Best Laid Plans of Lions and Men

• Published in 2017
• Added on 2017-03-22

We answer 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 to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that the answer to the question is not always "yes" by giving an example of a region $R$ in the plane where the man has a strategy to survive forever. $R$ is a polygonal region with holes and the exterior and interior boundaries are pairwise disjoint, simple polygons. Our construction is the first truly two-dimensional example where the man can survive. Next, we consider the following game played on the entire plane instead of a bounded area: There is any finite number of unit speed lions and one fast man who can run with speed $1+\varepsilon$ for some value $\varepsilon>0$. Can the man always survive? We answer the question in the affirmative for any constant $\varepsilon>0$.

Links

• http://arxiv.org/abs/1703.03687v1
• http://arxiv.org/pdf/1703.03687v1

Other information

BibTeX entry

@article{BestLaidPlansofLionsandMen,
	key = {BestLaidPlansofLionsandMen},
	type = {article},
	title = {Best Laid Plans of Lions and Men},
	author = {Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-Nilsen},
	abstract = {We answer 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 to avoid pathological examples where the man survives
forever because any path to the lions is infinitely long. We show that the
answer to the question is not always "yes" by giving an example of a region {\$}R{\$}
in the plane where the man has a strategy to survive forever. {\$}R{\$} is a
polygonal region with holes and the exterior and interior boundaries are
pairwise disjoint, simple polygons. Our construction is the first truly
two-dimensional example where the man can survive.
  Next, we consider the following game played on the entire plane instead of a
bounded area: There is any finite number of unit speed lions and one fast man
who can run with speed {\$}1+\varepsilon{\$} for some value {\$}\varepsilon>0{\$}. Can the
man always survive? We answer the question in the affirmative for any constant
{\$}\varepsilon>0{\$}.},
	comment = {},
	date_added = {2017-03-22},
	date_published = {2017-12-07},
	urls = {http://arxiv.org/abs/1703.03687v1,http://arxiv.org/pdf/1703.03687v1},
	collections = {Attention-grabbing titles,Protocols and strategies,Animals,Fun maths facts},
	url = {http://arxiv.org/abs/1703.03687v1 http://arxiv.org/pdf/1703.03687v1},
	urldate = {2017-03-22},
	archivePrefix = {arXiv},
	eprint = {1703.03687},
	primaryClass = {cs.CG},
	year = 2017
}