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Densities for stationary random sets and point processes

Published online by Cambridge University Press:  01 July 2016

Wolfgang Weil  and
John A. Wieacker
Wolfgang Weil*
Affiliation:
Universität Karlsruhe
John A. Wieacker*
Affiliation:
Universität Freiburg
*
∗Postal address: Mathematisches Institut II, Universität Karlsruhe, Englerstrasse 2, 7500 Karlsruhe 1, West Germany.
∗∗ Postal address: Mathematisches Institut, Universität Freiburg, Hebelstrasse 29, 7800 Freiburg, West Germany.
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Abstract

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

Keywords

QUERMASSINTEGRALSINTEGRAL GEOMETRIC FORMULAPOISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 2 , June 1984 , pp. 324 - 346
Copyright
Copyright © Applied Probability Trust 1984 

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References

Castaing, C. and Valadier, M. (1977) Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Davy, P. (1976) Projected thick sections through multi-dimensional particle aggregates. J. Appl. Prob. 13, 714–722. Correction: J. Appl. Prob. 15 (1978), 456.CrossRefGoogle Scholar
Davy, P. (1978) Stereology–A Statistical Viewpoint. , Australian National University, Canberra.CrossRefGoogle Scholar
Eckhoff, J. (1980) Die Euler-Charakteristik von Vereinigungen konvexer Mengen im Rd . Abh. Math. Sem. Univ. Hamburg 50, 135–146.CrossRefGoogle Scholar
Groemer, H. (1978) On the extension of additive functionals on classes of convex sets. Pacific J. Math. 75, 397–410.CrossRefGoogle Scholar
Hadwiger, H. (1956) Integralsätze im Konvexring. Abh. Math. Sem. Univ. Hamburg 20, 136–154.CrossRefGoogle Scholar
Kellerer, H. G. (1984) Minkowski functionals of Poisson processes. To appear.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
McMullen, P. (1977) Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3) 35, 113–135.CrossRefGoogle Scholar
McMullen, P. and Schneider, R. (1983) Valuations on convex bodies. In Convexity and its Applications, ed. Grubber, P. and Wills, J. M., Birkhäuser, Basel, 170–247.CrossRefGoogle Scholar
Neveu, J. (1977) Processus ponctuels. In Lecture Notes in Mathematics 598, Springer-Verlag, Berlin, 249–447.Google Scholar
Nguyen, X. X. and Zessin, H. (1979) Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.CrossRefGoogle Scholar
Schneider, R. (1978) Curvature measures of convex bodies. Ann. Mat. Pura Appl. 116, 101–134.CrossRefGoogle Scholar
Streit, F. (1970) On multiple integral geometric integrals and their applications to probability theory. Canad. J. Math. 22, 151–163.CrossRefGoogle Scholar
Weil, W. (1983) Stereology–A survey for geometers. In Convexity and its Applications, ed. Gruber, P. and Wills, J. M., Birkhäuser, Basel, 360–412.CrossRefGoogle Scholar
Weil, W. (1984) Densities of quermassintegrals for stationary random sets. In Stochastic Geometry, Geometric Statistics, Stereology, ed. Ambartzumian, R. V. and Weil, W.. Teubner, Leipzig.Google Scholar
Wieacker, J. A. (1982) Translative stochastische Geometrie der konvexen Körper. , Albert-Ludwigs-Universität, Freiburg.Google Scholar