.11.m6.10.10.3.2.cmml"><mo id="S4.17.p2.11.m6.10.10.3.1.2" stretchy="false" xref="S4.17.p2.11.m6.10.10.3.2.cmml">{</mo><mrow id="S4.17.p2.11.m6.10.10.3.1.1" xref="S4.17.p2.11.m6.10.10.3.1.1.cmml"><mo id="S4.17.p2.11.m6.10.10.3.1.1a" xref="S4.17.p2.11.m6.10.10.3.1.1.cmml">−</mo><mn id="S4.17.p2.11.m6.10.10.3.1.1.2" xref="S4.17.p2.11.m6.10.10.3.1.1.2.cmml">1</mn></mrow><mo id="S4.17.p2.11.m6.10.10.3.1.3" xref="S4.17.p2.11.m6.10.10.3.2.cmml">,</mo><mn id="S4.17.p2.11.m6.7.7" xref="S4.17.p2.11.m6.7.7.cmml">1</mn><mo id="S4.17.p2.11.m6.10.10.3.1.4" stretchy="false" xref="S4.17.p2.11.m6.10.10.3.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.11.m6.10b"><apply id="S4.17.p2.11.m6.10.10.cmml" xref="S4.17.p2.11.m6.10.10"><in id="S4.17.p2.11.m6.10.10.4.cmml" xref="S4.17.p2.11.m6.10.10.4"></in><list id="S4.17.p2.11.m6.9.9.2.3.cmml" xref="S4.17.p2.11.m6.9.9.2.2"><apply id="S4.17.p2.11.m6.8.8.1.1.1.cmml" xref="S4.17.p2.11.m6.8.8.1.1.1"><csymbol cd="ambiguous" id="S4.17.p2.11.m6.8.8.1.1.1.1.cmml" xref="S4.17.p2.11.m6.8.8.1.1.1">superscript</csymbol><apply id="S4.17.p2.11.m6.8.8.1.1.1.2.cmml" xref="S4.17.p2.11.m6.8.8.1.1.1"><csymbol cd="ambiguous" id="S4.17.p2.11.m6.8.8.1.1.1.2.1.cmml" xref="S4.17.p2.11.m6.8.8.1.1.1">subscript</csymbol><ci id="S4.17.p2.11.m6.8.8.1.1.1.2.2.cmml" xref="S4.17.p2.11.m6.8.8.1.1.1.2.2">𝜎</ci><list id="S4.17.p2.11.m6.2.2.2.3.cmml" xref="S4.17.p2.11.m6.2.2.2.4"><ci id="S4.17.p2.11.m6.1.1.1.1.cmml" xref="S4.17.p2.11.m6.1.1.1.1">𝑣</ci><ci id="S4.17.p2.11.m6.2.2.2.2.cmml" xref="S4.17.p2.11.m6.2.2.2.2">𝑛</ci></list></apply><cn id="S4.17.p2.11.m6.3.3.1.1.cmml" type="integer" xref="S4.17.p2.11.m6.3.3.1.1">1</cn></apply><apply id="S4.17.p2.11.m6.9.9.2.2.2.cmml" xref="S4.17.p2.11.m6.9.9.2.2.2"><csymbol cd="ambiguous" id="S4.17.p2.11.m6.9.9.2.2.2.1.cmml" xref="S4.17.p2.11.m6.9.9.2.2.2">superscript</csymbol><apply id="S4.17.p2.11.m6.9.9.2.2.2.2.cmml" xref="S4.17.p2.11.m6.9.9.2.2.2"><csymbol cd="ambiguous" id="S4.17.p2.11.m6.9.9.2.2.2.2.1.cmml" xref="S4.17.p2.11.m6.9.9.2.2.2">subscript</csymbol><ci id="S4.17.p2.11.m6.9.9.2.2.2.2.2.cmml" xref="S4.17.p2.11.m6.9.9.2.2.2.2.2">𝜎</ci><list id="S4.17.p2.11.m6.5.5.2.3.cmml" xref="S4.17.p2.11.m6.5.5.2.4"><ci id="S4.17.p2.11.m6.4.4.1.1.cmml" xref="S4.17.p2.11.m6.4.4.1.1">𝑣</ci><ci id="S4.17.p2.11.m6.5.5.2.2.cmml" xref="S4.17.p2.11.m6.5.5.2.2">𝑛</ci></list></apply><cn id="S4.17.p2.11.m6.6.6.1.1.cmml" type="integer" xref="S4.17.p2.11.m6.6.6.1.1">2</cn></apply></list><set id="S4.17.p2.11.m6.10.10.3.2.cmml" xref="S4.17.p2.11.m6.10.10.3.1"><apply id="S4.17.p2.11.m6.10.10.3.1.1.cmml" xref="S4.17.p2.11.m6.10.10.3.1.1"><minus id="S4.17.p2.11.m6.10.10.3.1.1.1.cmml" xref="S4.17.p2.11.m6.10.10.3.1.1"></minus><cn id="S4.17.p2.11.m6.10.10.3.1.1.2.cmml" type="integer" xref="S4.17.p2.11.m6.10.10.3.1.1.2">1</cn></apply><cn id="S4.17.p2.11.m6.7.7.cmml" type="integer" xref="S4.17.p2.11.m6.7.7">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.11.m6.10c">\sigma_{v,n}^{(1)},\sigma_{v,n}^{(2)}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.11.m6.10d">italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math> and affine functions <math alttext="f_{v,n}^{(1)},f_{v,n}^{(2)},f_{v,n}^{(3)}" class="ltx_Math" display="inline" id="S4.17.p2.12.m7.12"><semantics id="S4.17.p2.12.m7.12a"><mrow id="S4.17.p2.12.m7.12.12.3" xref="S4.17.p2.12.m7.12.12.4.cmml"><msubsup id="S4.17.p2.12.m7.10.10.1.1" xref="S4.17.p2.12.m7.10.10.1.1.cmml"><mi id="S4.17.p2.12.m7.10.10.1.1.2.2" xref="S4.17.p2.12.m7.10.10.1.1.2.2.cmml">f</mi><mrow id="S4.17.p2.12.m7.2.2.2.4" xref="S4.17.p2.12.m7.2.2.2.3.cmml"><mi id="S4.17.p2.12.m7.1.1.1.1" xref="S4.17.p2.12.m7.1.1.1.1.cmml">v</mi><mo id="S4.17.p2.12.m7.2.2.2.4.1" xref="S4.17.p2.12.m7.2.2.2.3.cmml">,</mo><mi id="S4.17.p2.12.m7.2.2.2.2" xref="S4.17.p2.12.m7.2.2.2.2.cmml">n</mi></mrow><mrow id="S4.17.p2.12.m7.3.3.1.3" xref="S4.17.p2.12.m7.10.10.1.1.cmml"><mo id="S4.17.p2.12.m7.3.3.1.3.1" stretchy="false" xref="S4.17.p2.12.m7.10.10.1.1.cmml">(</mo><mn id="S4.17.p2.12.m7.3.3.1.1" xref="S4.17.p2.12.m7.3.3.1.1.cmml">1</mn><mo id="S4.17.p2.12.m7.3.3.1.3.2" stretchy="false" xref="S4.17.p2.12.m7.10.10.1.1.cmml">)</mo></mrow></msubsup><mo id="S4.17.p2.12.m7.12.12.3.4" xref="S4.17.p2.12.m7.12.12.4.cmml">,</mo><msubsup id="S4.17.p2.12.m7.11.11.2.2" xref="S4.17.p2.12.m7.11.11.2.2.cmml"><mi id="S4.17.p2.12.m7.11.11.2.2.2.2" xref="S4.17.p2.12.m7.11.11.2.2.2.2.cmml">f</mi><mrow id="S4.17.p2.12.m7.5.5.2.4" xref="S4.17.p2.12.m7.5.5.2.3.cmml"><mi id="S4.17.p2.12.m7.4.4.1.1" xref="S4.17.p2.12.m7.4.4.1.1.cmml">v</mi><mo id="S4.17.p2.12.m7.5.5.2.4.1" xref="S4.17.p2.12.m7.5.5.2.3.cmml">,</mo><mi id="S4.17.p2.12.m7.5.5.2.2" xref="S4.17.p2.12.m7.5.5.2.2.cmml">n</mi></mrow><mrow id="S4.17.p2.12.m7.6.6.1.3" xref="S4.17.p2.12.m7.11.11.2.2.cmml"><mo id="S4.17.p2.12.m7.6.6.1.3.1" stretchy="false" xref="S4.17.p2.12.m7.11.11.2.2.cmml">(</mo><mn id="S4.17.p2.12.m7.6.6.1.1" xref="S4.17.p2.12.m7.6.6.1.1.cmml">2</mn><mo id="S4.17.p2.12.m7.6.6.1.3.2" stretchy="false" xref="S4.17.p2.12.m7.11.11.2.2.cmml">)</mo></mrow></msubsup><mo id="S4.17.p2.12.m7.12.12.3.5" xref="S4.17.p2.12.m7.12.12.4.cmml">,</mo><msubsup id="S4.17.p2.12.m7.12.12.3.3" xref="S4.17.p2.12.m7.12.12.3.3.cmml"><mi id="S4.17.p2.12.m7.12.12.3.3.2.2" xref="S4.17.p2.12.m7.12.12.3.3.2.2.cmml">f</mi><mrow id="S4.17.p2.12.m7.8.8.2.4" xref="S4.17.p2.12.m7.8.8.2.3.cmml"><mi id="S4.17.p2.12.m7.7.7.1.1" xref="S4.17.p2.12.m7.7.7.1.1.cmml">v</mi><mo id="S4.17.p2.12.m7.8.8.2.4.1" xref="S4.17.p2.12.m7.8.8.2.3.cmml">,</mo><mi id="S4.17.p2.12.m7.8.8.2.2" xref="S4.17.p2.12.m7.8.8.2.2.cmml">n</mi></mrow><mrow id="S4.17.p2.12.m7.9.9.1.3" xref="S4.17.p2.12.m7.12.12.3.3.cmml"><mo id="S4.17.p2.12.m7.9.9.1.3.1" stretchy="false" xref="S4.17.p2.12.m7.12.12.3.3.cmml">(</mo><mn id="S4.17.p2.12.m7.9.9.1.1" xref="S4.17.p2.12.m7.9.9.1.1.cmml">3</mn><mo id="S4.17.p2.12.m7.9.9.1.3.2" stretchy="false" xref="S4.17.p2.12.m7.12.12.3.3.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.17.p2.12.m7.12b"><list id="S4.17.p2.12.m7.12.12.4.cmml" xref="S4.17.p2.12.m7.12.12.3"><apply id="S4.17.p2.12.m7.10.10.1.1.cmml" xref="S4.17.p2.12.m7.10.10.1.1"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.10.10.1.1.1.cmml" xref="S4.17.p2.12.m7.10.10.1.1">superscript</csymbol><apply id="S4.17.p2.12.m7.10.10.1.1.2.cmml" xref="S4.17.p2.12.m7.10.10.1.1"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.10.10.1.1.2.1.cmml" xref="S4.17.p2.12.m7.10.10.1.1">subscript</csymbol><ci id="S4.17.p2.12.m7.10.10.1.1.2.2.cmml" xref="S4.17.p2.12.m7.10.10.1.1.2.2">𝑓</ci><list id="S4.17.p2.12.m7.2.2.2.3.cmml" xref="S4.17.p2.12.m7.2.2.2.4"><ci id="S4.17.p2.12.m7.1.1.1.1.cmml" xref="S4.17.p2.12.m7.1.1.1.1">𝑣</ci><ci id="S4.17.p2.12.m7.2.2.2.2.cmml" xref="S4.17.p2.12.m7.2.2.2.2">𝑛</ci></list></apply><cn id="S4.17.p2.12.m7.3.3.1.1.cmml" type="integer" xref="S4.17.p2.12.m7.3.3.1.1">1</cn></apply><apply id="S4.17.p2.12.m7.11.11.2.2.cmml" xref="S4.17.p2.12.m7.11.11.2.2"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.11.11.2.2.1.cmml" xref="S4.17.p2.12.m7.11.11.2.2">superscript</csymbol><apply id="S4.17.p2.12.m7.11.11.2.2.2.cmml" xref="S4.17.p2.12.m7.11.11.2.2"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.11.11.2.2.2.1.cmml" xref="S4.17.p2.12.m7.11.11.2.2">subscript</csymbol><ci id="S4.17.p2.12.m7.11.11.2.2.2.2.cmml" xref="S4.17.p2.12.m7.11.11.2.2.2.2">𝑓</ci><list id="S4.17.p2.12.m7.5.5.2.3.cmml" xref="S4.17.p2.12.m7.5.5.2.4"><ci id="S4.17.p2.12.m7.4.4.1.1.cmml" xref="S4.17.p2.12.m7.4.4.1.1">𝑣</ci><ci id="S4.17.p2.12.m7.5.5.2.2.cmml" xref="S4.17.p2.12.m7.5.5.2.2">𝑛</ci></list></apply><cn id="S4.17.p2.12.m7.6.6.1.1.cmml" type="integer" xref="S4.17.p2.12.m7.6.6.1.1">2</cn></apply><apply id="S4.17.p2.12.m7.12.12.3.3.cmml" xref="S4.17.p2.12.m7.12.12.3.3"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.12.12.3.3.1.cmml" xref="S4.17.p2.12.m7.12.12.3.3">superscript</csymbol><apply id="S4.17.p2.12.m7.12.12.3.3.2.cmml" xref="S4.17.p2.12.m7.12.12.3.3"><csymbol cd="ambiguous" id="S4.17.p2.12.m7.12.12.3.3.2.1.cmml" xref="S4.17.p2.12.m7.12.12.3.3">subscript</csymbol><ci id="S4.17.p2.12.m7.12.12.3.3.2.2.cmml" xref="S4.17.p2.12.m7.12.12.3.3.2.2">𝑓</ci><list id="S4.17.p2.12.m7.8.8.2.3.cmml" xref="S4.17.p2.12.m7.8.8.2.4"><ci id="S4.17.p2.12.m7.7.7.1.1.cmml" xref="S4.17.p2.12.m7.7.7.1.1">𝑣</ci><ci id="S4.17.p2.12.m7.8.8.2.2.cmml" xref="S4.17.p2.12.m7.8.8.2.2">𝑛</ci></list></apply><cn id="S4.17.p2.12.m7.9.9.1.1.cmml" type="integer" xref="S4.17.p2.12.m7.9.9.1.1">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.17.p2.12.m7.12c">f_{v,n}^{(1)},f_{v,n}^{(2)},f_{v,n}^{(3)}</annotation><annotation encoding="application/x-llamapun" id="S4.17.p2.12.m7.12d">italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex53"> <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_{v,n}=\sigma_{v,n}^{(1)}\max(f_{v,n}^{(1)},\sigma_{v,n}^{(2)}\max(f_{v,n}^{(% 2)},f_{v,n}^{(3)}))." class="ltx_Math" display="block" id="S4.Ex53.m1.20"><semantics id="S4.Ex53.m1.20a"><mrow id="S4.Ex53.m1.20.20.1" xref="S4.Ex53.m1.20.20.1.1.cmml"><mrow id="S4.Ex53.m1.20.20.1.1" xref="S4.Ex53.m1.20.20.1.1.cmml"><msub id="S4.Ex53.m1.20.20.1.1.4" 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id="S4.Ex53.m1.17.17.1.1.cmml" type="integer" xref="S4.Ex53.m1.17.17.1.1">3</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex53.m1.20c">f_{v,n}=\sigma_{v,n}^{(1)}\max(f_{v,n}^{(1)},\sigma_{v,n}^{(2)}\max(f_{v,n}^{(% 2)},f_{v,n}^{(3)})).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex53.m1.20d">italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.17.p2.13">Therefore,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E39"> <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^{v}=\sum_{n=1}^{d_{v}}\sigma_{v,n}^{(1)}\max(f_{v,n}^{(1)},\sigma_{v,n}^{(2)% }\max(f_{v,n}^{(2)},f_{v,n}^{(3)}))." class="ltx_Math" display="block" id="S4.E39.m1.18"><semantics id="S4.E39.m1.18a"><mrow id="S4.E39.m1.18.18.1" xref="S4.E39.m1.18.18.1.1.cmml"><mrow id="S4.E39.m1.18.18.1.1" xref="S4.E39.m1.18.18.1.1.cmml"><msup id="S4.E39.m1.18.18.1.1.4" xref="S4.E39.m1.18.18.1.1.4.cmml"><mi id="S4.E39.m1.18.18.1.1.4.2" xref="S4.E39.m1.18.18.1.1.4.2.cmml">f</mi><mi id="S4.E39.m1.18.18.1.1.4.3" 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id="S4.E39.m1.18c">f^{v}=\sum_{n=1}^{d_{v}}\sigma_{v,n}^{(1)}\max(f_{v,n}^{(1)},\sigma_{v,n}^{(2)% }\max(f_{v,n}^{(2)},f_{v,n}^{(3)})).</annotation><annotation encoding="application/x-llamapun" id="S4.E39.m1.18d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) 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">(39)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.18.p3"> <p class="ltx_p" id="S4.18.p3.4">Note, that <math alttext="\max(h_{1},h_{2})+h=\max(h_{1}+h,h_{2}+h)" class="ltx_Math" display="inline" id="S4.18.p3.1.m1.6"><semantics id="S4.18.p3.1.m1.6a"><mrow id="S4.18.p3.1.m1.6.6" xref="S4.18.p3.1.m1.6.6.cmml"><mrow id="S4.18.p3.1.m1.4.4.2" xref="S4.18.p3.1.m1.4.4.2.cmml"><mrow id="S4.18.p3.1.m1.4.4.2.2.2" xref="S4.18.p3.1.m1.4.4.2.2.3.cmml"><mi id="S4.18.p3.1.m1.1.1" xref="S4.18.p3.1.m1.1.1.cmml">max</mi><mo id="S4.18.p3.1.m1.4.4.2.2.2a" xref="S4.18.p3.1.m1.4.4.2.2.3.cmml">⁡</mo><mrow id="S4.18.p3.1.m1.4.4.2.2.2.2" xref="S4.18.p3.1.m1.4.4.2.2.3.cmml"><mo 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xref="S4.18.p3.1.m1.6.6.5.cmml">=</mo><mrow id="S4.18.p3.1.m1.6.6.4.2" xref="S4.18.p3.1.m1.6.6.4.3.cmml"><mi id="S4.18.p3.1.m1.2.2" xref="S4.18.p3.1.m1.2.2.cmml">max</mi><mo id="S4.18.p3.1.m1.6.6.4.2a" xref="S4.18.p3.1.m1.6.6.4.3.cmml">⁡</mo><mrow id="S4.18.p3.1.m1.6.6.4.2.2" xref="S4.18.p3.1.m1.6.6.4.3.cmml"><mo id="S4.18.p3.1.m1.6.6.4.2.2.3" stretchy="false" xref="S4.18.p3.1.m1.6.6.4.3.cmml">(</mo><mrow id="S4.18.p3.1.m1.5.5.3.1.1.1" xref="S4.18.p3.1.m1.5.5.3.1.1.1.cmml"><msub id="S4.18.p3.1.m1.5.5.3.1.1.1.2" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2.cmml"><mi id="S4.18.p3.1.m1.5.5.3.1.1.1.2.2" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2.2.cmml">h</mi><mn id="S4.18.p3.1.m1.5.5.3.1.1.1.2.3" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.18.p3.1.m1.5.5.3.1.1.1.1" xref="S4.18.p3.1.m1.5.5.3.1.1.1.1.cmml">+</mo><mi id="S4.18.p3.1.m1.5.5.3.1.1.1.3" xref="S4.18.p3.1.m1.5.5.3.1.1.1.3.cmml">h</mi></mrow><mo id="S4.18.p3.1.m1.6.6.4.2.2.4" xref="S4.18.p3.1.m1.6.6.4.3.cmml">,</mo><mrow id="S4.18.p3.1.m1.6.6.4.2.2.2" xref="S4.18.p3.1.m1.6.6.4.2.2.2.cmml"><msub id="S4.18.p3.1.m1.6.6.4.2.2.2.2" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2.cmml"><mi id="S4.18.p3.1.m1.6.6.4.2.2.2.2.2" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2.2.cmml">h</mi><mn id="S4.18.p3.1.m1.6.6.4.2.2.2.2.3" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.18.p3.1.m1.6.6.4.2.2.2.1" xref="S4.18.p3.1.m1.6.6.4.2.2.2.1.cmml">+</mo><mi id="S4.18.p3.1.m1.6.6.4.2.2.2.3" xref="S4.18.p3.1.m1.6.6.4.2.2.2.3.cmml">h</mi></mrow><mo id="S4.18.p3.1.m1.6.6.4.2.2.5" stretchy="false" xref="S4.18.p3.1.m1.6.6.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.18.p3.1.m1.6b"><apply id="S4.18.p3.1.m1.6.6.cmml" xref="S4.18.p3.1.m1.6.6"><eq id="S4.18.p3.1.m1.6.6.5.cmml" xref="S4.18.p3.1.m1.6.6.5"></eq><apply id="S4.18.p3.1.m1.4.4.2.cmml" xref="S4.18.p3.1.m1.4.4.2"><plus id="S4.18.p3.1.m1.4.4.2.3.cmml" xref="S4.18.p3.1.m1.4.4.2.3"></plus><apply id="S4.18.p3.1.m1.4.4.2.2.3.cmml" xref="S4.18.p3.1.m1.4.4.2.2.2"><max id="S4.18.p3.1.m1.1.1.cmml" xref="S4.18.p3.1.m1.1.1"></max><apply id="S4.18.p3.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.18.p3.1.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.18.p3.1.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.18.p3.1.m1.3.3.1.1.1.1.1">subscript</csymbol><ci id="S4.18.p3.1.m1.3.3.1.1.1.1.1.2.cmml" xref="S4.18.p3.1.m1.3.3.1.1.1.1.1.2">ℎ</ci><cn id="S4.18.p3.1.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S4.18.p3.1.m1.3.3.1.1.1.1.1.3">1</cn></apply><apply id="S4.18.p3.1.m1.4.4.2.2.2.2.2.cmml" xref="S4.18.p3.1.m1.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.18.p3.1.m1.4.4.2.2.2.2.2.1.cmml" xref="S4.18.p3.1.m1.4.4.2.2.2.2.2">subscript</csymbol><ci id="S4.18.p3.1.m1.4.4.2.2.2.2.2.2.cmml" xref="S4.18.p3.1.m1.4.4.2.2.2.2.2.2">ℎ</ci><cn id="S4.18.p3.1.m1.4.4.2.2.2.2.2.3.cmml" type="integer" xref="S4.18.p3.1.m1.4.4.2.2.2.2.2.3">2</cn></apply></apply><ci id="S4.18.p3.1.m1.4.4.2.4.cmml" xref="S4.18.p3.1.m1.4.4.2.4">ℎ</ci></apply><apply id="S4.18.p3.1.m1.6.6.4.3.cmml" xref="S4.18.p3.1.m1.6.6.4.2"><max id="S4.18.p3.1.m1.2.2.cmml" xref="S4.18.p3.1.m1.2.2"></max><apply id="S4.18.p3.1.m1.5.5.3.1.1.1.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1"><plus id="S4.18.p3.1.m1.5.5.3.1.1.1.1.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1.1"></plus><apply id="S4.18.p3.1.m1.5.5.3.1.1.1.2.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.18.p3.1.m1.5.5.3.1.1.1.2.1.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2">subscript</csymbol><ci id="S4.18.p3.1.m1.5.5.3.1.1.1.2.2.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2.2">ℎ</ci><cn id="S4.18.p3.1.m1.5.5.3.1.1.1.2.3.cmml" type="integer" xref="S4.18.p3.1.m1.5.5.3.1.1.1.2.3">1</cn></apply><ci id="S4.18.p3.1.m1.5.5.3.1.1.1.3.cmml" xref="S4.18.p3.1.m1.5.5.3.1.1.1.3">ℎ</ci></apply><apply id="S4.18.p3.1.m1.6.6.4.2.2.2.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2"><plus id="S4.18.p3.1.m1.6.6.4.2.2.2.1.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2.1"></plus><apply id="S4.18.p3.1.m1.6.6.4.2.2.2.2.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.18.p3.1.m1.6.6.4.2.2.2.2.1.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2">subscript</csymbol><ci id="S4.18.p3.1.m1.6.6.4.2.2.2.2.2.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2.2">ℎ</ci><cn id="S4.18.p3.1.m1.6.6.4.2.2.2.2.3.cmml" type="integer" xref="S4.18.p3.1.m1.6.6.4.2.2.2.2.3">2</cn></apply><ci id="S4.18.p3.1.m1.6.6.4.2.2.2.3.cmml" xref="S4.18.p3.1.m1.6.6.4.2.2.2.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.18.p3.1.m1.6c">\max(h_{1},h_{2})+h=\max(h_{1}+h,h_{2}+h)</annotation><annotation encoding="application/x-llamapun" id="S4.18.p3.1.m1.6d">roman_max ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_h = roman_max ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_h , italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_h )</annotation></semantics></math>, and <math alttext="h_{i}+h" class="ltx_Math" display="inline" id="S4.18.p3.2.m2.1"><semantics id="S4.18.p3.2.m2.1a"><mrow id="S4.18.p3.2.m2.1.1" xref="S4.18.p3.2.m2.1.1.cmml"><msub id="S4.18.p3.2.m2.1.1.2" xref="S4.18.p3.2.m2.1.1.2.cmml"><mi id="S4.18.p3.2.m2.1.1.2.2" xref="S4.18.p3.2.m2.1.1.2.2.cmml">h</mi><mi id="S4.18.p3.2.m2.1.1.2.3" xref="S4.18.p3.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S4.18.p3.2.m2.1.1.1" xref="S4.18.p3.2.m2.1.1.1.cmml">+</mo><mi id="S4.18.p3.2.m2.1.1.3" xref="S4.18.p3.2.m2.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.18.p3.2.m2.1b"><apply id="S4.18.p3.2.m2.1.1.cmml" xref="S4.18.p3.2.m2.1.1"><plus id="S4.18.p3.2.m2.1.1.1.cmml" xref="S4.18.p3.2.m2.1.1.1"></plus><apply id="S4.18.p3.2.m2.1.1.2.cmml" xref="S4.18.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.18.p3.2.m2.1.1.2.1.cmml" xref="S4.18.p3.2.m2.1.1.2">subscript</csymbol><ci id="S4.18.p3.2.m2.1.1.2.2.cmml" xref="S4.18.p3.2.m2.1.1.2.2">ℎ</ci><ci id="S4.18.p3.2.m2.1.1.2.3.cmml" xref="S4.18.p3.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S4.18.p3.2.m2.1.1.3.cmml" xref="S4.18.p3.2.m2.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.18.p3.2.m2.1c">h_{i}+h</annotation><annotation encoding="application/x-llamapun" id="S4.18.p3.2.m2.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_h</annotation></semantics></math> is affine if <math alttext="h_{i}" class="ltx_Math" display="inline" id="S4.18.p3.3.m3.1"><semantics id="S4.18.p3.3.m3.1a"><msub id="S4.18.p3.3.m3.1.1" xref="S4.18.p3.3.m3.1.1.cmml"><mi id="S4.18.p3.3.m3.1.1.2" xref="S4.18.p3.3.m3.1.1.2.cmml">h</mi><mi id="S4.18.p3.3.m3.1.1.3" xref="S4.18.p3.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.18.p3.3.m3.1b"><apply id="S4.18.p3.3.m3.1.1.cmml" xref="S4.18.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.18.p3.3.m3.1.1.1.cmml" xref="S4.18.p3.3.m3.1.1">subscript</csymbol><ci id="S4.18.p3.3.m3.1.1.2.cmml" xref="S4.18.p3.3.m3.1.1.2">ℎ</ci><ci id="S4.18.p3.3.m3.1.1.3.cmml" xref="S4.18.p3.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.18.p3.3.m3.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.18.p3.3.m3.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is. Therefore, the affine function <math alttext="h" class="ltx_Math" display="inline" id="S4.18.p3.4.m4.1"><semantics id="S4.18.p3.4.m4.1a"><mi id="S4.18.p3.4.m4.1.1" xref="S4.18.p3.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.18.p3.4.m4.1b"><ci id="S4.18.p3.4.m4.1.1.cmml" xref="S4.18.p3.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.18.p3.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.18.p3.4.m4.1d">italic_h</annotation></semantics></math> can be integrated into one of the summands of either (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E38" title="Equation 38 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">38</span></a>) or (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E39" title="Equation 39 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">39</span></a>). Thus, using (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E38" title="Equation 38 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">38</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E39" title="Equation 39 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">39</span></a>), we can write (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S4.E37" title="Equation 37 ‣ Proof. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">37</span></a>) as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx13"> <tbody id="S4.Ex54"><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)" class="ltx_Math" display="inline" id="S4.Ex54.m1.1"><semantics id="S4.Ex54.m1.1a"><mrow id="S4.Ex54.m1.1.2" xref="S4.Ex54.m1.1.2.cmml"><mi id="S4.Ex54.m1.1.2.2" xref="S4.Ex54.m1.1.2.2.cmml">f</mi><mo id="S4.Ex54.m1.1.2.1" xref="S4.Ex54.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex54.m1.1.2.3.2" xref="S4.Ex54.m1.1.2.cmml"><mo id="S4.Ex54.m1.1.2.3.2.1" stretchy="false" xref="S4.Ex54.m1.1.2.cmml">(</mo><mi id="S4.Ex54.m1.1.1" xref="S4.Ex54.m1.1.1.cmml">x</mi><mo id="S4.Ex54.m1.1.2.3.2.2" stretchy="false" xref="S4.Ex54.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex54.m1.1b"><apply id="S4.Ex54.m1.1.2.cmml" xref="S4.Ex54.m1.1.2"><times id="S4.Ex54.m1.1.2.1.cmml" xref="S4.Ex54.m1.1.2.1"></times><ci id="S4.Ex54.m1.1.2.2.cmml" xref="S4.Ex54.m1.1.2.2">𝑓</ci><ci id="S4.Ex54.m1.1.1.cmml" xref="S4.Ex54.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex54.m1.1c">\displaystyle f(x)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex54.m1.1d">italic_f ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{v\in V}\sum_{n=1}^{d_{v}}\sigma_{v,n}^{(1)}\max(f_{v,n}^{(% 1)},\sigma_{v,n}^{(2)}\max(f_{v,n}^{(2)},f_{v,n}^{(3)}))" class="ltx_Math" display="inline" id="S4.Ex54.m2.19"><semantics id="S4.Ex54.m2.19a"><mrow id="S4.Ex54.m2.19.19" xref="S4.Ex54.m2.19.19.cmml"><mi id="S4.Ex54.m2.19.19.4" xref="S4.Ex54.m2.19.19.4.cmml"></mi><mo id="S4.Ex54.m2.19.19.3" xref="S4.Ex54.m2.19.19.3.cmml">=</mo><mrow id="S4.Ex54.m2.19.19.2" xref="S4.Ex54.m2.19.19.2.cmml"><mstyle displaystyle="true" id="S4.Ex54.m2.19.19.2.3" xref="S4.Ex54.m2.19.19.2.3.cmml"><munder id="S4.Ex54.m2.19.19.2.3a" xref="S4.Ex54.m2.19.19.2.3.cmml"><mo id="S4.Ex54.m2.19.19.2.3.2" 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xref="S4.Ex54.m2.19.19.2.2.2.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex54.m2.19.19.2.2.2.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.Ex54.m2.19.19.2.2.2.2.2.2.2.2.2.2.2.2.2">𝑓</ci><list id="S4.Ex54.m2.14.14.2.3.cmml" xref="S4.Ex54.m2.14.14.2.4"><ci id="S4.Ex54.m2.13.13.1.1.cmml" xref="S4.Ex54.m2.13.13.1.1">𝑣</ci><ci id="S4.Ex54.m2.14.14.2.2.cmml" xref="S4.Ex54.m2.14.14.2.2">𝑛</ci></list></apply><cn id="S4.Ex54.m2.15.15.1.1.cmml" type="integer" xref="S4.Ex54.m2.15.15.1.1">3</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex54.m2.19c">\displaystyle=\sum_{v\in V}\sum_{n=1}^{d_{v}}\sigma_{v,n}^{(1)}\max(f_{v,n}^{(% 1)},\sigma_{v,n}^{(2)}\max(f_{v,n}^{(2)},f_{v,n}^{(3)}))</annotation><annotation encoding="application/x-llamapun" id="S4.Ex54.m2.19d">= ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_v , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex55"><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 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start_POSTSUPERSCRIPT | start_ARG italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG | + | start_ARG italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_ARG | end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.19.p4"> <p class="ltx_p" id="S4.19.p4.4">It remains to show that <math alttext="\sum_{v\in V}d_{v}+|E_{l}|+|E_{b}|\leq 9p" class="ltx_Math" display="inline" id="S4.19.p4.1.m1.2"><semantics id="S4.19.p4.1.m1.2a"><mrow id="S4.19.p4.1.m1.2.2" xref="S4.19.p4.1.m1.2.2.cmml"><mrow id="S4.19.p4.1.m1.2.2.2" xref="S4.19.p4.1.m1.2.2.2.cmml"><mrow id="S4.19.p4.1.m1.2.2.2.4" xref="S4.19.p4.1.m1.2.2.2.4.cmml"><msub id="S4.19.p4.1.m1.2.2.2.4.1" xref="S4.19.p4.1.m1.2.2.2.4.1.cmml"><mo id="S4.19.p4.1.m1.2.2.2.4.1.2" xref="S4.19.p4.1.m1.2.2.2.4.1.2.cmml">∑</mo><mrow id="S4.19.p4.1.m1.2.2.2.4.1.3" xref="S4.19.p4.1.m1.2.2.2.4.1.3.cmml"><mi id="S4.19.p4.1.m1.2.2.2.4.1.3.2" xref="S4.19.p4.1.m1.2.2.2.4.1.3.2.cmml">v</mi><mo id="S4.19.p4.1.m1.2.2.2.4.1.3.1" xref="S4.19.p4.1.m1.2.2.2.4.1.3.1.cmml">∈</mo><mi id="S4.19.p4.1.m1.2.2.2.4.1.3.3" xref="S4.19.p4.1.m1.2.2.2.4.1.3.3.cmml">V</mi></mrow></msub><msub id="S4.19.p4.1.m1.2.2.2.4.2" xref="S4.19.p4.1.m1.2.2.2.4.2.cmml"><mi id="S4.19.p4.1.m1.2.2.2.4.2.2" xref="S4.19.p4.1.m1.2.2.2.4.2.2.cmml">d</mi><mi id="S4.19.p4.1.m1.2.2.2.4.2.3" xref="S4.19.p4.1.m1.2.2.2.4.2.3.cmml">v</mi></msub></mrow><mo id="S4.19.p4.1.m1.2.2.2.3" xref="S4.19.p4.1.m1.2.2.2.3.cmml">+</mo><mrow id="S4.19.p4.1.m1.1.1.1.1.1" xref="S4.19.p4.1.m1.1.1.1.1.2.cmml"><mo id="S4.19.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="S4.19.p4.1.m1.1.1.1.1.2.1.cmml">|</mo><msub id="S4.19.p4.1.m1.1.1.1.1.1.1" xref="S4.19.p4.1.m1.1.1.1.1.1.1.cmml"><mi id="S4.19.p4.1.m1.1.1.1.1.1.1.2" xref="S4.19.p4.1.m1.1.1.1.1.1.1.2.cmml">E</mi><mi id="S4.19.p4.1.m1.1.1.1.1.1.1.3" xref="S4.19.p4.1.m1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S4.19.p4.1.m1.1.1.1.1.1.3" stretchy="false" xref="S4.19.p4.1.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.19.p4.1.m1.2.2.2.3a" xref="S4.19.p4.1.m1.2.2.2.3.cmml">+</mo><mrow id="S4.19.p4.1.m1.2.2.2.2.1" xref="S4.19.p4.1.m1.2.2.2.2.2.cmml"><mo id="S4.19.p4.1.m1.2.2.2.2.1.2" stretchy="false" xref="S4.19.p4.1.m1.2.2.2.2.2.1.cmml">|</mo><msub id="S4.19.p4.1.m1.2.2.2.2.1.1" xref="S4.19.p4.1.m1.2.2.2.2.1.1.cmml"><mi id="S4.19.p4.1.m1.2.2.2.2.1.1.2" xref="S4.19.p4.1.m1.2.2.2.2.1.1.2.cmml">E</mi><mi id="S4.19.p4.1.m1.2.2.2.2.1.1.3" xref="S4.19.p4.1.m1.2.2.2.2.1.1.3.cmml">b</mi></msub><mo id="S4.19.p4.1.m1.2.2.2.2.1.3" stretchy="false" xref="S4.19.p4.1.m1.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S4.19.p4.1.m1.2.2.3" xref="S4.19.p4.1.m1.2.2.3.cmml">≤</mo><mrow id="S4.19.p4.1.m1.2.2.4" xref="S4.19.p4.1.m1.2.2.4.cmml"><mn id="S4.19.p4.1.m1.2.2.4.2" xref="S4.19.p4.1.m1.2.2.4.2.cmml">9</mn><mo id="S4.19.p4.1.m1.2.2.4.1" xref="S4.19.p4.1.m1.2.2.4.1.cmml">⁢</mo><mi id="S4.19.p4.1.m1.2.2.4.3" xref="S4.19.p4.1.m1.2.2.4.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.19.p4.1.m1.2b"><apply id="S4.19.p4.1.m1.2.2.cmml" xref="S4.19.p4.1.m1.2.2"><leq id="S4.19.p4.1.m1.2.2.3.cmml" xref="S4.19.p4.1.m1.2.2.3"></leq><apply id="S4.19.p4.1.m1.2.2.2.cmml" xref="S4.19.p4.1.m1.2.2.2"><plus id="S4.19.p4.1.m1.2.2.2.3.cmml" xref="S4.19.p4.1.m1.2.2.2.3"></plus><apply id="S4.19.p4.1.m1.2.2.2.4.cmml" xref="S4.19.p4.1.m1.2.2.2.4"><apply id="S4.19.p4.1.m1.2.2.2.4.1.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1"><csymbol cd="ambiguous" id="S4.19.p4.1.m1.2.2.2.4.1.1.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1">subscript</csymbol><sum id="S4.19.p4.1.m1.2.2.2.4.1.2.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1.2"></sum><apply id="S4.19.p4.1.m1.2.2.2.4.1.3.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1.3"><in id="S4.19.p4.1.m1.2.2.2.4.1.3.1.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1.3.1"></in><ci id="S4.19.p4.1.m1.2.2.2.4.1.3.2.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1.3.2">𝑣</ci><ci id="S4.19.p4.1.m1.2.2.2.4.1.3.3.cmml" xref="S4.19.p4.1.m1.2.2.2.4.1.3.3">𝑉</ci></apply></apply><apply id="S4.19.p4.1.m1.2.2.2.4.2.cmml" xref="S4.19.p4.1.m1.2.2.2.4.2"><csymbol cd="ambiguous" id="S4.19.p4.1.m1.2.2.2.4.2.1.cmml" xref="S4.19.p4.1.m1.2.2.2.4.2">subscript</csymbol><ci id="S4.19.p4.1.m1.2.2.2.4.2.2.cmml" xref="S4.19.p4.1.m1.2.2.2.4.2.2">𝑑</ci><ci id="S4.19.p4.1.m1.2.2.2.4.2.3.cmml" xref="S4.19.p4.1.m1.2.2.2.4.2.3">𝑣</ci></apply></apply><apply id="S4.19.p4.1.m1.1.1.1.1.2.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1"><abs id="S4.19.p4.1.m1.1.1.1.1.2.1.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1.2"></abs><apply id="S4.19.p4.1.m1.1.1.1.1.1.1.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.19.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.19.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1.1.2">𝐸</ci><ci id="S4.19.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="S4.19.p4.1.m1.1.1.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S4.19.p4.1.m1.2.2.2.2.2.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1"><abs id="S4.19.p4.1.m1.2.2.2.2.2.1.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1.2"></abs><apply id="S4.19.p4.1.m1.2.2.2.2.1.1.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.19.p4.1.m1.2.2.2.2.1.1.1.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1.1">subscript</csymbol><ci id="S4.19.p4.1.m1.2.2.2.2.1.1.2.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1.1.2">𝐸</ci><ci id="S4.19.p4.1.m1.2.2.2.2.1.1.3.cmml" xref="S4.19.p4.1.m1.2.2.2.2.1.1.3">𝑏</ci></apply></apply></apply><apply id="S4.19.p4.1.m1.2.2.4.cmml" xref="S4.19.p4.1.m1.2.2.4"><times id="S4.19.p4.1.m1.2.2.4.1.cmml" xref="S4.19.p4.1.m1.2.2.4.1"></times><cn id="S4.19.p4.1.m1.2.2.4.2.cmml" type="integer" xref="S4.19.p4.1.m1.2.2.4.2">9</cn><ci id="S4.19.p4.1.m1.2.2.4.3.cmml" xref="S4.19.p4.1.m1.2.2.4.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.19.p4.1.m1.2c">\sum_{v\in V}d_{v}+|E_{l}|+|E_{b}|\leq 9p</annotation><annotation encoding="application/x-llamapun" id="S4.19.p4.1.m1.2d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT + | italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT | + | italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | ≤ 9 italic_p</annotation></semantics></math>. Since we chose the vertex and edge sets according to <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.1</span></a>, we have <math alttext="|E|\leq 3p" class="ltx_Math" display="inline" id="S4.19.p4.2.m2.1"><semantics id="S4.19.p4.2.m2.1a"><mrow id="S4.19.p4.2.m2.1.2" xref="S4.19.p4.2.m2.1.2.cmml"><mrow id="S4.19.p4.2.m2.1.2.2.2" xref="S4.19.p4.2.m2.1.2.2.1.cmml"><mo id="S4.19.p4.2.m2.1.2.2.2.1" stretchy="false" xref="S4.19.p4.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S4.19.p4.2.m2.1.1" xref="S4.19.p4.2.m2.1.1.cmml">E</mi><mo id="S4.19.p4.2.m2.1.2.2.2.2" stretchy="false" xref="S4.19.p4.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.19.p4.2.m2.1.2.1" xref="S4.19.p4.2.m2.1.2.1.cmml">≤</mo><mrow id="S4.19.p4.2.m2.1.2.3" xref="S4.19.p4.2.m2.1.2.3.cmml"><mn id="S4.19.p4.2.m2.1.2.3.2" xref="S4.19.p4.2.m2.1.2.3.2.cmml">3</mn><mo id="S4.19.p4.2.m2.1.2.3.1" xref="S4.19.p4.2.m2.1.2.3.1.cmml">⁢</mo><mi id="S4.19.p4.2.m2.1.2.3.3" xref="S4.19.p4.2.m2.1.2.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.19.p4.2.m2.1b"><apply id="S4.19.p4.2.m2.1.2.cmml" xref="S4.19.p4.2.m2.1.2"><leq id="S4.19.p4.2.m2.1.2.1.cmml" xref="S4.19.p4.2.m2.1.2.1"></leq><apply id="S4.19.p4.2.m2.1.2.2.1.cmml" xref="S4.19.p4.2.m2.1.2.2.2"><abs id="S4.19.p4.2.m2.1.2.2.1.1.cmml" xref="S4.19.p4.2.m2.1.2.2.2.1"></abs><ci id="S4.19.p4.2.m2.1.1.cmml" xref="S4.19.p4.2.m2.1.1">𝐸</ci></apply><apply id="S4.19.p4.2.m2.1.2.3.cmml" xref="S4.19.p4.2.m2.1.2.3"><times id="S4.19.p4.2.m2.1.2.3.1.cmml" xref="S4.19.p4.2.m2.1.2.3.1"></times><cn id="S4.19.p4.2.m2.1.2.3.2.cmml" type="integer" xref="S4.19.p4.2.m2.1.2.3.2">3</cn><ci id="S4.19.p4.2.m2.1.2.3.3.cmml" xref="S4.19.p4.2.m2.1.2.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.19.p4.2.m2.1c">|E|\leq 3p</annotation><annotation encoding="application/x-llamapun" id="S4.19.p4.2.m2.1d">| italic_E | ≤ 3 italic_p</annotation></semantics></math>. This directly implies <math alttext="|E_{l}|+|E_{b}|\leq 3p" class="ltx_Math" display="inline" id="S4.19.p4.3.m3.2"><semantics id="S4.19.p4.3.m3.2a"><mrow id="S4.19.p4.3.m3.2.2" xref="S4.19.p4.3.m3.2.2.cmml"><mrow id="S4.19.p4.3.m3.2.2.2" xref="S4.19.p4.3.m3.2.2.2.cmml"><mrow id="S4.19.p4.3.m3.1.1.1.1.1" xref="S4.19.p4.3.m3.1.1.1.1.2.cmml"><mo id="S4.19.p4.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.19.p4.3.m3.1.1.1.1.2.1.cmml">|</mo><msub id="S4.19.p4.3.m3.1.1.1.1.1.1" xref="S4.19.p4.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.19.p4.3.m3.1.1.1.1.1.1.2" xref="S4.19.p4.3.m3.1.1.1.1.1.1.2.cmml">E</mi><mi id="S4.19.p4.3.m3.1.1.1.1.1.1.3" xref="S4.19.p4.3.m3.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S4.19.p4.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.19.p4.3.m3.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.19.p4.3.m3.2.2.2.3" xref="S4.19.p4.3.m3.2.2.2.3.cmml">+</mo><mrow id="S4.19.p4.3.m3.2.2.2.2.1" xref="S4.19.p4.3.m3.2.2.2.2.2.cmml"><mo id="S4.19.p4.3.m3.2.2.2.2.1.2" stretchy="false" xref="S4.19.p4.3.m3.2.2.2.2.2.1.cmml">|</mo><msub id="S4.19.p4.3.m3.2.2.2.2.1.1" xref="S4.19.p4.3.m3.2.2.2.2.1.1.cmml"><mi id="S4.19.p4.3.m3.2.2.2.2.1.1.2" xref="S4.19.p4.3.m3.2.2.2.2.1.1.2.cmml">E</mi><mi id="S4.19.p4.3.m3.2.2.2.2.1.1.3" xref="S4.19.p4.3.m3.2.2.2.2.1.1.3.cmml">b</mi></msub><mo id="S4.19.p4.3.m3.2.2.2.2.1.3" stretchy="false" xref="S4.19.p4.3.m3.