Menger curvatures and $C^{1,\alpha}$ rectifiability of measures

Abstract

We further develop the relationship between $\beta$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^{\alpha}_{\mu;2}(x,r)$ at $\mu$- a.e. $x \in \mathbb{R}^{m}$ implies that $\mu$ is $C^{1,\alpha}$ $n$-rectifiable.