Inner Lipschitz approximation in o-minimal structures

TL;DR

This paper proves that Lipschitz mappings definable in o-minimal structures can be approximated by smooth or $ ext{C}^ ext{infty}$ mappings with controlled Lipschitz bounds, using partitions of unity.

Contribution

It introduces a method to approximate Lipschitz definable mappings in o-minimal structures with smooth ones, extending to outer Lipschitz mappings when $ ext{C}^ extinfty$ cell decomposition exists.

Findings

Approximation by $ ext{C}^1$ mappings with close Lipschitz bounds.

Extension to $ ext{C}^ extinfty$ mappings under certain conditions.

Construction of partitions of unity with sharp derivative bounds.

Abstract

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.