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Characterizations of rectifiable metric measure spaces

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Bate, David and Li, Sean (2017) Characterizations of rectifiable metric measure spaces. Annales Scientifiques de l'Ecole Normale Superieure, 50 (1). pp. 1-37. doi:10.24033/asens.2314 ISSN 0012-9593.

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Official URL: https://doi.org/10.24033/asens.2314

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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 modified versions of previously known ‘biLipschitz pieces' results on the charts.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: Annales Scientifiques de l'Ecole Normale Superieure
Publisher: Societe Mathematique de France
ISSN: 0012-9593
Official Date: 2017
Dates:
DateEvent
2017Published
2016Available
4 February 2016Accepted
Volume: 50
Number: 1
Page Range: pp. 1-37
DOI: 10.24033/asens.2314
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Open Access Version:
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