mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

to investigate the existence of an equilibrium. Another Ky Fan inequality is an

inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

involving the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

and

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

of two sets of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s of the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

Statement

Suppose that

E

is a convex compact subset of a Hilbert space and that

f

is a function from

E\times E

\mathbb

satisfying *

x\mapsto f(x,y)

is lower semicontinuous for every

y\in E

and *

y\mapsto f(x,y)

is concave for every

x\in E

. Then there exists

e\in E

such that :

\sup_f(e,y)\le \sup_ f(y,y).

References

*{{cite journal , last = Aubin , first = Jean-Pierre , title = Optima and Equilibria , publisher = Springer-Verlag , series = Graduate Texts in Mathematics , year = 1998 , volume = 140 , doi = 10.1007/978-3-662-03539-9 Game theory