Convex Set of Doubly Substochastic Matrices

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