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

Silvia Ghinassi, Max Goering

Abstract

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

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

Abstract

We further develop the relationship between -numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature at - a.e. implies that is -rectifiable.

Paper Structure

This paper contains 3 sections, 7 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Proof of Theorem \ref{['t:maintheorem']}

Key Result

Theorem 1.1

Let $\mu$ be a Radon measure on $\mathbb{R}^{m}$, with $0 < \Theta^{n}_{*}(\mu,x) \le \Theta^{n,*}(\mu,x) < \infty$, for $\mu$-a.e. $x \in \mathbb{R}^{m}$ and let $1 \le p < \infty$, $0 < \alpha \le 1$. If for $\mu$- a.e. $x \in \mathbb{R}^{m}$ then $\mu$ is $C^{1,\alpha}$$n$-rectifiable.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem I
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2: $\beta_{p}$-numbers
  • Definition 2.3
  • Theorem 2.4: ghinassi2017sufficient
  • Remark 2.5
  • Definition 2.6: Classical Menger curvature
  • Definition 2.7: Simplices
  • ...and 10 more