Minkowski's theorem

Statement

Let $n$ be a positive integer, and let $L \subset \mathbb{R}^n$ be a lattice with determinant $d(L)$. Let $S \subseteq \mathbb{R}^n$ be a convex set that is symmetric about the origin: $x \in S$ implies $-x \in S$. If the volume of $S$ is greater than $2^n d(L)$, then $S$ contains a point of $L$ other than $0$.

Remarks

The most common case is $L = \mathbb{Z}^n$, with $d(L) = 1$: a symmetric convex set of volume greater than $2^n$ contains a nonzero integer point. The bound is sharp: the open cube $(-1, 1)^n$ has volume $2^n$ and contains no nonzero integer point. Applications include proofs of Fermat's theorem on sums of two squares and Lagrange's four-square theorem.

History

Hermann Minkowski proved the theorem in 1889. It started the geometry of numbers.

Explanations

L1 human

A large, round, centered region in the plane must contain a lattice point other than the origin. In the plane with $L = \mathbb{Z}^2$, the area must be more than $4$.