Lattice coverings and gaussian measures of n-dimensional convex bodies

Abstract

Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and $γ_n$ the (standard) Gaussian measure on ${\bf R}^n$ with density $(2 π)^{-n/2} e^{- \| x\|^2 /2}$. Let $\vartheta$ ($ \simeq 1.3489795$) be defined by $γ_1 ([ - \vartheta /2, \vartheta /2]) = 1/2$ and let $L$ be a lattice in ${\bf R}^n$ generated by vectors of norm $\leq \vartheta$. Then, for any closed convex set $V$ in ${\bf R}^n$ with $γ_n (V) \geq \frac{1}{2}$ and for any $a \in {\bf R}^n$, $(a +L) \cap V \neq ϕ$. The above statement can be viewed as a ``nonsymmetric'' version of Minkowski Theorem.