@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-RZ3QL167-D
  skos:prefLabel "espace vectoriel topologique"@fr, "topological vector space"@en ;
  a skos:Concept ;
  skos:narrower psr:-SVQC69PF-C .

psr:-HX2VX066-P
  skos:prefLabel "functional analysis"@en, "analyse fonctionnelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SVQC69PF-C .

psr:-ZTD7VMDS-3
  skos:prefLabel "analyse convexe"@fr, "convex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-SVQC69PF-C .

psr: a skos:ConceptScheme .
psr:-XJ7K95G7-L
  skos:prefLabel "optimization"@en, "optimisation"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SVQC69PF-C .

psr:-SVQC69PF-C
  skos:prefLabel "fonctionnelle de Minkowski"@fr, "Minkowski functional"@en ;
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Fonctionnelle_de_Minkowski>, <https://en.wikipedia.org/wiki/Minkowski_functional> ;
  skos:altLabel "jauge"@fr, "gauge function"@en ;
  skos:broader psr:-RZ3QL167-D, psr:-XJ7K95G7-L, psr:-ZTD7VMDS-3, psr:-HX2VX066-P ;
  skos:definition """En géométrie, la notion de <b>jauge</b> généralise celle de semi-norme. À toute partie <span class="texhtml"><i>C</i></span> d'un ℝ-espace vectoriel <span class="texhtml"><i>E</i></span> on associe sa jauge, ou <b>fonctionnelle de Minkowski</b> <span class="texhtml"><i>p<sub>C</sub></i></span>, qui est une application de <span class="texhtml"><i>E</i></span> dans <span class="texhtml">[0, +∞]</span> mesurant, pour chaque vecteur, par quel rapport il faut dilater <span class="texhtml"><i>C</i></span> pour englober ce vecteur. Dès que <span class="texhtml"><i>C</i></span> contient l'origine, <span class="texhtml"><i>p<sub>C</sub></i></span> est positivement homogène&nbsp;; si <span class="texhtml"><i>C</i></span> est étoilée par rapport <span class="nowrap">à <span class="texhtml">0</span>,</span> <span class="texhtml"><i>p<sub>C</sub></i></span> possède d'autres propriétés élémentaires. Si <span class="texhtml"><i>C</i></span> est <b>convexe</b> — cas le plus souvent étudié — <span class="texhtml"><i>p<sub>C</sub></i></span> est même sous-linéaire, mais elle n'est pas nécessairement symétrique et elle peut prendre des valeurs infinies. Sous certaines hypothèses supplémentaires, <span class="texhtml"><i>p<sub>C</sub></i></span> est une semi-norme dont <span class="texhtml"><i>C</i></span> est la boule unité.
<br/>Cette notion intervient en analyse fonctionnelle (démonstration de la forme analytique du théorème de Hahn-Banach), en optimisation (problème de recouvrement par jauge, optimisation conique), en apprentissage automatique, en géométrie des nombres (second théorème de Minkowski),&nbsp;<abbr class="abbr" title="et cetera">etc.</abbr> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonctionnelle_de_Minkowski">https://fr.wikipedia.org/wiki/Fonctionnelle_de_Minkowski</a>)"""@fr, """In mathematics, in the field of functional analysis, a <b>Minkowski functional</b> (after Hermann Minkowski) or <b>gauge function</b> is a function that recovers a notion of distance on a linear space.
<br/>If <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 K}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>K</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="K"></span> is a subset of a real or complex vector space <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 X,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>X</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="X,"></span> then the <em>Minkowski functional</em> or <em>gauge</em> of <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 K}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>K</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="K"></span> is defined to be the function <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 p_{K}:X\\	o [0,\\\\infty ],}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>K</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>:</mo>
<br/>        <mi>X</mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle p_{K}:X\\	o [0,\\\\infty ],}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a71cfffc84a9ec4987dc33f00ae04e5e69794c1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.944ex; height:2.843ex;" alt="{\\\\displaystyle p_{K}:X\\	o [0,\\\\infty ],}"></span> valued in the extended real numbers, defined by
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle p_{K}(x):=\\\\inf\\\\{r\\\\in \\\\mathbb {R} :r>0{\\	ext{ and }}x\\\\in rK\\\\}\\\\quad {\\	ext{ for every }}x\\\\in X,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>K</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>:=</mo>
<br/>        <mo movablelimits="true" form="prefix">inf</mo>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>r</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo>:</mo>
<br/>        <mi>r</mi>
<br/>        <mo>&gt;</mo>
<br/>        <mn>0</mn>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>&nbsp;and&nbsp;</mtext>
<br/>        </mrow>
<br/>        <mi>x</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>r</mi>
<br/>        <mi>K</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mspace width="1em"></mspace>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>&nbsp;for every&nbsp;</mtext>
<br/>        </mrow>
<br/>        <mi>x</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>X</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle p_{K}(x):=\\\\inf\\\\{r\\\\in \\\\mathbb {R} :r&gt;0{\\	ext{ and }}x\\\\in rK\\\\}\\\\quad {\\	ext{ for every }}x\\\\in X,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c5d2f5e9bdee3a46182a34d6b011667233afc4" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:58.899ex; height:2.843ex;" alt="{\\\\displaystyle p_{K}(x):=\\\\inf\\\\{r\\\\in \\\\mathbb {R} :r>0{\\	ext{ and }}x\\\\in rK\\\\}\\\\quad {\\	ext{ for every }}x\\\\in X,}"></div>
<br/>where the infimum of the empty set is defined to be positive infinity <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 \\\\,\\\\infty \\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\,\\\\infty \\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d6272e6a9c114e487435e05a6f322f4c43b6f2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:1.676ex;" alt="{\\\\displaystyle \\\\,\\\\infty \\\\,}"></span> (which is <em>not</em> a real number so that <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 p_{K}(x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>K</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle p_{K}(x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422a847956d41736514023da5233b057ae1b5229" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.091ex; height:2.843ex;" alt="{\\\\displaystyle p_{K}(x)}"></span> would then <em>not</em> be real-valued).  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Minkowski_functional">https://en.wikipedia.org/wiki/Minkowski_functional</a>)"""@en ;
  dc:modified "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  dc:created "2023-08-16"^^xsd:date .