Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909053038362624 |
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| author | Morin, Ludovic |
| author_facet | Morin, Ludovic |
| contents | Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_K(n)$ when $n\to\infty$. This improves on a famous result of Bárány (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results Morin, Ludovic Probability Combinatorics Primary 52A22, 60D05 Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_K(n)$ when $n\to\infty$. This improves on a famous result of Bárány (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon. |
| title | Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results |
| topic | Probability Combinatorics Primary 52A22, 60D05 |
| url | https://arxiv.org/abs/2410.11706 |