Local connectedness of support points (Q919229)

When C is a closed convex subset of a Banach space X, the connectedness (more precisely, simple connectedness) properties of its support points supp(C) are examined. The set supp(C) is k-connected if and only if continuous maps of the k-sphere \(S^ k\) into supp(C) are null-homotopic, \(C^ k\) denotes this property and \(C^{\infty}\) denotes that supp(C) is \(C^ k\) for all integers k. Analogous local connectedness properties \(LC^ k\) and \(LC^{\infty}\) are defined. The basic result is that if C has weakly compact intersection with all closed balls then supp(C) is \(LC^{\infty}\); if, in addition, C contains no hyperplane, then supp(C) is arcwise connected, and if it contains no linear variety of finite codimension it is \(LC^{\infty}\). Addition of separability of C to the lst property yields contractibility of supp(C).