Handbook of Hilbert Geometry

This volume contains surveys on the various aspects of Hilbert geometry including the classical and the modern aspects. The subject is considered from several points of view: Finsler geometry, the calculus of variations, projective geometry, dynamical systems, etc. At several places, the fruitful relations between Hilbert geometry and other subjects in mathematics are highlighted, including Teichm\"uller spaces, convexity theory, Perron-Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups and discrete group actions. The Handbook is addressed at the same time to the student who wants to learn the theory and to the confirmed researcher and the specialist in the field. Chapter 1: Weak Minkowski Spaces (A. Papadopoulos and M. Troyanov); Chapter 2: From Funk to Hilbert Geometry (A. Papadopoulos and M. Troyanov); Chapter 3: Funk and Hilbert Geometries from the Finslerian Viewpoint (M.Troyanov); Chapter 4: On the Hilbert Geometry of Convex Polytopes (C. Vernicos); Chapter 5: The horofunction boundary and isometry group of the Hilbert geometry (C. Walsh); Chapter 6: Characterizations of hyperbolic geometry among Hilbert geometries (Ren Guo); Chapter 7: The geodesic flow of Finsler and Hilbert geometries (M. Crampon) Chapter 8: Around groups in Hilbert Geometry (L. Marquis); Chapter 9: The Dynamics of Hilbert nonexpansive maps (A. Karlsson); Chapter 10: Birkhoff's version of Hilbert's metric and its applications in analysis (B. Lemmens and R. Nussbaum); Chapter 11: Convex real projective structures (I. Kim and A. Papadopoulos); Chapter 12: The Weil-Petersson Funk metric on Teichm\"uller space (H. Miyachi, K. Ohshika, and S. Yamada); Chapter 13: A survey of the Funk and Hilbert geometries of convex sets in spaces of constant curvature (A. Papadopoulos and S. Yamada); Chapter 14: On the origin of Hilbert Geometry (M. Troyanov); Chapter 15: On Hilbert's fourth problem (A. Papadopoulos). Open problems (ed. A. Papadopoulos and M. Troyanov).