The Hilbert metric and Gromov hyperbolicity.
type: Item
instance of: scholarly article
title: The Hilbert metric and Gromov hyperbolicity.
zbMATH Open document ID: 1046.53026
author: Anders Karlsson
author: Gennady A. Noskov
published in: L'Enseignement Mathématique. 2e Série
publication date: 2002-01-01T00:00:00Z
review text: Given a convex domain \(D\) in the Euclidean space, for any pair of points \(x\) and \(y\) in \(D\) let us denote by \(x^\prime\) and \(y^\prime\) the intersections of the line through \(x\) and \(y\) with the boundary of \(D\) closest to \(x\) and \(y\). The logarithm of the crossratio of these four points defines the Hilbert metric on \(D\): \(h(x,y) = \log \frac{| yx^\prime| | xy^\prime| }{| xx^\prime| | yy^\prime| }\). Such a metric in the unit disk gives a model of hyperbolic space. It is said that a domain has the intersecting chords property if there is a positive constant \(M\) such that for any intersecting chords \(c_1\) and \(c_2\) in \(D\) we have \(M^{-1} \leq \frac{l_1 l_1^\prime}{l_2 l_2^\prime} \leq M\) where the intersection point divides the chord \(c_j\) into segments of the lengths \(l_j\) and \(l_j^\prime\), \(j=1,2\). The authors prove that if a convex domain \(D\) has the intersections chords property then this domain endowed with the Hilbert metric is Gromov hyperbolic.
reviewed by: I. A. Taimanov
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 2028619
zbMATH Keywords: convex domains
zbMATH Keywords: Hilbert metric
zbMATH Keywords: Gromov hyperbolicity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2028619