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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


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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