Real analytic nonexpansive maps on polyhedral normed spaces

If a real analytic nonexpansive map on a polyhedral normed space has a nonempty fixed point set, then we show that there is an isometry from an affine subspace onto the fixed point set. As a corollary, we prove that for any real analytic 1-norm or $\infty$-norm nonexpansive map on $\mathbb{R}^n$, there is a positive integer $q$ such that the period of any periodic orbit divides $q$ and $q$ is the order, or twice the order, of a permutation on $n$ letters. This confirms Nussbaum's $2^n$ Conjecture for $\infty$-norm nonexpansive maps in the special case where the maps are also real analytic.