Hyperbolic convex sets and quasisymmetric functions
type: Item
instance of: scholarly article
title: Hyperbolic convex sets and quasisymmetric functions
zbMATH Open document ID: 1049.53027
author: Yves Benoist
published in: Publications Mathématiques
publication date: 2004-03-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104189
review text: 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 \(C^2\).
reviewed by: Ruth Kellerhals
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 2053221
zbMATH Keywords: hyperbolic convex sets
zbMATH Keywords: quasisymmetric functions
MaRDI profile type: Publication
OpenAlex ID: W2029673425
cites work: Q5512337
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DOI: 10.1007/S10240-003-0012-4
description: scientific article; zbMATH DE number 2053221