Approaching the isoperimetric problem in $$H_{{\mathbb {C}}}^m$$ via the hyperbolic log-convex density conjecture

Abstract

AbstractWe prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space $$H_{{\mathbb {R}}}^n$$ H R n endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on $${\mathbb {R}}^n$$ R n . As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are uniquely isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.