S-Convexity: The Variable Radius Case

We generalize, to the variable radius case, the main results contained in a recent paper of the authors [A new class of sets regularity, J. Convex Analysis 25/4 (2018) 1059–1074], where a new regularity class, called S-convexity, is studied and shown to cover well-known regularity properties, including $\varphi_0$-convexity, $\theta_0$-exterior sphere condition, and $\psi_0$-union of closed balls property. Furthermore, we provide using S-convexity, new additional characterizations of the variable and the constant radius form of these three regularity properties.