  We reinterpret the 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 $\mathbb{H}^3$. Finally, we include the classification of stable
constant mean curvature surfaces in regions bounded by two geodesic planes in
$\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.
