Local vs. global Lipschitz geometry

TL;DR

This paper investigates the relationship between local and global Lipschitz geometry of singularities, establishing equivalences and invariances under stereographic and inversion transformations for definable sets in o-minimal structures.

Contribution

It proves the equivalence of inner distances on links and sets in o-minimal structures and explores invariance of Lipschitz properties under stereographic and inversion transformations.

Findings

Inner distance of the link is equivalent to that of the set restricted to the link.

Outer lipeomorphic sets are invariant under stereographic modifications and inversions.

Several relations between local and global Lipschitz geometry of singularities are established.

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.