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space</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Metric_epsilon-net.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metric_epsilon-net.svg/330px-Metric_epsilon-net.svg.png" decoding="async" width="300" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metric_epsilon-net.svg/450px-Metric_epsilon-net.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metric_epsilon-net.svg/600px-Metric_epsilon-net.svg.png 2x" data-file-width="468" data-file-height="360" /></a><figcaption>The red points form part of an <span class="texhtml mvar" style="font-style:italic;">ε</span>-net for the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, where <span class="texhtml mvar" style="font-style:italic;">ε</span> is the radius of the large yellow disks. The blue disks of half the radius are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a>, and the yellow disks together cover the whole plane, satisfying the two definitional requirements on an <span class="texhtml mvar" style="font-style:italic;">ε</span>-net.</figcaption></figure> <p>In the mathematical theory of <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>, <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-nets</b>, <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-packings</b>, <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-coverings</b>, <b>uniformly discrete sets</b>, <b>relatively dense sets</b>, and <b>Delone sets</b> (named after <a href="/wiki/Boris_Delone" class="mw-redirect" title="Boris Delone">Boris Delone</a>) are several closely related definitions of well-spaced <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> of points, and the <b>packing radius</b> and <b>covering radius</b> of these sets measure how well-spaced they are. These sets have applications in <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>, <a href="/wiki/Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a>, and the theory of <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml">(<i>M</i>, <i>d</i>)</span> is a metric space, and <span class="texhtml mvar" style="font-style:italic;">X</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">M</span>, then the <b>packing radius</b>, <span class="texhtml mvar" style="font-style:italic;">r</span>, of <span class="texhtml mvar" style="font-style:italic;">X</span> is half of the smallest distance between distinct members of <span class="texhtml mvar" style="font-style:italic;">X</span>. <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">Open balls</a> of radius <span class="texhtml mvar" style="font-style:italic;">r</span> centered at the points of <span class="texhtml mvar" style="font-style:italic;">X</span> will all be disjoint from each other. The <b>covering radius</b>, <span class="texhtml mvar" style="font-style:italic;">R</span>, of <span class="texhtml mvar" style="font-style:italic;">X</span> is the smallest distance such that every point of <span class="texhtml mvar" style="font-style:italic;">M</span> is within distance <span class="texhtml mvar" style="font-style:italic;">R</span> of at least one point in <span class="texhtml mvar" style="font-style:italic;">X</span>; that is, <span class="texhtml mvar" style="font-style:italic;">R</span> is the smallest radius such that <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">closed balls</a> of that radius centered at the points of <span class="texhtml mvar" style="font-style:italic;">X</span> have all of <span class="texhtml mvar" style="font-style:italic;">M</span> as their union. </p><p>An <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-packing</b> is a set <span class="texhtml mvar" style="font-style:italic;">X</span> of packing radius <span class="texhtml"><i>r</i> ≥ <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>ε</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> (equivalently, minimum distance <span class="texhtml">≥ <i>ε</i></span>), an <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-covering</b> is a set <span class="texhtml mvar" style="font-style:italic;">X</span> of covering radius <span class="texhtml"><i>R</i> ≤ <i>ε</i></span>, and an <b> <span class="texhtml mvar" style="font-style:italic;">ε</span>-net</b> is a set that is both an <span class="texhtml mvar" style="font-style:italic;">ε</span>-packing and an <span class="texhtml mvar" style="font-style:italic;">ε</span>-covering (<span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><i>ε</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> ≤ <i>r</i> ≤ <i>R</i> ≤ <i>ε</i></span>). </p><p>A set is <b>uniformly discrete</b> if it has a nonzero packing radius (<span class="texhtml">0 < <i>r</i></span>), and <b>relatively dense</b> if it has a finite covering radius (<span class="texhtml"><i>R</i> < ∞</span>). </p><p>A <b>Delone set</b> is a set that is both uniformly discrete and relatively dense (<span class="texhtml">0 < <i>r</i> ≤ <i>R</i> < ∞</span>). Thus every <span class="texhtml mvar" style="font-style:italic;">ε</span>-net is Delone, but not vice versa.