Lipschitz-Killing curvatures of self-similar random fractals

Zähle, M.

doi:10.1090/S0002-9947-2010-05198-0

Lipschitz-Killing curvatures of self-similar random fractals
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by M. Zähle

Trans. Amer. Math. Soc. 363 (2011), 2663-2684

DOI: https://doi.org/10.1090/S0002-9947-2010-05198-0

Published electronically: December 2, 2010

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Abstract:

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

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