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74
Boundary complexes of convex polytopes cannot be characterized locally
 BULL. LONDON MATH. SOC
, 1987
"... It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial spher ..."
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Cited by 3 (1 self)
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It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial
On the Even Permutation Polytope
, 2004
"... Abstract We consider the convex hull of the even permutations on a set of n elements. We define a class of valid inequalities and prove that they induce a large class of distinct facets of the polytope. Using the inequalities, we characterize the polytope for n = 4, and we confirm a conjecture of Br ..."
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of Brualdi and Liu that, unlike the convex hull of all permutations, this polytope cannot be described as the solution set of polynomially many linear inequalities. We also discuss the difficulty of determining whether a given point is in the polytope.
Cutting a Polytope
, 1990
"... We show that given two vertices of a polytope one cannot in general find a hyperplane containing the vertices, that has two or more facets of the polytope in one closed halfspace. Our result refutes a longstanding conjecture. We prove the result by constructing a 4dimensional polytope that provi ..."
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We show that given two vertices of a polytope one cannot in general find a hyperplane containing the vertices, that has two or more facets of the polytope in one closed halfspace. Our result refutes a longstanding conjecture. We prove the result by constructing a 4dimensional polytope
On Perfect 4Polytopes
"... Abstract. The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P) on its facelattice F (P). By Rostami’s conj ..."
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Abstract. The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P) on its facelattice F (P). By Rostami’s
The arithmetic of rational polytopes
, 2000
"... We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of the dilates of a polytope. We focus on applications of this theory to several problems in combinato ..."
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Cited by 2 (0 self)
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We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of the dilates of a polytope. We focus on applications of this theory to several problems
Connectivity in the regular polytope representation
, 2008
"... # The Author(s) 2009. This article is published with open access at Springerlink.com Abstract In order to be able to draw inferences about real world phenomena from a representation expressed in a digital computer, it is essential that the representation should have a rigorously correct algebraic st ..."
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, and cannot directly model a mathematical abstraction of space based on real numbers. This paper describes a basis for the robust geometrical construction of spatial objects in computer applications using a complex called the “Regular Polytope”. In contrast to most other spatial data types, this definition
The brick polytope of a sorting network
 EUROPEAN J. COMBIN
"... The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud & Pocchiola in their study of flip graphs on pseudoline arrang ..."
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Cited by 15 (9 self)
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networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.
OBSTRUCTIONS TO WEAK DECOMPOSABILITY FOR SIMPLICIAL POLYTOPES
, 2012
"... Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facetridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that a ..."
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that are not weakly vertexdecomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these ddimensional polytopes are not even weakly O ( √ d)decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial
COLOURING POLYTOPIC PARTITIONS IN Ê d
"... Abstract. We consider facetoface partitions of bounded polytopes into convex polytopes in Ê d for arbitrary d � 1 and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed d + 1. Partitions of polyhedra in Ê 3 into pentahedra and ..."
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Abstract. We consider facetoface partitions of bounded polytopes into convex polytopes in Ê d for arbitrary d � 1 and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed d + 1. Partitions of polyhedra in Ê 3 into pentahedra
On approximations by projections of polytopes with few facets
 Israel J. Math
"... Abstract We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an ndimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an ..."
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Cited by 2 (1 self)
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Abstract We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an ndimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists
Results 1  10
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74