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

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 $0<m\le k-k\mathrm{/}(n+k-1)$, that is, if they are stable (and $m>0$). The implicit unjustified assumption is that the generating curve is convex.