  A central question in geometric measure theory is whether geometric
properties of a set translate into analytical ones. In 1960, E. R. Reifenberg
proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well
approximated by $n$-planes at every point and at every scale, then $M$ is a
locally bi-H\"older image of an $n$-plane. Since then, Reifenberg's theorem has
been refined in several ways in order to ensure that $M$ is a bi-Lipschitz
image of an $n$-plane. In this paper, we show that a Carleson condition on the
oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$
of $\mathbb{R}^{n+1}$ satisfying a Poincar\'e-type inequality is sufficient to
prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional
affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincar\'e-type
inequality encodes geometrical information about $M$, namely it implies that
$M$ is quasiconvex.
