Minkowski's theorem

From Wikipedia, the free encyclopedia

In mathematics, Minkowski's theorem is the statement that any convex set in Rⁿ which is symmetric with respect to the origin and with volume greater than 2ⁿ contains a non-zero lattice point. The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called geometry of numbers.

1 Formulation
2 Example
3 Applications
4 Notes
5 See also
6 References

[edit] Formulation

Suppose that L is a lattice of determinant d(L) in the n-dimensional real vector space Rⁿ and S is a convex subset of Rⁿ that is symmetric with respect to the origin, meaning that if x is in S then −x is also in S. Minkowski's theorem states that if the volume of S is strictly greater than 2ⁿ d(L), then S must contain at least one lattice point other than the origin.^[1]

[edit] Example

The simplest example of a lattice is the set Zⁿ of all points with integer coefficients; its determinant is 1. For n = 2 the theorem claims that a convex figure in the plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp: if S is the interior of the square with vertices (±1, ±1) then S is symmetric and convex, has area 4, but the only lattice point it contains is the origin. This observation generalizes to every dimension n.

[edit] Applications

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.