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On the Dimensions of the Realization Spaces of Polytopes
Laith Rastanawi, Rainer Sinn, Günter M. Ziegler
Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full ra…
Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes
Joseph Doolittle, Jean-Philippe Labbé, Carsten E. M. C. Lange +3
For -dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hype…
Plus minus analogues for affine Tverberg type results
Pavle V. M. Blagojevic, Günter M. Ziegler
The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as w…
Characterizing face and flag vector pairs for polytopes
Hannah Sjöberg, Günter M. Ziegler
Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs of face numbers of -dimensional polytopes. Here we obtain a complete characterizatio…
Semi-algebraic sets of f-vectors
Hannah Sjöberg, Günter M. Ziegler
Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe tha…
Additive structures on -vector sets of polytopes
Günter M. Ziegler
We show that the -vector sets of -polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover,…
Simple polytopes without small separators, II: Thurston's bound
Lauri Loiskekoski, Günter M. Ziegler
We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least . This establishes a strong form of a cl…
Area difference bounds for dissections of a square into an odd number of triangles
Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler
Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences th…
Small f-vectors of 3-spheres and of 4-polytopes
Philip Brinkmann, Günter M. Ziegler
We present a new algorithmic approach that can be used to determine whether a given quadruple is the f-vector of any convex 4-dimensional polytope. By implement…
Simple polytopes without small separators
Lauri Loiskekoski, Günter M. Ziegler
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…
A flag vector of a 3-sphere that is not the flag vector of a 4-polytope
Philip Brinkmann, Günter M. Ziegler
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…
Barycenters of points that are constrained to a polytope skeleton
Pavle V. M. Blagojević, Florian Frick, Günter M. Ziegler
We give a short and simple proof of a recent result of Dobbins that any point in an -polytope is the barycenter of points in the -skeleton. This new proof builds on the…
Foldable Triangulations of Lattice Polygons
Michael Joswig, Günter M. Ziegler
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 certai…
Face numbers of centrally symmetric polytopes from split graphs
Ragnar Freij, Matthias Henze, Moritz W. Schmitt +1
We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3…
Many non-equivalent realizations of the associahedron
Cesar Ceballos, Francisco Santos, Günter M. Ziegler
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 n…
Three non-equivalent realizations of the associahedron
Cesar Ceballos, Günter M. Ziegler
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 pa…
Polyhedral Surfaces in Wedge Products
Thilo Rörig, Günter M. Ziegler
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, fr…
A Lost Counterexample and a Problem on Illuminated Polytopes
Ronald F. Wotzlaw, Günter M. Ziegler
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 s…
Non-rational configurations, polytopes, and surfaces
Günter M. Ziegler
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 cann…
Construction and Analysis of Projected Deformed Products
Raman Sanyal, Günter M. Ziegler
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 th…
Zonotopes With Large 2D Cuts
Thilo Rörig, Nikolaus Witte, Günter M. Ziegler
There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has Ω(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian e…
A Small Polyhedral Z-Acyclic 2-Complex in R4
Frank H. Lutz, Günter M. Ziegler
We present a small 4-dimensional polyhedral realization of a 2-dimensional Z-acyclic but non-contractible simplicial complex.
Polyhedral surfaces of high genus
Günter M. Ziegler
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 ``M…
Convex Polytopes: Extremal Constructions and f-Vector Shapes
Günter M. Ziegler
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 firs…