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Convexes hyperboliques et fonctions quasisymétriques

Yves Benoist (2003)

Publications Mathématiques de l'IHÉS

Every bounded convex open set Ω of Rm is endowed with its Hilbert metric dΩ. 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 C1 strictly convex function whose derivative is quasisymmetric.

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