Euclidean balls solve some isoperimetric problems with nonradial weights

TL;DR

This paper demonstrates that Euclidean balls centered at the origin minimize isoperimetric quotients in convex cones when using certain nonradial, homogeneous weights, extending classical results to weighted settings.

Contribution

It extends isoperimetric problem solutions to nonradial weighted measures in convex cones, showing Euclidean balls are optimal under broad conditions.

Findings

Euclidean balls minimize the weighted isoperimetric quotient.

The result applies to nonnegative homogeneous weights with a concavity condition.

Special case recovers classical results when the weight is constant.

Abstract

In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990.