Probability that $n$ points are in convex position in a regular $\kappa$-gon : Asymptotic results

Let ${\mathbb{P}}_{\kappa }(n)$ be the probability that $n$ points $z_1,\dots ,z_n$ picked uniformly and independently in ${\mathfrak{C}}_{\kappa }$, a regular $\kappa$-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 ${\mathbb{P}}_{\kappa }(n)$ for all $\kappa \ge 3$, 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 $n\to +\mathrm{\infty }$. Finally, we give an algorithm asymptotically exact for the random generation of $z_1,\dots ,z_n$, conditioned to be in convex position in ${\mathfrak{C}}_{\kappa }$.