Strongly log-concave measures and lattices

Source abstract

Let $Λ\subset \mathbb{R}^n$ be a full rank lattice with the property that $\det Γ\ge 1$ for all sublattices $Γ\subseteq Λ^*$. 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 $α$-strongly log-concave, where $α= 2π/e$, and $X + Λ$ is uniform on $\mathbb{R}^n/Λ$. Moreover, we show that the existence of such a density $f$ characterizes the assumption on $Γ$ 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.