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 $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the generating curve is convex. 
Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman,   Weitao Zhu
math.MG math.DG