2014/05/24 by Bruno Galerne, Galerne, Bruno, Raphaël Lachièze-Rey +2 · 2 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1405.6333
35pp
openalex publication_date 2014/05/24 · arxiv created 2015/02/28 · arxiv updated 2015/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.