Fractal curvature measures of self-similar sets

TL;DR

This paper computes fractal curvature measures for self-similar sets, revealing their geometric fine structure through limits of classical measures and linking them to normalized Hausdorff measures.

Contribution

It introduces a method to determine fractal Lipschitz-Killing curvature measures for self-similar sets, connecting classical geometric measures with fractal geometry.

Findings

Fractal curvature measures are limits of classical measures on parallel sets.

Limit measures are proportional to normalized Hausdorff measures.

Constants match total fractal curvatures, revealing geometric structure.

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.