@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:-BQ41NXBG-B
  skos:inScheme psr: ;
  skos:prefLabel "Minkowski's theorem"@en, "théorème de Minkowski"@fr ;
  skos:broader psr:-PW35VMXC-2, psr:-F7SFNL4R-1 ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Minkowski%27s_theorem>, <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Minkowski> ;
  dc:modified "2023-08-17"^^xsd:date ;
  dc:created "2023-08-17"^^xsd:date ;
  skos:definition """In mathematics, <b>Minkowski's theorem</b> is the statement that every convex set in <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 \\\\mathbb {R} ^{n}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
         <mi>n</mi>
         </mrow>
         </msup>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="\\\\mathbb {R} ^{n}"></span> which is symmetric with respect to the origin and which has volume greater than <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^{n}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mn>2</mn>
         <mrow class="MJX-TeXAtom-ORD">
         <mi>n</mi>
         </mrow>
         </msup>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2^{n}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="2^{n}"></span> contains a non-zero integer point (meaning a point in <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 \\\\mathbb {Z} ^{n}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">Z</mi>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
         <mi>n</mi>
         </mrow>
         </msup>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {Z} ^{n}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="\\\\mathbb{Z } ^{n}"></span> that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice <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> and to any symmetric convex set with volume greater than <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^{n}\\\\,d(L)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mn>2</mn>
         <mrow class="MJX-TeXAtom-ORD">
         <mi>n</mi>
         </mrow>
         </msup>
         <mspace width="thinmathspace" />
         <mi>d</mi>
         <mo stretchy="false">(</mo>
         <mi>L</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle 2^{n}\\\\,d(L)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39afdd57d370b7d800df22b204a0064ba9458af5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.376ex; height:2.843ex;" alt="{\\\\displaystyle 2^{n}\\\\,d(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 d(L)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>d</mi>
         <mo stretchy="false">(</mo>
         <mi>L</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle d(L)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27762bb5805f7b8c5013dab5eb94d9f2379e761" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.608ex; height:2.843ex;" alt="{\\\\displaystyle d(L)}"></span> denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Minkowski%27s_theorem">https://en.wikipedia.org/wiki/Minkowski%27s_theorem</a>)"""@en, """En mathématiques, le <b>théorème de Minkowski</b> concerne les réseaux de l'espace euclidien ℝ<sup><i>d</i></sup>. Étant donné un tel réseau Λ, il garantit l'existence, dans tout convexe symétrique de volume suffisant, d'un vecteur non nul de Λ. Hermann Minkowski a découvert ce théorème en 1891 et l'a publié en 1896, dans son livre fondateur de la géométrie des nombres. Ce résultat est en particulier utilisé en théorie algébrique des nombres. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Minkowski">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Minkowski</a>)"""@fr ;
  a skos:Concept .

psr:-PW35VMXC-2
  skos:prefLabel "convex set"@en, "ensemble convexe"@fr ;
  a skos:Concept ;
  skos:narrower psr:-BQ41NXBG-B .

psr:-F7SFNL4R-1
  skos:prefLabel "algebraic number theory"@en, "théorie algébrique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-BQ41NXBG-B .