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Random measurable sets and covariogram realizability problems

  • Bruno Galerne (a1) and Raphael Lachièze-Rey (a1)

We provide a characterization of 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 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 S 2 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.

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Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints-pères, 75006 Paris, France.
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Advances in Applied Probability
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  • EISSN: 1475-6064
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