Related papers: A lower bound theorem for $d$-polytopes with at mo…

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 study lower bounds for the number of vertices in a PL-triangulation of a given manifold $M$. While most of the previous estimates are based on the dimension and the connectivity of $M$, we show that further information can be extracted…

A convex body $R$ in $\mathbb R^d$ is called reduced if the minimal width $\Delta(R')$ of each convex body $R'\subset R$ different from $R$ is strictly smaller than the minimal width $\Delta(R)$ of $R$. In this article we construct a…

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position…

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove…

This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter…

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of…

For a $(d-1)$-dimensional simplicial complex $\Delta$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $\Delta$ and $m_i$ be the number of missing $i$-faces of $\Delta$. In the nineties, Kalai asked for a…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined)…

For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in…

We show that the graph of a simplicial polytope of dimension $d \ge 3$ has no nontrivial minimum edge cut with fewer than $d(d+1)/2$ edges, hence the graph is $\min\{\delta, d(d+1)/2\}$-edge-connected where $\delta$ denotes the minimum…

This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r \leq 2d$. A lower bound of $f_d(C_{r + 1}(n)) =…

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans…

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…

We show that the size of a minimal simplicial cover of a polytope $P$ is a lower bound for the size of a minimal triangulation of $P$, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and…

We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the…

In this paper, we provide the upper bound and the lower bound of the sum of the number of spherical face-partition pair of simple spherical polytope $P$ with $p$ facets.

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked…