The Extended Exterior Sphere Condition

TL;DR

This paper proves that the complement of a set satisfying an extended exterior sphere condition is a union of closed balls, generalizing previous results for prox-regular sets and providing conditions for their equivalence.

Contribution

It extends the exterior sphere condition framework, broadening the class of sets for which the complement can be characterized as a union of balls.

Findings

Complement of sets with extended exterior sphere condition is a union of closed balls.

Provides a sufficient condition for equivalence between prox-regularity and the extended exterior sphere condition.

Generalizes previous theorems to non-regular closed sets.

Abstract

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.