Minkowski’s theorem
Let ℒ∈ℝ2 be a lattice in the sense of
number theory, i.e. a 2-dimensional free group
over ℤ
which generates ℝ2 over ℝ. Let w1,w2 be
generators
of the lattice ℒ. A set ℱ of
the form
ℱ={(x,y)∈ℝ2:(x,y)=αw1+βw2,0≤α<1,0≤β<1} |
is usually called a fundamental domain or fundamental parallelogram for the lattice ℒ.
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 |