Minkowski's theorem
Minkowski's theorem is a statement in the field of geometry of numbers about convex symmetric sets and lattices. It relates the number of contained lattice points to the volume of such a set.
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).
Referenced By
Geometry of numbers | Herman Minkowski | Hermann Minkowski | Lattice (group) | List of geometry topics | List of group theory topics | List of mathematical topics (M-O) | List of number theory topics | Minkowski | Number Theory | Theory of numbers
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