New Characterizations of Strong Convexity

Parallel to the main results of C. Nour, R. J. Stern and J. Takche in " Proximal smoothness and the exterior sphere condition" [J. Convex Analysis 16/2 (2009) 501–514] and of C. Nour and J. Takche in "A new class of sets regularity" [J. Convex Analysis 25/4 (2018) 1059–1074], which explore the equivalence between prox-regularity, the exterior sphere condition, and S-convexity, we present novel characterizations of the r-strong convexity property, namely, of the sets that can be expressed as the intersection of closed balls with the same radius r > 0