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Extended formulations are an important tool to obtain small (even compact) formulations of polytopes by representing them as projections of higher dimensional ones. It is an important question whether a polytope admits a small extended…
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…
Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by…
We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be…
Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented…
A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented…
Gr\"unbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $(f_0,f_{03})$ for…
Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we…
We completely characterize the faces of the root polytope $\tilde Q_G = \text{conv}\{\mathbf 0, \mathbf e_i - \mathbf e_j\: (i,j) \in E(G)\}$ combinatorially. Our results specialize to state of the art results in a straightforward way.
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with…
We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually…
This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans…
We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and…
The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on…
Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense…
A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize…
An Ingletonian polymatroid satisfies, in addition to the polymatroid axioms, the inequalities of Ingleton (Combin. Math. Appln., 1971). These inequalities are required for a polymatroid to be representable. It is has been an open question…
A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope…
A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set…
We analyze the facial structures of the convex set consisting of all two qubit separable states. One of faces is a four dimensional convex body generated by the trigonometric moment curve arising from polyhedral combinatorics. Another one…