Lipschitz-Killing curvatures of self-similar random fractals

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(ε)), the surface area C_{d-1}(F(ε)) and the integrals of general mean curvatures over the unit normal bundles C_k(F(ε)) of the parallel sets F(ε) of distance ε, rescaled by ε^{D-k}, as ε\rightarrow 0. Here D equals the a.s. Hausdorff dimension of F. The corresponding results for the expectations are also proved.

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