A remark on two notions of flatness for sets in the Euclidean space
Abstract
In this note we compare two ways of measuring the n-dimensional “flatness” of a set S⊂RdS⊂ℝd , where n∈Nn∈ℕ and d>nd>n . The first is to consider the classical Reifenberg-flat numbers α(x,r)α(x,r) ( x∈Sx∈S , r>0r>0 ), which measure the minimal scaling-invariant Hausdorff distances in Br(x)Br(x) between S and n-dimensional affine subspaces of Rdℝd . The second is an “intrinsic” approach in which we view the same set S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers a(x,r)𝖺(x,r) that are the scaling-invariant Gromov–Hausdorff distances between balls centered at x of radius r in S and the n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers a(x,r)𝖺(x,r) behaves as the square of the numbers α(x,r)α(x,r) . Moreover, we show how this result finds application in extending the Cheeger–Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones’ numbers β (i.e. the one-sided version of the numbers α).
Main Author
Format
Articles
Research article
Published
2022
Series
Subjects
Publication in research information system
Publisher
Walter de Gruyter GmbH
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202211015047Use this for linking
Review status
Peer reviewed
ISSN
0075-4102
DOI
https://doi.org/10.1515/crelle-2022-0043
Language
English
Published in
Journal fur die reine und angewandte Mathematik
Citation
- Violo, I. Y. (2022). A remark on two notions of flatness for sets in the Euclidean space. Journal fur die reine und angewandte Mathematik, 2022(791), 157-171. https://doi.org/10.1515/crelle-2022-0043
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