Continuity properties of midconvex set-valued maps

Abstract

Some new results on the continuity of K-midconvex set-valued maps are given. In particular it is shown that (under suitable assumptions on D, Y and K) a K-midconvex set-valued map \( F:D \longrightarrow cc(Y) \) is K-continuous if and only if for every linear continuous positive functional \( y^{\ast}:Y\longrightarrow \Bbb R \) the functional \( f_{y^{\ast}}(x)={\rm inf}\,y^{\ast} (F(x)) \), \( x \in D \), is continuous. As a consequence of this result we obtain the equality of some set-classes connected with additive functionals and K-midconvex set-valued maps.