Higher order rectifiability of measures via averaged discrete curvatures

TL;DR

This paper establishes a geometric criterion involving discrete curvature integrals that ensures a Radon measure in Euclidean space is countably rectifiable with a certain smoothness class, extending geometric measure theory.

Contribution

It introduces a new sufficient condition based on averaged discrete curvatures for measures to be rectifiable of class C^{1,α}, advancing the understanding of geometric regularity of measures.

Findings

Provides a new geometric condition for rectifiability

Connects discrete curvature integrals with smoothness of measures

Extends rectifiability criteria to measures with positive lower density

Abstract

We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and finite upper density $\mu$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplices spanned by the parameters.