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The E_tConstruction for Lattices, Spheres and Polytopes
"... We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction ..."
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Cited by 9 (7 self)
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We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction
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 ..."
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Cited by 5 (0 self)
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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.
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR
"... mathématicienne diplômée, informaticienne diplômée de l'Université de Genève originaire de Genève (GE) acceptée sur proposition du jury: Prof. A. Strohmeier, directeur de thèse Dr D. Buchs, rapporteur ..."
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mathématicienne diplômée, informaticienne diplômée de l'Université de Genève originaire de Genève (GE) acceptée sur proposition du jury: Prof. A. Strohmeier, directeur de thèse Dr D. Buchs, rapporteur
Constructions of Cubical Polytopes vorgelegt von
"... In this thesis we consider cubical dpolytopes, convex bounded ddimensional polyhedra all of whose facets are combinatorially isomorphic to the (d − 1)dimensional standard cube. It is known that every cubical dpolytope P determines a PL immersion of an abstract closed cubical (d−2)manifold into t ..."
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In this thesis we consider cubical dpolytopes, convex bounded ddimensional polyhedra all of whose facets are combinatorially isomorphic to the (d − 1)dimensional standard cube. It is known that every cubical dpolytope P determines a PL immersion of an abstract closed cubical (d−2)manifold into the polytope boundary ∂P ∼ = S d−1. The immersed manifold is orientable if and only if the 2skeleton of the cubical dpolytope (d ≥ 3) is “edge orientable ” in the sense of Hetyei. He conjectured that there are cubical 4polytopes that are not edgeorientable. In the more general setting of cubical PL (d − 1)spheres, Babson and Chan have observed that every type of normal crossing PL immersion of a closed PL (d−2)manifold into an (d−1)sphere appears among the dual manifolds of some cubical PL (d − 1)sphere. No similar general result was available for cubical polytopes. The reason for this may be blamed to a lack of flexible construction techniques for cubical polytopes, and for more general cubical complexes (such as the “hexahedral meshes ” that are of great interest in CAD and in Numerical Analysis). In this thesis, we develop a number of new and improved construction techniques for cubical polytopes. We try to demonstrate that it always pays off to carry along convex lifting functions of high symmetry. The most complicated and subtle one of our constructions generalizes the “Hexhoop template ” which is a wellknown technique in the domain of hexahedral meshes. Using the constructions developed here, we achieve the following results: • A rather simple construction yields a cubical 4polytope (with 72 vertices and 62 facets) for which the immersed dual 2manifold is not orientable: One of its components is a Klein bottle. Apparently this is the first example of a cubical polytope with a nonorientable dual manifold. Its existence confirms the conjecture of Hetyei mentioned above.
SOME GEOMETRIC CONSTRUCTION TECHNIQUES IN POLYTOPES
"... A convex polytope is the convex hull of a finite set of points in the Euclidean space. When its affine hull is of dimension d, we call it a dpolytope. In a dpolytope P we denote the number of jdimension face as fj(P), and call f(P) = (f0(P), f1(P),..., fd−1(P)) the fvector ..."
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A convex polytope is the convex hull of a finite set of points in the Euclidean space. When its affine hull is of dimension d, we call it a dpolytope. In a dpolytope P we denote the number of jdimension face as fj(P), and call f(P) = (f0(P), f1(P),..., fd−1(P)) the fvector