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Lipschitz normally embedded sets do not need to have Lipschitz normally embedded medial axis

Michał Kosiba

Abstract

The aim of this paper is to study the Lipschitz normally embedded property for a set and its medial axis. We consider if and when a non-LNE set implies non-LNE medial axis and converse. We present an example a of Lipschitz normally set that has medial axis which is not Lipschitz normally emebedded. At the end we discuss special case when a set is a one dimensional germ on a plane.

Lipschitz normally embedded sets do not need to have Lipschitz normally embedded medial axis

Abstract

The aim of this paper is to study the Lipschitz normally embedded property for a set and its medial axis. We consider if and when a non-LNE set implies non-LNE medial axis and converse. We present an example a of Lipschitz normally set that has medial axis which is not Lipschitz normally emebedded. At the end we discuss special case when a set is a one dimensional germ on a plane.
Paper Structure (5 sections, 14 theorems, 20 equations)

This paper contains 5 sections, 14 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Lipschitz normally embedded sets - general informations
  3. Link Lipschitz normally embedded sets
  4. Lipschitz normally embedded sets and medial axis
  5. Plane case

Key Result

Theorem 2.5

Let $X \subset \mathbb{R}^{n}$ be a closed semialgebraic set. Then there exists a finite family of subsets $\{X_{i}\}_{i=1}^{k}$ called pancakes such that: 1) all $X_{i}$ are definable closed subsets of $X$; 2) $X=\bigcup_{i=1}^{k} X_{i}$; 3) $\text{dim}(X_{i} \cap X_{j}) < \min(\text{dim}X_{i},\te

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: Pancake decomposition, normal, kurdyka, kurdmost, paru
  • Definition 2.6
  • Theorem 2.7: normal Theorem 3.1
  • Proposition 2.8: normal Lemma 3.2
  • Theorem 2.9: Kurdyka-Orro, see also normal Theorem 3.2
  • Definition 2.10
  • ...and 32 more