Rigidity of Hilbert metrics

Abstract

We study the groups of isometries for Hilbert metrics on bounded open convex domains in ℝn and show that if  is such a set with a strictly convex boundary, the Hilbert geometry is asymptotically Riemannian at infinity. As a consequence of this result, we prove there are no Hausdorff quotients of  by isometry subgroups with finite volume except when ∂ is an ellipsoid.