On the link of Lipschitz normally embedded sets

TL;DR

This paper investigates the conditions under which subanalytic sets in Euclidean space are Lipschitz normally embedded, introducing a new concept called link Lipschitz normal embedding and establishing its equivalence to LNE for sets with connected links.

Contribution

It introduces the notion of link Lipschitz normally embedding and proves its equivalence to LNE for sets with connected links, advancing understanding of LNE properties.

Findings

Criteria for a subanalytic set to be LNE

Introduction of link Lipschitz normally embedding

Equivalence of link LNE and LNE for connected links

Abstract

A path-connected subanalytic subset in $\mathbb{R}^n$ is naturally equipped with two metrics: the inner and the outer metrics. We say that a subset is Lipschitz normally embedded (LNE) if these two metrics are equivalent. In this article, we give some criteria for a subanalytic set to be LNE. It is a fundamental question to know if the LNE property is conical, i.e., if it is possible to describe the LNE property of a germ of a subanalytic set in terms of the properties of its link. We answer this question by introducing a new notion called link Lipschitz normally embedding. We prove that this notion is equivalent to the LNE notion in the case of sets with connected links.