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Menger curvature and regularity of fractals

Authors:	Yong Lin Pertti Mattila

Institution:	Department of Mathematics, Renmin University of China, Information School, Beijing, 100872, China ; Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

Abstract:	We show that if is an -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 is finite and if $0<s\le 1/2$ , then $\mathcal{H}^{s}$ almost all of can be covered with countably many $C^{1}$ curves. We give an example to show that this is false for .

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