The Smallest Parallelepiped of n Random Points and Peeling

Abstract

Let n random points be given with uniform distribution in the d-dimensional unit cube [0,1]d. The smallest parallelepiped A which includes all the n random points is dealt with. We investigate the asymptotic behavior of the volume of A as n tends to ∞. Using a point process approach, we derive also the asymptotic behavior of the volumes of the k-th smallest parallelepipeds An(k) which are defined by iteration. Let An = An(1). Given An(k,-,1) delete the random points Xi which are on the boundary ∂An(k,-,1), and construct the smallest parallelepiped which includes the inner points of An(k,-,1), this defines An(k). This procedure is known as peeling of the parallelepiped An.