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 $\Gamma \subseteq \Lambda^*$. 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 \[ \mu(\Lambda, K) = \max_{x \in \mathbb{R}^n} \min_{\lambda \in \Lambda} \|x-\lambda\|_K \le \sqrt{\frac{2e}{\pi}} \cdot \mathbb{E}_G\, \|G\|_K \] 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.