Free ebooks ==> www.ebook777.com IRMA Lectures in Mathematics and Theoretical Physics 22 Edited by Christian Kassel and Vladimir G Turaev Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg rue René Descartes 67084 Strasbourg Cedex France www.ebook777.com Free ebooks ==> www.ebook777.com IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France) The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines Previously published in this series: Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D Bertrand, B Enriquez, C Mitschi, C Sabbah and R Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de) Free ebooks ==> www.ebook777.com Handbook of Hilbert Geometry Athanase Papadopoulos Marc Troyanov Editors www.ebook777.com Free ebooks ==> www.ebook777.com Editors: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg Rue René Descartes 67084 Strasbourg Cedex France Marc Troyanov Section de mathématiques École Polytechnique Fédérale de Lausanne SMA-Station 1015 Lausanne Switzerland 2010 Mathematics Subject Classification: 01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07 Key words: Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism ISBN 978-3-03719-147-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks For any kind of use permission of the copyright owner must be obtained © 2014 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321 Free ebooks ==> www.ebook777.com Foreword The idea of collecting the surveys that constitute this Handbook came out of a desire to present in a single volume the foundations as well as the modern developments of Hilbert geometry In the last two decades the subject has grown into a very active field of research The Handbook will allow the student to learn this theory, to understand the questions and problems that it leads to, and to acquire the tools that can be used to approach them It should also be useful to the confirmed researcher and to the specialist, for it contains an exposition and an update of the most recent developments Thus, some chapters contain classical material, highlighting works of Beltrami, Klein, Hilbert, Berwald, Funk, Busemann, Benzécri and the other founders of the theory, and other chapters present recent developments Hilbert geometry can be regarded from different points of view: the calculus of variations, Finsler geometry, projective geometry, dynamical systems, etc At several places in this volume, the fruitful relations between Hilbert geometry and other subjects in mathematics are reported on These subjects include Teichmüller 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 All these important topics appear in one way or another in this book We would like to take this opportunity to thank Gérard Besson who helped us at an early stage of this project Our warm thanks go to Manfred Karbe from the European Mathematical Society who encouraged the project and to Irene Zimmermann for a very efficient collaboration and for the seriousness of her work This work was supported in part by the French research program ANR FINSLER and by the Swiss National Science Foundation Strasbourg and Lausanne, September 2014 www.ebook777.com Athanase Papadopoulos Marc Troyanov Free ebooks ==> www.ebook777.com Free ebooks ==> www.ebook777.com Contents Foreword v Introduction Part I Minkowski, Hilbert and Funk geometries Chapter Weak Minkowski spaces by Athanase Papadopoulos and Marc Troyanov 11 Chapter From Funk to Hilbert geometry by Athanase Papadopoulos and Marc Troyanov 33 Chapter Funk and Hilbert geometries from the Finslerian viewpoint by Marc Troyanov 69 Chapter On the Hilbert geometry of convex polytopes by Constantin Vernicos 111 Chapter The horofunction boundary and isometry group of the Hilbert geometry by Cormac Walsh 127 Chapter Characterizations of hyperbolic geometry among Hilbert geometries by Ren Guo 147 Part II Groups and dynamics