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Almost midconvex and almost convex set-valued functions. (English) Zbl 0989.26006
The paper is devoted to $$\operatorname{Im}$$-almost midconvex and $$\operatorname{Im}$$-almost convex multifunctions defined on a convex and open subset of a real topological vector space, values of which are convex compact subsets of separable normed space. Under some invariance properties (called (a) and (b) in the paper) of $$\sigma$$-ideals considered, it is shown that an $$\operatorname{Im} _2$$-almost midconvex multifunction is $$\operatorname{Im} _1$$-almost everywhere equal to a midconvex multifunction on the same domain of definition. The main result of the paper reads as follows.
Theorem 2. Let $$X$$ be a topological vector space, $$D$$ be a convex and open subset of $$X$$ and $$Y$$ be a separable normed space. Denote by $$\operatorname{Im} _1$$ and $$\operatorname{Im} _2$$ proper linearly invariant conjugate $$\sigma$$-ideals in $$X$$ and $$X\times X$$, respectively, satisfying conditions (a) and (b). If a set-valued function $$F\: D\rightarrow cc(Y)$$ is $$\operatorname{Im} _2$$-almost midconvex, then there exists a unique midconvex set-valued function $$G\:D\to cc(Y)$$ such that $$F(x)=G(x)\;\;\operatorname{Im} _1$$-almost everywhere in $$D$$.
A similar result is obtained for convex multifunctions.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations 54C60 Set-valued maps in general topology 26E25 Set-valued functions 39B72 Systems of functional equations and inequalities
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##### References:
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