On Lipschitz partitions of unity and the AssouadNagata dimension
Abstract
We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant $\max(1,M1)/\mathcal{L}$, 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 $(M1)/(2\mathcal{L})$. 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 AssouadNagata 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2023
 DOI:
 10.48550/arXiv.2310.02865
 arXiv:
 arXiv:2310.02865
 Bibcode:
 2023arXiv231002865L
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Geometric Topology;
 51F30;
 54E35;
 54F45
 EPrint:
 20 pages, all comments welcome