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Construction_of_ε-nets"><span id="Construction_of_.CE.B5-nets"></span>Construction of <i>ε</i>-nets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=2" title="Edit section: Construction of ε-nets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the most restrictive of the definitions above, <span class="texhtml mvar" style="font-style:italic;">ε</span>-nets are at least as difficult to construct as <span class="texhtml mvar" style="font-style:italic;">ε</span>-packings, <span class="texhtml mvar" style="font-style:italic;">ε</span>-coverings, and Delone sets. However, whenever the points of <span class="texhtml mvar" style="font-style:italic;">M</span> have a <a href="/wiki/Well-ordering" class="mw-redirect" title="Well-ordering">well-ordering</a>, <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a> shows that it is possible to construct an <span class="texhtml mvar" style="font-style:italic;">ε</span>-net <span class="texhtml mvar" style="font-style:italic;">N</span>, by including in <span class="texhtml mvar" style="font-style:italic;">N</span> every point for which the infimum of distances to the set of earlier points in the ordering is at least <span class="texhtml mvar" style="font-style:italic;">ε</span>. For finite sets of points in a Euclidean space of bounded dimension, each point may be tested in constant time by mapping it to a grid of cells of diameter <span class="texhtml mvar" style="font-style:italic;">ε</span>, and using a <a href="/wiki/Hash_table" title="Hash table">hash table</a> to test which nearby cells already contain points of <span class="texhtml mvar" style="font-style:italic;">N</span>; thus, in this case, an <span class="texhtml mvar" style="font-style:italic;">ε</span>-net can be constructed in <a href="/wiki/Linear_time" class="mw-redirect" title="Linear time">linear time</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>For more general finite or <a href="/wiki/Compact_space" title="Compact space">compact</a> metric spaces, an alternative algorithm of <a href="/wiki/Teofilo_F._Gonzalez" title="Teofilo F. Gonzalez">Teo Gonzalez</a> based on the <a href="/wiki/Farthest-first_traversal" title="Farthest-first traversal">farthest-first traversal</a> can be used to construct a finite <span class="texhtml mvar" style="font-style:italic;">ε</span>-net. This algorithm initializes the net <span class="texhtml mvar" style="font-style:italic;">N</span> to be empty, and then repeatedly adds to <span class="texhtml mvar" style="font-style:italic;">N</span> the farthest point in <span class="texhtml mvar" style="font-style:italic;">M</span> from <span class="texhtml mvar" style="font-style:italic;">N</span>, breaking ties arbitrarily and stopping when all points of <span class="texhtml mvar" style="font-style:italic;">M</span> are within distance  <span class="texhtml mvar" style="font-style:italic;">ε</span> of <span class="texhtml mvar" style="font-style:italic;">N</span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Doubling_space" title="Doubling space">spaces of bounded doubling dimension</a>, Gonzalez' algorithm can be implemented in <span class="texhtml">O(<i>n</i> log <i>n</i>)</span> time for point sets with a polynomial ratio between their farthest and closest distances, and approximated in the same time bound for arbitrary point sets.<sup id="cite_ref-HM06_6-0" class="reference"><a href="#cite_note-HM06-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Coding_theory">Coding theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=4" title="Edit section: Coding theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hamming_bound#Covering_radius_and_packing_radius" title="Hamming bound">Hamming bound § Covering radius and packing radius</a></div> <p>In the theory of <a href="/wiki/Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting codes</a>, the metric space containing a <a href="/wiki/Block_code" title="Block code">block code</a> <span class="texhtml mvar" style="font-style:italic;">C</span> consists of strings of a fixed length, say <span class="texhtml mvar" style="font-style:italic;">n</span>, taken over an alphabet of size <span class="texhtml mvar" style="font-style:italic;">q</span> (can be thought of as <a href="/wiki/Coordinate_vector" title="Coordinate vector">vectors</a>), with the <a href="/wiki/Hamming_metric" class="mw-redirect" title="Hamming metric">Hamming metric</a>. This space is denoted by <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}_{q}^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}_{q}^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a83b9a1b3e638e931f7d2e2da9e29cc721ba88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.783ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}_{q}^{n}.