Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T00:28:10.441Z Has data issue: false hasContentIssue false
  • English
  • Français

The packing dimension of projections and sections of measures

Published online by Cambridge University Press:  24 October 2008

Kenneth J. Falconer  and
Pertti Mattila
Kenneth J. Falconer
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland
Pertti Mattila
Affiliation:
Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for a probability measure μ on ℝn

for almost all m–dimensional subspaces V, provided dimH μ≤m. Here projv denotes orthogonal projection onto V, and dimH and dimp denote the Hausdorff and packing dimension of a measure. In the case dimH μ > m we show that at μ-almost all points x the slices of μ by almost all (n − m)-planes Vx through x satisfy

We give examples to show that these inequalities are sharp.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 695 - 713
Copyright
Copyright © Cambridge Philosophical Society 1996

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

REFERENCES

[1]Falconer, K. J.. Fractal geometry - mathematical foundations and applications (Wiley, 1990).CrossRefGoogle Scholar
[2]Falconer, K. J.. Sets with large intersections. J. London Math. Soc. 49 (1994), 267–280.CrossRefGoogle Scholar
[3]Falconer, K. J. and Howroyd, J. D.. Projection theorems for box and packing dimensions, to appear in Math. Proc. Cambridge Philos. Soc.Google Scholar
[4]Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in ℝd. Math. Proc. Cambridge Philos. Soc. 115 (1994), 527–544.CrossRefGoogle Scholar
[5]Järvenpää, M.. On the upper Minkowski dimension, packing dimension, and orthogonal projections. Ann. Acad. Sei. Fenn. Ser. A I Math. Dissertationes 99 (1994).Google Scholar
[6]Marstrand, J. M.. Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc. (3) 4 (1954), 257–302.CrossRefGoogle Scholar
[7]Mattila, P.. Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A I Math, 1 (1975), 227–244.CrossRefGoogle Scholar
[8]Mattila, P.. Integralgeometric properties of capacities. Trans. Amer. Math. Soc. 266 (1981), 539–554.Google Scholar
[9]Mattila, P.. Geometry of sets and measures in Euclidean spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
[10]Preiss, D.. Geometry of measures in ℝn: distribution, rectifiability, and densities. Ann. of Math. 125 (1987), 537–643.CrossRefGoogle Scholar
[11]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
[12]Falconer, K. J. and Howroyd, J. D.. Packing dimensions of projections and dimension profiles, to appear.Google Scholar