Essential connectedness and the rigidity problem for Gaussian symmetrization

TL;DR

This paper characterizes when equality holds in Gaussian symmetrization inequalities using a new measure-theoretic connectedness concept, advancing understanding of geometric and measure-theoretic properties in Gaussian spaces.

Contribution

It introduces a novel measure-theoretic connectedness notion to characterize rigidity in Gaussian symmetrization equality cases.

Findings

Provides a geometric criterion for equality cases in Ehrhard's inequality.

Introduces a new measure-theoretic connectedness concept inspired by Federer.

Advances the understanding of geometric structures in Gaussian perimeter problems.

Abstract

We provide a geometric characterization of rigidity of equality cases in Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer's definition of indecomposable current.