}" /></span>⁠</span> The covering radius and packing radius of this metric space are related to the code's ability to correct errors. An example is provided by the <a href="/wiki/Berlekamp_switching_game" title="Berlekamp switching game">Berlekamp switching game</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Approximation_algorithms">Approximation algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=5" title="Edit section: Approximation algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFHar-PeledRaichel2013">Har-Peled & Raichel (2013)</a> describe an algorithmic paradigm that they call "net and prune" for designing <a href="/wiki/Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a> for certain types of geometric optimization problems defined on sets of points in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>. An algorithm of this type works by performing the following steps: </p> <ol><li>Choose a random point <span class="texhtml mvar" style="font-style:italic;">p</span> from the point set, find its nearest neighbor <span class="texhtml mvar" style="font-style:italic;">q</span>, and set <span class="texhtml mvar" style="font-style:italic;">ε</span> to the distance between <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span>.</li> <li>Test whether <span class="texhtml mvar" style="font-style:italic;">ε</span> is (approximately) bigger than or smaller than the optimal solution value (using a technique specific to the particular optimization problem being solved) <ul><li>If it is bigger, remove from the input the points whose closest neighbor is farther than <span class="texhtml mvar" style="font-style:italic;">ε</span></li> <li>If it is smaller, construct an <span class="texhtml mvar" style="font-style:italic;">ε</span>-net <span class="texhtml mvar" style="font-style:italic;">N</span>, and remove from the input the points that are not in <span class="texhtml mvar" style="font-style:italic;">N</span>.</li></ul></li></ol> <p>In both cases, the expected number of remaining points decreases by a constant factor, so the time is dominated by the testing step. As they show, this paradigm can be used to construct fast approximation algorithms for <a href="/wiki/K-center" class="mw-redirect" title="K-center">k-center</a> clustering, finding a pair of points with median distance, and several related problems. </p><p>A hierarchical system of nets, called a <i>net-tree</i>, may be used in <a href="/wiki/Doubling_space" title="Doubling space">spaces of bounded doubling dimension</a> to construct <a href="/wiki/Well-separated_pair_decomposition" title="Well-separated pair decomposition">well-separated pair decompositions</a>, <a href="/wiki/Geometric_spanner" title="Geometric spanner">geometric spanners</a>, and approximate <a href="/wiki/Nearest_neighbor_search" title="Nearest neighbor search">nearest neighbors</a>.<sup id="cite_ref-HM06_6-1" class="reference"><a href="#cite_note-HM06-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Crystallography">Crystallography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=6" title="Edit section: Crystallography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For points in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, a set <span class="texhtml mvar" style="font-style:italic;">X</span> is a <a href="/wiki/Meyer_set" title="Meyer set">Meyer set</a> if it is relatively dense and its <a href="/wiki/Minkowski_difference" class="mw-redirect" title="Minkowski difference">difference set</a> <span class="texhtml"><i>X</i> − <i>X</i></span> is uniformly discrete. Equivalently, <span class="texhtml mvar" style="font-style:italic;">X</span> is a Meyer set if both <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml"><i>X</i> − <i>X</i></span> are Delone sets. Meyer sets are named after <a href="/wiki/Yves_Meyer" title="Yves Meyer">Yves Meyer</a>, who introduced them (with a different but equivalent definition based on <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>) as a mathematical model for <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a>. They include the point sets of <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattices</a>, <a href="/wiki/Penrose_tiling" title="Penrose tiling">Penrose tilings</a>, and the <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sums</a> of these sets with finite sets.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Voronoi_cell" class="mw-redirect" title="Voronoi cell">Voronoi cells</a> of symmetric Delone sets form <a href="/wiki/Space-filling_polyhedron" title="Space-filling polyhedron">space-filling polyhedra</a> called <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedra</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClarkson2006" class="citation cs2">Clarkson, Kenneth L. (2006), "Building triangulations using <i>ε</i>-nets", <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing"><i>STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing</i></a>, New York: ACM, pp. <span class="nowrap">326–</span>335, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1132516.1132564">10.1145/1132516.1132564</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1595931341" title="Special:BookSources/1595931341"><bdi>1595931341</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2277158">2277158</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14132888">14132888</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Building+triangulations+using+%CE%B5-nets&rft.btitle=STOC%2706%3A+Proceedings+of+the+38th+Annual+ACM+Symposium+on+Theory+of+Computing&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E326-%3C%2Fspan%3E335&rft.pub=ACM&rft.date=2006&rft_id=info%3Adoi%2F10.1145%2F1132516.1132564&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2277158%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14132888%23id-name%3DS2CID&rft.isbn=1595931341&rft.aulast=Clarkson&rft.aufirst=Kenneth+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Some sources use " <span class="texhtml mvar" style="font-style:italic;">ε</span>-net" for what is here called an " <span class="texhtml mvar" style="font-style:italic;">ε</span>-covering"; see, e.g. