Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element of $\cal A$ and $f\circ g = g\circ f$ for any $f, g \in \cal A$. I am trying to show that elements of $\cal A$ have a common fixed point which is in $X$.
Any idea or references would be helpful.