Lipschitz-Killing curvatures of self-similar random fractals

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For a large class of self-similar random sets F in Rᵈ geometric parameters Cₖ(F), k=0,...,d, are introduced. They arise as a.s. (average or essential) limits of the volume Cd(F(ε)), the surface area Cd₋₁(F(ε)) and the integrals of general mean curvatures over the unit normal bundles Cₖ(F(ε)) of the parallel sets F(ε) of distance ε, rescaled by εᵈ⁻ᵏ, as ε→ 0. Here D equals the a.s. Hausdorff dimension of F. The corresponding results for the expectations are also proved.