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Menger curvature and $C^{1}$ regularity of fractals


Authors: Yong Lin and Pertti Mattila
Journal: Proc. Amer. Math. Soc. 129 (2001), 1755-1762
MSC (2000): Primary 28A75
DOI: https://doi.org/10.1090/S0002-9939-00-05814-7
Published electronically: October 31, 2000
MathSciNet review: 1814107
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Abstract:

We show that if $E$ is an $s$-regular set in $\mathbf{R}^{2}$ for which the triple integral $\int _{E}\int _{E}\int _{E}c(x,y,z)^{2s}\,d\mathcal{H}^{s}x\,d\mathcal{H}^{s}y\,d \mathcal{H}^{s}z$of the Menger curvature $c$ is finite and if $0<s\le 1/2$, then $\mathcal{H}^{s}$almost all of $E$ can be covered with countably many $C^{1}$ curves. We give an example to show that this is false for $1/2<s<1$.


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Additional Information

Yong Lin
Affiliation: Department of Mathematics, Renmin University of China, Information School, Beijing, 100872, China
Email: liny9@263.net

Pertti Mattila
Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
Email: pmattila@math.jyu.fi

DOI: https://doi.org/10.1090/S0002-9939-00-05814-7
Received by editor(s): September 27, 1999
Published electronically: October 31, 2000
Additional Notes: The authors gratefully acknowledge the hospitality of CRM at Universitat Autònoma de Barcelona where part of this work was done. The first author also wants to thank the Academy of Finland for financial support.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

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