Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let $$T=\{\triangle _1,\ldots ,\triangle _n\}$$ be a set of n triangles in $$\mathbb {R}^3$$ with pairwise-disjoint interiors, and let B convex polytope constant number faces. For each i, $$C_i = \triangle _i \oplus r_i B$$ denote the Minkowski sum $$\triangle _i$$ copy scaled by $$r_i>0$$ . We show that if scaling factors $$r_1, \ldots , r_n$$ are chosen randomly then expected complexity union $$C_1, C_n$$ is $$O(n^{2+{\varepsilon }})$$ for any $${\varepsilon }> 0$$ ; proportionality depends on }$$ B. The worst-case bound can $$\Theta (n^3)$$ also consider special case this problem which T points unit cube i.e., $$C_i$$ side-length $$2r_i$$ cubes $$O(n\log ^2n)$$ it improves to n)$$ from “well-behaved” probability density function (pdf). extend latter results higher dimensions. fixed odd value $$d>3$$ we hypercubes $$O(n^{{\lfloor d/2\rfloor }}\log pdf. bounds (n^2)$$ (n^{\lceil d/2\rceil })$$