I wonder whether it possible to generalize the inequality $$ \sum_i \min \{ x_{i,1}, x_{i,2} \} \cdot \sum_i \max \{ x_{i,1}, x_{i,2} \} \le \sum_i x_{i,1} \cdot \sum_i x_{i,2} $$ to higher dimensions as follows $$ \sum_i \min \{ x_{i,1}, \dots, x_{i,J} \} \cdot \sum_i \frac{\prod_j x_{i,j}}{\min \{ x_{i,1}, \dots, x_{i,J} \}} \le \prod_j \sum_i x_{i,j} $$ where $0 \le x_{i,j} \le 1$, for all $i$ and $j$.
Addendum 1:
Are both (bivariate and multivariate) cases simply a consequence of the following inequality? $$ \sum_i \prod_j x_{i,j} \le \prod_j \sum_i x_{i,j} $$