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A planar set that contains a unit segment in every direction is called aKakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. Ahunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. Ateach step, the hunter moves to a neighbouring vertex or stays in place, whilethe rabbit is free to jump to any node. Adler et al (2003) provide strategiesfor hunter and rabbit that are optimal up to constant factors and achieveprobability of capture in the first $n$ steps of order $1/\log n$. We showthese strategies yield a Kakeya set consisting of $4n$ triangles with minimalarea, (up to constant), namely $\Theta(1/\log n)$. As far as we know, this isthe first non-iterative construction of a boundary-optimal Kakeya set.Considering the continuum analog of the game yields a construction of a randomKakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$and $\{W(s): s \ge 0\}$. Let $\tau_t:=\min\{s \ge 0: B(s)=t\}$. Then$X_t=W(\tau_t)$ is a Cauchy process, and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ isa Kakeya set of zero area. The area of the $\epsilon$-neighborhood of $K$ is assmall as possible, i.e., almost surely of order $\Theta(1/|\log \epsilon|)$.

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets