35 research outputs found
Polygon dissections and some generalizations of cluster complexes
Let be a Weyl group corresponding to the root system or .
We define a simplicial complex in terms of polygon dissections
for such a group and any positive integer . For , is
isomorphic to the cluster complex corresponding to , defined in \cite{FZ}.
We enumerate the faces of and show that the entries of its
-vector are given by the generalized Narayana numbers , defined
in \cite{Atha3}. We also prove that for any the complex is shellable and hence Cohen-Macaulay.Comment: 9 pages, 3 figures, the type D case has been removed, some
corrections on the proof of Theorem 3.1 have been made. To appear in JCT
The maximum number of faces of the Minkowski sum of two convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of two
-dimensional convex polytopes and , as a function of the number
of vertices of the polytopes.
For even dimensions , the maximum values are attained when and
are cyclic -polytopes with disjoint vertex sets. For odd dimensions
, the maximum values are attained when and are
-neighborly -polytopes, whose vertex sets are
chosen appropriately from two distinct -dimensional moment-like curves.Comment: 37 pages, 8 figures, conference version to appear at SODA 2012; v2:
fixed typos, made stylistic changes, added figure
A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of convex
-polytopes in , where and , as a
(recursively defined) function on the number of vertices of the polytopes.
Our results coincide with those recently proved by Adiprasito and Sanyal [2].
In contrast to Adiprasito and Sanyal's approach, which uses tools from
Combinatorial Commutative Algebra, our approach is purely geometric and uses
basic notions such as - and -vector calculus and shellings, and
generalizes the methodology used in [15] and [14] for proving upper bounds on
the -vector of the Minkowski sum of two and three convex polytopes,
respectively.
The key idea behind our approach is to express the Minkowski sum
as a section of the Cayley polytope of the
summands; bounding the -faces of reduces to bounding the
subset of the -faces of that contain vertices from each
of the polytopes.
We end our paper with a sketch of an explicit construction that establishes
the tightness of the upper bounds.Comment: 43 pages; minor changes (mostly typos
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
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 -dimensional convex polytopes lying on parallel hyperplanes
in , where 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
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
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
.Comment: 22 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
Counting Shi regions with a fixed separating wall
Athanasiadis introduced separating walls for a region in the extended Shi
arrangement and used them to generalize the Narayana numbers. In this paper, we
fix a hyperplane in the extended Shi arrangement for type A and calculate the
number of dominant regions which have the fixed hyperplane as a separating
wall; that is, regions where the hyperplane supports a facet of the region and
separates the region from the origin.Comment: To appear in Annals of Combinatoric
Enumerating regions of Shi arrangements per Weyl Cone
Given a Shi arrangement , it is well-known that the total
number of regions is counted by the parking number of type and the total
number of regions in the dominant cone is given by the Catalan number of type
. In the case of the latter, Shi gave a bijection between antichains in
the root poset of and the regions in the dominant cone. This result was
later extended by Armstrong, Reiner and Rhoades where they gave a bijection
between the number of regions contained in an arbitrary Weyl cone in
and certain subposets of the root poset. In this article we
expand on these results by giving a determinental formula for the precise
number of regions in using paths in certain digraphs related to Shi
diagrams.Comment: 30 pages, 9 figure
The maximum number of faces of the Minkowski sum of three convex polytopes
We derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves