A remark on two notions of flatness for sets in the Euclidean space

Abstract

Abstract In this note we compare two ways of measuring the n-dimensional “flatness” of a set S ⊂ ℝ d {S\subset\mathbb{R}^{d}} , where n ∈ ℕ {n\in\mathbb{N}} and d > n {d>n} . The first is to consider the classical Reifenberg-flat numbers α ⁢ ( x , r ) {\alpha(x,r)} ( x ∈ S {x\in S} , r > 0 {r>0} ), which measure the minimal scaling-invariant Hausdorff distances in B r ⁢ ( x ) {B_{r}(x)} between S and n-dimensional affine subspaces of ℝ d {\mathbb{R}^{d}} . 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 ) {{\mathsf{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 ) {{\mathsf{a}}(x,r)} behaves as the square of the numbers α ⁢ ( x , r ) {\alpha(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).