Regular Vectors and Bi-Lipschitz Trivial Stratifications in O-Minimal Structures

These notes deal with semialgebraic and subanalytic subsets of ℝ n $${\mathbb R}^n$$ , and more generally with all the sets that are definable in a polynomially bounded o-minimal structure expanding ℝ $${\mathbb R}$$ , establishing existence of definably bi-Lipschitz trivial stratifications for these sets (Corollary 7.6.9). We start with basic definitions about o-minimal structures and Lipschitz geometry, and give a short survey of some historical results, such as existence of Mostowski’s Lipschitz stratifications or the Preparation Theorem for definable functions, which gives us the opportunity to state some needed theorems as well as to describe the way the results presented in the second part fit in the landscape. Our stratification theorem (Corollary 7.6.9) is obtained as a byproduct of two foregoing results of the author that are proved in Sects. 7.5 and 7.6 respectively. 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.