Ky Fan's inequality

In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.

Statement

Suppose that [math]\displaystyle{ E }[/math] is a convex compact subset of a Hilbert space and that [math]\displaystyle{ f }[/math] is a function from [math]\displaystyle{ E\times E }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] satisfying

• [math]\displaystyle{ x\mapsto f(x,y) }[/math] is lower semicontinuous for every [math]\displaystyle{ y\in E }[/math] and
• [math]\displaystyle{ y\mapsto f(x,y) }[/math] is concave for every [math]\displaystyle{ x\in E }[/math].

Then there exists [math]\displaystyle{ e\in E }[/math] such that

[math]\displaystyle{ \sup_{y\in E}f(e,y)\le \sup_{y\in E} f(y,y). }[/math]

References

• Aubin, Jean-Pierre. Optima and Equilibria. Graduate Texts in Mathematics. 140. Springer-Verlag. doi:10.1007/978-3-662-03539-9.