Multifunctions with selections of convex and concave type (Q2714290)

Let \({\mathcal F}\) denote one of the families of convex, midconvex, subadditive, sublinear, quasiconvex, and non-decreasing functions and \(D\) denote, in most cases, a convex subset of a real vector space. Multifunctions \( H : D \to \operatorname{cc} ({\mathbf{R}}) \) which have selections \( h_1 \in {\mathcal F} \) and \( h_2 \in -{\mathcal F} \) are characterized via intersection properties. The problem of the existence of selections belonging to \( {\mathcal F} \cap -{\mathcal F} \) for such multifunctions is also discussed. In several cases, counterexamples are presented to indicate that the two selection properties are not equivalent.