Open Access
2018 Tangents, rectifiability, and corkscrew domains
Jonas Azzam
Publ. Mat. 62(1): 161-176 (2018). DOI: 10.5565/PUBLMAT6211808


In a recent paper, Csörnyei 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. As an application, we show that if the dimension of harmonic measure for an NTA domain in $\mathbb{R}^{d+1}$ is less than $d$, then the boundary domain does not have $\sigma$-finite $\mathscr{H}^{d}$-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.


Download Citation

Jonas Azzam. "Tangents, rectifiability, and corkscrew domains." Publ. Mat. 62 (1) 161 - 176, 2018.


Received: 23 May 2016; Revised: 9 November 2016; Published: 2018
First available in Project Euclid: 16 December 2017

zbMATH: 06848690
MathSciNet: MR3738187
Digital Object Identifier: 10.5565/PUBLMAT6211808

Primary: 28A75 , 28A78 , 31A15

Keywords: Absolute continuity , contingent , corkscrew domains , harmonic measure , Semmes surfaces , tangent , uniform rectifiability

Rights: Copyright © 2018 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.62 • No. 1 • 2018
Back to Top