Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-26T06:30:32.435Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hausdorff and packing dimension of product spaces

Published online by Cambridge University Press:  24 October 2008

J. D. Howroyd
J. D. Howroyd
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for arbitrary metric spaces X and Y the following dimension inequalities hold:

where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 715 - 727
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]Besicovitch, A. S. and Moran, P. A. P.. The measure of product and cylinder sets. J. London Math. Soc. 20 (1945), 110–120.Google Scholar
[2]Cutler, C. D.. Strong and weak duality principles for fractal dimension in Euclidean space. Technical report series, University of Waterloo STAT-93–14 (1993).Google Scholar
[3]Federer, H.. Geometric Measure Theory (Springer-Verlag, 1969).Google Scholar
[4]Hasse, H.. On the dimension of product measures. Mathematika 37 (1990), 316–323.CrossRefGoogle Scholar
[5]Howroyd, J. D.. On dimension and on the existence of subsets of finite positive Hausdorff measure. Proc. London Math. Soc., to appear.Google Scholar
[6]Howroyd, J. D.. On the theory of Hausdorff measure in metric spaces. Ph.D. thesis, University College London, 1994.Google Scholar
[7]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
[8]Joyce, H. and Preiss, D.. On the existence of subsets of finite positive packing measure. Mathematika, to appear.Google Scholar
[9]Kelly, J. D.. A method for constructing measures appropriate for the study of Cartesian products. Proc. London Math. Soc. (3) 26 (1973), 521–546.CrossRefGoogle Scholar
[10]Larman, D. G.. On Hausdorff measure in finite-dimensional compact metric spaces. Proc. London Math. Soc. (3) 17 (1967), 193–206.CrossRefGoogle Scholar
[11]Marstrand, J. M.. The dimension of the Cartesian product sets. Proc. Cambridge Philos. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[12]Raymond, X. Saint and Tricot, C.. Packing regularity of sets in n-space. Math. Proc. Cambridge Philos. Soc. 103 (1988), 133–145.CrossRefGoogle Scholar
[13]Tricot, C.. Rarefaction indices. Mathematika 27 (1980), 46–57.CrossRefGoogle Scholar
[14]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
[15]Wegmann, H.. Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine Angew. Math. 234 (1969), 163–171.Google Scholar