Results 1  10
of
12
Construction and analysis of projected deformed products
, 2007
"... We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly p ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the kfaces) are “strictly preserved ” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)polytope Q on n−1 vertices we construct a deformed ncube, whose projection to the last d coordinates yields a neighborly cubical dpolytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs), which have a projection to dspace that retains the complete ( ⌊ d 2 ⌋ − 1)skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the projected deformed products of polygons that were announced by Ziegler (2004), a family of 4polytopes whose “fatness ” gets arbitrarily close to 9. 1
Flipping Cubical Meshes
 ACM Computer Science Archive June
, 2001
"... We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
NONPROJECTABILITY OF POLYTOPE SKELETA
, 2009
"... We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the giv ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to espace strictly preserves the kskeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of Rörig & Ziegler (2009) and complement their results.
Reconstructing a Simple Polytope from its Graph
 Combinatorial Optimization – Eureka, You Shrink
, 2002
"... Blind and Mani [2] proved that the entire combinatorial structure (the vertexfacet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai [14] found a short, elegant, and algorithmic proof of that result. However, his algorithm has an exp ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Blind and Mani [2] proved that the entire combinatorial structure (the vertexfacet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai [14] found a short, elegant, and algorithmic proof of that result. However, his algorithm has an exponential running time. We show that the problem to reconstruct the vertexfacet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of nding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertexfacet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joined work with Michael Joswig and Friederike Körner [11].
Polytopality and Cartesian products of graphs
 Accepted in Israel Journal of Mathematics
"... Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of po ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of nonpolytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (nonsimple) polytopal products whose factors are not polytopal. Even though graphs are perhaps the most prominent feature of polytopes, we are still far from being able to answer several basic questions regarding them. For applications, one of the most important ones is to bound the diameter of the graph in terms of the number of variables and inequalities defining the polytope [San10]. From a theoretical point of view, it is striking that we cannot even efficiently decide whether a given graph occurs as the graph of a polytope or not [RG96].
PRODSIMPLICIALNEIGHBORLY POLYTOPES
, 908
"... Abstract. We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products ” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we moreover require the PSN polytope to be obtained as a projection of a polytope combinatorially equivalent to the product of r simplices, when the sum of their dimensions is at least 2k. 1.
unknown title
, 2002
"... Abstract. We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation. ..."
Abstract
 Add to MetaCart
Abstract. We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
unknown title
, 2002
"... Abstract. We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation. ..."
Abstract
 Add to MetaCart
Abstract. We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.