Minkowski's theorem

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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).

The simplest application of the theorem concerns a convex figure symmetric about the origin and demonstrates that the figure will enclose at least one lattice point in addition to the origin if the area of the figure is greater than 4.

A corollary of this theorem is the fact that every class in the ideal class group of a number field $K$ contains an integral ideal of norm not exceeding a certain bound, depending on $K$ , called Minkowski's bound.