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For questions related to optimization problems for choosing the optimal locations of facilities such as warehouses, factories, fire stations, etc.

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Without loss of generality take $m=2$. Then $$x_i \leq y\implies\sum_{i=1}^m x_i \leq my$$ is proven by direct summation as in the OP. On the other hand $$ \sum_{i=1}^m x_i \leq my\quad\not\!\!\!\!\im …
answered Dec 11 '20 by Konstantin