# Minkowski's theorem

 related topics {math, number, function} {area, part, region}

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 d(L) 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 the geometry of numbers.

## Contents

### 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]

### 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.

### Proof

The following argument proves Minkowski's theorem for the special case of L=Z2. It can be generalized to arbitrary lattices in arbitrary dimensions.

Consider the map $f: S \to \mathbb{R}^2, (x,y) \mapsto (x \bmod 2, y \bmod 2)$. Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly f(S) has area ≤ 4. Suppose f were injective. Then each of the stacked squares would be non-overlapping, so f would be area-preserving, and the area of f(S) would be greater than 4, since S has area greater than 4. That is not the case, so f(p1) = f(p2) for some pair of points p1,p2 in S. Moreover, we know from the definition of f that p2 = p1 + (2i,2j) for some integers i and j, where i and j are not both zero.