Winter, Steffen; Zähle, Martina Fractal curvature measures of self-similar sets. (English) Zbl 1268.28002 Adv. Geom. 13, No. 2, 229-244 (2013). 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. Cited in 14 Documents MSC: 28A75 Length, area, volume, other geometric measure theory 28A80 Fractals 28A78 Hausdorff and packing measures 53C65 Integral geometry Keywords:self-similar set; parallel set; curvature measures; Lipschitz-Killing curvature measures; Minkowski content; Minkowski dimension; fractal curvature measures PDFBibTeX XMLCite \textit{S. Winter} and \textit{M. Zähle}, Adv. Geom. 13, No. 2, 229--244 (2013; Zbl 1268.28002) Full Text: DOI arXiv