The relative Chebyshev centers of finite sets in geodesic spaces
type: Item
instance of: scholarly article
title: The relative Chebyshev centers of finite sets in geodesic spaces
zbMATH Open document ID: 1169.53058
DOI: 10.3103/S1066369X08040075
published in: Russian Mathematics
publication date: 2009-03-26T00:00:00Z
review text: A metric space \(X\) is a \textit{geodesic~space} if any two points \(x,y\) of \(X\) may be joined by a continuous curve whose length equals the distance \(d(x,y)\). If \(M\) is a bounded subset of \(X\), then its \textit{Chebychev~radius} \(R(M)\) is the infimum of the radii \(R\) of all metric balls \(B(x,R)\) that contain \(M\), where \(x\) ranges over all points of \(X\). A point \(x\) of \(X\) is a \textit{Chebychev~center} of \(M\) if \(M\) is contained in \(B(x,R(M))\). Let \(Z(M)\) denote the set of all Chebychev centers of \(M\). One may also define relative versions \(R_{W}(M)\) and \(Z_{W}(M)\), where \(W\) is a bounded subset of \(M\) and the point \(x\) in the definitions above ranges over \(W\) instead of \(X\). The collection of bounded subsets of \(X\) is a metric space with respect to the Hausdorff metric. In this paper the author considers finite or compact subsets \(M\) of \(X\) and describes how \(R(M)\) and \(Z(M)\) vary as \(M\) varies. There are three main results, two of which assume that the metric space \(X\) has convex metric balls and unique shortest geodesic segments between any two points. These extra properties are satisfied by complete, simply connected, Riemannian manifolds of nonpositive sectional curvature. The main results are somewhat complicated to state, and we omit a precise formulation.
reviewed by: Patrick Eberlein
Mathematics Subject Classification ID: 53C70
zbMATH DE Number: 5533470
zbMATH Keywords: geodesic metric space
zbMATH Keywords: convexity of metric balls
zbMATH Keywords: finite or bounded subsets
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: relative Chebychev radius
zbMATH Keywords: relative Chebychev center
author: Evgenii N. Sosov
MaRDI profile type: Publication
cites work: Q3843864
cites work: On relative Hausdorff measures of noncompactness and relative Chebyshev radii in Banach spaces
cites work: Existence of Chebyshev Centers, Best n-Nets and Best Compact Approximants
cites work: On Metric Spaces with Uniform Normal Structure
cites work: Metric spaces, convexity and nonpositive curvature
cites work: Extension of Lipschitz maps into 3-manifolds
full work available at URL: s1066369x08040075
OpenAlex ID: W1986941081
description: scientific article; zbMATH DE number 5533470