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 ${\mathrm{curv}}_{\mu ;2}^{\alpha }(x,r)$ at $\mu$- a.e. $x\in {\mathbb{R}}^m$ implies that $\mu$ is $C^{1,\alpha }$ $n$-rectifiable.

To appear in Archiv der Mathematik

Cited by in corpus (1)

• Geometric criteria for $C^{1,\alpha }$ rectifiability