Strongly log-concave measures and lattices

Abstract

Let $Λ\subseteq \mathbb{R}^n$ 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 (1) $f$ is $α$-strongly log-concave, where $α= 2π/e$, and (2) $X + Λ$ is uniform on $\mathbb{R}^n/Λ$. Moreover, the existence of such a density characterizes this property up to a universal dilation. There are a few consequences of the above. We list some. First, for such lattices we can bound the covering radius \[ μ(Λ, K) = \max_{x\in\mathbb{R}^n}\min_{λ\inΛ} \left\|x-λ\right\|_K \le \sqrt{\frac{2e}π} \cdot \mathbb{E}\,\left\|G\right\|_K \] of a convex body $K \subseteq \mathbb{R}^n$ containing $0$ in its interior, where $G \sim N(0,\mathrm{Id})$. 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. We also compare the entire ordered spectrum of the associated flat torus on $Λ^*$ with that of a Gaussian, yielding an alternative proof of the reverse Minkowski theorem.