Minkowski’s theorem


Let 2 be a lattice in the sense of number theoryMathworldPlanetmathPlanetmath, i.e. a 2-dimensional free groupMathworldPlanetmath over which generates 2 over . Let w1,w2 be generatorsPlanetmathPlanetmathPlanetmath of the lattice . A set of the form

={(x,y)2:(x,y)=αw1+βw2,0α<1,0β<1}

is usually called a fundamental domain or fundamental parallelogram for the lattice .

Theorem 1 (Minkowski’s Theorem).

Let L be an arbitrary lattice in R2 and let Δ be the area of a fundamental parallelogram. Any convex region K symmetrical about the origin and of area greater than 4Δ contains points of the lattice L other than the origin.

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

Theorem 2.

Let L be an arbitrary lattice in Rn and let Δ be the area of a fundamental parallelopiped. Any convex region K symmetrical about the origin and of volume greater than 2nΔ contains points of the lattice 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