Nonsymmetric conical upper density and 𝑘-porosity

Käenmäki, Antti; Suomala, Ville

doi:10.1090/S0002-9947-2010-04869-X

Nonsymmetric conical upper density and $k$-porosity
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by Antti Käenmäki and Ville Suomala

Trans. Amer. Math. Soc. 363 (2011), 1183-1195

DOI: https://doi.org/10.1090/S0002-9947-2010-04869-X

Published electronically: October 21, 2010

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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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