# 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

 $\mathcal{F}=\{(x,y)\in\mathbb{R}^{2}:(x,y)=\alpha w_{1}+\beta w_{2},\quad 0% \leq\alpha<1,\quad 0\leq\beta<1\}$

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

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

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

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

###### Theorem 2.

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

Title Minkowski’s theorem MinkowskisTheorem 2013-03-22 13:51:42 2013-03-22 13:51:42 alozano (2414) alozano (2414) 8 alozano (2414) Theorem msc 11H06 Minkowski’s theorem on convex bodies LatticeInMathbbRn ProofOfMinkowskisBound Minkowski’s theorem fundamental parallelogram