A1 Journal article (refereed)
A remark on two notions of flatness for sets in the Euclidean space (2022)

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

JYU authors or editors

Publication details

All authors or editors: Violo, Ivan Yuri

Journal or series: Journal fur die reine und angewandte Mathematik

ISSN: 0075-4102

eISSN: 1435-5345

Publication year: 2022

Publication date: 10/08/2022

Volume: 2022

Issue number: 791

Pages range: 157-171

Publisher: Walter de Gruyter GmbH

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1515/crelle-2022-0043

Publication open access: Not open

Publication channel open access: 

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/83742

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2102.12910

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).

Keywords: mathematics; mathematical analysis; Euclidean geometry

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2022

JUFO rating: 3