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

In this note we compare two ways of measuring the n-dimensional "flatness" of a set S subset of R-d, where n is an element of N and d > n. The first is to consider the classical Reifenberg-flat numbers alpha(x, r) (x is an element of S, r > 0), which measure the minimal scaling-invariant Hausdorff distances in B-r(x) between S and n-dimensional affine subspaces of 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) 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 alpha(x, r) behaves as the square of the numbers 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 beta (i.e. the one-sided version of the numbers alpha).