# Minkowski’s theorem

Let $\mathcal{L}\in {\mathbb{R}}^{2}$ be a lattice in the sense of
number theory^{}, i.e. a 2-dimensional free group^{} over $\mathbb{Z}$
which generates ${\mathbb{R}}^{2}$ over $\mathbb{R}$. Let ${w}_{1},{w}_{2}$ be
generators^{} of the lattice $\mathcal{L}$. A set $\mathcal{F}$ of
the form

$$ |

is usually called a *fundamental domain* or *fundamental parallelogram* for the lattice $\mathcal{L}$.

###### Theorem 1 (Minkowski’s Theorem).

Let $\mathrm{L}$ be an arbitrary lattice in ${\mathrm{R}}^{\mathrm{2}}$ and let $\mathrm{\Delta}$ be the area of a fundamental parallelogram. Any convex region $\mathrm{K}$ symmetrical about the origin and of area greater than $\mathrm{4}\mathit{}\mathrm{\Delta}$ contains points of the lattice $\mathrm{L}$ other than the origin.

More generally, there is the following $n$-dimensional analogue.

###### Theorem 2.

Let $\mathrm{L}$ be an arbitrary lattice in ${\mathrm{R}}^{n}$ and let $\mathrm{\Delta}$ be the area of a fundamental parallelopiped. Any convex region $\mathrm{K}$ symmetrical about the origin and of volume greater than ${\mathrm{2}}^{n}\mathit{}\mathrm{\Delta}$ contains points of the lattice $\mathrm{L}$ other than the origin.

Title | Minkowski’s theorem |
---|---|

Canonical name | MinkowskisTheorem |

Date of creation | 2013-03-22 13:51:42 |

Last modified on | 2013-03-22 13:51:42 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 8 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11H06 |

Synonym | Minkowski’s theorem on convex bodies |

Related topic | LatticeInMathbbRn |

Related topic | ProofOfMinkowskisBound |

Defines | Minkowski’s theorem |

Defines | fundamental parallelogram |