Fixed points of the ball

Brouwer’s theorem shows that any map of the unit ball to itself has a fixed point. Evidently the collection of fixed points is a closed subset of the ball. What closed subsets are realized as the fixed point set of a continuous map of the ball into itself?

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One thought on “Fixed points of the ball”

Claim: every non-empty closed subset.
By Brower’s theorem we know that such a set must be non empty and as already stated we know that the set must be closed.
For the converse let $F$ be a non-empty closed set. Let $p \in F$ . The map $f(x) = d(x,F)$ is continuous.
Consider the map $x \mapsto \frac{x-p}{f(x) + 1}+p = (1+f(x))^{-1} x + (1-(1+f(x))^{-1}) p$ . By the RHS it is clear that this is a well defined map, since the ball is convex. By the LHS it clear that this fixes a point $x$ if and only if $x = p$ or $f(x) = 0$ , and thus if and only if $x \in F$ .