Union of Hypercubes and 3D Minkowski Sums with Random Sizes.

Let T={triangle_1,...,triangle_n} be a set of of n pairwise-disjoint triangles in R^3, and let B be a convex polytope in R^3 with a constant number of faces. For each i, let C_i = triangle_i oplus r_i B denote the Minkowski sum of triangle_i with a copy of B scaled by r_i>0. We show that if the scaling factors r_1, ..., r_n are chosen randomly then the expected complexity of the union of C_1, ..., C_n is O(n^{2+epsilon), for any epsilon > 0; the...