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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonsymmetric conical upper density and $ k$-porosity


Authors: Antti Käenmäki and Ville Suomala
Journal: Trans. Amer. Math. Soc. 363 (2011), 1183-1195
MSC (2000): Primary 28A75; Secondary 28A78, 28A80
Published electronically: October 21, 2010
MathSciNet review: 2737262
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Abstract: We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in $ \mathbb{R}^n$. As an application, we find an upper bound close to $ n-k$ for the Hausdorff dimension of sets with large $ k$-porosity. With $ k$-porous sets we mean sets which have holes in $ k$ different directions on every small scale.


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Additional Information

Antti Käenmäki
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
Email: antti.kaenmaki@jyu.fi

Ville Suomala
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
Email: ville.suomala@jyu.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04869-X
PII: S 0002-9947(2010)04869-X
Keywords: Conical density, porosity, Hausdorff dimension.
Received by editor(s): May 1, 2004
Received by editor(s) in revised form: July 4, 2008
Published electronically: October 21, 2010
Additional Notes: The first author acknowledges the support of the Academy of Finland (project #114821)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.