Isoperimetric interpretation for the Renormalized volume of convex co-compact hyperbolic 3-manifolds

We reinterpret renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in H³. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in H³ or in cyclic quotients of H³.