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

Abstract Let P_ (n) be the probability that n points z₁, , zₙ picked uniformly and independently in C_, a regular -gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute P_ (n) up to asymptotic equivalence, as n+, for all 3, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n -tuple of points in convex position when n+. Finally, we give an asymptotically exact algorithm for the random generation of z₁, , zₙ, conditioned to be in convex position in C_.