A nonempty compact convex set in a finite-dimensional space and an upper hemicontinuous set-valued mapping with nonempty closed convex values admit . A closed graph suffices when the whole output space is compact.
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The Kakutani Fixed-Point Theorem is an important result in the field of mathematical analysis and game theory, particularly in the study of convex sets and continuous functions. It generalizes the Brouwer Fixed-Point Theorem, which is applicable in Euclidean spaces, to more complex scenarios involving multi-valued functions.