The isoperimetric problem in the plane with the sum of two Gaussian densities

TL;DR

This paper investigates the isoperimetric problem involving the sum of two Gaussian densities in the plane and line, establishing the shape of optimal regions under these conditions.

Contribution

It characterizes the isoperimetric regions for the sum of two Gaussian densities in both the line and the plane, identifying rays and half-spaces as optimal shapes.

Findings

In the line, the isoperimetric regions are rays.

In the plane, if regions are half-spaces, they are bounded by vertical lines.

Provides conditions under which these shapes are optimal.

Abstract

We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the plane are half-spaces, then they must be bounded by vertical lines.