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Hausdorff dimension and the exceptional set of projections

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer
K. J. Falconer
Affiliation:
The School of Mathematics, University of Bristol, University Walk, Bristol. BS8 1TW
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Extract

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory

Information

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 109 - 115
Copyright
Copyright © University College London 1982

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