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSutherland1975" class="citation cs2"><a href="/wiki/Wilson_Sutherland" title="Wilson Sutherland">Sutherland, W. A.</a> (1975), <i>Introduction to metric and topological spaces</i>, Oxford University Press, p. 110, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853161-3" title="Special:BookSources/0-19-853161-3"><bdi>0-19-853161-3</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0304.54002">0304.54002</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+metric+and+topological+spaces&rft.pages=110&rft.pub=Oxford+University+Press&rft.date=1975&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0304.54002%23id-name%3DZbl&rft.isbn=0-19-853161-3&rft.aulast=Sutherland&rft.aufirst=W.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHar-Peled2004" class="citation cs2"><a href="/wiki/Sariel_Har-Peled" title="Sariel Har-Peled">Har-Peled, S.</a> (2004), "Clustering motion", <i><a href="/wiki/Discrete_and_Computational_Geometry" class="mw-redirect" title="Discrete and Computational Geometry">Discrete and Computational Geometry</a></i>, <b>31</b> (4): <span class="nowrap">545–</span>565, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00454-004-2822-7">10.1007/s00454-004-2822-7</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2053498">2053498</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+and+Computational+Geometry&rft.atitle=Clustering+motion&rft.volume=31&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E545-%3C%2Fspan%3E565&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2Fs00454-004-2822-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2053498%23id-name%3DMR&rft.aulast=Har-Peled&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHar-PeledRaichel2013" class="citation cs2">Har-Peled, S.; Raichel, B. (2013), "Net and prune: A linear time algorithm for Euclidean distance problems", <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing"><i>STOC'13: Proceedings of the 45th Annual ACM Symposium on Theory of Computing</i></a>, pp. <span class="nowrap">605–</span>614, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1409.7425">1409.7425</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Net+and+prune%3A+A+linear+time+algorithm+for+Euclidean+distance+problems&rft.btitle=STOC%2713%3A+Proceedings+of+the+45th+Annual+ACM+Symposium+on+Theory+of+Computing&rft.pages=%3Cspan+class%3D%22nowrap%22%3E605-%3C%2Fspan%3E614&rft.date=2013&rft_id=info%3Aarxiv%2F1409.7425&rft.aulast=Har-Peled&rft.aufirst=S.&rft.au=Raichel%2C+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGonzalez1985" class="citation cs2">Gonzalez, T. F. (1985), "Clustering to minimize the maximum intercluster distance", <i><a href="/wiki/Theoretical_Computer_Science_(journal)" title="Theoretical Computer Science (journal)">Theoretical Computer Science</a></i>, <b>38</b> (<span class="nowrap">2–</span>3): <span class="nowrap">293–</span>306, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2885%2990224-5">10.1016/0304-3975(85)90224-5</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0807927">0807927</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Clustering+to+minimize+the+maximum+intercluster+distance&rft.volume=38&rft.issue=%3Cspan+class%3D%22nowrap%22%3E2%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E293-%3C%2Fspan%3E306&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2885%2990224-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D807927%23id-name%3DMR&rft.aulast=Gonzalez&rft.aufirst=T.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-HM06-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-HM06_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HM06_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHar-PeledMendel2006" class="citation cs2">Har-Peled, S.; Mendel, M. (2006), "Fast construction of nets in low-dimensional metrics, and their applications", <i><a href="/wiki/SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a></i>, <b>35</b> (5): <span class="nowrap">1148–</span>1184, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0409057">cs/0409057</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539704446281">10.1137/S0097539704446281</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2217141">2217141</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:37346335">37346335</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=Fast+construction+of+nets+in+low-dimensional+metrics%2C+and+their+applications&rft.volume=35&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1148-%3C%2Fspan%3E1184&rft.date=2006&rft_id=info%3Aarxiv%2Fcs%2F0409057&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2217141%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A37346335%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1137%2FS0097539704446281&rft.aulast=Har-Peled&rft.aufirst=S.