On the link of Lipschitz normally embedded sets

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.

arXiv commentThe previous version had two sets of different results. We have divided our pre-print into two, and this version contains the first set of results, which was sections 1-4. The second part, which was Section 5, will appear in a new arXiv submission. The title was changed