### Abstract

Minkowski's theorem relates a subset of
ℝ^{n}, the Lebesgue measure, and the
integer lattice ℤ^{n}: It states that
any convex subset of ℝ^{n} with volume
greater than 2^{n} contains at least one lattice
point from ℤ^{n}\{0}, i. e. a
non-zero point with integer coefficients.

A
related theorem which directly implies this is Blichfeldt's
theorem, which states that any subset of
ℝ^{n} with a volume greater than 1
contains two different points whose difference vector has integer
components.

The entry contains a proof of both theorems.