Fractal curvature measures of self-similar sets
Metric Geometry
2010-09-29 v2 Differential Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.1007.0696,
title = {Fractal curvature measures of self-similar sets},
author = {Steffen Winter and Martina Zähle},
journal= {arXiv preprint arXiv:1007.0696},
year = {2010}
}
Comments
13 pages