# Kakutani theorem

Let $X$ be a non-empty compact subset of $\mathbb{R}^{n}$, let $X^{*}$ be the set of its subsets, and let $f: X \to X^{*}$ be an upper semi-continuous mapping such that for each $x \in X$, the set $f(x)$ is non-empty, closed and convex. The theorem then states that $f$ has a fixed point (i.e., there is a point $x \in X$ such that $x \in f(x)$). S. Kakutani showed in [1] that from his theorem, the minimax principle for finite games does follow.