Exercise 27, § 5

Let $E, F$ be two Hausdorff topological vector spaces, and $X$ (resp. $Y$) a compact convex set in $E$ (resp. $F$). Let $f$ be a real valued function defined in $X \times Y$ with the following properties : (i) For all $x \in X$, the mapping $y \mapsto f(x, y)$ is lower semi-continuous in $Y$, and for all $c \in \mathbf{R}$, the set of the $y \in Y$ such that $f(x, y) \leq c$ is convex. (ii) For all $y \in Y$, the mapping $x \mapsto f(x, y)$ is upper semi-continuous in $X$, and for all $c \in \mathbf{R}$, the set of the $x \in X$ such that $f(x, y) \geq c$ is convex. Show that, in these conditions, we have

(Argue by reductio ad absurdum, supposing that there exists a number $c$ such that For all $x \in X$ (resp. all $y \in Y$) let $A_x$ be the set of $y \in Y$ such that $f(x, y) > c$ (resp. $B_y$ the set of the $x \in X$ such that $f(x, y) < c$), which is open in $Y$ (resp. in $X$); the $A_x$ (resp. the $B_y$) form a covering of $Y$ (resp. $X$) when $x$ varies in $X$ (resp. $y$ varies in $Y$). Show that there exist two finite sets $X_0 \subset X, Y_0 \subset Y$ such that : 1° for all $y$ belonging to the convex envelope $B_0$ of $Y_0$, there exists $x \in X_0$ such that $f(x, y) > c$, and $X_0$ is minimal for this property ; 2° for all $x$ belonging to the convex envelope $A_0$ of $X_0$, there exists $y \in Y_0$ such that $f(x, y) < c$, and $Y_0$ is minimal for this property. Then for all $y \in Y_0$, let $C_y$ be the set of $x \in A_0$ such that $f(x, y) \geq c$; using exerc. 21, a) of II, p. 79, show that the intersection of the $C_y$ for $y \in Y_0$ is not empty. Proceed in the same way in $B_0$ and obtain a contradiction.)