Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Abstract

Given a set Σ of spheres in Ed, with d≥3 and d odd, having a fixed number of m distinct radii ρ1,ρ2,...,ρm, we show that the worst-case combinatorial complexity of the convex hull CHd(Σ) of Σ is Θ(Σ{1≤i≠j≤m}ninj⌊ d/2 ⌋), where ni is the number of spheres in Σ with radius ρi. Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of CHd(Σ) for all odd d≥3.To prove the lower bound, we construct a set of Θ(n1+n2) spheres in Ed, with d≥3 odd, where ni spheres have radius ρi, i=1,2, and ρ_2≠ρ1, such that their convex hull has combinatorial complexity Ω(n1n2⌊ d/2 ⌋+n2n1⌊ d/2 ⌋). 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 d-dimensional convex polytopes lying on m parallel hyperplanes in Ed+1, 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 P{1,P2,...,Pm} of m d-dimensional convex polytopes lying on m parallel hyperplanes of Ed+1 is O(Σ1≤i≠j≤mninj⌊ d/2 ⌋), where ni is the number of vertices of Pi. This bound is an improvement over the worst-case bound on the combinatorial complexity of the convex hull of a point set where we impose no restriction on the points' configuration; using the lower bound construction for the sphere convex hull problem, it is also shown to be tight for all odd d≥3.Finally: (1) we briefly discuss how to compute convex hulls of spheres with a fixed number of distinct radii, or convex hulls of a fixed number of polytopes lying on parallel hyperplanes; (2) we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in Ed; and (3) we state some open problems and directions for future work.