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Validity of the Strong Version of the Union of Uniform Closed Balls Conjecture in the Plane

Chadi Nour, Jean Takche

Abstract

We prove the validity of the strong version of the union of uniform closed balls conjecture, formulated in 2011 as [4, Conjecture 2.5], in the plane.

Validity of the Strong Version of the Union of Uniform Closed Balls Conjecture in the Plane

Abstract

We prove the validity of the strong version of the union of uniform closed balls conjecture, formulated in 2011 as [4, Conjecture 2.5], in the plane.
Paper Structure (3 sections, 3 theorems, 126 equations, 7 figures)

This paper contains 3 sections, 3 theorems, 126 equations, 7 figures.

Table of Contents

  1. Preliminaries
  2. Proof of Theorem \ref{['mainthm']}
  3. Conclusion and perspectives

Key Result

Lemma 1.1

Let $A\subset\mathbb{R}^n$ be nonempty and closed and let $r>0$. Then $A$ satisfies the interior $r$-sphere condition if and only if $A$ is regular closed and for any $a\in\textnormal{bdry}\, A$, there exists at least one unit proximal normal $\zeta\in N_{A'}^P(a)$ which is realized by an $r$-sphere

Figures (7)

  • Figure 1: Geometric configuration of Lemma \ref{['geolem']}
  • Figure 2: Proof of Lemma \ref{['secondlem']}
  • Figure 3: The boundary point $s'_0$ is not regular
  • Figure 4: The ball $\overline{B}(x_2;r_2)$
  • Figure 5: $s"_0\not=s_0$
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1