A characterization of midconvex set-valued functions.(English)Zbl 0712.39027

Let Y be a Hausdorff topological vector space over $${\mathbb{R}}$$. Denote by C(Y) the family of all compact nonempty subsets of Y; and by CC(Y) the family of all compact, convex and nonempty subsets of Y. The main result reported is the following Theorem: Let $$I\subset {\mathbb{R}}$$ be an open interval and Y be a locally convex space. A set-valued function F: $$I\to C(Y)$$ is midconvex if and only if there exist an additive function a: $${\mathbb{R}}\to Y$$ and a convex continuous set-valued function G: $$I\to CC(Y)$$ such that $$F(x)=a(x)+G(x)$$ for all $$x\in I$$. The continuity of G is with respect to the Hausdorff topology on CC(Y). It is an extension of a result of H. Rådström on additive set-valued functions [Ark. Mat. 4, 87-97 (1960; Zbl 0093.304)]. The proof contains an idea due to A. Smajdor and W. Smajdor on Jensen selections.
Reviewer: C.T.Ng

MSC:

 39B72 Systems of functional equations and inequalities 26B25 Convexity of real functions of several variables, generalizations 26E25 Set-valued functions

Zbl 0093.304
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