Kakutani's theorem (geometry)
In geometry, Kakutani's theorem, also known as Kakutani's fixed-point theorem, states that for a non-empty, compact, and convex subset S of a Euclidean space, if F: S → 2S is a set-valued function such that for every x in S, F(x) is a non-empty, convex set, and the graph of F is closed (or equivalently, F is upper hemicontinuous), then there exists a point x* in S such that x* is an element of F(x*). This point x* is called a fixed point of the set-valued function F.
In simpler terms, the theorem guarantees the existence of a point that, when mapped by the function F, results in a set that contains the original point itself. It is a generalization of the Brouwer fixed-point theorem, which applies to single-valued continuous functions. Kakutani's theorem extends the Brouwer fixed-point theorem to set-valued functions under specific convexity and continuity conditions.
The theorem has significant applications in various fields, including game theory (where it is used to prove the existence of Nash equilibria), economics, and mathematical optimization. Its importance stems from its ability to address situations where the mapping between points is not necessarily a single value but rather a set of possible values, making it suitable for modeling scenarios involving uncertainty or multiple options.
The upper hemicontinuity condition (or closed graph condition) is crucial for the theorem's validity. It ensures a certain level of "continuity" for the set-valued function, preventing abrupt jumps in the set of possible outputs as the input changes. Without this condition, the existence of a fixed point cannot be guaranteed.