On the boundary structure of the convex hull of random points (Q2891075)

When investigating the number of vertices of the convex hull of \(n\) points chosen independently and uniformly from the interior of a convex polygon, it is essential to observe that the vertices are concentrated close to the vertices of the polygon as \(n\) tends to infinity. Here concentration means that the number of vertices of the convex hull outside of the neighborhoods of the vertices of the polygon is negligible asymptotically. The author investigates a random variable \(N_n\), defined as follows: Assume that \(n\) points \(P_1, P_2,\dots ,P_n\) are distributed independently and uniformly in the interior of the triangle with vertices \((0, 1)\), \((0, 0)\), and \((1, 0)\). Consider the convex hull of \((0, 1)\), \(P_1, P_2, \dots ,P_n\), and \((1, 0)\). Denote by \(N_n\) the number of those of the points \(P_1, P_2,\dots,P_n\), which are vertices of the convex hull.NEWLINENEWLINEThe present definition of \(N_n\) turns out not only to be suitable to obtain the required asymptotic formulae, but also to be exactly the right one in order to obtain explicit formulae for the number of vertices of the convex hull of a fixed number of random points. The first moment for \(N_n\) was obtained in [[1]: \textit{A. Rényi} and \textit{R. Sulanke}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 75--84 (1963; Zbl 0118.13701)]. The second moment for \(N_n\) was achieved in [[2]: \textit{P. Groeneboom}, Probab. Theory Related Fields 79, 327--368 (1988; Zbl 0635.60012)] using a Poisson point process approximation technique. Based on purely geometric approach, which avoids stochastic processes, the author of the present paper derives the moments of all orders. The classical result for the first moment in [1] and the Groeneboom's result for the variance in [2] are particular cases of these results.