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Menger curvature and C^1 regularity of fractals

Yong Lin Department of Mathematics, Renmin University of China, Information School, Beijing, 100872, China Pertti Mattila Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

TBD mathscidoc:2207.43003

Proceeding of American Mathematics Society, 129, (6), 1755-1762, 2000.10
We show that if E is an s-regular set in R^2 for which the triple integral \int_E \int_E \int_E c(x, y, z)^{2s} dH^sx dH^sy dH^sz of the Menger curvature c is finite and if 0 < s ≤ 1/2, then 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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@inproceedings{yong2000menger,
  title={Menger curvature and C^1 regularity of fractals},
  author={Yong Lin, and Pertti Mattila},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707110012179075547},
  booktitle={Proceeding of American Mathematics Society},
  volume={129},
  number={6},
  pages={1755-1762},
  year={2000},
}
Yong Lin, and Pertti Mattila. Menger curvature and C^1 regularity of fractals. 2000. Vol. 129. In Proceeding of American Mathematics Society. pp.1755-1762. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707110012179075547.
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