Strongly log-concave measures and lattices

Let $\mathrm{\Lambda }\subset {\mathbb{R}}^n$ be a full rank lattice with the property that $\det \mathrm{\Gamma }\ge 1$ for all sublattices $\mathrm{\Gamma }\subseteq {\mathrm{\Lambda }}^{\ast }$. We prove that there exists a symmetric random vector $X\in {\mathbb{R}}^n$ with a smooth, strictly positive density $f:{\mathbb{R}}^n\to (0,\mathrm{\infty })$ satisfying that $f$ is $\alpha$-strongly log-concave, where $\alpha =2\pi \mathrm{/}e$, and $X+\mathrm{\Lambda }$ is uniform on ${\mathbb{R}}^n\mathrm{/}\mathrm{\Lambda }$. Moreover, we show that the existence of such a density $f$ characterizes the assumption on $\mathrm{\Gamma }$ up to a constant. There are a few consequences of the above. First, for such lattices we can bound the covering radius $$\mu (\mathrm{\Lambda },K)=\underset{x\in {\mathbb{R}}^n}{\max }\underset{\lambda \in \mathrm{\Lambda }}{\min }\mathrm{\parallel }x-\lambda {\mathrm{\parallel }}_K\le \sqrt{\frac{2e}{\pi }}\cdot {\mathbb{E}}_G\mathrm{\parallel }G{\mathrm{\parallel }}_K$$ of a convex body $K\subset {\mathbb{R}}^n$ containing $0$ in its interior, where $G\sim N(0,I)$. This implies an $O(n\sqrt{\log n})$ bound for the flatness constant and an $O({\log }^{3\mathrm{/}2}n)$ bound for the subspace flatness constant. Second, we establish as a consequence an alternative proof of the reverse Minkowski theorem with an explicit constant.