View source for 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|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 [[#References|[1]]] that from his theorem, the [[Minimax principle|minimax principle]] for finite games does follow.

====References====

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

====References====

<table>
<TR><TD valign="top">[a1]</TD><TD valign="top">
J. Dugundji, A. Granas, “Fixed point theory”, '''1''', PWN (1982).</TD></TR>
</table>