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  • Cited by 5
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    Oberlin, Richard 2010. Two bounds for the X-ray transform. Mathematische Zeitschrift, Vol. 266, Issue. 3, p. 623.

    Steprāns, Juris 2005. Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle. Bulletin of Symbolic Logic, Vol. 11, Issue. 04, p. 517.

    Falconer, K. J. 1982. Hausdorff dimension and the exceptional set of projections. Mathematika, Vol. 29, Issue. 01, p. 109.

    Falconer, K. J. 1980. Sections of Sets of zero Lebesgue measure. Mathematika, Vol. 27, Issue. 01, p. 90.

    Marstrand, J. M. 1979. Packing planes in 3. Mathematika, Vol. 26, Issue. 02, p. 180.


Packing planes in ℝ3

  • J. M. Marstrand (a1)
  • DOI:
  • Published online: 01 February 2010

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
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