Best Laid Plans of Lions and Men
release_v3e2jlywubehpfkiayansyxyly
by
Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian
Wulff-Nilsen
2017
Abstract
We study the following question dating back to J.E. Littlewood (1885-1977):
Can two lions catch a man in a bounded area with rectifiable lakes? The lions
and the man are all assumed to be points moving with at most unit speed. That
the lakes are rectifiable means that their boundaries are finitely long. This
requirement is necessary to avoid pathological examples where the man survives
forever because any path to the lions is infinitely long.
We show that three lions have a winning strategy against a man in a bounded
region with finitely many rectifiable lakes. This is "tight" in the sense that
there exists a region R in the plane where the man has a strategy to survive
forever. We give a rigorous description of such a region R; a polygonal
region with holes whose exterior and interior boundaries are pairwise disjoint,
simple polygons.
Finally, we consider the following game played on the entire plane instead of
a compact region: There is any finite number of unit speed lions and one fast
man who can run with speed 1+ε for some value ε>0. Can
the man always survive? We answer the question in the affirmative for any
ε>0. By letting the number of lions tend to infinity, we
furthermore show that the man can survive against any countably infinite set of
lions.
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