Journal of Convex Analysis 21 (2014), No. 4, 925--950
Copyright Heldermann Verlag 2014
Asymptotic Order of the Parallel Volume Difference in Minkowski Spaces
Jürgen Kampf
Institut für Stochastik, Universität Ulm, Helmholtzstr. 18, 89069 Ulm, Germany
jurgen.kampf@uni-ulm.de
[Abstract-pdf]
We investigate the asymptotic behavior 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, which is called parallel volume difference, can at most have order $r^{d-2}$ in a $d$-dimensional Minkowski space. Then we will show that in certain Minkowski spaces (and in particular in Euclidean spaces) this difference can at most have order $r^{d-3}$. We will characterize the $2$-dimensional Minkowski spaces in which the parallel volume difference has always at most order $r^{-1}$. Finally we present applications concerning Brownian paths and Boolean models.
Keywords: Convex geometry, parallel volume, non-convex body, random body.
MSC: 52A20, 52A21, 52A22, 52A38
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