Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

TL;DR

This paper investigates the isoperimetric problem in Euclidean space with specific volume and perimeter densities, identifying conditions under which balls are uniquely optimal, and discusses a gap related to the convexity assumption of generating curves.

Contribution

The paper reveals a gap in the proof of a conjecture about isoperimetric regions in weighted Euclidean spaces, specifically concerning the convexity assumption of generating curves.

Findings

Identifies a gap in the proof of the isoperimetric conjecture.

Clarifies conditions for balls to be uniquely isoperimetric.

Highlights the importance of convexity in the generating curve assumption.

Abstract

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the generating curve is convex.