Balls Isoperimetric in ℝⁿ with Volume and Perimeter Densities r^m and r^k

2016/10/19 by Di Giosia, Leonardo, Habib, Jahangir, Kenigsberg, Lea +2
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1610.05830

Abstract

We have discovered a "little" gap in our proof of the sharp conjecture that in ℝⁿ with volume and perimeter densities r^m and r^k, balls about the origin are uniquely isoperimetric if 0 < m ≤ 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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