**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.

**inspirations:** Some references on the web to start with

- Monge-Kantorovitch duality (transportation theory) http://en.wikipedia.org/wiki/Transportation_theory#Monge_and_Kantorovich_formulations
- Decision Theory http://en.wikipedia.org/wiki/Minimax
- I don't have it anymore but I remenber the book of Simons was so interesting http://www.amazon.com/Minimax-Monotonicity-Lecture-Notes-Mathematics/dp/3540647554 (I can't remember the proof of Banach Alaoglu Theorem that was given there using 3 lines and Min max theorem)