&rft.au=Mendel%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKrauthgamerLee2004" class="citation cs2">Krauthgamer, Robert; Lee, James R. (2004), "Navigating nets: simple algorithms for proximity search", <a rel="nofollow" class="external text" href="http://dl.acm.org/citation.cfm?id=982792.982913"><i>Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '04)</i></a>, Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, pp. <span class="nowrap">798–</span>807, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-558-X" title="Special:BookSources/0-89871-558-X"><bdi>0-89871-558-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Navigating+nets%3A+simple+algorithms+for+proximity+search&rft.btitle=Proceedings+of+the+15th+Annual+ACM-SIAM+Symposium+on+Discrete+Algorithms+%28SODA+%2704%29&rft.place=Philadelphia%2C+PA%2C+USA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E798-%3C%2Fspan%3E807&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2004&rft.isbn=0-89871-558-X&rft.aulast=Krauthgamer&rft.aufirst=Robert&rft.au=Lee%2C+James+R.&rft_id=http%3A%2F%2Fdl.acm.org%2Fcitation.cfm%3Fid%3D982792.982913&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMoody1997" class="citation cs2"><a href="/wiki/Robert_Moody" title="Robert Moody">Moody, Robert V.</a> (1997), "Meyer sets and their duals", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303174632/http://www.math.ualberta.ca/~rvmoody/psFiles/moody.ps"><i>The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995)</i></a>, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489, Dordrecht: Kluwer Academic Publishers, pp. <span class="nowrap">403–</span>441, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1460032">1460032</a>, archived from <a rel="nofollow" class="external text" href="http://www.math.ualberta.ca/~rvmoody/psFiles/moody.ps">the original</a> on 2016-03-03<span class="reference-accessdate">, retrieved <span class="nowrap">2013-07-10</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Meyer+sets+and+their+duals&rft.btitle=The+Mathematics+of+Long-Range+Aperiodic+Order+%28Waterloo%2C+ON%2C+1995%29&rft.place=Dordrecht&rft.series=NATO+Advanced+Science+Institutes+Series+C%3A+Mathematical+and+Physical+Sciences&rft.pages=%3Cspan+class%3D%22nowrap%22%3E403-%3C%2Fspan%3E441&rft.pub=Kluwer+Academic+Publishers&rft.date=1997&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1460032%23id-name%3DMR&rft.aulast=Moody&rft.aufirst=Robert+V.&rft_id=http%3A%2F%2Fwww.math.ualberta.ca%2F~rvmoody%2FpsFiles%2Fmoody.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrünbaumShephard1980" class="citation cs2"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a>; <a href="/wiki/Geoffrey_Colin_Shephard" title="Geoffrey Colin Shephard">Shephard, G. C.</a> (1980), "Tilings with congruent tiles", <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, New Series, <b>3</b> (3): <span class="nowrap">951–</span>973, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14827-2">10.1090/S0273-0979-1980-14827-2</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0585178">0585178</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Tilings+with+congruent+tiles&rft.volume=3&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E951-%3C%2Fspan%3E973&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1980-14827-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D585178%23id-name%3DMR&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rft.au=Shephard%2C+G.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADelone+set" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Delone_set&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304095756/http://tilings.math.uni-bielefeld.de/glossary/delone_set">Delone set</a> – Tilings Encyclopedia</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output 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href="/wiki/Metric_space" title="Metric space">Metric spaces</a> (<a href="/wiki/Category:Metric_spaces" title="Category:Metric spaces">Category</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Metric_space" title="Metric space">Metric space</a></li> <li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Complete_metric_space" title="Complete metric space">Completeness</a></li> <li><a href="/wiki/Equivalence_of_metrics" title="Equivalence of metrics">Equivalent metrics</a></li> <li><a href="/wiki/Metrizable_space" title="Metrizable space">Metrizable space</a></li> <li><a href="/wiki/Triangle_inequality" title="Triangle inequality">Triangle inequality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></li> <li><a href="/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point</a></li> <li><a href="/wiki/Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a></li> <li><a href="/wiki/Lebesgue%27s_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a></li> <li><a href="/wiki/Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorems</a>: <ul><li><a href="/wiki/Bing_metrization_theorem" title="Bing metrization theorem">Bing</a></li> <li><a