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Convexes hyperboliques et fonctions quasisymétriques

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Abstract

Hyperbolic convex sets and quasisymmetric functions

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 C1 strictly convex function whose derivative is quasisymmetric.

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Authors and Affiliations

  1. Ecole Normale Supérieure-CNRS, 45 rue d’Ulm, 75230, Paris, France

    Yves Benoist

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  1. Yves Benoist
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Correspondence to Yves Benoist.

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Benoist, Y. Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. 97, 181–237 (2003). https://doi.org/10.1007/s10240-003-0012-4

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  • DOI: https://doi.org/10.1007/s10240-003-0012-4

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