The Variable Radius Form of the Extended Exterior Sphere Condition

We consider a variable radius form of the extended exterior sphere condition recently introduced by the authors [The extended exterior sphere condition, J. Convex Analysis 31/1 (2024) 39--50], and then, we prove that the complement of a closed set satisfying this new property is nothing but the union of closed balls with lower semicontinuous radius function. This generalizes, to the variable radius case, the main result of the paper cited above, namely, Theorem 1.2. On the other hand, as it is shown in two further papers of the authors [A new class of sets regularity, ibid. 25/4 (2018) 1059--1074; S-convexity: the variable radius case, ibid. 29/4 (2022) 1167--1192] for prox-regularity, the exterior sphere condition, and the union of closed balls property, we prove that the constant and the variable radius forms of the extended exterior sphere condition belong to the S-convexity regularity class.