Union of hypercubes and 3D minkowski sums with random sizes

Let T = (_{1 n}) be a set of of n pairwise-disjoint triangles in R³, and let B be a convex polytope in R³ with a constant number of faces. For each i, let C_i =_i r_iB denote the Minkowski sum of_i with a copy of B scaled by r_i > 0. We show that if the scaling factors r₁, . . ., r_n are chosen randomly then the expected complexity of the union of C₁, . . ., C_n is O(n^2+ε), for any ε > 0; the constant of proportionality depends on ε and the complexity of B. The worst-case bound can be (n³). We also consider a special case of this problem in which T is a set of points in R³ and B is a unit cube in R³, i.e., each C_i is a cube of side-length 2r_i. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log² n), and it improves to O(n log n) if the scaling factors are chosen randomly from a “well-behaved” probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of d, we show that the expected complexity of the union of the hypercubes is O(nd/2 log n) and the bound improves to O(nd/2) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are (n²) in R³, and (nd/2) in higher dimensions.