Convex Set of Doubly Substochastic Matrices

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 $\mathcal{B}_k$.