Fractal curvature measures of self-similar sets

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Fractal Lipschitz-Killing curvature measures Cᶠₖ(F,.), k = 0,..., d, are determined for a large class of self-similar sets F in Rᵈ. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures Cₖ(Fᵣ,.) from geometric measure theory of parallel sets Fᵣ 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 Cᶠₖ(F). This provides information on the 'second order' geometric fine structure of such fractals.