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Random measurable sets and covariogram realizability problems

Part of: Geometric probability and stochastic geometry Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  21 March 2016

Bruno Galerne  and
Raphael Lachièze-Rey
Bruno Galerne*
Affiliation:
Université Paris Descartes
Raphael Lachièze-Rey*
Affiliation:
Université Paris Descartes
*
∗ Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints-pères, 75006 Paris, France.
∗ 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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Abstract

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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.

Keywords

Random measurable setrealizabilityS2 problemcovariogramperimetertruncated moment problem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 611 - 639
Copyright
Copyright © Applied Probability Trust 2015 

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