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174

Mar 20, 2007
03/07

by
M. Ziegler

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Todd doing the Swissmas 2006 dance in Grindelwald.

Topics: Grindelwald, Switzerland, Swissmas

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Sep 23, 2013
09/13

by
Günter M. Ziegler

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These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e.g. using {\tt polymake}). However, any intuition...

Source: http://arxiv.org/abs/math/9909177v1

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30

Sep 19, 2013
09/13

by
Günter M. Ziegler

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In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many more edges and 2-faces than there are vertices and facets, while complexity C(P) := (f_{03}-20)/(f_0+f_3-10) is large if every facet has many vertices, and every vertex is in many facets. Recent results suggest that these parameters might allow one to...

Source: http://arxiv.org/abs/math/0208073v2

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0.0

Jan 25, 2020
01/20

by
A M Ziegler

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29

Sep 24, 2013
09/13

by
Günter M. Ziegler

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The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization...

Source: http://arxiv.org/abs/math/0103146v1

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Sep 24, 2013
09/13

by
Günter M. Ziegler

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The construction of the COMBINATORIAL data for a surface with n vertices of maximal genus is a classical problem: The maximal genus g=[(n-3)(n-4)/12] was achieved in the famous ``Map Color Theorem'' by Ringel et al. (1968). We present the nicest one of Ringel's constructions, for the case when n is congruent to 7 mod 12, but also an alternative construction, essentially due to Heffter (1898), which easily and explicitly yields surfaces of genus g ~ 1/16 n^2. For GEOMETRIC (polyhedral) surfaces...

Source: http://arxiv.org/abs/math/0412093v1

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42

Sep 23, 2013
09/13

by
Günter M. Ziegler

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These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction...

Source: http://arxiv.org/abs/math/0411400v2

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Sep 23, 2013
09/13

by
Günter M. Ziegler

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It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes'': combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple construction of non-rational polytopes that does not need duality (Perles' ``Gale diagrams''): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational...

Source: http://arxiv.org/abs/0710.4453v2

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Sep 19, 2013
09/13

by
Günter M. Ziegler

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Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a classical result by Orlik \& Solomon asserts that the cohomology algebra of the complement can be constructed from the combinatorial data that are given by the intersection lattice. If $B'$ is, more generally, a $2$-arrangement in $R^{2d}$ (an arrangement of real subspaces of codimension $2$ with even-dimensional intersections), then the intersection lattice still determines the cohomology {\it groups} of the...

Source: http://arxiv.org/abs/alg-geom/9202005v1

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26

Sep 21, 2013
09/13

by
Günter M. Ziegler

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We construct a 2-parameter family of 4-dimensional polytopes with extreme combinatorial structure: In this family, the ``fatness'' of the f-vector gets arbitrarily close to 9, the ``complexity'' (given by the flag vector) gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to four-space.

Source: http://arxiv.org/abs/math/0407042v1

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2.0

Jan 25, 2020
01/20

by
A M Ziegler

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26

Sep 22, 2013
09/13

by
Thilo Rörig; Günter M. Ziegler

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We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be...

Source: http://arxiv.org/abs/0908.3159v1

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104

Sep 22, 2013
09/13

by
Benjamin Nill; Günter M. Ziegler

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We show that up to unimodular equivalence there are only finitely many d-dimensional lattice polytopes without interior lattice points that do not admit a lattice projection onto a (d-1)-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein. As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner and Weismantel, namely the finiteness of the number of maximal lattice polytopes without interior lattice points. Moreover, we...

Source: http://arxiv.org/abs/1101.4292v2

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0.0

Jun 28, 2018
06/18

by
Emerson León; Günter M. Ziegler

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We construct and study the space C(\R^d,n) of all partitions of \R^d into n non-empty open convex regions (n-partitions). A representation on the upper hemisphere of an n-sphere is used to obtain a metric and thus a topology on this space. We show that the space of partitions into possibly empty regions C(\R^d,\le n) yields a compactification with respect to this metric. We also describe faces and face lattices, combinatorial types, and adjacency graphs for $n$-partitions, and use these...

Topics: Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1511.02904

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2.0

Jun 28, 2018
06/18

by
Lauri Loiskekoski; Günter M. Ziegler

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We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log^{3/2}n)$. This disproves a conjecture by Kalai from 1991/2004.

Topics: Combinatorics, Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1510.00511

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50

Sep 22, 2013
09/13

by
Alexander Schwartz; Guenter M. Ziegler

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We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R^3 PL-equivalent to a dual manifold immersion of a...

