KLUEDO RSS FeedKLUEDO Dokumente/documents
https://kluedo.ub.uni-kl.de/index/index/
Mon, 27 Feb 2012 10:09:25 +0100Mon, 27 Feb 2012 10:09:25 +0100Asymptotic Order of the Parallel Volume Difference
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2912
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.Jürgen Kampfpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2912Mon, 27 Feb 2012 10:09:25 +0100Asymptotic order of the parallel volume difference
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2327
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.Jürgen Kampfpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2327Thu, 26 May 2011 11:11:40 +0200