Minkowski’s theorem

Let $\\mathcal{L}\\in\\mathbb{R}^{2}$ be a lattice in the sense ofnumber theory, i.e. a 2-dimensional free group over ${\\mathbb{Z}}$ which generates $\\mathbb{R}^{2}$ over $\\mathbb{R}$ . Let $w_{1},w_{2}$ begenerators of the lattice $\\mathcal{L}$ . A set $\\mathcal{F}$ ofthe 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 $\\mathbb{R}^{2}$ andlet $\\Delta$ be the area of a fundamental parallelogram. Anyconvex region $\\mathfrak{K}$ symmetrical about the origin and ofarea 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 $\\mathbb{R}^{n}$ andlet $\\Delta$ be the area of a fundamental parallelopiped. Anyconvex region $\\mathfrak{K}$ symmetrical about the origin and ofvolume greater than $2^{n}\\Delta$ contains points of the lattice $\\mathcal{L}$ other than the origin.