Brouwer fixed-point theorem
Convexity

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball D n {\displaystyle D^{n}}. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).

The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function f (x) = − x {\displaystyle f(x)=-x}, which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n -dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex). The function f does have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.
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