We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$Q = \left\{(q_{ij})_{(i,j) \in I \times J} \in \mathbb{R}^{I \times J} \mid q_{ij} \geq 0 \text{ for all } (i,j) \in I \times J\right\}$$ and $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ Verify that $F(\alpha, \beta)$ is a convex set of the vector space $E = \mathbb{R}^{I \times J}$.
We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote
$$Q = \left\{(q_{ij})_{(i,j) \in I \times J} \in \mathbb{R}^{I \times J} \mid q_{ij} \geq 0 \text{ for all } (i,j) \in I \times J\right\}$$
and
$$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$
Verify that $F(\alpha, \beta)$ is a convex set of the vector space $E = \mathbb{R}^{I \times J}$.