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S4.19.p4.3.m3.2.2.3" xref="S4.19.p4.3.m3.2.2.3.cmml">≤</mo><mrow id="S4.19.p4.3.m3.2.2.4" xref="S4.19.p4.3.m3.2.2.4.cmml"><mn id="S4.19.p4.3.m3.2.2.4.2" xref="S4.19.p4.3.m3.2.2.4.2.cmml">3</mn><mo id="S4.19.p4.3.m3.2.2.4.1" xref="S4.19.p4.3.m3.2.2.4.1.cmml">⁢</mo><mi id="S4.19.p4.3.m3.2.2.4.3" xref="S4.19.p4.3.m3.2.2.4.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.19.p4.3.m3.2b"><apply id="S4.19.p4.3.m3.2.2.cmml" xref="S4.19.p4.3.m3.2.2"><leq id="S4.19.p4.3.m3.2.2.3.cmml" xref="S4.19.p4.3.m3.2.2.3"></leq><apply id="S4.19.p4.3.m3.2.2.2.cmml" xref="S4.19.p4.3.m3.2.2.2"><plus id="S4.19.p4.3.m3.2.2.2.3.cmml" xref="S4.19.p4.3.m3.2.2.2.3"></plus><apply id="S4.19.p4.3.m3.1.1.1.1.2.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1"><abs id="S4.19.p4.3.m3.1.1.1.1.2.1.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1.2"></abs><apply id="S4.19.p4.3.m3.1.1.1.1.1.1.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.19.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S4.19.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1.1.2">𝐸</ci><ci id="S4.19.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.19.p4.3.m3.1.1.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S4.19.p4.3.m3.2.2.2.2.2.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1"><abs id="S4.19.p4.3.m3.2.2.2.2.2.1.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1.2"></abs><apply id="S4.19.p4.3.m3.2.2.2.2.1.1.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.19.p4.3.m3.2.2.2.2.1.1.1.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1.1">subscript</csymbol><ci id="S4.19.p4.3.m3.2.2.2.2.1.1.2.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1.1.2">𝐸</ci><ci id="S4.19.p4.3.m3.2.2.2.2.1.1.3.cmml" xref="S4.19.p4.3.m3.2.2.2.2.1.1.3">𝑏</ci></apply></apply></apply><apply id="S4.19.p4.3.m3.2.2.4.cmml" xref="S4.19.p4.3.m3.2.2.4"><times id="S4.19.p4.3.m3.2.2.4.1.cmml" xref="S4.19.p4.3.m3.2.2.4.1"></times><cn id="S4.19.p4.3.m3.2.2.4.2.cmml" type="integer" xref="S4.19.p4.3.m3.2.2.4.2">3</cn><ci id="S4.19.p4.3.m3.2.2.4.3.cmml" xref="S4.19.p4.3.m3.2.2.4.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.19.p4.3.m3.2c">|E_{l}|+|E_{b}|\leq 3p</annotation><annotation encoding="application/x-llamapun" id="S4.19.p4.3.m3.2d">| italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT | + | italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | ≤ 3 italic_p</annotation></semantics></math>, since <math alttext="E_{l}\cup E_{b}\subseteq E" class="ltx_Math" display="inline" id="S4.19.p4.4.m4.1"><semantics id="S4.19.p4.4.m4.1a"><mrow id="S4.19.p4.4.m4.1.1" xref="S4.19.p4.4.m4.1.1.cmml"><mrow id="S4.19.p4.4.m4.1.1.2" xref="S4.19.p4.4.m4.1.1.2.cmml"><msub id="S4.19.p4.4.m4.1.1.2.2" xref="S4.19.p4.4.m4.1.1.2.2.cmml"><mi id="S4.19.p4.4.m4.1.1.2.2.2" xref="S4.19.p4.4.m4.1.1.2.2.2.cmml">E</mi><mi id="S4.19.p4.4.m4.1.1.2.2.3" xref="S4.19.p4.4.m4.1.1.2.2.3.cmml">l</mi></msub><mo id="S4.19.p4.4.m4.1.1.2.1" xref="S4.19.p4.4.m4.1.1.2.1.cmml">∪</mo><msub id="S4.19.p4.4.m4.1.1.2.3" xref="S4.19.p4.4.m4.1.1.2.3.cmml"><mi id="S4.19.p4.4.m4.1.1.2.3.2" xref="S4.19.p4.4.m4.1.1.2.3.2.cmml">E</mi><mi id="S4.19.p4.4.m4.1.1.2.3.3" xref="S4.19.p4.4.m4.1.1.2.3.3.cmml">b</mi></msub></mrow><mo id="S4.19.p4.4.m4.1.1.1" xref="S4.19.p4.4.m4.1.1.1.cmml">⊆</mo><mi id="S4.19.p4.4.m4.1.1.3" xref="S4.19.p4.4.m4.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.19.p4.4.m4.1b"><apply id="S4.19.p4.4.m4.1.1.cmml" xref="S4.19.p4.4.m4.1.1"><subset id="S4.19.p4.4.m4.1.1.1.cmml" xref="S4.19.p4.4.m4.1.1.1"></subset><apply id="S4.19.p4.4.m4.1.1.2.cmml" xref="S4.19.p4.4.m4.1.1.2"><union id="S4.19.p4.4.m4.1.1.2.1.cmml" xref="S4.19.p4.4.m4.1.1.2.1"></union><apply id="S4.19.p4.4.m4.1.1.2.2.cmml" xref="S4.19.p4.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S4.19.p4.4.m4.1.1.2.2.1.cmml" xref="S4.19.p4.4.m4.1.1.2.2">subscript</csymbol><ci id="S4.19.p4.4.m4.1.1.2.2.2.cmml" xref="S4.19.p4.4.m4.1.1.2.2.2">𝐸</ci><ci id="S4.19.p4.4.m4.1.1.2.2.3.cmml" xref="S4.19.p4.4.m4.1.1.2.2.3">𝑙</ci></apply><apply id="S4.19.p4.4.m4.1.1.2.3.cmml" xref="S4.19.p4.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S4.19.p4.4.m4.1.1.2.3.1.cmml" xref="S4.19.p4.4.m4.1.1.2.3">subscript</csymbol><ci id="S4.19.p4.4.m4.1.1.2.3.2.cmml" xref="S4.19.p4.4.m4.1.1.2.3.2">𝐸</ci><ci id="S4.19.p4.4.m4.1.1.2.3.3.cmml" xref="S4.19.p4.4.m4.1.1.2.3.3">𝑏</ci></apply></apply><ci id="S4.19.p4.4.m4.1.1.3.cmml" xref="S4.19.p4.4.m4.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.19.p4.4.m4.1c">E_{l}\cup E_{b}\subseteq E</annotation><annotation encoding="application/x-llamapun" id="S4.19.p4.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ∪ italic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊆ italic_E</annotation></semantics></math> is a disjoint union. Moreover,</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex56"> <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_{v\in V}d_{v}\leq\sum_{v\in V}\deg(v)\leq 2\absolutevalue{E}\leq 6p," class="ltx_Math" display="block" id="S4.Ex56.m1.4"><semantics id="S4.Ex56.m1.4a"><mrow id="S4.Ex56.m1.4.4.1" xref="S4.Ex56.m1.4.4.1.1.cmml"><mrow id="S4.Ex56.m1.4.4.1.1" xref="S4.Ex56.m1.4.4.1.1.cmml"><mrow id="S4.Ex56.m1.4.4.1.1.2" xref="S4.Ex56.m1.4.4.1.1.2.cmml"><munder id="S4.Ex56.m1.4.4.1.1.2.1" xref="S4.Ex56.m1.4.4.1.1.2.1.cmml"><mo id="S4.Ex56.m1.4.4.1.1.2.1.2" movablelimits="false" xref="S4.Ex56.m1.4.4.1.1.2.1.2.cmml">∑</mo><mrow id="S4.Ex56.m1.4.4.1.1.2.1.3" xref="S4.Ex56.m1.4.4.1.1.2.1.3.cmml"><mi id="S4.Ex56.m1.4.4.1.1.2.1.3.2" xref="S4.Ex56.m1.4.4.1.1.2.1.3.2.cmml">v</mi><mo id="S4.Ex56.m1.4.4.1.1.2.1.3.1" xref="S4.Ex56.m1.4.4.1.1.2.1.3.1.cmml">∈</mo><mi id="S4.Ex56.m1.4.4.1.1.2.1.3.3" xref="S4.Ex56.m1.4.4.1.1.2.1.3.3.cmml">V</mi></mrow></munder><msub id="S4.Ex56.m1.4.4.1.1.2.2" xref="S4.Ex56.m1.4.4.1.1.2.2.cmml"><mi id="S4.Ex56.m1.4.4.1.1.2.2.2" xref="S4.Ex56.m1.4.4.1.1.2.2.2.cmml">d</mi><mi id="S4.Ex56.m1.4.4.1.1.2.2.3" xref="S4.Ex56.m1.4.4.1.1.2.2.3.cmml">v</mi></msub></mrow><mo id="S4.Ex56.m1.4.4.1.1.3" rspace="0.111em" xref="S4.Ex56.m1.4.4.1.1.3.cmml">≤</mo><mrow id="S4.Ex56.m1.4.4.1.1.4" xref="S4.Ex56.m1.4.4.1.1.4.cmml"><munder id="S4.Ex56.m1.4.4.1.1.4.1" xref="S4.Ex56.m1.4.4.1.1.4.1.cmml"><mo id="S4.Ex56.m1.4.4.1.1.4.1.2" movablelimits="false" xref="S4.Ex56.m1.4.4.1.1.4.1.2.cmml">∑</mo><mrow id="S4.Ex56.m1.4.4.1.1.4.1.3" xref="S4.Ex56.m1.4.4.1.1.4.1.3.cmml"><mi id="S4.Ex56.m1.4.4.1.1.4.1.3.2" xref="S4.Ex56.m1.4.4.1.1.4.1.3.2.cmml">v</mi><mo id="S4.Ex56.m1.4.4.1.1.4.1.3.1" xref="S4.Ex56.m1.4.4.1.1.4.1.3.1.cmml">∈</mo><mi id="S4.Ex56.m1.4.4.1.1.4.1.3.3" xref="S4.Ex56.m1.4.4.1.1.4.1.3.3.cmml">V</mi></mrow></munder><mrow id="S4.Ex56.m1.4.4.1.1.4.2.2" xref="S4.Ex56.m1.4.4.1.1.4.2.1.cmml"><mi id="S4.Ex56.m1.2.2" xref="S4.Ex56.m1.2.2.cmml">deg</mi><mo id="S4.Ex56.m1.4.4.1.1.4.2.2a" xref="S4.Ex56.m1.4.4.1.1.4.2.1.cmml">⁡</mo><mrow id="S4.Ex56.m1.4.4.1.1.4.2.2.1" xref="S4.Ex56.m1.4.4.1.1.4.2.1.cmml"><mo id="S4.Ex56.m1.4.4.1.1.4.2.2.1.1" stretchy="false" xref="S4.Ex56.m1.4.4.1.1.4.2.1.cmml">(</mo><mi id="S4.Ex56.m1.3.3" xref="S4.Ex56.m1.3.3.cmml">v</mi><mo id="S4.Ex56.m1.4.4.1.1.4.2.2.1.2" stretchy="false" xref="S4.Ex56.m1.4.4.1.1.4.2.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex56.m1.4.4.1.1.5" xref="S4.Ex56.m1.4.4.1.1.5.cmml">≤</mo><mrow id="S4.Ex56.m1.4.4.1.1.6" xref="S4.Ex56.m1.4.4.1.1.6.cmml"><mn id="S4.Ex56.m1.4.4.1.1.6.2" xref="S4.Ex56.m1.4.4.1.1.6.2.cmml">2</mn><mo id="S4.Ex56.m1.4.4.1.1.6.1" xref="S4.Ex56.m1.4.4.1.1.6.1.cmml">⁢</mo><mrow id="S4.Ex56.m1.1.1.3" xref="S4.Ex56.m1.1.1.2.cmml"><mo id="S4.Ex56.m1.1.1.3.1" xref="S4.Ex56.m1.1.1.2.1.cmml">|</mo><mi id="S4.Ex56.m1.1.1.1.1.1" xref="S4.Ex56.m1.1.1.1.1.1.cmml">E</mi><mo id="S4.Ex56.m1.1.1.3.2" xref="S4.Ex56.m1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex56.m1.4.4.1.1.7" xref="S4.Ex56.m1.4.4.1.1.7.cmml">≤</mo><mrow id="S4.Ex56.m1.4.4.1.1.8" xref="S4.Ex56.m1.4.4.1.1.8.cmml"><mn id="S4.Ex56.m1.4.4.1.1.8.2" xref="S4.Ex56.m1.4.4.1.1.8.2.cmml">6</mn><mo id="S4.Ex56.m1.4.4.1.1.8.1" xref="S4.Ex56.m1.4.4.1.1.8.1.cmml">⁢</mo><mi id="S4.Ex56.m1.4.4.1.1.8.3" xref="S4.Ex56.m1.4.4.1.1.8.3.cmml">p</mi></mrow></mrow><mo id="S4.Ex56.m1.4.4.1.2" xref="S4.Ex56.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex56.m1.4b"><apply id="S4.Ex56.m1.4.4.1.1.cmml" xref="S4.Ex56.m1.4.4.1"><and id="S4.Ex56.m1.4.4.1.1a.cmml" xref="S4.Ex56.m1.4.4.1"></and><apply id="S4.Ex56.m1.4.4.1.1b.cmml" xref="S4.Ex56.m1.4.4.1"><leq id="S4.Ex56.m1.4.4.1.1.3.cmml" xref="S4.Ex56.m1.4.4.1.1.3"></leq><apply id="S4.Ex56.m1.4.4.1.1.2.cmml" xref="S4.Ex56.m1.4.4.1.1.2"><apply id="S4.Ex56.m1.4.4.1.1.2.1.cmml" 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id="S4.Ex56.m1.4.4.1.1.8.2.cmml" type="integer" xref="S4.Ex56.m1.4.4.1.1.8.2">6</cn><ci id="S4.Ex56.m1.4.4.1.1.8.3.cmml" xref="S4.Ex56.m1.4.4.1.1.8.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex56.m1.4c">\sum_{v\in V}d_{v}\leq\sum_{v\in V}\deg(v)\leq 2\absolutevalue{E}\leq 6p,</annotation><annotation encoding="application/x-llamapun" id="S4.Ex56.m1.4d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≤ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT roman_deg ( italic_v ) ≤ 2 | start_ARG italic_E end_ARG | ≤ 6 italic_p ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.19.p4.5">which completes the proof. ∎</p> </div> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Neural Network Representation of CPA Functions</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.2">The following theorem translates the <math alttext="\max" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">max</mi><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><max id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"></max></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\max</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">roman_max</annotation></semantics></math>-representation of <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mi id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><ci id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">roman_CPA</annotation></semantics></math> functions, provided in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 4.5</span></a>, into a neural network representation.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.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="S5.Thmtheorem1.1.1.1">Theorem 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.5.5">Any function <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem1.p1.1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S5.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.1.m1.1b"><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1"><in id="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.1"></in><ci 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id="S5.Thmtheorem1.p1.5.5.m5.3.3.3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.3.1.1.2">45</cn><ci id="S5.Thmtheorem1.p1.5.5.m5.3.3.3.1.1.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.3.3.3.1.1.3">𝑝</ci></apply><apply id="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2"><times id="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.1.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.1"></times><cn id="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.2">27</cn><ci id="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.3.cmml" xref="S5.Thmtheorem1.p1.5.5.m5.4.4.4.2.2.3">𝑝</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.5.m5.4c">(s_{1},s_{2})\leq(45p,27p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.5.m5.4d">( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≤ ( 45 italic_p , 27 italic_p )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">The proof relies on the following definition, which can be interpreted as the direct sum of affine functions.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.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="S5.Thmtheorem2.1.1.1">Definition 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.5">For <math alttext="w\in\mathds{N}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.m1.1a"><mrow id="S5.Thmtheorem2.p1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.1.m1.1.1.2" xref="S5.Thmtheorem2.p1.1.m1.1.1.2.cmml">w</mi><mo id="S5.Thmtheorem2.p1.1.m1.1.1.1" xref="S5.Thmtheorem2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S5.Thmtheorem2.p1.1.m1.1.1.3" xref="S5.Thmtheorem2.p1.1.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.m1.1b"><apply id="S5.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1"><in id="S5.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1.1"></in><ci id="S5.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1.2">𝑤</ci><ci id="S5.Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.m1.1c">w\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.m1.1d">italic_w ∈ blackboard_N</annotation></semantics></math> and <math alttext="i\in[w]" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.m2.1a"><mrow id="S5.Thmtheorem2.p1.2.m2.1.2" xref="S5.Thmtheorem2.p1.2.m2.1.2.cmml"><mi id="S5.Thmtheorem2.p1.2.m2.1.2.2" xref="S5.Thmtheorem2.p1.2.m2.1.2.2.cmml">i</mi><mo id="S5.Thmtheorem2.p1.2.m2.1.2.1" xref="S5.Thmtheorem2.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S5.Thmtheorem2.p1.2.m2.1.2.3.2" xref="S5.Thmtheorem2.p1.2.m2.1.2.3.1.cmml"><mo id="S5.Thmtheorem2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S5.Thmtheorem2.p1.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.cmml">w</mi><mo id="S5.Thmtheorem2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem2.p1.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.m2.1.2.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.2"><in id="S5.Thmtheorem2.p1.2.m2.1.2.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.2.1"></in><ci id="S5.Thmtheorem2.p1.2.m2.1.2.2.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.2.2">𝑖</ci><apply id="S5.Thmtheorem2.p1.2.m2.1.2.3.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.2.3.2"><csymbol cd="latexml" id="S5.Thmtheorem2.p1.2.m2.1.2.3.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.m2.1c">i\in[w]</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.m2.1d">italic_i ∈ [ italic_w ]</annotation></semantics></math>, let <math alttext="T_{i}:\mathds{R}^{n_{i}}\to\mathds{R}^{m_{i}}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.m3.1a"><mrow id="S5.Thmtheorem2.p1.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.cmml"><msub id="S5.Thmtheorem2.p1.3.m3.1.1.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.2.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.2.cmml">T</mi><mi id="S5.Thmtheorem2.p1.3.m3.1.1.2.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem2.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S5.Thmtheorem2.p1.3.m3.1.1.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml"><msup id="S5.Thmtheorem2.p1.3.m3.1.1.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.2.cmml">ℝ</mi><msub id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.2.cmml">n</mi><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.3.cmml">i</mi></msub></msup><mo id="S5.Thmtheorem2.p1.3.m3.1.1.3.1" stretchy="false" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.1.cmml">→</mo><msup id="S5.Thmtheorem2.p1.3.m3.1.1.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.2.cmml">ℝ</mi><msub id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.2.cmml">m</mi><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.3.cmml">i</mi></msub></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.m3.1b"><apply id="S5.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1"><ci id="S5.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1">:</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.2.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.2">𝑇</ci><ci id="S5.Thmtheorem2.p1.3.m3.1.1.2.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3"><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.1">→</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.2">ℝ</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.2">𝑛</ci><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.2.3.3">𝑖</ci></apply></apply><apply id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3">superscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.2">ℝ</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.2">𝑚</ci><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.m3.1c">T_{i}:\mathds{R}^{n_{i}}\to\mathds{R}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be affine functions. With <math alttext="n:=\sum_{i\in[w]}n_{i}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.4.m4.1"><semantics id="S5.Thmtheorem2.p1.4.m4.1a"><mrow id="S5.Thmtheorem2.p1.4.m4.1.2" xref="S5.Thmtheorem2.p1.4.m4.1.2.cmml"><mi id="S5.Thmtheorem2.p1.4.m4.1.2.2" xref="S5.Thmtheorem2.p1.4.m4.1.2.2.cmml">n</mi><mo id="S5.Thmtheorem2.p1.4.m4.1.2.1" lspace="0.278em" rspace="0.111em" xref="S5.Thmtheorem2.p1.4.m4.1.2.1.cmml">:=</mo><mrow id="S5.Thmtheorem2.p1.4.m4.1.2.3" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.cmml"><msub id="S5.Thmtheorem2.p1.4.m4.1.2.3.1" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.1.cmml"><mo id="S5.Thmtheorem2.p1.4.m4.1.2.3.1.2" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.1.2.cmml">∑</mo><mrow id="S5.Thmtheorem2.p1.4.m4.1.1.1" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.4.m4.1.1.1.3" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.3.cmml">i</mi><mo id="S5.Thmtheorem2.p1.4.m4.1.1.1.2" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem2.p1.4.m4.1.1.1.4.2" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.4.1.cmml"><mo id="S5.Thmtheorem2.p1.4.m4.1.1.1.4.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.4.1.1.cmml">[</mo><mi id="S5.Thmtheorem2.p1.4.m4.1.1.1.1" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.1.cmml">w</mi><mo id="S5.Thmtheorem2.p1.4.m4.1.1.1.4.2.2" stretchy="false" xref="S5.Thmtheorem2.p1.4.m4.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msub id="S5.Thmtheorem2.p1.4.m4.1.2.3.2" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2.cmml"><mi id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.2" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2.2.cmml">n</mi><mi id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.3" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.4.m4.1b"><apply id="S5.Thmtheorem2.p1.4.m4.1.2.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2"><csymbol cd="latexml" id="S5.Thmtheorem2.p1.4.m4.1.2.1.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2.1">assign</csymbol><ci id="S5.Thmtheorem2.p1.4.m4.1.2.2.cmml" 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xref="S5.Thmtheorem2.p1.4.m4.1.1.1.1">𝑤</ci></apply></apply></apply><apply id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.1.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.2.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2.2">𝑛</ci><ci id="S5.Thmtheorem2.p1.4.m4.1.2.3.2.3.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.2.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.4.m4.1c">n:=\sum_{i\in[w]}n_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.4.m4.1d">italic_n := ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_w ] end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="m:=\sum_{i\in[w]}m_{i}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.5.m5.1"><semantics 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xref="S5.Thmtheorem2.p1.5.m5.1.2.3"><apply id="S5.Thmtheorem2.p1.5.m5.1.2.3.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.2.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.5.m5.1.2.3.1.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.2.3.1">subscript</csymbol><sum id="S5.Thmtheorem2.p1.5.m5.1.2.3.1.2.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.2.3.1.2"></sum><apply id="S5.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1"><in id="S5.Thmtheorem2.p1.5.m5.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1.2"></in><ci id="S5.Thmtheorem2.p1.5.m5.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1.3">𝑖</ci><apply id="S5.Thmtheorem2.p1.5.m5.1.1.1.4.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1.4.2"><csymbol cd="latexml" id="S5.Thmtheorem2.p1.5.m5.1.1.1.4.1.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem2.p1.5.m5.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1.1.1">𝑤</ci></apply></apply></apply><apply id="S5.Thmtheorem2.p1.5.m5.1.2.3.2.cmml" 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alttext="T:=\begin{bmatrix}T_{1}\\ \vdots\\ T_{w}\end{bmatrix}:\mathds{R}^{n}\to\mathds{R}^{m},\quad T(x_{1},\dots,x_{w}):% =\begin{pmatrix}T_{1}(x_{1})\\ \vdots\\ T_{w}(x_{w})\end{pmatrix}\text{ for }x_{i}\in\mathds{R}^{n_{i}}." class="ltx_Math" display="block" id="S5.Ex57.m1.4"><semantics id="S5.Ex57.m1.4a"><mrow id="S5.Ex57.m1.4.4.1" xref="S5.Ex57.m1.4.4.1.1.cmml"><mrow id="S5.Ex57.m1.4.4.1.1" xref="S5.Ex57.m1.4.4.1.1.cmml"><mrow id="S5.Ex57.m1.4.4.1.1.4" xref="S5.Ex57.m1.4.4.1.1.4.cmml"><mi id="S5.Ex57.m1.4.4.1.1.4.2" xref="S5.Ex57.m1.4.4.1.1.4.2.cmml">T</mi><mo id="S5.Ex57.m1.4.4.1.1.4.1" lspace="0.278em" rspace="0.278em" xref="S5.Ex57.m1.4.4.1.1.4.1.cmml">:=</mo><mrow id="S5.Ex57.m1.1.1.3" xref="S5.Ex57.m1.1.1.2.cmml"><mo id="S5.Ex57.m1.1.1.3.1" xref="S5.Ex57.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.Ex57.m1.1.1.1.1" rowspacing="0pt" xref="S5.Ex57.m1.1.1.1.1.cmml"><mtr id="S5.Ex57.m1.1.1.1.1a" xref="S5.Ex57.m1.1.1.1.1.cmml"><mtd id="S5.Ex57.m1.1.1.1.1b" xref="S5.Ex57.m1.1.1.1.1.cmml"><msub id="S5.Ex57.m1.1.1.1.1.1.1.1" xref="S5.Ex57.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex57.m1.1.1.1.1.1.1.1.2" xref="S5.Ex57.m1.1.1.1.1.1.1.1.2.cmml">T</mi><mn id="S5.Ex57.m1.1.1.1.1.1.1.1.3" xref="S5.Ex57.m1.1.1.1.1.1.1.1.3.cmml">1</mn></msub></mtd></mtr><mtr id="S5.Ex57.m1.1.1.1.1c" xref="S5.Ex57.m1.1.1.1.1.cmml"><mtd id="S5.Ex57.m1.1.1.1.1d" xref="S5.Ex57.m1.1.1.1.1.cmml"><mi id="S5.Ex57.m1.1.1.1.1.2.1.1" mathvariant="normal" xref="S5.Ex57.m1.1.1.1.1.2.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S5.Ex57.m1.1.1.1.1e" xref="S5.Ex57.m1.1.1.1.1.cmml"><mtd id="S5.Ex57.m1.1.1.1.1f" xref="S5.Ex57.m1.1.1.1.1.cmml"><msub id="S5.Ex57.m1.1.1.1.1.3.1.1" xref="S5.Ex57.m1.1.1.1.1.3.1.1.cmml"><mi id="S5.Ex57.m1.1.1.1.1.3.1.1.2" xref="S5.Ex57.m1.1.1.1.1.3.1.1.2.cmml">T</mi><mi id="S5.Ex57.m1.1.1.1.1.3.1.1.3" xref="S5.Ex57.m1.1.1.1.1.3.1.1.3.cmml">w</mi></msub></mtd></mtr></mtable><mo id="S5.Ex57.m1.1.1.3.2" rspace="0.278em" 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}x_{i}\in\mathds{R}^{n_{i}}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex57.m1.4d">italic_T := [ start_ARG start_ROW start_CELL italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_T start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] : blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_T ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) := ( start_ARG start_ROW start_CELL italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_T start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ) for italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem2.p1.7">The function <math alttext="T" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.6.m1.1"><semantics id="S5.Thmtheorem2.p1.6.m1.1a"><mi id="S5.Thmtheorem2.p1.6.m1.1.1" xref="S5.Thmtheorem2.p1.6.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.6.m1.1b"><ci id="S5.Thmtheorem2.p1.6.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.6.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.6.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.6.m1.1d">italic_T</annotation></semantics></math> is referred to as the affine function that <em class="ltx_emph ltx_font_italic" 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id="S5.Thmtheorem2.p1.7.m2.3.3.2.2.3" xref="S5.Thmtheorem2.p1.7.m2.3.3.2.2.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.7.m2.3b"><list id="S5.Thmtheorem2.p1.7.m2.3.3.3.cmml" xref="S5.Thmtheorem2.p1.7.m2.3.3.2"><apply id="S5.Thmtheorem2.p1.7.m2.2.2.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.7.m2.2.2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m2.2.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.7.m2.2.2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m2.2.2.1.1.2">𝑇</ci><cn id="S5.Thmtheorem2.p1.7.m2.2.2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.7.m2.2.2.1.1.3">1</cn></apply><ci id="S5.Thmtheorem2.p1.7.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m2.1.1">…</ci><apply id="S5.Thmtheorem2.p1.7.m2.3.3.2.2.cmml" xref="S5.Thmtheorem2.p1.7.m2.3.3.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.7.m2.3.3.2.2.1.cmml" xref="S5.Thmtheorem2.p1.7.m2.3.3.2.2">subscript</csymbol><ci id="S5.Thmtheorem2.p1.7.m2.3.3.2.2.2.cmml" xref="S5.Thmtheorem2.p1.7.m2.3.3.2.2.2">𝑇</ci><ci id="S5.Thmtheorem2.p1.7.m2.3.3.2.2.3.cmml" xref="S5.Thmtheorem2.p1.7.m2.3.3.2.2.3">𝑤</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.7.m2.3c">T_{1},\dots,T_{w}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.7.m2.3d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S5.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 5.1</span></a>.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.1">Using <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 4.5</span></a>, <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.1b"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">italic_f</annotation></semantics></math> can be expressed as</p> <table class="ltx_equation ltx_eqn_table" id="S5.E40"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.1.1.1.1.2.3">𝑛</ci></apply><cn id="S5.E40.m1.4.4.1.1.cmml" type="integer" xref="S5.E40.m1.4.4.1.1">2</cn></apply><apply id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.1.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2">superscript</csymbol><apply id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.1.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.2.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.2">𝑓</ci><ci id="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.3.cmml" xref="S5.E40.m1.9.9.1.1.2.2.2.2.2.2.2.2.2.2.2.3">𝑛</ci></apply><cn id="S5.E40.m1.5.5.1.1.cmml" type="integer" xref="S5.E40.m1.5.5.1.1">3</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E40.m1.9c">f(x)=\sum_{n=1}^{9p}\sigma_{n}^{(1)}\max(f_{n}^{(1)},\sigma_{n}^{(2)}\max(f_{n% }^{(2)},f_{n}^{(3)})),</annotation><annotation encoding="application/x-llamapun" id="S5.E40.m1.9d">italic_f ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 9 italic_p end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) 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">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.4">where <math alttext="f_{n}^{(k)}" class="ltx_Math" display="inline" id="S5.1.p1.2.m1.1"><semantics id="S5.1.p1.2.m1.1a"><msubsup id="S5.1.p1.2.m1.1.2" xref="S5.1.p1.2.m1.1.2.cmml"><mi id="S5.1.p1.2.m1.1.2.2.2" xref="S5.1.p1.2.m1.1.2.2.2.cmml">f</mi><mi id="S5.1.p1.2.m1.1.2.2.3" xref="S5.1.p1.2.m1.1.2.2.3.cmml">n</mi><mrow id="S5.1.p1.2.m1.1.1.1.3" xref="S5.1.p1.2.m1.1.2.cmml"><mo id="S5.1.p1.2.m1.1.1.1.3.1" stretchy="false" xref="S5.1.p1.2.m1.1.2.cmml">(</mo><mi id="S5.1.p1.2.m1.1.1.1.1" xref="S5.1.p1.2.m1.1.1.1.1.cmml">k</mi><mo id="S5.1.p1.2.m1.1.1.1.3.2" stretchy="false" xref="S5.1.p1.2.m1.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m1.1b"><apply id="S5.1.p1.2.m1.1.2.cmml" xref="S5.1.p1.2.m1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.2.m1.1.2.1.cmml" xref="S5.1.p1.2.m1.1.2">superscript</csymbol><apply id="S5.1.p1.2.m1.1.2.2.cmml" xref="S5.1.p1.2.m1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.2.m1.1.2.2.1.cmml" xref="S5.1.p1.2.m1.1.2">subscript</csymbol><ci id="S5.1.p1.2.m1.1.2.2.2.cmml" xref="S5.1.p1.2.m1.1.2.2.2">𝑓</ci><ci id="S5.1.p1.2.m1.1.2.2.3.cmml" xref="S5.1.p1.2.m1.1.2.2.3">𝑛</ci></apply><ci id="S5.1.p1.2.m1.1.1.1.1.cmml" xref="S5.1.p1.2.m1.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m1.1c">f_{n}^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m1.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> are affine functions and <math alttext="\sigma_{n}^{(k)}\in\{-1,1\}" class="ltx_Math" display="inline" id="S5.1.p1.3.m2.3"><semantics id="S5.1.p1.3.m2.3a"><mrow id="S5.1.p1.3.m2.3.3" xref="S5.1.p1.3.m2.3.3.cmml"><msubsup id="S5.1.p1.3.m2.3.3.3" xref="S5.1.p1.3.m2.3.3.3.cmml"><mi id="S5.1.p1.3.m2.3.3.3.2.2" xref="S5.1.p1.3.m2.3.3.3.2.2.cmml">σ</mi><mi id="S5.1.p1.3.m2.3.3.3.2.3" xref="S5.1.p1.3.m2.3.3.3.2.3.cmml">n</mi><mrow id="S5.1.p1.3.m2.1.1.1.3" xref="S5.1.p1.3.m2.3.3.3.cmml"><mo id="S5.1.p1.3.m2.1.1.1.3.1" stretchy="false" xref="S5.1.p1.3.m2.3.3.3.cmml">(</mo><mi id="S5.1.p1.3.m2.1.1.1.1" xref="S5.1.p1.3.m2.1.1.1.1.cmml">k</mi><mo id="S5.1.p1.3.m2.1.1.1.3.2" stretchy="false" xref="S5.1.p1.3.m2.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S5.1.p1.3.m2.3.3.2" xref="S5.1.p1.3.m2.3.3.2.cmml">∈</mo><mrow id="S5.1.p1.3.m2.3.3.1.1" xref="S5.1.p1.3.m2.3.3.1.2.cmml"><mo id="S5.1.p1.3.m2.3.3.1.1.2" stretchy="false" xref="S5.1.p1.3.m2.3.3.1.2.cmml">{</mo><mrow id="S5.1.p1.3.m2.3.3.1.1.1" xref="S5.1.p1.3.m2.3.3.1.1.1.cmml"><mo id="S5.1.p1.3.m2.3.3.1.1.1a" xref="S5.1.p1.3.m2.3.3.1.1.1.cmml">−</mo><mn id="S5.1.p1.3.m2.3.3.1.1.1.2" xref="S5.1.p1.3.m2.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S5.1.p1.3.m2.3.3.1.1.3" xref="S5.1.p1.3.m2.3.3.1.2.cmml">,</mo><mn id="S5.1.p1.3.m2.2.2" xref="S5.1.p1.3.m2.2.2.cmml">1</mn><mo id="S5.1.p1.3.m2.3.3.1.1.4" stretchy="false" xref="S5.1.p1.3.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m2.3b"><apply id="S5.1.p1.3.m2.3.3.cmml" xref="S5.1.p1.3.m2.3.3"><in id="S5.1.p1.3.m2.3.3.2.cmml" xref="S5.1.p1.3.m2.3.3.2"></in><apply id="S5.1.p1.3.m2.3.3.3.cmml" xref="S5.1.p1.3.m2.3.3.3"><csymbol cd="ambiguous" id="S5.1.p1.3.m2.3.3.3.1.cmml" xref="S5.1.p1.3.m2.3.3.3">superscript</csymbol><apply id="S5.1.p1.3.m2.3.3.3.2.cmml" xref="S5.1.p1.3.m2.3.3.3"><csymbol cd="ambiguous" id="S5.1.p1.3.m2.3.3.3.2.1.cmml" xref="S5.1.p1.3.m2.3.3.3">subscript</csymbol><ci id="S5.1.p1.3.m2.3.3.3.2.2.cmml" xref="S5.1.p1.3.m2.3.3.3.2.2">𝜎</ci><ci id="S5.1.p1.3.m2.3.3.3.2.3.cmml" xref="S5.1.p1.3.m2.3.3.3.2.3">𝑛</ci></apply><ci id="S5.1.p1.3.m2.1.1.1.1.cmml" xref="S5.1.p1.3.m2.1.1.1.1">𝑘</ci></apply><set id="S5.1.p1.3.m2.3.3.1.2.cmml" xref="S5.1.p1.3.m2.3.3.1.1"><apply id="S5.1.p1.3.m2.3.3.1.1.1.cmml" xref="S5.1.p1.3.m2.3.3.1.1.1"><minus id="S5.1.p1.3.m2.3.3.1.1.1.1.cmml" xref="S5.1.p1.3.m2.3.3.1.1.1"></minus><cn id="S5.1.p1.3.m2.3.3.1.1.1.2.cmml" type="integer" xref="S5.1.p1.3.m2.3.3.1.1.1.2">1</cn></apply><cn id="S5.1.p1.3.m2.2.2.cmml" type="integer" xref="S5.1.p1.3.m2.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m2.3c">\sigma_{n}^{(k)}\in\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m2.3d">italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ { - 1 , 1 }</annotation></semantics></math>. To obtain a neural network representation of <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.4.m3.1"><semantics id="S5.1.p1.4.m3.1a"><mi id="S5.1.p1.4.m3.1.1" xref="S5.1.p1.4.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m3.1b"><ci id="S5.1.p1.4.m3.1.1.cmml" xref="S5.1.p1.4.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m3.1d">italic_f</annotation></semantics></math>, we first construct individual neural networks for each summand in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S5.E40" title="Equation 40 ‣ Proof of Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">40</span></a>) and then stack the affine functions corresponding to the same layer.</p> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.4">Let <math alttext="f_{1},f_{2},f_{3}" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.3"><semantics id="S5.2.p2.1.m1.3a"><mrow id="S5.2.p2.1.m1.3.3.3" xref="S5.2.p2.1.m1.3.3.4.cmml"><msub id="S5.2.p2.1.m1.1.1.1.1" xref="S5.2.p2.1.m1.1.1.1.1.cmml"><mi id="S5.2.p2.1.m1.1.1.1.1.2" xref="S5.2.p2.1.m1.1.1.1.1.2.cmml">f</mi><mn id="S5.2.p2.1.m1.1.1.1.1.3" xref="S5.2.p2.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.2.p2.1.m1.3.3.3.4" xref="S5.2.p2.1.m1.3.3.4.cmml">,</mo><msub id="S5.2.p2.1.m1.2.2.2.2" xref="S5.2.p2.1.m1.2.2.2.2.cmml"><mi id="S5.2.p2.1.m1.2.2.2.2.2" xref="S5.2.p2.1.m1.2.2.2.2.2.cmml">f</mi><mn id="S5.2.p2.1.m1.2.2.2.2.3" xref="S5.2.p2.1.m1.2.2.2.2.3.cmml">2</mn></msub><mo id="S5.2.p2.1.m1.3.3.3.5" xref="S5.2.p2.1.m1.3.3.4.cmml">,</mo><msub id="S5.2.p2.1.m1.3.3.3.3" xref="S5.2.p2.1.m1.3.3.3.3.cmml"><mi id="S5.2.p2.1.m1.3.3.3.3.2" xref="S5.2.p2.1.m1.3.3.3.3.2.cmml">f</mi><mn id="S5.2.p2.1.m1.3.3.3.3.3" xref="S5.2.p2.1.m1.3.3.3.