### Abstract

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.

### License

BSD License

### Topics

- Mathematics/Geometry
- Mathematics/Number theory

### Session Minkowskis_Theorem

- Minkowskis_Theorem

## Cite

×

Minkowskis_Theorem-AFP

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

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