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

TL;DR

This paper establishes geometric criteria for $C^{1,eta}$ rectifiability of sets in Euclidean space using tangent paraboloids and $eta$ numbers, linking smoothness and geometric approximation.

Contribution

It introduces new criteria for $C^{1,eta}$ rectifiability based on tangent paraboloids and $eta$ numbers, extending previous geometric measure theory results.

Findings

Criteria involving approximate tangent paraboloids for $C^{1,eta}$ rectifiability.

A version of the criteria without a priori tangent planes, using dyadic scales.

Connection between $eta$ numbers and $C^{1,eta}$ rectifiability, with boundedness conditions.

Abstract

We prove criteria for $\mathcal{H}^k$-rectifiability of subsets of $\mathbb{R}^n$ with $C^{1,\alpha}$ maps, $0<\alpha\leq 1$, in terms of suitable approximate tangent paraboloids. We also provide a version for the case when there is not an a priori tangent plane, measuring on dyadic scales how close the set is to lying in a $k$-plane. We then discuss the relation with similar criteria involving Peter Jones' $\beta$ numbers, in particular proving that a sufficient condition is the boundedness for small $r$ of $r^{-\alpha}\beta_p(x,r)$ for $\mathcal{H}^k$-a.e. $x$ and for any $1\leq p\leq \infty$.