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 compute $\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as $n\to+\infty$ , for all $\kappa\geq 3$ , 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 $n\to+\infty$ . Finally, we give an asymptotically exact algorithm for the random generation of $z_1,\ldots,z_n$ , conditioned to be in convex position in $\mathfrak{C}_\kappa$ .