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Minkowski's theorem
In mathematics, Minkowski's theorem in the geometry of numbers applies to convex symmetric sets and lattices; it relates the number of contained lattice points to the volume of such a set. This relationship was discovered by Hermann Minkowski in 1889.
Let L be a lattice in Rn with determinant d(L). The simplest example is the lattice Zn of all points with integer coefficients; its determinant is 1.
Consider a convex subset S of Rn that is symmetric with respect to the origin, meaning that x in S implies −x in S. Minkowski's theorem states that if the volume of S is bigger than 2nd(L), then S must contain at least 3 lattice points (the origin, another point, and its negative).
Last updated: 05-23-2005 19:27:15
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details