Continuous selections of multifunctions with weakly convex values
type: Item
instance of: scholarly article
title: Continuous selections of multifunctions with weakly convex values
zbMATH Open document ID: 1154.54010
author: G. E. Ivanov
published in: Topology and its Applications
publication date: 2008-05-15T00:00:00Z
review text: The author shows that given a paracompact space \(X\), numbers \(0<\varepsilon<R\) and a lower semicontinuous set-valued map \(F\) defined on \(X\) with closed values in a Hilbert space \(H\) such that for any \(x\in X\), \(F(x)\) is weakly convex with respect to \(R\), \(F\) admits a continuous selection provided it has an \(\varepsilon\)-selection. A set \(A\subset H\) is weakly convex if (roughly speaking) for any \(x_0,x_1\in A\), the segment \([x_0,x_1]\) meets \(A\) at a point \(x\) different from \(x_0\) and \(x_1\). In a similar spirit the author shows the existence of a continuous selection of a lower semicontinuous map \(F\) defined on a paracompact space \(X\) of dimension \(n+1\), provided the values of \(F\) are \(n\)-connected, closed and weakly convex with respect to \(R>0\).
reviewed by: Wojciech Kryszewski
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 5275707
zbMATH Keywords: set-valued map
zbMATH Keywords: continuous selection
zbMATH Keywords: weakly convex set
MaRDI profile type: Publication
full work available at URL: j.topol.2007.03.017
OpenAlex ID: W1997002846
cites work: Continuous selections. I
cites work: Continuous selections. II
cites work: Paraconvex Sets.
cites work: Functionally paraconvex sets
cites work: Q4219111
cites work: Strong and Weak Convexity of Sets and Functions
cites work: Weak convexity in the senses of Vial and Efimov-Stechkin
cites work: Properties of the metric projection on weakly Vial-convex sets and parametrization of set-valued mappings with weakly convex images
cites work: Weakly convex sets and their properties
DOI: 10.1016/J.TOPOL.2007.03.017
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description: scientific article; zbMATH DE number 5275707