# Minimax theorem

### From Glossary

Proven by von Neumann in 1928, this is a cornerstone of duality (and of game theory). It states that there exists and such that is a saddlepoint of the bilinear form:

This extends to the following.

Let be such that and are non-empty, convex, compact sets, is convex on for each , and is concave on for each . Then, there exists a saddlepoint, such that