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# 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.

## Mathematics Subject Classification

11H06*no label found*

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