Convex Hulls of Spheres and Convex Hulls of Convex Polytopes Lying on Parallel Hyperplanes

Given a set Sigma of spheres in E(d), with d >= 3 and d odd, having a fixed number of in distinct radii rho(1), rho(2), ... ,rho(m), We show that the worst-case combinatorial complexity of the convex hull CH(d)(Sigma) of Sigma is Theta(Sigma(1 <= i not equal j <= m) n(i)n(j)(left perpendiculard/2right perpendicular)), where n(i) is the number of spheres in Sigma with radius pi. Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of CH(d)(Sigma) for all odd d >= 3. To prove the lower bound, we construct a set of Theta(n(1) + n(2)) spheres in E(d), with d >= 3 odd, where ni spheres have radius rho(i) , i = 1, 2, and rho(2) not equal rho(1), such that their convex hull has combinatorial complexity Omega(n(1)n(2)(left perpendiculard/2right perpendicular) + n(2)n(1)(left perpendiculard/2right perpendicular)). 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 in parallel hyperplanes in E(d+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 {Pi, P2, ... , P(m)} of m d-dimensional convex polytopes lying on m parallel hyperpla.nes of E(d+1) is O(Sigma(1 <= i not equal j <= m) n(i)n(j)(left perpendiculard/2right perpendicular)), where ni is the number of vertices of P(i). 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 E(d); and (3) we state some open problems and directions for future work.