Minkowski’s theorem
Let be a lattice in the sense of number theory, i.e. a 2-dimensional free group over which generates over . Let be generators of the lattice . A set of the form
is usually called a fundamental domain or fundamental parallelogram for the lattice .
Theorem 1 (Minkowski’s Theorem).
Let be an arbitrary lattice in and let be the area of a fundamental parallelogram. Any convex region symmetrical about the origin and of area greater than contains points of the lattice other than the origin.
More generally, there is the following -dimensional analogue.
Theorem 2.
Let be an arbitrary lattice in and let be the area of a fundamental parallelopiped. Any convex region symmetrical about the origin and of volume greater than contains points of the lattice 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 |