On Lipschitz partitions of unity and the Assouad--Nagata dimension
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909207424401408 |
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| author | Licht, Martin W. |
| author_facet | Licht, Martin W. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02865 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Lipschitz partitions of unity and the Assouad--Nagata dimension Licht, Martin W. Metric Geometry Geometric Topology 51F30, 54E35, 54F45 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. |
| title | On Lipschitz partitions of unity and the Assouad--Nagata dimension |
| topic | Metric Geometry Geometric Topology 51F30, 54E35, 54F45 |
| url | https://arxiv.org/abs/2310.02865 |