Minkowski's theoremMinkowskisTheorem2003-08-15 16:53:252005-11-24 11:22:40TheoremMinkowski's theoremfundamental parallelogramMinkowskiconvex% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}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 \emph{fundamental domain} or \emph{fundamental parallelogram} for the lattice $\mathcal{L}$.
\begin{thm}[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.
\end{thm}
More generally, there is the following $n$-dimensional analogue.
\begin{thm}
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.
\end{thm}