Favard length and quantitative rectifiability
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914904372412416 |
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| author | Dąbrowski, Damian |
| author_facet | Dąbrowski, Damian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Favard length and quantitative rectifiability Dąbrowski, Damian Classical Analysis and ODEs Metric Geometry 28A75 (primary) 28A78 (secondary) 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. |
| title | Favard length and quantitative rectifiability |
| topic | Classical Analysis and ODEs Metric Geometry 28A75 (primary) 28A78 (secondary) |
| url | https://arxiv.org/abs/2408.03919 |