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Dilation volumes of sets of finite perimeter

Part of: Functions of several variables Geometric probability and stochastic geometry Classical measure theory

Published online by Cambridge University Press:  29 November 2018

Markus Kiderlen  and
Jan Rataj
Markus Kiderlen*
Affiliation:
University of Aarhus
Jan Rataj*
Affiliation:
Charles University, Prague
*
* Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. Email address: kiderlen@math.au.dk
** Postal address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. Email address: rataj@karlin.mff.cuni.cz
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Abstract

In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.

Keywords

Bounded variationcontact distribution functiondilation volumedirectional variationsets of finite perimeterstationary random setsurface area measure

MSC classification

Primary: 26B30: Absolutely continuous functions, functions of bounded variation
Secondary: 28A75: Length, area, volume, other geometric measure theory 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1095 - 1118
Copyright
Copyright © Applied Probability Trust 2018 

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