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Some remarks on the Kakeya problem

  • Roy O. Davies (a1)

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

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(2) A. S. Besicovitch On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312320.

(3) O. Perron Über emen Satz von Besicovitch. Math. Z. 28 (1928), 383386.

(4) A. S. Besicovitch The Kakeya problem. Amer. Math. Monthly 70 (1963), 697706.

(8) J. R. Kinney A thin set of circles. Amer. Math. Monthly 75 (1968), 10771081.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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