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Lipschitz-Killing curvatures of self-similar random fractals


Author: M. Zähle
Journal: Trans. Amer. Math. Soc. 363 (2011), 2663-2684
MSC (2000): Primary 28A80, 60D05; Secondary 28A75, 28A78, 53C65, 60J80, 60J85
DOI: https://doi.org/10.1090/S0002-9947-2010-05198-0
Published electronically: December 2, 2010
MathSciNet review: 2763731
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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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Additional Information

M. Zähle
Affiliation: Mathematical Institute, University of Jena, D-07737 Jena, Germany
Email: martina.zaehle@uni-jena.de

DOI: https://doi.org/10.1090/S0002-9947-2010-05198-0
Keywords: Self-similar random fractals, curvatures, Minkowski content, branching random walks
Received by editor(s): April 7, 2009
Received by editor(s) in revised form: July 20, 2009, and September 4, 2009
Published electronically: December 2, 2010
Dedicated: Dedicated to Herbert Federer
Article copyright: © Copyright 2010 American Mathematical Society

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