## Asymptotic order of the parallel volume difference

• In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [13]. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order $$r^{d-2}$$ in dimension $$d$$. Then we will show that in the Euclidean case this difference can at most have order $$r^{d-3}$$. We will also examine the asymptotic behavior of the derivative of this difference as the distance tends to infinity. After this we will compute the derivative of $$f_\mu (rK)$$ in $$r$$, where $$f_\mu$$ is the Wills functional or a similar functional, $$K$$ is a fixed body and $$rK$$ is the Minkowski-product of $$r$$ and $$K$$. Finally we will use these results to examine Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

Author: Jürgen Kampf urn:nbn:de:hbz:386-kluedo-17006 Report in Wirtschaftsmathematik (WIMA Report) (139) Preprint English 2011 2011 Technische Universität Kaiserslautern Parallel volume ; Wills functional; non-convex body A newer version of this document is available on KLUEDO: urn:nbn:de:hbz:386-kluedo-29122 Fachbereich Mathematik 510 Mathematik
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