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