@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: a skos:ConceptScheme .
psr:-V0G085HP-P
  skos:prefLabel "differential geometry"@en, "géométrie différentielle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-ZNXZ6LDT-1 .

psr:-ZNXZ6LDT-1
  skos:inScheme psr: ;
  skos:definition """In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least 2π . Equivalently, the average curvature is at least <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 2\\\\pi /L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mn>2</mn>
         <mi>π<!-- π --></mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>/</mo>
         </mrow>
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2\\\\pi /L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93371c1513249eb6bfcc7092f41da0984aff05f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.24ex; height:2.843ex;" alt="{\\\\displaystyle 2\\\\pi /L}"></span>, where <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 L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="L"></span>  is the length of the curve. The only curves of this type whose total absolute curvature equals 2π and whose average curvature equals <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 2\\\\pi /L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mn>2</mn>
         <mi>π<!-- π --></mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>/</mo>
         </mrow>
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2\\\\pi /L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93371c1513249eb6bfcc7092f41da0984aff05f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.24ex; height:2.843ex;" alt="{\\\\displaystyle 2\\\\pi /L}"></span> are the plane convex curves. The theorem is named after Werner Fenchel, who published it in 1929. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Fenchel%27s_theorem">https://en.wikipedia.org/wiki/Fenchel%27s_theorem</a>)"""@en, """En géométrie différentielle, le théorème de Fenchel stipule que la courbure moyenne de toute courbe convexe fermée dans le plan euclidien est égale à <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 2\\\\pi /L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mn>2</mn>
         <mi>π<!-- π --></mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>/</mo>
         </mrow>
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2\\\\pi /L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93371c1513249eb6bfcc7092f41da0984aff05f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.24ex; height:2.843ex;" alt="{\\\\displaystyle 2\\\\pi /L}"></span>, où <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 L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="L"></span> est la longueur de la courbe. Il porte le nom de Werner Fenchel, qui l'a publié en 1929. Plus généralement, pour une courbe fermée arbitraire de l'espace, la courbure moyenne est ≥ <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 2\\\\pi /L}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mn>2</mn>
         <mi>π<!-- π --></mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>/</mo>
         </mrow>
         <mi>L</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2\\\\pi /L}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93371c1513249eb6bfcc7092f41da0984aff05f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.24ex; height:2.843ex;" alt="{\\\\displaystyle 2\\\\pi /L}"></span> avec égalité uniquement pour les courbes planes convexes. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fenchel">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fenchel</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fenchel>, <https://en.wikipedia.org/wiki/Fenchel%27s_theorem> ;
  dc:created "2023-07-19"^^xsd:date ;
  skos:prefLabel "théorème de Fenchel"@fr, "Fenchel's theorem"@en ;
  dc:modified "2023-07-19"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-V0G085HP-P .