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 φ 0 -convexity, \theta_0 θ 0 -exterior sphere condition, and \psi_0 ψ 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.