Random Measurable Sets and Covariogram Realisability Problems

Bruno Galerne (1) , Raphaël Lachièze-Rey (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Keywords: covariogram; $S_{2}$ problem; realisability; perimeter; truncated moment problem; Random measurable sets

Domains: Probability [math.PR]

Bruno Galerne, Raphaël Lachièze-Rey. Random Measurable Sets and Covariogram Realisability Problems. Advances in Applied Probability, 2015, 47 (3), pp.611-639. ⟨hal-00995853v3⟩