Lipschitz-Killing curvatures of self-similar random fractals

TL;DR

This paper introduces geometric parameters called Lipschitz-Killing curvatures for self-similar random fractals, derived as limits of rescaled curvature measures of their parallel sets, extending geometric analysis to fractal structures.

Contribution

It defines and analyzes Lipschitz-Killing curvatures for a broad class of self-similar random fractals, establishing their existence as limits of curvature measures of parallel sets.

Findings

Existence of Lipschitz-Killing curvatures as limits for self-similar fractals.

Results hold almost surely, in expectation, and in the essential sense.

Provides a geometric framework for fractals extending classical curvature measures.

Abstract

For a large class of self-similar random sets F in R^d geometric parameters C_k(F), k=0,...,d, are introduced. They arise as a.s. (average or essential) limits of the volume C_d(F(\epsilon)), the surface area C_{d-1}(F(\epsilon)) and the integrals of general mean curvatures over the unit normal bundles C_k(F(\epsilon)) of the parallel sets F(\epsilon) of distance \epsilon, rescaled by \epsilon^{D-k}, as \epsilon\rightarrow 0. Here D equals the a.s. Hausdorff dimension of F. The corresponding results for the expectations are also proved.