Fractal curvature measures of self-similar sets

Abstract

Abstract Fractal Lipschitz-Killing curvature measures C k f (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(Fε; ·) from geometric measure theory of parallel sets Fε for small distances ε. 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 Ck f (F) This provides information on the ’second order’ geometric fine structure of such fractals.