On the Geometry of Rectifiable Sets with Carleson and Poincar\'e-type Conditions

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.