The following Theorem is well-known:
Schauder-Tychnonov fixed-point theorem: Let $K$ be a compact convex subset of a Banach space, $E$. If $T:K\to K$ is continuous, then $T$ has a fixed point.
I'm trying to see that if I relax convexity, then the Theorem does not hold. I'm thinking of a sphere in the space, $\left(C[0,1],\|\|\right)$ as counter-example.
However, I don't know how to get it together. Any help in putting it together, will be appreciated. Thanks