  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.
