Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, convex subset $X$ of $\mathbb{R}^n$, that would mean that there exists a fixed point $x \in X$ such that $B_{\epsilon}(x) \in X$. I think that the usual proofs could be extended to show this. Is that so?
Edit : It seems the assumption that $f(x)$ is not a point may be required.
Edit 2 : As Mike pointed out the above assumption is not good enough.