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# Applications of minmax theorem(s)

Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions, $$\inf_Y \sup_X f = \sup_X \inf_Y f$$.

This theorem is full of applications in a lot of different fiels of mathematics, applied mathematics, statistics, economy, ...

Question: what is the application of this theorem you prefer? (the deepest, the most tricky ...) what is the interpretation of minmax duality that strikes you the most ?

Companion document: 11 different formulations of minimax theorem (with different assumptions, context ...) can be found in http://www.math.ucsb.edu/~simons/preprints/Eoo.pdf and we should refer to these formulations in answers to be more rigorous.