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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:Hunter\, Cauchy Rabbit\, and Optimal Kakeya Sets -
Perla Sousi\, DPMMS
DTSTART;TZID=Europe/London:20150506T143000
DTEND;TZID=Europe/London:20150506T153000
UID:TALK59277AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/59277
DESCRIPTION: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 the
ory and harmonic analysis since the work of Besico
vich (1928)\; we find a new connection to game the
ory 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 neighb
oring 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 capt
ure time. The known optimal randomized strategies
for hunter and rabbit achieve expected capture tim
e of order n log n. We show that every rabbit stra
tegy yields a Kakeya set\; the optimal rabbit stra
tegy is based on a discretized Cauchy random walk\
, and it yields a Kakeya set K consisting of 4n tr
iangles\, that has minimal area among such sets (t
he 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 zero area from two Brownian
motions. (Joint work with Y. Babichenko\, Y. Per
es\, R. Peretz and P. Winkler).
LOCATION:MR4\, Centre for Mathematical Sciences
CONTACT:Dominic Dold
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