Invariance of regularity conditions under definable, locally Lipschitz, weakly bi-Lipschitz mappings

TL;DR

This paper explores how certain regularity conditions in stratified spaces, like Whitney (B) and Verdier conditions, remain invariant under specific classes of Lipschitz and bi-Lipschitz mappings, enhancing understanding of geometric stability.

Contribution

It introduces the concept of weak Lipschitzianity on $C^{q}$ stratifications and identifies a class of regularity conditions invariant under definable, locally Lipschitz, and weakly bi-Lipschitz homeomorphisms.

Findings

Regularity conditions like Whitney (B) and Verdier are invariant under certain Lipschitz mappings.

The paper defines weak Lipschitzianity for mappings on $C^{q}$ stratifications.

Identifies a class of conditions stable under definable, locally Lipschitz, and weakly bi-Lipschitz transformations.

Abstract

In this paper we describe the notion of a weak lipschitzianity of a mapping on a $C^{q}$ stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the Verdier condition.