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

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.