Metrics on convex hyperbolic polyhedra

Using the "Hilbert metric", we define a (complex) distance on the "exterior" of the hyperbolic space, as well as between a point in Hn and an "exterior" point. The resulting space has a natural notion of polyhedra; we describe the metrics induced on the convex polyhedra in dimension 3. This extends results of Alexandrov, Andreev and Rivin concerning the induced and the dual metrics on convex hyperbolic polyhedra. Several lemmas - rigidity of convex polyhedra, description of degenerations - extend to higher dimensions or to analoguous situations.