Right inverses of linear maps on convex sets
type: Item
instance of: scholarly article
title: Right inverses of linear maps on convex sets
zbMATH Open document ID: 1178.54007
author: J. Martínez
published in: Topology and its Applications
publication date: 2009-04-23T00:00:00Z
review text: Let \(X\), \(Y\) be Banach spaces and \(L: X\to Y\) a continuous linear surjection. If \(M\) is a convex, non-open subset of \(X\), then the restriction of \(L\) to \(M\) generally fails to be an open map onto \(L(M)\) even when \(X\), \(Y\) are finite-dimensional and \(M\) has non-empty interior. Results connected to conditions guaranteeing that such a restriction has a continuous right inverse are presented. The cases when \(L(M)\) is a polyhedral body and \(M\) is bounded and \(X\), \(Y\) finite-dimensional and \(M\) a compact convex subset of \(X\) strictly convex along \(\text{Ker\,}L\) are particularly considered. It is deduced, as a corollary, the possibility of splitting a continuous function with values in the Minkowski sum of two convex bodies. The results are related to a problem posed by D. Repovš and P. Semenov.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 5546602
zbMATH Keywords: convex-valued multifunction
zbMATH Keywords: continuous selection
zbMATH Keywords: polyhedral body
zbMATH Keywords: self-similarity
zbMATH Keywords: right-inverse
zbMATH Keywords: open mapping theorem
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.028
OpenAlex ID: W2071682792
cites work: Q3994935
cites work: Q4023720
cites work: Q4219111
cites work: Q5638112
DOI: 10.1016/J.TOPOL.2008.12.028
description: scientific article; zbMATH DE number 5546602