On a random variable related to a system of convex bodies in the Euclidean space E n

Abstract

Let us consider, in the Euclidean space E n , a fixed n-dimensional convex body K 0 of volume V 0 and a system K 1,…,K m of m n-dimensional convex bodies, congruent to a convex set K. Assume that the sets K i (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of E n and denote by V m the volume of the convex body K m = K 0 ∩ (K 1 ∩ … ∩ K m ). The aim of this paper is the evaluation of the second moment of the random variable V m .