On a form of KKM principle and SupInfSup inequalities of von Neumann and of Ky Fan type
This article deals with some modification of the KKM-principle (Knaster- Kuratowski-Maurkiewicz): If X is a compact convex set in a Hausdorff topological vector space, Y is a topological space and S,A: \(X\to Y\) are two mappings for which S is upper semicontinuous and Ax is closed for each \(x\in X\) then either there exists \(x_ 0\in X\) such that \(Ax\cap Sx_ 0\neq \emptyset\) for all \(x\in X\) or S is trappable by \(A^ cx=Y\setminus Ax\). The latter means there are \(x_ 1,...,x_ m\in X\) and \(x\in co\{x_ 1,...,x_ n\}\) such that \(Sx\subseteq A^ cx_ i\) \((i=1,...,n)\). This result allows the author to obtain in a unique way the well-known minimaximal inequality of von Neumann and Ky Fan and some of their analogues and generalizations.
minimaximal inequality of von Neumann and Ky Fan
scientific article; zbMATH DE number 4214774
scientific article