The integral convexity of sets and functionals in Banach spaces
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.
scientific article; zbMATH DE number 2093805
scientific article