Minkowski's Theorem

Manuel Eberl 🌐

July 13, 2017

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


Minkowski's theorem relates a subset of ℝn, the Lebesgue measure, and the integer lattice ℤn: It states that any convex subset of ℝn with volume greater than 2n contains at least one lattice point from ℤn\{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 ℝn with a volume greater than 1 contains two different points whose difference vector has integer components.

The entry contains a proof of both theorems.


BSD License


Session Minkowskis_Theorem