Strongly log-concave measures and lattices

Abstract

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

arXiv comment19 pages, 1 figure