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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 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.

Keywords

Random measurable setrealizabilityS 2 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

Information

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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