In this note we compare two ways of measuring the\ n-dimensional "flatness" of a set\ S⊂Rd, where\ n∈N\ and\ d\>n. The first one is to consider the classical Reifenberg-flat numbers\ α(x,r)\ (x∈S,\ r\>0), which measure the minimal scaling-invariant Hausdorff distances in\ Br(x)\ between\ S\ and\ n-dimensional affine subspaces of\ Rd. The second is an {\textquoteleft}intrinsic{\textquoteright} 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){\textquoteright}s, 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){\textquoteright}s behaves as the square of the numbers\ α(x,r){\textquoteright}s. 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{\textquoteright} numbers\ β{\textquoteright}s (i.e. the one-sided version of the numbers\ α{\textquoteright}s).

}, author = {Ivan Yuri Violo} }