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Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were…
We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of…
In this paper, we describe the structural properties of the cone of $\mathcal{Z}$-transformations on the second order cone in terms of the semidefinite cone and copositive/completely positive cones induced by the second order cone and its…
We show that every thick subcategory of the singularity category of a complete intersection ring is self dual. We also prove the analogous statement for thick subcategories of the bounded derived category and give applications to the…
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)…
Our goal is twofold. On one hand we show that the cones of divisors ample in codimension $k$ on a Mori dream space are rational polyhedral. On the other hand we study the duality between such cones and the cones of $k$-moving curves by…
We study an extreme scenario of the Mastrand projection theorem for which a fractal has the property that its orthogonal projection is the same as the orthogonal projection of its convex hull. We extend results in current literature and…
A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit…
This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive). We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for…
We study mapping cones and their dual cones of positive maps of the n by n matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular…
We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite…
Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In…
Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f \in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it does not have a root whose vector of the imaginary parts is contained in the interior…
Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an {\epsilon}-factorised differential equation is persymmetric. It was observed…
We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of…
The strict complementary slackness condition (SCSC) is an important concept in the duality theory of linear programming (LP). The current study aims at extending this concept to the framework of linear fractional programming (LFP). First,…
We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…
In this note, we work out a simple inductive proof showing that every polyhedral cone K is the conic hull of a finite set X of vectors. The base cases of the induction are linear subspaces and linear halfspaces of linear subspaces. The…
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…