in Hilbert geometry Chapter The geodesic flow of Finsler and Hilbert geometries by Mickaël Crampon 161 Chapter Around groups in Hilbert geometry by Ludovic Marquis 207 www.ebook777.com Free ebooks ==> www.ebook777.com viii Contents Chapter Dynamics of Hilbert nonexpansive maps by Anders Karlsson 263 Chapter 10 Birkhoff’s version of Hilbert’s metric and and its applications in analysis by Bas Lemmens and Roger Nussbaum 275 Part III Developments and applications Chapter 11 Convex real projective structures and Hilbert metrics by Inkang Kim and Athanase Papadopoulos 307 Chapter 12 Weil–Petersson Funk metric on Teichmüller space by Hideki Miyachi, Ken’ichi Ohshika and Sumio Yamada 339 Chapter 13 Funk and Hilbert geometries in spaces of constant curvature by Athanase Papadopoulos and Sumio Yamada 353 Part IV History of the subject Chapter 14 On the origin of Hilbert geometry by Marc Troyanov 383 Chapter 15 Hilbert’s fourth problem by Athanase Papadopoulos 391 Open problems 433 List of Contributors 443 Index 445 Free ebooks ==> www.ebook777.com Introduction The project of editing this Handbook arose from the observation that Hilbert geometry is today a very active field of research, and that no comprehensive reference exists for it, except for results which are spread in various papers and a few classical (and very inspiring) pages in books of Busemann We hope that this Handbook will serve as an introduction and a reference for both beginners and experts in the field Hilbert geometry is a natural geometry defined in an arbitrary convex subset of real affine space The notion of convex set is certainly one of the most basic notions in mathematics, and convexity is a rich theory, offering a large supply of refined concepts and deep results Besides being interesting in themselves, convex sets are ubiquitous; they are used in a number of areas of pure and applied mathematics, such as number theory, mathematical analysis, geometry, dynamical systems and optimization In 1894 Hilbert discovered how to associate a length to each segment in a convex set by way of an elementary geometric construction and using the cross ratio In fact, Hilbert defined a canonical metric in the relative interior of an arbitrary convex set Hilbert geometry is the geometric study of this canonical metric The special case where the convex set is a ball, or more generally an ellipsoid, gives the Beltrami–Klein model of hyperbolic geometry In this sense, Hilbert geometry is a generalization of hyperbolic geometry Hilbert geometry gives new insights into classical questions from convexity theory, and it also provides a rich class of examples of geometries that can be studied from the point of view of metric geometry or differential geometry (in particular Finsler geometry) Let us recall Hilbert’s construction The line joining two distinct points x and y in a bounded convex domain intersects the boundary of that domain in two other points p and q Assuming that y lies between x and p, the Hilbert distance from x to y is the logarithm of the cross ratio of these four points: p y x d.x; y/ D jx log jy pj jy pj jx qj : qj q This distance was defined in a letter to Klein written by Hilbert in 1894 It is a distance in the usual sense, and the relative interior of the convex set is a complete metric space for this distance The Hilbert metric is invariant under projective transformations and depends in a monotonic way on the domain: a larger domain induces a smaller Hilbert distance The Hilbert metric is projective in the sense that the straight www.ebook777.com Free ebooks ==> www.ebook777.com Introduction lines are geodesics In other words, d.x; y/ D d.x; z/ C d.z; y/ whenever z Œx; y Furthermore, if the convex domain is strictly convex, then the affine segment is the unique geodesic joining two points The fourth Hilbert problem asks for a