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Fractal curvature measures of self-similar sets. (English) Zbl 1268.28002

Summary: Fractal Lipschitz-Killing curvature measures \(C_{k}^{f }(F, \cdot )\), \(k = 0, \dotsc , d\), are determined for a large class of self-similar sets \(F\) in \(\mathbb R ^{d}\). They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures \(C_{k}(F _{\varepsilon }, \cdot )\) from geometric measure theory of parallel sets \(F_{\varepsilon }\) for small distances \(\varepsilon \). 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 \(C_{k} ^{f }(F)\). This provides information on the “second-order” geometric fine-structure of such fractals.

MSC:

28A75 Length, area, volume, other geometric measure theory
28A80 Fractals
28A78 Hausdorff and packing measures
53C65 Integral geometry
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