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Characterising rectifiable metric spaces using tangent spaces
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Bate, David (2022) Characterising rectifiable metric spaces using tangent spaces. Inventiones Mathematicae, 230 . pp. 995-1070. doi:10.1007/s00222-022-01136-7 ISSN 0020-9910.
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Official URL: https://doi.org/10.1007/s00222-022-01136-7
Abstract
We characterise rectifiable subsets of a complete metric space X in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an n-dimensional Banach space. In fact, if EāX with Hn(E)<ā and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, E is approximated by a bi-Lipschitz image of Euclidean space.
We also introduce a generalisation of Preiss's tangent measures that is suitable for the setting of arbitrary metric spaces and formulate our characterisation in terms of tangent measures. This definition is equivalent to that of Preiss when the ambient space is Euclidean, and equivalent to the measured Gromov-Hausdorff tangent space when the measure is doubling.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Metric spaces, Geometric measure theory, Hausdorff measures | ||||||||
Journal or Publication Title: | Inventiones Mathematicae | ||||||||
Publisher: | Springer | ||||||||
ISSN: | 0020-9910 | ||||||||
Official Date: | December 2022 | ||||||||
Dates: |
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Volume: | 230 | ||||||||
Page Range: | pp. 995-1070 | ||||||||
DOI: | 10.1007/s00222-022-01136-7 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 21 June 2022 | ||||||||
Date of first compliant Open Access: | 13 July 2022 | ||||||||
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