On Lipschitz partitions of unity and the Assouad--Nagata dimension

TL;DR

This paper establishes optimal Lipschitz bounds for partitions of unity subordinate to open covers in metric spaces, relates these bounds to the Assouad--Nagata dimension, and characterizes spaces with finite Assouad--Nagata dimension.

Contribution

It provides sharp Lipschitz estimates for partitions of unity and characterizes metric spaces with finite Assouad--Nagata dimension via open covers and refinements.

Findings

Lipschitz constant bounds depend on cover multiplicity and Lebesgue number.

Improved bounds for spaces with the approximate midpoint property.

Characterization of spaces with finite Assouad--Nagata dimension.

Abstract

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant $\max(1,M-1)/\mathcal{L}$, where $\mathcal{L}$ is the Lebesgue number and $M$ is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, such as length spaces do, then the upper bound improves to $(M-1)/(2\mathcal{L})$. These Lipschitz estimates are optimal. We also address the Lipschitz analysis of $\ell^{p}$-generalizations of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad--Nagata dimension $n$ as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity $n+1$ while reducing the Lebesgue number by at most a constant factor.