Tangents, rectifiability, and corkscrew domains

TL;DR

This paper extends the understanding of tangents and rectifiability in Euclidean spaces, showing conditions under which sets have tangent points of positive measure and providing simplified proofs for properties of Semmes surfaces and corkscrew domains.

Contribution

It establishes new conditions for the existence of tangent points in higher dimensions and offers shorter proofs for rectifiability and domain boundary properties.

Findings

Sets with certain geometric properties have positive measure tangent points.

Semmes surfaces are proven to be uniformly rectifiable with simplified methods.

Exterior corkscrew domains with finite measure boundaries contain large Lipschitz subdomains.

Abstract

In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $\Sigma\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $\Sigma$ contains two large balls in different components of $\Sigma^{c}$ and $\Sigma$ has $\sigma$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if $\Omega\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.