Sharp Poincar\'e-type inequality for the Gaussian measure on the boundary of convex sets

TL;DR

This paper establishes a sharp Poincaré-type inequality for Gaussian measures on convex set boundaries, leading to new inequalities related to Gaussian mean curvature and second variations, and connects to Ehrhard's inequality.

Contribution

It introduces a novel sharp inequality for Gaussian measures on convex boundaries, extending the understanding of Gaussian isoperimetric and curvature properties.

Findings

Derives a sharp Poincaré-type inequality for Gaussian boundary measures

Establishes Gaussian mean-curvature and second-variation inequalities

Links the new inequality to Ehrhard's Gaussian inequality

Abstract

A sharp Poincar\'e-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.