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

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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.

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     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://geodesic.mathdoc.fr/articles/10.1007/s10240-003-0012-4/}
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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

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