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arXiv · 1409.4242

Characterizations of rectifiable metric measure spaces

David Bate·Sean Li
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Abstract

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply previously known "biLipschitz pieces" results on the charts.

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David Bate, Sean Li. 2015-01-10. Characterizations of rectifiable metric measure spaces. https://doi.org/10.24033/asens.2314

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