Dilation volumes of sets of finite perimeter

• Published in: Advances in applied probability Vol. 50; no. 4; pp. 1095 - 1118
• Main Authors: Kiderlen, Markus, Rataj, Jan
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2018; Applied Probability Trust
• Subjects: Dilation; Distribution functions; Euclidean geometry; Euclidean space; Original Article; Probability; 60D05; Bounded variation; sets of finite perimeter; stationary random set; Primary 26B30; surface area measure; directional variation; Secondary 28A75; contact distribution function; dilation volume
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2018.52

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.