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arXiv · 1610.05830

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

Leonardo Di Giosia·Jahangir Habib·Lea Kenigsberg·Dylanger Pittman·Weitao Zhu
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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.

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Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu. 2016-10-19. Balls Isoperimetric in $\mathbb{R}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$. https://arxiv.org/abs/1610.05830

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