Hyperbolic geometry and disks
type: Item
instance of: scholarly article
title: Hyperbolic geometry and disks
zbMATH Open document ID: 0943.30028
DOI: 10.1016/S0377-0427(99)00016-3
published in: Journal of Computational and Applied Mathematics
publication date: 2000-06-29T00:00:00Z
review text: The paper contains two characterizations for a Euclidean disk in terms of hyperbolic geometry. In order to state the first one we need a couple of simple definitions. A plane domain \(G\) is circularly accessible at a point \(w\in\partial G\) if there exists a disk \(B_w\subset G\) with \(w\in \partial B_w\). For a bounded domain \(G\) the points \(w_1,w_2\) are diametral points if \(w_1,w_2\in \partial G\) and \(|w_1-w_2|=\text{diam}(G)\). The first characterization is: A bounded simply connected domain \(D\) is a disk if and only if \(D\) is circularly accessible at a pair of diametral points and \(h_D(z_1,z_2)\leq j_D(z_1,z_2)\), for \(z_1,z_2\in D\). Here \(h_D\) denotes the hyperbolic distance in \(D\) and \(j_D\) is another metric in \(D\) that involves only Euclidean quantities (distances between \(z_1, z_2\) and \(\partial D\)). The second characterization involves convexity. A simply connected domain \(D\subset \bar{\mathbb C}\) with \(\bar{D}\neq \bar{\mathbb C}\) is a disk or half-plane if it is a hyperbolically convex subset of each simply connected domain which contains it. This result was originally proved by \textit{B. Brown Flinn} [Indiana Univ. Math. J. 32, 831-841 (1983; 522.30005)].
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 1334869
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: quasidisk
zbMATH Keywords: convexity
author: F. W. Gehring
author: Kari Hag
reviewed by: Dimitrios Betsakos
MaRDI profile type: Publication
cites work: Q4139907
cites work: Q3958719
cites work: Q4263243
cites work: Quasiconformally homogeneous domains
cites work: Q3869782
cites work: On an inequality for the hyperbolic measure and its applications in the theory of functions
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description: scientific article; zbMATH DE number 1334869