Hostname: page-component-5cf477f64f-54txb Total loading time: 0 Render date: 2025-04-04T23:04:26.830Z Has data issue: false hasContentIssue false
  • English
  • Français

Constancy results for special families of projections

Published online by Cambridge University Press:  07 February 2013

KATRIN FÄSSLER  and
TUOMAS ORPONEN
KATRIN FÄSSLER
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, 00014University of Helsinki, Finland e-mail: katrin.fassler@helsinki.fi, tuomas.orponen@helsinki.fi
TUOMAS ORPONEN
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, 00014University of Helsinki, Finland e-mail: katrin.fassler@helsinki.fi, tuomas.orponen@helsinki.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

Let { = V × ℝl : V ∈ G(n−l,m−l)} be the family of m-dimensional subspaces of ℝn containing {0} × ℝl, and let : ℝn → be the orthogonal projection onto . We prove that the mapping V ↦ Dim (B) is almost surely constant for any analytic set B ⊂ ℝn, where Dim denotes either Hausdorff or packing dimension.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 549 - 568
Copyright
Copyright © Cambridge Philosophical Society 2013

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]Bogachev, V. I.Measure Theory (Springer, 2006).Google Scholar
[2]Falconer, K. J. and Howroyd, J.Packing dimensions of projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 121, Issue 2 (1997), pp. 269–286.CrossRefGoogle Scholar
[3]Federer, H.Geometric Measure Theory (Springer, 1969).Google Scholar
[4]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
[5]Järvenpää, E., Järvenpää, M., and Keleti, T. Hausdorff dimension and non-degenerate families of projections. arXiv: 1203.5296v1.Google Scholar
[6]Järvenpää, E., Järvenpää, M., Ledrappier, F. and Leikas, M.One-dimensional families of projections. Nonlinearity 21 (2008), pp. 453–463.CrossRefGoogle Scholar
[7]Joyce, H. and Preiss, D.On the existence of subsets of finite positive packing measure. Mathematika 42 (1995), pp. 15–24.CrossRefGoogle Scholar
[8]Kaufman, R.On Hausdorff dimension of projections. Mathematika 15 (1968), pp. 153–155.CrossRefGoogle Scholar
[9]Lubin, A.Extensions of measures and the von Neumann selection theorem. Proc. Amer. Math. Soc. 43 (1) (1974), pp. 118–122.CrossRefGoogle Scholar
[10]Mattila, P.Orthogonal projections, Riesz capacities, and Minkowski content. Indiana Univ. Math. J. 39, Issue 1 (1990), pp. 185–198.CrossRefGoogle Scholar
[11]Mattila, P.Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
[12]Orponen, T. On the packing dimension and category of exceptional sets of orthogonal projections. arXiv:1204.2121.Google Scholar
[13]Whitney, H.Geometric Integration Theory (Princeton University Press, 1957).CrossRefGoogle Scholar