description of all projective metrics in a convex region, that is, metrics for which the straight lines are geodesics At the beginning of the twentieth century, Hamel, who was a student of Hilbert, worked on this problem; he discovered new examples of projective metrics and found some deep results using differential calculus and the calculus of variations The subject of Finsler geometry gradually emerged as an independent topic, and at the end of the 1920s, Funk and Berwald gave a differential geometric characterization of Hilbert metrics among all Finsler metrics on a domain with a smooth and strongly convex boundary All these facts and several others which we describe below are reported on in this Handbook The deepest and most thorough studies in Hilbert geometry during the twentieth century are due to Busemann and his students and collaborators During the period from the 1940s to the 1990s this school investigated Hilbert geometry from the viewpoint of metric geometry A variety of questions regarding these metrics were studied, concerning their geodesics, their convexity theory (convexity of the distance function, of the spheres, etc.), their curvature, area, asymptotic geometry, limit cycles (horocycles) and limit spheres (horospheres), and several other features For instance, these authors gave several characterizations of the ellipsoid in terms of its Hilbert geometry They noticed (like Hilbert did before them) that the special case of the simplex is particularly interesting and they studied it in detail They worked on the metrical properties of the Hilbert metric as well as on the axiomatic theory, making relations with the axioms and the basic notions of Euclidean and of non-Euclidean geometries, in particular the theory of parallels They established relations between Hilbert geometry and other fields, including the foundations of mathematics, the calculus of variations, convex geometry, Minkowski geometry, geometric group theory and projective geometry They also developed the basics of the closely related Funk metric Busemann formulated and initiated the study of several problems of which he (and his collaborators) gave only partial solutions, and a large amount of the research on Hilbert geometry that was done after him is directly or indirectly inspired by his work Let us briefly mention two further important directions in which the subject developed in the last century In the late 1950s, Birkhoff found a new proof of the classical Perron–Frobenius theorem on eigenvectors of non-negative matrices based on the Hilbert metric in the positive cone This new proof brought a new point of view on the subject and initiated a rich generalization of Perron–Frobenius theory During the same period, Benzécri initiated the theory of divisible convex domains, that is, convex domains admitting a discrete cocompact group of projective transformations The quotient manifold or orbifold naturally carries a Finsler structure whose universal cover is a Hilbert geometry The reader will find more information on twentieth Free ebooks ==> www.ebook777.com List of Contributors Mickaël Crampon, Universidad de Santiago de Chile, Av El Belloto 3580, Estación Central, Santiago, Chile email: mickael.crampon@usach.cl Ren Guo, Department of Mathematics, Oregon State University, Corvallis, OR 973314605, U.S.A email: guore@math.oregonstate.edu Anders Karlsson, Section de mathématiques, Université de Genève, 2-4 Rue du Lièvre, Case Postale 64, 211 Genève 4, Switzerland email: Anders.Karlsson@unige.ch Inkang Kim, School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu Seoul, 130-722, Republic of Korea email: inkang@kias.re.kr Bas Lemmens, School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, UK email: B.Lemmens@kent.ac.uk Ludovic Marquis, Institut de Recherche Mathématique de Rennes, Rue Sine 190, 31062 Toulouse Cedex 