Convex Set of Doubly Substochastic Matrices

TL;DR

This paper proves that the set of all $1/k$-bounded doubly substochastic matrices is the convex hull of matrices with entries only 0 or 1/k, providing a geometric characterization of these matrix sets.

Contribution

It establishes that the set of $1/k$-bounded doubly substochastic matrices is exactly the convex hull of matrices with entries 0 or 1/k, revealing their convex structure.

Findings

$ ext{A}_k$ is the convex hull of $ ext{B}_k$.

Characterization of doubly substochastic matrices.

Geometric insight into matrix set structure.

Abstract

Denote $\mathcal{A}$ as the set of all doubly substochastic $m \times n$ matrices and let $k$ be a positive integer. Let $\mathcal{A}_k$ be the set of all $1/k$-bounded doubly substochastic $m \times n$ matrices, i.e., $\mathcal{A}_k \triangleq \{E \in \mathcal{A}: e_{i,j} \in [0, 1/k], \forall i=1,2,\cdots,m, j = 1,2,\cdots, n\}$. Denote $\mathcal{B}_k$ as the set of all matrices in $\mathcal{A}_k$ whose entries are either $0$ or $1/k$. We prove that $\mathcal{A}_k$ is the convex hull of all matrices in $\mathcal{B}_k$.