Related papers: A lower bound theorem for $d$-polytopes with $2d+2…

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 $d$.

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the…

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…

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…

It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2…

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as…

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two $d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number of vertices of the polytopes.…

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of…

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$.…

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…

It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for…

We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new…

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$-skeleton. Call a vertex of a $d$-polytope \emph{nonsimple} if the number of edges incident to it is more than $d$.…

We consider the hypergraph Tur\'an problem of determining $\mathrm{ex}(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a…

It is known that in the Minkowski sum of $r$ polytopes in dimension $d$, with $r<d$, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand. However, the number of vertices for…

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…

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 $k\ge 2$,…

Let $d \geq 3$ be an integer. It is known that the number of edges of the edge polytope of the complete graph with $d$ vertices is $d(d-1)(d-2)/2$. In this paper, we study the maximum possible number $\mu_d$ of edges of the edge polytope…

We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $\Delta$ that represents a connected normal pseudomanifold of dimension $d\geq 3$ is at least as large as ${d+2 \choose 2}m(\Delta)$, where…