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.3b"><list id="S5.2.p2.1.m1.3.3.4.cmml" xref="S5.2.p2.1.m1.3.3.3"><apply id="S5.2.p2.1.m1.1.1.1.1.cmml" xref="S5.2.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.1.1.1.1.1.cmml" xref="S5.2.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S5.2.p2.1.m1.1.1.1.1.2.cmml" xref="S5.2.p2.1.m1.1.1.1.1.2">𝑓</ci><cn id="S5.2.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="S5.2.p2.1.m1.1.1.1.1.3">1</cn></apply><apply id="S5.2.p2.1.m1.2.2.2.2.cmml" xref="S5.2.p2.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.2.2.2.2.1.cmml" xref="S5.2.p2.1.m1.2.2.2.2">subscript</csymbol><ci id="S5.2.p2.1.m1.2.2.2.2.2.cmml" xref="S5.2.p2.1.m1.2.2.2.2.2">𝑓</ci><cn id="S5.2.p2.1.m1.2.2.2.2.3.cmml" type="integer" xref="S5.2.p2.1.m1.2.2.2.2.3">2</cn></apply><apply id="S5.2.p2.1.m1.3.3.3.3.cmml" xref="S5.2.p2.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.3.3.3.3.1.cmml" xref="S5.2.p2.1.m1.3.3.3.3">subscript</csymbol><ci id="S5.2.p2.1.m1.3.3.3.3.2.cmml" xref="S5.2.p2.1.m1.3.3.3.3.2">𝑓</ci><cn id="S5.2.p2.1.m1.3.3.3.3.3.cmml" type="integer" xref="S5.2.p2.1.m1.3.3.3.3.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.3c">f_{1},f_{2},f_{3}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.3d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> be affine functions. Using the <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.1"><semantics id="S5.2.p2.2.m2.1a"><mi id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.1b"><ci id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.1d">roman_ReLU</annotation></semantics></math> activation function <math alttext="\rho" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.1"><semantics id="S5.2.p2.3.m3.1a"><mi id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.1b"><ci id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.1d">italic_ρ</annotation></semantics></math>, one can ’skip’ the affine function <math alttext="f_{1}" class="ltx_Math" display="inline" id="S5.2.p2.4.m4.1"><semantics id="S5.2.p2.4.m4.1a"><msub id="S5.2.p2.4.m4.1.1" xref="S5.2.p2.4.m4.1.1.cmml"><mi id="S5.2.p2.4.m4.1.1.2" xref="S5.2.p2.4.m4.1.1.2.cmml">f</mi><mn id="S5.2.p2.4.m4.1.1.3" xref="S5.2.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m4.1b"><apply id="S5.2.p2.4.m4.1.1.cmml" xref="S5.2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.2.p2.4.m4.1.1.1.cmml" xref="S5.2.p2.4.m4.1.1">subscript</csymbol><ci id="S5.2.p2.4.m4.1.1.2.cmml" xref="S5.2.p2.4.m4.1.1.2">𝑓</ci><cn id="S5.2.p2.4.m4.1.1.3.cmml" type="integer" xref="S5.2.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m4.1c">f_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m4.1d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> through one layer using</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex58"> <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_{1}=\max(0,f_{1})-\max(0,-f_{1})=\rho(f_{1})-\rho(-f_{1})." class="ltx_Math" display="block" id="S5.Ex58.m1.5"><semantics id="S5.Ex58.m1.5a"><mrow id="S5.Ex58.m1.5.5.1" xref="S5.Ex58.m1.5.5.1.1.cmml"><mrow id="S5.Ex58.m1.5.5.1.1" xref="S5.Ex58.m1.5.5.1.1.cmml"><msub id="S5.Ex58.m1.5.5.1.1.6" xref="S5.Ex58.m1.5.5.1.1.6.cmml"><mi id="S5.Ex58.m1.5.5.1.1.6.2" xref="S5.Ex58.m1.5.5.1.1.6.2.cmml">f</mi><mn id="S5.Ex58.m1.5.5.1.1.6.3" xref="S5.Ex58.m1.5.5.1.1.6.3.cmml">1</mn></msub><mo id="S5.Ex58.m1.5.5.1.1.7" xref="S5.Ex58.m1.5.5.1.1.7.cmml">=</mo><mrow id="S5.Ex58.m1.5.5.1.1.2" xref="S5.Ex58.m1.5.5.1.1.2.cmml"><mrow id="S5.Ex58.m1.5.5.1.1.1.1.1" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml"><mi id="S5.Ex58.m1.1.1" xref="S5.Ex58.m1.1.1.cmml">max</mi><mo id="S5.Ex58.m1.5.5.1.1.1.1.1a" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml">⁡</mo><mrow id="S5.Ex58.m1.5.5.1.1.1.1.1.1" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml"><mo id="S5.Ex58.m1.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml">(</mo><mn id="S5.Ex58.m1.2.2" xref="S5.Ex58.m1.2.2.cmml">0</mn><mo id="S5.Ex58.m1.5.5.1.1.1.1.1.1.3" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml">,</mo><msub id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.2" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.2.cmml">f</mi><mn id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.3" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Ex58.m1.5.5.1.1.1.1.1.1.4" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Ex58.m1.5.5.1.1.2.3" xref="S5.Ex58.m1.5.5.1.1.2.3.cmml">−</mo><mrow id="S5.Ex58.m1.5.5.1.1.2.2.1" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml"><mi id="S5.Ex58.m1.3.3" xref="S5.Ex58.m1.3.3.cmml">max</mi><mo id="S5.Ex58.m1.5.5.1.1.2.2.1a" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml">⁡</mo><mrow id="S5.Ex58.m1.5.5.1.1.2.2.1.1" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml"><mo id="S5.Ex58.m1.5.5.1.1.2.2.1.1.2" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml">(</mo><mn id="S5.Ex58.m1.4.4" xref="S5.Ex58.m1.4.4.cmml">0</mn><mo id="S5.Ex58.m1.5.5.1.1.2.2.1.1.3" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml">,</mo><mrow id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.cmml"><mo id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1a" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.cmml">−</mo><msub id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.cmml"><mi id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.2" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.2.cmml">f</mi><mn id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.3" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S5.Ex58.m1.5.5.1.1.2.2.1.1.4" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex58.m1.5.5.1.1.8" xref="S5.Ex58.m1.5.5.1.1.8.cmml">=</mo><mrow id="S5.Ex58.m1.5.5.1.1.4" xref="S5.Ex58.m1.5.5.1.1.4.cmml"><mrow id="S5.Ex58.m1.5.5.1.1.3.1" xref="S5.Ex58.m1.5.5.1.1.3.1.cmml"><mi id="S5.Ex58.m1.5.5.1.1.3.1.3" xref="S5.Ex58.m1.5.5.1.1.3.1.3.cmml">ρ</mi><mo id="S5.Ex58.m1.5.5.1.1.3.1.2" xref="S5.Ex58.m1.5.5.1.1.3.1.2.cmml">⁢</mo><mrow id="S5.Ex58.m1.5.5.1.1.3.1.1.1" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.cmml"><mo id="S5.Ex58.m1.5.5.1.1.3.1.1.1.2" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.cmml">(</mo><msub id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.cmml"><mi id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.2" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.2.cmml">f</mi><mn id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.3" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Ex58.m1.5.5.1.1.3.1.1.1.3" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex58.m1.5.5.1.1.4.3" xref="S5.Ex58.m1.5.5.1.1.4.3.cmml">−</mo><mrow id="S5.Ex58.m1.5.5.1.1.4.2" xref="S5.Ex58.m1.5.5.1.1.4.2.cmml"><mi id="S5.Ex58.m1.5.5.1.1.4.2.3" xref="S5.Ex58.m1.5.5.1.1.4.2.3.cmml">ρ</mi><mo id="S5.Ex58.m1.5.5.1.1.4.2.2" xref="S5.Ex58.m1.5.5.1.1.4.2.2.cmml">⁢</mo><mrow id="S5.Ex58.m1.5.5.1.1.4.2.1.1" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml"><mo id="S5.Ex58.m1.5.5.1.1.4.2.1.1.2" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml">(</mo><mrow id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml"><mo id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1a" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml">−</mo><msub id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.cmml"><mi id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.2" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.2.cmml">f</mi><mn id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.3" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S5.Ex58.m1.5.5.1.1.4.2.1.1.3" stretchy="false" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.Ex58.m1.5.5.1.2" lspace="0em" xref="S5.Ex58.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex58.m1.5b"><apply id="S5.Ex58.m1.5.5.1.1.cmml" xref="S5.Ex58.m1.5.5.1"><and id="S5.Ex58.m1.5.5.1.1a.cmml" xref="S5.Ex58.m1.5.5.1"></and><apply id="S5.Ex58.m1.5.5.1.1b.cmml" xref="S5.Ex58.m1.5.5.1"><eq id="S5.Ex58.m1.5.5.1.1.7.cmml" xref="S5.Ex58.m1.5.5.1.1.7"></eq><apply id="S5.Ex58.m1.5.5.1.1.6.cmml" xref="S5.Ex58.m1.5.5.1.1.6"><csymbol cd="ambiguous" id="S5.Ex58.m1.5.5.1.1.6.1.cmml" xref="S5.Ex58.m1.5.5.1.1.6">subscript</csymbol><ci id="S5.Ex58.m1.5.5.1.1.6.2.cmml" xref="S5.Ex58.m1.5.5.1.1.6.2">𝑓</ci><cn id="S5.Ex58.m1.5.5.1.1.6.3.cmml" type="integer" xref="S5.Ex58.m1.5.5.1.1.6.3">1</cn></apply><apply id="S5.Ex58.m1.5.5.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.2"><minus id="S5.Ex58.m1.5.5.1.1.2.3.cmml" xref="S5.Ex58.m1.5.5.1.1.2.3"></minus><apply id="S5.Ex58.m1.5.5.1.1.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.1.1.1"><max id="S5.Ex58.m1.1.1.cmml" xref="S5.Ex58.m1.1.1"></max><cn id="S5.Ex58.m1.2.2.cmml" type="integer" xref="S5.Ex58.m1.2.2">0</cn><apply id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.2">𝑓</ci><cn id="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex58.m1.5.5.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Ex58.m1.5.5.1.1.2.2.2.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1"><max id="S5.Ex58.m1.3.3.cmml" xref="S5.Ex58.m1.3.3"></max><cn id="S5.Ex58.m1.4.4.cmml" type="integer" xref="S5.Ex58.m1.4.4">0</cn><apply id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1"><minus id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1"></minus><apply id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.1.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2">subscript</csymbol><ci id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.2.cmml" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.2">𝑓</ci><cn id="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.3.cmml" type="integer" xref="S5.Ex58.m1.5.5.1.1.2.2.1.1.1.2.3">1</cn></apply></apply></apply></apply></apply><apply id="S5.Ex58.m1.5.5.1.1c.cmml" xref="S5.Ex58.m1.5.5.1"><eq id="S5.Ex58.m1.5.5.1.1.8.cmml" xref="S5.Ex58.m1.5.5.1.1.8"></eq><share href="https://arxiv.org/html/2503.13001v1#S5.Ex58.m1.5.5.1.1.2.cmml" id="S5.Ex58.m1.5.5.1.1d.cmml" xref="S5.Ex58.m1.5.5.1"></share><apply id="S5.Ex58.m1.5.5.1.1.4.cmml" xref="S5.Ex58.m1.5.5.1.1.4"><minus id="S5.Ex58.m1.5.5.1.1.4.3.cmml" xref="S5.Ex58.m1.5.5.1.1.4.3"></minus><apply id="S5.Ex58.m1.5.5.1.1.3.1.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1"><times id="S5.Ex58.m1.5.5.1.1.3.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1.2"></times><ci id="S5.Ex58.m1.5.5.1.1.3.1.3.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1.3">𝜌</ci><apply id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1"><csymbol cd="ambiguous" id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1">subscript</csymbol><ci id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.2">𝑓</ci><cn id="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.3.cmml" type="integer" xref="S5.Ex58.m1.5.5.1.1.3.1.1.1.1.3">1</cn></apply></apply><apply id="S5.Ex58.m1.5.5.1.1.4.2.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2"><times id="S5.Ex58.m1.5.5.1.1.4.2.2.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.2"></times><ci id="S5.Ex58.m1.5.5.1.1.4.2.3.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.3">𝜌</ci><apply id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1"><minus id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.1.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1"></minus><apply id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.1.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2">subscript</csymbol><ci id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.2.cmml" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.2">𝑓</ci><cn id="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.3.cmml" type="integer" xref="S5.Ex58.m1.5.5.1.1.4.2.1.1.1.2.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex58.m1.5c">f_{1}=\max(0,f_{1})-\max(0,-f_{1})=\rho(f_{1})-\rho(-f_{1}).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex58.m1.5d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_max ( 0 , italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - roman_max ( 0 , - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_ρ ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ρ ( - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.2.p2.8">This can be written as <math alttext="f_{1}=T^{(2)}_{1}\circ\rho\circ T^{(1)}_{1}" class="ltx_Math" display="inline" id="S5.2.p2.5.m1.2"><semantics id="S5.2.p2.5.m1.2a"><mrow id="S5.2.p2.5.m1.2.3" xref="S5.2.p2.5.m1.2.3.cmml"><msub id="S5.2.p2.5.m1.2.3.2" xref="S5.2.p2.5.m1.2.3.2.cmml"><mi id="S5.2.p2.5.m1.2.3.2.2" xref="S5.2.p2.5.m1.2.3.2.2.cmml">f</mi><mn id="S5.2.p2.5.m1.2.3.2.3" xref="S5.2.p2.5.m1.2.3.2.3.cmml">1</mn></msub><mo id="S5.2.p2.5.m1.2.3.1" xref="S5.2.p2.5.m1.2.3.1.cmml">=</mo><mrow id="S5.2.p2.5.m1.2.3.3" xref="S5.2.p2.5.m1.2.3.3.cmml"><msubsup id="S5.2.p2.5.m1.2.3.3.2" xref="S5.2.p2.5.m1.2.3.3.2.cmml"><mi id="S5.2.p2.5.m1.2.3.3.2.2.2" xref="S5.2.p2.5.m1.2.3.3.2.2.2.cmml">T</mi><mn id="S5.2.p2.5.m1.2.3.3.2.3" xref="S5.2.p2.5.m1.2.3.3.2.3.cmml">1</mn><mrow id="S5.2.p2.5.m1.1.1.1.3" xref="S5.2.p2.5.m1.2.3.3.2.cmml"><mo id="S5.2.p2.5.m1.1.1.1.3.1" stretchy="false" xref="S5.2.p2.5.m1.2.3.3.2.cmml">(</mo><mn id="S5.2.p2.5.m1.1.1.1.1" xref="S5.2.p2.5.m1.1.1.1.1.cmml">2</mn><mo id="S5.2.p2.5.m1.1.1.1.3.2" stretchy="false" xref="S5.2.p2.5.m1.2.3.3.2.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.5.m1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.2.p2.5.m1.2.3.3.1.cmml">∘</mo><mi id="S5.2.p2.5.m1.2.3.3.3" xref="S5.2.p2.5.m1.2.3.3.3.cmml">ρ</mi><mo id="S5.2.p2.5.m1.2.3.3.1a" lspace="0.222em" rspace="0.222em" xref="S5.2.p2.5.m1.2.3.3.1.cmml">∘</mo><msubsup id="S5.2.p2.5.m1.2.3.3.4" xref="S5.2.p2.5.m1.2.3.3.4.cmml"><mi id="S5.2.p2.5.m1.2.3.3.4.2.2" xref="S5.2.p2.5.m1.2.3.3.4.2.2.cmml">T</mi><mn id="S5.2.p2.5.m1.2.3.3.4.3" xref="S5.2.p2.5.m1.2.3.3.4.3.cmml">1</mn><mrow id="S5.2.p2.5.m1.2.2.1.3" xref="S5.2.p2.5.m1.2.3.3.4.cmml"><mo id="S5.2.p2.5.m1.2.2.1.3.1" stretchy="false" xref="S5.2.p2.5.m1.2.3.3.4.cmml">(</mo><mn id="S5.2.p2.5.m1.2.2.1.1" xref="S5.2.p2.5.m1.2.2.1.1.cmml">1</mn><mo id="S5.2.p2.5.m1.2.2.1.3.2" stretchy="false" xref="S5.2.p2.5.m1.2.3.3.4.cmml">)</mo></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.5.m1.2b"><apply id="S5.2.p2.5.m1.2.3.cmml" xref="S5.2.p2.5.m1.2.3"><eq id="S5.2.p2.5.m1.2.3.1.cmml" xref="S5.2.p2.5.m1.2.3.1"></eq><apply id="S5.2.p2.5.m1.2.3.2.cmml" xref="S5.2.p2.5.m1.2.3.2"><csymbol cd="ambiguous" id="S5.2.p2.5.m1.2.3.2.1.cmml" xref="S5.2.p2.5.m1.2.3.2">subscript</csymbol><ci id="S5.2.p2.5.m1.2.3.2.2.cmml" xref="S5.2.p2.5.m1.2.3.2.2">𝑓</ci><cn id="S5.2.p2.5.m1.2.3.2.3.cmml" type="integer" xref="S5.2.p2.5.m1.2.3.2.3">1</cn></apply><apply id="S5.2.p2.5.m1.2.3.3.cmml" xref="S5.2.p2.5.m1.2.3.3"><compose id="S5.2.p2.5.m1.2.3.3.1.cmml" xref="S5.2.p2.5.m1.2.3.3.1"></compose><apply id="S5.2.p2.5.m1.2.3.3.2.cmml" xref="S5.2.p2.5.m1.2.3.3.2"><csymbol cd="ambiguous" id="S5.2.p2.5.m1.2.3.3.2.1.cmml" xref="S5.2.p2.5.m1.2.3.3.2">subscript</csymbol><apply id="S5.2.p2.5.m1.2.3.3.2.2.cmml" xref="S5.2.p2.5.m1.2.3.3.2"><csymbol cd="ambiguous" id="S5.2.p2.5.m1.2.3.3.2.2.1.cmml" xref="S5.2.p2.5.m1.2.3.3.2">superscript</csymbol><ci id="S5.2.p2.5.m1.2.3.3.2.2.2.cmml" xref="S5.2.p2.5.m1.2.3.3.2.2.2">𝑇</ci><cn id="S5.2.p2.5.m1.1.1.1.1.cmml" type="integer" xref="S5.2.p2.5.m1.1.1.1.1">2</cn></apply><cn id="S5.2.p2.5.m1.2.3.3.2.3.cmml" type="integer" xref="S5.2.p2.5.m1.2.3.3.2.3">1</cn></apply><ci id="S5.2.p2.5.m1.2.3.3.3.cmml" xref="S5.2.p2.5.m1.2.3.3.3">𝜌</ci><apply id="S5.2.p2.5.m1.2.3.3.4.cmml" xref="S5.2.p2.5.m1.2.3.3.4"><csymbol cd="ambiguous" id="S5.2.p2.5.m1.2.3.3.4.1.cmml" xref="S5.2.p2.5.m1.2.3.3.4">subscript</csymbol><apply id="S5.2.p2.5.m1.2.3.3.4.2.cmml" xref="S5.2.p2.5.m1.2.3.3.4"><csymbol cd="ambiguous" id="S5.2.p2.5.m1.2.3.3.4.2.1.cmml" xref="S5.2.p2.5.m1.2.3.3.4">superscript</csymbol><ci id="S5.2.p2.5.m1.2.3.3.4.2.2.cmml" xref="S5.2.p2.5.m1.2.3.3.4.2.2">𝑇</ci><cn id="S5.2.p2.5.m1.2.2.1.1.cmml" type="integer" xref="S5.2.p2.5.m1.2.2.1.1">1</cn></apply><cn id="S5.2.p2.5.m1.2.3.3.4.3.cmml" type="integer" xref="S5.2.p2.5.m1.2.3.3.4.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.5.m1.2c">f_{1}=T^{(2)}_{1}\circ\rho\circ T^{(1)}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.5.m1.2d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∘ italic_ρ ∘ italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="T^{(1)}_{1}=(f_{1},-f_{1})^{T}" class="ltx_Math" display="inline" id="S5.2.p2.6.m2.3"><semantics id="S5.2.p2.6.m2.3a"><mrow id="S5.2.p2.6.m2.3.3" xref="S5.2.p2.6.m2.3.3.cmml"><msubsup id="S5.2.p2.6.m2.3.3.4" xref="S5.2.p2.6.m2.3.3.4.cmml"><mi id="S5.2.p2.6.m2.3.3.4.2.2" xref="S5.2.p2.6.m2.3.3.4.2.2.cmml">T</mi><mn id="S5.2.p2.6.m2.3.3.4.3" xref="S5.2.p2.6.m2.3.3.4.3.cmml">1</mn><mrow id="S5.2.p2.6.m2.1.1.1.3" xref="S5.2.p2.6.m2.3.3.4.cmml"><mo id="S5.2.p2.6.m2.1.1.1.3.1" stretchy="false" xref="S5.2.p2.6.m2.3.3.4.cmml">(</mo><mn id="S5.2.p2.6.m2.1.1.1.1" xref="S5.2.p2.6.m2.1.1.1.1.cmml">1</mn><mo id="S5.2.p2.6.m2.1.1.1.3.2" stretchy="false" xref="S5.2.p2.6.m2.3.3.4.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.6.m2.3.3.3" xref="S5.2.p2.6.m2.3.3.3.cmml">=</mo><msup id="S5.2.p2.6.m2.3.3.2" xref="S5.2.p2.6.m2.3.3.2.cmml"><mrow id="S5.2.p2.6.m2.3.3.2.2.2" xref="S5.2.p2.6.m2.3.3.2.2.3.cmml"><mo id="S5.2.p2.6.m2.3.3.2.2.2.3" stretchy="false" xref="S5.2.p2.6.m2.3.3.2.2.3.cmml">(</mo><msub id="S5.2.p2.6.m2.2.2.1.1.1.1" xref="S5.2.p2.6.m2.2.2.1.1.1.1.cmml"><mi id="S5.2.p2.6.m2.2.2.1.1.1.1.2" xref="S5.2.p2.6.m2.2.2.1.1.1.1.2.cmml">f</mi><mn id="S5.2.p2.6.m2.2.2.1.1.1.1.3" xref="S5.2.p2.6.m2.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.2.p2.6.m2.3.3.2.2.2.4" xref="S5.2.p2.6.m2.3.3.2.2.3.cmml">,</mo><mrow id="S5.2.p2.6.m2.3.3.2.2.2.2" xref="S5.2.p2.6.m2.3.3.2.2.2.2.cmml"><mo id="S5.2.p2.6.m2.3.3.2.2.2.2a" xref="S5.2.p2.6.m2.3.3.2.2.2.2.cmml">−</mo><msub id="S5.2.p2.6.m2.3.3.2.2.2.2.2" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2.cmml"><mi id="S5.2.p2.6.m2.3.3.2.2.2.2.2.2" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2.2.cmml">f</mi><mn id="S5.2.p2.6.m2.3.3.2.2.2.2.2.3" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2.3.cmml">1</mn></msub></mrow><mo id="S5.2.p2.6.m2.3.3.2.2.2.5" stretchy="false" xref="S5.2.p2.6.m2.3.3.2.2.3.cmml">)</mo></mrow><mi id="S5.2.p2.6.m2.3.3.2.4" xref="S5.2.p2.6.m2.3.3.2.4.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.6.m2.3b"><apply id="S5.2.p2.6.m2.3.3.cmml" xref="S5.2.p2.6.m2.3.3"><eq id="S5.2.p2.6.m2.3.3.3.cmml" xref="S5.2.p2.6.m2.3.3.3"></eq><apply id="S5.2.p2.6.m2.3.3.4.cmml" xref="S5.2.p2.6.m2.3.3.4"><csymbol cd="ambiguous" id="S5.2.p2.6.m2.3.3.4.1.cmml" xref="S5.2.p2.6.m2.3.3.4">subscript</csymbol><apply id="S5.2.p2.6.m2.3.3.4.2.cmml" xref="S5.2.p2.6.m2.3.3.4"><csymbol cd="ambiguous" id="S5.2.p2.6.m2.3.3.4.2.1.cmml" xref="S5.2.p2.6.m2.3.3.4">superscript</csymbol><ci id="S5.2.p2.6.m2.3.3.4.2.2.cmml" xref="S5.2.p2.6.m2.3.3.4.2.2">𝑇</ci><cn id="S5.2.p2.6.m2.1.1.1.1.cmml" type="integer" xref="S5.2.p2.6.m2.1.1.1.1">1</cn></apply><cn id="S5.2.p2.6.m2.3.3.4.3.cmml" type="integer" xref="S5.2.p2.6.m2.3.3.4.3">1</cn></apply><apply id="S5.2.p2.6.m2.3.3.2.cmml" xref="S5.2.p2.6.m2.3.3.2"><csymbol cd="ambiguous" id="S5.2.p2.6.m2.3.3.2.3.cmml" xref="S5.2.p2.6.m2.3.3.2">superscript</csymbol><interval closure="open" id="S5.2.p2.6.m2.3.3.2.2.3.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2"><apply id="S5.2.p2.6.m2.2.2.1.1.1.1.cmml" xref="S5.2.p2.6.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.6.m2.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.6.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S5.2.p2.6.m2.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.6.m2.2.2.1.1.1.1.2">𝑓</ci><cn id="S5.2.p2.6.m2.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.2.p2.6.m2.2.2.1.1.1.1.3">1</cn></apply><apply id="S5.2.p2.6.m2.3.3.2.2.2.2.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2.2"><minus id="S5.2.p2.6.m2.3.3.2.2.2.2.1.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2.2"></minus><apply id="S5.2.p2.6.m2.3.3.2.2.2.2.2.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.2.p2.6.m2.3.3.2.2.2.2.2.1.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2">subscript</csymbol><ci id="S5.2.p2.6.m2.3.3.2.2.2.2.2.2.cmml" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2.2">𝑓</ci><cn id="S5.2.p2.6.m2.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S5.2.p2.6.m2.3.3.2.2.2.2.2.3">1</cn></apply></apply></interval><ci id="S5.2.p2.6.m2.3.3.2.4.cmml" xref="S5.2.p2.6.m2.3.3.2.4">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.6.m2.3c">T^{(1)}_{1}=(f_{1},-f_{1})^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.6.m2.3d">italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="T^{(2)}_{1}(x)=(1,-1)x" class="ltx_Math" display="inline" id="S5.2.p2.7.m3.4"><semantics id="S5.2.p2.7.m3.4a"><mrow id="S5.2.p2.7.m3.4.4" xref="S5.2.p2.7.m3.4.4.cmml"><mrow id="S5.2.p2.7.m3.4.4.3" xref="S5.2.p2.7.m3.4.4.3.cmml"><msubsup id="S5.2.p2.7.m3.4.4.3.2" xref="S5.2.p2.7.m3.4.4.3.2.cmml"><mi id="S5.2.p2.7.m3.4.4.3.2.2.2" xref="S5.2.p2.7.m3.4.4.3.2.2.2.cmml">T</mi><mn id="S5.2.p2.7.m3.4.4.3.2.3" xref="S5.2.p2.7.m3.4.4.3.2.3.cmml">1</mn><mrow id="S5.2.p2.7.m3.1.1.1.3" xref="S5.2.p2.7.m3.4.4.3.2.cmml"><mo id="S5.2.p2.7.m3.1.1.1.3.1" stretchy="false" xref="S5.2.p2.7.m3.4.4.3.2.cmml">(</mo><mn id="S5.2.p2.7.m3.1.1.1.1" xref="S5.2.p2.7.m3.1.1.1.1.cmml">2</mn><mo id="S5.2.p2.7.m3.1.1.1.3.2" stretchy="false" xref="S5.2.p2.7.m3.4.4.3.2.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.7.m3.4.4.3.1" xref="S5.2.p2.7.m3.4.4.3.1.cmml">⁢</mo><mrow id="S5.2.p2.7.m3.4.4.3.3.2" xref="S5.2.p2.7.m3.4.4.3.cmml"><mo id="S5.2.p2.7.m3.4.4.3.3.2.1" stretchy="false" xref="S5.2.p2.7.m3.4.4.3.cmml">(</mo><mi id="S5.2.p2.7.m3.2.2" xref="S5.2.p2.7.m3.2.2.cmml">x</mi><mo id="S5.2.p2.7.m3.4.4.3.3.2.2" stretchy="false" xref="S5.2.p2.7.m3.4.4.3.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.7.m3.4.4.2" xref="S5.2.p2.7.m3.4.4.2.cmml">=</mo><mrow id="S5.2.p2.7.m3.4.4.1" xref="S5.2.p2.7.m3.4.4.1.cmml"><mrow id="S5.2.p2.7.m3.4.4.1.1.1" xref="S5.2.p2.7.m3.4.4.1.1.2.cmml"><mo id="S5.2.p2.7.m3.4.4.1.1.1.2" stretchy="false" xref="S5.2.p2.7.m3.4.4.1.1.2.cmml">(</mo><mn id="S5.2.p2.7.m3.3.3" xref="S5.2.p2.7.m3.3.3.cmml">1</mn><mo id="S5.2.p2.7.m3.4.4.1.1.1.3" xref="S5.2.p2.7.m3.4.4.1.1.2.cmml">,</mo><mrow id="S5.2.p2.7.m3.4.4.1.1.1.1" xref="S5.2.p2.7.m3.4.4.1.1.1.1.cmml"><mo id="S5.2.p2.7.m3.4.4.1.1.1.1a" xref="S5.2.p2.7.m3.4.4.1.1.1.1.cmml">−</mo><mn id="S5.2.p2.7.m3.4.4.1.1.1.1.2" xref="S5.2.p2.7.m3.4.4.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.2.p2.7.m3.4.4.1.1.1.4" stretchy="false" xref="S5.2.p2.7.m3.4.4.1.1.2.cmml">)</mo></mrow><mo id="S5.2.p2.7.m3.4.4.1.2" xref="S5.2.p2.7.m3.4.4.1.2.cmml">⁢</mo><mi id="S5.2.p2.7.m3.4.4.1.3" xref="S5.2.p2.7.m3.4.4.1.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.7.m3.4b"><apply id="S5.2.p2.7.m3.4.4.cmml" xref="S5.2.p2.7.m3.4.4"><eq id="S5.2.p2.7.m3.4.4.2.cmml" xref="S5.2.p2.7.m3.4.4.2"></eq><apply id="S5.2.p2.7.m3.4.4.3.cmml" xref="S5.2.p2.7.m3.4.4.3"><times id="S5.2.p2.7.m3.4.4.3.1.cmml" xref="S5.2.p2.7.m3.4.4.3.1"></times><apply id="S5.2.p2.7.m3.4.4.3.2.cmml" xref="S5.2.p2.7.m3.4.4.3.2"><csymbol cd="ambiguous" id="S5.2.p2.7.m3.4.4.3.2.1.cmml" xref="S5.2.p2.7.m3.4.4.3.2">subscript</csymbol><apply id="S5.2.p2.7.m3.4.4.3.2.2.cmml" xref="S5.2.p2.7.m3.4.4.3.2"><csymbol cd="ambiguous" id="S5.2.p2.7.m3.4.4.3.2.2.1.cmml" xref="S5.2.p2.7.m3.4.4.3.2">superscript</csymbol><ci id="S5.2.p2.7.m3.4.4.3.2.2.2.cmml" xref="S5.2.p2.7.m3.4.4.3.2.2.2">𝑇</ci><cn id="S5.2.p2.7.m3.1.1.1.1.cmml" type="integer" xref="S5.2.p2.7.m3.1.1.1.1">2</cn></apply><cn id="S5.2.p2.7.m3.4.4.3.2.3.cmml" type="integer" xref="S5.2.p2.7.m3.4.4.3.2.3">1</cn></apply><ci id="S5.2.p2.7.m3.2.2.cmml" xref="S5.2.p2.7.m3.2.2">𝑥</ci></apply><apply id="S5.2.p2.7.m3.4.4.1.cmml" xref="S5.2.p2.7.m3.4.4.1"><times id="S5.2.p2.7.m3.4.4.1.2.cmml" xref="S5.2.p2.7.m3.4.4.1.2"></times><interval closure="open" id="S5.2.p2.7.m3.4.4.1.1.2.cmml" xref="S5.2.p2.7.m3.4.4.1.1.1"><cn id="S5.2.p2.7.m3.3.3.cmml" type="integer" xref="S5.2.p2.7.m3.3.3">1</cn><apply id="S5.2.p2.7.m3.4.4.1.1.1.1.cmml" xref="S5.2.p2.7.m3.4.4.1.1.1.1"><minus id="S5.2.p2.7.m3.4.4.1.1.1.1.1.cmml" xref="S5.2.p2.7.m3.4.4.1.1.1.1"></minus><cn id="S5.2.p2.7.m3.4.4.1.1.1.1.2.cmml" type="integer" xref="S5.2.p2.7.m3.4.4.1.1.1.1.2">1</cn></apply></interval><ci id="S5.2.p2.7.m3.4.4.1.3.cmml" xref="S5.2.p2.7.m3.4.4.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.7.m3.4c">T^{(2)}_{1}(x)=(1,-1)x</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.7.m3.4d">italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) = ( 1 , - 1 ) italic_x</annotation></semantics></math>, which is a neural network with one hidden layer of width two. Next, for the function <math alttext="\max(f_{2},f_{3})" class="ltx_Math" display="inline" id="S5.2.p2.8.m4.3"><semantics id="S5.2.p2.8.m4.3a"><mrow id="S5.2.p2.8.m4.3.3.2" xref="S5.2.p2.8.m4.3.3.3.cmml"><mi id="S5.2.p2.8.m4.1.1" xref="S5.2.p2.8.m4.1.1.cmml">max</mi><mo id="S5.2.p2.8.m4.3.3.2a" xref="S5.2.p2.8.m4.3.3.3.cmml">⁡</mo><mrow id="S5.2.p2.8.m4.3.3.2.2" xref="S5.2.p2.8.m4.3.3.3.cmml"><mo id="S5.2.p2.8.m4.3.3.2.2.3" stretchy="false" xref="S5.2.p2.8.m4.3.3.3.cmml">(</mo><msub id="S5.2.p2.8.m4.2.2.1.1.1" xref="S5.2.p2.8.m4.2.2.1.1.1.cmml"><mi id="S5.2.p2.8.m4.2.2.1.1.1.2" xref="S5.2.p2.8.m4.2.2.1.1.1.2.cmml">f</mi><mn id="S5.2.p2.8.m4.2.2.1.1.1.3" xref="S5.2.p2.8.m4.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S5.2.p2.8.m4.3.3.2.2.4" xref="S5.2.p2.8.m4.3.3.3.cmml">,</mo><msub id="S5.2.p2.8.m4.3.3.2.2.2" xref="S5.2.p2.8.m4.3.3.2.2.2.cmml"><mi id="S5.2.p2.8.m4.3.3.2.2.2.2" xref="S5.2.p2.8.m4.3.3.2.2.2.2.cmml">f</mi><mn id="S5.2.p2.8.m4.3.3.2.2.2.3" xref="S5.2.p2.8.m4.3.3.2.2.2.3.cmml">3</mn></msub><mo id="S5.2.p2.8.m4.3.3.2.2.5" stretchy="false" xref="S5.2.p2.8.m4.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.8.m4.3b"><apply id="S5.2.p2.8.m4.3.3.3.cmml" xref="S5.2.p2.8.m4.3.3.2"><max id="S5.2.p2.8.m4.1.1.cmml" xref="S5.2.p2.8.m4.1.1"></max><apply id="S5.2.p2.8.m4.2.2.1.1.1.cmml" xref="S5.2.p2.8.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.8.m4.2.2.1.1.1.1.cmml" xref="S5.2.p2.8.m4.2.2.1.1.1">subscript</csymbol><ci id="S5.2.p2.8.m4.2.2.1.1.1.2.cmml" xref="S5.2.p2.8.m4.2.2.1.1.1.2">𝑓</ci><cn id="S5.2.p2.8.m4.2.2.1.1.1.3.cmml" type="integer" xref="S5.2.p2.8.m4.2.2.1.1.1.3">2</cn></apply><apply id="S5.2.p2.8.m4.3.3.2.2.2.cmml" xref="S5.2.p2.8.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.2.p2.8.m4.3.3.2.2.2.1.cmml" xref="S5.2.p2.8.m4.3.3.2.2.2">subscript</csymbol><ci id="S5.2.p2.8.m4.3.3.2.2.2.2.cmml" xref="S5.2.p2.8.m4.3.3.2.2.2.2">𝑓</ci><cn id="S5.2.p2.8.m4.3.3.2.2.2.3.cmml" type="integer" xref="S5.2.p2.8.m4.3.3.2.2.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.8.m4.3c">\max(f_{2},f_{3})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.8.m4.3d">roman_max ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex59"> <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="\max(f_{2},f_{3})=\max(0,f_{2}-f_{3})+f_{3}=\rho(f_{2}-f_{3})+\rho(f_{3})-\rho% (-f_{3})," class="ltx_Math" display="block" id="S5.Ex59.m1.4"><semantics id="S5.Ex59.m1.4a"><mrow id="S5.Ex59.m1.4.4.1" xref="S5.Ex59.m1.4.4.1.1.cmml"><mrow id="S5.Ex59.m1.4.4.1.1" xref="S5.Ex59.m1.4.4.1.1.cmml"><mrow id="S5.Ex59.m1.4.4.1.1.2.2" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml"><mi id="S5.Ex59.m1.1.1" xref="S5.Ex59.m1.1.1.cmml">max</mi><mo id="S5.Ex59.m1.4.4.1.1.2.2a" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">⁡</mo><mrow id="S5.Ex59.m1.4.4.1.1.2.2.2" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml"><mo id="S5.Ex59.m1.4.4.1.1.2.2.2.3" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">(</mo><msub id="S5.Ex59.m1.4.4.1.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S5.Ex59.m1.4.4.1.1.1.1.1.1.2" xref="S5.Ex59.m1.4.4.1.1.1.1.1.1.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.1.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S5.Ex59.m1.4.4.1.1.2.2.2.4" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">,</mo><msub id="S5.Ex59.m1.4.4.1.1.2.2.2.2" xref="S5.Ex59.m1.4.4.1.1.2.2.2.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.2.2.2.2.2" xref="S5.Ex59.m1.4.4.1.1.2.2.2.2.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.2.2.2.2.3" xref="S5.Ex59.m1.4.4.1.1.2.2.2.2.3.cmml">3</mn></msub><mo id="S5.Ex59.m1.4.4.1.1.2.2.2.5" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.1.8" xref="S5.Ex59.m1.4.4.1.1.8.cmml">=</mo><mrow id="S5.Ex59.m1.4.4.1.1.3" xref="S5.Ex59.m1.4.4.1.1.3.cmml"><mrow id="S5.Ex59.m1.4.4.1.1.3.1.1" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml"><mi id="S5.Ex59.m1.2.2" xref="S5.Ex59.m1.2.2.cmml">max</mi><mo id="S5.Ex59.m1.4.4.1.1.3.1.1a" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml">⁡</mo><mrow id="S5.Ex59.m1.4.4.1.1.3.1.1.1" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml"><mo id="S5.Ex59.m1.4.4.1.1.3.1.1.1.2" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml">(</mo><mn id="S5.Ex59.m1.3.3" xref="S5.Ex59.m1.3.3.cmml">0</mn><mo id="S5.Ex59.m1.4.4.1.1.3.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml">,</mo><mrow id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.cmml"><msub id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2.2" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2.3" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3.cmml"><mi id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3.2" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3.3" xref="S5.Ex59.m1.4.4.1.1.3.1.1.1.1.3.3.cmml">3</mn></msub></mrow><mo id="S5.Ex59.m1.4.4.1.1.3.1.1.1.4" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.3.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.1.3.2" xref="S5.Ex59.m1.4.4.1.1.3.2.cmml">+</mo><msub id="S5.Ex59.m1.4.4.1.1.3.3" xref="S5.Ex59.m1.4.4.1.1.3.3.cmml"><mi id="S5.Ex59.m1.4.4.1.1.3.3.2" xref="S5.Ex59.m1.4.4.1.1.3.3.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.3.3.3" xref="S5.Ex59.m1.4.4.1.1.3.3.3.cmml">3</mn></msub></mrow><mo id="S5.Ex59.m1.4.4.1.1.9" xref="S5.Ex59.m1.4.4.1.1.9.cmml">=</mo><mrow id="S5.Ex59.m1.4.4.1.1.6" xref="S5.Ex59.m1.4.4.1.1.6.cmml"><mrow id="S5.Ex59.m1.4.4.1.1.5.2" xref="S5.Ex59.m1.4.4.1.1.5.2.cmml"><mrow id="S5.Ex59.m1.4.4.1.1.4.1.1" xref="S5.Ex59.m1.4.4.1.1.4.1.1.cmml"><mi id="S5.Ex59.m1.4.4.1.1.4.1.1.3" xref="S5.Ex59.m1.4.4.1.1.4.1.1.3.cmml">ρ</mi><mo id="S5.Ex59.m1.4.4.1.1.4.1.1.2" xref="S5.Ex59.m1.4.4.1.1.4.1.1.2.cmml">⁢</mo><mrow id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.cmml"><mo id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.2" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.cmml"><msub id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2.2" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2.3" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.1" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3.cmml"><mi id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3.2" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3.3" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.3.3.cmml">3</mn></msub></mrow><mo id="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.3" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.1.5.2.3" xref="S5.Ex59.m1.4.4.1.1.5.2.3.cmml">+</mo><mrow id="S5.Ex59.m1.4.4.1.1.5.2.2" xref="S5.Ex59.m1.4.4.1.1.5.2.2.cmml"><mi id="S5.Ex59.m1.4.4.1.1.5.2.2.3" xref="S5.Ex59.m1.4.4.1.1.5.2.2.3.cmml">ρ</mi><mo id="S5.Ex59.m1.4.4.1.1.5.2.2.2" xref="S5.Ex59.m1.4.4.1.1.5.2.2.2.cmml">⁢</mo><mrow id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.cmml"><mo id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.2" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.cmml">(</mo><msub id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.cmml"><mi id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.2" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.2.cmml">f</mi><mn id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.3" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.3.cmml">3</mn></msub><mo id="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.3" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.5.