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,
on
.
In this paper we show that the illumination number of
is equal
to
,
and hence provide a sharp lower bound for the running time of the algorithm.
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