Minkowski's theorem

Minkowski's theorem is a statement in the field of geometry of numbers about convex symmetric sets and latticess. 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).