Abstract:

Suppose that a positive radial log-convex density is given on Euclidean space. This is used to weight volume and perimeter. It was conjectured in [Rosales et al (2006)] that balls are minimisers for the weighted isoperimetric problem (with some additional requirements). The conjecture was proved in large part in [Chambers (2013)]. We obtain an independent proof of the conjecture under weaker hypotheses in the planar case. The argument is more analytic in flavour. It includes a comparison theorem for solutions to a Ricatti equation as well as a reverse Hermite-Hadamard inequality.