---
tag: 0K387S2PKQ
kind: statement
role: theorem
name: Minkowski's theorem
aliases: [minkowski-convex-body-theorem]
msc: [11H06]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski%27s_theorem
    locator: Formulation
same_as:
  wikidata: Q1097021
  wikipedia: 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|043B3ENSTY]] and [[Lagrange's four-square theorem|0KYYSMEVHH]].

## 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 $4$.