Related papers: Polytopes with few vertices and few facets

Kalai's $3^d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3^d$ faces. We give short proofs for two special cases: if $P$ is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane),…

We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and…

We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors.…

In this paper, we investigate two properties concerning the unimodality of the $\delta$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove…

We show that the $f$-vector sets of $d$-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or…

What is the minimal closed cone containing all $f$-vectors of cubical $d$-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant, thus…

Using equivariant topology, we prove that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold…

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a…

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…

Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and…

Let P be a random $d$-dimensional 0/1-polytope with $n(d)$ vertices, and denote by $\phi_k(P)$ the \emph{$k$-face density} of $P$, i.e., the quotient of the number of $k$-dimensional faces of $P$ and $\binom{n(d)}{k+1}$. For each $k\ge 2$,…

The rank of a $d$-dimensional polytope $P$ is defined by $F-(d+1)$, where $F$ denotes the number of facets of $P$. In this paper, We focus on the toric rings of $(0,1)$-polytopes with small rank. We study their normality, the…

When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…

Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis…

We prove a combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there are at most $2^{d+1}-2$ nearly neighbourly simplices in $\mathbb R^d$.

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we…

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local…

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of $k$-faces of $d$-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic…

Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…

Although the Unimodality Conjecture holds for some certain classes of cubical polytopes (e.g. cubes, capped cubical polytopes, neighborly cubical polytopes), it fails for cubical polytopes in general. A 12-dimensional cubical polytope with…