  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.
