Random compact sets related to the Kakeya problem
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- by Ralph Alexander PDF
- Proc. Amer. Math. Soc. 53 (1975), 415-419 Request permission
Abstract:
A $B$-set is defined to be a compact planar set of zero measure which contains a translate of any line segment lying in a disk of diameter one. A construction is given which associates a unique compact planar set with each sequence in a closed interval, and it is shown that for almost all such sequences a $B$-set is obtained. The construction depends on the measure properties of certain perfect linear sets. Several related problems of a subtler nature are also considered.References
- A. S. Besicovitch, On Kakeya’s problem and a similar one, Math. Z. 27 (1928), no. 1, 312–320. MR 1544912, DOI 10.1007/BF01171101
- A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points (III), Math. Ann. 116 (1939), no. 1, 349–357. MR 1513231, DOI 10.1007/BF01597361
- A. S. Besicovitch, The Kakeya problem, Amer. Math. Monthly 70 (1963), 697–706. MR 157266, DOI 10.2307/2312249
- William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 0088081
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 53 (1975), 415-419
- MSC: Primary 28A75
- DOI: https://doi.org/10.1090/S0002-9939-1975-0393427-3
- MathSciNet review: 0393427