By Athanase Papadapoulous, Marc Troyanov

This quantity offers surveys, written via specialists within the box, on a number of classical and glossy features of Hilbert geometry. They think a number of issues of view: Finsler geometry, calculus of diversifications, projective geometry, dynamical structures, and others. a few fruitful family members among Hilbert geometry and different topics in arithmetic are emphasised, together with Teichmüller areas, convexity thought, Perron-Frobenius idea, illustration thought, partial differential equations, coarse geometry, ergodic thought, algebraic teams, Coxeter teams, geometric crew idea, Lie teams and discrete staff activities. This ebook is addressed to either scholars who are looking to examine the idea and researchers during this region.

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Q/. q 0 ; y 0 / 2 Œa; b. p 0 ; ı 0 /. p 0 ; ı 0 / is not geodesically convex. 9. 8 and the corresponding result concerning the geodesic segments of a Minkowski metric on Rn : if the unit ball of a Minkowski metric is strictly convex, then the only geodesic segments of this metric are the affine segments. 8 Nearest points in Funk geometry Let Rn be a convex set equipped with its Funk metric F . 1. Let x be a point in and let A be a subset of . x; z/: z2A 54 Athanase Papadopoulos and Marc Troyanov It is clear from the continuity of the function y 7!

2. x; x/ D 0 for all x; y 2 . y; z/ for all x; y; z 2 . y; z/ whenever z is a point on the affine segment Œx; y. y; x/ in general. x; y/ > 0, if and only if the domain is bounded. (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.

0. x; R/ is the Euclidean ball of radius R centered at the origin. It is easy to check that F R converges uniformly as R ! 1. The continuity of F follows to F on every compact subset of therefore from the proof for bounded convex domains. 3. Let ı be a weak metric defined on a set X . xk ; xm / < 1 where the supremum is taken over all pairs k, m satisfying m k. Note that this definition corresponds to the usual notion in the case of a usual (symmetric) metric space. X; ı/ is forward proper, or forward boundedly compact if every forward bounded sequence has a converging subsequence.