Tangents, rectifiability, and corkscrew domains

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $σ$-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 $Σ\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $Σ$ contains two large balls in different components of $Σ^{c}$ and $Σ$ has $σ$-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 $Ω\subseteq \mathbb{R}^{d+1}$ 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.

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