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Markus Kiderlen

Dilation volumes of sets of finite perimeter

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In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation AastQ 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.

Original languageEnglish
JournalAdvances in Applied Probability
Volume50
Issue4
Pages (from-to)1095-1118
Number of pages24
ISSN0001-8678
DOIs
Publication statusPublished - 1 Dec 2018

    Research areas

  • 2010 Mathematics subject classification:Primary 26B30Secondary 28A7560D05

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