4, France email: ludovic.marquis@univ-rennes1.fr Hideki Miyachi, Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan email: miyachi@math.sci.osaka-u.ac.jp Roger Nussbaum, Department of Mathematics, Rutgers, The State University Of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, U.S.A email: nussbaum@math.rutgers.edu Ken’ichi Ohshika, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, 560-0043, Osaka, Japan email: ohshika@math.sci.osaka-u.ac.jp Athanase Papadopoulos, Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, rue René Descartes, 67084 Strasbourg Cedex, France email: papadopoulos@math.u-strasbg.fr Marc Troyanov, Section de mathématiques, École Polytechnique Fédérale de Lausanne, SMA–Station 8, 1015 Lausanne, Switzerland email: marc.troyanov@epfl.ch www.ebook777.com Free ebooks ==> www.ebook777.com 444 List of Contributors Constantin Vernicos, Institut de mathématique et de modélisation de Montpellier, Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34395 Montpellier Cedex, France email: Constantin.Vernicos@um2.fr Cormac Walsh, INRIA Saclay & Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France e-mail: cormac.walsh@inria.fr Sumio Yamada, Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima, Tokyo 171-8588, Japan email: yamada@math.gakushuin.ac.jp Free ebooks ==> www.ebook777.com Index absolutely continuous measure, 193 area centro-projective, 439 affine chart, 209 affine metric, 316, 318 affine midpoint, 21 affine normal, 316 affine patch, 58, 310 affine ratio, 37, 61 affine sphere, 316 affinely convex, 52 Alexandrov’s theorem, 202 algebraic horosphere, 216 almost-geodesic, 130 ˛-conformal density, 332 amenable Hilbert metric, 438 Anosov decomposition, 189 Anosov flow, 189, 327 approximability convex body, 440 arithmetic symmetrization metric, 34 asymptotic convex hypersurface, 231 asymptotically harmonic, 195 attractor, 298 automorphism group, 166 axioms continuity, 385 Hilbert, 385 incidence, 385 order, 385 forward, 43 left, 43 right, 43 Beltrami Theorem, 99 Beltrami, Eugenio, 393 Beltrami–Cayley–Klein model, 59,308, 310, 394 Benzécri, Jean-Paul, 308, 312 Berwald metric, 74 Berwald, Ludwig, 405 ˇ-convexity, 186 bi-, 213 Binet–Legendre metric, 436 biproximal, 213 Birkhoff theorem, 282 Birkhoff version of Hilbert’s metric, 276 Blaschke connection, 316 bottom of the spectrum, 437 boundary horofunction, 130 Bowen–Margulis measure, 194, 332 bulging deformation, 324 Busemann G-space, 27 Busemann cocycle, 330 Busemann function, 127, 178, 215, 266, 330 Busemann point, 131 Busemann volume, 210, 311, 438 Busemann zero curvature, 28 Busemann, Herbert, 392, 401, 408, 409 Busemann–Hausdorff volume, 166 backward complete Finsler manifold, 73 backward complete metric, 47 backward open ball, 43 backward proper metric, 47 backward sphere, 43 ball backward, 43 Cayley, Arthur, 393 centered John ellipsoid, 20 centro-projective area, 439 character variety, 321 characteristic function, 143 closed orbit, 170 collineation, 115, 140 www.ebook777.com Free ebooks ==> www.ebook777.com 446 cometric, 172 compactification horofunction, 130 complete Finsler manifold, 73 cone, 277, 317 above , 212 ice-cream, 143 Lorentz, 143 normal, 277 symmetric, 143 total, 277 cone linear mapping, 283 conformal density, 332 conformal dimension, 437 conical face, 113 conical flag, 114 connection, 173 convex, 209 with C boundary, 211 affinely, 52 ˇ-convex, 246 class C 1C" , 246 cone, 222 divisible, 208 geodesically, 52 Menger, 411 projective structure, 167, 198 properly, 163, 209 quasi-divisible, 208 round, 211 strictly, 211 strongly, 246 convex body, 209 approximability, 440 convex cone, 222, 317 decomposable, 222 indecomposable, 222 sharp, 217 convex domain proper, 35 sharp, 60 convex hull, 218 convex hypersurface, 231 Index asymptotic, 231 convex polytope, 42 convex projective structure, 308 deformation space, 321 