September 2015 Random measurable sets and covariogram realizability problems
Bruno Galerne, Raphael Lachièze-Rey
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Adv. in Appl. Probab. 47(3): 611-639 (September 2015). DOI: 10.1239/aap/1444308874


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.


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Bruno Galerne. Raphael Lachièze-Rey. "Random measurable sets and covariogram realizability problems." Adv. in Appl. Probab. 47 (3) 611 - 639, September 2015.


Published: September 2015
First available in Project Euclid: 8 October 2015

zbMATH: 1353.60013
MathSciNet: MR3406600
Digital Object Identifier: 10.1239/aap/1444308874

Primary: 60D05
Secondary: 28C05

Keywords: covariogram , perimeter , Random measurable set , realizability , S_2 problem , truncated moment problem

Rights: Copyright © 2015 Applied Probability Trust


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Vol.47 • No. 3 • September 2015
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