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Favard length and quantitative rectifiability
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abstract
The Favard length of a Borel set $E\subset\mathbb{R}^2$ is the average length of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of the Besicovitch projection theorem. As a corollary, we answer questions of David and Semmes and of Peres and Solomyak. We also make progress on Vitushkin's conjecture.
Forward citations
Cited by 3 Pith papers
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Power Laws for the Favard Length Problem in $\mathbb{R}^d$
For rational product Cantor sets in R^d (d≥2) with cyclotomic conditions on digit sets, the Favard length of N^{-1}-neighborhoods decays as N^{-ε}, new for d≥3 and for fibered digit sets.
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An isoperimetric inequality for mean shadow
The circle minimizes Favard length among unit-area planar Borel sets.
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Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions
A field survey showing that quantitative rectifiability underpins results on Riesz transforms, removable sets, harmonic measure, and L^p boundary value problems for the Laplacian.
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