# On the local geometry of definably stratified sets

## TL;DR

This paper extends Pawlucki's theorem on Whitney regularity and stratified sets from subanalytic to definable sets in polynomially bounded o-minimal structures, providing new insights into their local geometry and counterexamples.

## Contribution

It generalizes Pawlucki's theorem to o-minimal structures, introduces a quantified Whitney (b)-regularity, and analyzes the continuity of density and normal cones in definably stratified sets.

## Key findings

- Pawlucki's theorem applies to polynomially bounded o-minimal structures.
- A refined version of the theorem with quantified Whitney (b)-regularity is established.
- Counterexamples show limitations of extending the theorem to all o-minimal structures.

## Abstract

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Pawlucki's theorem for arbitrary o-minimal structures, replacing Whitney (b)-regularity by a quantified version, and we prove related results concerning normal cones and continuity of the density. We analyse two counterexamples to the extension of Pawlucki's theorem to definable sets in general o-minimal structures, and to several other statements valid for subanalytic sets. In particular we give the first example of a Whitney (b)-regular definably stratified set for which the density is not continuous along a stratum.