On lower semicontinuity of the restricted center multifunction

Abstract

Given a finite dimensional subspace \(V\) and a certain family \({\mathcal F}\) of nonempty closed and bounded subsets of \( {\mathcal C}_0(T,U)\), where \(T\) is a locally compact Hausdorff space and \(U\) is a strictly convex Banach space, we investigate here lower semicontinuity of the restricted center multifunction \(C_{V}:{\mathcal F} \rightrightarrows V.\) In particular, we establish a Haar-like intrinsic characterization of finite dimensional subspaces \(V\) of \({\mathcal C}_0(T,U)\) which yields lower semicontinuity of \(C_{V}.\)