Construction theorems for polytopes

Contemporary Mathematics, 2006

Geometriae Dedicata, 2012

Let be a set of n-dimensional polytopes. A set of n-dimensional polytopes is said to be an element set for if each polytope in is the union of a finite number of polytopes in identified along (n − 1)-dimensional faces. The element number of the set of polyhedra, denoted by e( ), is the minimum cardinality of the element sets for , where the minimum is taken over all possible element sets ∈ E( ). It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ≥ 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ≥ 2.

2001

In 1982, I. Shemer introduced the sewing construction for neighbourly 2m-polytopes. We extend the sewing to simplicial neighbourly d-polytopes via a verification that is not dependent on the parity of the dimension. We present also descibable classes of 4-polyopes and 5-polytopes generated by the construction.

Monatshefte f�r Mathematik, 1990

We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.

SIAM Journal on Discrete Mathematics

2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P)f d−1 (P) ≤ d2 d+1 for a large collection of families of such polytopes P. Here f0(P) (resp. f d−1 (P)) is the number of vertices (resp. facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.

Journal of Combinatorial Theory, Series B, 1971

Journal of Combinatorial Theory, Series A, 2006

We construct a new 2-parameter family E mn , m, n ≥ 3, of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E 44 is the 24-cell. For large m, n the f -vectors have "fatness" close to 6.

Contributions to Algebra and Geometry, 2004

Various facts about triangulations of simplicial polytopes, particularly those pertaining to the equality case in the generalized lower bound conjecture, are collected together here. They include an apparently weaker restriction on the kind of triangulation which needs to be found, and an inductive argument which reduces the number of cases to be established.

2005

The Regular Polytope has been shown to be a promising candidate for the rigorous representation of geometric objects, in a form that is computable using the finite arithmetic available on digital computers. It is also apparent that the approach can be used in the formulation of repeatable and verifiable definitions of geometric objects with the aim of defining robust information interchange protocols. In the definition and investigation of the properties of the Regular Polytope representation, several propositions have been made, and the proofs developed. These proofs can be quite long and complex, and not suitable for presentation at conferences or publication in journals. They are gathered here to make them available for scrutiny in conjunction with the various papers and publications that refer to them. The major assertions that this document addresses are: The set of Regular Polytopes forms and spans a Topological Space. There is a useful correspondence between the definition of equality of Regular Polytopes and the natural "point set" definition of equality.. A simplified "programming shortcut" can be used to reduce significantly the algorithmic complexity of implementation.

2008

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new.

TOP, 2013

The Hirsch conjecture, posed in 1957, stated that the graph of a ddimensional polytope or polyhedron with n facets cannot have diameter greater than n − d. 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.

Journal of information processing, 2017

In this paper, an n-dimensional polytope is called Wythoffian if it is derived by the Wythoff construction from an n-dimensional regular polytope whose finite reflection group belongs to A n , B n , C n , F 4 , G 2 , H 3 , H 4 or I 2 (p). Based on combinatorial and topological arguments, we give a matrix-form recursive algorithm that calculates the number of k-faces (k = 0, 1,. .. , n) of all the Wythoffian-n-polytopes using Schläfli-Wythoff symbols. The correctness of the algorithm is reconfirmed by the method of exhaustion using a computer.

Lecture Notes in Computer Science, 2016

2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. We investigate upper bounds on the product of the number of facets f d−1 (P) and the number of vertices f0(P), where d is the dimension of a 2-level polytope P. This question was first posed in [3], where experimental results showed f0(P)f d−1 (P) ≤ d2 d+1 up to d = 6. We show that this bound holds for all known (to the best of our knowledge) 2-level polytopes coming from combinatorial settings, including stable set polytopes of perfect graphs and all 2-level base polytopes of matroids. For the latter family, we also give a simple description of the facet-defining inequalities. These results are achieved by an investigation of related combinatorial objects, that could be of independent interest.

Annals of the New York Academy of Sciences, 1989

Periodica Mathematica Hungarica, 2006

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2006.