Convexes Hyperboliques et Quasiisométries

For any m ≥ 3, we construct properly convex open sets Ω in the real projective space whose Hilbert metric is Gromov hyperbolic but is not quasiisometric to the hyperbolic space . We show that such examples cannot exist for m = 2. Some of our examples are divisible, i.e. there exists a discrete group Г of projective transformations preserving Ω with a compact quotient Г\Ω. The open set Ω is strictly convex but the group Г is not isomorphic to any cocompact lattice in the isometry group of .