The equivalence of three regularity properties of a set: \(\varphi_0\)-convexity, \(\theta_0\)-exterior sphere condition, \(\psi_0\)-union of closed balls under suitable conditions where the functions \(\varphi_0, \theta_0\) and \(\psi_0\) are constant is well-known (see, e.g., [\textit{C. Nour} et al., J. Convex Anal. 16, No. 2, 501--514 (2009; Zbl 1179.49019); \textit{C. Nour} and \textit{J. Takche}, J. Convex Anal. 25, No. 4, 1059--1074 (2018; Zbl 1404.49009)]). The concept \(S\)-convexity, introduced and thoroughly studied in [\textit{C. Nour} et al., J. Convex Anal. 25, No. 1, 1--19 (2018; Zbl 1394.49015); Nour and Takche, 2018, loc. cit.], covers the above three regularity properties. The current papers extends the equivalence result to the variable case, i.e., the considered functions \(\varphi, \theta\) and \(\psi\) are non-constant and continuous. In addition, new characterizations of the \(\varphi\)-convexity, \(\theta\)-exterior sphere condition, \(\psi\)-union of closed balls using \(S\)-convexity is given.