On Convex Functions of Gaussian Variables

TL;DR

This work develops a unified convexity framework for normalized log moment generating functions of random variables, tying convexity of $\Lambda_X$ to Ehrhard-type concavity properties of the distribution. The authors prove that $\Lambda_X$ is convex, and strictly so unless $X$ is Gaussian, whenever $t \mapsto \Phi^{-1}(\mathbb{P}[X>t])$ is concave, thereby characterizing convex Gaussian images and extending Chen’s 1D results to higher dimensions. They apply these results to obtain sharp Renyi-divergence comparisons between Gaussianity and strongly log-concave variables, derive sub-Gaussian deviation bounds for conic intrinsic volumes, and establish a reversal of McMullen's inequality via a Wills functional bound. Collectively, the paper links probabilistic convexity, optimal transport, and geometric functional inequalities to yield precise concentration and information-theoretic bounds in Gaussian-analytic settings.

Abstract

We investigate a convexity properties for normalized log moment generating function continuing a recent investigation of Chen of convex images of Gaussians. We show that any variable satisfying a ``Ehrhard-like'' property for its distribution function has a strictly convex normalized log moment generating function, unless the variable is Gaussian, in which case affine-ness is achieved. Moreover we characterize variables that satisfy the Ehrhard-like property as the convex images of Gaussians. As applications, we derive sharp comparisons between Rényi divergences for a Gaussian and a strongly log-concave variable, and characterize the equality case. We also demonstrate essentially optimal concentration bounds for the sequence of conic intrinsic volumes associated to convex cone and we obtain a reversal of McMullen's inequality between the sum of the (Euclidean) intrinsic volumes associated to a convex body and the body's mean width that generalizes and sharpens a result of Alonso-Hernandez-Yepes.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Ehrhard ehrhard1983symetrisation
• proof
• Lemma 3.1: Ehrhard ehrhard1984inegalites
• proof
• Theorem 3.2: Chen chen2023gaussian
• proof