Regular vectors and bi-Lipschitz trivial stratifications in o-minimal structures

TL;DR

This paper proves that sets definable in polynomially bounded o-minimal structures can be stratified with locally definably bi-Lipschitz trivial structures, advancing Lipschitz geometry understanding in o-minimal contexts.

Contribution

It introduces a stratification result for definable sets with bi-Lipschitz triviality, based on new bi-Lipschitz versions of existing theorems, avoiding the real spectrum.

Findings

Existence of a regular vector for definable families

Bi-Lipschitz version of Hardt's theorem

Stratification with bi-Lipschitz triviality in o-minimal structures

Abstract

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which is locally definably bi-Lipschitz trivial along the strata. This result is obtained as a byproduct of two foregoing results of the author. The first one asserts that, given a family definable in an o-minimal structure, there is a regular vector, up to a definable family of bi-Lipschitz homeomorphisms. The second one is a bi-Lipschitz version of the famous Hardt's theorem. We give proofs of these two theorems that avoid the use of the real spectrum. The article recalls the basic facts and the results about Lipschitz geometry that are needed to understand the proofs, providing references.