  In recent work with Lins and Nussbaum the first author gave an algorithm that
can detect the existence of a positive eigenvector for order-preserving
homogeneous maps on the standard positive cone. The main goal of this paper is
to determine the minimum number of iterations this algorithm requires. It is
known that this number is equal to the illumination number of the unit ball of
the variation norm. In this paper we determine its illumination number, and
hence provide a sharp lower bound for the running time of the algorithm.
