The Log Convex Density Conjecture in Hyperbolic Space

Abstract

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on ${\mathbb{R}}^n$ the answer is a sphere about the origin. We generalize his results from ${\mathbb{R}}^n$ to ${\mathbb{H}}^n$ with related but different volume and perimeter densities.