Sion's Minimax Theorem

In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem.

It states:

Let be a compact convex subset of a linear topological space and a convex subset of a linear topological space. If is a real-valued function on with

upper semicontinuous and quasiconcave on, and

is lower semicontinuous and quasi-convex on,

then,