  We give necessary and sufficient conditions for a nonexpansive map on a
finite dimensional normed space to have a nonempty, bounded set of fixed
points. Among other results we show that if $f : V \rightarrow V$ is a
nonexpansive map on a finite dimensional normed space $V$, then the fixed point
set of $f$ is nonempty and bounded if and only if there exist $w_1, \ldots ,
w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ illuminates the
unit ball. This yields a numerical procedure for detecting fixed points of
nonexpansive maps on finite dimensional spaces. We also discuss applications of
this procedure to certain nonlinear eigenvalue problems arising in game theory
and mathematical biology.
