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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 $$t \in T$$.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46B20 Geometry and structure of normed linear spaces
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