Approaching the isoperimetric problem in HCm via the hyperbolic log-convex density conjecture

We prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space 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 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.