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Celebrating Loday's Associahedron
Vincent Pilaud, Francisco Santos, Günter M. Ziegler
We survey Jean-Louis Loday's vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry and algebra. We present in particular four…
Cutting a part from many measures
Pavle V. M. Blagojević, Nevena Palić, Pablo Soberón +1
Holmsen, Kynčl and Valculescu recently conjectured that if a finite set with points in that is colored by different colors can be partitioned into $…
Tverberg-type theorems for matroids: A counterexample and a proof
Pavle V. M. Blagojević, Albert Haase, Günter M. Ziegler
Bárány, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to…
Convex equipartitions of colored point sets
Pavle V. M. Blagojević, Günter Rote, Johanna K. Steinmeyer +1
We show that any -colored set of points in general position in can be partitioned into subsets with disjoint convex hulls such that the set of points and all…
Barycenters of Polytope Skeleta and Counterexamples to the Topological Tverberg Conjecture, via Constraints
Pavle V. M. Blagojević, Florian Frick, Günter M. Ziegler
Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skelet…
Using Brouwer's fixed point theorem
Anders Björner, Jiří Matoušek, Günter M. Ziegler
Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, r…
Tverberg plus constraints
Pavle V. M. Blagojević, Florian Frick, Günter M. Ziegler
Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof…
Extremal edge polytopes
Tuan Tran, Günter M. Ziegler
The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3,…
Many projectively unique polytopes
Karim A. Adiprasito, Günter M. Ziegler
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 Stein…
Realizing the associahedron: Mysteries and questions
Cesar Ceballos, Günter M. Ziegler
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 realiza…
Projecting lattice polytopes without interior lattice points
Benjamin Nill, Günter M. Ziegler
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…
Optimal bounds for the colored Tverberg problem
Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler
We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of B…
On locally constructible spheres and balls
Bruno Benedetti, Günter M. Ziegler
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-s…
On Kalai's conjectures concerning centrally symmetric polytopes
Raman Sanyal, Axel Werner, Günter M. Ziegler
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known a…
Combinatorial Stokes formulas via minimal resolutions
Bernhard Hanke, Raman Sanyal, Carsten Schultz +1
We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induce…
On generalized Kneser hypergraph colorings
Carsten Lange, Guenter M. Ziegler
In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs $\KG{r}{\pmb s}{\calS}$, "generalized -uniform Kneser hypergraphs with inte…
The Topological Tverberg Problem and winding numbers
Torsten Schöneborn, Günter M. Ziegler
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…
Bier spheres and posets
Anders Björner, Andreas Paffenholz, Jonas Sjöstrand +1
In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices…
Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds
Alexander Schwartz, Guenter M. Ziegler
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…
The Simplex Algorithm in Dimension Three
Volker Kaibel, Rafael Mechtel, Micha Sharir +1
We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we c…
Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors
Thomas Voigt, Günter M. Ziegler
Let be the probability that a random 0/1-matrix of size is singular, and let be the expected number of 0/1-vectors in the linear subspace spanned by d-1…
The Random Edge Rule on Three-Dimensional Linear Programs
Volker Kaibel, Raphael Mechtel, Micha Sharir +1
The worst-case expected length f(n) of the path taken by the simplex algorithm with the Random Edge pivot rule on a 3-dimensional linear program with n constraints is shown to be b…
Counting Lattice Triangulations
Volker Kaibel, Günter M. Ziegler
We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size . Among other tools, we employ…
Topological lower bounds for the chromatic number: A hierarchy
Jiri Matousek, Günter M. Ziegler
This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the \em…