Symmetry of minimizers of a Gaussian isoperimetric problem

TL;DR

This paper characterizes the minimizers of a Gaussian isoperimetric functional with a barycenter term, showing that near full volume, solutions are either half-spaces or symmetric strips, depending on the repulsive strength.

Contribution

It provides a complete characterization of minimizers for a Gaussian isoperimetric problem with a barycenter penalty near volume one.

Findings

Minimizers are either half-spaces or symmetric strips near volume one.

The symmetric strip is optimal among symmetric sets for large volume.

The barycenter term influences the shape of minimizers depending on its strength.

Abstract

We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one.