The Extended Exterior Sphere Condition

Journal of Convex Analysis Volume 31 (2024), No. 1, 039--050 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 [11, Theorem 3] 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 [13, Corollary 3.12] to the case in which S is not necessarily regular closed.