Fractal curvature measures of self-similar sets

Details Fractal curvature measures of self-similar sets Fractal curvature measures of self-similar sets

2010-07-05

arxiv.org/abs/1007.0696v2

Mathematics

Metric Geometry

13 pages

Scientific paper

Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures C_k(F_r,.) from geometric measure theory of parallel sets F_r for small distances r>0. Due to self-similarity the limit measures appear to be constant multiples of the normalized Hausdorff measures on F, and the constants agree with the corresponding total fractal curvatures C^f_k(F). This provides information on the 'second order' geometric fine structure of such fractals.

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