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Dilation volumes of sets of finite perimeter. (English) Zbl 1440.60012

Summary: In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation \(A\oplus tQ\) 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.

MSC:

60D05 Geometric probability and stochastic geometry
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
28A75 Length, area, volume, other geometric measure theory
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