Comparison of \(P\)-convexity, \(O\)-convexity and other geometrical properties
type: Item
instance of: scholarly article
title: Comparison of \(P\)-convexity, \(O\)-convexity and other geometrical properties
zbMATH Open document ID: 1270.46014
author: Helga Fetter
author: Enrique Llorens-Fuster
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-10-19T00:00:00Z
review text: A real Banach space \(X\) is \(P(n,\epsilon)\)-convex, \( n \geq 2\), \(\epsilon > 0\), if, for every \(x_1, \dots,x_n\) in the closed unit ball of \(X\), there exist \(i \neq j\) such that \(\|x_i - x_j\|\;\leq 2 -\epsilon\). \(X\) is \(P(n)\)-convex if it is \(P(n,\epsilon)\)-convex for some \(\epsilon > 0\), and \(X\) is \(P\)-convex if it is \(P(n)\)-convex for some \(n \in N\). A real Banach space \(X\) is \(O(n,2\epsilon)\)-convex, \( n \geq 2\), \(\epsilon > 0\), if, for every \(x_1,\dots,x_n\) in the closed unit ball of \(X\), there exist \(i \neq j\) such that \(\min (\|x_i + x_j\|, \|x_i - x_j\|) \leq 2 (1- \epsilon)\). \(X\) is \(O(n)\)-convex if it is \(O(n, 2\epsilon)\)-convex for some \(\epsilon > 1\), and it is \(O\)-convex if it is \(O(n)\)-convex for some \(n\). Some properties of these spaces are discussed. Fixed point theorems are proved and some examples are provided.
reviewed by: Srinivasa Swaminathan
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6096070
zbMATH Keywords: \(P\)-convexity
zbMATH Keywords: \(O\)-convexity
zbMATH Keywords: fixed points
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.07.021
OpenAlex ID: W2073352085
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DOI: 10.1016/J.JMAA.2012.07.021
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description: scientific article; zbMATH DE number 6096070