Source: http://arxiv.org/abs/math/0310269v3

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40

Sep 23, 2013
09/13

by
Cesar Ceballos; Günter M. Ziegler

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We review three realizations of the associahedron that arise as secondary polytopes, from cluster algebras, and as Minkowski sums of simplices, and show that under any choice of parameters, the resulting associahedra are affinely non-equivalent.

Source: http://arxiv.org/abs/1006.3487v2

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Sep 20, 2013
09/13

by
Michael Joswig; Günter M. Ziegler

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We give a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. This yields lower bounds on the number of real roots of certain of systems of polynomial equations known as "Wronski systems".

Source: http://arxiv.org/abs/1207.6865v2

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86

Sep 23, 2013
09/13

by
Bernd Gonska; Günter M. Ziegler

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We characterize the combinatorial types of stacked d-polytopes that are inscribable. Equivalently, we identify the triangulations of a simplex by stellar subdivisions that can be realized as Delaunay triangulations.

Source: http://arxiv.org/abs/1111.5322v1

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30

Sep 23, 2013
09/13

by
Cesar Ceballos; Günter M. Ziegler

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There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe three -- concerning special realizations with the vertices on a sphere, the space of all possible realizations, and possible realizations of the multiassociahedron.

Source: http://arxiv.org/abs/1110.4059v1

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Sep 22, 2013
09/13

by
Torsten Schöneborn; Günter M. Ziegler

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The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d=1, and if q is a prime power, but not yet in general.) The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a ``Winding Number Conjecture'' that concerns only maps of the (d-1)-skeleton of a (q-1)(d+1)-simplex to \R^d. ``Many Tverberg...

Source: http://arxiv.org/abs/math/0409081v2

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1.0

Jun 28, 2018
06/18

by
Philip Brinkmann; Günter M. Ziegler

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We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two...

Topics: Metric Geometry, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1506.08148

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70

Sep 22, 2013
09/13

by
Julian Pfeifle; Günter M. Ziegler

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The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)

Source: http://arxiv.org/abs/math/0308186v1

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116

Jul 20, 2013
07/13

by
Francisco Santos; Günter M. Ziegler

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A seminal result in the theory of toric varieties, due to Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds showing that: 1. It contains all composite...

Source: http://arxiv.org/abs/1304.7296v1

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1.0

Jun 28, 2018
06/18

by
Arnau Padrol; Günter M. Ziegler

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Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six particular topics related to inscribable polytopes. Along the way we collect a list of (new and old) open questions.

Topics: Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1511.03458

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51

Sep 22, 2013
09/13

by
Christian Haase; Günter M. Ziegler

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The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of...

Source: http://arxiv.org/abs/math/0011170v2

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28

Sep 19, 2013
09/13

by
Jiri Matousek; Günter M. Ziegler

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This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} $\KG_{m,n}$, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by B\'ar\'any,...

Source: http://arxiv.org/abs/math/0208072v3

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38

Sep 19, 2013
09/13

by
Andreas Paffenholz; Günter M. Ziegler

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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 does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein, Kuperberg...

Source: http://arxiv.org/abs/math/0304492v2

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37

Sep 18, 2013
09/13

by
Carsten Lange; Guenter M. Ziegler

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In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs $\KG{r}{\pmb s}{\calS}$, "generalized $r$-uniform Kneser hypergraphs with intersection multiplicities $\pmb s$." It generalized previous lower bounds by Kriz (1992/2000) for the case ${\pmb s}=(1,...,1)$ without intersection multiplicities, and by Sarkaria (1990) for $\calS=\tbinom{[n]}k$. Here we discuss subtleties and difficulties that arise for intersection multiplicities...

Source: http://arxiv.org/abs/math/0504607v2

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0.0

Jun 29, 2018
06/18

by
Philip Brinkmann; Günter M. Ziegler

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We present a new algorithmic approach that can be used to determine whether a given quadruple $(f_0,f_1,f_2,f_3)$ is the f-vector of any convex 4-dimensional polytope. By implementing this approach, we classify the f-vectors of 4-polytopes in the range $f_0+f_3\le22$. In particular, we thus prove that there are f-vectors of cellular 3-spheres with the intersection property that are not f-vectors of any convex 4-polytopes, thus answering a question that may be traced back to the works of...

Topics: Metric Geometry, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1610.01028

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40

Sep 18, 2013
09/13

by
Julian Pfeifle; Günter M. Ziegler

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We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about d-spheres and (d+1)-polytopes for fixed d...

