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Intersection densities of nonstationary Poisson processes of hypersurfaces

Part of: General convexity Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Lars Michael Hoffmann
Lars Michael Hoffmann*
Affiliation:
Universität Karlsruhe (TH)
*
∗ Postal address: Institut für Algebra und Geometrie, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany. Email address: [email protected]
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Abstract

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Intersection densities are introduced for a large class of nonstationary Poisson processes of hypersurfaces and inequalities for them are proved. In doing so, similar results from both Wieacker (1986) and Schneider (2003) are summarized in one theorem and the concept of an associated zonoid of a Poisson process of hypersurfaces is generalized to a nonstationary setting.

Keywords

Poisson processintersection processhypersurfaceassociated zonoidSteiner convex set

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 307 - 317
Copyright
Copyright © Applied Probability Trust 2007 

References

Fallert, H. (1996). Quermass dichten für Punktprozesse konvexer Körper und Boolesche Modelle. Math. Nachr. 181, 165–184.Google Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.Google Scholar
Goodey, P. and Weil, W. (1993). Zonoids and generalisations. In Handbook of Convex Geometry, eds Gruber, P. M. and Wills, J. M., North-Holland, Amsterdam, pp. 1297–1326.Google Scholar
Kiêu, K. (1992). A coarea formula for multiple geometric integrals. Math. Nachr. 156, 75–86.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. (2003). Nonstationary Poisson hyperplanes and their induced tesselations. Adv. Appl. Prob. 35, 139–158.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Wieacker, J. A. (1984). Translative Poincaré formulae for Hausdorff rectifiable sets. Geometriae Dedicata 16, 231–248.Google Scholar
Wieacker, J. A. (1986). Intersections of random hypersurfaces and visibility. Prob. Theory Relat. Fields 71, 405–433.Google Scholar
Wieacker, J. A. (1989). Geometric inequalities for random surfaces. Math. Nachr. 142, 73–106.Google Scholar
Zähle, M. (1982). Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72.CrossRefGoogle Scholar