Probability that n points are in convex position in a general convex polygon: Asymptotic results

Abstract Let PK (n) be the probability that n points z₁, , zₙ picked uniformly and independently in K, a compact convex polygon in R² with non-empty interior, are in convex position, that is, form the vertex set of a convex polygon. In this paper, the exact asymptotic behaviour of PK (n) is determined as n+. This improves on a famous result of Bárány 1999 Ann. Probab. 27, 2020–2034 (yet valid for a general convex set K) and a result initiated in the case where K is a regular convex polygon (Morin 2025 Adv. Appl. Probab. 57, 811–870).