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex59.m1.4.4.1.1.6.4" xref="S5.Ex59.m1.4.4.1.1.6.4.cmml">−</mo><mrow id="S5.Ex59.m1.4.4.1.1.6.3" xref="S5.Ex59.m1.4.4.1.1.6.3.cmml"><mi id="S5.Ex59.m1.4.4.1.1.6.3.3" xref="S5.Ex59.m1.4.4.1.1.6.3.3.cmml">ρ</mi><mo id="S5.Ex59.m1.4.4.1.1.6.3.2" xref="S5.Ex59.m1.4.4.1.1.6.3.2.cmml">⁢</mo><mrow id="S5.Ex59.m1.4.4.1.1.6.3.1.1" xref="S5.Ex59.m1.4.4.1.1.6.3.1.1.1.cmml"><mo id="S5.Ex59.m1.4.4.1.1.6.3.1.1.2" stretchy="false" xref="S5.Ex59.m1.4.4.1.1.6.3.1.1.1.cmml">(</mo><mrow id="S5.Ex59.m1.4.4.1.1.6.3.1.1.1" xref="S5.Ex59.m1.4.4.1.1.6.3.1.1.1.cmml"><mo id="S5.Ex59.m1.4.4.1.1.6.3.1.1.1a" xref="S5.Ex59.m1.4.4.1.1.6.3.1.1.1.cmml">−</mo><msub id="S5.Ex59.m1.4.4.1.1.6.3.1.1.1.2" 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id="S5.Ex59.m1.4c">\max(f_{2},f_{3})=\max(0,f_{2}-f_{3})+f_{3}=\rho(f_{2}-f_{3})+\rho(f_{3})-\rho% (-f_{3}),</annotation><annotation encoding="application/x-llamapun" id="S5.Ex59.m1.4d">roman_max ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = roman_max ( 0 , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_ρ ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_ρ ( italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - italic_ρ ( - italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.2.p2.13">which allows us to represent <math alttext="\sigma_{2}\max(f_{2},f_{3})" class="ltx_Math" display="inline" 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xref="S5.2.p2.9.m1.3.3.2.2"><max id="S5.2.p2.9.m1.1.1.cmml" xref="S5.2.p2.9.m1.1.1"></max><apply id="S5.2.p2.9.m1.2.2.1.1.1.1.cmml" xref="S5.2.p2.9.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.9.m1.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.9.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S5.2.p2.9.m1.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.9.m1.2.2.1.1.1.1.2">𝑓</ci><cn id="S5.2.p2.9.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.2.p2.9.m1.2.2.1.1.1.1.3">2</cn></apply><apply id="S5.2.p2.9.m1.3.3.2.2.2.2.cmml" xref="S5.2.p2.9.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.2.p2.9.m1.3.3.2.2.2.2.1.cmml" xref="S5.2.p2.9.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S5.2.p2.9.m1.3.3.2.2.2.2.2.cmml" xref="S5.2.p2.9.m1.3.3.2.2.2.2.2">𝑓</ci><cn id="S5.2.p2.9.m1.3.3.2.2.2.2.3.cmml" type="integer" xref="S5.2.p2.9.m1.3.3.2.2.2.2.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.9.m1.3c">\sigma_{2}\max(f_{2},f_{3})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.9.m1.3d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> by a neural network with one hidden layer of width three, given by <math alttext="T^{(1)}_{2}=(f_{2}-f_{3},f_{3},-f_{3})^{T}" class="ltx_Math" display="inline" id="S5.2.p2.10.m2.4"><semantics id="S5.2.p2.10.m2.4a"><mrow id="S5.2.p2.10.m2.4.4" xref="S5.2.p2.10.m2.4.4.cmml"><msubsup id="S5.2.p2.10.m2.4.4.5" xref="S5.2.p2.10.m2.4.4.5.cmml"><mi id="S5.2.p2.10.m2.4.4.5.2.2" xref="S5.2.p2.10.m2.4.4.5.2.2.cmml">T</mi><mn id="S5.2.p2.10.m2.4.4.5.3" xref="S5.2.p2.10.m2.4.4.5.3.cmml">2</mn><mrow id="S5.2.p2.10.m2.1.1.1.3" xref="S5.2.p2.10.m2.4.4.5.cmml"><mo id="S5.2.p2.10.m2.1.1.1.3.1" stretchy="false" xref="S5.2.p2.10.m2.4.4.5.cmml">(</mo><mn id="S5.2.p2.10.m2.1.1.1.1" xref="S5.2.p2.10.m2.1.1.1.1.cmml">1</mn><mo id="S5.2.p2.10.m2.1.1.1.3.2" stretchy="false" xref="S5.2.p2.10.m2.4.4.5.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.10.m2.4.4.4" xref="S5.2.p2.10.m2.4.4.4.cmml">=</mo><msup id="S5.2.p2.10.m2.4.4.3" xref="S5.2.p2.10.m2.4.4.3.cmml"><mrow id="S5.2.p2.10.m2.4.4.3.3.3" xref="S5.2.p2.10.m2.4.4.3.3.4.cmml"><mo id="S5.2.p2.10.m2.4.4.3.3.3.4" stretchy="false" xref="S5.2.p2.10.m2.4.4.3.3.4.cmml">(</mo><mrow id="S5.2.p2.10.m2.2.2.1.1.1.1" xref="S5.2.p2.10.m2.2.2.1.1.1.1.cmml"><msub id="S5.2.p2.10.m2.2.2.1.1.1.1.2" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2.cmml"><mi id="S5.2.p2.10.m2.2.2.1.1.1.1.2.2" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2.2.cmml">f</mi><mn id="S5.2.p2.10.m2.2.2.1.1.1.1.2.3" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S5.2.p2.10.m2.2.2.1.1.1.1.1" xref="S5.2.p2.10.m2.2.2.1.1.1.1.1.cmml">−</mo><msub id="S5.2.p2.10.m2.2.2.1.1.1.1.3" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3.cmml"><mi id="S5.2.p2.10.m2.2.2.1.1.1.1.3.2" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3.2.cmml">f</mi><mn id="S5.2.p2.10.m2.2.2.1.1.1.1.3.3" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3.3.cmml">3</mn></msub></mrow><mo id="S5.2.p2.10.m2.4.4.3.3.3.5" xref="S5.2.p2.10.m2.4.4.3.3.4.cmml">,</mo><msub id="S5.2.p2.10.m2.3.3.2.2.2.2" xref="S5.2.p2.10.m2.3.3.2.2.2.2.cmml"><mi id="S5.2.p2.10.m2.3.3.2.2.2.2.2" xref="S5.2.p2.10.m2.3.3.2.2.2.2.2.cmml">f</mi><mn id="S5.2.p2.10.m2.3.3.2.2.2.2.3" xref="S5.2.p2.10.m2.3.3.2.2.2.2.3.cmml">3</mn></msub><mo id="S5.2.p2.10.m2.4.4.3.3.3.6" xref="S5.2.p2.10.m2.4.4.3.3.4.cmml">,</mo><mrow id="S5.2.p2.10.m2.4.4.3.3.3.3" xref="S5.2.p2.10.m2.4.4.3.3.3.3.cmml"><mo id="S5.2.p2.10.m2.4.4.3.3.3.3a" xref="S5.2.p2.10.m2.4.4.3.3.3.3.cmml">−</mo><msub id="S5.2.p2.10.m2.4.4.3.3.3.3.2" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2.cmml"><mi id="S5.2.p2.10.m2.4.4.3.3.3.3.2.2" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2.2.cmml">f</mi><mn id="S5.2.p2.10.m2.4.4.3.3.3.3.2.3" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2.3.cmml">3</mn></msub></mrow><mo id="S5.2.p2.10.m2.4.4.3.3.3.7" stretchy="false" xref="S5.2.p2.10.m2.4.4.3.3.4.cmml">)</mo></mrow><mi id="S5.2.p2.10.m2.4.4.3.5" xref="S5.2.p2.10.m2.4.4.3.5.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.10.m2.4b"><apply id="S5.2.p2.10.m2.4.4.cmml" xref="S5.2.p2.10.m2.4.4"><eq id="S5.2.p2.10.m2.4.4.4.cmml" xref="S5.2.p2.10.m2.4.4.4"></eq><apply id="S5.2.p2.10.m2.4.4.5.cmml" xref="S5.2.p2.10.m2.4.4.5"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.4.4.5.1.cmml" xref="S5.2.p2.10.m2.4.4.5">subscript</csymbol><apply id="S5.2.p2.10.m2.4.4.5.2.cmml" xref="S5.2.p2.10.m2.4.4.5"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.4.4.5.2.1.cmml" xref="S5.2.p2.10.m2.4.4.5">superscript</csymbol><ci id="S5.2.p2.10.m2.4.4.5.2.2.cmml" xref="S5.2.p2.10.m2.4.4.5.2.2">𝑇</ci><cn id="S5.2.p2.10.m2.1.1.1.1.cmml" type="integer" xref="S5.2.p2.10.m2.1.1.1.1">1</cn></apply><cn id="S5.2.p2.10.m2.4.4.5.3.cmml" type="integer" xref="S5.2.p2.10.m2.4.4.5.3">2</cn></apply><apply id="S5.2.p2.10.m2.4.4.3.cmml" xref="S5.2.p2.10.m2.4.4.3"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.4.4.3.4.cmml" xref="S5.2.p2.10.m2.4.4.3">superscript</csymbol><vector id="S5.2.p2.10.m2.4.4.3.3.4.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3"><apply id="S5.2.p2.10.m2.2.2.1.1.1.1.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1"><minus id="S5.2.p2.10.m2.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.1"></minus><apply id="S5.2.p2.10.m2.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.2.2.1.1.1.1.2.1.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S5.2.p2.10.m2.2.2.1.1.1.1.2.2.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2.2">𝑓</ci><cn id="S5.2.p2.10.m2.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S5.2.p2.10.m2.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S5.2.p2.10.m2.2.2.1.1.1.1.3.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.2.2.1.1.1.1.3.1.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S5.2.p2.10.m2.2.2.1.1.1.1.3.2.cmml" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3.2">𝑓</ci><cn id="S5.2.p2.10.m2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S5.2.p2.10.m2.2.2.1.1.1.1.3.3">3</cn></apply></apply><apply id="S5.2.p2.10.m2.3.3.2.2.2.2.cmml" xref="S5.2.p2.10.m2.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.3.3.2.2.2.2.1.cmml" xref="S5.2.p2.10.m2.3.3.2.2.2.2">subscript</csymbol><ci id="S5.2.p2.10.m2.3.3.2.2.2.2.2.cmml" xref="S5.2.p2.10.m2.3.3.2.2.2.2.2">𝑓</ci><cn id="S5.2.p2.10.m2.3.3.2.2.2.2.3.cmml" type="integer" xref="S5.2.p2.10.m2.3.3.2.2.2.2.3">3</cn></apply><apply id="S5.2.p2.10.m2.4.4.3.3.3.3.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3.3"><minus id="S5.2.p2.10.m2.4.4.3.3.3.3.1.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3.3"></minus><apply id="S5.2.p2.10.m2.4.4.3.3.3.3.2.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2"><csymbol cd="ambiguous" id="S5.2.p2.10.m2.4.4.3.3.3.3.2.1.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2">subscript</csymbol><ci id="S5.2.p2.10.m2.4.4.3.3.3.3.2.2.cmml" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2.2">𝑓</ci><cn id="S5.2.p2.10.m2.4.4.3.3.3.3.2.3.cmml" type="integer" xref="S5.2.p2.10.m2.4.4.3.3.3.3.2.3">3</cn></apply></apply></vector><ci id="S5.2.p2.10.m2.4.4.3.5.cmml" xref="S5.2.p2.10.m2.4.4.3.5">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.10.m2.4c">T^{(1)}_{2}=(f_{2}-f_{3},f_{3},-f_{3})^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.10.m2.4d">italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , - italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="T^{(2)}_{2}(x)=\sigma_{2}(1,1,-1)x" class="ltx_Math" display="inline" id="S5.2.p2.11.m3.5"><semantics id="S5.2.p2.11.m3.5a"><mrow id="S5.2.p2.11.m3.5.5" xref="S5.2.p2.11.m3.5.5.cmml"><mrow id="S5.2.p2.11.m3.5.5.3" xref="S5.2.p2.11.m3.5.5.3.cmml"><msubsup id="S5.2.p2.11.m3.5.5.3.2" xref="S5.2.p2.11.m3.5.5.3.2.cmml"><mi id="S5.2.p2.11.m3.5.5.3.2.2.2" xref="S5.2.p2.11.m3.5.5.3.2.2.2.cmml">T</mi><mn id="S5.2.p2.11.m3.5.5.3.2.3" xref="S5.2.p2.11.m3.5.5.3.2.3.cmml">2</mn><mrow id="S5.2.p2.11.m3.1.1.1.3" xref="S5.2.p2.11.m3.5.5.3.2.cmml"><mo id="S5.2.p2.11.m3.1.1.1.3.1" stretchy="false" xref="S5.2.p2.11.m3.5.5.3.2.cmml">(</mo><mn id="S5.2.p2.11.m3.1.1.1.1" xref="S5.2.p2.11.m3.1.1.1.1.cmml">2</mn><mo id="S5.2.p2.11.m3.1.1.1.3.2" stretchy="false" xref="S5.2.p2.11.m3.5.5.3.2.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.11.m3.5.5.3.1" xref="S5.2.p2.11.m3.5.5.3.1.cmml">⁢</mo><mrow id="S5.2.p2.11.m3.5.5.3.3.2" xref="S5.2.p2.11.m3.5.5.3.cmml"><mo id="S5.2.p2.11.m3.5.5.3.3.2.1" stretchy="false" xref="S5.2.p2.11.m3.5.5.3.cmml">(</mo><mi id="S5.2.p2.11.m3.2.2" xref="S5.2.p2.11.m3.2.2.cmml">x</mi><mo id="S5.2.p2.11.m3.5.5.3.3.2.2" stretchy="false" xref="S5.2.p2.11.m3.5.5.3.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.11.m3.5.5.2" xref="S5.2.p2.11.m3.5.5.2.cmml">=</mo><mrow id="S5.2.p2.11.m3.5.5.1" xref="S5.2.p2.11.m3.5.5.1.cmml"><msub id="S5.2.p2.11.m3.5.5.1.3" xref="S5.2.p2.11.m3.5.5.1.3.cmml"><mi id="S5.2.p2.11.m3.5.5.1.3.2" xref="S5.2.p2.11.m3.5.5.1.3.2.cmml">σ</mi><mn id="S5.2.p2.11.m3.5.5.1.3.3" xref="S5.2.p2.11.m3.5.5.1.3.3.cmml">2</mn></msub><mo id="S5.2.p2.11.m3.5.5.1.2" xref="S5.2.p2.11.m3.5.5.1.2.cmml">⁢</mo><mrow id="S5.2.p2.11.m3.5.5.1.1.1" xref="S5.2.p2.11.m3.5.5.1.1.2.cmml"><mo id="S5.2.p2.11.m3.5.5.1.1.1.2" stretchy="false" xref="S5.2.p2.11.m3.5.5.1.1.2.cmml">(</mo><mn id="S5.2.p2.11.m3.3.3" xref="S5.2.p2.11.m3.3.3.cmml">1</mn><mo id="S5.2.p2.11.m3.5.5.1.1.1.3" xref="S5.2.p2.11.m3.5.5.1.1.2.cmml">,</mo><mn id="S5.2.p2.11.m3.4.4" xref="S5.2.p2.11.m3.4.4.cmml">1</mn><mo id="S5.2.p2.11.m3.5.5.1.1.1.4" xref="S5.2.p2.11.m3.5.5.1.1.2.cmml">,</mo><mrow id="S5.2.p2.11.m3.5.5.1.1.1.1" xref="S5.2.p2.11.m3.5.5.1.1.1.1.cmml"><mo id="S5.2.p2.11.m3.5.5.1.1.1.1a" xref="S5.2.p2.11.m3.5.5.1.1.1.1.cmml">−</mo><mn id="S5.2.p2.11.m3.5.5.1.1.1.1.2" xref="S5.2.p2.11.m3.5.5.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.2.p2.11.m3.5.5.1.1.1.5" stretchy="false" xref="S5.2.p2.11.m3.5.5.1.1.2.cmml">)</mo></mrow><mo id="S5.2.p2.11.m3.5.5.1.2a" xref="S5.2.p2.11.m3.5.5.1.2.cmml">⁢</mo><mi id="S5.2.p2.11.m3.5.5.1.4" xref="S5.2.p2.11.m3.5.5.1.4.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.11.m3.5b"><apply id="S5.2.p2.11.m3.5.5.cmml" xref="S5.2.p2.11.m3.5.5"><eq id="S5.2.p2.11.m3.5.5.2.cmml" xref="S5.2.p2.11.m3.5.5.2"></eq><apply id="S5.2.p2.11.m3.5.5.3.cmml" xref="S5.2.p2.11.m3.5.5.3"><times id="S5.2.p2.11.m3.5.5.3.1.cmml" xref="S5.2.p2.11.m3.5.5.3.1"></times><apply id="S5.2.p2.11.m3.5.5.3.2.cmml" xref="S5.2.p2.11.m3.5.5.3.2"><csymbol cd="ambiguous" id="S5.2.p2.11.m3.5.5.3.2.1.cmml" xref="S5.2.p2.11.m3.5.5.3.2">subscript</csymbol><apply id="S5.2.p2.11.m3.5.5.3.2.2.cmml" xref="S5.2.p2.11.m3.5.5.3.2"><csymbol cd="ambiguous" id="S5.2.p2.11.m3.5.5.3.2.2.1.cmml" xref="S5.2.p2.11.m3.5.5.3.2">superscript</csymbol><ci id="S5.2.p2.11.m3.5.5.3.2.2.2.cmml" xref="S5.2.p2.11.m3.5.5.3.2.2.2">𝑇</ci><cn id="S5.2.p2.11.m3.1.1.1.1.cmml" type="integer" xref="S5.2.p2.11.m3.1.1.1.1">2</cn></apply><cn id="S5.2.p2.11.m3.5.5.3.2.3.cmml" type="integer" xref="S5.2.p2.11.m3.5.5.3.2.3">2</cn></apply><ci id="S5.2.p2.11.m3.2.2.cmml" xref="S5.2.p2.11.m3.2.2">𝑥</ci></apply><apply id="S5.2.p2.11.m3.5.5.1.cmml" xref="S5.2.p2.11.m3.5.5.1"><times id="S5.2.p2.11.m3.5.5.1.2.cmml" xref="S5.2.p2.11.m3.5.5.1.2"></times><apply id="S5.2.p2.11.m3.5.5.1.3.cmml" xref="S5.2.p2.11.m3.5.5.1.3"><csymbol cd="ambiguous" id="S5.2.p2.11.m3.5.5.1.3.1.cmml" xref="S5.2.p2.11.m3.5.5.1.3">subscript</csymbol><ci id="S5.2.p2.11.m3.5.5.1.3.2.cmml" xref="S5.2.p2.11.m3.5.5.1.3.2">𝜎</ci><cn id="S5.2.p2.11.m3.5.5.1.3.3.cmml" type="integer" xref="S5.2.p2.11.m3.5.5.1.3.3">2</cn></apply><vector id="S5.2.p2.11.m3.5.5.1.1.2.cmml" xref="S5.2.p2.11.m3.5.5.1.1.1"><cn id="S5.2.p2.11.m3.3.3.cmml" type="integer" xref="S5.2.p2.11.m3.3.3">1</cn><cn id="S5.2.p2.11.m3.4.4.cmml" type="integer" xref="S5.2.p2.11.m3.4.4">1</cn><apply id="S5.2.p2.11.m3.5.5.1.1.1.1.cmml" xref="S5.2.p2.11.m3.5.5.1.1.1.1"><minus id="S5.2.p2.11.m3.5.5.1.1.1.1.1.cmml" xref="S5.2.p2.11.m3.5.5.1.1.1.1"></minus><cn id="S5.2.p2.11.m3.5.5.1.1.1.1.2.cmml" type="integer" xref="S5.2.p2.11.m3.5.5.1.1.1.1.2">1</cn></apply></vector><ci id="S5.2.p2.11.m3.5.5.1.4.cmml" xref="S5.2.p2.11.m3.5.5.1.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.11.m3.5c">T^{(2)}_{2}(x)=\sigma_{2}(1,1,-1)x</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.11.m3.5d">italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 , 1 , - 1 ) italic_x</annotation></semantics></math>. Similarly, <math alttext="\sigma_{1}\max({\mkern 2.0mu\cdot\mkern 2.0mu},{\mkern 2.0mu\cdot\mkern 2.0mu})" class="ltx_Math" display="inline" id="S5.2.p2.12.m4.3"><semantics id="S5.2.p2.12.m4.3a"><mrow id="S5.2.p2.12.m4.3.4" xref="S5.2.p2.12.m4.3.4.cmml"><msub id="S5.2.p2.12.m4.3.4.2" xref="S5.2.p2.12.m4.3.4.2.cmml"><mi id="S5.2.p2.12.m4.3.4.2.2" xref="S5.2.p2.12.m4.3.4.2.2.cmml">σ</mi><mn id="S5.2.p2.12.m4.3.4.2.3" xref="S5.2.p2.12.m4.3.4.2.3.cmml">1</mn></msub><mo id="S5.2.p2.12.m4.3.4.1" lspace="0.167em" xref="S5.2.p2.12.m4.3.4.1.cmml">⁢</mo><mrow id="S5.2.p2.12.m4.3.4.3.2" xref="S5.2.p2.12.m4.3.4.3.1.cmml"><mi id="S5.2.p2.12.m4.1.1" xref="S5.2.p2.12.m4.1.1.cmml">max</mi><mo id="S5.2.p2.12.m4.3.4.3.2a" xref="S5.2.p2.12.m4.3.4.3.1.cmml">⁡</mo><mrow id="S5.2.p2.12.m4.3.4.3.2.1" xref="S5.2.p2.12.m4.3.4.3.1.cmml"><mo id="S5.2.p2.12.m4.3.4.3.2.1.1" stretchy="false" xref="S5.2.p2.12.m4.3.4.3.1.cmml">(</mo><mo id="S5.2.p2.12.m4.2.2" lspace="0.110em" rspace="0.110em" xref="S5.2.p2.12.m4.2.2.cmml">⋅</mo><mo id="S5.2.p2.12.m4.3.4.3.2.1.2" rspace="0.055em" xref="S5.2.p2.12.m4.3.4.3.1.cmml">,</mo><mo id="S5.2.p2.12.m4.3.3" lspace="0.055em" rspace="0.110em" xref="S5.2.p2.12.m4.3.3.cmml">⋅</mo><mo id="S5.2.p2.12.m4.3.4.3.2.1.3" stretchy="false" xref="S5.2.p2.12.m4.3.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.12.m4.3b"><apply id="S5.2.p2.12.m4.3.4.cmml" xref="S5.2.p2.12.m4.3.4"><times id="S5.2.p2.12.m4.3.4.1.cmml" xref="S5.2.p2.12.m4.3.4.1"></times><apply id="S5.2.p2.12.m4.3.4.2.cmml" xref="S5.2.p2.12.m4.3.4.2"><csymbol cd="ambiguous" id="S5.2.p2.12.m4.3.4.2.1.cmml" xref="S5.2.p2.12.m4.3.4.2">subscript</csymbol><ci id="S5.2.p2.12.m4.3.4.2.2.cmml" xref="S5.2.p2.12.m4.3.4.2.2">𝜎</ci><cn id="S5.2.p2.12.m4.3.4.2.3.cmml" type="integer" xref="S5.2.p2.12.m4.3.4.2.3">1</cn></apply><apply id="S5.2.p2.12.m4.3.4.3.1.cmml" xref="S5.2.p2.12.m4.3.4.3.2"><max id="S5.2.p2.12.m4.1.1.cmml" xref="S5.2.p2.12.m4.1.1"></max><ci id="S5.2.p2.12.m4.2.2.cmml" xref="S5.2.p2.12.m4.2.2">⋅</ci><ci id="S5.2.p2.12.m4.3.3.cmml" xref="S5.2.p2.12.m4.3.3">⋅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.12.m4.3c">\sigma_{1}\max({\mkern 2.0mu\cdot\mkern 2.0mu},{\mkern 2.0mu\cdot\mkern 2.0mu})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.12.m4.3d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_max ( ⋅ , ⋅ )</annotation></semantics></math> can be represented by a neural network <math alttext="(T^{(1)}_{3},T^{(2)}_{3})" class="ltx_Math" display="inline" id="S5.2.p2.13.m5.4"><semantics id="S5.2.p2.13.m5.4a"><mrow id="S5.2.p2.13.m5.4.4.2" xref="S5.2.p2.13.m5.4.4.3.cmml"><mo id="S5.2.p2.13.m5.4.4.2.3" stretchy="false" xref="S5.2.p2.13.m5.4.4.3.cmml">(</mo><msubsup id="S5.2.p2.13.m5.3.3.1.1" xref="S5.2.p2.13.m5.3.3.1.1.cmml"><mi id="S5.2.p2.13.m5.3.3.1.1.2.2" xref="S5.2.p2.13.m5.3.3.1.1.2.2.cmml">T</mi><mn id="S5.2.p2.13.m5.3.3.1.1.3" xref="S5.2.p2.13.m5.3.3.1.1.3.cmml">3</mn><mrow id="S5.2.p2.13.m5.1.1.1.3" xref="S5.2.p2.13.m5.3.3.1.1.cmml"><mo id="S5.2.p2.13.m5.1.1.1.3.1" stretchy="false" xref="S5.2.p2.13.m5.3.3.1.1.cmml">(</mo><mn id="S5.2.p2.13.m5.1.1.1.1" xref="S5.2.p2.13.m5.1.1.1.1.cmml">1</mn><mo id="S5.2.p2.13.m5.1.1.1.3.2" stretchy="false" xref="S5.2.p2.13.m5.3.3.1.1.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.13.m5.4.4.2.4" xref="S5.2.p2.13.m5.4.4.3.cmml">,</mo><msubsup id="S5.2.p2.13.m5.4.4.2.2" xref="S5.2.p2.13.m5.4.4.2.2.cmml"><mi id="S5.2.p2.13.m5.4.4.2.2.2.2" xref="S5.2.p2.13.m5.4.4.2.2.2.2.cmml">T</mi><mn id="S5.2.p2.13.m5.4.4.2.2.3" xref="S5.2.p2.13.m5.4.4.2.2.3.cmml">3</mn><mrow id="S5.2.p2.13.m5.2.2.1.3" xref="S5.2.p2.13.m5.4.4.2.2.cmml"><mo id="S5.2.p2.13.m5.2.2.1.3.1" stretchy="false" xref="S5.2.p2.13.m5.4.4.2.2.cmml">(</mo><mn id="S5.2.p2.13.m5.2.2.1.1" xref="S5.2.p2.13.m5.2.2.1.1.cmml">2</mn><mo id="S5.2.p2.13.m5.2.2.1.3.2" stretchy="false" xref="S5.2.p2.13.m5.4.4.2.2.cmml">)</mo></mrow></msubsup><mo id="S5.2.p2.13.m5.4.4.2.5" stretchy="false" xref="S5.2.p2.13.m5.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.13.m5.4b"><interval closure="open" id="S5.2.p2.13.m5.4.4.3.cmml" xref="S5.2.p2.13.m5.4.4.2"><apply id="S5.2.p2.13.m5.3.3.1.1.cmml" xref="S5.2.p2.13.m5.3.3.1.1"><csymbol cd="ambiguous" id="S5.2.p2.13.m5.3.3.1.1.1.cmml" xref="S5.2.p2.13.m5.3.3.1.1">subscript</csymbol><apply id="S5.2.p2.13.m5.3.3.1.1.2.cmml" xref="S5.2.p2.13.m5.3.3.1.1"><csymbol cd="ambiguous" id="S5.2.p2.13.m5.3.3.1.1.2.1.cmml" xref="S5.2.p2.13.m5.3.3.1.1">superscript</csymbol><ci id="S5.2.p2.13.m5.3.3.1.1.2.2.cmml" xref="S5.2.p2.13.m5.3.3.1.1.2.2">𝑇</ci><cn id="S5.2.p2.13.m5.1.1.1.1.cmml" type="integer" xref="S5.2.p2.13.m5.1.1.1.1">1</cn></apply><cn id="S5.2.p2.13.m5.3.3.1.1.3.cmml" type="integer" xref="S5.2.p2.13.m5.3.3.1.1.3">3</cn></apply><apply id="S5.2.p2.13.m5.4.4.2.2.cmml" xref="S5.2.p2.13.m5.4.4.2.2"><csymbol cd="ambiguous" id="S5.2.p2.13.m5.4.4.2.2.1.cmml" xref="S5.2.p2.13.m5.4.4.2.2">subscript</csymbol><apply id="S5.2.p2.13.m5.4.4.2.2.2.cmml" xref="S5.2.p2.13.m5.4.4.2.2"><csymbol cd="ambiguous" id="S5.2.p2.13.m5.4.4.2.2.2.1.cmml" xref="S5.2.p2.13.m5.4.4.2.2">superscript</csymbol><ci id="S5.2.p2.13.m5.4.4.2.2.2.2.cmml" xref="S5.2.p2.13.m5.4.4.2.2.2.2">𝑇</ci><cn id="S5.2.p2.13.m5.2.2.1.1.cmml" type="integer" xref="S5.2.p2.13.m5.2.2.1.1">2</cn></apply><cn id="S5.2.p2.13.m5.4.4.2.2.3.cmml" type="integer" xref="S5.2.p2.13.m5.4.4.2.2.3">3</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.13.m5.4c">(T^{(1)}_{3},T^{(2)}_{3})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.13.m5.4d">( italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> of the same width.</p> </div> <div class="ltx_para" id="S5.3.p3"> <p class="ltx_p" id="S5.3.p3.4">In total, we get</p> <table class="ltx_equation ltx_eqn_table" id="S5.E41"> <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="\sigma_{1}\max(f_{1},\sigma_{2}\max(f_{2},f_{3}))=T^{(2)}_{3}\circ\rho\circ% \underbrace{T^{(1)}_{3}\circ\begin{bmatrix}T^{(2)}_{1}\\ T^{(2)}_{2}\end{bmatrix}}_{\mathds{R}^{5}\to\mathds{R}^{3}}\circ\rho\circ% \underbrace{\begin{bmatrix}T^{(1)}_{1}\\ T^{(1)}_{2}\end{bmatrix}}_{\mathds{R}^{2}\to\mathds{R}^{5}}." class="ltx_Math" display="block" id="S5.E41.m1.7"><semantics id="S5.E41.m1.7a"><mrow id="S5.E41.m1.7.7.1" xref="S5.E41.m1.7.7.1.1.cmml"><mrow id="S5.E41.m1.7.7.1.1" xref="S5.E41.m1.7.7.1.1.cmml"><mrow id="S5.E41.m1.7.7.1.1.2" xref="S5.E41.m1.7.7.1.1.2.cmml"><msub id="S5.E41.m1.7.7.1.1.2.4" xref="S5.E41.m1.7.7.1.1.2.4.cmml"><mi id="S5.E41.m1.7.7.1.1.2.4.2" xref="S5.E41.m1.7.7.1.1.2.4.2.cmml">σ</mi><mn id="S5.E41.m1.7.7.1.1.2.4.3" xref="S5.E41.m1.7.7.1.1.2.4.3.cmml">1</mn></msub><mo id="S5.E41.m1.7.7.1.1.2.3" lspace="0.167em" xref="S5.E41.m1.7.7.1.1.2.3.cmml">⁢</mo><mrow id="S5.E41.m1.7.7.1.1.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml"><mi id="S5.E41.m1.6.6" xref="S5.E41.m1.6.6.cmml">max</mi><mo id="S5.E41.m1.7.7.1.1.2.2.2a" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml">⁡</mo><mrow id="S5.E41.m1.7.7.1.1.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml"><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.3" stretchy="false" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml">(</mo><msub id="S5.E41.m1.7.7.1.1.1.1.1.1.1" xref="S5.E41.m1.7.7.1.1.1.1.1.1.1.cmml"><mi id="S5.E41.m1.7.7.1.1.1.1.1.1.1.2" xref="S5.E41.m1.7.7.1.1.1.1.1.1.1.2.cmml">f</mi><mn id="S5.E41.m1.7.7.1.1.1.1.1.1.1.3" xref="S5.E41.m1.7.7.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.4" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml">,</mo><mrow id="S5.E41.m1.7.7.1.1.2.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.cmml"><msub id="S5.E41.m1.7.7.1.1.2.2.2.2.2.4" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.4.cmml"><mi id="S5.E41.m1.7.7.1.1.2.2.2.2.2.4.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.4.2.cmml">σ</mi><mn id="S5.E41.m1.7.7.1.1.2.2.2.2.2.4.3" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.4.3.cmml">2</mn></msub><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.2.3" lspace="0.167em" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml"><mi id="S5.E41.m1.5.5" xref="S5.E41.m1.5.5.cmml">max</mi><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2a" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml">⁡</mo><mrow id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml"><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.3" stretchy="false" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml">(</mo><msub id="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1.cmml"><mi id="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1.2.cmml">f</mi><mn id="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1.3" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.1.1.1.1.3.cmml">2</mn></msub><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.4" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml">,</mo><msub id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.cmml"><mi id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.2" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.2.cmml">f</mi><mn id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.3" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.2.2.5" stretchy="false" xref="S5.E41.m1.7.7.1.1.2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E41.m1.7.7.1.1.2.2.2.2.5" stretchy="false" xref="S5.E41.m1.7.7.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E41.m1.7.7.1.1.3" xref="S5.E41.m1.7.7.1.1.3.cmml">=</mo><mrow id="S5.E41.m1.7.7.1.1.4" xref="S5.E41.m1.7.7.1.1.4.cmml"><msubsup id="S5.E41.m1.7.7.1.1.4.2" xref="S5.E41.m1.7.7.1.1.4.2.cmml"><mi id="S5.E41.m1.7.7.1.1.4.2.2.2" xref="S5.E41.m1.7.7.1.1.4.2.2.2.cmml">T</mi><mn id="S5.E41.m1.7.7.1.1.4.2.3" xref="S5.E41.m1.7.7.1.1.4.2.3.cmml">3</mn><mrow id="S5.E41.m1.3.3.1.3" xref="S5.E41.m1.7.7.1.1.4.2.cmml"><mo id="S5.E41.m1.3.3.1.3.1" stretchy="false" xref="S5.E41.m1.7.7.1.1.4.2.cmml">(</mo><mn id="S5.E41.m1.3.3.1.1" xref="S5.E41.m1.3.3.1.1.cmml">2</mn><mo id="S5.E41.m1.3.3.1.3.2" stretchy="false" xref="S5.E41.m1.7.7.1.1.4.2.cmml">)</mo></mrow></msubsup><mo id="S5.E41.m1.7.7.1.1.4.1" lspace="0.222em" rspace="0.222em" xref="S5.E41.m1.7.7.1.1.4.1.cmml">∘</mo><mi id="S5.E41.m1.7.7.1.1.4.3" xref="S5.E41.m1.7.7.1.1.4.3.cmml">ρ</mi><mo id="S5.E41.m1.7.7.1.1.4.1a" lspace="0.222em" rspace="0.222em" xref="S5.E41.m1.7.7.1.1.4.1.cmml">∘</mo><munder id="S5.E41.m1.7.7.1.1.4.4" xref="S5.E41.m1.7.7.1.1.4.4.cmml"><munder accentunder="true" id="S5.E41.m1.4.4" xref="S5.E41.m1.4.4.cmml"><mrow id="S5.E41.m1.4.4.2" xref="S5.E41.m1.4.4.2.cmml"><msubsup id="S5.E41.m1.4.4.2.4" xref="S5.E41.m1.4.4.2.4.cmml"><mi id="S5.E41.m1.4.4.2.4.2.2" xref="S5.E41.m1.4.4.2.4.2.2.cmml">T</mi><mn id="S5.E41.m1.4.4.2.4.3" xref="S5.E41.m1.4.4.2.4.3.cmml">3</mn><mrow id="S5.E41.m1.4.4.2.2.1.3" xref="S5.E41.m1.4.4.2.4.cmml"><mo id="S5.E41.m1.4.4.2.2.1.3.1" stretchy="false" xref="S5.E41.m1.4.4.2.4.cmml">(</mo><mn id="S5.E41.m1.4.4.2.2.1.1" xref="S5.E41.m1.4.4.2.2.1.1.cmml">1</mn><mo id="S5.E41.m1.4.4.2.2.1.3.2" stretchy="false" xref="S5.E41.m1.4.4.2.4.cmml">)</mo></mrow></msubsup><mo id="S5.E41.m1.4.4.2.3" lspace="0.222em" rspace="0.222em" xref="S5.E41.m1.4.4.2.3.cmml">∘</mo><mrow id="S5.E41.m1.1.1.1.1.3" xref="S5.E41.m1.1.1.1.1.2.cmml"><mo id="S5.E41.m1.1.1.1.1.3.1" xref="S5.E41.m1.1.1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.E41.m1.1.1.1.1.1.1" rowspacing="0pt" xref="S5.E41.m1.1.1.1.1.1.1.cmml"><mtr id="S5.E41.m1.1.1.1.1.1.1a" xref="S5.E41.m1.1.1.1.1.1.1.cmml"><mtd id="S5.E41.m1.1.1.1.1.1.1b" xref="S5.E41.m1.1.1.1.1.1.1.cmml"><msubsup id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">T</mi><mn id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.4" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">1</mn><mrow id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.1" stretchy="false" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">2</mn><mo id="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.2" stretchy="false" xref="S5.E41.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr><mtr id="S5.E41.m1.1.1.1.1.1.1c" xref="S5.E41.m1.1.1.1.1.1.1.cmml"><mtd id="S5.E41.m1.1.1.1.1.1.1d" xref="S5.E41.m1.1.1.1.1.1.1.cmml"><msubsup id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.cmml"><mi id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.3.2" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.3.2.cmml">T</mi><mn id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.4" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.4.cmml">2</mn><mrow id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.1.1.3" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.cmml"><mo id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.1.1.3.1" stretchy="false" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.cmml">(</mo><mn id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.1.1.1" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.1.1.1.cmml">2</mn><mo id="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.1.1.3.2" stretchy="false" xref="S5.E41.m1.1.1.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr></mtable><mo id="S5.E41.m1.1.1.1.1.3.2" rspace="0.055em" xref="S5.E41.m1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.E41.m1.4.4.3" xref="S5.E41.m1.4.4.3.cmml">⏟</mo></munder><mrow id="S5.E41.m1.7.7.1.1.4.4.2" xref="S5.E41.m1.7.7.1.1.4.4.2.cmml"><msup id="S5.E41.m1.7.7.1.1.4.4.2.2" xref="S5.E41.m1.7.7.1.1.4.4.2.2.cmml"><mi id="S5.E41.m1.7.7.1.1.4.4.2.2.2" xref="S5.E41.m1.7.7.1.1.4.4.2.2.2.cmml">ℝ</mi><mn id="S5.E41.m1.7.7.1.1.4.4.2.2.3" xref="S5.E41.m1.7.7.1.1.4.4.2.2.3.cmml">5</mn></msup><mo id="S5.E41.m1.7.7.1.1.4.4.2.1" stretchy="false" xref="S5.E41.m1.7.7.1.1.4.4.2.1.cmml">→</mo><msup id="S5.E41.m1.7.7.1.1.4.4.2.3" xref="S5.E41.m1.7.7.1.1.4.4.2.3.cmml"><mi id="S5.E41.m1.7.7.1.1.4.4.2.3.2" xref="S5.E41.m1.7.7.1.1.4.4.2.3.2.cmml">ℝ</mi><mn id="S5.E41.m1.7.7.1.1.4.4.2.3.3" xref="S5.E41.m1.7.7.1.1.4.4.2.3.3.cmml">3</mn></msup></mrow></munder><mo id="S5.E41.m1.7.7.1.1.4.1b" rspace="0.222em" xref="S5.E41.m1.7.7.1.1.4.1.cmml">∘</mo><mi id="S5.E41.m1.7.7.1.1.4.5" xref="S5.E41.m1.7.7.1.1.4.5.cmml">ρ</mi><mo id="S5.E41.m1.7.7.1.1.4.1c" lspace="0.222em" rspace="0.222em" xref="S5.E41.m1.7.7.1.1.4.1.cmml">∘</mo><munder id="S5.E41.m1.7.7.1.1.4.6" xref="S5.E41.m1.7.7.1.1.4.6.cmml"><munder accentunder="true" id="S5.E41.m1.2.2" xref="S5.E41.m1.2.2.cmml"><mrow id="S5.E41.m1.2.2.1.3" xref="S5.E41.m1.2.2.1.2.cmml"><mo id="S5.E41.m1.2.2.1.3.1" xref="S5.E41.m1.2.2.1.2.1.cmml">[</mo><mtable 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xref="S5.E41.m1.2.2.1.1.1.1.cmml"><mtd id="S5.E41.m1.2.2.1.1.1.1d" xref="S5.E41.m1.2.2.1.1.1.1.cmml"><msubsup id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.cmml"><mi id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.3.2" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.3.2.cmml">T</mi><mn id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.4" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.4.cmml">2</mn><mrow id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.1.1.3" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.cmml"><mo id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.1.1.3.1" stretchy="false" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.cmml">(</mo><mn id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.1.1.1" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.1.1.1.cmml">1</mn><mo id="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.1.1.3.2" stretchy="false" xref="S5.E41.m1.2.2.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></msubsup></mtd></mtr></mtable><mo id="S5.E41.m1.2.2.1.3.2" xref="S5.E41.m1.2.2.1.2.1.cmml">]</mo></mrow><mo id="S5.E41.m1.2.2.2" xref="S5.E41.m1.2.2.2.cmml">⏟</mo></munder><mrow id="S5.E41.m1.7.7.1.1.4.6.2" 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\underbrace{\begin{bmatrix}T^{(1)}_{1}\\ T^{(1)}_{2}\end{bmatrix}}_{\mathds{R}^{2}\to\mathds{R}^{5}}.</annotation><annotation encoding="application/x-llamapun" id="S5.E41.m1.7d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) = italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∘ italic_ρ ∘ under⏟ start_ARG italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∘ [ start_ARG start_ROW start_CELL italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_T start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] end_ARG start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∘ italic_ρ ∘ under⏟ start_ARG [ start_ARG start_ROW start_CELL italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_T start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] end_ARG start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT 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">(41)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.3.p3.3">Therefore, each summand of the form <math 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id="S5.3.p3.1.m1.9c">\sigma_{n}^{(1)}\max(f_{n}^{(1)},\sigma_{n}^{(2)}\max(f_{n}^{(2)},f_{n}^{(3)}))</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.1.m1.9d">italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT roman_max ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ) )</annotation></semantics></math> can be represented by a neural network with two hidden layers of width <math alttext="5" class="ltx_Math" display="inline" id="S5.3.p3.2.m2.1"><semantics id="S5.3.p3.2.m2.1a"><mn id="S5.3.p3.2.m2.1.1" xref="S5.3.p3.2.m2.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S5.3.p3.2.m2.1b"><cn id="S5.3.p3.2.m2.1.1.cmml" type="integer" xref="S5.3.p3.2.m2.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.2.m2.1c">5</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.2.m2.1d">5</annotation></semantics></math> and <math alttext="3" class="ltx_Math" display="inline" id="S5.3.p3.3.m3.1"><semantics id="S5.3.p3.3.m3.1a"><mn id="S5.3.p3.3.m3.1.1" xref="S5.3.p3.3.m3.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S5.3.p3.3.m3.1b"><cn id="S5.3.p3.3.m3.1.1.cmml" type="integer" xref="S5.3.p3.3.m3.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.3.m3.1c">3</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.3.m3.1d">3</annotation></semantics></math> respectively.