Funk and Hilbert geometries from the Finslerian viewpoint
Marc TroyanovEcole Polytechnique Fédérale de Lausanne, Switzerland
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In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a strongly convex bounded domain of is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to . The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk–Berwald characterization theorem.