Convex hulls of spheres and convex hulls of disjoint convex polytopes

Given a set Σ of spheres in E, with d ≥ 3 and d odd, having a constant number of m distinct radii ρ1, ρ2, . . . , ρm, we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ( ∑ 1≤i6=j≤m nin ⌊ d 2 ⌋ j ), where ni is the number of spheres in Σ with radius ρi. To prove the lower bound, we construct a set of Θ(n1+n2) spheres in E , with d ≥ 3 odd, where ni spheres have radius ρi, i = 1, 2, and ρ2 6= ρ1, such that their convex hull has combinatorial complexity Ω(n1n ⌊ d 2 ⌋ 2 +n2n ⌊ d 2 ⌋ 1 ). Our construction is then generalized to the case where the spheres have m ≥ 3 distinct radii. For the upper bound, we reduce the sphere convex hull problem to the problem of computing the worst-case combinatorial complexity of the convex hull of a set of m disjoint d-dimensional convex polytopes in E, where d ≥ 3 odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the convex hull of a set of m disjoint d-dimensional convex polytopes in E is O( ∑ 1≤i6=j≤m nin ⌊ d 2 ⌋ j ), where ni is the number of vertices of the i-th polytope. Using the lower bound construction for the sphere convex hull problem, it is also shown to be tight for all odd d ≥ 3. Finally, we discuss how to compute convex hulls of spheres with a constant number of distinct radii, or convex hulls of a constant number of disjoint convex polytopes.