Definable triangulations with regularity conditions

TL;DR

This paper proves that every definable set can be triangulated in a way that satisfies certain regularity conditions, including Whitney (B) and Verdier, with a universal construction method.

Contribution

It introduces a universal method to produce definable triangulations satisfying a broad class of regularity conditions, extending previous results.

Findings

Triangulations are locally Lipschitz and weakly bi-Lipschitz.

Includes a universal construction for triangulations with regularity conditions.

Covers Whitney (B) and Verdier conditions.

Abstract

In this paper we prove that every definable set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of the simplicial complex. We also distinguish a class T of regularity conditions and give a universal construction of a definable triangulation with a T condition of a definable set. This class includes the Whitney (B) condition and the Verdier condition.