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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)
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