Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Zn. A hunter and a rabbit move on the nodes of Zn without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the rst n steps of order 1= logn. We show these strategies yield a Kakeya set consisting of 4n triangles with minimal area, (up to constant), namely

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