  We use the inverse mean curvature flow to prove a sharp
Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain
locally hyperbolic manifolds. As an application we derive an optimal Penrose
inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq
3$. When the horizon has the topology of a compact surface of genus at least
one, this provides an affirmative answer, for this class of initial data sets,
to a question posed by Gibbons, Chru\'sciel and Simon on the validity of a
Penrose-type inequality for black hole solutions carrying a higher genus
horizon.
