Probability that n points are in convex position in a regular κ-gon : Asymptotic results

2024/01/29 by Ludovic Morin, Morin, Ludovic
Mathematics · #60D05 #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Point processes and geometric inequalities #Primary 52A22 #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2401.16207

openalex publication_date 2024/01/29 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28

Abstract

Let ℙ_κ(n) be the probability that n points z₁,…,z_n picked uniformly and independently in \mathfrakC_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 ℙ_κ(n) for all κ≥ 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→+∞. Finally, we give an algorithm asymptotically exact for the random generation of z₁,…,z_n, conditioned to be in convex position in \mathfrakC_κ.

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