Cone and paraboloid points of arbitrary subsets of Euclidean space

TL;DR

This paper characterizes cone and paraboloid points of arbitrary subsets in Euclidean space using integrability conditions of Jones-type coefficients, extending classical results to higher dimensions and more general sets.

Contribution

It provides a high-dimensional generalization of Bishop and Jones's theorem, linking geometric point classifications to integrability of specialized Jones coefficients.

Findings

Characterization of cone points via integrability of $eta_E^{d,2}$ coefficients.

Extension of classical theorems to arbitrary subsets in Euclidean space.

Establishment of conditions for $ ext{C}^{1,eta}$ rectifiability using $eta$-coefficients.

Abstract

In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given $E \subset \mathbb{R}^n$, $x \in E$ is a cone point of $E$ if and only if \begin{align*} \int_{0}^1 \beta_{E}^{d,2}(B(x,r))^2 \frac{dr}{r} < \infty, \end{align*} up to a set of zero $d$-measure. The coefficients $\beta_E^{d,2}$ are a variation of the Jones coefficients. This is a high dimensional counterpart of a theorem of Bishop and Jones from 1994. We also prove similar results for $\alpha$-paraboloid points, which are the $C^{1,\alpha}$ rectifiability counterparts to cone points: $x \in E$ is an $\alpha$-paraboloid point if and only if \begin{align*} \int_0^1 \frac{\overline{\beta}_{E}^{d,2}(B(x,r))^2}{r^{2\alpha}} \, \frac{dr}{r} < \infty \end{align*} up to a set of zero $d$-measure. Here, $\overline{\beta}^{d,2}_E$ is another variant of the Jones coefficients, introduced by Azzam and Schul.