zbMATH — the first resource for mathematics

Strongly exposed points of decomposable sets in spaces of Bochner-integrable functions. (English. Russian original) Zbl 1033.46030
Math. Notes 71, No. 2, 267-275 (2002); translation from Mat. Zametki 71, No. 2, 298-306 (2002).
Let \(K\) be a bounded convex subset of a Banach space \(Y\). Then \(y_0 \in K\) is said to be a strongly exposed point if there exists a \(y^\ast \in Y^\ast\) such that its supremum on \(K\) is attained at \(y_0\) and for any sequence \(\{y_n\} \subset K\), \(y^\ast(y_n) \rightarrow y^\ast(y_0)\) implies \(\| y_n -y_0\| \rightarrow 0\). For \(1\leq p < \infty\),let \(L_p(T,Y)\) denote the space of Bochner integrable functions w.r.t. a non-negative Radon measure \(\mu \) on a locally compact Hausdorff space \(T\) which is countable at infinity.
A set \(\Gamma \subset L_p(T,Y)\) is said to be decomposable if for any \(f,\phi \in \Gamma\), \(f \chi _E + \phi \chi _{T-E} \in \Gamma\), for all \(\mu\)-measurable sets \(E\). It is known that given a bounded decomposable closed set \(\Gamma\), there exists a measurable, \(p\)-integrally bounded closed-valued map \(F: T \rightarrow 2_0^Y\) such that \(\Gamma = \{f \in L_p(T,Y) : f(t) \in F(t)\) almost everywhere} [F. Hiai and H. Umegaki, J. Multivariate Anal. 7, 149–182 (1977; Zbl 0368.60006)]. Extremal structures in this setup have been studied earlier by several authors (see the references cited in this paper).
For a separable Banach space \(Y\), the main result of this paper characterizes strongly exposed points of a decomposable bounded convex set \(\Gamma \subset L_p(T,Y)\) as those \(f\) for which \(f(t)\) is a strongly exposed point of the set \(F(t)\) for almost all \(t \in T\).

46E40 Spaces of vector- and operator-valued functions
46B20 Geometry and structure of normed linear spaces
Full Text: DOI