UNIVERSITÄTSBIBLIOTHEK
  • search hit 1 of 1
Back to Result List

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.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar
Metadaten
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
Date of the Publication (Server):2012/02/27
Tag:Convex geometry; Non-convex body; Parallel volume; Random body; Wills functional
Number of page:32
Note:
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:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):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]
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 15.02.2012