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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.
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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

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References

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