Statement
The theorem has several formulations, depending on the context in which it is used. The simplest is sometimes given as follows:
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- In the plane
- Every continuous function f from a closed disk to itself has at least one fixed point.
This can be generalized to an arbitrary finite dimension:
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- In Euclidean space
- Every continuous function from a closed ball of a Euclidean space to itself has a fixed point.
A slightly more general version is as follows:
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- Convex compact set
- Every continuous function f from a convex compact subset K of a Euclidean space to K itself has a fixed point.
An even more general form is better known under a different name:
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- Schauder fixed point theorem
- Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.
Read more about this topic: Brouwer Fixed-point Theorem
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