Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space

Abstract

Given even strongly log-concave random vectors $X_0$ and $X_1$ in ${\mathbb{R}}^n$, we show that a natural joint distribution $(X_0,X_1)$ satisfies, \begin{equation} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{equation} where $Z$ is distributed according to the standard Gaussian measure $\gamma$ on ${\mathbb{R}}^n$, $t\in [0,1]$, and $D(\cdot \Vert Z)$ is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis, namely that \begin{equation} γ\left( (1-t) K_{0} + t K_{1} \right)^{\frac{1}{n}} \geq (1-t) γ(K_{0})^{\frac{1}{n}} + t γ(K_{1})^{\frac{1}{n}}, \end{equation} when $K_0,K_1\subseteq {\mathbb{R}}^n$ are origin-symmetric convex bodies. As an application, using Donsker--Varadhan duality, we obtain Gaussian Borell--Brascamp--Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis--Moschidis inequality.

Revised final version, 18 pages