On the stability of closed-convex-valued mappings and the associated boundaries (Q2484268)

Let \(Y\) be a locally metrizable space and \(F: Y\rightrightarrows \mathbb{R}^n\) a multifunction satisfying \(F(y)= conv\, bd\,F(y)\), \(y\in Y\). Denote \(B(y):= bd\,F(y)\), \(F_r y:= F(y)\cap\overline K(0, r)\). Some conditions under which \(B\) inherits from \(F\) or \(F_r\) (and vice versa) various kinds of continuity are exhibited by 5 propositions and 9 interesting counterexamples.