Minkowski's Theorem


Title: Minkowski's Theorem
Author: Manuel Eberl
Submission date: 2017-07-13

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.

  author  = {Manuel Eberl},
  title   = {Minkowski's Theorem},
  journal = {Archive of Formal Proofs},
  month   = jul,
  year    = 2017,
  note    = {\url{https://isa-afp.org/entries/Minkowskis_Theorem.html},
            Formal proof development},
  ISSN    = {2150-914x},
License: BSD License