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 $z_1,\ldots,z_n$ 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 $\kappa\geq 3$, which improves on a famous result of B\'ar\'any. 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 $n\to+\infty$. Finally, we give an algorithm asymptotically exact for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_\kappa$.