href="/wiki/Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov</a></li> <li><a href="/wiki/Urysohn%27s_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contraction_mapping" title="Contraction mapping">Contraction</a> <ul><li><a href="/wiki/Metric_map" title="Metric map">Metric map</a></li></ul></li> <li><a href="/wiki/Dilation_(metric_space)" title="Dilation (metric space)">Dilation</a></li> <li><a href="/wiki/Equicontinuity" title="Equicontinuity">Equicontinuity</a></li> <li>(<a href="/wiki/Quasi-isometry" title="Quasi-isometry">Quasi-</a>) <a href="/wiki/Isometry" title="Isometry">Isometry</a></li> <li><a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></li> <li><a href="/wiki/Metric_derivative" title="Metric derivative">Metric derivative</a></li> <li><a href="/wiki/Metric_outer_measure" title="Metric outer measure">Metric outer measure</a></li> <li><a href="/wiki/Metric_projection" title="Metric projection">Metric projection</a></li> <li><a href="/wiki/Motion_(geometry)" title="Motion (geometry)">Motion</a></li> <li><a href="/wiki/Quasisymmetric_map" title="Quasisymmetric map">Quasisymmetric</a></li> <li><a href="/wiki/Stretch_factor" title="Stretch factor">Stretch factor</a></li> <li><a href="/wiki/Uniform_continuity" title="Uniform continuity">Uniform continuity</a> <ul><li><a href="/wiki/Uniform_isomorphism" title="Uniform isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />metric spaces</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complete_metric_space" title="Complete metric space">Complete</a></li> <li><a href="/wiki/Convex_metric_space" title="Convex metric space">Convex</a></li> <li><a href="/wiki/Doubling_space" title="Doubling space">Doubling</a></li> <li><a href="/wiki/Hyperbolic_metric_space" title="Hyperbolic metric space">Hyperbolic</a></li> <li><a href="/wiki/Injective_metric_space" title="Injective metric space">Injective</a></li> <li><a href="/wiki/Length_metric_space" class="mw-redirect" title="Length metric space">Length metric space</a></li> <li><a href="/wiki/Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">Metric space aimed at its subspace</a></li> <li><a href="/wiki/Polish_space" title="Polish space">Polish</a></li> <li><a href="/wiki/Totally_bounded_space" title="Totally bounded space">Totally bounded</a></li> <li><a href="/wiki/Tree-graded_space" title="Tree-graded space">Tree-graded</a></li> <li><a href="/wiki/Ultrametric_space" title="Ultrametric space">Ultrametric space</a></li> <li><a href="/wiki/Uniformly_disconnected_space" title="Uniformly disconnected space">Uniformly disconnected</a></li> <li><a href="/wiki/Urysohn_universal_space" title="Urysohn universal space">Urysohn universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">Balls</a></li> <li><a href="/wiki/Borel_set" title="Borel set">Borel</a></li> <li><a href="/wiki/Bounded_set" title="Bounded set">Bounded</a></li> <li><a class="mw-selflink selflink">Delone</a></li> <li><a href="/wiki/Diameter_of_a_set" title="Diameter of a set">Diameter</a></li> <li><a href="/wiki/Distance_set" title="Distance set">Distance set</a></li> <li><a href="/wiki/Gromov_product" title="Gromov product">Gromov product</a></li> <li><a href="/wiki/Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff convergence</a></li> <li><a href="/wiki/Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a></li> <li><a href="/wiki/Kuratowski_convergence" title="Kuratowski convergence">Kuratowski convergence</a></li> <li><a href="/wiki/Meyer_set" title="Meyer set">Meyer</a></li> <li><a href="/wiki/Packing_dimension" title="Packing dimension">Packing dimension</a></li> <li><a href="/wiki/Porous_set" title="Porous set">Porous</a></li> <li><a href="/wiki/Positively_separated_sets" title="Positively separated sets">Positively separated sets</a></li> <li><a href="/wiki/Tight_span" title="Tight span">Tight span</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Manifold" title="Manifold">Manifolds</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a><br />and <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/L%C3%A9vy_metric" title="Lévy metric">Lévy metric</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed space</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li> <li><a href="/wiki/Wasserstein_metric" title="Wasserstein metric">Wasserstein metric</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/General_topology" title="General topology">General topology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_space" title="Discrete space">Discrete space</a></li> <li><a href="/wiki/Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></li> <li><a href="/wiki/Laakso_space" title="Laakso space">Laakso space</a></li> <li><a href="/wiki/Product_metric" title="Product metric">Product metric</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_of_metric_spaces" title="Category of metric spaces">Category of metric spaces</a></li> <li><a href="/wiki/Cantor_space" title="Cantor space">Cantor space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approach_space" title="Approach space">Approach space</a></li> <li><a href="/wiki/Cauchy_space" title="Cauchy space">Cauchy space</a></li> 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