Inequalities for convex hulls of random points (Q1841253)

This paper is concerned with sharp estimates of some geometric probabilities associated with random points in a plane convex body \(K\) of unit area. Let \(P_1,\dots,P_j\) and \(Q_1,\dots,Q_k\) be independently and uniformly distributed random points in \(K\). Denote by \(p_{jk}(K)\) the probability that the convex hulls {conv}\(\{P_1,\dots,P_j\}\) and {conv}\(\{Q_1,\dots,Q_k\}\) are disjoint. The main result of the paper is a sharp upper bound for \(p_{jk}(K)\) in terms of the integral \(K_{[s]}:=1/8\int_0^{2\pi}l^2(s,\varphi) d\varphi\) for \(s=k/(j+k)\), where \(l(s,\varphi)\) is the length of the chord in direction \(\varphi+\pi/2\) cutting off from \(K\) a segment of area \(s\in [0,1]\). The main tools used in the proof are the new established fact that \(s\mapsto K_{[s]}\) is a concave function and an integral representation of \(p_{jk}(K)\) involving \(K_{[s]}\) proved previously by the author and \textit{C. Buchta} [Probab. Theory Relat. Fields 108, No. 3, 385-415 (1997; Zbl 0882.52004)]. Other results of the paper are related to the expected number \(E_n(K)=np_{n-1,1}(K)\) of vertices of the convex hull of \(n\) i.i.d. uniform random points in \(K\).