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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2501.05920 |
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| _version_ | 1866912182880436224 |
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| author | Bate, David |
| author_facet | Bate, David |
| contents | A metric measure space $(X,μ)$ is 1-regular if \[0< \lim_{r\to 0} \frac{μ(B(x,r))}{r}<\infty\] for $μ$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces.
A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,ν)$, which satisfies $ν(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05920 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On 1-regular and 1-uniform metric measure spaces Bate, David Metric Geometry A metric measure space $(X,μ)$ is 1-regular if \[0< \lim_{r\to 0} \frac{μ(B(x,r))}{r}<\infty\] for $μ$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces. A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,ν)$, which satisfies $ν(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces. |
| title | On 1-regular and 1-uniform metric measure spaces |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2501.05920 |