Random Measurable Sets and Covariogram Realisability Problems

TL;DR

This paper characterizes the realisable set covariograms using random measurable sets (RAMS), offering a new framework for solving geometric problems like the $S_2$ problem in materials science.

Contribution

It introduces a rigorous characterization of covariogram realisability via RAMS and extends the theory of random measurable sets, especially regarding perimeter approximation.

Findings

Provided a characterization of realisable set covariograms.

Extended the theory of random measurable sets.

Demonstrated RAMS as a useful framework for geometric problems.

Abstract

We provide a characterization of the realisable set covariograms, bringing a rigorous yet abstract solution to the $S\_2$ problem in materials science. Our method is based on the covariogram functional for random mesurable sets (RAMS) and on a result about the representation of positive operators in a locally compact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, they provide a weaker framework allowing to manipulate more irregular functionals, such as the perimeter. We therefore use the illustration provided by the $S\_{2}$ problem to advocate the use of RAMS for solving theoretical problems of geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.