The Apollonian inner metric
type: Item
instance of: scholarly article
title: The Apollonian inner metric
zbMATH Open document ID: 1146.30025
published in: Communications in Analysis and Geometry
publication date: 2005-03-21T00:00:00Z
review text: Let \(G\) be a domain in \(\mathbb R^n\), where \(G\) is a proper subset of \(\mathbb R^n\). For \(x,y \in G\), the Apollonian metric is defined by \(\alpha_G(x,y)=\sup_{a,b\in \partial G}\log\frac{| a-x |}{| a-y |}\frac{| b-y |}{| b-x |}\). This metric is known for analysts as the Apollonian metric, introduced and first studied by \textit{A. F. Beardon} [in: Quasiconformal mappings and analysis, New York, Springer, 91--108 (1998; Zbl 0890.30030)], and for geometers as the Barbilian metric. To any metric \(d\) on \(G\) one may associate an {inner metric} \(\tilde{d}\), with \(\tilde{d}(x,y)\) being defined as the infimum of lengths of curves joining \(x\) and \(y\) inside \(G\). Some highlights of this study of the inner metric \(\tilde{\alpha}_G\) associated with the Apollonian metric \(\alpha_G\): (1) \(\tilde{\alpha}_G\) is a pseudometric for every proper domain of \(\mathbb R^n\), and a metric if and only if the complement \(G^c\) of \(G\) is not contained in an \((n-2)\)-dimensional plane; (2) if \(G^c\) is not contained in a hyperplane, then there exists a geodesic connecting any two points in \(G\).
reviewed by: Victor V. Pambuccian
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 2148048
zbMATH Keywords: Apollonian metric
zbMATH Keywords: geodesics
author: Peter Hästö
MaRDI profile type: Publication
DOI: 10.4310/CAG.2004.V12.N4.A7
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description: scientific article; zbMATH DE number 2148048