The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization

TL;DR

This paper establishes the isoperimetric inequality for anisotropic Gaussian measures, explores the limitations of Ehrhard symmetrization in this context, and introduces a new inequality with an error term to advance understanding of anisotropic symmetrization.

Contribution

It proves the anisotropic Gaussian isoperimetric inequality, characterizes equality cases, and proposes a modified inequality addressing Ehrhard symmetrization failure.

Findings

Proved the anisotropic Gaussian isoperimetric inequality.

Identified cases where Ehrhard symmetrization fails to decrease perimeter.

Derived a new inequality with an error term for anisotropic Gaussian measures.

Abstract

In this paper, we prove the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. We also find an example that shows Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and gives a new inequality that includes an error term. This new inequality, in particular, gives us a hint to prove a uniqueness result for the anisotropic Ehrhard symmetrization.