Uniform convexity, normal structure and the fixed point property of metric spaces
type: Item
instance of: scholarly article
title: Uniform convexity, normal structure and the fixed point property of metric spaces
zbMATH Open document ID: 1331.54060
author: N. E. Zubov
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: A metric space \(X\) is said to have the fixed point property (FPP) if the existence of a bounded orbit under a group which acts by isometries on \(X\) implies the presence of a fixed point. The paper introduces a definition of uniform convexity that is applicable beyond geodesic metric spaces. It guarantees a complete metric space satisfies the FPP, as the circumcenter (known to be unique) of the bounded orbit is fixed. The same result is proved for a more general uniform convexity, called \textit{uniform convexity in the wider sense}, which generalizes the related notion in Banach spaces to metric ones. Both uniform convexity and uniform convexity in the wider sense imply compactness and normal structure in the family of admissible subsets of \(X\), properties which are shown to secure the FPP. The author also weakens uniform convexity to the notion of \textit{uniform pseudo-convexity} and shows it implies the FPP as well.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6522022
zbMATH Keywords: fixed point property
zbMATH Keywords: uniform convexity
zbMATH Keywords: normal structure
zbMATH Keywords: isometric action
zbMATH Keywords: bounded orbit
zbMATH Keywords: geodesic metric space
reviewed by: Luis Hernández Corbato
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.039
OpenAlex ID: W351822810
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DOI: 10.1016/J.TOPOL.2015.05.039
description: scientific article; zbMATH DE number 6522022