Related papers: Convex Set of Doubly Substochastic Matrices

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…

The type C_n full root polytope is the convex hull in R^n of the origin and the points e_i-e_j, e_i+e_j, 2e_k for 1 <= i < j <= n, k \in [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector…

It is well known that the number of tilting modules over a path algebra of type A_n coincides with the Catalan number C(n). Moreover, the number of support tilting modules of type A_n is the Catalan number C(n+1). We show that the convex…

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$…

We determine the extreme points and facets of the convex hull of all dual degree partitions of simple graphs on $n$ vertices.

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I^n can be…

We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary…

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to $1$. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if $d$…

In this paper we derive strong linear inequalities for sets of the form {(x, q) \in Rd \times R : q \geq Q(x), x \in Rd - int(P)}, where Q(x) : Rd \rightarrow R is a quadratic function, P \subset Rd and "int" denotes interior. Of particular…

Suppose that the set ${\mathcal{T}}= \{T_1, T_2,...,T_q \} $ of real $n\times n$ matrices has joint spectral radius less than $1$. Then for any digit set $ D= \{d_1, \cdots, d_q\} \subset {\Bbb R}^n$, there exists a unique nonempty compact…

A set $S \subset \mathbb{Z}^d$ is digital convex if $conv(S) \cap \mathbb{Z}^d = S$, where $conv(S)$ denotes the convex hull of $S$. In this paper, we consider the algorithmic problem of testing whether a given set $S$ of $n$ lattice points…

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…

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the…

For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots,a_k)$ of positive integers, let $V'_S(\mathbf{a})=\{ a_1s_1+\cdots+a_ks_k : s_i\in S\} \setminus \{0\}$. For a positive integer $n$, let $\mathcal T(n)$ be the set…

Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that…

Consider a regular triangulation of the convex-hull $P$ of a set $\mathcal A$ of $n$ points in $\mathbb R^d$, and a real matrix $C$ of size $d \times n$. A version of Viro's method allows to construct from these data an unmixed polynomial…

In this paper we present a complete characterization of the smallest sets which block all the simple perfect matchings in a complete convex geometric graph on $2m$ vertices. In particular, we show that all these sets are caterpillar graphs…

We prove P. Alexandersson's conjecture that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. We further prove that the equality $p^{-1}(H_p) =…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…