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.


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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