Convex polytopes in restricted point sets in $\mathbb{R}^d$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910803687374848 |
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| author | Bukh, Boris Dong, Zichao |
| author_facet | Bukh, Boris Dong, Zichao |
| contents | For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, α}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < α\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, α}(n)$ as $n \to \infty$ by showing the existence of positive constants $β= β(d, α)$ and $γ= γ(d)$ such that $βn^{\frac{d-1}{d+1}} \le c_{d, α}(n) \le γn^{\frac{d-1}{d+1}}$ for $α\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_02487 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convex polytopes in restricted point sets in $\mathbb{R}^d$ Bukh, Boris Dong, Zichao Combinatorics Metric Geometry 52A20, 52C07 For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, α}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < α\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, α}(n)$ as $n \to \infty$ by showing the existence of positive constants $β= β(d, α)$ and $γ= γ(d)$ such that $βn^{\frac{d-1}{d+1}} \le c_{d, α}(n) \le γn^{\frac{d-1}{d+1}}$ for $α\geq 2$. |
| title | Convex polytopes in restricted point sets in $\mathbb{R}^d$ |
| topic | Combinatorics Metric Geometry 52A20, 52C07 |
| url | https://arxiv.org/abs/2204.02487 |