Abstract

We investigate the asymptotic behavior of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance rr 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 rd−2r^{d-2} in a dd-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 rd−3r^{d-3}. We will characterize the 22-dimensional Minkowski spaces in which the parallel volume difference has always at most order r−1r^{-1}. Finally we present applications concerning Brownian paths and Boolean models.

Author information

Contact details are reproduced from the original publication and may be historical.

Jürgen Kampf

Institut für Stochastik, Universität Ulm, Helmholtzstr. 18, 89069 Ulm, Germany

jurgen.kampf@uni-ulm.de

Keywords

  • Convex geometry
  • parallel volume
  • non-convex body
  • random body

Mathematics Subject Classification

52A20, 52A21, 52A22, 52A38

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Suggested citation

J. Kampf. “Asymptotic Order of the Parallel Volume Difference in Minkowski Spaces.” Journal of Convex Analysis 21 (2014), No. 4, 925–950. https://doi.org/10.68381/jca21050

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.