Let's consider a $(m\times n)$ matrix with real values. Let's name some sets:
$S_1$: the set of all maxima of each column of the above matrix
$S_2$: the set of all minima of each row of the above matrix
I would like to compare $A=\min(S_1)$ and $B=\max(S_2)$.
My multiple tries have let me think that $A\geq B$ holds, but I cannot formally prove it. Any help with that formal proof? Or does there no comparative relationship between $A, B$ generally exist?
For fixed $i$ you have that $$ \min_j a_{ij} \leq \min_j \max_k a_{kj} $$ but then as the RHS does not depend upon $i$: $$ \max_i \min_j a_{ij} \leq \min_j \max_k a_{kj} $$ Examples show that there need not be equality.