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.

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Author:Jürgen Kampf
URN (permanent link):urn:nbn:de:hbz:386-kluedo-29122
Serie (Series number):Report in Wirtschaftsmathematik (WIMA Report) (139a)
Document Type:Preprint
Language of publication:English
Publication Date:2012/02/27
Year of Publication:2012
Publishing Institute:Technische Universität Kaiserslautern
Tag:Convex geometry; Non-convex body; Parallel volume; Random body; Wills functional
Number of page:32
This document is an updated version of WiMa Report 139. A part of the content was removed and will become an own article.
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:516 Geometrie
519 Wahrscheinlichkeiten, angewandte Mathematik
MSC-Classification (mathematics):52A20 Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
52A22 Random convex sets and integral geometry [See also 53C65, 60D05]
52A38 Length, area, volume [See also 26B15, 28A75, 49Q20]

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