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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.05920 |
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Table of 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.