Between quasi-convex and convex set-valued mappings.
type: Item
instance of: scholarly article
title: Between quasi-convex and convex set-valued mappings.
zbMATH Open document ID: 1074.47023
DOI: 10.1016/S0893-9659(04)90058-4
author: Nicolae Popovici
author: Joël Benoist
published in: Applied Mathematics Letters
publication date: 2005-02-25T00:00:00Z
review text: Let \(X\) be a linear space, \(D \subset X\) be a nonempty convex set, and let \(Y\) be a topological linear space partially ordered by a closed convex cone \(K\) with nonempty interior. A set-valued mapping \(F: D \rightarrow 2^Y\) is said to be \(K\)-convex if \(tF(x)+(1-t)F(x') \subset F(tx+(1-t)x')+K\) for all \(x,x' \in D\) and \(t \in [0,1]\). It is said to be \(K\)-quasi-convex if \(\{x \in D : y \in F(x) + K\}\) is convex for all \(y \in Y\). Any \(K\)-convex map is \(K\)-quasi-convex. In this paper, the authors give a sufficient condition for a quasi-convex map to be convex. The condition is that for every \(x \in D\), \(F(x)+K\) is closed and the \(K\)-convex condition holds on a nonempty compact set in \([0,1]\).
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 2138577
zbMATH Keywords: quasi-convex maps
zbMATH Keywords: set-valued mappings
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
cites work: On some class of midconvex functions
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cites work: A characterization of convex function
cites work: Another Proof that Convex Functions are Locally Lipschitz
cites work: Q4347619
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cites work: Generalized convexities of lower semicontinuous functions
full work available at URL: s0893-9659(04)90058-4
OpenAlex ID: W2005700899
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description: scientific article; zbMATH DE number 2138577