Rigidity of Hilbert metrics

Colbois, Bruno ; Verovic, Patrick

In: Bulletin of the Australian Mathematical Society, 2002, vol. 65, no. 1, p. 23-34

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    We study the groups of isometries for Hilbert metrics on bounded open convex domains in n 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