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The Extended Exterior Sphere Condition

Chadi Nour, Jean Takche

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.

The Extended Exterior Sphere Condition

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.
Paper Structure (4 sections, 5 theorems, 39 equations, 3 figures)

This paper contains 4 sections, 5 theorems, 39 equations, 3 figures.

Table of Contents

  1. Proofs of the main results
  2. Proof of Theorem \ref{['th1']}
  3. Proof of Theorem \ref{['th2']}
  4. Proof of Theorem \ref{['th3']}

Key Result

Theorem 1

Let $S\subset \mathbb{R}^n$ be a closed set such that $\textnormal{cl}\,(\textnormal{int}\, S)$ satisfies the exterior $r$-sphere condition for some $r>0$. Then $(\textnormal{int}\, S)^c$ is the union of closed $\frac{r}{2}$-balls. If in addition $S$ is regular closed, then $S^c$ is the union of clo

Figures (3)

  • Figure 1: Case 3: $\|y_\varepsilon-x \|\geq r$
  • Figure 2: Example \ref{['ex3']}
  • Figure 3: Example of Remark \ref{['newlastplease']}

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1.1
  • proof
  • Example 1.1
  • Proposition 1.2
  • Example 1.2
  • Claim 1
  • Claim 2
  • ...and 5 more