Going Beyond Variation of Sets

TL;DR

This paper investigates integral geometric representations of variations of general sets in Euclidean space without regularity assumptions, focusing on measure-theoretic boundaries when classical theorems do not apply.

Contribution

It introduces a measure-theoretic boundary concept to represent variations of sets with minimal regularity, extending classical geometric measure theory results.

Findings

Established integral geometric representations using measure-theoretic boundaries.

Extended the theory of set variations beyond classical regularity assumptions.

Identified suitable notions of measure-theoretic boundary for general sets.

Abstract

We study integralgeometric representations of variations of general sets $A$ in the Euclidean n-space without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function $\chi^A$ is a signed Borel measure with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss-Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a `measure-theoretic boundary' plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of `measure-theoretic boundary' and one can address the question to find notions of measure-theoretic boundary that are as fine as possible.