PlanetMath: Minkowski's theorem

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Minkowski's theorem

(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 be generators of the lattice $\mathcal{L}$ . A set $\mathcal{F}$ of the form

$\displaystyle \mathcal{F}=\{(x,y)\in\mathbb{R}^2: (x,y)=\alpha w_1+\beta w_2,\quad 0\leq \alpha < 1,\quad 0\leq \beta <1 \}$

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 -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.

"Minkowski's theorem" is owned by alozano.

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