Covering points by hyperplanes and related problems
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| Format: | Preprint |
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2024
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| _version_ | 1866915796640333824 |
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| author | Patáková, Zuzana Sharir, Micha |
| author_facet | Patáková, Zuzana Sharir, Micha |
| contents | For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of $k$-rich hyperplanes in $\mathbb R^d$, $d \geq 3$, is at least $Ω(n^d/k^α+ n/k)$, with a sufficiently large constant of proportionality and with $d\le α< 2d-1$, then there exists a $(d-2)$-flat that contains $Ω(k^{(2d-1-α)/(d-1)})$ points of $P$. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for $k$-rich spheres or $k$-rich flats. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covering points by hyperplanes and related problems Patáková, Zuzana Sharir, Micha Combinatorics Computational Geometry For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of $k$-rich hyperplanes in $\mathbb R^d$, $d \geq 3$, is at least $Ω(n^d/k^α+ n/k)$, with a sufficiently large constant of proportionality and with $d\le α< 2d-1$, then there exists a $(d-2)$-flat that contains $Ω(k^{(2d-1-α)/(d-1)})$ points of $P$. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for $k$-rich spheres or $k$-rich flats. |
| title | Covering points by hyperplanes and related problems |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2412.05157 |