Menger curvature and $C^{1}$ regularity of fractals
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- by Yong Lin and Pertti Mattila PDF
- Proc. Amer. Math. Soc. 129 (2001), 1755-1762 Request permission
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 $1/2<s<1$.References
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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
- MR Author ID: 121505
- Email: pmattila@math.jyu.fi
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1755-1762
- MSC (2000): Primary 28A75
- DOI: https://doi.org/10.1090/S0002-9939-00-05814-7
- MathSciNet review: 1814107