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

Nikodem, Kazimierz

Acta Univ. Carol., Math. Phys. 30, No. 2, 125-129 (1989).

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

Keywords:

convex set-valued functions; Hausdorff topological vector space; locally convex space; midconvex

Citations:

Zbl 0093.304