| Hausdorff Dimension and mean porosity (1997) | |||||||||||||
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| Math Ann Mathematische Annalen Springer Verlag Hausdorff Dimension and mean porosity Pekka Koskela Steffen Rohde Department Mathematics University Michigan Ann Arbor USA Received June Revised version July Introduction recent result Jones and Makarov states that the Hausdorff dimension the Minkowski dimension the boundary the image the disk under uniformly lder continuous univalent function does not exceed where universal constant Moreover they show that this statement sharp the sense that there exist constants for which this estimate fails even for small The lder continuity univalent function equivalent appropriate logarithmic growth condition the hyperbolic metric the image domain was shown Becker and Pommerenke The hyperbolic metric comparable the quasihyperbolic metric see Section for the nition simply connected plane domain the Koebe distortion theorem and hence the result Jones and Makarov can interpreted sharp dimension estimate for boundaries simply connected plane domains that satisfy logarithmic growth condition their quasihyperbolic metric This growth condition can ned for any bounded domain the Euclidean space Smith and Stegenga produced estimate for the dimension such general situation modifying the ideas Jones and Makarov However their estimate weaker than the result Jones and Makarov when applied the case simply connected plane domain The main purpose this paper establish sharp dimension estimates for sets satisfying certain porosity condition Roughly speaking r. http://deepblue.lib.umich.edu/bitstream/2027.42/41928/1/208-309-4-593_73090593.pdf | |||||||||||||
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