  We investigate necessary and sufficient conditions for a nonexpansive map $f$
on a Banach space $X$ to have surjective displacement, that is, for $f -
\mathrm{id}$ to map onto $X$. In particular, we give a computable necessary and
sufficient condition when $X$ is a finite dimensional space with a polyhedral
norm. We give a similar computable necessary and sufficient condition for a
fixed point of a polyhedral norm nonexpansive map to be unique. We also
consider applications to nonlinear Perron-Frobenius theory and suggest some
additional computable sufficient conditions for surjective displacement and
uniqueness of fixed points.
