Remark on a theorem of Lindström

Let Ω denote the set of all n by n doubly stochastic matrices and let m be a positive integer. Then the set $\text{Ω}{}_{\mathrm{m}}\text{= {A ? Ω : 0 ? a}{}_{\mathrm{ij}}\text{?}\text{1}\text{m}\text{, 1 ? i, j ≤ n}}$ is the convex hull of the matrices in $\text{Ω}{}_{\mathrm{m}}$ having exactly m entries equal to $\text{1}\text{m}$ in each row and column and the other entries equal to zero.