Number theory Quadratic forms and geometry of numbers Theorem

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.

Where this leads

9 pages build on this one.

Tag	`0K387S2PKQ`
URI	`m2://0K387S2PKQ`
Aliases	`minkowski-convex-body-theorem`
Kind	statement, theorem
Writing	W0 stub
MSC	11H06
Status	This copy of the site has no build data. Run `m2 status 0K387S2PKQ`.
Same as	Wikidata Q1097021 Wikipedia Minkowski's theorem

Minkowski's theorem

Statement

Remarks

History

Where this leads

Further reading

What this page needs for W1

What links here

Used by