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SUMMARY:Hunter\, Cauchy Rabbit\, and Optimal Kakeya Sets - Perla Sousi\, D
 PMMS
DTSTART:20150506T133000Z
DTEND:20150506T143000Z
UID:TALK59277@talks.cam.ac.uk
CONTACT:Dominic Dold
DESCRIPTION:A planar set that contains a unit segment in every direction i
 s called a Kakeya set. These sets have been studied intensively in geometr
 ic 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. A
 t 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 ra
 ndomized strategies for hunter and rabbit achieve expected capture time of
  order n log n. We show that every rabbit strategy yields a Kakeya set\; t
 he 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 th
 e scaling limit yields a simple construction of a random Kakeya set with z
 ero area from two Brownian motions.  (Joint work with Y. Babichenko\, Y. P
 eres\, R. Peretz and P. Winkler).
LOCATION:MR4\, Centre for Mathematical Sciences
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