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

TL;DR

This paper establishes a link between Menger curvatures and $C^{1,eta}$ rectifiability of measures, providing a new proof that finiteness of a discrete curvature quantity implies rectifiability under weak density conditions.

Contribution

It introduces a novel proof connecting discrete curvature finiteness to $C^{1,eta}$ rectifiability, enhancing understanding of geometric measure theory.

Findings

Finiteness of pointwise discrete curvature implies $C^{1,eta}$ rectifiability.

Develops a new proof method based on $eta$-numbers and discrete curvatures.

Strengthens the theoretical framework relating curvature and rectifiability.

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.