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The probability that \(n\) random points in a disk are in convex position. (English) Zbl 1372.52009

Let \(x_1\), \(x_2\), …, \(x_n\) 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: [I. Bárány, Ann. Probab. 27, No. 4, 2020–2034 (1999; Zbl 0959.60006)] and [J.-F. Marckert, “Maple program and first values for \(P_D^{n,m}\)”, http:www.labri.fr/perso/marckert/computations.tar].

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
60D05 Geometric probability and stochastic geometry
53C65 Integral geometry

Citations:

Zbl 0959.60006

Software:

Maple