In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of σ-finite length have tangents on a set of positive ℋ1-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⊆ℝd+1 has the property that each ball centered on Σ contains two large balls in different components of Σc and Σ has σ-finite ℋd-measure, then it has d-dimensional tangent points in a set of positive ℋd-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if Ω⊆ℝd+1 is an exterior corkscrew domain whose boundary has locally finite ℋd-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.
|Early online date||16 Dec 2017|
|Publication status||Published - 2018|