Loterre: Mathématiques: Minkowski's theorem

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Minkowski's theorem

In mathematics, Minkowski's theorem is the statement that every convex set in ${\\displaystyle \\mathbb {R} ^{n}}$ which is symmetric with respect to the origin and which has volume greater than ${\\displaystyle 2^{n}}$ contains a non-zero integer point (meaning a point in ${\\displaystyle \\mathbb {Z} ^{n}}$ 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 ${\\displaystyle L}$ and to any symmetric convex set with volume greater than ${\\displaystyle 2^{n}\\,d(L)}$ , where ${\\displaystyle d(L)}$ denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Minkowski%27s_theorem)

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