Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
Abstract
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 ℤ_n. A hunter and a rabbit move on the nodes of ℤ_n 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 first n steps of order 1/ log n. We show these strategies yield a Kakeya set consisting of 4n triangles with minimal area (up to constant), namely Θ(1/ log n). As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions {B(s) : s ≥ 0} and {W(s) : s ≥ 0}. Let τ_t := min{s ≥ 0 : B(s) = t}. Then X_t = W(τ_t) is a Cauchy process and K := {(ɑ,X_t + ɑt) : ɑ, t ∈ [0, 1]} is a Kakeya set of zero area. The area of the ε-neighbourhood of K is as small as possible, i.e., almost surely of order Θ(1/| log ε|).
Additional Information
© 2014 American Mathematical Society. Received by the editors February 13, 2013. Article electronically published on June 10, 2014. The first author would like to acknowledge financial support by ERC 030-7950-411 and ISF 039-7679-411. The third author's work was supported in part by grant #212/09 of the Israel Science Foundation and by the Google Inter-university center for Electronic Markets and Auctions. The fifth author's work was supported by Microsoft Research, by a Simons Professorship at MSRI, and by NSF grant DMS-0901475. The authors also thank MSRI, Berkeley, where part of this work was completed. The authors thank Abraham Neyman for useful discussions.Attached Files
Submitted - 1207.6389v1.pdf
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Additional details
- Eprint ID
- 49838
- Resolver ID
- CaltechAUTHORS:20140918-150052882
- 039-7679-411
- European Research Council (ERC)
- 212/09
- Israel Science Foundation
- Google Inter-University Center for Electronic Markets and Auctions
- Microsoft Research
- MSRI Simons Professorship
- DMS-0901475
- NSF
- Created
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2014-09-18Created from EPrint's datestamp field
- Updated
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2021-11-02Created from EPrint's last_modified field