Related papers: Polytopes of Minimum Positive Semidefinite Rank

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 $\mathcal{P}$.…

There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful…

The Fine interior $F(P)$ of a $d$-dimensional lattice polytope $P \subset {\Bbb R}^d$ is the set of all points $y \in P$ having integral distance at least $1$ to any integral supporting hyperplane of $P$. We call a lattice polytope…

In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

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 prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs…

If P is a rational polytope in R^d, then $i_P(t):=#(tP\cap Z^d)$ is a quasi-polynomial in t, called the Ehrhart quasi-polynomial of P. A period of i_P(t) is D(P), the smallest positive integer D such that D*P has integral vertices. Often,…

We prove that if $M \longrightarrow P$ is a small cover of a compact right-angled hyperbolic polyhedron $P$ then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if $\pi_1(M)$ is commensurable…

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…

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…

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show…

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…

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A=BB^T. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In…

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev $p$-norm ($1<p<\infty$) for the case $p=1$. Some relevant examples are indicated. The second part deals…