Strict convexity in M-spaces
type: Item
instance of: scholarly article
title: Strict convexity in M-spaces
zbMATH Open document ID: 0684.54019
DOI: 10.1007/BF01224232
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: A metric space (X,d) is called an M-space if for every \(p,q\in X\) with \(d(p,q)=\lambda\), and every \(\alpha\in [0,\lambda]\), there is a unique \(r_{\alpha}\in X\) such that \(d(p,r_{\alpha})=\alpha\) and \(d(r_{\alpha},q)=\lambda -\alpha\). The authors prove that the strict convexity of an M-space (X,d) is equivalent to the fact that every proximinal convex subset of X is a Chebyshev set and, also, to a monotonicity property of d. These characterizations remain valid in any complete, convex metric space. Finally, the concept of local M-space is introduced and, for this structure, local versions of the above results are proved.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 4121001
zbMATH Keywords: monotonicity of the distance function
zbMATH Keywords: M-space
zbMATH Keywords: strict convexity
zbMATH Keywords: proximinal convex subset
zbMATH Keywords: Chebyshev set
author: Raymond W. Freese
author: Grattan P. Murphy
author: Edward Z. Andalafte
MaRDI profile type: Publication
cites work: Criteria for Unique Metric Lines in Banach Spaces
cites work: Characterization of inner product spaces in V- and L-spaces
cites work: Q5818996
cites work: Q5719717
cites work: Criteria for unique metric lines
cites work: Strictly convex spheres in V-spaces
cites work: Best Approximation in Metric Spaces
full work available at URL: https://doi.org/10.1007/bf01224232
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description: scientific article; zbMATH DE number 4121001