Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

TL;DR

This paper proves that in Euclidean space with a density proportional to the radius raised to a power, the only isoperimetric hypersurfaces passing through the origin are spheres.

Contribution

It establishes the uniqueness of spheres passing through the origin as isoperimetric hypersurfaces in $ abla^n$ with density $r^p$ for $n \u2265 3$ and $p>0$.

Findings

Spheres passing through the origin are the unique isoperimetric hypersurfaces.

The result applies to dimensions $n \u2265 3$ and densities with $p>0$.

The proof characterizes the geometry of isoperimetric regions with density $r^p$.

Abstract

We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.