On a link criterion for Lipschitz normal embeddings among definable sets

TL;DR

This paper extends the characterization of Lipschitz normal embedding via links from subanalytic germs to all definable germs in o-minimal structures, and discusses limitations of MD-homology in detecting such embeddings.

Contribution

It generalizes the link criterion for Lipschitz normal embeddings to definable germs in o-minimal structures and examines the limitations of MD-homology in this context.

Findings

The link criterion for Lipschitz normal embedding applies to definable germs in o-minimal structures.

An example shows isomorphic MD-homologies do not guarantee Lipschitz normal embedding.

The result broadens the understanding of Lipschitz geometry in o-minimal settings.

Abstract

It is known by a result of Mendes and Sampaio that the Lipschitz normal embedding of a subanalytic germ is fully characterized by the Lipschitz normal embedding of its link. In this note, we show that the result still holds for definable germs in any o-minimal structure on $(\mathbb{R}, + , .)$. We also give an example showing that for homomorphisms between MD-homologies induced by the identity map, being isomorphic is not enough to ensure that the given germ is Lipschitz normally embedded. This is a negative answer to the question asked by Bobadilla et al. in their paper about Moderately Discontinuous Homology.