Goldman parameters, 321 marked, 313 convex real projective manifold, 313 convex set divisible, 313 hyperbolic space, 358 projective automorphism group, 310 projective space, 310 properly, 310 sphere, 358 symmetric, 223 convex structure dual, 326 convex structures deformation space, 313 moduli space, 313 C 1C regularity, 186 Crofton formula, 115 cross ratio Euclidean, 364 hyperbolic, 364 spherical, 365 current, 329 curvature of a Finsler metric, 175 of Hilbert geometry, 177 of the boundary, 202 pseudo-Gaussian, 439 curvature zero (Busemann), 28 decomposable convex cone, 222 Dehn invariant, 404 Dehn, Max, 404 Denjoy–Wolff theorem, 298 density function, 154 Desargues’ property, 27 Desarguesian space, 27, 411 detour cost, 131 detour metric, 132 Free ebooks ==> www.ebook777.com Index developing map, 198 divisible, 167, 226, 308 convex set, 313 divisible convex, 208 division ratio, 37, 61 dominate, 277 dual, 217 dual convex structure, 326 dual wedge, 277 dyadic number, 21 earthquake deformation, 324 Ehresmann, Charles, 308, 311 eigenvalues, 293 ellipsoid, 220 centered John, 20 John, 20 elliptic, 214 energy of a curve, 84 entropy measure-theoretical, 193 topological, 193, 198, 328 volume, 197, 331 epigraph topology, 134 equiaffine metric, 316 ergodic measure, 192 Euclidean Jordan algebra, 288 Euler lemma, 25 Euler–Lagrange equations, 85 exposed face, 49 exposed point, 49 extreme set, 135 face, 113, 214, 278 dimension, 214 open, 214 support, 214 face (of a convex set), 48 Fenchel–Nielsen parameters, 320 Finsler cometric, 172 Finsler metric, 163, 310 for Hilbert geometry, 165 regular, 164 Ricci curvature, 96 447 Finsler structure, 72, 151 Euclidean tautological, 362 hyperbolic tautological, 362 spherical tautological, 363 tautological, 34 Finsler, Paul, 404 fixed point, 296 fixed point theorem, 264 flag curvature, 96 Hilbert metric, 100 flip map, 198 flow Anosov, 327 topologically mixing, 328 foot, 53 formal Christoffel symbols, 86 forward bounded metric, 47 forward boundedly compact metric, 47 forward Cauchy sequence, 47 forward open ball, 43 forward proper metric, 47 forward sphere, 43 functional Minkowski, 19 fundamental tensor (of a Finsler metric), 83 Funk metric, 36, 76, 128, 280, 340 hyperbolic, 359 relative, 58 reverse, 41 spherical, 360 Funk, Paul, 404 gauge, 136 gauge-preserving map, 144 gauge-reversing map, 142 Genocchi, Angelo, 394 geodesic, 85 geodesic current, 329 www.ebook777.com Free ebooks ==> www.ebook777.com 448 Index geodesic flow, 327 for regular Finsler metrics, 164 for Hilbert geometries, 169 geodesic metric space, 210, 328 Gromov-hyperbolic, 237 geodesically convex, 51, 52 geodesics, 164 in Hilbert geometry, 166 geometry Hilbert, 148 non-Archimedean, 405 non-Arguesian, 405 non-Euclidean, 405 non-Legendrian, 406 non-Pascalian, 405 semi-Euclidean, 406 Goldman parameters, 321 Gromov product, 269, 292 Gromov-hyperbolic, 237 group, 168 space, 168 G-space, 27, 410 Hilbert, David, 383 Hitchin component, 308, 325 Hoüel, Guillaume Jules, 395 holonomy map, 198 homogeneous Hilbert geometry, 166 homogeneous tangent bundle, 165 homography, 115 horoball, 129, 132, 178, 266 horofunction, 127, 130, 266 horofunction boundary, 130 horofunction compactification, 130 horosphere, 178, 215 algebraic, 216 hyperbolic, 214 isometry, 168 orbit, 186 space, 165 hyperbolic metric, 307 hyperbolic metric space, 328 hypermetric, 417 hyperplane supporting, 211 Hölder cocycle, 331 Hölder regularity, 186 Hamel potential, 93 Hamel, Georg, 404 harmonic measure, 195 harmonic symmetrization, 81 Hessian, 164, 202 Hilbert 1-form, 171 Hilbert form, 92 Hilbert measure, 155 Hilbert metric, 58, 128, 147, 148, 163, 209, 276, 310 amenable, 438 Birkhoff version, 278 Finsler norm, 310 flag curvature, 100 hyperbolic, 366 spherical, 367 Hilbert Problem IV, 422 Hilbert volume, 311 ice-cream cone, 143 ideal triangle, 155 idempotents, 293 