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Hausdorff and packing dimensions and sections of measures

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Maarit Järvenpää  and
Pertti Mattila
Maarit Järvenpää
Affiliation:
Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351, Jyväskylä, Finland.
Pertti Mattila
Affiliation:
Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351, Jyväskylä, Finland.
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Abstract

Let m and n be integers with 0<m<n and let μ be a Radon measure on ℝn with compact support. For the Hausdorff dimension, dimH, of sections of measures we have the following equality: for almost all (n − m)-dimensional linear subspaces V

provided that dimH μ > m. Here μv,a is the sliced measure and V⊥ is the orthogonal complement of V. If the (m + d)-energy of the measure μ is finite for some d>0, then for almost all (n − m)-dimensional linear subspaces V we have

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 45 , Issue 1 , June 1998 , pp. 55 - 77
Copyright
Copyright © University College London 1998

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References

1.Cutler, C. D.. Strong and weak duality principles for fractal dimension in Euclidean space. Math. Proc. Cambridge Phil. Soc, 118 (1995), 393–410.CrossRefGoogle Scholar
2.Falconer, K. J.. Techniques in fractal geometry (John Wiley & Sons, 1997).Google Scholar
3.Falconer, K. J. Sets with large intersections. J. London Math. Soc, 49 (1994), 267–280.CrossRefGoogle Scholar
4.Falconer, K. J. and Howroyd, J. D.. Projection theorems for box and packing dimensions. Math. Proc. Cambridge Phil. Soc, 119 (1996), 287–295.CrossRefGoogle Scholar
5.Falconer, K. J. and Howroyd, J. D.. Packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Phil. Soc, 121 (1997), 269–286.CrossRefGoogle Scholar
6.Falconer, K. J. and Mattila, P.. The packing dimension of projections and sections of measures. Math. Proc. Cambridge Phil. Soc, 119 (1996), 695–713.CrossRefGoogle Scholar
7.Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in Ud. Math. Proc. Cambridge Phil. Soc, 115 (1994), 527–544.CrossRefGoogle Scholar
8.Järvenpää, M.. On the upper Minkowski dimension, the packing dimension, and orthogonal projections. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 99 (1994).Google Scholar
9.Marstrand, J. M. Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc (3), 4 (1954), 257–302.CrossRefGoogle Scholar
10.Mattila, P.. Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A I Math., 1 (1975), 227–244.CrossRefGoogle Scholar
11.Mattila, P.. Integralgeometric properties of capacities. Trans. Amer. Math. Soc, 266 (1981), 539–554.Google Scholar
12.Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar