Lipschitz-Killing curvatures of self-similar random fractals

Abstract

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