Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes · arXivDesk
0911.5086Nov 26, 200922 pages, 5 figures, new proof of upper bound for the complexity of the convex hull of parallel polytopes (the new proof gives upper bounds for all face numbers of the convex hull of the parallel polytopes)
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set Σ of spheres in Ed, with d≥3 and d odd, having a fixed number of m
Nearby in the stack
distinct radii
ρ1,ρ2,...,ρm
, we show that the worst-case combinatorial complexity of the convex hull
CHd(Σ)
of
Σ
is
Θ(∑1≤i=j≤mninj⌊2d⌋)
, 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
Ed
, with
d≥3
odd, where
ni
spheres have radius
ρi
,
i=1,2
, and
ρ2=ρ1
, such that their convex hull has combinatorial complexity
Ω(n1n2⌊2d⌋+n2n1⌊2d⌋)
. 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
{P1,P2,...,Pm}
of
m
d
-dimensional convex polytopes lying on
m
parallel hyperplanes of
Ed+1
is
O(∑1≤i=j≤mninj⌊2d⌋)
, where
ni
is the number of vertices of
Pi
. We end with algorithmic considerations, and 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