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Hyperbolic convex sets and quasisymmetric functions. (Convexes hyperboliques et fonctions quasisymétriques.) (French) Zbl 1049.53027
Consider a bounded convex open set $$\Omega\subset{\mathbb R}^m$$ endowed with its Hilbert metric $$d_{\Omega}$$. The aim of the article is a characterisation of (Gromov-)hyperbolicity of $$\Omega$$. To this end, the author introduces the notion of quasisymmetric convexity. In dimension two, this notion says that, with respect to affine coordinates, the boundary $$\partial\Omega$$ is the graph of a strictly convex $$C^1-$$function whose derivative is quasisymmetric. The main result of the work is the proof of the equivalence of hyperbolicity and quasisymmetric convexity of $$\Omega$$. As a corollary, $$\Omega$$ is always hyperbolic if $$\partial\Omega$$ is analytic. This work extends previous results of the author. In [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 5, 387–390 (2001; Zbl 1010.37014)] it is shown that hyperbolicity implies strict convexity of $$\Omega$$, and that such convex sets $$\Omega$$ exist whose boundary is nowhere of class $$C^2$$.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
##### Keywords:
hyperbolic convex sets; quasisymmetric functions
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##### References:
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