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arXiv · 2501.05920

On 1-regular and 1-uniform metric measure spaces

David Bate
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Abstract

A metric measure space $(X,\mu)$ is 1-regular if \[0< \lim_{r\to 0} \frac{\mu(B(x,r))}{r}<\infty\] for $\mu$-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 $(Y,\nu)$, which satisfies $\nu(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces.

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David Bate. 2025-01-10. On 1-regular and 1-uniform metric measure spaces. https://arxiv.org/abs/2501.05920

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