Lipschitz-Killing curvatures of self-similar random fractals

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.