Weakly convex and proximally smooth sets in Banach spaces

We establish interconnections between the conditions of weak convexity in the sense of Vial, weak convexity in the sense of Efimov–Stechkin, and proximal smoothness of sets in Banach spaces. We prove a theorem on the separation by a sphere of two disjoint sets, one of which is weakly convex in the sense of Vial and the other is strongly convex. We also prove that weakly convex and proximally smooth sets are locally connected, and study questions related to the preservation of the conditions of weak convexity and proximal smoothness under passage to the limit.