Related papers: A lower bound theorem for centrally symmetric simp…

In 1967, Gr\"unbaum conjectured that the function $$ \phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a…

We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a…

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d<4. We solve the problem up…

The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$, there is a point $c$, not necessarily from $P$, such that each halfspace containing…

Let $\mu_{\text{2n}}(d,v)$ (respectively, $\mu^{\text{s}}_{\text{2n}}(d,v)$) be the minimal number of facets of a (simplicial) 2-neighborly $d$-polytope with $v$ vertices, $v > d \ge 4$. It is known that $\mu_{\text{2n}}(4,v) = v (v-3)/2$,…

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…

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…

We prove a lower bound on the number of ordinary conics determined by a finite point set in $\mathbb{R}^2$. An ordinary conic for a subset $S$ of $\mathbb{R}^2$ is a conic that is determined by five points of $S$, and contains no other…

For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of…

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most…

Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of…

The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the…

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 prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case…

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that…

For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we…

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)) =…

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$ with…

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and…

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…