627 research outputs found
Vertex-Facet Incidences of Unbounded Polyhedra
How much of the combinatorial structure of a pointed polyhedron is contained
in its vertex-facet incidences? Not too much, in general, as we demonstrate by
examples. However, one can tell from the incidence data whether the polyhedron
is bounded. In the case of a polyhedron that is simple and "simplicial," i.e.,
a d-dimensional polyhedron that has d facets through each vertex and d vertices
on each facet, we derive from the structure of the vertex-facet incidence
matrix that the polyhedron is necessarily bounded. In particular, this yields a
characterization of those polyhedra that have circulants as vertex-facet
incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
A New Algorithm in Geometry of Numbers
A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the
only ellipsoid circumscribed about P. We present a new algorithm for finding
perfect Delaunay polytopes. Our method overcomes the major shortcomings of the
previously used method. We have implemented and used our algorithm for finding
perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new
conjecture that sheds light on the structure of lattice Delaunay tilings.Comment: 7 pages, 3 figures; Proceedings of ISVD-07, International Symposium
on Voronoi diagrams in Science and Engineering held in July of 2007 in Wales,
U
Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
We study the combinatorial complexity of D-dimensional polyhedra defined as
the intersection of n halfspaces, with the property that the highest dimension
of any bounded face is much smaller than D. We show that, if d is the maximum
dimension of a bounded face, then the number of vertices of the polyhedron is
O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For
inputs in general position the number of bounded faces is O(n^d). For any fixed
d, we show how to compute the set of all vertices, how to determine the maximum
dimension of a bounded face of the polyhedron, and how to compute the set of
bounded faces in polynomial time, by solving a polynomial number of linear
programs
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Valuative invariants for polymatroids
Many important invariants for matroids and polymatroids, such as the Tutte
polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant
introduced by the first author, are valuative. In this paper we
construct the -modules of all -valued valuative functions for labeled
matroids and polymatroids on a fixed ground set, and their unlabeled
counterparts, the -modules of valuative invariants. We give explicit bases
for these modules and for their dual modules generated by indicator functions
of polytopes, and explicit formulas for their ranks. Our results confirm a
conjecture of the first author that is universal for valuative
invariants.Comment: 54 pp, 9 figs. Mostly minor changes; Cor 10.5 and formula for
products of s corrected; Prop 7.2 is new. To appear in Advances in
Mathematic
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