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Local vs. global Lipschitz geometry

José Edson Sampaio

Abstract

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we obtain several consequences. We present also several relations between the local and the global Lipschitz geometry of singularities. For instance, we prove that two sets in Euclidean spaces, not necessarily definable in an o-minimal structure, are outer lipeomorphic if and only if their stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.

Local vs. global Lipschitz geometry

Abstract

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we obtain several consequences. We present also several relations between the local and the global Lipschitz geometry of singularities. For instance, we prove that two sets in Euclidean spaces, not necessarily definable in an o-minimal structure, are outer lipeomorphic if and only if their stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.
Paper Structure (8 sections, 20 theorems, 33 equations, 2 figures)

This paper contains 8 sections, 20 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Some key results
  4. Lipschitz normally embedding theorem
  5. Lipschitz arc-criterion
  6. LNE vs. LLNE
  7. Outer Lipschitz geometry: local vs. global
  8. Outer and inner definable Lipschitz geometry: local vs. global

Key Result

Theorem 1.1

Let $X\subset\mathbb{R}^n$ and $Y\subset\mathbb{R}^m$ be closed semi-algebraic surfaces with isolated inner Lipschitz singularities. Then, $X$ and $Y$ are inner lipeomorphic if and only if the pointed stereographic compactifications $(\widehat{X},e_{n+1})$ and $(\widehat{Y},e_{m+1})$ are inner lipeo

Figures (2)

  • Figure 1: Obtuse triangle.
  • Figure 2: Obtuse angle ($\theta>\frac{\pi}{2}$).

Theorems & Definitions (38)

  • Theorem 1.1: Corollary 5.7 in FernandesS:2022
  • Theorem \ref{main_theorem}
  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4: Isosceles property at infinity
  • proof
  • Definition 3.1
  • Definition 3.2: See BirbrairM:2000
  • ...and 28 more