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.

References

[1]	S. Kakutani, “A generalization of Brouwer's fixed point theorem”, Duke Math. J., 8: 3 (1941), pp. 457–459.
[2]	Ky Fan, “Fixed point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad. Sci. USA, 38 (1952), pp. 121–126,
[3]	H. Nikaido, “Convex structures and economic theory”, Acad. Press (1968).

References

[a1]	J. Dugundji, A. Granas, “Fixed point theory”, 1, PWN (1982).