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

Chadi Nour, Jean Takche

TL;DR

This work addresses the structure of the complement $S^c$ of a nonempty closed set $S\subset \mathbb{R}^n$ under the extended exterior $r(\cdot)$-sphere condition. It introduces a new radius function $\varrho_\gamma$ parameterized by $\gamma \in [\frac{1}{2\sqrt{3}-2},1)$, built from the projection-based radius $\rho(x)$ and the extended sphere radius $\rho(x)$, and proves that $S^c$ is the union of closed balls with radius function $\varrho_\gamma(\cdot)$, with $\varrho_\gamma \geq \rho$ so the result generalizes NT2025. The proof is analytical, relying on a sharp characterization of the extended exterior sphere condition and of the union-of-closed-balls property, and uses a constructive, case-based argument to establish the covering. This provides a more flexible and potentially tighter geometric description of the exterior of such sets, with implications for proximal analysis and nonsmooth geometry in optimization contexts.

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].

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

TL;DR

This work addresses the structure of the complement of a nonempty closed set under the extended exterior -sphere condition. It introduces a new radius function parameterized by , built from the projection-based radius and the extended sphere radius , and proves that is the union of closed balls with radius function , with so the result generalizes NT2025. The proof is analytical, relying on a sharp characterization of the extended exterior sphere condition and of the union-of-closed-balls property, and uses a constructive, case-based argument to establish the covering. This provides a more flexible and potentially tighter geometric description of the exterior of such sets, with implications for proximal analysis and nonsmooth geometry in optimization contexts.

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].
Paper Structure (7 sections, 3 theorems, 55 equations, 1 figure)

This paper contains 7 sections, 3 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries $-$ Main Result
  3. Basic Notations and Definitions
  4. The Extended Exterior Sphere Condition
  5. The Union of Closed Balls Property
  6. Statement of the Main Result
  7. Proof of the Main Result

Key Result

Proposition 1

Let $\mathcal{O}\subset\mathbb{R}^n$ be nonempty and open. For $\varrho\colon\mathcal{O}\longrightarrow(0,+\infty]$ a lower semicontinuous function, the set $\mathcal{O}$ is the union of closed balls with radius function $\varrho(\cdot)$ if and only if for every $x\in\mathcal{O}$ there exists a unit

Figures (1)

  • Figure 1: The set $S$ of Example \ref{['remexam']}

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Example 1
  • Theorem 1
  • Corollary 1: NT2025
  • Example 2