On support points and support functionals of convex sets
type: Item
instance of: scholarly article
title: On support points and support functionals of convex sets
zbMATH Open document ID: 1181.46010
published in: Israel Journal of Mathematics
publication date: 2009-08-31T00:00:00Z
review text: Let \(K\) be a bounded convex set in a Banach space \(X\). If \(x\in K\) and \(x^*\in X^*\backslash\{0\}\) are such that \(x^*(x)=\sup x^*(K)\), then \(x\) is said to be a support point and \(x^*\) a support functional of \(K\). The authors prove that the set of all support points is pathwise connected and the set \(\Sigma_1(K)\) of the norm-one support functionals is uncountable in each nonempty open set that intersects the dual unit sphere. Before, these result were known under different additional conditions. The second result answers a question of L.\,Zajíček. Reviewer's remark. The well-known problem whether \(\Sigma_1(K)\) is always norm connected is still open, even for balls. \textit{L.\,Veselý} [J.~Math.\ Anal.\ Appl.\ 350, 550--561 (2009; Zbl 1162.49006)] proved that, if the Banach space admits a Fréchet smooth renorming, then \(\Sigma_1(K)\) is pathwise connected and locally pathwise connected.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 5600317
zbMATH Keywords: support points
zbMATH Keywords: support functionals
zbMATH Keywords: pathwise connected set
author: Carlo Alberto De Bernardi
author: Libor Vesely
reviewed by: A. M. Plichko
MaRDI profile type: Publication
full work available at URL: s11856-009-0037-6
OpenAlex ID: W2014933711
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DOI: 10.1007/S11856-009-0037-6
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description: scientific article; zbMATH DE number 5600317