The hunter, Cauchy Rabbit, and optimal Kakeya set – Bruno Schapira

Bruno Schapira
I2M, Aix-Marseille Université

Date(s) : 06/11/2015   iCal
11 h 00 min - 12 h 00 min

“A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on the integer points in [0,n) without seeing each other. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions.”

A work by Y. Babichenko, Y. Peres, R. Peretz, P. Sousi and P. Winkler.




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