  We introduce a variable radius form of the extended exterior sphere condition
of [16], 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 semicontinous
radius function. This generalizes, to the variable radius case, the main result
of [16], namely, [16, Theorem 1.2]. On the other hand, as it is shown in
[14,15] 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.
