Probability that $n$ points are in convex position in a regular $κ$-gon : Asymptotic results
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909348656054272 |
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| author | Morin, Ludovic |
| author_facet | Morin, Ludovic |
| contents | Let $\mathbb{P}_κ(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_κ$, a regular $κ$-gon with area $1$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_κ(n)$ for all $κ\geq 3$, which improves on a famous result of Bárány. A second aim of the paper is to establish a limit theorem which describes the fluctuations around the limit shape of a $n$-tuple of points in convex position when $n\to+\infty$. Finally, we give an algorithm asymptotically exact for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16207 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probability that $n$ points are in convex position in a regular $κ$-gon : Asymptotic results Morin, Ludovic Probability Combinatorics Primary 52A22, 60D05 Let $\mathbb{P}_κ(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_κ$, a regular $κ$-gon with area $1$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_κ(n)$ for all $κ\geq 3$, which improves on a famous result of Bárány. A second aim of the paper is to establish a limit theorem which describes the fluctuations around the limit shape of a $n$-tuple of points in convex position when $n\to+\infty$. Finally, we give an algorithm asymptotically exact for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_κ$. |
| title | Probability that $n$ points are in convex position in a regular $κ$-gon : Asymptotic results |
| topic | Probability Combinatorics Primary 52A22, 60D05 |
| url | https://arxiv.org/abs/2401.16207 |