Abstract
AbstractIn this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilationA⊕tQastconverges to 0. HereAandQare subsets ofn-dimensional Euclidean space,Ahas finite perimeter, andQis finite. IfQconsists of two points only,nandn+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram ofAin directionu. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure ofA. We extend this result to finite setsQand 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 ofAby smooth functions of bounded variation.
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