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).