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Dilation volumes of sets of finite perimeter

Markus Kiderlen, Jan Rataj

TL;DR

This paper investigates the first-order, right-sided behavior of dilation volumes $\lambda_n(A\oplus tQ)$ as $t\to0^+$ for sets $A\subset\mathbb{R}^n$ of finite perimeter and finite test sets $Q$. By extending the covariogram derivative from two-point tests to general finite $Q$, the authors derive a first-order formula for dilation volumes in terms of a generalized surface area measure $S_{n-1}^*(A;\cdot)$, using smooth BV-function approximations of $\mathbf{1}_A$. They introduce the $Q$-variation $V^Q(f)$ and establish $V^Q(f)=|Df|_Q$ with an explicit integral representation, enabling a rigorous first-order expansion. The results yield a precise relation between dilation volumes and the surface area measure, generalizing the cosine-transform connection to finite $Q$, and they apply the framework to the derivative of the contact distribution function for stationary random sets. The approach avoids classical regularity restrictions by working with sets of finite perimeter and random measurable sets (RAMS), offering broad applicability in stochastic geometry and geometric measure theory.

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.

Dilation volumes of sets of finite perimeter

TL;DR

This paper investigates the first-order, right-sided behavior of dilation volumes as for sets of finite perimeter and finite test sets . By extending the covariogram derivative from two-point tests to general finite , the authors derive a first-order formula for dilation volumes in terms of a generalized surface area measure , using smooth BV-function approximations of . They introduce the -variation and establish with an explicit integral representation, enabling a rigorous first-order expansion. The results yield a precise relation between dilation volumes and the surface area measure, generalizing the cosine-transform connection to finite , and they apply the framework to the derivative of the contact distribution function for stationary random sets. The approach avoids classical regularity restrictions by working with sets of finite perimeter and random measurable sets (RAMS), offering broad applicability in stochastic geometry and geometric measure theory.

Abstract

This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation as converges to zero. Here and are subsets of -dimensional Euclidean space, has finite perimeter and is finite. If consists of two points only, and , say, this derivative coincides up to sign with the directional derivative of the covariogram of in direction . By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of . We extend this result to finite sets 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 by smooth functions of bounded variation and showing corresponding formulas for them.

Paper Structure

This paper contains 5 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The variation with respect to a compact set
  4. Dilation volumes
  5. An application: Contact distributions of stationary random sets

Key Result

Theorem 1.1

Assume that $A\subset{\mathbb R}^n$ has finite perimeter. If $\emptyset \ne Q\subset{\mathbb R}^n$ is finite then If, in addition, $A$ has bounded volume then also

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Basic properties of the variation
  • Lemma 2.2
  • proof
  • Proposition 2.3: Basic properties of the directional variation
  • proof
  • Lemma 2.4: AFP00
  • Remark 2.5
  • Lemma 3.1: Ordinary variation and variation with respect to $Q$
  • ...and 22 more