Abstract
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.
Publisher
Cambridge University Press (CUP)
Reference15 articles.
1. Sur deux questions de l'intégrabiité des fonctions;Besicovitch;J. Soc. Phys. Math. (Perm),1919
2. A Plane Set of Measure Zero Containing Circumferences of Every Radius
3. (10) Ward J. A set of plane measure zero containing all finite polygonal arcs. Caned. J. Math., forthcoming.
4. Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions
5. On the intersections of transforms of linear sets
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