Lattice coverings and Gaussian measures of n-dimensional convex bodies

Abstract: Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and
$\gamma_n$ the
(standard)
Gaussian measure on ${\bf R}^n$ with density $(2 \pi )^{-n/2} e^{- \|
x\|^2
/2}$. Let
$\vartheta$ ($ \simeq 1.3489795$) be defined by $\gamma_1 ([ -
\vartheta
/2, \vartheta /2]) = 1/2$ and let $L$ be a
lattice in ${\bf R}^n$ generated by vectors of norm $\leq \vartheta$.
Then, for any closed
convex set $V$ in ${\bf R}^n$ with $\gamma_n (V) \geq 1/2$ and
for any
$a \in {\bf R}^n$, $(a +L) \cap V \neq \phi$. The above statement can
be
viewed as a
``nonsymmetric'' version of Minkowski Theorem.
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