Random measurable sets and covariogram realizability problems

Abstract

We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S₂ problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S₂ problem to advocate the use of RAMS for solving theoretical problems of a 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.