Prescribed projections and efficient coverings by curves in the plane
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910884628004864 |
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| author | Chang, Alan McDonald, Alex Taylor, Krystal |
| author_facet | Chang, Alan McDonald, Alex Taylor, Krystal |
| contents | Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as $W$. This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set $W$ and a curve $Γ=\{(t,f(t)): t\in [a,b]\}$, where $f$ is $C^1$ with strictly monotone derivative, we show that $W$ can be covered by translations of $Γ$ in such a way that the union of the translated curves has the same measure as $W$. This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08776 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prescribed projections and efficient coverings by curves in the plane Chang, Alan McDonald, Alex Taylor, Krystal Classical Analysis and ODEs Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as $W$. This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set $W$ and a curve $Γ=\{(t,f(t)): t\in [a,b]\}$, where $f$ is $C^1$ with strictly monotone derivative, we show that $W$ can be covered by translations of $Γ$ in such a way that the union of the translated curves has the same measure as $W$. This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction. |
| title | Prescribed projections and efficient coverings by curves in the plane |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2310.08776 |