Isoperimetry in Surfaces of Revolution with Density

TL;DR

This paper extends isoperimetric results with density to surfaces of revolution and dual densities, providing new existence, boundedness, and proof techniques for spheres about the origin.

Contribution

It generalizes Chambers' results to other revolution spaces and dual densities, introducing novel methods for establishing isoperimetric solutions.

Findings

Existence and boundedness of isoperimetric regions established.

New approach to proving circles about the origin are isoperimetric.

Generalization of log-convex density results to broader spaces.

Abstract

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric.