Related papers: Lower bound for the maximal number of facets of a …

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, for these values of $m$.…

In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension complexity is…

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…

We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies…

We give a complete enumeration of all 2-neighborly $d$-polytopes with $d+9$ and less facets. All of them are realized as 0/1-polytopes, except a 6-polytope $P_{6,10,15}$ with 10 vertices and 15 facets, and pyramids over $P_{6,10,15}$. In…

Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…

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.…

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We…

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…

A $0/1$-polytope in $\mathbb{R}^n$ is the convex hull of a subset of $\{0,1\}^n$. The graph of a polytope $P$ is the graph whose vertices are the zero-dimensional faces of $P$ and whose edges are the one-dimensional faces of $P$. A…

The Monotone Upper Bound Problem (Klee, 1965) asks if the number M(d,n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d,n) = M_{ubt}(d,n), the maximal…

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 $d$.

Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of $d$-dimensional twinned chain polytopes is at most…

Assume that $rB_{2}^{n} \subset P$ for some polytope $P \subset \mathbb{R}^n$, where $r \in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball…

Symmetric edge polytopes of graphs are important object in Ehrhart theory,and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of…

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes…

Let $\mathscr{O}(P)$ and $\mathscr{C}(P)$ denote the order polytope and chain polytope, respectively, associated with a finite poset $P$. We prove the following result: if $P$ is a maximal ranked poset, then the number of triangular…

We prove that there are 0/1 polytopes P that do not admit a compact LP formulation. More precisely we show that for every n there is a sets X \subseteq {0,1}^n such that conv(X) must have extension complexity at least 2^{n/2 * (1-o(1))}. In…

Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound…