TY - INPR
A1 - Kampf, Jürgen
T1 - Asymptotic Order of the Parallel Volume Difference
N2 - 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.
T3 - Report in Wirtschaftsmathematik (WIMA Report) - 139a
KW - Convex geometry
KW - Parallel volume
KW - Non-convex body
KW - Wills functional
KW - Random body
Y1 - 2012
UR - https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2912
UR - https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-29122
N1 - This document is an updated version of WiMa Report 139. A part of the content was removed and will become an own article.
ER -