The probability that \(n\) random points in a disk are in convex position (Q2357058)

Let \(x_1\), \(x_2\), \dots, \(x_n\) be independent and uniformly distributed points in a disk \(D\). Denote by \(C\) the convex hull of the set \(\{x_1, x_2,\dots,x_n\}\). Let \(P_D^{n, m}\) be the probability that exactly \(m\) points among \(\{x_1, x_2, \dots, x_n\}\) are on the boundary of \(C\). The main result of the paper is to establish a formula for \(P_D^{n,m}\). In particular, the author computes \(P_D^{n, n}\). For completeness, see also the article: [\textit{I. Bárány}, Ann. Probab. 27, No. 4, 2020--2034 (1999; Zbl 0959.60006)] and [\textit{J.-F. Marckert}, ``Maple program and first values for \(P_D^{n,m}\)'', \url{http:www.labri.fr/perso/marckert/computations.tar}].