On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type

TL;DR

This paper investigates a conjecture about Gaussian measure concavity for symmetric convex sets, proving a near version of the conjecture and extending log-concavity to p-concavity under certain conditions.

Contribution

It proves a near version of the Gardner-Zvavitch conjecture for Gaussian measure and extends log-concavity to p-concavity with symmetric convex sets.

Findings

Proved Gaussian measure is approximately 1/n-concave for symmetric convex sets.

Extended log-concavity to p-concavity under Hessian bounds.

Established dimension-free bounds for measure concavity.

Abstract

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ \gamma(\lambda K+(1-\lambda)L)^{\frac{1}{2n}}\geq \lambda \gamma(K)^{\frac{1}{2n}}+(1-\lambda)\gamma(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>0,$ with respect to the addition of symmetric convex sets.