</p> </div> <div class="ltx_para" id="S5.4.p4"> <p class="ltx_p" id="S5.4.p4.5">Now, we can represent a vector-valued function, whose <math alttext="i" class="ltx_Math" display="inline" id="S5.4.p4.1.m1.1"><semantics id="S5.4.p4.1.m1.1a"><mi id="S5.4.p4.1.m1.1.1" xref="S5.4.p4.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.1.m1.1b"><ci id="S5.4.p4.1.m1.1.1.cmml" xref="S5.4.p4.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.1.m1.1d">italic_i</annotation></semantics></math>-th component is the <math alttext="i" class="ltx_Math" display="inline" id="S5.4.p4.2.m2.1"><semantics id="S5.4.p4.2.m2.1a"><mi id="S5.4.p4.2.m2.1.1" xref="S5.4.p4.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.2.m2.1b"><ci id="S5.4.p4.2.m2.1.1.cmml" xref="S5.4.p4.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.2.m2.1d">italic_i</annotation></semantics></math>-th summand of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S5.E40" title="Equation 40 ‣ Proof of Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">40</span></a>), with a neural network by stacking the affine functions of the neural networks that represent the individual summands for each layer separately. Finally, to represent <math alttext="f" class="ltx_Math" display="inline" id="S5.4.p4.3.m3.1"><semantics id="S5.4.p4.3.m3.1a"><mi id="S5.4.p4.3.m3.1.1" xref="S5.4.p4.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.3.m3.1b"><ci id="S5.4.p4.3.m3.1.1.cmml" xref="S5.4.p4.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.3.m3.1d">italic_f</annotation></semantics></math>, we need to replace the affine function of the last layer with its concatenation with the linear sum operation. Since there are <math alttext="9p" class="ltx_Math" display="inline" id="S5.4.p4.4.m4.1"><semantics id="S5.4.p4.4.m4.1a"><mrow id="S5.4.p4.4.m4.1.1" xref="S5.4.p4.4.m4.1.1.cmml"><mn id="S5.4.p4.4.m4.1.1.2" xref="S5.4.p4.4.m4.1.1.2.cmml">9</mn><mo id="S5.4.p4.4.m4.1.1.1" xref="S5.4.p4.4.m4.1.1.1.cmml">⁢</mo><mi id="S5.4.p4.4.m4.1.1.3" xref="S5.4.p4.4.m4.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.4.m4.1b"><apply id="S5.4.p4.4.m4.1.1.cmml" xref="S5.4.p4.4.m4.1.1"><times id="S5.4.p4.4.m4.1.1.1.cmml" xref="S5.4.p4.4.m4.1.1.1"></times><cn id="S5.4.p4.4.m4.1.1.2.cmml" type="integer" xref="S5.4.p4.4.m4.1.1.2">9</cn><ci id="S5.4.p4.4.m4.1.1.3.cmml" xref="S5.4.p4.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.4.m4.1c">9p</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.4.m4.1d">9 italic_p</annotation></semantics></math> summands, the width vector is <math alttext="(s_{1},s_{2})=9p(5,3)=(45p,27p)" class="ltx_Math" display="inline" id="S5.4.p4.5.m5.6"><semantics id="S5.4.p4.5.m5.6a"><mrow id="S5.4.p4.5.m5.6.6" xref="S5.4.p4.5.m5.6.6.cmml"><mrow id="S5.4.p4.5.m5.4.4.2.2" xref="S5.4.p4.5.m5.4.4.2.3.cmml"><mo id="S5.4.p4.5.m5.4.4.2.2.3" stretchy="false" xref="S5.4.p4.5.m5.4.4.2.3.cmml">(</mo><msub id="S5.4.p4.5.m5.3.3.1.1.1" xref="S5.4.p4.5.m5.3.3.1.1.1.cmml"><mi id="S5.4.p4.5.m5.3.3.1.1.1.2" xref="S5.4.p4.5.m5.3.3.1.1.1.2.cmml">s</mi><mn id="S5.4.p4.5.m5.3.3.1.1.1.3" xref="S5.4.p4.5.m5.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S5.4.p4.5.m5.4.4.2.2.4" xref="S5.4.p4.5.m5.4.4.2.3.cmml">,</mo><msub id="S5.4.p4.5.m5.4.4.2.2.2" xref="S5.4.p4.5.m5.4.4.2.2.2.cmml"><mi id="S5.4.p4.5.m5.4.4.2.2.2.2" xref="S5.4.p4.5.m5.4.4.2.2.2.2.cmml">s</mi><mn id="S5.4.p4.5.m5.4.4.2.2.2.3" xref="S5.4.p4.5.m5.4.4.2.2.2.3.cmml">2</mn></msub><mo id="S5.4.p4.5.m5.4.4.2.2.5" stretchy="false" xref="S5.4.p4.5.m5.4.4.2.3.cmml">)</mo></mrow><mo id="S5.4.p4.5.m5.6.6.6" xref="S5.4.p4.5.m5.6.6.6.cmml">=</mo><mrow id="S5.4.p4.5.m5.6.6.7" xref="S5.4.p4.5.m5.6.6.7.cmml"><mn id="S5.4.p4.5.m5.6.6.7.2" xref="S5.4.p4.5.m5.6.6.7.2.cmml">9</mn><mo id="S5.4.p4.5.m5.6.6.7.1" xref="S5.4.p4.5.m5.6.6.7.1.cmml">⁢</mo><mi id="S5.4.p4.5.m5.6.6.7.3" xref="S5.4.p4.5.m5.6.6.7.3.cmml">p</mi><mo id="S5.4.p4.5.m5.6.6.7.1a" xref="S5.4.p4.5.m5.6.6.7.1.cmml">⁢</mo><mrow id="S5.4.p4.5.m5.6.6.7.4.2" xref="S5.4.p4.5.m5.6.6.7.4.1.cmml"><mo id="S5.4.p4.5.m5.6.6.7.4.2.1" stretchy="false" xref="S5.4.p4.5.m5.6.6.7.4.1.cmml">(</mo><mn id="S5.4.p4.5.m5.1.1" xref="S5.4.p4.5.m5.1.1.cmml">5</mn><mo id="S5.4.p4.5.m5.6.6.7.4.2.2" xref="S5.4.p4.5.m5.6.6.7.4.1.cmml">,</mo><mn id="S5.4.p4.5.m5.2.2" xref="S5.4.p4.5.m5.2.2.cmml">3</mn><mo id="S5.4.p4.5.m5.6.6.7.4.2.3" stretchy="false" xref="S5.4.p4.5.m5.6.6.7.4.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p4.5.m5.6.6.8" xref="S5.4.p4.5.m5.6.6.8.cmml">=</mo><mrow id="S5.4.p4.5.m5.6.6.4.2" xref="S5.4.p4.5.m5.6.6.4.3.cmml"><mo id="S5.4.p4.5.m5.6.6.4.2.3" stretchy="false" xref="S5.4.p4.5.m5.6.6.4.3.cmml">(</mo><mrow id="S5.4.p4.5.m5.5.5.3.1.1" xref="S5.4.p4.5.m5.5.5.3.1.1.cmml"><mn id="S5.4.p4.5.m5.5.5.3.1.1.2" xref="S5.4.p4.5.m5.5.5.3.1.1.2.cmml">45</mn><mo id="S5.4.p4.5.m5.5.5.3.1.1.1" xref="S5.4.p4.5.m5.5.5.3.1.1.1.cmml">⁢</mo><mi id="S5.4.p4.5.m5.5.5.3.1.1.3" xref="S5.4.p4.5.m5.5.5.3.1.1.3.cmml">p</mi></mrow><mo id="S5.4.p4.5.m5.6.6.4.2.4" xref="S5.4.p4.5.m5.6.6.4.3.cmml">,</mo><mrow id="S5.4.p4.5.m5.6.6.4.2.2" xref="S5.4.p4.5.m5.6.6.4.2.2.cmml"><mn id="S5.4.p4.5.m5.6.6.4.2.2.2" xref="S5.4.p4.5.m5.6.6.4.2.2.2.cmml">27</mn><mo id="S5.4.p4.5.m5.6.6.4.2.2.1" xref="S5.4.p4.5.m5.6.6.4.2.2.1.cmml">⁢</mo><mi id="S5.4.p4.5.m5.6.6.4.2.2.3" xref="S5.4.p4.5.m5.6.6.4.2.2.3.cmml">p</mi></mrow><mo id="S5.4.p4.5.m5.6.6.4.2.5" stretchy="false" xref="S5.4.p4.5.m5.6.6.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.5.m5.6b"><apply id="S5.4.p4.5.m5.6.6.cmml" xref="S5.4.p4.5.m5.6.6"><and id="S5.4.p4.5.m5.6.6a.cmml" xref="S5.4.p4.5.m5.6.6"></and><apply id="S5.4.p4.5.m5.6.6b.cmml" xref="S5.4.p4.5.m5.6.6"><eq id="S5.4.p4.5.m5.6.6.6.cmml" xref="S5.4.p4.5.m5.6.6.6"></eq><interval closure="open" id="S5.4.p4.5.m5.4.4.2.3.cmml" xref="S5.4.p4.5.m5.4.4.2.2"><apply id="S5.4.p4.5.m5.3.3.1.1.1.cmml" xref="S5.4.p4.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.3.3.1.1.1.1.cmml" xref="S5.4.p4.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S5.4.p4.5.m5.3.3.1.1.1.2.cmml" xref="S5.4.p4.5.m5.3.3.1.1.1.2">𝑠</ci><cn id="S5.4.p4.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S5.4.p4.5.m5.3.3.1.1.1.3">1</cn></apply><apply id="S5.4.p4.5.m5.4.4.2.2.2.cmml" xref="S5.4.p4.5.m5.4.4.2.2.2"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.4.4.2.2.2.1.cmml" xref="S5.4.p4.5.m5.4.4.2.2.2">subscript</csymbol><ci id="S5.4.p4.5.m5.4.4.2.2.2.2.cmml" xref="S5.4.p4.5.m5.4.4.2.2.2.2">𝑠</ci><cn id="S5.4.p4.5.m5.4.4.2.2.2.3.cmml" type="integer" xref="S5.4.p4.5.m5.4.4.2.2.2.3">2</cn></apply></interval><apply id="S5.4.p4.5.m5.6.6.7.cmml" xref="S5.4.p4.5.m5.6.6.7"><times id="S5.4.p4.5.m5.6.6.7.1.cmml" xref="S5.4.p4.5.m5.6.6.7.1"></times><cn id="S5.4.p4.5.m5.6.6.7.2.cmml" type="integer" xref="S5.4.p4.5.m5.6.6.7.2">9</cn><ci id="S5.4.p4.5.m5.6.6.7.3.cmml" xref="S5.4.p4.5.m5.6.6.7.3">𝑝</ci><interval closure="open" id="S5.4.p4.5.m5.6.6.7.4.1.cmml" xref="S5.4.p4.5.m5.6.6.7.4.2"><cn id="S5.4.p4.5.m5.1.1.cmml" type="integer" xref="S5.4.p4.5.m5.1.1">5</cn><cn id="S5.4.p4.5.m5.2.2.cmml" type="integer" xref="S5.4.p4.5.m5.2.2">3</cn></interval></apply></apply><apply id="S5.4.p4.5.m5.6.6c.cmml" xref="S5.4.p4.5.m5.6.6"><eq id="S5.4.p4.5.m5.6.6.8.cmml" xref="S5.4.p4.5.m5.6.6.8"></eq><share href="https://arxiv.org/html/2503.13001v1#S5.4.p4.5.m5.6.6.7.cmml" id="S5.4.p4.5.m5.6.6d.cmml" xref="S5.4.p4.5.m5.6.6"></share><interval closure="open" id="S5.4.p4.5.m5.6.6.4.3.cmml" xref="S5.4.p4.5.m5.6.6.4.2"><apply id="S5.4.p4.5.m5.5.5.3.1.1.cmml" xref="S5.4.p4.5.m5.5.5.3.1.1"><times id="S5.4.p4.5.m5.5.5.3.1.1.1.cmml" xref="S5.4.p4.5.m5.5.5.3.1.1.1"></times><cn id="S5.4.p4.5.m5.5.5.3.1.1.2.cmml" type="integer" xref="S5.4.p4.5.m5.5.5.3.1.1.2">45</cn><ci id="S5.4.p4.5.m5.5.5.3.1.1.3.cmml" xref="S5.4.p4.5.m5.5.5.3.1.1.3">𝑝</ci></apply><apply id="S5.4.p4.5.m5.6.6.4.2.2.cmml" xref="S5.4.p4.5.m5.6.6.4.2.2"><times id="S5.4.p4.5.m5.6.6.4.2.2.1.cmml" xref="S5.4.p4.5.m5.6.6.4.2.2.1"></times><cn id="S5.4.p4.5.m5.6.6.4.2.2.2.cmml" type="integer" xref="S5.4.p4.5.m5.6.6.4.2.2.2">27</cn><ci id="S5.4.p4.5.m5.6.6.4.2.2.3.cmml" xref="S5.4.p4.5.m5.6.6.4.2.2.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.5.m5.6c">(s_{1},s_{2})=9p(5,3)=(45p,27p)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.5.m5.6d">( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 9 italic_p ( 5 , 3 ) = ( 45 italic_p , 27 italic_p )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">The provided constants are meant to show that they are reasonable and could be slightly improved. The following corollary highlights the efficiency of the neural network representation by addressing the sparsity of the model parameters.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.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="S5.Thmtheorem3.1.1.1">Corollary 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.5.5">Any <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p1.1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><apply id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1"><in id="S5.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.1"></in><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.2">CPA</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="p" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.2.m2.1a"><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.1b"><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.1d">italic_p</annotation></semantics></math> pieces can be represented by a <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.3.m3.1a"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.1b"><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.1d">roman_ReLU</annotation></semantics></math> neural network of depth <math alttext="3" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.4.m4.1a"><mn id="S5.Thmtheorem3.p1.4.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.4.m4.1b"><cn id="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem3.p1.4.4.m4.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.4.m4.1c">3</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.4.m4.1d">3</annotation></semantics></math> with <math alttext="O(p)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.5.m5.1"><semantics id="S5.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.cmml"><mi id="S5.Thmtheorem3.p1.5.5.m5.1.2.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.2.cmml">O</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.2.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.2.3.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.cmml"><mo id="S5.Thmtheorem3.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.cmml">(</mo><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml">p</mi><mo id="S5.Thmtheorem3.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.5.5.m5.1b"><apply id="S5.Thmtheorem3.p1.5.5.m5.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.2"><times id="S5.Thmtheorem3.p1.5.5.m5.1.2.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.1"></times><ci id="S5.Thmtheorem3.p1.5.5.m5.1.2.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.2">𝑂</ci><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.5.5.m5.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.5.m5.1d">italic_O ( italic_p )</annotation></semantics></math> non-zero parameters.</span></p> </div> </div> <div class="ltx_proof" id="S5.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.5.p1"> <p class="ltx_p" id="S5.5.p1.5">For <math alttext="i\in[w]" class="ltx_Math" display="inline" id="S5.5.p1.1.m1.1"><semantics id="S5.5.p1.1.m1.1a"><mrow id="S5.5.p1.1.m1.1.2" xref="S5.5.p1.1.m1.1.2.cmml"><mi id="S5.5.p1.1.m1.1.2.2" xref="S5.5.p1.1.m1.1.2.2.cmml">i</mi><mo id="S5.5.p1.1.m1.1.2.1" xref="S5.5.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S5.5.p1.1.m1.1.2.3.2" xref="S5.5.p1.1.m1.1.2.3.1.cmml"><mo id="S5.5.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.5.p1.1.m1.1.2.3.1.1.cmml">[</mo><mi id="S5.5.p1.1.m1.1.1" xref="S5.5.p1.1.m1.1.1.cmml">w</mi><mo id="S5.5.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.5.p1.1.m1.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.1.m1.1b"><apply id="S5.5.p1.1.m1.1.2.cmml" xref="S5.5.p1.1.m1.1.2"><in id="S5.5.p1.1.m1.1.2.1.cmml" xref="S5.5.p1.1.m1.1.2.1"></in><ci id="S5.5.p1.1.m1.1.2.2.cmml" xref="S5.5.p1.1.m1.1.2.2">𝑖</ci><apply id="S5.5.p1.1.m1.1.2.3.1.cmml" xref="S5.5.p1.1.m1.1.2.3.2"><csymbol cd="latexml" id="S5.5.p1.1.m1.1.2.3.1.1.cmml" xref="S5.5.p1.1.m1.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.5.p1.1.m1.1.1.cmml" xref="S5.5.p1.1.m1.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.1.m1.1c">i\in[w]</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.1.m1.1d">italic_i ∈ [ italic_w ]</annotation></semantics></math>, let <math alttext="T_{i}:\,\mathds{R}^{n_{i}}\to\mathds{R}^{m_{i}}" class="ltx_Math" display="inline" id="S5.5.p1.2.m2.1"><semantics id="S5.5.p1.2.m2.1a"><mrow id="S5.5.p1.2.m2.1.1" xref="S5.5.p1.2.m2.1.1.cmml"><msub id="S5.5.p1.2.m2.1.1.2" xref="S5.5.p1.2.m2.1.1.2.cmml"><mi id="S5.5.p1.2.m2.1.1.2.2" xref="S5.5.p1.2.m2.1.1.2.2.cmml">T</mi><mi id="S5.5.p1.2.m2.1.1.2.3" xref="S5.5.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.448em" xref="S5.5.p1.2.m2.1.1.1.cmml">:</mo><mrow id="S5.5.p1.2.m2.1.1.3" xref="S5.5.p1.2.m2.1.1.3.cmml"><msup id="S5.5.p1.2.m2.1.1.3.2" xref="S5.5.p1.2.m2.1.1.3.2.cmml"><mi id="S5.5.p1.2.m2.1.1.3.2.2" xref="S5.5.p1.2.m2.1.1.3.2.2.cmml">ℝ</mi><msub id="S5.5.p1.2.m2.1.1.3.2.3" xref="S5.5.p1.2.m2.1.1.3.2.3.cmml"><mi id="S5.5.p1.2.m2.1.1.3.2.3.2" xref="S5.5.p1.2.m2.1.1.3.2.3.2.cmml">n</mi><mi id="S5.5.p1.2.m2.1.1.3.2.3.3" xref="S5.5.p1.2.m2.1.1.3.2.3.3.cmml">i</mi></msub></msup><mo id="S5.5.p1.2.m2.1.1.3.1" stretchy="false" xref="S5.5.p1.2.m2.1.1.3.1.cmml">→</mo><msup id="S5.5.p1.2.m2.1.1.3.3" xref="S5.5.p1.2.m2.1.1.3.3.cmml"><mi id="S5.5.p1.2.m2.1.1.3.3.2" xref="S5.5.p1.2.m2.1.1.3.3.2.cmml">ℝ</mi><msub id="S5.5.p1.2.m2.1.1.3.3.3" xref="S5.5.p1.2.m2.1.1.3.3.3.cmml"><mi id="S5.5.p1.2.m2.1.1.3.3.3.2" xref="S5.5.p1.2.m2.1.1.3.3.3.2.cmml">m</mi><mi id="S5.5.p1.2.m2.1.1.3.3.3.3" xref="S5.5.p1.2.m2.1.1.3.3.3.3.cmml">i</mi></msub></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.2.m2.1b"><apply id="S5.5.p1.2.m2.1.1.cmml" xref="S5.5.p1.2.m2.1.1"><ci id="S5.5.p1.2.m2.1.1.1.cmml" xref="S5.5.p1.2.m2.1.1.1">:</ci><apply id="S5.5.p1.2.m2.1.1.2.cmml" xref="S5.5.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.2.1.cmml" xref="S5.5.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.2.2.cmml" xref="S5.5.p1.2.m2.1.1.2.2">𝑇</ci><ci id="S5.5.p1.2.m2.1.1.2.3.cmml" xref="S5.5.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.2.m2.1.1.3.cmml" xref="S5.5.p1.2.m2.1.1.3"><ci id="S5.5.p1.2.m2.1.1.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.1">→</ci><apply id="S5.5.p1.2.m2.1.1.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.2.1.cmml" xref="S5.5.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.2.2.cmml" xref="S5.5.p1.2.m2.1.1.3.2.2">ℝ</ci><apply id="S5.5.p1.2.m2.1.1.3.2.3.cmml" xref="S5.5.p1.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.2.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.2.3">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.2.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.2.3.2">𝑛</ci><ci id="S5.5.p1.2.m2.1.1.3.2.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.2.3.3">𝑖</ci></apply></apply><apply id="S5.5.p1.2.m2.1.1.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.3.2">ℝ</ci><apply id="S5.5.p1.2.m2.1.1.3.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.3.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.3.3">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.3.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.3.3.2">𝑚</ci><ci id="S5.5.p1.2.m2.1.1.3.3.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.2.m2.1c">T_{i}:\,\mathds{R}^{n_{i}}\to\mathds{R}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be affine functions, with <math alttext="T_{i}(x)=A_{i}x+b_{i}" class="ltx_Math" display="inline" id="S5.5.p1.3.m3.1"><semantics id="S5.5.p1.3.m3.1a"><mrow id="S5.5.p1.3.m3.1.2" xref="S5.5.p1.3.m3.1.2.cmml"><mrow id="S5.5.p1.3.m3.1.2.2" xref="S5.5.p1.3.m3.1.2.2.cmml"><msub id="S5.5.p1.3.m3.1.2.2.2" xref="S5.5.p1.3.m3.1.2.2.2.cmml"><mi id="S5.5.p1.3.m3.1.2.2.2.2" xref="S5.5.p1.3.m3.1.2.2.2.2.cmml">T</mi><mi id="S5.5.p1.3.m3.1.2.2.2.3" xref="S5.5.p1.3.m3.1.2.2.2.3.cmml">i</mi></msub><mo id="S5.5.p1.3.m3.1.2.2.1" xref="S5.5.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.5.p1.3.m3.1.2.2.3.2" xref="S5.5.p1.3.m3.1.2.2.cmml"><mo id="S5.5.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.5.p1.3.m3.1.2.2.cmml">(</mo><mi id="S5.5.p1.3.m3.1.1" xref="S5.5.p1.3.m3.1.1.cmml">x</mi><mo id="S5.5.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.5.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.3.m3.1.2.1" xref="S5.5.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S5.5.p1.3.m3.1.2.3" xref="S5.5.p1.3.m3.1.2.3.cmml"><mrow id="S5.5.p1.3.m3.1.2.3.2" xref="S5.5.p1.3.m3.1.2.3.2.cmml"><msub id="S5.5.p1.3.m3.1.2.3.2.2" xref="S5.5.p1.3.m3.1.2.3.2.2.cmml"><mi id="S5.5.p1.3.m3.1.2.3.2.2.2" xref="S5.5.p1.3.m3.1.2.3.2.2.2.cmml">A</mi><mi id="S5.5.p1.3.m3.1.2.3.2.2.3" xref="S5.5.p1.3.m3.1.2.3.2.2.3.cmml">i</mi></msub><mo id="S5.5.p1.3.m3.1.2.3.2.1" xref="S5.5.p1.3.m3.1.2.3.2.1.cmml">⁢</mo><mi id="S5.5.p1.3.m3.1.2.3.2.3" xref="S5.5.p1.3.m3.1.2.3.2.3.cmml">x</mi></mrow><mo id="S5.5.p1.3.m3.1.2.3.1" xref="S5.5.p1.3.m3.1.2.3.1.cmml">+</mo><msub id="S5.5.p1.3.m3.1.2.3.3" xref="S5.5.p1.3.m3.1.2.3.3.cmml"><mi id="S5.5.p1.3.m3.1.2.3.3.2" xref="S5.5.p1.3.m3.1.2.3.3.2.cmml">b</mi><mi id="S5.5.p1.3.m3.1.2.3.3.3" xref="S5.5.p1.3.m3.1.2.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.3.m3.1b"><apply id="S5.5.p1.3.m3.1.2.cmml" xref="S5.5.p1.3.m3.1.2"><eq id="S5.5.p1.3.m3.1.2.1.cmml" xref="S5.5.p1.3.m3.1.2.1"></eq><apply id="S5.5.p1.3.m3.1.2.2.cmml" xref="S5.5.p1.3.m3.1.2.2"><times id="S5.5.p1.3.m3.1.2.2.1.cmml" xref="S5.5.p1.3.m3.1.2.2.1"></times><apply id="S5.5.p1.3.m3.1.2.2.2.cmml" xref="S5.5.p1.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S5.5.p1.3.m3.1.2.2.2.1.cmml" xref="S5.5.p1.3.m3.1.2.2.2">subscript</csymbol><ci id="S5.5.p1.3.m3.1.2.2.2.2.cmml" xref="S5.5.p1.3.m3.1.2.2.2.2">𝑇</ci><ci id="S5.5.p1.3.m3.1.2.2.2.3.cmml" xref="S5.5.p1.3.m3.1.2.2.2.3">𝑖</ci></apply><ci id="S5.5.p1.3.m3.1.1.cmml" xref="S5.5.p1.3.m3.1.1">𝑥</ci></apply><apply id="S5.5.p1.3.m3.1.2.3.cmml" xref="S5.5.p1.3.m3.1.2.3"><plus id="S5.5.p1.3.m3.1.2.3.1.cmml" xref="S5.5.p1.3.m3.1.2.3.1"></plus><apply id="S5.5.p1.3.m3.1.2.3.2.cmml" xref="S5.5.p1.3.m3.1.2.3.2"><times id="S5.5.p1.3.m3.1.2.3.2.1.cmml" xref="S5.5.p1.3.m3.1.2.3.2.1"></times><apply id="S5.5.p1.3.m3.1.2.3.2.2.cmml" xref="S5.5.p1.3.m3.1.2.3.2.2"><csymbol cd="ambiguous" id="S5.5.p1.3.m3.1.2.3.2.2.1.cmml" xref="S5.5.p1.3.m3.1.2.3.2.2">subscript</csymbol><ci id="S5.5.p1.3.m3.1.2.3.2.2.2.cmml" xref="S5.5.p1.3.m3.1.2.3.2.2.2">𝐴</ci><ci id="S5.5.p1.3.m3.1.2.3.2.2.3.cmml" xref="S5.5.p1.3.m3.1.2.3.2.2.3">𝑖</ci></apply><ci id="S5.5.p1.3.m3.1.2.3.2.3.cmml" xref="S5.5.p1.3.m3.1.2.3.2.3">𝑥</ci></apply><apply id="S5.5.p1.3.m3.1.2.3.3.cmml" xref="S5.5.p1.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S5.5.p1.3.m3.1.2.3.3.1.cmml" xref="S5.5.p1.3.m3.1.2.3.3">subscript</csymbol><ci id="S5.5.p1.3.m3.1.2.3.3.2.cmml" xref="S5.5.p1.3.m3.1.2.3.3.2">𝑏</ci><ci id="S5.5.p1.3.m3.1.2.3.3.3.cmml" xref="S5.5.p1.3.m3.1.2.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.3.m3.1c">T_{i}(x)=A_{i}x+b_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x + italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="A_{i}\in\mathds{R}^{m_{i}\times n_{i}}" class="ltx_Math" display="inline" id="S5.5.p1.4.m4.1"><semantics id="S5.5.p1.4.m4.1a"><mrow id="S5.5.p1.4.m4.1.1" xref="S5.5.p1.4.m4.1.1.cmml"><msub id="S5.5.p1.4.m4.1.1.2" xref="S5.5.p1.4.m4.1.1.2.cmml"><mi id="S5.5.p1.4.m4.1.1.2.2" xref="S5.5.p1.4.m4.1.1.2.2.cmml">A</mi><mi id="S5.5.p1.4.m4.1.1.2.3" xref="S5.5.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.4.m4.1.1.1" xref="S5.5.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S5.5.p1.4.m4.1.1.3" xref="S5.5.p1.4.m4.1.1.3.cmml"><mi id="S5.5.p1.4.m4.1.1.3.2" xref="S5.5.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S5.5.p1.4.m4.1.1.3.3" xref="S5.5.p1.4.m4.1.1.3.3.cmml"><msub id="S5.5.p1.4.m4.1.1.3.3.2" xref="S5.5.p1.4.m4.1.1.3.3.2.cmml"><mi id="S5.5.p1.4.m4.1.1.3.3.2.2" xref="S5.5.p1.4.m4.1.1.3.3.2.2.cmml">m</mi><mi id="S5.5.p1.4.m4.1.1.3.3.2.3" xref="S5.5.p1.4.m4.1.1.3.3.2.3.cmml">i</mi></msub><mo id="S5.5.p1.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.5.p1.4.m4.1.1.3.3.1.cmml">×</mo><msub id="S5.5.p1.4.m4.1.1.3.3.3" xref="S5.5.p1.4.m4.1.1.3.3.3.cmml"><mi id="S5.5.p1.4.m4.1.1.3.3.3.2" xref="S5.5.p1.4.m4.1.1.3.3.3.2.cmml">n</mi><mi id="S5.5.p1.4.m4.1.1.3.3.3.3" xref="S5.5.p1.4.m4.1.1.3.3.3.3.cmml">i</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.4.m4.1b"><apply id="S5.5.p1.4.m4.1.1.cmml" xref="S5.5.p1.4.m4.1.1"><in id="S5.5.p1.4.m4.1.1.1.cmml" xref="S5.5.p1.4.m4.1.1.1"></in><apply id="S5.5.p1.4.m4.1.1.2.cmml" xref="S5.5.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.1.2.1.cmml" xref="S5.5.p1.4.m4.1.1.2">subscript</csymbol><ci id="S5.5.p1.4.m4.1.1.2.2.cmml" xref="S5.5.p1.4.m4.1.1.2.2">𝐴</ci><ci id="S5.5.p1.4.m4.1.1.2.3.cmml" xref="S5.5.p1.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.4.m4.1.1.3.cmml" xref="S5.5.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.1.3.1.cmml" xref="S5.5.p1.4.m4.1.1.3">superscript</csymbol><ci id="S5.5.p1.4.m4.1.1.3.2.cmml" xref="S5.5.p1.4.m4.1.1.3.2">ℝ</ci><apply id="S5.5.p1.4.m4.1.1.3.3.cmml" xref="S5.5.p1.4.m4.1.1.3.3"><times id="S5.5.p1.4.m4.1.1.3.3.1.cmml" xref="S5.5.p1.4.m4.1.1.3.3.1"></times><apply id="S5.5.p1.4.m4.1.1.3.3.2.cmml" xref="S5.5.p1.4.m4.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.1.3.3.2.1.cmml" xref="S5.5.p1.4.m4.1.1.3.3.2">subscript</csymbol><ci id="S5.5.p1.4.m4.1.1.3.3.2.2.cmml" xref="S5.5.p1.4.m4.1.1.3.3.2.2">𝑚</ci><ci id="S5.5.p1.4.m4.1.1.3.3.2.3.cmml" xref="S5.5.p1.4.m4.1.1.3.3.2.3">𝑖</ci></apply><apply id="S5.5.p1.4.m4.1.1.3.3.3.cmml" xref="S5.5.p1.4.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.1.3.3.3.1.cmml" xref="S5.5.p1.4.m4.1.1.3.3.3">subscript</csymbol><ci id="S5.5.p1.4.m4.1.1.3.3.3.2.cmml" xref="S5.5.p1.4.m4.1.1.3.3.3.2">𝑛</ci><ci id="S5.5.p1.4.m4.1.1.3.3.3.3.cmml" xref="S5.5.p1.4.m4.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.4.m4.1c">A_{i}\in\mathds{R}^{m_{i}\times n_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT × italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="b_{i}\in\mathds{R}^{m_{i}}" class="ltx_Math" display="inline" id="S5.5.p1.5.m5.1"><semantics id="S5.5.p1.5.m5.1a"><mrow id="S5.5.p1.5.m5.1.1" xref="S5.5.p1.5.m5.1.1.cmml"><msub id="S5.5.p1.5.m5.1.1.2" xref="S5.5.p1.5.m5.1.1.2.cmml"><mi id="S5.5.p1.5.m5.1.1.2.2" xref="S5.5.p1.5.m5.1.1.2.2.cmml">b</mi><mi id="S5.5.p1.5.m5.1.1.2.3" xref="S5.5.p1.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.5.m5.1.1.1" xref="S5.5.p1.5.m5.1.1.1.cmml">∈</mo><msup id="S5.5.p1.5.m5.1.1.3" xref="S5.5.p1.5.m5.1.1.3.cmml"><mi id="S5.5.p1.5.m5.1.1.3.2" xref="S5.5.p1.5.m5.1.1.3.2.cmml">ℝ</mi><msub id="S5.5.p1.5.m5.1.1.3.3" xref="S5.5.p1.5.m5.1.1.3.3.cmml"><mi id="S5.5.p1.5.m5.1.1.3.3.2" xref="S5.5.p1.5.m5.1.1.3.3.2.cmml">m</mi><mi id="S5.5.p1.5.m5.1.1.3.3.3" xref="S5.5.p1.5.m5.1.1.3.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.5.m5.1b"><apply id="S5.5.p1.5.m5.1.1.cmml" xref="S5.5.p1.5.m5.1.1"><in id="S5.5.p1.5.m5.1.1.1.cmml" xref="S5.5.p1.5.m5.1.1.1"></in><apply id="S5.5.p1.5.m5.1.1.2.cmml" xref="S5.5.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.1.1.2.1.cmml" xref="S5.5.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.5.p1.5.m5.1.1.2.2.cmml" xref="S5.5.p1.5.m5.1.1.2.2">𝑏</ci><ci id="S5.5.p1.5.m5.1.1.2.3.cmml" xref="S5.5.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.5.m5.1.1.3.cmml" xref="S5.5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.1.1.3.1.cmml" xref="S5.5.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.5.p1.5.m5.1.1.3.2.cmml" xref="S5.5.p1.5.m5.1.1.3.2">ℝ</ci><apply id="S5.5.p1.5.m5.1.1.3.3.cmml" xref="S5.5.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.1.1.3.3.1.cmml" xref="S5.5.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S5.5.p1.5.m5.1.1.3.3.2.cmml" xref="S5.5.p1.5.m5.1.1.3.3.2">𝑚</ci><ci id="S5.5.p1.5.m5.1.1.3.3.3.cmml" xref="S5.5.p1.5.m5.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.5.m5.1c">b_{i}\in\mathds{R}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.5.m5.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex60"> <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="\begin{bmatrix}T_{1}\\ \vdots\\ T_{w}\end{bmatrix}(x)=\begin{pmatrix}A_{1}&0&0\\ 0&\ddots&0\\ 0&0&A_{w}\end{pmatrix}x+\begin{pmatrix}b_{1}\\ \vdots\\ b_{w}\end{pmatrix},\quad\forall x\in\mathds{R}^{n_{1}+\dots+n_{w}}." class="ltx_Math" display="block" id="S5.Ex60.m1.5"><semantics id="S5.Ex60.m1.5a"><mrow id="S5.Ex60.m1.5.5.1"><mrow id="S5.Ex60.m1.5.5.1.1.2" xref="S5.Ex60.m1.5.5.1.1.3.cmml"><mrow id="S5.Ex60.m1.5.5.1.1.1.1" xref="S5.Ex60.m1.5.5.1.1.1.1.cmml"><mrow id="S5.Ex60.m1.5.5.1.1.1.1.2" xref="S5.Ex60.m1.5.5.1.1.1.1.2.cmml"><mrow id="S5.Ex60.m1.1.1.3" xref="S5.Ex60.m1.1.1.2.cmml"><mo id="S5.Ex60.m1.1.1.3.1" xref="S5.Ex60.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S5.Ex60.m1.1.1.1.1" rowspacing="0pt" xref="S5.Ex60.m1.1.1.1.1.cmml"><mtr id="S5.Ex60.m1.1.1.1.1a" xref="S5.Ex60.m1.1.1.1.1.cmml"><mtd id="S5.Ex60.m1.1.1.1.1b" xref="S5.Ex60.m1.1.1.1.1.cmml"><msub id="S5.Ex60.m1.1.1.1.1.1.1.1" xref="S5.Ex60.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex60.m1.1.1.1.1.1.1.1.2" xref="S5.Ex60.m1.1.1.1.1.1.1.1.2.cmml">T</mi><mn id="S5.Ex60.m1.1.1.1.1.1.1.1.3" xref="S5.Ex60.m1.1.1.1.1.1.1.1.3.cmml">1</mn></msub></mtd></mtr><mtr id="S5.Ex60.m1.1.1.1.1c" xref="S5.Ex60.m1.1.1.1.1.cmml"><mtd id="S5.Ex60.m1.1.1.1.1d" xref="S5.Ex60.m1.1.1.1.1.cmml"><mi id="S5.Ex60.m1.1.1.1.1.2.1.1" mathvariant="normal" xref="S5.Ex60.m1.1.1.1.1.2.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S5.Ex60.m1.1.1.1.1e" xref="S5.Ex60.m1.1.1.1.1.cmml"><mtd id="S5.Ex60.m1.1.1.1.1f" xref="S5.Ex60.m1.1.1.1.1.cmml"><msub id="S5.Ex60.m1.1.1.1.1.3.1.1" xref="S5.Ex60.m1.1.1.1.1.3.1.1.cmml"><mi id="S5.Ex60.m1.1.1.1.1.3.1.1.2" xref="S5.Ex60.m1.1.1.1.1.3.1.1.2.cmml">T</mi><mi id="S5.Ex60.m1.1.1.1.1.3.1.1.3" xref="S5.Ex60.m1.1.1.1.1.3.1.1.3.cmml">w</mi></msub></mtd></mtr></mtable><mo id="S5.Ex60.m1.1.1.3.2" xref="S5.Ex60.m1.1.1.2.1.cmml">]</mo></mrow><mo id="S5.Ex60.m1.5.5.1.1.1.1.2.1" xref="S5.Ex60.m1.5.5.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.Ex60.m1.5.5.1.1.1.1.2.2.2" xref="S5.Ex60.m1.5.5.1.1.1.1.2.cmml"><mo id="S5.Ex60.m1.5.5.1.1.1.1.2.2.2.1" stretchy="false" xref="S5.Ex60.m1.5.5.1.1.1.1.2.cmml">(</mo><mi id="S5.Ex60.m1.4.4" xref="S5.Ex60.m1.4.4.cmml">x</mi><mo id="S5.Ex60.m1.5.5.1.1.1.1.2.2.2.2" stretchy="false" xref="S5.Ex60.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Ex60.m1.5.5.1.1.1.1.1" xref="S5.Ex60.m1.5.5.1.1.1.1.1.cmml">=</mo><mrow id="S5.Ex60.m1.5.5.1.1.1.1.3" xref="S5.Ex60.m1.5.5.1.1.1.1.3.cmml"><mrow id="S5.Ex60.m1.5.5.1.1.1.1.3.2" xref="S5.Ex60.m1.5.5.1.1.1.1.3.2.cmml"><mrow id="S5.Ex60.m1.2.2.3" xref="S5.Ex60.m1.2.2.2.cmml"><mo id="S5.Ex60.m1.2.2.3.1" xref="S5.Ex60.m1.2.2.2.1.cmml">(</mo><mtable columnspacing="5pt" displaystyle="true" id="S5.Ex60.m1.2.2.1.1" 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xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.2.3">1</cn></apply><ci id="S5.Ex60.m1.5.5.1.1.2.2.3.3.3.cmml" xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.3">⋯</ci><apply id="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.cmml" xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.4"><csymbol cd="ambiguous" id="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.1.cmml" xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.4">subscript</csymbol><ci id="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.2.cmml" xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.2">𝑛</ci><ci id="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.3.cmml" xref="S5.Ex60.m1.5.5.1.1.2.2.3.3.4.3">𝑤</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex60.m1.5c">\begin{bmatrix}T_{1}\\ \vdots\\ T_{w}\end{bmatrix}(x)=\begin{pmatrix}A_{1}&0&0\\ 0&\ddots&0\\ 0&0&A_{w}\end{pmatrix}x+\begin{pmatrix}b_{1}\\ \vdots\\ b_{w}\end{pmatrix},\quad\forall x\in\mathds{R}^{n_{1}+\dots+n_{w}}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex60.m1.5d">[ start_ARG start_ROW start_CELL italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_T start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ( italic_x ) = ( start_ARG start_ROW start_CELL italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL ⋱ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_A start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) italic_x + ( start_ARG start_ROW start_CELL italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , ∀ italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_n start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p1.10">Thus, since <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 5.1</span></a> stacks <math alttext="9p" class="ltx_Math" display="inline" id="S5.5.p1.6.m1.1"><semantics id="S5.5.p1.6.m1.1a"><mrow id="S5.5.p1.6.m1.1.1" xref="S5.5.p1.6.m1.1.1.cmml"><mn id="S5.5.p1.6.m1.1.1.2" xref="S5.5.p1.6.m1.1.1.2.cmml">9</mn><mo id="S5.5.p1.6.m1.1.1.1" xref="S5.5.p1.6.m1.1.1.1.cmml">⁢</mo><mi id="S5.5.p1.6.m1.1.1.3" xref="S5.5.p1.6.m1.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.6.m1.1b"><apply id="S5.5.p1.6.m1.1.1.cmml" xref="S5.5.p1.6.m1.1.1"><times id="S5.5.p1.6.m1.1.1.1.cmml" xref="S5.5.p1.6.m1.1.1.1"></times><cn id="S5.5.p1.6.m1.1.1.2.cmml" type="integer" xref="S5.5.p1.6.m1.1.1.2">9</cn><ci id="S5.5.p1.6.m1.1.1.3.cmml" xref="S5.5.p1.6.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.6.m1.1c">9p</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.6.m1.1d">9 italic_p</annotation></semantics></math> affine functions of the same form given by (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S5.E41" title="Equation 41 ‣ Proof of Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">41</span></a>) for each layer, the total number of non-zero parameters is <math alttext="9p" class="ltx_Math" display="inline" id="S5.5.p1.7.m2.1"><semantics id="S5.5.p1.7.m2.1a"><mrow id="S5.5.p1.7.m2.1.1" xref="S5.5.p1.7.m2.1.1.cmml"><mn id="S5.5.p1.7.m2.1.1.2" xref="S5.5.p1.7.m2.1.1.2.cmml">9</mn><mo id="S5.5.p1.7.m2.1.1.1" xref="S5.5.p1.7.m2.1.1.1.cmml">⁢</mo><mi id="S5.5.p1.7.m2.1.1.3" xref="S5.5.p1.7.m2.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.7.m2.1b"><apply id="S5.5.p1.7.m2.1.1.cmml" xref="S5.5.p1.7.m2.1.1"><times id="S5.5.p1.7.m2.1.1.1.cmml" xref="S5.5.p1.7.m2.1.1.1"></times><cn id="S5.5.p1.7.m2.1.1.2.cmml" type="integer" xref="S5.5.p1.7.m2.1.1.2">9</cn><ci id="S5.5.p1.7.m2.1.1.3.cmml" xref="S5.5.p1.7.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.7.m2.1c">9p</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.7.m2.1d">9 italic_p</annotation></semantics></math> times the number of parameters required in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13001v1#S5.E41" title="Equation 41 ‣ Proof of Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">41</span></a>). We have <math alttext="A_{i}\in\mathds{R}^{5\times 2}" class="ltx_Math" display="inline" id="S5.5.p1.8.m3.1"><semantics id="S5.5.p1.8.m3.1a"><mrow id="S5.5.p1.8.m3.1.1" xref="S5.5.p1.8.m3.1.1.cmml"><msub id="S5.5.p1.8.m3.1.1.2" xref="S5.5.p1.8.m3.1.1.2.cmml"><mi id="S5.5.p1.8.m3.1.1.2.2" xref="S5.5.p1.8.m3.1.1.2.2.cmml">A</mi><mi id="S5.5.p1.8.m3.1.1.2.3" xref="S5.5.p1.8.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.8.m3.1.1.1" xref="S5.5.p1.8.m3.1.1.1.cmml">∈</mo><msup id="S5.5.p1.8.m3.1.1.3" xref="S5.5.p1.8.m3.1.1.3.cmml"><mi id="S5.5.p1.8.m3.1.1.3.2" xref="S5.5.p1.8.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S5.5.p1.8.m3.1.1.3.3" xref="S5.5.p1.8.m3.1.1.3.3.cmml"><mn id="S5.5.p1.8.m3.1.1.3.3.2" xref="S5.5.p1.8.m3.1.1.3.3.2.cmml">5</mn><mo id="S5.5.p1.8.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.5.p1.8.m3.1.1.3.3.1.cmml">×</mo><mn id="S5.5.p1.8.m3.1.1.3.3.3" xref="S5.5.p1.8.m3.1.1.3.3.3.