Related papers: Face numbers of centrally symmetric polytopes from…

We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the…

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this…

We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplexes in the classical results of triangulation of semi-algebraic sets over real closed…

In a d-simplex every facet is a (d-1)-simplex. We consider as generalized simplices other combinatorial classes of polytopes, all of whose facets are in the class. Cubes and multiplexes are two such classes of generalized simplices. In this…

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…

We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as…

It will be proved that a $k$-clique in the $1$-skeleton of either the order polytope or the chain polytope corresponds to the $(k-1)$-face, which is a simplex, in each polytope. These results generalize the known explicit descriptions of…

We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics…

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gr\"obner basis techniques, half-open decompositions and methods for…

In this note we prove that the number of combinatorial types of $d$-polytopes with $d+1+\alpha$ vertices and $d+1+\beta$ facets is bounded by a constant independent of $d$.

Given a finite quiver (directed graph) without loops and multiedges, the convex hull of the column vector of the incidence matrix is called the directed edge polytope and is an interesting example of lattice polytopes. In this paper, we…

The polytope $ASM_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $ASM_n$ corresponds to an elementary flow grid defined by Striker, and each elementary flow…

We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal…

Consider the question: Given integers $k<d<n$, does there exist a simple $d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers $G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only if…

A nearly platonic graph is a k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for a finite graph to have exactly one disparate face, and offer some conjectures,…

A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so…

Barnette's conjecture states that every cubic, bipartite, planar and 3-connected graph is Hamiltonian. Goodey verified Barnette's conjecture for all graphs with faces of size up to 6. We substantially strengthen Goodey's result by proving…

Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating…

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence,…

For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the…