A note on König's minimax theorem (Q1320488)

The following generalization of König's minimax theorem [\textit{H. König}, Arch. Math. 19, 482-487 (1968; Zbl 0179.210)] is deduced from \textit{Ky Fan's} minimax theorem [Proc. Nat. Acad. Sci. USA 39, 42-47 (1953; Zbl 0050.065)] by the function lifting technique of \textit{I. Joó} and the author [Acta Math. Acad. Sci. Hung. 39, 401-407 (1982; Zbl 0493.49013)]. Let \(X\) be a compact space, \(Y\neq \emptyset\) any set and \(f: X\times Y\to\mathbb{R}\) be a real-valued function. By identifying the points of \(Y\) with their indicator functions, we embed \(Y\) into the space \(\overline Y\) of all functions \(Y\to \mathbb{R}\) with finite support and we define the lifted function \(\overline f: X\to \overline Y\) as the affine continuation of \(f\) in the second variable. Then we have \(\inf_{y\in Y} \sup_{x\in X} f(x,y)= \sup_{x\in X} \inf_{y\in Y} f(x,y)\) whenever the subfunctions \(x\mapsto f(x,y)\) are upper semicontinuous and concave in Ky Fan's sense and the set \(\{\overline y\in \text{co}(Y): \exists y\in Y,\;f(\cdot,y)\leq \overline f(\cdot,\overline y)\}\) is dense in the convex hull \(\text{co}(Y)\) of \(Y\) with respect to the finest locally convex topology in \(\overline Y\).