Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let T= { ▵₁, … , ▵_n} be a set of n triangles in R³ with pairwise-disjoint interiors, 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 on the complexity of B. The worst-case bound can be Θ (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 2 r_i. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(nlog ²n) , and it improves to O(nlog 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> 3 , we show that the expected complexity of the union of the hypercubes is O(n^⌊d/2⌋log n) and the bound improves to O(n^⌊d/2⌋) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are Θ (n²) in R³, and Θ (n^⌈d/2⌉) in higher odd dimensions.