## Asymptotic Order of the Parallel Volume Difference

• In this paper we investigate the asymptotic behaviour of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance $$r$$ tends to infinity. 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 a $$d$$-dimensional space. Then we will show that in Euclidean spaces this difference can at most have order $$r^{d-3}$$. These results have several applications, e.g. we will use them to compute the derivative of $$f_\mu(rK)$$ in $$r = 0$$, where $$f_\mu$$ is the Wills functional or a similar functional, $$K$$ is a body and $$rK$$ is the Minkowski-product of $$r$$ and $$K$$. Finally we present applications concerning Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

Verfasserangaben: Jürgen Kampf urn:nbn:de:hbz:386-kluedo-29122 Report in Wirtschaftsmathematik (WIMA Report) (139a) Preprint Englisch 27.02.2012 2012 Technische Universität Kaiserslautern 27.02.2012 Convex geometry; Non-convex body; Parallel volume; Random body; Wills functional 32 This document is an updated version of WiMa Report 139. A part of the content was removed and will become an own article. Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 52-XX CONVEX AND DISCRETE GEOMETRY / 52Axx General convexity / 52A20 Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52-XX CONVEX AND DISCRETE GEOMETRY / 52Axx General convexity / 52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 52-XX CONVEX AND DISCRETE GEOMETRY / 52Axx General convexity / 52A22 Random convex sets and integral geometry [See also 53C65, 60D05] 52-XX CONVEX AND DISCRETE GEOMETRY / 52Axx General convexity / 52A38 Length, area, volume [See also 26B15, 28A75, 49Q20] Standard gemäß KLUEDO-Leitlinien vom 15.02.2012
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