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Dilation volumes of sets of finite perimeter

  • Markus Kiderlen (a1) and Jan Rataj (a2)


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


Corresponding author

* Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. Email address:
** Postal address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. Email address:


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Dilation volumes of sets of finite perimeter

  • Markus Kiderlen (a1) and Jan Rataj (a2)


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