The integral convexity of sets and functionals in Banach spaces
type: Item
instance of: scholarly article
title: The integral convexity of sets and functionals in Banach spaces
zbMATH Open document ID: 1057.46006
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-20T00:00:00Z
review text: In this rather badly written paper, the authors study the notion of an integral convex set in a Banach space. \(A \subset X\) is said to be integral convex if for every \(f : [0,1] \rightarrow X \) which is Bochner integrable and a.e \(A\)-valued (with respect to the Lebesgue measure \(\mu\)) \(\int f d\mu \in A\). \(x_0 \in A\) is said to be an \(\int\)-extreme point if for every one-sided continuous (i.e., either left or right continuous at every point) Bochner integrable function \(f : [0,1] \rightarrow A\), \(x_0 = \int f d\mu\) implies \(f \equiv x_0\). With these notions, it is proved that every nonempty compact set has an \(\int\)-extreme point. No connection has been made with the known integral representation theorems. An interested reader should consult Chapter 6 of \textit{R. D. Bourgin}'s monograph [``Geometric aspects of convex sets with the Radon-Nikodým property'' (Lect. Notes Math. 993, Springer-Verlag, Berlin--Heidelberg--New York--Tokyo)(1983; Zbl 0512.46017)]. No care has apparently been taken by the journal to improve the presentation, bibliography style, or language of this paper. Thus one can find strange expressions like `applications are obtained at last', `In this second and the next', among other mishaps.
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 2093805
zbMATH Keywords: integral convexity
zbMATH Keywords: integral extreme point
author: Jianyong Wang
author: Yumei Ma
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.033
OpenAlex ID: W2072138345
cites work: Q4148368
cites work: Geometry of Banach spaces. Selected topics
cites work: Q3234755
cites work: Q3692267
cites work: Q4104377
DOI: 10.1016/J.JMAA.2004.03.033
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description: scientific article; zbMATH DE number 2093805