The Extended Exterior Sphere Condition

We prove that the complement of a closed set S satisfying an extended exterior sphere condition is nothing but the union of closed balls with common radius. This generalizes Theorem 3 of F. Nacry and L. Thibault [Distance function associated to a prox-regular set, Set-Valued Var. Analysis 30 (2022) 731--750] where the set S is assumed to be prox-regular, a property stronger than the extended exterior sphere condition. We also provide a sufficient condition for the equivalence between prox-regularity and the extended exterior sphere condition that generalizes previous work of C. Nour, R. J. Stern and J. Takche [Proximal smoothness and the exterior sphere condition, J. Convex Analysis 16/2 (2009) 501--514] to the case in which S is not necessarily regular closed.