Convexity is crucial in Schauder-Tychnonov fixed-point theorem.

Question

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

Answer 1

Consider the map: $T: S^1 \to S^1$ given by quarter-clockwise rotation (here $E$ is $\mathbb{R}^2$), where $S^1 = \{x \in \mathbb{R}^2 : \|x\| =1\}$.

This is a fixed-point free, continuous map from a compact (albeit not convex) subset to itself.

Edit: What really is needed is some sort of 'no holes' type property for your domain. Convexity is one condition that ensures this and as a condition gives you a lot of fine analytic structure to work with, but in the spirit of the question, it is 'far' from necessary.