The Kakutani Fixed-Point Theorem states that a set-valued function satisfying certain convexity and continuity conditions on a compact convex subset of Euclidean space must have a fixed point, a point that the function maps to a set containing that same point. Proved by Shizuo Kakutani, it generalizes the Brouwer Fixed-Point Theorem to set-valued maps and was later used by John Nash to prove the existence of equilibria in game theory.
Facts
StatementIf S is a non-empty, compact and convex subset of a Euclidean space, and a set-valued function on S has a closed graph and maps every point of S to a non-empty convex subset of S, then the function has a fixed point. Shizuo Kakutani proved this generalization of the Brouwer fixed point theorem in 1941, and John Nash later used it to establish the existence of Nash equilibria. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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1. Kakutani Fixed-Point Theorem (Wikipedia)
Wikimedia Foundationlead section, first and second paragraphsQuote, lead section, first and second paragraphs
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria.
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