  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 $\mathfrak{C}_\kappa$.
