The Complement of a Closed Set Satisfying the Extended Exterior Sphere Condition

TL;DR

This paper presents an improved analytical proof demonstrating that the complement of a closed set satisfying the extended exterior sphere condition can be represented as a union of closed balls with a lower semicontinuous radius function, enhancing previous results.

Contribution

It provides a novel, improved analytical proof showing the complement of such sets is a union of closed balls with a larger, lower semicontinuous radius function.

Findings

The complement of the set is a union of closed balls.

The radius function is lower semicontinuous and larger than previous versions.

The proof is analytical and improves upon earlier theorems.

Abstract

We provide a novel analytical proof of an improved version of [10, Theorem 3.1], showing that the complement of a closed set satisfying the extended exterior sphere condition is nothing but the union of closed balls with lower semicontinuous radius function. The improvement lies in the radius function, which is now larger than the one used in [10, Theorem 3.1].