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.8.m3.1b"><apply id="S5.5.p1.8.m3.1.1.cmml" xref="S5.5.p1.8.m3.1.1"><in id="S5.5.p1.8.m3.1.1.1.cmml" xref="S5.5.p1.8.m3.1.1.1"></in><apply id="S5.5.p1.8.m3.1.1.2.cmml" xref="S5.5.p1.8.m3.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.8.m3.1.1.2.1.cmml" xref="S5.5.p1.8.m3.1.1.2">subscript</csymbol><ci id="S5.5.p1.8.m3.1.1.2.2.cmml" xref="S5.5.p1.8.m3.1.1.2.2">𝐴</ci><ci id="S5.5.p1.8.m3.1.1.2.3.cmml" xref="S5.5.p1.8.m3.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.8.m3.1.1.3.cmml" xref="S5.5.p1.8.m3.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.8.m3.1.1.3.1.cmml" xref="S5.5.p1.8.m3.1.1.3">superscript</csymbol><ci id="S5.5.p1.8.m3.1.1.3.2.cmml" xref="S5.5.p1.8.m3.1.1.3.2">ℝ</ci><apply id="S5.5.p1.8.m3.1.1.3.3.cmml" xref="S5.5.p1.8.m3.1.1.3.3"><times id="S5.5.p1.8.m3.1.1.3.3.1.cmml" xref="S5.5.p1.8.m3.1.1.3.3.1"></times><cn id="S5.5.p1.8.m3.1.1.3.3.2.cmml" type="integer" xref="S5.5.p1.8.m3.1.1.3.3.2">5</cn><cn id="S5.5.p1.8.m3.1.1.3.3.3.cmml" type="integer" xref="S5.5.p1.8.m3.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.8.m3.1c">A_{i}\in\mathds{R}^{5\times 2}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.8.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 5 × 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in the first hidden layer, <math alttext="A_{i}\in\mathds{R}^{3\times 5}" class="ltx_Math" display="inline" id="S5.5.p1.9.m4.1"><semantics id="S5.5.p1.9.m4.1a"><mrow id="S5.5.p1.9.m4.1.1" xref="S5.5.p1.9.m4.1.1.cmml"><msub id="S5.5.p1.9.m4.1.1.2" xref="S5.5.p1.9.m4.1.1.2.cmml"><mi id="S5.5.p1.9.m4.1.1.2.2" xref="S5.5.p1.9.m4.1.1.2.2.cmml">A</mi><mi id="S5.5.p1.9.m4.1.1.2.3" xref="S5.5.p1.9.m4.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.9.m4.1.1.1" xref="S5.5.p1.9.m4.1.1.1.cmml">∈</mo><msup id="S5.5.p1.9.m4.1.1.3" xref="S5.5.p1.9.m4.1.1.3.cmml"><mi id="S5.5.p1.9.m4.1.1.3.2" xref="S5.5.p1.9.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S5.5.p1.9.m4.1.1.3.3" xref="S5.5.p1.9.m4.1.1.3.3.cmml"><mn id="S5.5.p1.9.m4.1.1.3.3.2" xref="S5.5.p1.9.m4.1.1.3.3.2.cmml">3</mn><mo id="S5.5.p1.9.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.5.p1.9.m4.1.1.3.3.1.cmml">×</mo><mn id="S5.5.p1.9.m4.1.1.3.3.3" xref="S5.5.p1.9.m4.1.1.3.3.3.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.9.m4.1b"><apply id="S5.5.p1.9.m4.1.1.cmml" xref="S5.5.p1.9.m4.1.1"><in id="S5.5.p1.9.m4.1.1.1.cmml" xref="S5.5.p1.9.m4.1.1.1"></in><apply id="S5.5.p1.9.m4.1.1.2.cmml" xref="S5.5.p1.9.m4.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.9.m4.1.1.2.1.cmml" xref="S5.5.p1.9.m4.1.1.2">subscript</csymbol><ci id="S5.5.p1.9.m4.1.1.2.2.cmml" xref="S5.5.p1.9.m4.1.1.2.2">𝐴</ci><ci id="S5.5.p1.9.m4.1.1.2.3.cmml" xref="S5.5.p1.9.m4.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.9.m4.1.1.3.cmml" xref="S5.5.p1.9.m4.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.9.m4.1.1.3.1.cmml" xref="S5.5.p1.9.m4.1.1.3">superscript</csymbol><ci id="S5.5.p1.9.m4.1.1.3.2.cmml" xref="S5.5.p1.9.m4.1.1.3.2">ℝ</ci><apply id="S5.5.p1.9.m4.1.1.3.3.cmml" xref="S5.5.p1.9.m4.1.1.3.3"><times id="S5.5.p1.9.m4.1.1.3.3.1.cmml" xref="S5.5.p1.9.m4.1.1.3.3.1"></times><cn id="S5.5.p1.9.m4.1.1.3.3.2.cmml" type="integer" xref="S5.5.p1.9.m4.1.1.3.3.2">3</cn><cn id="S5.5.p1.9.m4.1.1.3.3.3.cmml" type="integer" xref="S5.5.p1.9.m4.1.1.3.3.3">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.9.m4.1c">A_{i}\in\mathds{R}^{3\times 5}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.9.m4.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 3 × 5 end_POSTSUPERSCRIPT</annotation></semantics></math> in the second hidden layer, and <math alttext="A_{i}\in\mathds{R}^{1\times 3}" class="ltx_Math" display="inline" id="S5.5.p1.10.m5.1"><semantics id="S5.5.p1.10.m5.1a"><mrow id="S5.5.p1.10.m5.1.1" xref="S5.5.p1.10.m5.1.1.cmml"><msub id="S5.5.p1.10.m5.1.1.2" xref="S5.5.p1.10.m5.1.1.2.cmml"><mi id="S5.5.p1.10.m5.1.1.2.2" xref="S5.5.p1.10.m5.1.1.2.2.cmml">A</mi><mi id="S5.5.p1.10.m5.1.1.2.3" xref="S5.5.p1.10.m5.1.1.2.3.cmml">i</mi></msub><mo id="S5.5.p1.10.m5.1.1.1" xref="S5.5.p1.10.m5.1.1.1.cmml">∈</mo><msup id="S5.5.p1.10.m5.1.1.3" xref="S5.5.p1.10.m5.1.1.3.cmml"><mi id="S5.5.p1.10.m5.1.1.3.2" xref="S5.5.p1.10.m5.1.1.3.2.cmml">ℝ</mi><mrow id="S5.5.p1.10.m5.1.1.3.3" xref="S5.5.p1.10.m5.1.1.3.3.cmml"><mn id="S5.5.p1.10.m5.1.1.3.3.2" xref="S5.5.p1.10.m5.1.1.3.3.2.cmml">1</mn><mo id="S5.5.p1.10.m5.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.5.p1.10.m5.1.1.3.3.1.cmml">×</mo><mn id="S5.5.p1.10.m5.1.1.3.3.3" xref="S5.5.p1.10.m5.1.1.3.3.3.cmml">3</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.10.m5.1b"><apply id="S5.5.p1.10.m5.1.1.cmml" xref="S5.5.p1.10.m5.1.1"><in id="S5.5.p1.10.m5.1.1.1.cmml" xref="S5.5.p1.10.m5.1.1.1"></in><apply id="S5.5.p1.10.m5.1.1.2.cmml" xref="S5.5.p1.10.m5.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.10.m5.1.1.2.1.cmml" xref="S5.5.p1.10.m5.1.1.2">subscript</csymbol><ci id="S5.5.p1.10.m5.1.1.2.2.cmml" xref="S5.5.p1.10.m5.1.1.2.2">𝐴</ci><ci id="S5.5.p1.10.m5.1.1.2.3.cmml" xref="S5.5.p1.10.m5.1.1.2.3">𝑖</ci></apply><apply id="S5.5.p1.10.m5.1.1.3.cmml" xref="S5.5.p1.10.m5.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.10.m5.1.1.3.1.cmml" xref="S5.5.p1.10.m5.1.1.3">superscript</csymbol><ci id="S5.5.p1.10.m5.1.1.3.2.cmml" xref="S5.5.p1.10.m5.1.1.3.2">ℝ</ci><apply id="S5.5.p1.10.m5.1.1.3.3.cmml" xref="S5.5.p1.10.m5.1.1.3.3"><times id="S5.5.p1.10.m5.1.1.3.3.1.cmml" xref="S5.5.p1.10.m5.1.1.3.3.1"></times><cn id="S5.5.p1.10.m5.1.1.3.3.2.cmml" type="integer" xref="S5.5.p1.10.m5.1.1.3.3.2">1</cn><cn id="S5.5.p1.10.m5.1.1.3.3.3.cmml" type="integer" xref="S5.5.p1.10.m5.1.1.3.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.10.m5.1c">A_{i}\in\mathds{R}^{1\times 3}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.10.m5.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 1 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math> in the output layer. Therefore, we need at most</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex61"> <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="9p\cdot((5\cdot 2+5)+(3\cdot 5+3)+(1\cdot 3+1))=O(p)" class="ltx_Math" display="block" id="S5.Ex61.m1.2"><semantics id="S5.Ex61.m1.2a"><mrow id="S5.Ex61.m1.2.2" xref="S5.Ex61.m1.2.2.cmml"><mrow id="S5.Ex61.m1.2.2.1" xref="S5.Ex61.m1.2.2.1.cmml"><mrow id="S5.Ex61.m1.2.2.1.3" xref="S5.Ex61.m1.2.2.1.3.cmml"><mn id="S5.Ex61.m1.2.2.1.3.2" xref="S5.Ex61.m1.2.2.1.3.2.cmml">9</mn><mo id="S5.Ex61.m1.2.2.1.3.1" xref="S5.Ex61.m1.2.2.1.3.1.cmml">⁢</mo><mi id="S5.Ex61.m1.2.2.1.3.3" xref="S5.Ex61.m1.2.2.1.3.3.cmml">p</mi></mrow><mo id="S5.Ex61.m1.2.2.1.2" lspace="0.222em" rspace="0.222em" xref="S5.Ex61.m1.2.2.1.2.cmml">⋅</mo><mrow id="S5.Ex61.m1.2.2.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.cmml"><mo id="S5.Ex61.m1.2.2.1.1.1.2" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.cmml"><mrow id="S5.Ex61.m1.2.2.1.1.1.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex61.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">5</mn><mo id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.1.cmml">⋅</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">2</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.3" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.3.cmml">5</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.4" xref="S5.Ex61.m1.2.2.1.1.1.1.4.cmml">+</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1.2.1" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.cmml"><mo id="S5.Ex61.m1.2.2.1.1.1.1.2.1.2" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.cmml"><mrow id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.cmml"><mn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.2" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.2.cmml">3</mn><mo id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.1.cmml">⋅</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.3" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.3.cmml">5</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.1.cmml">+</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.3" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.3.cmml">3</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.2.1.3" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.cmml">)</mo></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.4a" xref="S5.Ex61.m1.2.2.1.1.1.1.4.cmml">+</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1.3.1" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.cmml"><mo id="S5.Ex61.m1.2.2.1.1.1.1.3.1.2" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.cmml">(</mo><mrow id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.cmml"><mrow id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.cmml"><mn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.2" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.2.cmml">1</mn><mo id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.1.cmml">⋅</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.3" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.3.cmml">3</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.1" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.1.cmml">+</mo><mn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.3" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.1.3.1.3" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex61.m1.2.2.1.1.1.3" stretchy="false" xref="S5.Ex61.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex61.m1.2.2.2" xref="S5.Ex61.m1.2.2.2.cmml">=</mo><mrow id="S5.Ex61.m1.2.2.3" xref="S5.Ex61.m1.2.2.3.cmml"><mi id="S5.Ex61.m1.2.2.3.2" xref="S5.Ex61.m1.2.2.3.2.cmml">O</mi><mo id="S5.Ex61.m1.2.2.3.1" xref="S5.Ex61.m1.2.2.3.1.cmml">⁢</mo><mrow id="S5.Ex61.m1.2.2.3.3.2" xref="S5.Ex61.m1.2.2.3.cmml"><mo id="S5.Ex61.m1.2.2.3.3.2.1" stretchy="false" xref="S5.Ex61.m1.2.2.3.cmml">(</mo><mi id="S5.Ex61.m1.1.1" xref="S5.Ex61.m1.1.1.cmml">p</mi><mo id="S5.Ex61.m1.2.2.3.3.2.2" stretchy="false" xref="S5.Ex61.m1.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex61.m1.2b"><apply id="S5.Ex61.m1.2.2.cmml" xref="S5.Ex61.m1.2.2"><eq id="S5.Ex61.m1.2.2.2.cmml" xref="S5.Ex61.m1.2.2.2"></eq><apply id="S5.Ex61.m1.2.2.1.cmml" xref="S5.Ex61.m1.2.2.1"><ci id="S5.Ex61.m1.2.2.1.2.cmml" xref="S5.Ex61.m1.2.2.1.2">⋅</ci><apply id="S5.Ex61.m1.2.2.1.3.cmml" xref="S5.Ex61.m1.2.2.1.3"><times id="S5.Ex61.m1.2.2.1.3.1.cmml" xref="S5.Ex61.m1.2.2.1.3.1"></times><cn id="S5.Ex61.m1.2.2.1.3.2.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.3.2">9</cn><ci id="S5.Ex61.m1.2.2.1.3.3.cmml" xref="S5.Ex61.m1.2.2.1.3.3">𝑝</ci></apply><apply id="S5.Ex61.m1.2.2.1.1.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1"><plus id="S5.Ex61.m1.2.2.1.1.1.1.4.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.4"></plus><apply id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1"><plus id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.1"></plus><apply id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2"><ci id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.1">⋅</ci><cn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.2">5</cn><cn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.2.3">2</cn></apply><cn id="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.1.1.1.3">5</cn></apply><apply id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1"><plus id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.1"></plus><apply id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2"><ci id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.1">⋅</ci><cn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.2.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.2">3</cn><cn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.2.3">5</cn></apply><cn id="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.2.1.1.3">3</cn></apply><apply id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1"><plus id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.1"></plus><apply id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2"><ci id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.1.cmml" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.1">⋅</ci><cn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.2.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.2">1</cn><cn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.2.3">3</cn></apply><cn id="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.3.cmml" type="integer" xref="S5.Ex61.m1.2.2.1.1.1.1.3.1.1.3">1</cn></apply></apply></apply><apply id="S5.Ex61.m1.2.2.3.cmml" xref="S5.Ex61.m1.2.2.3"><times id="S5.Ex61.m1.2.2.3.1.cmml" xref="S5.Ex61.m1.2.2.3.1"></times><ci id="S5.Ex61.m1.2.2.3.2.cmml" xref="S5.Ex61.m1.2.2.3.2">𝑂</ci><ci id="S5.Ex61.m1.1.1.cmml" xref="S5.Ex61.m1.1.1">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex61.m1.2c">9p\cdot((5\cdot 2+5)+(3\cdot 5+3)+(1\cdot 3+1))=O(p)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex61.m1.2d">9 italic_p ⋅ ( ( 5 ⋅ 2 + 5 ) + ( 3 ⋅ 5 + 3 ) + ( 1 ⋅ 3 + 1 ) ) = italic_O ( italic_p )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p1.11">non-zero parameters. ∎</p> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Discussion</h2> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.1 </span>Comparison to the Result of <span class="ltx_ERROR undefined" id="S6.SS1.1.1">\citet</span>Koutschan2023</h3> <div class="ltx_para" id="S6.SS1.p1"> <span class="ltx_ERROR undefined" id="S6.SS1.p1.1">\citet</span> <p class="ltx_p" id="S6.SS1.p1.2">Koutschan2023 showed the following result:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S6.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="S6.Thmtheorem1.1.1.1">Theorem 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p1.5.5">Any CPA function <math alttext="f:\mathds{R}^{d}\to\mathds{R}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.1.m1.1"><semantics id="S6.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S6.Thmtheorem1.p1.1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">f</mi><mo id="S6.Thmtheorem1.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><msup id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml"><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.2" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.3" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml">d</mi></msup><mo id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml">→</mo><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.1.m1.1b"><apply id="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1"><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.1">:</ci><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.2">𝑓</ci><apply id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3"><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.1">→</ci><apply id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.2">ℝ</ci><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.2.3">𝑑</ci></apply><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.1.1.m1.1c">f:\mathds{R}^{d}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.1.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> with <math alttext="n" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.2.2.m2.1"><semantics id="S6.Thmtheorem1.p1.2.2.m2.1a"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.2.2.m2.1b"><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.2.2.m2.1d">italic_n</annotation></semantics></math> affine components can be represented by a <math alttext="\operatorname{ReLU}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.3.3.m3.1"><semantics id="S6.Thmtheorem1.p1.3.3.m3.1a"><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml">ReLU</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.3.3.m3.1b"><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1">ReLU</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.3.3.m3.1c">\operatorname{ReLU}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.3.3.m3.1d">roman_ReLU</annotation></semantics></math> neural network of depth <math alttext="\lceil\log_{2}(d+1)\rceil+1" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.4.4.m4.1"><semantics id="S6.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S6.Thmtheorem1.p1.4.4.m4.1.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mrow id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.2.cmml"><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.2.1.cmml">⌈</mo><mrow id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml"><msub id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2a" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml">⁡</mo><mrow id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml"><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.2" stretchy="false" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml">(</mo><mrow id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.cmml"><mi id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.2" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.2.cmml">d</mi><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.1.cmml">+</mo><mn id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.3" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.2.1.cmml">⌉</mo></mrow><mo id="S6.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">+</mo><mn id="S6.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.4.4.m4.1b"><apply id="S6.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1"><plus id="S6.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.2"></plus><apply id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1"><ceiling id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.2"></ceiling><apply id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2"><apply id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1">subscript</csymbol><log id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.2"></log><cn id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1"><plus id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.1"></plus><ci id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.2.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.2">𝑑</ci><cn id="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.1.3">1</cn></apply></apply></apply><cn id="S6.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.4.4.m4.1c">\lceil\log_{2}(d+1)\rceil+1</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.4.4.m4.1d">⌈ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_d + 1 ) ⌉ + 1</annotation></semantics></math> with <math alttext="O(n^{d+1})" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.5.5.m5.1"><semantics id="S6.Thmtheorem1.p1.5.5.m5.1a"><mrow id="S6.Thmtheorem1.p1.5.5.m5.1.1" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.cmml"><mi id="S6.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.3.cmml">O</mi><mo id="S6.Thmtheorem1.p1.5.5.m5.1.1.2" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml">(</mo><msup id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.cmml">n</mi><mrow id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.2.cmml">d</mi><mo id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.1" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.1.cmml">+</mo><mn id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.3" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.5.5.m5.1b"><apply id="S6.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1"><times id="S6.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.2"></times><ci id="S6.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.3">𝑂</ci><apply id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2">𝑛</ci><apply id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3"><plus id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.1"></plus><ci id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.2">𝑑</ci><cn id="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.5.5.m5.1c">O(n^{d+1})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.5.5.m5.1d">italic_O ( italic_n start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> neurons per layer.</span></p> </div> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.4">Compared to <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 5.1</span></a>, their proof follows a more algebraic approach, leading to dependence on the number of affine components rather than on the number of pieces. For a CPA function <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S6.SS1.p2.1.m1.1"><semantics id="S6.SS1.p2.1.m1.1a"><mrow id="S6.SS1.p2.1.m1.1.1" xref="S6.SS1.p2.1.m1.1.1.cmml"><mi id="S6.SS1.p2.1.m1.1.1.2" xref="S6.SS1.p2.1.m1.1.1.2.cmml">f</mi><mo id="S6.SS1.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS1.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S6.SS1.p2.1.m1.1.1.3" xref="S6.SS1.p2.1.m1.1.1.3.cmml"><msup id="S6.SS1.p2.1.m1.1.1.3.2" xref="S6.SS1.p2.1.m1.1.1.3.2.cmml"><mi id="S6.SS1.p2.1.m1.1.1.3.2.2" xref="S6.SS1.p2.1.m1.1.1.3.2.2.cmml">ℝ</mi><mn id="S6.SS1.p2.1.m1.1.1.3.2.3" xref="S6.SS1.p2.1.m1.1.1.3.2.3.cmml">2</mn></msup><mo id="S6.SS1.p2.1.m1.1.1.3.1" stretchy="false" xref="S6.SS1.p2.1.m1.1.1.3.1.cmml">→</mo><mi id="S6.SS1.p2.1.m1.1.1.3.3" xref="S6.SS1.p2.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.1b"><apply id="S6.SS1.p2.1.m1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1"><ci id="S6.SS1.p2.1.m1.1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1.1">:</ci><ci id="S6.SS1.p2.1.m1.1.1.2.cmml" xref="S6.SS1.p2.1.m1.1.1.2">𝑓</ci><apply id="S6.SS1.p2.1.m1.1.1.3.cmml" xref="S6.SS1.p2.1.m1.1.1.3"><ci id="S6.SS1.p2.1.m1.1.1.3.1.cmml" xref="S6.SS1.p2.1.m1.1.1.3.1">→</ci><apply id="S6.SS1.p2.1.m1.1.1.3.2.cmml" xref="S6.SS1.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.p2.1.m1.1.1.3.2.1.cmml" xref="S6.SS1.p2.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.SS1.p2.1.m1.1.1.3.2.2.cmml" xref="S6.SS1.p2.1.m1.1.1.3.2.2">ℝ</ci><cn id="S6.SS1.p2.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.p2.1.m1.1.1.3.2.3">2</cn></apply><ci id="S6.SS1.p2.1.m1.1.1.3.3.cmml" xref="S6.SS1.p2.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.1.m1.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> with <math alttext="p" class="ltx_Math" display="inline" id="S6.SS1.p2.2.m2.1"><semantics id="S6.SS1.p2.2.m2.1a"><mi id="S6.SS1.p2.2.m2.1.1" xref="S6.SS1.p2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.1b"><ci id="S6.SS1.p2.2.m2.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.2.m2.1d">italic_p</annotation></semantics></math> pieces and <math alttext="n" class="ltx_Math" display="inline" id="S6.SS1.p2.3.m3.1"><semantics id="S6.SS1.p2.3.m3.1a"><mi id="S6.SS1.p2.3.m3.1.1" xref="S6.SS1.p2.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.3.m3.1b"><ci id="S6.SS1.p2.3.m3.1.1.cmml" xref="S6.SS1.p2.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.3.m3.1d">italic_n</annotation></semantics></math> affine components, <span class="ltx_ERROR undefined" id="S6.SS1.p2.4.1">\citet</span>zanotti2025pieces showed that a constant <math alttext="c\in\mathds{R}" class="ltx_Math" display="inline" id="S6.SS1.p2.4.m4.1"><semantics id="S6.SS1.p2.4.m4.1a"><mrow id="S6.SS1.p2.4.m4.1.1" xref="S6.SS1.p2.4.m4.1.1.cmml"><mi id="S6.SS1.p2.4.m4.1.1.2" xref="S6.SS1.p2.4.m4.1.1.2.cmml">c</mi><mo id="S6.SS1.p2.4.m4.1.1.1" xref="S6.SS1.p2.4.m4.1.1.1.cmml">∈</mo><mi id="S6.SS1.p2.4.m4.1.1.3" xref="S6.SS1.p2.4.m4.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.4.m4.1b"><apply id="S6.SS1.p2.4.m4.1.1.cmml" xref="S6.SS1.p2.4.m4.1.1"><in id="S6.SS1.p2.4.m4.1.1.1.cmml" xref="S6.SS1.p2.4.m4.1.1.1"></in><ci id="S6.SS1.p2.4.m4.1.1.2.cmml" xref="S6.SS1.p2.4.m4.1.1.2">𝑐</ci><ci id="S6.SS1.p2.4.m4.1.1.3.cmml" xref="S6.SS1.p2.4.m4.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.4.m4.1c">c\in\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.4.m4.1d">italic_c ∈ blackboard_R</annotation></semantics></math> exists such that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex62"> <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="n\leq p\leq cn^{3}." class="ltx_Math" display="block" id="S6.Ex62.m1.1"><semantics id="S6.Ex62.m1.1a"><mrow id="S6.Ex62.m1.1.1.1" xref="S6.Ex62.m1.1.1.1.1.cmml"><mrow id="S6.Ex62.m1.1.1.1.1" xref="S6.Ex62.m1.1.1.1.1.cmml"><mi id="S6.Ex62.m1.1.1.1.1.2" xref="S6.Ex62.m1.1.1.1.1.2.cmml">n</mi><mo id="S6.Ex62.m1.1.1.1.1.3" xref="S6.Ex62.m1.1.1.1.1.3.cmml">≤</mo><mi id="S6.Ex62.m1.1.1.1.1.4" xref="S6.Ex62.m1.1.1.1.1.4.cmml">p</mi><mo id="S6.Ex62.m1.1.1.1.1.5" xref="S6.Ex62.m1.1.1.1.1.5.cmml">≤</mo><mrow id="S6.Ex62.m1.1.1.1.1.6" xref="S6.Ex62.m1.1.1.1.1.6.cmml"><mi id="S6.Ex62.m1.1.1.1.1.6.2" xref="S6.Ex62.m1.1.1.1.1.6.2.cmml">c</mi><mo id="S6.Ex62.m1.1.1.1.1.6.1" xref="S6.Ex62.m1.1.1.1.1.6.1.cmml">⁢</mo><msup id="S6.Ex62.m1.1.1.1.1.6.3" xref="S6.Ex62.m1.1.1.1.1.6.3.cmml"><mi id="S6.Ex62.m1.1.1.1.1.6.3.2" xref="S6.Ex62.m1.1.1.1.1.6.3.2.cmml">n</mi><mn id="S6.Ex62.m1.1.1.1.1.6.3.3" xref="S6.Ex62.m1.1.1.1.1.6.3.3.cmml">3</mn></msup></mrow></mrow><mo id="S6.Ex62.m1.1.1.1.2" lspace="0em" xref="S6.Ex62.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex62.m1.1b"><apply id="S6.Ex62.m1.1.1.1.1.cmml" xref="S6.Ex62.m1.1.1.1"><and id="S6.Ex62.m1.1.1.1.1a.cmml" xref="S6.Ex62.m1.1.1.1"></and><apply id="S6.Ex62.m1.1.1.1.1b.cmml" xref="S6.Ex62.m1.1.1.1"><leq id="S6.Ex62.m1.1.1.1.1.3.cmml" xref="S6.Ex62.m1.1.1.1.1.3"></leq><ci id="S6.Ex62.m1.1.1.1.1.2.cmml" xref="S6.Ex62.m1.1.1.1.1.2">𝑛</ci><ci id="S6.Ex62.m1.1.1.1.1.4.cmml" xref="S6.Ex62.m1.1.1.1.1.4">𝑝</ci></apply><apply id="S6.Ex62.m1.1.1.1.1c.cmml" xref="S6.Ex62.m1.1.1.1"><leq id="S6.Ex62.m1.1.1.1.1.5.cmml" xref="S6.Ex62.m1.1.1.1.1.5"></leq><share href="https://arxiv.org/html/2503.13001v1#S6.Ex62.m1.1.1.1.1.4.cmml" id="S6.Ex62.m1.1.1.1.1d.cmml" xref="S6.Ex62.m1.1.1.1"></share><apply id="S6.Ex62.m1.1.1.1.1.6.cmml" xref="S6.Ex62.m1.1.1.1.1.6"><times id="S6.Ex62.m1.1.1.1.1.6.1.cmml" xref="S6.Ex62.m1.1.1.1.1.6.1"></times><ci id="S6.Ex62.m1.1.1.1.1.6.2.cmml" xref="S6.Ex62.m1.1.1.1.1.6.2">𝑐</ci><apply id="S6.Ex62.m1.1.1.1.1.6.3.cmml" xref="S6.Ex62.m1.1.1.1.1.6.3"><csymbol cd="ambiguous" id="S6.Ex62.m1.1.1.1.1.6.3.1.cmml" xref="S6.Ex62.m1.1.1.1.1.6.3">superscript</csymbol><ci id="S6.Ex62.m1.1.1.1.1.6.3.2.cmml" xref="S6.Ex62.m1.1.1.1.1.6.3.2">𝑛</ci><cn id="S6.Ex62.m1.1.1.1.1.6.3.3.cmml" type="integer" xref="S6.Ex62.m1.1.1.1.1.6.3.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex62.m1.1c">n\leq p\leq cn^{3}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex62.m1.1d">italic_n ≤ italic_p ≤ italic_c italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.p2.14">Thus, <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 5.1</span></a> implies a neural network representation with width <math alttext="O(n^{3})" class="ltx_Math" display="inline" id="S6.SS1.p2.5.m1.1"><semantics id="S6.SS1.p2.5.m1.1a"><mrow id="S6.SS1.p2.5.m1.1.1" xref="S6.SS1.p2.5.m1.1.1.cmml"><mi id="S6.SS1.p2.5.m1.1.1.3" xref="S6.SS1.p2.5.m1.1.1.3.cmml">O</mi><mo id="S6.SS1.p2.5.m1.1.1.2" xref="S6.SS1.p2.5.m1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p2.5.m1.1.1.1.1" xref="S6.SS1.p2.5.m1.1.1.1.1.1.cmml"><mo id="S6.SS1.p2.5.m1.1.1.1.1.2" stretchy="false" xref="S6.SS1.p2.5.m1.1.1.1.1.1.cmml">(</mo><msup id="S6.SS1.p2.5.m1.1.1.1.1.1" xref="S6.SS1.p2.5.m1.1.1.1.1.1.cmml"><mi id="S6.SS1.p2.5.m1.1.1.1.1.1.2" xref="S6.SS1.p2.5.m1.1.1.1.1.1.2.cmml">n</mi><mn id="S6.SS1.p2.5.m1.1.1.1.1.1.3" xref="S6.SS1.p2.5.m1.1.1.1.1.1.3.cmml">3</mn></msup><mo id="S6.SS1.p2.5.m1.1.1.1.1.3" stretchy="false" xref="S6.SS1.p2.5.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.5.m1.1b"><apply id="S6.SS1.p2.5.m1.1.1.cmml" xref="S6.SS1.p2.5.m1.1.1"><times id="S6.SS1.p2.5.m1.1.1.2.cmml" xref="S6.SS1.p2.5.m1.1.1.2"></times><ci id="S6.SS1.p2.5.m1.1.1.3.cmml" xref="S6.SS1.p2.5.m1.1.1.3">𝑂</ci><apply id="S6.SS1.p2.5.m1.1.1.1.1.1.cmml" xref="S6.SS1.p2.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.5.m1.1.1.1.1.1.1.cmml" xref="S6.SS1.p2.5.m1.1.1.1.1">superscript</csymbol><ci id="S6.SS1.p2.5.m1.1.1.1.1.1.2.cmml" xref="S6.SS1.p2.5.m1.1.1.1.1.1.2">𝑛</ci><cn id="S6.SS1.p2.5.m1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p2.5.m1.1.1.1.1.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.5.m1.1c">O(n^{3})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.5.m1.1d">italic_O ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math> in terms of affine components, which matches the result of <span class="ltx_ERROR undefined" id="S6.SS1.p2.14.1">\citet</span>Koutschan2023. Moreover, <span class="ltx_ERROR undefined" id="S6.SS1.p2.14.2">\citet</span>zanotti2025pieces also showed that there exist <math alttext="\beta,c>0" class="ltx_Math" display="inline" id="S6.SS1.p2.6.m2.2"><semantics id="S6.SS1.p2.6.m2.2a"><mrow id="S6.SS1.p2.6.m2.2.3" xref="S6.SS1.p2.6.m2.2.3.cmml"><mrow id="S6.SS1.p2.6.m2.2.3.2.2" xref="S6.SS1.p2.6.m2.2.3.2.1.cmml"><mi id="S6.SS1.p2.6.m2.1.1" xref="S6.SS1.p2.6.m2.1.1.cmml">β</mi><mo id="S6.SS1.p2.6.m2.2.3.2.2.1" xref="S6.SS1.p2.6.m2.2.3.2.1.cmml">,</mo><mi id="S6.SS1.p2.6.m2.2.2" xref="S6.SS1.p2.6.m2.2.2.cmml">c</mi></mrow><mo id="S6.SS1.p2.6.m2.2.3.1" xref="S6.SS1.p2.6.m2.2.3.1.cmml">></mo><mn id="S6.SS1.p2.6.m2.2.3.3" xref="S6.SS1.p2.6.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.6.m2.2b"><apply id="S6.SS1.p2.6.m2.2.3.cmml" xref="S6.SS1.p2.6.m2.2.3"><gt id="S6.SS1.p2.6.m2.2.3.1.cmml" xref="S6.SS1.p2.6.m2.2.3.1"></gt><list id="S6.SS1.p2.6.m2.2.3.2.1.cmml" xref="S6.SS1.p2.6.m2.2.3.2.2"><ci id="S6.SS1.p2.6.m2.1.1.cmml" xref="S6.SS1.p2.6.m2.1.1">𝛽</ci><ci id="S6.SS1.p2.6.m2.2.2.cmml" xref="S6.SS1.p2.6.m2.2.2">𝑐</ci></list><cn id="S6.SS1.p2.6.m2.2.3.3.cmml" type="integer" xref="S6.SS1.p2.6.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.6.m2.2c">\beta,c>0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.6.m2.2d">italic_β , italic_c > 0</annotation></semantics></math> such that, for every <math alttext="n\in\mathds{N}" class="ltx_Math" display="inline" id="S6.SS1.p2.7.m3.1"><semantics id="S6.SS1.p2.7.m3.1a"><mrow id="S6.SS1.p2.7.m3.1.1" xref="S6.SS1.p2.7.m3.1.1.cmml"><mi id="S6.SS1.p2.7.m3.1.1.2" xref="S6.SS1.p2.7.m3.1.1.2.cmml">n</mi><mo id="S6.SS1.p2.7.m3.1.1.1" xref="S6.SS1.p2.7.m3.1.1.1.cmml">∈</mo><mi id="S6.SS1.p2.7.m3.1.1.3" xref="S6.SS1.p2.7.m3.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.7.m3.1b"><apply id="S6.SS1.p2.7.m3.1.1.cmml" xref="S6.SS1.p2.7.m3.1.1"><in id="S6.SS1.p2.7.m3.1.1.1.cmml" xref="S6.SS1.p2.7.m3.1.1.1"></in><ci id="S6.SS1.p2.7.m3.1.1.2.cmml" xref="S6.SS1.p2.7.m3.1.1.2">𝑛</ci><ci id="S6.SS1.p2.7.m3.1.1.3.cmml" xref="S6.SS1.p2.7.m3.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.7.m3.1c">n\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.7.m3.1d">italic_n ∈ blackboard_N</annotation></semantics></math>, there exists a <math alttext="\operatorname{CPA}" class="ltx_Math" display="inline" id="S6.SS1.p2.8.m4.1"><semantics id="S6.SS1.p2.8.m4.1a"><mi id="S6.SS1.p2.8.m4.1.1" xref="S6.SS1.p2.8.m4.1.1.cmml">CPA</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.8.m4.1b"><ci id="S6.SS1.p2.8.m4.1.1.cmml" xref="S6.SS1.p2.8.m4.1.1">CPA</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.8.m4.1c">\operatorname{CPA}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.8.m4.1d">roman_CPA</annotation></semantics></math> function <math alttext="f:\mathds{R}^{2}\to\mathds{R}" class="ltx_Math" display="inline" id="S6.SS1.p2.9.m5.1"><semantics id="S6.SS1.p2.9.m5.1a"><mrow id="S6.SS1.p2.9.m5.1.1" xref="S6.SS1.p2.9.m5.1.1.cmml"><mi id="S6.SS1.p2.9.m5.1.1.2" xref="S6.SS1.p2.9.m5.1.1.2.cmml">f</mi><mo id="S6.SS1.p2.9.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS1.p2.9.m5.1.1.1.cmml">:</mo><mrow id="S6.SS1.p2.9.m5.1.1.3" xref="S6.SS1.p2.9.m5.1.1.3.cmml"><msup id="S6.SS1.p2.9.m5.1.1.3.2" xref="S6.SS1.p2.9.m5.1.1.3.2.cmml"><mi id="S6.SS1.p2.9.m5.1.1.3.2.2" xref="S6.SS1.p2.9.m5.1.1.3.2.2.cmml">ℝ</mi><mn id="S6.SS1.p2.9.m5.1.1.3.2.3" xref="S6.SS1.p2.9.m5.1.1.3.2.3.cmml">2</mn></msup><mo id="S6.SS1.p2.9.m5.1.1.3.1" stretchy="false" xref="S6.SS1.p2.9.m5.1.1.3.1.cmml">→</mo><mi id="S6.SS1.p2.9.m5.1.1.3.3" xref="S6.SS1.p2.9.m5.