New characterizations of minimal cusco maps

Abstract

We give new characterizations of minimal cusco maps in the class of all set-valued maps extending results from \cite{BZ1, GM}. Let $X$ be a topological space and $Y$ a Hausdorff locally convex linear topological space. Let $F: X \to Y$ be a set-valued map. The following are equivalent: (1)~$F$ is minimal cusco; (2)~$F$ has nonempty compact values, there is a quasicontinuous, subcontinuous selection $f$ of $F$ such that $F(x) = \overline{co\,}\overline f(x)$ for every $x \in X$; (3)~$F$ has nonempty compact values, there is a densely defined subcontinuous, quasicontinuous selection $f$ of $F$ such that $F(x) = \overline{co}\,\overline f(x)$ for every $x \in X$; (4)~$F$ has nonempty compact convex values, $F$ has a closed graph, every extreme function of $F$ is quasicontinuous, subcontinuous and any two extreme functions of $F$ have the same closures of their graphs. Some applications to known results are given.