Structure of sets with nearly maximal Favard length
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910453233352704 |
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| author | Chang, Alan Dąbrowski, Damian Orponen, Tuomas Villa, Michele |
| author_facet | Chang, Alan Dąbrowski, Damian Orponen, Tuomas Villa, Michele |
| contents | Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - δ, $$ the set $E$ can be covered by an $ε$-Lipschitz graph, up to a set of length $ε$. The dependence between $ε$ and $δ$ is polynomial: in fact, the conclusions hold with $ε= Cδ^{1/70}$ for an absolute constant $C > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01279 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Structure of sets with nearly maximal Favard length Chang, Alan Dąbrowski, Damian Orponen, Tuomas Villa, Michele Classical Analysis and ODEs Metric Geometry 28A75, 28A78 Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - δ, $$ the set $E$ can be covered by an $ε$-Lipschitz graph, up to a set of length $ε$. The dependence between $ε$ and $δ$ is polynomial: in fact, the conclusions hold with $ε= Cδ^{1/70}$ for an absolute constant $C > 0$. |
| title | Structure of sets with nearly maximal Favard length |
| topic | Classical Analysis and ODEs Metric Geometry 28A75, 28A78 |
| url | https://arxiv.org/abs/2203.01279 |