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 R n \mathbb {R}^n , there is a positive integer q q such that the period of any periodic orbit divides q q and q q is the order, or twice the order, of a permutation on n n letters. This confirms Nussbaum’s 2 n 2^n Conjecture for ∞ \infty -norm nonexpansive maps in the special case where the maps are also real analytic.