A remark on two notions of flatness for sets in the Euclidean space
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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Journal fur die reine und angewandte MathematikAuthors
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2022Copyright
© De Gruyter 2022
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).
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