By Athanase Papadapoulous, Marc Troyanov

This quantity provides surveys, written through specialists within the box, on a number of classical and smooth features of Hilbert geometry. They suppose a number of issues of view: Finsler geometry, calculus of adaptations, projective geometry, dynamical structures, and others. a few fruitful kin among Hilbert geometry and different topics in arithmetic are emphasised, together with Teichmüller areas, convexity idea, Perron-Frobenius concept, illustration idea, partial differential equations, coarse geometry, ergodic idea, algebraic teams, Coxeter teams, geometric staff concept, Lie teams and discrete staff activities. This booklet is addressed to either scholars who are looking to research the speculation and researchers during this zone.

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16] D. C. Kay, The Ptolemaic inequality in Hilbert geometries. Pacific J. Math. 21 (1967), 293–301. [17] F. Klein, Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz). Math. Ann. VI (1873), 112–145. [18] B. Nica, The Mazur–Ulam theorem. Expo. Math. 30 (2012), no. 4, 397–398. [19] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 391 (1988). [20] H. Martini, K. Swanepoel and G. Weiss, The geometry of Minkowski spaces – a survey. I. Expo.

F) The weak metric F is unbounded. Property (a) and (b) say that F is a weak metric. Proof. x; y/j 1 37 Chapter 2. From Funk to Hilbert geometry and we have equality if y D x. The triangle inequality (b) is not completely obvious. The classical proof by Hilbert (who wrote it for the Hilbert metric) is given in Appendix B and a new proof is given in Section 7. y; z/. x; z/: aj Property (d) is easy to check, see Section 3 for more details. 13. To prove (f), we recall that we always assume ¤ Rn , and therefore @ ¤ ;.

Geometry 83 (2005), no. 1–2, 22–31. [14] P. de la Harpe, On Hilbert’s metric for simplices. In Geometric group theory (Graham A. ,) Vol. 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge 1993, 97–119. [15] D. Hilbert, Mathematische Probleme. Göttinger Nachrichten 1900, 253–297, reprinted in Archiv der Mathematik und Physik, 3d. , vol. 1 (1901) 44–63 and 213–237; English version, “Mathematical problems”, reprinted also in Bull. Amer. Math. Soc. ) 37 (2000), no. 4, 407–436.