Abstract
Given a set Ξ£ of spheres in Ed, with dβ₯3 and d odd, having a fixed number of m distinct radii Ο1,Ο2,...,Οm, we show that the worst-case combinatorial complexity of the convex hull CHd(Ξ£) of Ξ£ is Ξ(Ξ£{1β€iβ jβ€m}ninjβ d/2 β), where ni is the number of spheres in Ξ£ with radius Οi. Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of CHd(Ξ£) for all odd dβ₯3.To prove the lower bound, we construct a set of Ξ(n1+n2) spheres in Ed, with dβ₯3 odd, where ni spheres have radius Οi, i=1,2, and Ο_2β Ο1, such that their convex hull has combinatorial complexity Ξ©(n1n2β d/2 β+n2n1β d/2 β). Our construction is then generalized to the case where the spheres have mβ₯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 dβ₯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(Ξ£1β€iβ jβ€mninjβ d/2 β), 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 dβ₯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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.