Convexes hyperboliques et fonctions quasisymétriques
Publications Mathématiques de l'IHÉS, Volume 97 (2003), p. 181-237

Every bounded convex open set Ω of 𝐑 m is endowed with its Hilbert metric d Ω . We give a necessary and sufficient condition, called quasisymmetric convexity, for this metric space to be hyperbolic. As a corollary, when the boundary is real analytic, Ω is always hyperbolic. In dimension 2, this condition is: in affine coordinates, the boundary Ω is locally the graph of a C 1 strictly convex function whose derivative is quasisymmetric.

@article{PMIHES_2003__97__181_0,
     author = {Benoist, Yves},
     title = {Convexes hyperboliques et fonctions quasisym\'etriques},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {97},
     year = {2003},
     pages = {181-237},
     doi = {10.1007/s10240-003-0012-4},
     zbl = {1049.53027},
     mrnumber = {2010741},
     language = {fr},
     url = {http://www.numdam.org/item/PMIHES_2003__97__181_0}
}
Benoist, Yves. Convexes hyperboliques et fonctions quasisymétriques. Publications Mathématiques de l'IHÉS, Volume 97 (2003) pp. 181-237. doi : 10.1007/s10240-003-0012-4. http://www.numdam.org/item/PMIHES_2003__97__181_0/

1. L. Ahlfors, Lectures on quasiconformal mappings, Wadthworth (1966). | MR 200442 | Zbl 0138.06002

2. L. Ahlfors, A. Beurling, The boundary correspondance under quasiconformal mappings, Acta Math. 96 (1956) 125-142. | MR 86869 | Zbl 0072.29602

3. Y. Benoist, Convexes divisibles I, preprint (2001) et Comp. Rend. Ac. Sc. 332 (2001), 387-390. | MR 1826621 | Zbl 1010.37014

4. J. P. Benzecri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. Fr. 88 (1960), 229-332. | Numdam | MR 124005 | Zbl 0098.35204

5. E. Bierstone, P. Milman, Semianalytic and subanalytics sets, Publ. IHES 67 (1988), 5-42. | Numdam | MR 972342 | Zbl 0674.32002

6. J. Faraut, A. Koranyi, Analysis on symmetric cones, Oxford Math. Mono. (1994). | MR 1446489 | Zbl 0841.43002

7. E. Ghys, P. De La Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, PM 83, Birkhäuser (1990). | Zbl 0731.20025

8. W. Goldman, Projective Geometry, Notes de cours a Maryland (1988).

9. M. Gromov, Hyperbolic groups, in “Essays in group theory”, MSRI Publ. 8 (1987), 75-263. | Zbl 0634.20015

10. F. John, Extremum problem with inequalities as subsidiary conditions, Courant anniversary volume (1948), 187-204. | MR 30135 | Zbl 0034.10503

11. A. Karlsson, G. Noskov, The Hilbert metric and Gromov hyperbolicity, l'Ens. Math. 48 (2002), 73-89. | Zbl 1046.53026

12. S. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math. 1 (1983), 193-260. | MR 782608 | Zbl 0518.46018