Source: http://arxiv.org/abs/math/0212004v2

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69

Sep 21, 2013
09/13

by
Bruno Benedetti; Günter M. Ziegler

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Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and...

Source: http://arxiv.org/abs/0902.0436v4

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Sep 23, 2013
09/13

by
Raman Sanyal; Günter M. Ziegler

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We introduce a deformed product construction for simple polytopes in terms of lower-triangular 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 k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields...

Source: http://arxiv.org/abs/0710.2162v1

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Sep 22, 2013
09/13

by
Thomas Voigt; Günter M. Ziegler

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Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on $P(d)$ are equivalent to bounds on $E(d)$: \[ P(d) = (2^{-d} E(d) + \frac{d^2}{2^{d+1}}) (1 + o(1)). \] We also report about computational...

Source: http://arxiv.org/abs/math/0308050v3

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1.2K

Jul 29, 2017
07/17

by
Martin Aigner, Günter M. Ziegler

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Book Proofs from THE BOOK, 5th edition. The one that says "-PRINT" has extra blank pages for printing.

Topics: math, proofs, demonstrations

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0.0

Jan 20, 2020
01/20

by
Harry B Weiss; Grace M Ziegler

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25

Sep 23, 2013
09/13

by
Karim A. Adiprasito; Günter M. Ziegler

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We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard...

Source: http://arxiv.org/abs/1212.5812v2

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1.0

Jan 20, 2020
01/20

by
Harry B Weiss; Grace M Ziegler

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0.0

Jan 20, 2020
01/20

by
Harry B Weiss; Grace M Ziegler

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54

Sep 20, 2013
09/13

by
Michael Joswig; G"unter M. Ziegler

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This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if...

Source: http://arxiv.org/abs/math/0301100v1

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2.0

Nov 19, 2019
11/19

by
Harry B Weiss; G M Ziegler

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0.0

Jan 20, 2020
01/20

by
Harry B Weiss; Grace M Ziegler

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57

Sep 17, 2013
09/13

by
M. Ziegler; P. Sievers; U. Straumann

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The triple GEM detector is a micropattern gas detector which consists of a primary ionisation gap and three consecutive gas electron multiplier (GEM) foils. A printed circuit board with readout strips detects the current induced by the drifting electron cloud originating from the last GEM stage. Thus the gas amplification and the signal readout are completely separated. Triple GEM detectors are being developed as a possible technology for the inner tracking in the LHCb experiment. In an earlier...

Source: http://arxiv.org/abs/hep-ex/0007007v1

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Sep 19, 2013
09/13

by
Michael Joswig; G"unter M. Ziegler

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Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary $\partial C^n_d$ of a neighborly cubical polytope $C^n_d$ maximizes the $f$-vector among all cubical $(d-1)$-spheres with $2^n$ vertices. While we show that this is true for polytopal spheres for $n\le d+1$, we also...

Source: http://arxiv.org/abs/math/9812033v2

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Sep 18, 2013
09/13

by
Jürgen Richter-Gebert; Günter M. Ziegler

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Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$....

Source: http://arxiv.org/abs/math/9510217v1

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Sep 22, 2013
09/13

by
Ronald F. Wotzlaw; Günter M. Ziegler

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In a Note added in proof to a 1984 paper, Daniel A. Marcus claimed to have a counterexample to his conjecture that a minimal positively k-spanning vector configuration in R^m has size at most 2km. However, the counterexample was never published, and seems to be lost. Independently, ten years earlier, Peter Mani in 1974 solved a problem by Hadwiger, disproving that every ``illuminated'' d-dimensional polytope must have at least 2d vertices. These two studies are related by Gale duality, an...

Source: http://arxiv.org/abs/0908.1698v1

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2.0

Jan 20, 2020
01/20

by
Harry B Weiss; G M Ziegler

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Sep 23, 2013
09/13

by
Thilo Rörig; Nikolaus Witte; Günter M. Ziegler

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There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs'' by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.

Source: http://arxiv.org/abs/0710.3116v3

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Sep 23, 2013
09/13

by
Cesar Ceballos; Francisco Santos; Günter M. Ziegler

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Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two different ways of constructing exponential families of realizations of the n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify the associahedra obtained by these constructions modulo linear equivalence of their normal fans and show, in particular, that the only realization that can be obtained with both methods is the...

Source: http://arxiv.org/abs/1109.5544v2

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Sep 24, 2013
09/13

by
Pavle V. M. Blagojević; Günter M. Ziegler

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We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

Source: http://arxiv.org/abs/1202.5504v3