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arXiv · 2310.02865

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

Martin W. Licht
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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.

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Martin W. Licht. 2024-05-21. On Lipschitz partitions of unity and the Assouad--Nagata dimension. https://arxiv.org/abs/2310.02865

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