Entropy and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space
Gautam Aishwarya, Dongbin Li
TL;DR
The paper addresses dimensional Brunn–Minkowski-type inequalities in Gaussian space for even strongly log-concave distributions by developing a transport-entropy framework. It proves an exponentiated Gaussian relative entropy inequality along a natural interpolation between two such random vectors, using Brenier transport maps and a velocity-field approach, with an explicit sine-type weight capturing the interpolation. By applying Donsker–Varadhan duality, it then derives Gaussian Borell–Brascamp–Lieb inequalities that recover the geometric dim‑BM bound for origin-symmetric convex sets as a corollary, and it extends these ideas to β‑homogeneous priors for broader functional forms. Overall, the work provides a transport-entropy route to Gaussian dim‑BM and related functional inequalities, offering new tools and potential extensions to broader even log-concave measures in convex geometry and analysis.
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 $γ$ 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.