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.9.m5.1b"><apply id="S6.SS1.p2.9.m5.1.1.cmml" xref="S6.SS1.p2.9.m5.1.1"><ci id="S6.SS1.p2.9.m5.1.1.1.cmml" xref="S6.SS1.p2.9.m5.1.1.1">:</ci><ci id="S6.SS1.p2.9.m5.1.1.2.cmml" xref="S6.SS1.p2.9.m5.1.1.2">𝑓</ci><apply id="S6.SS1.p2.9.m5.1.1.3.cmml" xref="S6.SS1.p2.9.m5.1.1.3"><ci id="S6.SS1.p2.9.m5.1.1.3.1.cmml" xref="S6.SS1.p2.9.m5.1.1.3.1">→</ci><apply id="S6.SS1.p2.9.m5.1.1.3.2.cmml" xref="S6.SS1.p2.9.m5.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.p2.9.m5.1.1.3.2.1.cmml" xref="S6.SS1.p2.9.m5.1.1.3.2">superscript</csymbol><ci id="S6.SS1.p2.9.m5.1.1.3.2.2.cmml" xref="S6.SS1.p2.9.m5.1.1.3.2.2">ℝ</ci><cn id="S6.SS1.p2.9.m5.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.p2.9.m5.1.1.3.2.3">2</cn></apply><ci id="S6.SS1.p2.9.m5.1.1.3.3.cmml" xref="S6.SS1.p2.9.m5.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.9.m5.1c">f:\mathds{R}^{2}\to\mathds{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.9.m5.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> with at most <math alttext="n" class="ltx_Math" display="inline" id="S6.SS1.p2.10.m6.1"><semantics id="S6.SS1.p2.10.m6.1a"><mi id="S6.SS1.p2.10.m6.1.1" xref="S6.SS1.p2.10.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.10.m6.1b"><ci id="S6.SS1.p2.10.m6.1.1.cmml" xref="S6.SS1.p2.10.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.10.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.10.m6.1d">italic_n</annotation></semantics></math> affine components and at least <math alttext="\beta\cdot n^{3-\frac{c}{\sqrt{\log_{2}(n)}}}" class="ltx_Math" display="inline" id="S6.SS1.p2.11.m7.2"><semantics id="S6.SS1.p2.11.m7.2a"><mrow id="S6.SS1.p2.11.m7.2.3" xref="S6.SS1.p2.11.m7.2.3.cmml"><mi id="S6.SS1.p2.11.m7.2.3.2" xref="S6.SS1.p2.11.m7.2.3.2.cmml">β</mi><mo id="S6.SS1.p2.11.m7.2.3.1" lspace="0.222em" 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xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.3.cmml">2</mn></msub><mo id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2a" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.3.cmml">⁡</mo><mrow id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.2" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.3.cmml"><mo id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.2.1" stretchy="false" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.3.cmml">(</mo><mi id="S6.SS1.p2.11.m7.1.1.1.1.1.1.1.1" xref="S6.SS1.p2.11.m7.1.1.1.1.1.1.1.1.cmml">n</mi><mo id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.2.2" stretchy="false" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></msqrt></mfrac></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.11.m7.2b"><apply id="S6.SS1.p2.11.m7.2.3.cmml" xref="S6.SS1.p2.11.m7.2.3"><ci id="S6.SS1.p2.11.m7.2.3.1.cmml" xref="S6.SS1.p2.11.m7.2.3.1">⋅</ci><ci id="S6.SS1.p2.11.m7.2.3.2.cmml" xref="S6.SS1.p2.11.m7.2.3.2">𝛽</ci><apply id="S6.SS1.p2.11.m7.2.3.3.cmml" xref="S6.SS1.p2.11.m7.2.3.3"><csymbol cd="ambiguous" 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id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.1.cmml" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1">subscript</csymbol><log id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.2.cmml" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.2"></log><cn id="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.3.cmml" type="integer" xref="S6.SS1.p2.11.m7.2.2.2.2.2.2.2.2.1.3">2</cn></apply><ci id="S6.SS1.p2.11.m7.1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.p2.11.m7.1.1.1.1.1.1.1.1">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.11.m7.2c">\beta\cdot n^{3-\frac{c}{\sqrt{\log_{2}(n)}}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.11.m7.2d">italic_β ⋅ italic_n start_POSTSUPERSCRIPT 3 - divide start_ARG italic_c end_ARG start_ARG square-root start_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_n ) end_ARG end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math> pieces. For such functions, the construction presented in this work does not provide a significant improvement over <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S6.Thmtheorem1" title="Theorem 6.1. ‣ 6.1 Comparison to the Result of \citetKoutschan2023 ‣ 6 Discussion ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 6.1</span></a>. However, there also exist CPA functions with <math alttext="p=n" class="ltx_Math" display="inline" id="S6.SS1.p2.12.m8.1"><semantics id="S6.SS1.p2.12.m8.1a"><mrow id="S6.SS1.p2.12.m8.1.1" xref="S6.SS1.p2.12.m8.1.1.cmml"><mi id="S6.SS1.p2.12.m8.1.1.2" xref="S6.SS1.p2.12.m8.1.1.2.cmml">p</mi><mo id="S6.SS1.p2.12.m8.1.1.1" xref="S6.SS1.p2.12.m8.1.1.1.cmml">=</mo><mi id="S6.SS1.p2.12.m8.1.1.3" xref="S6.SS1.p2.12.m8.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.12.m8.1b"><apply id="S6.SS1.p2.12.m8.1.1.cmml" xref="S6.SS1.p2.12.m8.1.1"><eq id="S6.SS1.p2.12.m8.1.1.1.cmml" xref="S6.SS1.p2.12.m8.1.1.1"></eq><ci id="S6.SS1.p2.12.m8.1.1.2.cmml" xref="S6.SS1.p2.12.m8.1.1.2">𝑝</ci><ci id="S6.SS1.p2.12.m8.1.1.3.cmml" xref="S6.SS1.p2.12.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.12.m8.1c">p=n</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.12.m8.1d">italic_p = italic_n</annotation></semantics></math>, for which <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Neural Network Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 5.1</span></a> gives a neural network representation with width <math alttext="O(n)" class="ltx_Math" display="inline" id="S6.SS1.p2.13.m9.1"><semantics id="S6.SS1.p2.13.m9.1a"><mrow id="S6.SS1.p2.13.m9.1.2" xref="S6.SS1.p2.13.m9.1.2.cmml"><mi id="S6.SS1.p2.13.m9.1.2.2" xref="S6.SS1.p2.13.m9.1.2.2.cmml">O</mi><mo id="S6.SS1.p2.13.m9.1.2.1" xref="S6.SS1.p2.13.m9.1.2.1.cmml">⁢</mo><mrow id="S6.SS1.p2.13.m9.1.2.3.2" xref="S6.SS1.p2.13.m9.1.2.cmml"><mo id="S6.SS1.p2.13.m9.1.2.3.2.1" stretchy="false" xref="S6.SS1.p2.13.m9.1.2.cmml">(</mo><mi id="S6.SS1.p2.13.m9.1.1" xref="S6.SS1.p2.13.m9.1.1.cmml">n</mi><mo id="S6.SS1.p2.13.m9.1.2.3.2.2" stretchy="false" xref="S6.SS1.p2.13.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.13.m9.1b"><apply id="S6.SS1.p2.13.m9.1.2.cmml" xref="S6.SS1.p2.13.m9.1.2"><times id="S6.SS1.p2.13.m9.1.2.1.cmml" xref="S6.SS1.p2.13.m9.1.2.1"></times><ci id="S6.SS1.p2.13.m9.1.2.2.cmml" xref="S6.SS1.p2.13.m9.1.2.2">𝑂</ci><ci id="S6.SS1.p2.13.m9.1.1.cmml" xref="S6.SS1.p2.13.m9.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.13.m9.1c">O(n)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.13.m9.1d">italic_O ( italic_n )</annotation></semantics></math>, improving upon the <math alttext="O(n^{3})" class="ltx_Math" display="inline" id="S6.SS1.p2.14.m10.1"><semantics id="S6.SS1.p2.14.m10.1a"><mrow id="S6.SS1.p2.14.m10.1.1" xref="S6.SS1.p2.14.m10.1.1.cmml"><mi id="S6.SS1.p2.14.m10.1.1.3" xref="S6.SS1.p2.14.m10.1.1.3.cmml">O</mi><mo id="S6.SS1.p2.14.m10.1.1.2" xref="S6.SS1.p2.14.m10.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p2.14.m10.1.1.1.1" xref="S6.SS1.p2.14.m10.1.1.1.1.1.cmml"><mo id="S6.SS1.p2.14.m10.1.1.1.1.2" stretchy="false" xref="S6.SS1.p2.14.m10.1.1.1.1.1.cmml">(</mo><msup id="S6.SS1.p2.14.m10.1.1.1.1.1" xref="S6.SS1.p2.14.m10.1.1.1.1.1.cmml"><mi id="S6.SS1.p2.14.m10.1.1.1.1.1.2" xref="S6.SS1.p2.14.m10.1.1.1.1.1.2.cmml">n</mi><mn id="S6.SS1.p2.14.m10.1.1.1.1.1.3" xref="S6.SS1.p2.14.m10.1.1.1.1.1.3.cmml">3</mn></msup><mo id="S6.SS1.p2.14.m10.1.1.1.1.3" stretchy="false" xref="S6.SS1.p2.14.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.14.m10.1b"><apply id="S6.SS1.p2.14.m10.1.1.cmml" xref="S6.SS1.p2.14.m10.1.1"><times id="S6.SS1.p2.14.m10.1.1.2.cmml" xref="S6.SS1.p2.14.m10.1.1.2"></times><ci id="S6.SS1.p2.14.m10.1.1.3.cmml" xref="S6.SS1.p2.14.m10.1.1.3">𝑂</ci><apply id="S6.SS1.p2.14.m10.1.1.1.1.1.cmml" xref="S6.SS1.p2.14.m10.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.14.m10.1.1.1.1.1.1.cmml" xref="S6.SS1.p2.14.m10.1.1.1.1">superscript</csymbol><ci id="S6.SS1.p2.14.m10.1.1.1.1.1.2.cmml" xref="S6.SS1.p2.14.m10.1.1.1.1.1.2">𝑛</ci><cn id="S6.SS1.p2.14.m10.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p2.14.m10.1.1.1.1.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.14.m10.1c">O(n^{3})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.14.m10.1d">italic_O ( italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math> bound of <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S6.Thmtheorem1" title="Theorem 6.1. ‣ 6.1 Comparison to the Result of \citetKoutschan2023 ‣ 6 Discussion ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Theorem 6.1</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.2 </span>Comparison with Convex Pieces</h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.1">Bounded polygons without holes can be triangulated without introducing additional vertices, ensuring that each edge is part of the triangulation <span class="ltx_ERROR undefined" id="S6.SS2.p1.1.1">\citep</span>ORourke1987ArtGallery. For unbounded polygons, each end can be closed by adding four vertices, as illustrated in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.F8" title="Figure 8 ‣ Proof. ‣ 3.1.2 Only arcs ‣ 3.1 Proof of 3.2 ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">Figure 8</span></a>. The bounded part of the polygon can then be triangulated, while the unbounded part of each end can be subdivided into three convex polygons. This procedure produces a refinement of any set of pieces into convex pieces.</p> </div> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.19">For <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S6.SS2.p2.1.m1.1"><semantics id="S6.SS2.p2.1.m1.1a"><mrow id="S6.SS2.p2.1.m1.1.1" xref="S6.SS2.p2.1.m1.1.1.cmml"><mi id="S6.SS2.p2.1.m1.1.1.2" xref="S6.SS2.p2.1.m1.1.1.2.cmml">f</mi><mo id="S6.SS2.p2.1.m1.1.1.1" xref="S6.SS2.p2.1.m1.1.1.1.cmml">∈</mo><msub id="S6.SS2.p2.1.m1.1.1.3" xref="S6.SS2.p2.1.m1.1.1.3.cmml"><mi id="S6.SS2.p2.1.m1.1.1.3.2" xref="S6.SS2.p2.1.m1.1.1.3.2.cmml">CPA</mi><mi id="S6.SS2.p2.1.m1.1.1.3.3" xref="S6.SS2.p2.1.m1.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.1.m1.1b"><apply id="S6.SS2.p2.1.m1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1"><in id="S6.SS2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1.1"></in><ci id="S6.SS2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.p2.1.m1.1.1.2">𝑓</ci><apply id="S6.SS2.p2.1.m1.1.1.3.cmml" xref="S6.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p2.1.m1.1.1.3.1.cmml" xref="S6.SS2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS2.p2.1.m1.1.1.3.2.cmml" xref="S6.SS2.p2.1.m1.1.1.3.2">CPA</ci><ci id="S6.SS2.p2.1.m1.1.1.3.3.cmml" xref="S6.SS2.p2.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.1.m1.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.1.m1.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.p2.2.m2.1"><semantics id="S6.SS2.p2.2.m2.1a"><mi id="S6.SS2.p2.2.m2.1.1" xref="S6.SS2.p2.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.2.m2.1b"><ci id="S6.SS2.p2.2.m2.1.1.cmml" xref="S6.SS2.p2.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.2.m2.1d">italic_p</annotation></semantics></math>, <math alttext="e" class="ltx_Math" display="inline" id="S6.SS2.p2.3.m3.1"><semantics id="S6.SS2.p2.3.m3.1a"><mi id="S6.SS2.p2.3.m3.1.1" xref="S6.SS2.p2.3.m3.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.3.m3.1b"><ci id="S6.SS2.p2.3.m3.1.1.cmml" xref="S6.SS2.p2.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.3.m3.1c">e</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.3.m3.1d">italic_e</annotation></semantics></math>, and <math alttext="v" class="ltx_Math" display="inline" id="S6.SS2.p2.4.m4.1"><semantics id="S6.SS2.p2.4.m4.1a"><mi id="S6.SS2.p2.4.m4.1.1" xref="S6.SS2.p2.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.4.m4.1b"><ci id="S6.SS2.p2.4.m4.1.1.cmml" xref="S6.SS2.p2.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.4.m4.1d">italic_v</annotation></semantics></math> denote the numbers of pieces, edges, and vertices corresponding to the admissible set of pieces given by <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.1</span></a>. Applying the aforementioned procedure results in a subdivision with <math alttext="p_{t}" class="ltx_Math" display="inline" id="S6.SS2.p2.5.m5.1"><semantics id="S6.SS2.p2.5.m5.1a"><msub id="S6.SS2.p2.5.m5.1.1" xref="S6.SS2.p2.5.m5.1.1.cmml"><mi id="S6.SS2.p2.5.m5.1.1.2" xref="S6.SS2.p2.5.m5.1.1.2.cmml">p</mi><mi id="S6.SS2.p2.5.m5.1.1.3" xref="S6.SS2.p2.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.5.m5.1b"><apply id="S6.SS2.p2.5.m5.1.1.cmml" xref="S6.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.1.cmml" xref="S6.SS2.p2.5.m5.1.1">subscript</csymbol><ci id="S6.SS2.p2.5.m5.1.1.2.cmml" xref="S6.SS2.p2.5.m5.1.1.2">𝑝</ci><ci id="S6.SS2.p2.5.m5.1.1.3.cmml" xref="S6.SS2.p2.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.5.m5.1c">p_{t}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.5.m5.1d">italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> convex pieces (primarily triangles), <math alttext="e_{t}" class="ltx_Math" display="inline" id="S6.SS2.p2.6.m6.1"><semantics id="S6.SS2.p2.6.m6.1a"><msub id="S6.SS2.p2.6.m6.1.1" xref="S6.SS2.p2.6.m6.1.1.cmml"><mi id="S6.SS2.p2.6.m6.1.1.2" xref="S6.SS2.p2.6.m6.1.1.2.cmml">e</mi><mi id="S6.SS2.p2.6.m6.1.1.3" xref="S6.SS2.p2.6.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.6.m6.1b"><apply id="S6.SS2.p2.6.m6.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.6.m6.1.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.p2.6.m6.1.1.2.cmml" xref="S6.SS2.p2.6.m6.1.1.2">𝑒</ci><ci id="S6.SS2.p2.6.m6.1.1.3.cmml" xref="S6.SS2.p2.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.6.m6.1c">e_{t}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.6.m6.1d">italic_e start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> edges, and <math alttext="v_{t}" class="ltx_Math" display="inline" id="S6.SS2.p2.7.m7.1"><semantics id="S6.SS2.p2.7.m7.1a"><msub id="S6.SS2.p2.7.m7.1.1" xref="S6.SS2.p2.7.m7.1.1.cmml"><mi id="S6.SS2.p2.7.m7.1.1.2" xref="S6.SS2.p2.7.m7.1.1.2.cmml">v</mi><mi id="S6.SS2.p2.7.m7.1.1.3" xref="S6.SS2.p2.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.7.m7.1b"><apply id="S6.SS2.p2.7.m7.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.7.m7.1.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.p2.7.m7.1.1.2.cmml" xref="S6.SS2.p2.7.m7.1.1.2">𝑣</ci><ci id="S6.SS2.p2.7.m7.1.1.3.cmml" xref="S6.SS2.p2.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.7.m7.1c">v_{t}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.7.m7.1d">italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> vertices. Since each end is bounded by at least one edge, and any edge bounds at most four ends, the pieces of <math alttext="f" class="ltx_Math" display="inline" id="S6.SS2.p2.8.m8.1"><semantics id="S6.SS2.p2.8.m8.1a"><mi id="S6.SS2.p2.8.m8.1.1" xref="S6.SS2.p2.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.8.m8.1b"><ci id="S6.SS2.p2.8.m8.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.8.m8.1d">italic_f</annotation></semantics></math> collectively have at most <math alttext="4e" class="ltx_Math" display="inline" id="S6.SS2.p2.9.m9.1"><semantics id="S6.SS2.p2.9.m9.1a"><mrow id="S6.SS2.p2.9.m9.1.1" xref="S6.SS2.p2.9.m9.1.1.cmml"><mn id="S6.SS2.p2.9.m9.1.1.2" xref="S6.SS2.p2.9.m9.1.1.2.cmml">4</mn><mo id="S6.SS2.p2.9.m9.1.1.1" xref="S6.SS2.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S6.SS2.p2.9.m9.1.1.3" xref="S6.SS2.p2.9.m9.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.9.m9.1b"><apply id="S6.SS2.p2.9.m9.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1"><times id="S6.SS2.p2.9.m9.1.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1.1"></times><cn id="S6.SS2.p2.9.m9.1.1.2.cmml" type="integer" xref="S6.SS2.p2.9.m9.1.1.2">4</cn><ci id="S6.SS2.p2.9.m9.1.1.3.cmml" xref="S6.SS2.p2.9.m9.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.9.m9.1c">4e</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.9.m9.1d">4 italic_e</annotation></semantics></math> ends. Consequently, the convex subdivision satisfies <math alttext="v_{t}\leq v+4\cdot 4e" class="ltx_Math" display="inline" id="S6.SS2.p2.10.m10.1"><semantics id="S6.SS2.p2.10.m10.1a"><mrow id="S6.SS2.p2.10.m10.1.1" xref="S6.SS2.p2.10.m10.1.1.cmml"><msub id="S6.SS2.p2.10.m10.1.1.2" xref="S6.SS2.p2.10.m10.1.1.2.cmml"><mi id="S6.SS2.p2.10.m10.1.1.2.2" xref="S6.SS2.p2.10.m10.1.1.2.2.cmml">v</mi><mi id="S6.SS2.p2.10.m10.1.1.2.3" xref="S6.SS2.p2.10.m10.1.1.2.3.cmml">t</mi></msub><mo id="S6.SS2.p2.10.m10.1.1.1" xref="S6.SS2.p2.10.m10.1.1.1.cmml">≤</mo><mrow id="S6.SS2.p2.10.m10.1.1.3" xref="S6.SS2.p2.10.m10.1.1.3.cmml"><mi id="S6.SS2.p2.10.m10.1.1.3.2" xref="S6.SS2.p2.10.m10.1.1.3.2.cmml">v</mi><mo id="S6.SS2.p2.10.m10.1.1.3.1" xref="S6.SS2.p2.10.m10.1.1.3.1.cmml">+</mo><mrow id="S6.SS2.p2.10.m10.1.1.3.3" xref="S6.SS2.p2.10.m10.1.1.3.3.cmml"><mrow id="S6.SS2.p2.10.m10.1.1.3.3.2" xref="S6.SS2.p2.10.m10.1.1.3.3.2.cmml"><mn id="S6.SS2.p2.10.m10.1.1.3.3.2.2" xref="S6.SS2.p2.10.m10.1.1.3.3.2.2.cmml">4</mn><mo id="S6.SS2.p2.10.m10.1.1.3.3.2.1" lspace="0.222em" rspace="0.222em" xref="S6.SS2.p2.10.m10.1.1.3.3.2.1.cmml">⋅</mo><mn id="S6.SS2.p2.10.m10.1.1.3.3.2.3" xref="S6.SS2.p2.10.m10.1.1.3.3.2.3.cmml">4</mn></mrow><mo id="S6.SS2.p2.10.m10.1.1.3.3.1" xref="S6.SS2.p2.10.m10.1.1.3.3.1.cmml">⁢</mo><mi id="S6.SS2.p2.10.m10.1.1.3.3.3" xref="S6.SS2.p2.10.m10.1.1.3.3.3.cmml">e</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.10.m10.1b"><apply id="S6.SS2.p2.10.m10.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1"><leq id="S6.SS2.p2.10.m10.1.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1.1"></leq><apply id="S6.SS2.p2.10.m10.1.1.2.cmml" xref="S6.SS2.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p2.10.m10.1.1.2.1.cmml" xref="S6.SS2.p2.10.m10.1.1.2">subscript</csymbol><ci id="S6.SS2.p2.10.m10.1.1.2.2.cmml" xref="S6.SS2.p2.10.m10.1.1.2.2">𝑣</ci><ci id="S6.SS2.p2.10.m10.1.1.2.3.cmml" xref="S6.SS2.p2.10.m10.1.1.2.3">𝑡</ci></apply><apply id="S6.SS2.p2.10.m10.1.1.3.cmml" xref="S6.SS2.p2.10.m10.1.1.3"><plus id="S6.SS2.p2.10.m10.1.1.3.1.cmml" xref="S6.SS2.p2.10.m10.1.1.3.1"></plus><ci id="S6.SS2.p2.10.m10.1.1.3.2.cmml" xref="S6.SS2.p2.10.m10.1.1.3.2">𝑣</ci><apply id="S6.SS2.p2.10.m10.1.1.3.3.cmml" xref="S6.SS2.p2.10.m10.1.1.3.3"><times id="S6.SS2.p2.10.m10.1.1.3.3.1.cmml" xref="S6.SS2.p2.10.m10.1.1.3.3.1"></times><apply id="S6.SS2.p2.10.m10.1.1.3.3.2.cmml" xref="S6.SS2.p2.10.m10.1.1.3.3.2"><ci id="S6.SS2.p2.10.m10.1.1.3.3.2.1.cmml" xref="S6.SS2.p2.10.m10.1.1.3.3.2.1">⋅</ci><cn id="S6.SS2.p2.10.m10.1.1.3.3.2.2.cmml" type="integer" xref="S6.SS2.p2.10.m10.1.1.3.3.2.2">4</cn><cn id="S6.SS2.p2.10.m10.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS2.p2.10.m10.1.1.3.3.2.3">4</cn></apply><ci id="S6.SS2.p2.10.m10.1.1.3.3.3.cmml" xref="S6.SS2.p2.10.m10.1.1.3.3.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.10.m10.1c">v_{t}\leq v+4\cdot 4e</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.10.m10.1d">italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ italic_v + 4 ⋅ 4 italic_e</annotation></semantics></math>. By <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 max-Representation of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">4.1</span></a>, <math alttext="v\leq 2p" class="ltx_Math" display="inline" id="S6.SS2.p2.11.m11.1"><semantics id="S6.SS2.p2.11.m11.1a"><mrow id="S6.SS2.p2.11.m11.1.1" xref="S6.SS2.p2.11.m11.1.1.cmml"><mi id="S6.SS2.p2.11.m11.1.1.2" xref="S6.SS2.p2.11.m11.1.1.2.cmml">v</mi><mo id="S6.SS2.p2.11.m11.1.1.1" xref="S6.SS2.p2.11.m11.1.1.1.cmml">≤</mo><mrow id="S6.SS2.p2.11.m11.1.1.3" xref="S6.SS2.p2.11.m11.1.1.3.cmml"><mn id="S6.SS2.p2.11.m11.1.1.3.2" xref="S6.SS2.p2.11.m11.1.1.3.2.cmml">2</mn><mo id="S6.SS2.p2.11.m11.1.1.3.1" xref="S6.SS2.p2.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S6.SS2.p2.11.m11.1.1.3.3" xref="S6.SS2.p2.11.m11.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.11.m11.1b"><apply id="S6.SS2.p2.11.m11.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1"><leq id="S6.SS2.p2.11.m11.1.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1.1"></leq><ci id="S6.SS2.p2.11.m11.1.1.2.cmml" xref="S6.SS2.p2.11.m11.1.1.2">𝑣</ci><apply id="S6.SS2.p2.11.m11.1.1.3.cmml" xref="S6.SS2.p2.11.m11.1.1.3"><times id="S6.SS2.p2.11.m11.1.1.3.1.cmml" xref="S6.SS2.p2.11.m11.1.1.3.1"></times><cn id="S6.SS2.p2.11.m11.1.1.3.2.cmml" type="integer" xref="S6.SS2.p2.11.m11.1.1.3.2">2</cn><ci id="S6.SS2.p2.11.m11.1.1.3.3.cmml" xref="S6.SS2.p2.11.m11.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.11.m11.1c">v\leq 2p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.11.m11.1d">italic_v ≤ 2 italic_p</annotation></semantics></math> and <math alttext="e\leq 3p" class="ltx_Math" display="inline" id="S6.SS2.p2.12.m12.1"><semantics id="S6.SS2.p2.12.m12.1a"><mrow id="S6.SS2.p2.12.m12.1.1" xref="S6.SS2.p2.12.m12.1.1.cmml"><mi id="S6.SS2.p2.12.m12.1.1.2" xref="S6.SS2.p2.12.m12.1.1.2.cmml">e</mi><mo id="S6.SS2.p2.12.m12.1.1.1" xref="S6.SS2.p2.12.m12.1.1.1.cmml">≤</mo><mrow id="S6.SS2.p2.12.m12.1.1.3" xref="S6.SS2.p2.12.m12.1.1.3.cmml"><mn id="S6.SS2.p2.12.m12.1.1.3.2" xref="S6.SS2.p2.12.m12.1.1.3.2.cmml">3</mn><mo id="S6.SS2.p2.12.m12.1.1.3.1" xref="S6.SS2.p2.12.m12.1.1.3.1.cmml">⁢</mo><mi id="S6.SS2.p2.12.m12.1.1.3.3" xref="S6.SS2.p2.12.m12.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.12.m12.1b"><apply id="S6.SS2.p2.12.m12.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1"><leq id="S6.SS2.p2.12.m12.1.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1.1"></leq><ci id="S6.SS2.p2.12.m12.1.1.2.cmml" xref="S6.SS2.p2.12.m12.1.1.2">𝑒</ci><apply id="S6.SS2.p2.12.m12.1.1.3.cmml" xref="S6.SS2.p2.12.m12.1.1.3"><times id="S6.SS2.p2.12.m12.1.1.3.1.cmml" xref="S6.SS2.p2.12.m12.1.1.3.1"></times><cn id="S6.SS2.p2.12.m12.1.1.3.2.cmml" type="integer" xref="S6.SS2.p2.12.m12.1.1.3.2">3</cn><ci id="S6.SS2.p2.12.m12.1.1.3.3.cmml" xref="S6.SS2.p2.12.m12.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.12.m12.1c">e\leq 3p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.12.m12.1d">italic_e ≤ 3 italic_p</annotation></semantics></math>. Using Euler’s formula, we find <math alttext="p_{t}\leq 2v_{t}" class="ltx_Math" display="inline" id="S6.SS2.p2.13.m13.1"><semantics id="S6.SS2.p2.13.m13.1a"><mrow id="S6.SS2.p2.13.m13.1.1" xref="S6.SS2.p2.13.m13.1.1.cmml"><msub id="S6.SS2.p2.13.m13.1.1.2" xref="S6.SS2.p2.13.m13.1.1.2.cmml"><mi id="S6.SS2.p2.13.m13.1.1.2.2" xref="S6.SS2.p2.13.m13.1.1.2.2.cmml">p</mi><mi id="S6.SS2.p2.13.m13.1.1.2.3" xref="S6.SS2.p2.13.m13.1.1.2.3.cmml">t</mi></msub><mo id="S6.SS2.p2.13.m13.1.1.1" xref="S6.SS2.p2.13.m13.1.1.1.cmml">≤</mo><mrow id="S6.SS2.p2.13.m13.1.1.3" xref="S6.SS2.p2.13.m13.1.1.3.cmml"><mn id="S6.SS2.p2.13.m13.1.1.3.2" xref="S6.SS2.p2.13.m13.1.1.3.2.cmml">2</mn><mo id="S6.SS2.p2.13.m13.1.1.3.1" xref="S6.SS2.p2.13.m13.1.1.3.1.cmml">⁢</mo><msub id="S6.SS2.p2.13.m13.1.1.3.3" xref="S6.SS2.p2.13.m13.1.1.3.3.cmml"><mi id="S6.SS2.p2.13.m13.1.1.3.3.2" xref="S6.SS2.p2.13.m13.1.1.3.3.2.cmml">v</mi><mi id="S6.SS2.p2.13.m13.1.1.3.3.3" xref="S6.SS2.p2.13.m13.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.13.m13.1b"><apply id="S6.SS2.p2.13.m13.1.1.cmml" xref="S6.SS2.p2.13.m13.1.1"><leq id="S6.SS2.p2.13.m13.1.1.1.cmml" xref="S6.SS2.p2.13.m13.1.1.1"></leq><apply id="S6.SS2.p2.13.m13.1.1.2.cmml" xref="S6.SS2.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p2.13.m13.1.1.2.1.cmml" xref="S6.SS2.p2.13.m13.1.1.2">subscript</csymbol><ci id="S6.SS2.p2.13.m13.1.1.2.2.cmml" xref="S6.SS2.p2.13.m13.1.1.2.2">𝑝</ci><ci id="S6.SS2.p2.13.m13.1.1.2.3.cmml" xref="S6.SS2.p2.13.m13.1.1.2.3">𝑡</ci></apply><apply id="S6.SS2.p2.13.m13.1.1.3.cmml" xref="S6.SS2.p2.13.m13.1.1.3"><times id="S6.SS2.p2.13.m13.1.1.3.1.cmml" xref="S6.SS2.p2.13.m13.1.1.3.1"></times><cn id="S6.SS2.p2.13.m13.1.1.3.2.cmml" type="integer" xref="S6.SS2.p2.13.m13.1.1.3.2">2</cn><apply id="S6.SS2.p2.13.m13.1.1.3.3.cmml" xref="S6.SS2.p2.13.m13.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS2.p2.13.m13.1.1.3.3.1.cmml" xref="S6.SS2.p2.13.m13.1.1.3.3">subscript</csymbol><ci id="S6.SS2.p2.13.m13.1.1.3.3.2.cmml" xref="S6.SS2.p2.13.m13.1.1.3.3.2">𝑣</ci><ci id="S6.SS2.p2.13.m13.1.1.3.3.3.cmml" xref="S6.SS2.p2.13.m13.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.13.m13.1c">p_{t}\leq 2v_{t}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.13.m13.1d">italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ 2 italic_v start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, implying that there exists <math alttext="c\in\mathds{N}" class="ltx_Math" display="inline" id="S6.SS2.p2.14.m14.1"><semantics id="S6.SS2.p2.14.m14.1a"><mrow id="S6.SS2.p2.14.m14.1.1" xref="S6.SS2.p2.14.m14.1.1.cmml"><mi id="S6.SS2.p2.14.m14.1.1.2" xref="S6.SS2.p2.14.m14.1.1.2.cmml">c</mi><mo id="S6.SS2.p2.14.m14.1.1.1" xref="S6.SS2.p2.14.m14.1.1.1.cmml">∈</mo><mi id="S6.SS2.p2.14.m14.1.1.3" xref="S6.SS2.p2.14.m14.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.14.m14.1b"><apply id="S6.SS2.p2.14.m14.1.1.cmml" xref="S6.SS2.p2.14.m14.1.1"><in id="S6.SS2.p2.14.m14.1.1.1.cmml" xref="S6.SS2.p2.14.m14.1.1.1"></in><ci id="S6.SS2.p2.14.m14.1.1.2.cmml" xref="S6.SS2.p2.14.m14.1.1.2">𝑐</ci><ci id="S6.SS2.p2.14.m14.1.1.3.cmml" xref="S6.SS2.p2.14.m14.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.14.m14.1c">c\in\mathds{N}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.14.m14.1d">italic_c ∈ blackboard_N</annotation></semantics></math> such that <math alttext="p_{t}\leq cp" class="ltx_Math" display="inline" id="S6.SS2.p2.15.m15.1"><semantics id="S6.SS2.p2.15.m15.1a"><mrow id="S6.SS2.p2.15.m15.1.1" xref="S6.SS2.p2.15.m15.1.1.cmml"><msub id="S6.SS2.p2.15.m15.1.1.2" xref="S6.SS2.p2.15.m15.1.1.2.cmml"><mi id="S6.SS2.p2.15.m15.1.1.2.2" xref="S6.SS2.p2.15.m15.1.1.2.2.cmml">p</mi><mi id="S6.SS2.p2.15.m15.1.1.2.3" xref="S6.SS2.p2.15.m15.1.1.2.3.cmml">t</mi></msub><mo id="S6.SS2.p2.15.m15.1.1.1" xref="S6.SS2.p2.15.m15.1.1.1.cmml">≤</mo><mrow id="S6.SS2.p2.15.m15.1.1.3" xref="S6.SS2.p2.15.m15.1.1.3.cmml"><mi id="S6.SS2.p2.15.m15.1.1.3.2" xref="S6.SS2.p2.15.m15.1.1.3.2.cmml">c</mi><mo id="S6.SS2.p2.15.m15.1.1.3.1" xref="S6.SS2.p2.15.m15.1.1.3.1.cmml">⁢</mo><mi id="S6.SS2.p2.15.m15.1.1.3.3" xref="S6.SS2.p2.15.m15.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.15.m15.1b"><apply id="S6.SS2.p2.15.m15.1.1.cmml" xref="S6.SS2.p2.15.m15.1.1"><leq id="S6.SS2.p2.15.m15.1.1.1.cmml" xref="S6.SS2.p2.15.m15.1.1.1"></leq><apply id="S6.SS2.p2.15.m15.1.1.2.cmml" xref="S6.SS2.p2.15.m15.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p2.15.m15.1.1.2.1.cmml" xref="S6.SS2.p2.15.m15.1.1.2">subscript</csymbol><ci id="S6.SS2.p2.15.m15.1.1.2.2.cmml" xref="S6.SS2.p2.15.m15.1.1.2.2">𝑝</ci><ci id="S6.SS2.p2.15.m15.1.1.2.3.cmml" xref="S6.SS2.p2.15.m15.1.1.2.3">𝑡</ci></apply><apply id="S6.SS2.p2.15.m15.1.1.3.cmml" xref="S6.SS2.p2.15.m15.1.1.3"><times id="S6.SS2.p2.15.m15.1.1.3.1.cmml" xref="S6.SS2.p2.15.m15.1.1.3.1"></times><ci id="S6.SS2.p2.15.m15.1.1.3.2.cmml" xref="S6.SS2.p2.15.m15.1.1.3.2">𝑐</ci><ci id="S6.SS2.p2.15.m15.1.1.3.3.cmml" xref="S6.SS2.p2.15.m15.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.15.m15.1c">p_{t}\leq cp</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.15.m15.1d">italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ italic_c italic_p</annotation></semantics></math>, where <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.p2.16.m16.1"><semantics id="S6.SS2.p2.16.m16.1a"><mi id="S6.SS2.p2.16.m16.1.1" xref="S6.SS2.p2.16.m16.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.16.m16.1b"><ci id="S6.SS2.p2.16.m16.1.1.cmml" xref="S6.SS2.p2.16.m16.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.16.m16.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.16.m16.1d">italic_c</annotation></semantics></math> is independent of <math alttext="f" class="ltx_Math" display="inline" id="S6.SS2.p2.17.m17.1"><semantics id="S6.SS2.p2.17.m17.1a"><mi id="S6.SS2.p2.17.m17.1.1" xref="S6.SS2.p2.17.m17.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.17.m17.1b"><ci id="S6.SS2.p2.17.m17.1.1.cmml" xref="S6.SS2.p2.17.m17.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.17.m17.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.17.m17.1d">italic_f</annotation></semantics></math>. Therefore, any <math alttext="f\in\operatorname{CPA}_{p}" class="ltx_Math" display="inline" id="S6.SS2.p2.18.m18.1"><semantics id="S6.SS2.p2.18.m18.1a"><mrow id="S6.SS2.p2.18.m18.1.1" xref="S6.SS2.p2.18.m18.1.1.cmml"><mi id="S6.SS2.p2.18.m18.1.1.2" xref="S6.SS2.p2.18.m18.1.1.2.cmml">f</mi><mo id="S6.SS2.p2.18.m18.1.1.1" xref="S6.SS2.p2.18.m18.1.1.1.cmml">∈</mo><msub id="S6.SS2.p2.18.m18.1.1.3" xref="S6.SS2.p2.18.m18.1.1.3.cmml"><mi id="S6.SS2.p2.18.m18.1.1.3.2" xref="S6.SS2.p2.18.m18.1.1.3.2.cmml">CPA</mi><mi id="S6.SS2.p2.18.m18.1.1.3.3" xref="S6.SS2.p2.18.m18.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.18.m18.1b"><apply id="S6.SS2.p2.18.m18.1.1.cmml" xref="S6.SS2.p2.18.m18.1.1"><in id="S6.SS2.p2.18.m18.1.1.1.cmml" xref="S6.SS2.p2.18.m18.1.1.1"></in><ci id="S6.SS2.p2.18.m18.1.1.2.cmml" xref="S6.SS2.p2.18.m18.1.1.2">𝑓</ci><apply id="S6.SS2.p2.18.m18.1.1.3.cmml" xref="S6.SS2.p2.18.m18.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p2.18.m18.1.1.3.1.cmml" xref="S6.SS2.p2.18.m18.1.1.3">subscript</csymbol><ci id="S6.SS2.p2.18.m18.1.1.3.2.cmml" xref="S6.SS2.p2.18.m18.1.1.3.2">CPA</ci><ci id="S6.SS2.p2.18.m18.1.1.3.3.cmml" xref="S6.SS2.p2.18.m18.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.18.m18.1c">f\in\operatorname{CPA}_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.18.m18.1d">italic_f ∈ roman_CPA start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> can be represented with an admissible set of at most <math alttext="c\cdot p" class="ltx_Math" display="inline" id="S6.SS2.p2.19.m19.1"><semantics id="S6.SS2.p2.19.m19.1a"><mrow id="S6.SS2.p2.19.m19.1.1" xref="S6.SS2.p2.19.m19.1.1.cmml"><mi id="S6.SS2.p2.19.m19.1.1.2" xref="S6.SS2.p2.19.m19.1.1.2.cmml">c</mi><mo id="S6.SS2.p2.19.m19.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.SS2.p2.19.m19.1.1.1.cmml">⋅</mo><mi id="S6.SS2.p2.19.m19.1.1.3" xref="S6.SS2.p2.19.m19.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.19.m19.1b"><apply id="S6.SS2.p2.19.m19.1.1.cmml" xref="S6.SS2.p2.19.m19.1.1"><ci id="S6.SS2.p2.19.m19.1.1.1.cmml" xref="S6.SS2.p2.19.m19.1.1.1">⋅</ci><ci id="S6.SS2.p2.19.m19.1.1.2.cmml" xref="S6.SS2.p2.19.m19.1.1.2">𝑐</ci><ci id="S6.SS2.p2.19.m19.1.1.3.cmml" xref="S6.SS2.p2.19.m19.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.19.m19.1c">c\cdot p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.19.m19.1d">italic_c ⋅ italic_p</annotation></semantics></math> convex pieces.</p> </div> <div class="ltx_para" id="S6.SS2.p3"> <p class="ltx_p" id="S6.SS2.p3.1">Starting with such a refinement into convex pieces would also yield results with linear complexity in <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.p3.1.m1.1"><semantics id="S6.SS2.p3.1.m1.1a"><mi id="S6.SS2.p3.1.m1.1.1" xref="S6.SS2.p3.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.1.m1.1b"><ci id="S6.SS2.p3.1.m1.1.1.cmml" xref="S6.SS2.p3.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.1.m1.1d">italic_p</annotation></semantics></math>. While this approach simplifies some proofs, it results in worse constants. But more importantly, it sacrifices insights into the natural notion of pieces introduced here, which has not been previously considered in the literature. For instance, the precise form of the conic decomposition of non-convex polygons (<a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem2" title="Lemma 3.2. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.2</span></a>) may be of independent interest.</p> </div> </section> <section class="ltx_subsection" id="S6.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.3 </span>Extension to Higher Dimensions</h3> <div class="ltx_para" id="S6.SS3.p1"> <p class="ltx_p" id="S6.SS3.p1.4">In the case of <math alttext="\mathds{R}^{2}" class="ltx_Math" display="inline" id="S6.SS3.p1.1.m1.1"><semantics id="S6.SS3.p1.1.m1.1a"><msup id="S6.SS3.p1.1.m1.1.1" xref="S6.SS3.p1.1.m1.1.1.cmml"><mi id="S6.SS3.p1.1.m1.1.1.2" xref="S6.SS3.p1.1.m1.1.1.2.cmml">ℝ</mi><mn id="S6.SS3.p1.1.m1.1.1.3" xref="S6.SS3.p1.