Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T23:00:38.736Z Has data issue: false hasContentIssue false
  • English
  • Français

Packing planes in ℝ3

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

J. M. Marstrand
J. M. Marstrand
Affiliation:
The School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW.
Get access

Extract

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.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 26 , Issue 2 , December 1979 , pp. 180 - 183
Copyright
Copyright © University College London 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Besicovitch, A. S.. “Sur deux questions d'entegrabilite des fonctions”, J. Soc. Phys.-Math., Perm', 2 (1919 (1920)), 105–123.Google Scholar
2.Besicovitch, A. S.. “On Kakeya's problem and a similar one”, Math. Zeit., 27 (1928), 312–320.CrossRefGoogle Scholar
3.Davies, Roy O.. “Some remarks on the Kakeya problem”, Proc. Camb. Phil. Soc., 69 (1971), 417–421.CrossRefGoogle Scholar
4.Marstrand, J. M.. “Packing smooth curves in Rq”, Mathematika, 26 (1979), 1–12.CrossRefGoogle Scholar
5.Stein, E. M. and Wainger, S.. “Problems in harmonic analysis related to curvature”, Bull. Amer. Math. Soc., 84 (1978), 1239–1295.CrossRefGoogle Scholar
6.Falconer, K. J.. “Continuity properties of k-plane integrals and Besicovitch sets”, to appear in Math. Proc. Camb. Phil. Soc.Google Scholar