complete system of orthogonal, 293 primitive, 293 indecomposable, 222 indecomposable convex cone, 222 indicatrix, 17, 73 invariant measure, 192 isometry group, 166 Jacobi field, 175 Hilbert geometry, 177 Jacobi operator, 175 Hilbert geometry, 177 John ellipsoid, 20, 218, 247 Klein model, 59, 81, 394 Krein–Milman theorem, 288 Laguerre, Edmond, 384 Free ebooks ==> www.ebook777.com Index metric left open ball, 43 affine, 316, 318 Legendre transform, 172 arithmetic symmetrization, 34 on HM , 172 backward complete, 47 Legendre–Clebsch condition, 83 backward proper, 47 Legendre–Fenchel transform, 137 Binet–Legendre, 436 Lemma Blaschke, 316 Euler, 25 detour, 132 Schur, 97 equiaffine, 316 Selberg, 239 forward bounded, 47 Zassenhaus–Kazhdan–Margulis, 251 forward boundedly compact, 47 length forward proper, 47 translation, 330 Funk, 36, 128, 340 length spectrum, 326 hyperbolic, 359 length spectrum metric, 61 spherical, 360 limit set, 296 geodesically convex, 51 Liouville measure, 171, 193 Gromov hyperbolic, 328 locally finite fundamental domain, 233 Hilbert, 128, 147, 209 locally Ptolemaic, 150 hyperbolic, 366 Lorentz cone, 143, 287 spherical, 367 Lyapunov decomposition, 188 hyperbolic, 307 Lyapunov exponents, 188 length spectrum, 61 of a periodic orbit, 188 max-symmetrization, 61 marked convex projective structure, 242, projective, 36, 70, 249 313 projectively flat, 13 marked length spectrum, 326 relative Funk, 58 marked projective structure, 242 reverse Funk, 41, 61 convex, 242, 313 reverse-Funk, 129 equivalent, 242 Sasaki, 176 marking, 313 Thurston, 33, 349 max-symmetrization variational formulation of metric, 61 Teichmüller, 348 Mazur–Ulam Theorem, 20 weak, 12, 33 measure weak Minkowski, 13 Bowen–Margulis, 332 Weil–Petersson, 341 Hilbert, 155 completion, 342 Patterson–Sullivan, 331 Weil–Petersson Funk, 60, 344 Sinai–Ruelle–Bowen, 435 metric space measure of maximal entropy, 194 geodesic, 210 measure-theoretical entropy, 193 proper, 210, 266 Menelaus, 372 uniquely geodesic, 210 Menelaus Theorem, 373 midpoint property, 21 Menger convexity, 411 minimal displacement, 265 www.ebook777.com 449 Free ebooks ==> www.ebook777.com 450 Index Minkowski functional, 19, 38 Funk metric Euclidean, 362 hyperbolic, 362 spherical, 363 Minkowski norm strictly convex, 25 strongly convex, 25 Minkowski rank, 434 mixing measure, 192 topologically, 190 natural, 249 nearest point, 53 non-Archimedian field, 404, 405 non-Arguesian geometry, 405 non-Euclidean geometry, 405 non-Legendrian geometry, 406 non-Pascalian field, 406 non-Pascalian geometry, 405, 406 nonexpansive map, 265, 433 orbit, 170 ordered geometry, 385 orthogonal representation, 226 osculating Riemannian metric, 94 Painlevé–Kuratowski topology, 132 pair of pants, 314 decomposition geodesic, 315 topological, 315 parabolic, 214 parallel transport, 174 parts, 278 parts of boundary, 132 Pasch, Moritz, 385 patch affine, 310 Patterson–Sullivan geodesic current, 333 Patterson–Sullivan measure, 331 period, 296 cocycle, 331 periodic point, 296 perpendicular, 55 Perron–Frobenius operator, 284 Perron–Frobenius theory, 264 perspectivity, 148 physical measure, 200 planar automorphism, 219 Pogorelov, Alexei Vasil’evich, 399, 409, 413 polytope, 113 positive hyperbolic transformation, 322 positively biproximal, 213 positively proximal, 213 positively semi-proximal, 213 power, 213 primitive element, 170 Problem Hilbert fourth, 422 product, 222 projective metric, 249 natural, 249 volume, 249 projective center, 152 projective diameter, 282 projective line, 309 projective manifold, 242, 311 convex, 313 strictly, 313 isomorphism, 311 projective metric, 36, 70, 249 natural, 249 projective structure, 311 convex, 308, 312 flat, 242 strictly convex, 308 projective volume, 249 natural, 249 projective weak metric, 13 projectively equivalent, 242 projectively flat metric, 13 Free ebooks ==> www.ebook777.com