Every bounded convex open set of is endowed with its Hilbert metric . 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, is always hyperbolic. In dimension 2, this condition is: in affine coordinates, the boundary is locally the graph of a C 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://www.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://www.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 -
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://www.numdam.org/articles/10.1007/s10240-003-0012-4/
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