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.p1.1.m1.1b"><apply id="S6.SS3.p1.1.m1.1.1.cmml" xref="S6.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS3.p1.1.m1.1.1.1.cmml" xref="S6.SS3.p1.1.m1.1.1">superscript</csymbol><ci id="S6.SS3.p1.1.m1.1.1.2.cmml" xref="S6.SS3.p1.1.m1.1.1.2">ℝ</ci><cn id="S6.SS3.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.SS3.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p1.1.m1.1c">\mathds{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p1.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> studied in this paper, the pieces of a CPA function form a subdivision of the plane consisting of vertices, edges, and regions. More generally, we can think of these as 0-, 1-, and 2-dimensional <em class="ltx_emph ltx_font_italic" id="S6.SS3.p1.4.1">faces</em>, respectively. In higher dimensions <math alttext="d>2" class="ltx_Math" display="inline" id="S6.SS3.p1.2.m2.1"><semantics id="S6.SS3.p1.2.m2.1a"><mrow id="S6.SS3.p1.2.m2.1.1" xref="S6.SS3.p1.2.m2.1.1.cmml"><mi id="S6.SS3.p1.2.m2.1.1.2" xref="S6.SS3.p1.2.m2.1.1.2.cmml">d</mi><mo id="S6.SS3.p1.2.m2.1.1.1" xref="S6.SS3.p1.2.m2.1.1.1.cmml">></mo><mn id="S6.SS3.p1.2.m2.1.1.3" xref="S6.SS3.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p1.2.m2.1b"><apply id="S6.SS3.p1.2.m2.1.1.cmml" xref="S6.SS3.p1.2.m2.1.1"><gt id="S6.SS3.p1.2.m2.1.1.1.cmml" xref="S6.SS3.p1.2.m2.1.1.1"></gt><ci id="S6.SS3.p1.2.m2.1.1.2.cmml" xref="S6.SS3.p1.2.m2.1.1.2">𝑑</ci><cn id="S6.SS3.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.SS3.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p1.2.m2.1c">d>2</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p1.2.m2.1d">italic_d > 2</annotation></semantics></math>, the pieces of a CPA function <math alttext="f" class="ltx_Math" display="inline" id="S6.SS3.p1.3.m3.1"><semantics id="S6.SS3.p1.3.m3.1a"><mi id="S6.SS3.p1.3.m3.1.1" xref="S6.SS3.p1.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p1.3.m3.1b"><ci id="S6.SS3.p1.3.m3.1.1.cmml" xref="S6.SS3.p1.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p1.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p1.3.m3.1d">italic_f</annotation></semantics></math> similarly induce a subdivision of the input space, consisting of faces of all dimensions <math alttext="0,...,d" class="ltx_Math" display="inline" id="S6.SS3.p1.4.m4.3"><semantics id="S6.SS3.p1.4.m4.3a"><mrow id="S6.SS3.p1.4.m4.3.4.2" xref="S6.SS3.p1.4.m4.3.4.1.cmml"><mn id="S6.SS3.p1.4.m4.1.1" xref="S6.SS3.p1.4.m4.1.1.cmml">0</mn><mo id="S6.SS3.p1.4.m4.3.4.2.1" xref="S6.SS3.p1.4.m4.3.4.1.cmml">,</mo><mi id="S6.SS3.p1.4.m4.2.2" mathvariant="normal" xref="S6.SS3.p1.4.m4.2.2.cmml">…</mi><mo id="S6.SS3.p1.4.m4.3.4.2.2" xref="S6.SS3.p1.4.m4.3.4.1.cmml">,</mo><mi id="S6.SS3.p1.4.m4.3.3" xref="S6.SS3.p1.4.m4.3.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p1.4.m4.3b"><list id="S6.SS3.p1.4.m4.3.4.1.cmml" xref="S6.SS3.p1.4.m4.3.4.2"><cn id="S6.SS3.p1.4.m4.1.1.cmml" type="integer" xref="S6.SS3.p1.4.m4.1.1">0</cn><ci id="S6.SS3.p1.4.m4.2.2.cmml" xref="S6.SS3.p1.4.m4.2.2">…</ci><ci id="S6.SS3.p1.4.m4.3.3.cmml" xref="S6.SS3.p1.4.m4.3.3">𝑑</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p1.4.m4.3c">0,...,d</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p1.4.m4.3d">0 , … , italic_d</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.p2"> <p class="ltx_p" id="S6.SS3.p2.9">Loosely speaking, analogously to the functions <math alttext="f^{v}" class="ltx_Math" display="inline" id="S6.SS3.p2.1.m1.1"><semantics id="S6.SS3.p2.1.m1.1a"><msup id="S6.SS3.p2.1.m1.1.1" xref="S6.SS3.p2.1.m1.1.1.cmml"><mi id="S6.SS3.p2.1.m1.1.1.2" xref="S6.SS3.p2.1.m1.1.1.2.cmml">f</mi><mi id="S6.SS3.p2.1.m1.1.1.3" xref="S6.SS3.p2.1.m1.1.1.3.cmml">v</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.1.m1.1b"><apply id="S6.SS3.p2.1.m1.1.1.cmml" xref="S6.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS3.p2.1.m1.1.1.1.cmml" xref="S6.SS3.p2.1.m1.1.1">superscript</csymbol><ci id="S6.SS3.p2.1.m1.1.1.2.cmml" xref="S6.SS3.p2.1.m1.1.1.2">𝑓</ci><ci id="S6.SS3.p2.1.m1.1.1.3.cmml" xref="S6.SS3.p2.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.1.m1.1c">f^{v}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.1.m1.1d">italic_f start_POSTSUPERSCRIPT italic_v end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f^{e}" class="ltx_Math" display="inline" id="S6.SS3.p2.2.m2.1"><semantics id="S6.SS3.p2.2.m2.1a"><msup id="S6.SS3.p2.2.m2.1.1" xref="S6.SS3.p2.2.m2.1.1.cmml"><mi id="S6.SS3.p2.2.m2.1.1.2" xref="S6.SS3.p2.2.m2.1.1.2.cmml">f</mi><mi id="S6.SS3.p2.2.m2.1.1.3" xref="S6.SS3.p2.2.m2.1.1.3.cmml">e</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.2.m2.1b"><apply id="S6.SS3.p2.2.m2.1.1.cmml" xref="S6.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS3.p2.2.m2.1.1.1.cmml" xref="S6.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S6.SS3.p2.2.m2.1.1.2.cmml" xref="S6.SS3.p2.2.m2.1.1.2">𝑓</ci><ci id="S6.SS3.p2.2.m2.1.1.3.cmml" xref="S6.SS3.p2.2.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.2.m2.1c">f^{e}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.2.m2.1d">italic_f start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT</annotation></semantics></math> in <a class="ltx_ref ltx_refmacro_autoref" href="https://arxiv.org/html/2503.13001v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Decomposition of CPA Functions ‣ Linear-Size Neural Network Representation of Piecewise Affine Functions in ℝ²"><span class="ltx_text ltx_ref_tag">3.1</span></a>, one can introduce a face-function for each face <math alttext="F" class="ltx_Math" display="inline" id="S6.SS3.p2.3.m3.1"><semantics id="S6.SS3.p2.3.m3.1a"><mi id="S6.SS3.p2.3.m3.1.1" xref="S6.SS3.p2.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.3.m3.1b"><ci id="S6.SS3.p2.3.m3.1.1.cmml" xref="S6.SS3.p2.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.3.m3.1d">italic_F</annotation></semantics></math> of the subdivision as a CPA function <math alttext="f^{F}" class="ltx_Math" display="inline" id="S6.SS3.p2.4.m4.1"><semantics id="S6.SS3.p2.4.m4.1a"><msup id="S6.SS3.p2.4.m4.1.1" xref="S6.SS3.p2.4.m4.1.1.cmml"><mi id="S6.SS3.p2.4.m4.1.1.2" xref="S6.SS3.p2.4.m4.1.1.2.cmml">f</mi><mi id="S6.SS3.p2.4.m4.1.1.3" xref="S6.SS3.p2.4.m4.1.1.3.cmml">F</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.4.m4.1b"><apply id="S6.SS3.p2.4.m4.1.1.cmml" xref="S6.SS3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS3.p2.4.m4.1.1.1.cmml" xref="S6.SS3.p2.4.m4.1.1">superscript</csymbol><ci id="S6.SS3.p2.4.m4.1.1.2.cmml" xref="S6.SS3.p2.4.m4.1.1.2">𝑓</ci><ci id="S6.SS3.p2.4.m4.1.1.3.cmml" xref="S6.SS3.p2.4.m4.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.4.m4.1c">f^{F}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.4.m4.1d">italic_f start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT</annotation></semantics></math> whose affine components agree on <math alttext="F" class="ltx_Math" display="inline" id="S6.SS3.p2.5.m5.1"><semantics id="S6.SS3.p2.5.m5.1a"><mi id="S6.SS3.p2.5.m5.1.1" xref="S6.SS3.p2.5.m5.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.5.m5.1b"><ci id="S6.SS3.p2.5.m5.1.1.cmml" xref="S6.SS3.p2.5.m5.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.5.m5.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.5.m5.1d">italic_F</annotation></semantics></math> and which satisfies <math alttext="f^{F}=f" class="ltx_Math" display="inline" id="S6.SS3.p2.6.m6.1"><semantics id="S6.SS3.p2.6.m6.1a"><mrow id="S6.SS3.p2.6.m6.1.1" xref="S6.SS3.p2.6.m6.1.1.cmml"><msup id="S6.SS3.p2.6.m6.1.1.2" xref="S6.SS3.p2.6.m6.1.1.2.cmml"><mi id="S6.SS3.p2.6.m6.1.1.2.2" xref="S6.SS3.p2.6.m6.1.1.2.2.cmml">f</mi><mi id="S6.SS3.p2.6.m6.1.1.2.3" xref="S6.SS3.p2.6.m6.1.1.2.3.cmml">F</mi></msup><mo id="S6.SS3.p2.6.m6.1.1.1" xref="S6.SS3.p2.6.m6.1.1.1.cmml">=</mo><mi id="S6.SS3.p2.6.m6.1.1.3" xref="S6.SS3.p2.6.m6.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.6.m6.1b"><apply id="S6.SS3.p2.6.m6.1.1.cmml" xref="S6.SS3.p2.6.m6.1.1"><eq id="S6.SS3.p2.6.m6.1.1.1.cmml" xref="S6.SS3.p2.6.m6.1.1.1"></eq><apply id="S6.SS3.p2.6.m6.1.1.2.cmml" xref="S6.SS3.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.p2.6.m6.1.1.2.1.cmml" xref="S6.SS3.p2.6.m6.1.1.2">superscript</csymbol><ci id="S6.SS3.p2.6.m6.1.1.2.2.cmml" xref="S6.SS3.p2.6.m6.1.1.2.2">𝑓</ci><ci id="S6.SS3.p2.6.m6.1.1.2.3.cmml" xref="S6.SS3.p2.6.m6.1.1.2.3">𝐹</ci></apply><ci id="S6.SS3.p2.6.m6.1.1.3.cmml" xref="S6.SS3.p2.6.m6.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.6.m6.1c">f^{F}=f</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.6.m6.1d">italic_f start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT = italic_f</annotation></semantics></math> locally around <math alttext="F" class="ltx_Math" display="inline" id="S6.SS3.p2.7.m7.1"><semantics id="S6.SS3.p2.7.m7.1a"><mi id="S6.SS3.p2.7.m7.1.1" xref="S6.SS3.p2.7.m7.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.7.m7.1b"><ci id="S6.SS3.p2.7.m7.1.1.cmml" xref="S6.SS3.p2.7.m7.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.7.m7.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.7.m7.1d">italic_F</annotation></semantics></math>. A higher-dimensional analogue of the presented approach then involves decomposing <math alttext="f" class="ltx_Math" display="inline" id="S6.SS3.p2.8.m8.1"><semantics id="S6.SS3.p2.8.m8.1a"><mi id="S6.SS3.p2.8.m8.1.1" xref="S6.SS3.p2.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.8.m8.1b"><ci id="S6.SS3.p2.8.m8.1.1.cmml" xref="S6.SS3.p2.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.8.m8.1d">italic_f</annotation></semantics></math> into a linear combination of all face-functions of dimension less than <math alttext="d" class="ltx_Math" display="inline" id="S6.SS3.p2.9.m9.1"><semantics id="S6.SS3.p2.9.m9.1a"><mi id="S6.SS3.p2.9.m9.1.1" xref="S6.SS3.p2.9.m9.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p2.9.m9.1b"><ci id="S6.SS3.p2.9.m9.1.1.cmml" xref="S6.SS3.p2.9.m9.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p2.9.m9.1c">d</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p2.9.m9.1d">italic_d</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.p3"> <span class="ltx_ERROR undefined" id="S6.SS3.p3.2">\citet</span> <p class="ltx_p" id="S6.SS3.p3.1">[Proposition 18 together with Proposition 6]Tran2024MinimalTRF proved that, up to an affine function, such a decomposition exists if <math alttext="f" class="ltx_Math" display="inline" id="S6.SS3.p3.1.m1.1"><semantics id="S6.SS3.p3.1.m1.1a"><mi id="S6.SS3.p3.1.m1.1.1" xref="S6.SS3.p3.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p3.1.m1.1b"><ci id="S6.SS3.p3.1.m1.1.1.cmml" xref="S6.SS3.p3.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p3.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p3.1.m1.1d">italic_f</annotation></semantics></math> is convex, using the formalism of tropical hypersurfaces. By refining the subdivision to consist of convex faces, this result can be extended to the non-convex case.</p> </div> <div class="ltx_para" id="S6.SS3.p4"> <p class="ltx_p" id="S6.SS3.p4.6">To construct a shallow neural network representation, the next step is to represent each face-function as a shallow network and then stack them into a single network implementing the linear combination. For a fixed dimension <math alttext="d" class="ltx_Math" display="inline" id="S6.SS3.p4.1.m1.1"><semantics id="S6.SS3.p4.1.m1.1a"><mi id="S6.SS3.p4.1.m1.1.1" xref="S6.SS3.p4.1.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.1.m1.1b"><ci id="S6.SS3.p4.1.m1.1.1.cmml" xref="S6.SS3.p4.1.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.1.m1.1c">d</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.1.m1.1d">italic_d</annotation></semantics></math>, there exist subdivisions with <math alttext="p" class="ltx_Math" display="inline" id="S6.SS3.p4.2.m2.1"><semantics id="S6.SS3.p4.2.m2.1a"><mi id="S6.SS3.p4.2.m2.1.1" xref="S6.SS3.p4.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.2.m2.1b"><ci id="S6.SS3.p4.2.m2.1.1.cmml" xref="S6.SS3.p4.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.2.m2.1d">italic_p</annotation></semantics></math> regions such that the number of vertices grows as <math alttext="\Theta(p^{\lfloor\frac{d+1}{2}\rfloor})" class="ltx_Math" display="inline" id="S6.SS3.p4.3.m3.2"><semantics id="S6.SS3.p4.3.m3.2a"><mrow id="S6.SS3.p4.3.m3.2.2" xref="S6.SS3.p4.3.m3.2.2.cmml"><mi id="S6.SS3.p4.3.m3.2.2.3" mathvariant="normal" xref="S6.SS3.p4.3.m3.2.2.3.cmml">Θ</mi><mo id="S6.SS3.p4.3.m3.2.2.2" xref="S6.SS3.p4.3.m3.2.2.2.cmml">⁢</mo><mrow id="S6.SS3.p4.3.m3.2.2.1.1" xref="S6.SS3.p4.3.m3.2.2.1.1.1.cmml"><mo id="S6.SS3.p4.3.m3.2.2.1.1.2" stretchy="false" xref="S6.SS3.p4.3.m3.2.2.1.1.1.cmml">(</mo><msup id="S6.SS3.p4.3.m3.2.2.1.1.1" xref="S6.SS3.p4.3.m3.2.2.1.1.1.cmml"><mi id="S6.SS3.p4.3.m3.2.2.1.1.1.2" xref="S6.SS3.p4.3.m3.2.2.1.1.1.2.cmml">p</mi><mrow id="S6.SS3.p4.3.m3.1.1.1.3" xref="S6.SS3.p4.3.m3.1.1.1.2.cmml"><mo id="S6.SS3.p4.3.m3.1.1.1.3.1" stretchy="false" xref="S6.SS3.p4.3.m3.1.1.1.2.1.cmml">⌊</mo><mfrac id="S6.SS3.p4.3.m3.1.1.1.1" xref="S6.SS3.p4.3.m3.1.1.1.1.cmml"><mrow id="S6.SS3.p4.3.m3.1.1.1.1.2" xref="S6.SS3.p4.3.m3.1.1.1.1.2.cmml"><mi id="S6.SS3.p4.3.m3.1.1.1.1.2.2" xref="S6.SS3.p4.3.m3.1.1.1.1.2.2.cmml">d</mi><mo id="S6.SS3.p4.3.m3.1.1.1.1.2.1" xref="S6.SS3.p4.3.m3.1.1.1.1.2.1.cmml">+</mo><mn id="S6.SS3.p4.3.m3.1.1.1.1.2.3" xref="S6.SS3.p4.3.m3.1.1.1.1.2.3.cmml">1</mn></mrow><mn id="S6.SS3.p4.3.m3.1.1.1.1.3" xref="S6.SS3.p4.3.m3.1.1.1.1.3.cmml">2</mn></mfrac><mo id="S6.SS3.p4.3.m3.1.1.1.3.2" stretchy="false" xref="S6.SS3.p4.3.m3.1.1.1.2.1.cmml">⌋</mo></mrow></msup><mo id="S6.SS3.p4.3.m3.2.2.1.1.3" stretchy="false" xref="S6.SS3.p4.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.3.m3.2b"><apply id="S6.SS3.p4.3.m3.2.2.cmml" xref="S6.SS3.p4.3.m3.2.2"><times id="S6.SS3.p4.3.m3.2.2.2.cmml" xref="S6.SS3.p4.3.m3.2.2.2"></times><ci id="S6.SS3.p4.3.m3.2.2.3.cmml" xref="S6.SS3.p4.3.m3.2.2.3">Θ</ci><apply id="S6.SS3.p4.3.m3.2.2.1.1.1.cmml" xref="S6.SS3.p4.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS3.p4.3.m3.2.2.1.1.1.1.cmml" xref="S6.SS3.p4.3.m3.2.2.1.1">superscript</csymbol><ci id="S6.SS3.p4.3.m3.2.2.1.1.1.2.cmml" xref="S6.SS3.p4.3.m3.2.2.1.1.1.2">𝑝</ci><apply id="S6.SS3.p4.3.m3.1.1.1.2.cmml" xref="S6.SS3.p4.3.m3.1.1.1.3"><floor id="S6.SS3.p4.3.m3.1.1.1.2.1.cmml" xref="S6.SS3.p4.3.m3.1.1.1.3.1"></floor><apply id="S6.SS3.p4.3.m3.1.1.1.1.cmml" xref="S6.SS3.p4.3.m3.1.1.1.1"><divide id="S6.SS3.p4.3.m3.1.1.1.1.1.cmml" xref="S6.SS3.p4.3.m3.1.1.1.1"></divide><apply id="S6.SS3.p4.3.m3.1.1.1.1.2.cmml" xref="S6.SS3.p4.3.m3.1.1.1.1.2"><plus id="S6.SS3.p4.3.m3.1.1.1.1.2.1.cmml" xref="S6.SS3.p4.3.m3.1.1.1.1.2.1"></plus><ci id="S6.SS3.p4.3.m3.1.1.1.1.2.2.cmml" xref="S6.SS3.p4.3.m3.1.1.1.1.2.2">𝑑</ci><cn id="S6.SS3.p4.3.m3.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.p4.3.m3.1.1.1.1.2.3">1</cn></apply><cn id="S6.SS3.p4.3.m3.1.1.1.1.3.cmml" type="integer" xref="S6.SS3.p4.3.m3.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.3.m3.2c">\Theta(p^{\lfloor\frac{d+1}{2}\rfloor})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.3.m3.2d">roman_Θ ( italic_p start_POSTSUPERSCRIPT ⌊ divide start_ARG italic_d + 1 end_ARG start_ARG 2 end_ARG ⌋ end_POSTSUPERSCRIPT )</annotation></semantics></math>. These subdivisions arise as Schlegel diagrams of the polars of cyclic (<math alttext="d+1" class="ltx_Math" display="inline" id="S6.SS3.p4.4.m4.1"><semantics id="S6.SS3.p4.4.m4.1a"><mrow id="S6.SS3.p4.4.m4.1.1" xref="S6.SS3.p4.4.m4.1.1.cmml"><mi id="S6.SS3.p4.4.m4.1.1.2" xref="S6.SS3.p4.4.m4.1.1.2.cmml">d</mi><mo id="S6.SS3.p4.4.m4.1.1.1" xref="S6.SS3.p4.4.m4.1.1.1.cmml">+</mo><mn id="S6.SS3.p4.4.m4.1.1.3" xref="S6.SS3.p4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.4.m4.1b"><apply id="S6.SS3.p4.4.m4.1.1.cmml" xref="S6.SS3.p4.4.m4.1.1"><plus id="S6.SS3.p4.4.m4.1.1.1.cmml" xref="S6.SS3.p4.4.m4.1.1.1"></plus><ci id="S6.SS3.p4.4.m4.1.1.2.cmml" xref="S6.SS3.p4.4.m4.1.1.2">𝑑</ci><cn id="S6.SS3.p4.4.m4.1.1.3.cmml" type="integer" xref="S6.SS3.p4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.4.m4.1c">d+1</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.4.m4.1d">italic_d + 1</annotation></semantics></math>)-polytopes. Schlegel diagrams are regular, meaning that there exist convex CPA functions whose pieces realise these diagrams, implying the existence of CPA functions with <math alttext="p" class="ltx_Math" display="inline" id="S6.SS3.p4.5.m5.1"><semantics id="S6.SS3.p4.5.m5.1a"><mi id="S6.SS3.p4.5.m5.1.1" xref="S6.SS3.p4.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.5.m5.1b"><ci id="S6.SS3.p4.5.m5.1.1.cmml" xref="S6.SS3.p4.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.5.m5.1d">italic_p</annotation></semantics></math> pieces and <math alttext="\Theta(p^{\lfloor\frac{d+1}{2}\rfloor})" class="ltx_Math" display="inline" id="S6.SS3.p4.6.m6.2"><semantics id="S6.SS3.p4.6.m6.2a"><mrow id="S6.SS3.p4.6.m6.2.2" xref="S6.SS3.p4.6.m6.2.2.cmml"><mi id="S6.SS3.p4.6.m6.2.2.3" mathvariant="normal" xref="S6.SS3.p4.6.m6.2.2.3.cmml">Θ</mi><mo id="S6.SS3.p4.6.m6.2.2.2" xref="S6.SS3.p4.6.m6.2.2.2.cmml">⁢</mo><mrow id="S6.SS3.p4.6.m6.2.2.1.1" xref="S6.SS3.p4.6.m6.2.2.1.1.1.cmml"><mo id="S6.SS3.p4.6.m6.2.2.1.1.2" stretchy="false" xref="S6.SS3.p4.6.m6.2.2.1.1.1.cmml">(</mo><msup id="S6.SS3.p4.6.m6.2.2.1.1.1" xref="S6.SS3.p4.6.m6.2.2.1.1.1.cmml"><mi id="S6.SS3.p4.6.m6.2.2.1.1.1.2" xref="S6.SS3.p4.6.m6.2.2.1.1.1.2.cmml">p</mi><mrow id="S6.SS3.p4.6.m6.1.1.1.3" xref="S6.SS3.p4.6.m6.1.1.1.2.cmml"><mo id="S6.SS3.p4.6.m6.1.1.1.3.1" stretchy="false" xref="S6.SS3.p4.6.m6.1.1.1.2.1.cmml">⌊</mo><mfrac id="S6.SS3.p4.6.m6.1.1.1.1" xref="S6.SS3.p4.6.m6.1.1.1.1.cmml"><mrow id="S6.SS3.p4.6.m6.1.1.1.1.2" xref="S6.SS3.p4.6.m6.1.1.1.1.2.cmml"><mi id="S6.SS3.p4.6.m6.1.1.1.1.2.2" xref="S6.SS3.p4.6.m6.1.1.1.1.2.2.cmml">d</mi><mo id="S6.SS3.p4.6.m6.1.1.1.1.2.1" xref="S6.SS3.p4.6.m6.1.1.1.1.2.1.cmml">+</mo><mn id="S6.SS3.p4.6.m6.1.1.1.1.2.3" xref="S6.SS3.p4.6.m6.1.1.1.1.2.3.cmml">1</mn></mrow><mn id="S6.SS3.p4.6.m6.1.1.1.1.3" xref="S6.SS3.p4.6.m6.1.1.1.1.3.cmml">2</mn></mfrac><mo id="S6.SS3.p4.6.m6.1.1.1.3.2" stretchy="false" xref="S6.SS3.p4.6.m6.1.1.1.2.1.cmml">⌋</mo></mrow></msup><mo id="S6.SS3.p4.6.m6.2.2.1.1.3" stretchy="false" xref="S6.SS3.p4.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p4.6.m6.2b"><apply id="S6.SS3.p4.6.m6.2.2.cmml" xref="S6.SS3.p4.6.m6.2.2"><times id="S6.SS3.p4.6.m6.2.2.2.cmml" xref="S6.SS3.p4.6.m6.2.2.2"></times><ci id="S6.SS3.p4.6.m6.2.2.3.cmml" xref="S6.SS3.p4.6.m6.2.2.3">Θ</ci><apply id="S6.SS3.p4.6.m6.2.2.1.1.1.cmml" xref="S6.SS3.p4.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS3.p4.6.m6.2.2.1.1.1.1.cmml" xref="S6.SS3.p4.6.m6.2.2.1.1">superscript</csymbol><ci id="S6.SS3.p4.6.m6.2.2.1.1.1.2.cmml" xref="S6.SS3.p4.6.m6.2.2.1.1.1.2">𝑝</ci><apply id="S6.SS3.p4.6.m6.1.1.1.2.cmml" xref="S6.SS3.p4.6.m6.1.1.1.3"><floor id="S6.SS3.p4.6.m6.1.1.1.2.1.cmml" xref="S6.SS3.p4.6.m6.1.1.1.3.1"></floor><apply id="S6.SS3.p4.6.m6.1.1.1.1.cmml" xref="S6.SS3.p4.6.m6.1.1.1.1"><divide id="S6.SS3.p4.6.m6.1.1.1.1.1.cmml" xref="S6.SS3.p4.6.m6.1.1.1.1"></divide><apply id="S6.SS3.p4.6.m6.1.1.1.1.2.cmml" xref="S6.SS3.p4.6.m6.1.1.1.1.2"><plus id="S6.SS3.p4.6.m6.1.1.1.1.2.1.cmml" xref="S6.SS3.p4.6.m6.1.1.1.1.2.1"></plus><ci id="S6.SS3.p4.6.m6.1.1.1.1.2.2.cmml" xref="S6.SS3.p4.6.m6.1.1.1.1.2.2">𝑑</ci><cn id="S6.SS3.p4.6.m6.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.p4.6.m6.1.1.1.1.2.3">1</cn></apply><cn id="S6.SS3.p4.6.m6.1.1.1.1.3.cmml" type="integer" xref="S6.SS3.p4.6.m6.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p4.6.m6.2c">\Theta(p^{\lfloor\frac{d+1}{2}\rfloor})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p4.6.m6.2d">roman_Θ ( italic_p start_POSTSUPERSCRIPT ⌊ divide start_ARG italic_d + 1 end_ARG start_ARG 2 end_ARG ⌋ end_POSTSUPERSCRIPT )</annotation></semantics></math> vertices.</p> </div> <div class="ltx_para" id="S6.SS3.p5"> <p class="ltx_p" id="S6.SS3.p5.6">It seems plausible that the total width required for the subnetworks representing vertex-functions cannot asymptotically be smaller than the number of vertices. As a result, the best possible upper bound on network width that a higher-dimensional analogue of the presented approach could achieve is <math alttext="O(p^{\lfloor\frac{d+1}{2}\rfloor})" class="ltx_Math" display="inline" id="S6.SS3.p5.1.m1.2"><semantics id="S6.SS3.p5.1.m1.2a"><mrow id="S6.SS3.p5.1.m1.2.2" xref="S6.SS3.p5.1.m1.2.2.cmml"><mi id="S6.SS3.p5.1.m1.2.2.3" xref="S6.SS3.p5.1.m1.2.2.3.cmml">O</mi><mo id="S6.SS3.p5.1.m1.2.2.2" xref="S6.SS3.p5.1.m1.2.2.2.cmml">⁢</mo><mrow id="S6.SS3.p5.1.m1.2.2.1.1" xref="S6.SS3.p5.1.m1.2.2.1.1.1.cmml"><mo id="S6.SS3.p5.1.m1.2.2.1.1.2" stretchy="false" xref="S6.SS3.p5.1.m1.2.2.1.1.1.cmml">(</mo><msup id="S6.SS3.p5.1.m1.2.2.1.1.1" xref="S6.SS3.p5.1.m1.2.2.1.1.1.cmml"><mi id="S6.SS3.p5.1.m1.2.2.1.1.1.2" xref="S6.SS3.p5.1.m1.2.2.1.1.1.2.cmml">p</mi><mrow id="S6.SS3.p5.1.m1.1.1.1.3" xref="S6.SS3.p5.1.m1.1.1.1.2.cmml"><mo id="S6.SS3.p5.1.m1.1.1.1.3.1" stretchy="false" xref="S6.SS3.p5.1.m1.1.1.1.2.1.cmml">⌊</mo><mfrac id="S6.SS3.p5.1.m1.1.1.1.1" xref="S6.SS3.p5.1.m1.1.1.1.1.cmml"><mrow id="S6.SS3.p5.1.m1.1.1.1.1.2" xref="S6.SS3.p5.1.m1.1.1.1.1.2.cmml"><mi id="S6.SS3.p5.1.m1.1.1.1.1.2.2" xref="S6.SS3.p5.1.m1.1.1.1.1.2.2.cmml">d</mi><mo id="S6.SS3.p5.1.m1.1.1.1.1.2.1" xref="S6.SS3.p5.1.m1.1.1.1.1.2.1.cmml">+</mo><mn id="S6.SS3.p5.1.m1.1.1.1.1.2.3" xref="S6.SS3.p5.1.m1.1.1.1.1.2.3.cmml">1</mn></mrow><mn id="S6.SS3.p5.1.m1.1.1.1.1.3" xref="S6.SS3.p5.1.m1.1.1.1.1.3.cmml">2</mn></mfrac><mo id="S6.SS3.p5.1.m1.1.1.1.3.2" stretchy="false" xref="S6.SS3.p5.1.m1.1.1.1.2.1.cmml">⌋</mo></mrow></msup><mo id="S6.SS3.p5.1.m1.2.2.1.1.3" stretchy="false" xref="S6.SS3.p5.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.1.m1.2b"><apply id="S6.SS3.p5.1.m1.2.2.cmml" xref="S6.SS3.p5.1.m1.2.2"><times id="S6.SS3.p5.1.m1.2.2.2.cmml" xref="S6.SS3.p5.1.m1.2.2.2"></times><ci id="S6.SS3.p5.1.m1.2.2.3.cmml" xref="S6.SS3.p5.1.m1.2.2.3">𝑂</ci><apply id="S6.SS3.p5.1.m1.2.2.1.1.1.cmml" xref="S6.SS3.p5.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS3.p5.1.m1.2.2.1.1.1.1.cmml" xref="S6.SS3.p5.1.m1.2.2.1.1">superscript</csymbol><ci id="S6.SS3.p5.1.m1.2.2.1.1.1.2.cmml" xref="S6.SS3.p5.1.m1.2.2.1.1.1.2">𝑝</ci><apply id="S6.SS3.p5.1.m1.1.1.1.2.cmml" xref="S6.SS3.p5.1.m1.1.1.1.3"><floor id="S6.SS3.p5.1.m1.1.1.1.2.1.cmml" xref="S6.SS3.p5.1.m1.1.1.1.3.1"></floor><apply id="S6.SS3.p5.1.m1.1.1.1.1.cmml" xref="S6.SS3.p5.1.m1.1.1.1.1"><divide id="S6.SS3.p5.1.m1.1.1.1.1.1.cmml" xref="S6.SS3.p5.1.m1.1.1.1.1"></divide><apply id="S6.SS3.p5.1.m1.1.1.1.1.2.cmml" xref="S6.SS3.p5.1.m1.1.1.1.1.2"><plus id="S6.SS3.p5.1.m1.1.1.1.1.2.1.cmml" xref="S6.SS3.p5.1.m1.1.1.1.1.2.1"></plus><ci id="S6.SS3.p5.1.m1.1.1.1.1.2.2.cmml" xref="S6.SS3.p5.1.m1.1.1.1.1.2.2">𝑑</ci><cn id="S6.SS3.p5.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.p5.1.m1.1.1.1.1.2.3">1</cn></apply><cn id="S6.SS3.p5.1.m1.1.1.1.1.3.cmml" type="integer" xref="S6.SS3.p5.1.m1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.1.m1.2c">O(p^{\lfloor\frac{d+1}{2}\rfloor})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.1.m1.2d">italic_O ( italic_p start_POSTSUPERSCRIPT ⌊ divide start_ARG italic_d + 1 end_ARG start_ARG 2 end_ARG ⌋ end_POSTSUPERSCRIPT )</annotation></semantics></math>. If <math alttext="p" class="ltx_Math" display="inline" id="S6.SS3.p5.2.m2.1"><semantics id="S6.SS3.p5.2.m2.1a"><mi id="S6.SS3.p5.2.m2.1.1" xref="S6.SS3.p5.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.2.m2.1b"><ci id="S6.SS3.p5.2.m2.1.1.cmml" xref="S6.SS3.p5.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.2.m2.1d">italic_p</annotation></semantics></math> is equal to the number of affine components <math alttext="n" class="ltx_Math" display="inline" id="S6.SS3.p5.3.m3.1"><semantics id="S6.SS3.p5.3.m3.1a"><mi id="S6.SS3.p5.3.m3.1.1" xref="S6.SS3.p5.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.3.m3.1b"><ci id="S6.SS3.p5.3.m3.1.1.cmml" xref="S6.SS3.p5.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.3.m3.1d">italic_n</annotation></semantics></math>, this would represent an improvement over the <math alttext="O(n^{d+1})" class="ltx_Math" display="inline" id="S6.SS3.p5.4.m4.1"><semantics id="S6.SS3.p5.4.m4.1a"><mrow id="S6.SS3.p5.4.m4.1.1" xref="S6.SS3.p5.4.m4.1.1.cmml"><mi id="S6.SS3.p5.4.m4.1.1.3" xref="S6.SS3.p5.4.m4.1.1.3.cmml">O</mi><mo id="S6.SS3.p5.4.m4.1.1.2" xref="S6.SS3.p5.4.m4.1.1.2.cmml">⁢</mo><mrow id="S6.SS3.p5.4.m4.1.1.1.1" xref="S6.SS3.p5.4.m4.1.1.1.1.1.cmml"><mo id="S6.SS3.p5.4.m4.1.1.1.1.2" stretchy="false" xref="S6.SS3.p5.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S6.SS3.p5.4.m4.1.1.1.1.1" xref="S6.SS3.p5.4.m4.1.1.1.1.1.cmml"><mi id="S6.SS3.p5.4.m4.1.1.1.1.1.2" xref="S6.SS3.p5.4.m4.1.1.1.1.1.2.cmml">n</mi><mrow id="S6.SS3.p5.4.m4.1.1.1.1.1.3" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.cmml"><mi id="S6.SS3.p5.4.m4.1.1.1.1.1.3.2" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.2.cmml">d</mi><mo id="S6.SS3.p5.4.m4.1.1.1.1.1.3.1" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.1.cmml">+</mo><mn id="S6.SS3.p5.4.m4.1.1.1.1.1.3.3" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S6.SS3.p5.4.m4.1.1.1.1.3" stretchy="false" xref="S6.SS3.p5.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.4.m4.1b"><apply id="S6.SS3.p5.4.m4.1.1.cmml" xref="S6.SS3.p5.4.m4.1.1"><times id="S6.SS3.p5.4.m4.1.1.2.cmml" xref="S6.SS3.p5.4.m4.1.1.2"></times><ci id="S6.SS3.p5.4.m4.1.1.3.cmml" xref="S6.SS3.p5.4.m4.1.1.3">𝑂</ci><apply id="S6.SS3.p5.4.m4.1.1.1.1.1.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.p5.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1">superscript</csymbol><ci id="S6.SS3.p5.4.m4.1.1.1.1.1.2.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1.1.2">𝑛</ci><apply id="S6.SS3.p5.4.m4.1.1.1.1.1.3.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3"><plus id="S6.SS3.p5.4.m4.1.1.1.1.1.3.1.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.1"></plus><ci id="S6.SS3.p5.4.m4.1.1.1.1.1.3.2.cmml" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.2">𝑑</ci><cn id="S6.SS3.p5.4.m4.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.SS3.p5.4.m4.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.4.m4.1c">O(n^{d+1})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.4.m4.1d">italic_O ( italic_n start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> bound on network width established by <span class="ltx_ERROR undefined" id="S6.SS3.p5.6.1">\citet</span>Koutschan2023. However, <span class="ltx_ERROR undefined" id="S6.SS3.p5.6.2">\citet</span>zanotti2025pieces presents a family of CPA functions for which <math alttext="p" class="ltx_Math" display="inline" id="S6.SS3.p5.5.m5.1"><semantics id="S6.SS3.p5.5.m5.1a"><mi id="S6.SS3.p5.5.m5.1.1" xref="S6.SS3.p5.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.5.m5.1b"><ci id="S6.SS3.p5.5.m5.1.1.cmml" xref="S6.SS3.p5.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.5.m5.1d">italic_p</annotation></semantics></math> behaves roughly like <math alttext="n^{d+1}" class="ltx_Math" display="inline" id="S6.SS3.p5.6.m6.1"><semantics id="S6.SS3.p5.6.m6.1a"><msup id="S6.SS3.p5.6.m6.1.1" xref="S6.SS3.p5.6.m6.1.1.cmml"><mi id="S6.SS3.p5.6.m6.1.1.2" xref="S6.SS3.p5.6.m6.1.1.2.cmml">n</mi><mrow id="S6.SS3.p5.6.m6.1.1.3" xref="S6.SS3.p5.6.m6.1.1.3.cmml"><mi id="S6.SS3.p5.6.m6.1.1.3.2" xref="S6.SS3.p5.6.m6.1.1.3.2.cmml">d</mi><mo id="S6.SS3.p5.6.m6.1.1.3.1" xref="S6.SS3.p5.6.m6.1.1.3.1.cmml">+</mo><mn id="S6.SS3.p5.6.m6.1.1.3.3" xref="S6.SS3.p5.6.m6.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.p5.6.m6.1b"><apply id="S6.SS3.p5.6.m6.1.1.cmml" xref="S6.SS3.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS3.p5.6.m6.1.1.1.cmml" xref="S6.SS3.p5.6.m6.1.1">superscript</csymbol><ci id="S6.SS3.p5.6.m6.1.1.2.cmml" xref="S6.SS3.p5.6.m6.1.1.2">𝑛</ci><apply id="S6.SS3.p5.6.m6.1.1.3.cmml" xref="S6.SS3.p5.6.m6.1.1.3"><plus id="S6.SS3.p5.6.m6.1.1.3.1.cmml" xref="S6.SS3.p5.6.m6.1.1.3.1"></plus><ci id="S6.SS3.p5.6.m6.1.1.3.2.cmml" xref="S6.SS3.p5.6.m6.1.1.3.2">𝑑</ci><cn id="S6.SS3.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S6.SS3.p5.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p5.6.m6.1c">n^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p5.6.m6.1d">italic_n start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. In these cases, the presented approach does not lead to any improvement.</p> </div> <div class="ltx_para" id="S6.SS3.p6"> <p class="ltx_p" id="S6.SS3.p6.2">Nevertheless, the presented ideas may be useful for representing specific subclasses of higher-dimensional CPA functions where the underlying subdivisions have manageable combinatorics. In particular, a geometric approach can be advantageous when the number of faces is <math alttext="O(p)" class="ltx_Math" display="inline" id="S6.SS3.p6.1.m1.1"><semantics id="S6.SS3.p6.1.m1.1a"><mrow id="S6.SS3.p6.1.m1.1.2" xref="S6.SS3.p6.1.m1.1.2.cmml"><mi id="S6.SS3.p6.1.m1.1.2.2" xref="S6.SS3.p6.1.m1.1.2.2.cmml">O</mi><mo id="S6.SS3.p6.1.m1.1.2.1" xref="S6.SS3.p6.1.m1.1.2.1.cmml">⁢</mo><mrow id="S6.SS3.p6.1.m1.1.2.3.2" xref="S6.SS3.p6.1.m1.1.2.cmml"><mo id="S6.SS3.p6.1.m1.1.2.3.2.1" stretchy="false" xref="S6.SS3.p6.1.m1.1.2.cmml">(</mo><mi id="S6.SS3.p6.1.m1.1.1" xref="S6.SS3.p6.1.m1.1.1.cmml">p</mi><mo id="S6.SS3.p6.1.m1.1.2.3.2.2" stretchy="false" xref="S6.SS3.p6.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p6.1.m1.1b"><apply id="S6.SS3.p6.1.m1.1.2.cmml" xref="S6.SS3.p6.1.m1.1.2"><times id="S6.SS3.p6.1.m1.1.2.1.cmml" xref="S6.SS3.p6.1.m1.1.2.1"></times><ci id="S6.SS3.p6.1.m1.1.2.2.cmml" xref="S6.SS3.p6.1.m1.1.2.2">𝑂</ci><ci id="S6.SS3.p6.1.m1.1.1.cmml" xref="S6.SS3.p6.1.m1.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p6.1.m1.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p6.1.m1.1d">italic_O ( italic_p )</annotation></semantics></math> and the number of pieces incident to each face is bounded by a constant. In this case, each face-function requires only constant width, allowing the linear combination expressing the original function to be represented by a neural network of width <math alttext="O(p)" class="ltx_Math" display="inline" id="S6.SS3.p6.2.m2.1"><semantics id="S6.SS3.p6.2.m2.1a"><mrow id="S6.SS3.p6.2.m2.1.2" xref="S6.SS3.p6.2.m2.1.2.cmml"><mi id="S6.SS3.p6.2.m2.1.2.2" xref="S6.SS3.p6.2.m2.1.2.2.cmml">O</mi><mo id="S6.SS3.p6.2.m2.1.2.1" xref="S6.SS3.p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.SS3.p6.2.m2.1.2.3.2" xref="S6.SS3.p6.2.m2.1.2.cmml"><mo id="S6.SS3.p6.2.m2.1.2.3.2.1" stretchy="false" xref="S6.SS3.p6.2.m2.1.2.cmml">(</mo><mi id="S6.SS3.p6.2.m2.1.1" xref="S6.SS3.p6.2.m2.1.1.cmml">p</mi><mo id="S6.SS3.p6.2.m2.1.2.3.2.2" stretchy="false" xref="S6.SS3.p6.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.p6.2.m2.1b"><apply id="S6.SS3.p6.2.m2.1.2.cmml" xref="S6.SS3.p6.2.m2.1.2"><times id="S6.SS3.p6.2.m2.1.2.1.cmml" xref="S6.SS3.p6.2.m2.1.2.1"></times><ci id="S6.SS3.p6.2.m2.1.2.2.cmml" xref="S6.SS3.p6.2.m2.1.2.2">𝑂</ci><ci id="S6.SS3.p6.2.m2.1.1.cmml" xref="S6.SS3.p6.2.m2.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p6.2.m2.1c">O(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p6.2.m2.1d">italic_O ( italic_p )</annotation></semantics></math>. Similar structural assumptions have already been employed by <span class="ltx_ERROR undefined" id="S6.SS3.p6.2.1">\citet</span>He2020FEM, who consider shape regular finite element triangulations.</p> </div> </section> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">I would like to thank Henning Bruhn-Fujimoto for helpful discussions and for introducing me to the topic of this paper.</p> </div> <div class="ltx_para" id="Sx1.p2"> <span class="ltx_ERROR undefined" id="Sx1.p2.1">\printbibliography</span> <div class="ltx_pagination ltx_role_newpage"></div> </div> </section><div about="" class="ltx_rdf" content="Leo Zanotti" property="dcterms:creator"></div> <div about="" class="ltx_rdf" content="Linear-Size Neural Network Representation of Piecewise Affine Functions in $\mathds{R}^{2}$" property="dcterms:title"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Mon Mar 17 09:48:17 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>