Index projectively flat Finsler manifold, 88 proper convex, 310 proper convex domain, 35 proper metric space, 210 properly convex, 163, 209 indecomposable, 222 product, 222 properly convex set, 310 proximal, 213, 225, 233 action, 233 bi, 213 group, 233 positively, 213 semi, 213 pseudo-Gaussian curvature, 439 Ptolemaic, 150 Ptolemaic inequality, 150 quadratic representation, 294 quasi-divisible, 226, 208 quasi-hyperbolic, 214 rank Hilbert geometry, 434 Randers metric, 73 Rayleigh quotient, 437 reductive, 234 regular orbit, 188 regular Finsler metric, 164 relative Funk metric, 58 representation equal mod 2, 226 highest restricted weight, 225 highest weight, 225 lattice of restricted weights, 225 orthogonal, 226 proximal, 225 restricted weight, 225 symplectic, 226 weight, 225 reverse Funk metric, 41, 61, 129 reversible norm, 310 reversible tautological 451 Finsler structure, 34 Ricci curvature of a Finsler metric, 96 right open ball, 43 round, 211 Ruelle’s inequality, 198, 201 Sasaki metric, 176 Schur Lemma, 97 Schwarzian derivative, 106 Selberg lemma, 239 semi-proximal, 213 semi-simple k-semi-simple, 213 S1 -semi-simple, 213 semicontraction, 265 sharp convex cone, 217 sharp convex domain, 60 simplicial cone, 289 Sinai–Ruelle–Bowen measures, 200 space G-, 410 Busemann G-, 27 Desarguesian, 27, 411 spectral radius, 213 spectrum, 293 sphere backward, 43 forward, 43 spherical metric projective model, 88 spray, 87 stable set, 179 standard positive cone, 287 -map, 143 strictly convex, 211 strictly convex Minkowski norm, 25 strictly convex projective structure, 308 strictly convex set, 49 strongly convex, 246 strongly convex Finsler metric, 83 strongly convex Minkowski norm, 25 subhomogeneous map, 282 support hyperplane, 38, 211 www.ebook777.com Free ebooks ==> www.ebook777.com 452 support of a face, 214 supporting functional, 38 symmetric cone, 143 symmetric convex set, 223 symmetric Hilbert geometry, 166 symplectic representation, 226 tautological Finsler structure, 34, 56 tautological Finsler structure Euclidean, 362 hyperbolic, 362 spherical, 363 Teichmüller component, 308 Teichmüller space, 341 tensor, 83 Theorem Beltrami, 99 Birkhoff, 282 Ceva, 63 Denjoy–Wolff, 298 Desargues, 405 Funk–Berwald, 70 Krein–Milman, 49, 288 Mazur–Ulam, 20 Menelaus, 61, 373 Pascal, 405 sector figure, 372 Thurston metric, 33, 349 topological entropy, 193, 198, 328 topologically mixing, 190, 328 transitive, 190 topology epigraph, 134 Painlevé–Kuratowski, 132 transitive flow, 190 translation distance, 168 translation length, 214, 265, Index 330 translation number, 265 triangle ı-thin, 328 unipotent, 213 uniquely geodesic metric space, 210 unstable set, 179 upper asymptotic volume, 116 variation norm, 289 variational formulation of Teichmüller metric, 348 variational principle, 194 Veronese, Guiseppe, 405 vertex, 113 Vinberg’s -map, 143 volume, 165 Busemann, 210, 438 Busemann–Hausdorff, 166 projective, 249 volume entropy, 197, 331 von Staudt, Karl Georg Christian, 383 weak Finsler structure, 72 weak metric, 12, 33 geodesic, 51 projective, 13 weak Minkowski metric, 13 wedge, 277 Weil–Petersson Funk metric, 60, 344 Weil–Petersson metric, 341 Zariski-dense, 168 Zermelo transform, 75 zero curvature Busemann, 28 zonoid, 417 zonotope, 417 ... domain in R2 which is symmetric around the origin and has area greater than four contains at least one non-zero point with integer coordinates One step in Minkowski’s proof amounts to considering... distance in the domain (see Definition 2.1 in Chapter of this volume [25]) It is also in the spirit of the following definition of Busemann ([4], Definition 17.1): A metric d.x; y/ in Rn is Minkowskian... from the point of view of metric geometry or differential geometry (in particular Finsler geometry) Let us recall Hilbert’s construction The line joining two distinct points x and y in a bounded