Related papers: Refined Estimates on the Dimensions of Maximal Fac…

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum…

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner)…

We generalized to higher dimensions the notions of optical orthogonal codes. We establish uper bounds on the capacity of general $ n $-dimensional OOCs, and on specific types of ideal codes (codes with zero off-peak autocorrelation). The…

Postive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$,…

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number…

Let $n \geq 5$ and let $u^1,\dots,u^n$ be nonnegative real $n$-vectors such that the indices of their positive elements form the sets $\{1,2,\dots,n-2\},\{2,3,\dots,n-1\},\dots,\{n,1,\dots,n-3\}$, respectively. Here each index set is…

Error bounds are a requisite for trusting or distrusting solutions in an informed way. Until recently, provable error bounds in the absence of constraint qualifications were unattainable for many classes of cones that do not admit…

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…

We compute the minimal exponent of the affine cone over a complete intersection of smooth projective hypersurfaces intersecting transversely. The upper bound for the minimal exponent is proved, more generally, in the weighted homogeneous…

A new lower bound for the maximal length of a multivector is obtained. It is much closer to the best known upper bound than previously reported lower bound estimates. The maximal length appears to be unexpectedly large for $n$-vectors, with…

We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone…

Motivated by the separability problem in quantum systems $2\otimes4$, $3\otimes3$ and $2\otimes2\otimes2$, we study the maximal (proper) faces of the convex body, $S_1$, of normalized separable states in an arbitrary quantum system with…

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in $\R^n$ in terms of the cone spectral radius. We also define weak upper and lower bounds for these maps. For a proper…

This article investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in 2019. Here this notion is extended to matrix convex sets in…

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally…

The problem of matrix completion and decomposition in the cone of positive semidefinite (PSD) matrices is a well-understood problem, with many important applications in areas such as linear algebra, optimization, and control theory. This…

In this work we will discuss the facial structure of the cone of nonnegative ternary quartics with real coefficients. We will establish an equivalence relation on the set of all faces, which preserves certain properties like dimension or…

We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points…

An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A…

In this paper, we study the facial structure of the linear image of a cone. We define the singularity degree of a face of a cone to be the minimum number of steps it takes to expose it using exposing vectors from the dual cone. We show that…