Lambda-hyperconvexity in metric spaces
type: Item
instance of: scholarly article
title: Lambda-hyperconvexity in metric spaces
zbMATH Open document ID: 0969.47036
DOI: 10.1016/S0362-546X(99)00171-6
author: R. Smith
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-09-23T00:00:00Z
review text: It is shown that every non-expansive map \(f:M\to M\) on a bounded \(\lambda\)-hyperconvex metric space \(M\), \(\lambda<2\), has a fixed point. A metric space \(M\) is defined to be \(\lambda\)-hyperconvex if for any non-empty set \(A\subset M\) which is an intersection of closed balls and any collection \(\{B(a_i,r_i)\}_{i\in\mathcal I}\) of pairwise intersecting closed balls of radii \(r_i\) centered at points \(a_i\in A\) the intersection \(A\cap\bigcap_{i\in\mathcal I}B(a_i,\lambda r_i)\) is not empty. If this condition holds for any finite (resp. countable) index set \(\mathcal I\), then the space \(M\) is called finitely (resp. sequentially) \(\lambda\)-hyperconvex. It is shown that each complete metric space is 2-hyperconvex and each uniformly convex Banach space is \(\lambda\)-hyperconvex for some \(\lambda<2\). The Banach space \(c_0\) is finitely 1-hyperconvex but not 1-hyperconvex while the Banach space \(\ell_\infty^c([0,1])\) (of all bounded real-valued functions on [0,1] with countable support) is sequentially 1-hyperconvex but not 1-hyperconvex.
reviewed by: Taras Banakh
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1539288
zbMATH Keywords: hyperconvex metric space
zbMATH Keywords: non-expansive map
zbMATH Keywords: fixed point
zbMATH Keywords: uniformly convex Banach space
MaRDI profile type: Publication
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full work available at URL: s0362-546x(99)00171-6
OpenAlex ID: W2014467334
description: scientific article; zbMATH DE number 1539288