Results 1  10
of
13
polymake: a Framework for Analyzing Convex Polytopes
, 1999
"... polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few ..."
Abstract

Cited by 95 (15 self)
 Add to MetaCart
polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tuberlin.de/diskregeom/polymake/.
Face Numbers of 4Polytopes and 3Spheres
 Proceedings of the international congress of mathematicians, ICM 2002
, 2002
"... Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to s ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Construction techniques for cubical complexes, odd cubical 4polytopes, and prescribed dual manifolds
, 2003
"... ..."
PROJECTED PRODUCTS OF POLYGONS
, 2004
"... It is an open problem to characterize the cone of fvectors of 4dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4polytope can be arbitrarily large is a key problem in this context. Here we construct a 2parameter family of 4dimensional polytopes π(P 2r n) w ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
It is an open problem to characterize the cone of fvectors of 4dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4polytope can be arbitrarily large is a key problem in this context. Here we construct a 2parameter family of 4dimensional polytopes π(P 2r n) with extreme combinatorial structure. In this family, the “fatness ” of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the “complexity, ” gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to R4.
Reconstructing a nonsimple polytope from its graph
 DMV Seminar 29, Birkhauser, Basel 2000
, 1999
"... A wellknown theorem of Blind and Mani [4] says that every simple polytope is uniquely determined by its graph. In [11] Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai’s proof can be suitably generalized without much effort. ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
A wellknown theorem of Blind and Mani [4] says that every simple polytope is uniquely determined by its graph. In [11] Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai’s proof can be suitably generalized without much effort. We apply our results to a special class of cubical polytopes.
NEIGHBORLY CUBICAL POLYTOPES AND SPHERES
, 2005
"... Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytop ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills [16] can be embedded into 3. 1.
Constructions for 4polytopes and the cone of flag vectors
 Cont. Math
, 2005
"... Abstract. We describe a construction for dpolytopes generalising the well known stacking operation. The construction is applied to produce 2simplicial and 2simple 4polytopes with g2 = 0 on any number of n ≥ 13 vertices. In particular, this implies that the ray ℓ1, described by Bayer (1987), is f ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We describe a construction for dpolytopes generalising the well known stacking operation. The construction is applied to produce 2simplicial and 2simple 4polytopes with g2 = 0 on any number of n ≥ 13 vertices. In particular, this implies that the ray ℓ1, described by Bayer (1987), is fully contained in the convex hull of all flag vectors of 4polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented. 1.
M.: Geometric Reasoning with polymake
, 2005
"... The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background. 1