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Valuations and Boolean models. (English) Zbl 1378.60033

Vedel Jensen, Eva B. (ed.) et al., Tensor valuations and their applications in stochastic geometry and imaging. Based on the presentations at the workshop, Sandbjerg Manor, Denmark, September 21–26, 2014. Cham: Springer (ISBN 978-3-319-51950-0/pbk; 978-3-319-51951-7/ebook). Lecture Notes in Mathematics 2177, 301-338 (2017).
The Boolean model, defined as the union set of a stationary Poisson process of convex particles in \(n\)-dimensional Euclidean space, is certainly the most tractable model of a random closed set. Since the seminal work of Matheron, Miles and Davy in the 1970s, the investigation of Boolean models has been widened and deepened considerably and is now an impressive amalgamation of the theories of Poisson processes, translative integral geometry, and valuations. Central themes are the expectation \({\mathbf E}\,\varphi(Z\cap K_0)\) for a Boolean model \(Z\), a valuation \(\varphi\) and a convex observation window \(K_0\), its limit (after division by the volume of \(K_0\)) for \(K_0\) expanding to the whole space, and the relation of such functional densities to densities of the underlying particle process. The present chapter gives a survey, up to most recent developments. It treats real-, measure-, and tensor-valued valuations, basic equations for the Boolean model, integral geometry for valuations, mean values for valuations, special valuations such as mixed volumes, support functions, area measures, flag measures, and tensor valuations on Boolean models. An outlook sheds light on non-stationary Boolean models and on the use of harmonic intrinsic volumes in the study of stationary non-isotropic Boolean models.
For the entire collection see [Zbl 1372.53001].

MSC:

60D05 Geometric probability and stochastic geometry
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
53C65 Integral geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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References:

[1] H. Fallert, Quermaßdichten für Punktprozesse konvexer Körper und Boolesche Modelle. Math. Nachr. 181, 37–48 (1998)
[2] P. Goodey, W. Weil, Translative and kinematic integral formulae for support functions II. Geom. Dedicata 99, 103–125 (2003) · Zbl 1031.52002 · doi:10.1023/A:1024912419608
[3] P. Goodey, W. Hinderer, D. Hug, J. Rataj, W. Weil, A flag representation of projection functions. Adv. Geom. (2017, in print) · Zbl 1401.52008
[4] H. Hadwiger, Translationsinvariante, additive und schwachstetige Polyederfunktionale. Arch. Math. 3, 387–394 (1952) · Zbl 0048.28801 · doi:10.1007/BF01899378
[5] W. Hinderer, Integral representations of projection functions. PhD Thesis, University of Karlsruhe, Karlsruhe (2002)
[6] J. Hörrmann, The method of densities for non-isotropic Boolean models. PhD Thesis, KIT Scientific Publishing, Karlsruhe (2015)
[7] J. Hörrmann, D. Hug, M. Klatt, K. Mecke, Minkowski tensor density formulas for Boolean models. Adv. Appl. Math. 55, 48–85 (2014) · Zbl 1308.60019 · doi:10.1016/j.aam.2014.01.001
[8] D. Hug, Measures, curvatures and currents in convex geometry. Habilitation Thesis, University of Freiburg, Freiburg (1999)
[9] D. Hug, G. Last, On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Probab. 28, 796–850 (2000) · Zbl 1044.60006 · doi:10.1214/aop/1019160261
[10] D. Hug, I. Türk, W. Weil, Flag measures for convex bodies, in Asymptotic Geometric Analysis, ed. by M. Ludwig, et al. Fields Institute Communications, vol. 68 (Springer, Berlin, 2013), pp. 145–187 · Zbl 1277.52003 · doi:10.1007/978-1-4614-6406-8_7
[11] S.C. Kapfer, Morphometry and physics of particulate and porous media. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen (2011)
[12] M. Kiderlen, W. Weil, Measure-valued valuations and mixed curvature measures of convex bodies. Geom. Dedicata 76, 291–329 (1999) · Zbl 0933.52014 · doi:10.1023/A:1005173927802
[13] M.A. Klatt, Morphometry of random spatial structures in physics. PhD Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen (2016)
[14] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes. Proc. Lond. Math. Soc. (3) 35, 113–135 (1977) · Zbl 0353.52001 · doi:10.1112/plms/s3-35.1.113
[15] P. McMullen, Continuous translation-invariant valuations on the space of compact convex sets. Arch. Math. 34, 377–384 (1980) · Zbl 0424.52003 · doi:10.1007/BF01224974
[16] P. McMullen, Valuations and dissections, in Handbook of Convex Geometry, ed. by P.M. Gruber, J.M. Wills, vol. B, (North-Holland, Amsterdam, 1993), pp. 933–988 · Zbl 0791.52014 · doi:10.1016/B978-0-444-89597-4.50010-X
[17] W. Mickel, S.C. Kapfer, G.E. Schröder-Turk, K. Mecke, Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. J. Chem. Phys. 138, 044501 (2013) · doi:10.1063/1.4774084
[18] R. Schneider, Mixed polytopes. Discrete Comput. Geom. 29, 575–593 (2003) · Zbl 1034.52012 · doi:10.1007/s00454-002-0788-x
[19] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, 2nd edn. (Cambridge University Press, Cambridge, 2014) · Zbl 1287.52001
[20] R. Schneider, W. Weil, Translative and kinematic integral formulae for curvature measures. Math. Nachr. 129, 67–80 (1986) · Zbl 0602.52003 · doi:10.1002/mana.19861290106
[21] R. Schneider, W. Weil, Stochastic and Integral Geometry (Springer, Heidelberg/New York, 2008) · Zbl 1175.60003 · doi:10.1007/978-3-540-78859-1
[22] G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F. Schaller, M.J.F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, K. Mecke, Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mater. 23, 2535–2553 (2011) · doi:10.1002/adma.201100562
[23] J. Schulte, Intensity estimation for non-isotropic Boolean models via harmonic intrinsic volumes (in preparation)
[24] W. Weil, Iterations of translative integral formulae and non-isotropic Poisson processes of particles. Math. Z. 205, 531–549 (1990) · Zbl 0693.52001 · doi:10.1007/BF02571261
[25] W. Weil, Translative and kinematic integral formulae for support functions. Geom. Dedicata 57, 91–103 (1995) · Zbl 0838.52004 · doi:10.1007/BF01264062
[26] W. Weil, Integral geometry of translation invariant functionals, I: The polytopal case. Adv. Appl. Math. 66, 46–79 (2015) · Zbl 1327.52008
[27] W. Weil, Integral geometry of translation invariant functionals, II: The case of general convex bodies. Adv. Appl. Math. 83, 145–171 (2017) · Zbl 1358.52010
[28] W. Weil, J.A. Wieacker, Densities for stationary random sets and point processes. Adv. Appl. Probab. 16, 324–346 (1984) · Zbl 0546.60014 · doi:10.1017/S0001867800022552
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