Editing Minimax theorem (section)

== Sion's minimax theorem ==

Sion's minimax theorem is a generalization of von Neumann's minimax theorem due to [[Maurice Sion]],<ref name=":1">{{cite journal |last=Sion |first=Maurice |year=1958 |title=On general minimax theorems |journal=[[Pacific Journal of Mathematics]] |volume=8 |issue=1 |pages=171–176 |doi=10.2140/pjm.1958.8.171 |mr=0097026 |zbl=0081.11502 |doi-access=free}}</ref> relaxing the requirement that X and Y be standard simplexes and that f be bilinear. It states:<ref name=":1" /><ref>{{cite journal |last=Komiya |first=Hidetoshi |year=1988 |title=Elementary proof for Sion's minimax theorem |url=http://projecteuclid.org/euclid.kmj/1138038812 |journal=[[Kodai Mathematical Journal]] |volume=11 |issue=1 |pages=5–7 |doi=10.2996/kmj/1138038812 |mr=0930413 |zbl=0646.49004}}</ref>

Let <math>X</math> be a [[Convex set|convex]] subset of a [[linear topological space]] and let <math>Y</math> be a [[compact space|compact]] [[Convex set|convex]] subset of a [[linear topological space]].  If <math>f</math> is a real-valued [[Function (mathematics)|function]] on <math>X\times Y</math> with

: <math>f(\cdot,y)</math> [[upper semicontinuous]]  and [[quasi-convex function|quasi-concave]] on <math>X</math>, for every fixed <math>y\in Y</math>, and
: <math>f(x,\cdot)</math> lower semicontinuous and quasi-convex on <math>Y</math>, for every fixed <math>x\in X</math>.

Then we have that

: <math>\sup_{x\in X}\min_{y\in Y} f(x,y) = \min_{y\in Y}\sup_{x\in X} f(x,y). </math>