Definable triangulations with regularity conditions

Abstract

We prove that every definable in an o-minimal structure set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of a simplicial complex. We also distinguish a class $\mathcal{T}$ of regularity conditions and give a universal construction of a definable triangulation with a $\mathcal{T}$ condition of a definable set. This class includes the Whitney (B) and the Verdier conditions.