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

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]).