Dilation volumes of sets of finite perimeter

TL;DR

This paper studies how the volume of a set with finite perimeter changes under dilation by small sets, extending known results to finite sets and applying findings to stochastic geometry.

Contribution

It extends the analysis of volume derivatives under dilation from two-point sets to finite sets and connects these results to the contact distribution function in stochastic geometry.

Findings

Derived the first order volume derivative for finite sets Q.

Connected the derivative to the cosine transform of the surface area measure.

Applied results to the contact distribution function of stationary random sets.

Abstract

This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation $A\oplus tQ$ as $t$ converges to zero. 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, $x$ and $x+u$, say, this derivative coincides up to 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 zero. The proofs are based on approximation of the characteristic function of $A$ by smooth functions of bounded variation and showing corresponding formulas for them.