Lattice coverings and gaussian measures of n-dimensional convex bodies

Abstract

Let $\Vert \cdot \Vert$ be the euclidean norm on ${\mathbf{R}}^n$ and ${\gamma }_n$ the (standard) Gaussian measure on ${\mathbf{R}}^n$ with density $(2\pi {)}^{-n/2}{e}^{-\Vert x{\Vert }^2/2}$. Let $\vartheta$ ($\simeq 1.3489795$) be defined by ${\gamma }_1([-\vartheta /2,\vartheta /2])=1/2$ and let $L$ be a lattice in ${\mathbf{R}}^n$ generated by vectors of norm $\le \vartheta$. Then, for any closed convex set $V$ in ${\mathbf{R}}^n$ with ${\gamma }_n(V)\ge \frac{1}{2}$ and for any $a\in {\mathbf{R}}^n$, $(a+L)\cap V\ne \varphi$. The above statement can be viewed as a ``nonsymmetric'' version of Minkowski Theorem.