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Packing planes in ℝ3

  • J. M. Marstrand (a1)

We denote by S the unit sphere in ℝ3, and µ is the rotationally invariant measure, generalizing surface area on S; thus µS = 4π. We identify directions (or unit vectors) in ℝ3 with points on S, and prove the following:

Theorem 1. If E is a subset of ℝ3 of Lebesgue measure zero, then for µ almost all directions α, every plane normal to α intersects E in a set of plane measure zero.

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

4. J. M. Marstrand . “Packing smooth curves in Rq”, Mathematika, 26 (1979), 112.

5. E. M. Stein and S. Wainger . “Problems in harmonic analysis related to curvature”, Bull. Amer. Math. Soc., 84 (1978), 12391295.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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