Lattice coverings and Gaussian measures of \(n\)-dimensional convex bodies (Q1355192)

Let \(|\cdot |\) be the euclidean norm on \(\mathbb{R}^n\) and let \(L\subset \mathbb{R}^n\) be a lattice generated by vectors of norm \(\leq \vartheta\), where \(\vartheta (=1.348 \dots)\) is well-defined. Then the authors show that for any closed convex set \(V\subset \mathbb{R}^n \), whose Gaussian measure on \(\mathbb{R}^n\) is at least \({1\over 2}\), one gets \(L+V = \mathbb{R}^n\). This result can be considered as a Minkowski-type lattice theorem. As a consequence of this theorem the authors obtain a good bound for a problem of \textit{P. McMullen} and \textit{J. M. Wills} [Mathematika 28, 255-264 (1981; Zbl 0464.52006)]: The minimal width of a lattice-point-free simplex in \(\mathbb{Z}^n\) is less than \(c(n(1+ \log n))^{1/2}\).