Kakutani Fixed-point Theorem - Alternative Statement

Alternative Statement

Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem:

Let S be a non-empty, compact and convex subset of some Euclidean space Rn. Let φ: S→2S be an upper hemicontinuous set-valued function on S with the property that φ(x) is non-empty, closed and convex for all x ∈ S. Then φ has a fixed point.

This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article.

We can show this by using the Closed graph theorem for set-valued functions, which says that a for a compact Hausdorff range space Y, a set-valued function φ: X→2Y has a closed graph if and only if it is upper hemicontinuous and φ(x) is a closed set for all x. Since all Euclidean spaces are Hausdorff (being metric spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent.