Abstract:
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.
Keywords:proximal smoothness, weak convexity, uniform convexity, uniform smoothness, generating set, separation by a sphere, supporting ball.