## Asymptotic order of the parallel volume difference

- 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.

Author: | Jürgen Kampf |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-17006 |

Serie (Series number): | Report in Wirtschaftsmathematik (WIMA Report) (139) |

Document Type: | Preprint |

Language of publication: | English |

Year of Completion: | 2011 |

Year of Publication: | 2011 |

Publishing Institute: | Technische Universität Kaiserslautern |

Date of the Publication (Server): | 2011/05/26 |

Tag: | Parallel volume ; Wills functional; non-convex body |

Note: | A newer version of this document is available on KLUEDO: urn:nbn:de:hbz:386-kluedo-29122 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011 |