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

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

Content Provider	ACM Digital Library
Author	Tzanaki, Eleni Karavelas, Menelaos I.
Abstract	Given a set Σ of spheres in $E^{d},$ with d≥3 and d odd, having a fixed number of m distinct radii $ρ_{1},ρ_{2},.,ρ_{m},$ we show that the worst-case combinatorial complexity of the convex hull $CH_{d}(Σ)$ of Σ is $Θ(Σ_{\{1≤i≠j≤m\}}n_{i}n_{j}^{⌊$ d/2 ⌋), where $n_{i}$ is the number of spheres in Σ with radius $ρ_{i}.$ Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of $CH_{d}(Σ)$ for all odd d≥3. To prove the lower bound, we construct a set of $Θ(n_{1}+n_{2})$ spheres in $E^{d},$ with d≥3 odd, where $n_{i}$ spheres have radius $ρ_{i},$ i=1,2, and $ρ_2≠ρ_{1},$ such that their convex hull has combinatorial complexity $Ω(n_{1}n_{2}^{⌊$ d/2 $⌋}+n_{2}n_{1}^{⌊$ 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 $E^{d+1},$ 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 $P\{_{1},P_{2},.,P_{m}\}$ of m d-dimensional convex polytopes lying on m parallel hyperplanes of $E^{d+1}$ is $O(Σ_{1≤i≠j≤m}n_{i}n_{j}^{⌊$ d/2 ⌋), where $n_{i}$ 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.
Starting Page	397
Ending Page	406
Page Count	10
File Format	PDF
ISBN	9781450306829
DOI	10.1145/1998196.1998262
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2011-06-13
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Combinatorial complexity Convex polytopes Discrete geometry Convex hull Minkowski sum High-dimensional geometry Combinatorial geometry Spheres Parallel hyperplanes
Content Type	Text
Resource Type	Article

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