Convexes hyperboliques et fonctions quasisymétriques
Publications Mathématiques de l'IHÉS, Tome 97 (2003), pp. 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},
     pages = {181--237},
     publisher = {Springer},
     volume = {97},
     year = {2003},
     doi = {10.1007/s10240-003-0012-4},
     mrnumber = {2010741},
     zbl = {1049.53027},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.1007/s10240-003-0012-4/}
}
TY  - JOUR
AU  - Benoist, Yves
TI  - Convexes hyperboliques et fonctions quasisymétriques
JO  - Publications Mathématiques de l'IHÉS
PY  - 2003
SP  - 181
EP  - 237
VL  - 97
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-003-0012-4/
DO  - 10.1007/s10240-003-0012-4
LA  - fr
ID  - PMIHES_2003__97__181_0
ER  - 
%0 Journal Article
%A Benoist, Yves
%T Convexes hyperboliques et fonctions quasisymétriques
%J Publications Mathématiques de l'IHÉS
%D 2003
%P 181-237
%V 97
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-003-0012-4/
%R 10.1007/s10240-003-0012-4
%G fr
%F PMIHES_2003__97__181_0
Benoist, Yves. Convexes hyperboliques et fonctions quasisymétriques. Publications Mathématiques de l'IHÉS, Tome 97 (2003), pp. 181-237. doi : 10.1007/s10240-003-0012-4. http://archive.numdam.org/articles/10.1007/s10240-003-0012-4/

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :