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Distance measurements on processes of flats

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

Published online by Cambridge University Press:  01 July 2016

Daniel Hug ,
Günter Last  and
Wolfgang Weil
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität Freiburg
Günter Last*
Affiliation:
Universität Karlsruhe (TH)
Wolfgang Weil*
Affiliation:
Universität Karlsruhe (TH)
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104 Freiburg im Breisgau, Germany.
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany.
∗∗∗ Mathematisches Institut II, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany. Email address: [email protected]
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Abstract

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd, the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.

Keywords

Stochastic geometrycontact distribution functionsupport (curvature) measurek-flatPoisson point processRadon transform

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60G57: Random measures 52A22: Random convex sets and integral geometry 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 70 - 95
Copyright
Copyright © Applied Probability Trust 2003 

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References

Arbeiter, E. and Zähle, M. (1991). Kinematic relations for Hausdorff moment measures in spherical spaces. Math. Nachr. 153, 333–348.CrossRefGoogle Scholar
Fallert, H. (1992). Intensitätsmaße und Quermaßdichten für (nichtstationäre) zufällige Mengen und geometrische Punktprozesse. Dissertation, Universität Karls-ruhe.Google Scholar
Fallert, H. (1996). Quermaßdichten für Punktprozesse konvexer Körper und Boole-sche Modelle. Math. Nachr. 181, 165–184.CrossRefGoogle Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.Google Scholar
Goodey, P. and Howard, R. (1990). Processes of flats induced by higher-dimensional processes. Adv. Math. 80, 92–109.CrossRefGoogle Scholar
Goodey, P. and Howard, R. (1990). Processes of flats induced by higher-dimensional processes II. Contemp. Math. 113, 111–119.CrossRefGoogle Scholar
Goodey, P. and Weil, W. (1991). Centrally symmetric convex bodies and Radon transforms on higher order Grassmannians. Mathematika 38, 117–133.CrossRefGoogle Scholar
Goodey, P. and Weil, W. (1993). Zonoids and generalisations. In Handbook of Convex Geometry, eds Gruber, P. M. and Wills, J. M., Elsevier, Amsterdam, pp. 1297–1326.CrossRefGoogle Scholar
Goodey, P., Howard, R. and Reeder, M. (1996). Processes of flats induced by higher-dimensional processes III. Geom. Dedicata 61, 257–269.CrossRefGoogle Scholar
Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press.CrossRefGoogle Scholar
Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Prob. 28, 796–850.CrossRefGoogle Scholar
Hug, D., Last, G. and Weil, W. (2002). Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Prob. 34, 21–47.CrossRefGoogle Scholar
Hug, D., Last, G. and Weil, W. (2002). A survey on contact distributions. To appear in In Morphology of Condensed Matter: Physics and Geometry of Spatially Complex Systems (Lecture Notes Phys. 600), eds Mecke, K. and Stoyan, D., Springer, Berlin, pp. 317–357.CrossRefGoogle Scholar
Kalf, H. (1995). On the expansion of a function in terms of spherical harmonics in arbitrary dimensions. Bull. Belgian Math. Soc. 2, 361–380.Google Scholar
Kiderlen, M. and Weil, W. (1999). Measure-valued valuations and mixed curvature measures of convex bodies. Geom. Dedicata 76, 291–329.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mecke, J., Schneider, R., Stoyan, D. and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.CrossRefGoogle Scholar
Miles, R. (1961). Random polytopes: the generalisation to n dimensions of the intervals of a Poisson process. Thesis, Cambridge University.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. (1999). A duality for Poisson flats. Adv. Appl. Prob. 31, 63–68.CrossRefGoogle Scholar
Schneider, R. (2003). Nonstationary Poisson hyperplanes and their induced tessellations. To appear in Adv. Appl. Prob. 35, 139–158.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
Schneider, R. and Wieacker, J. A. (1993). Integral geometry. In Handbook of Convex Geometry, eds Gruber, P. M. and Wills, J. M., Elsevier, Amsterdam, pp. 1349–1390.CrossRefGoogle Scholar
Spodarev, E. (2001). Selected topics in the theory of spatial stationary flat processes. Dissertation, Universität Jena.Google Scholar
Spodarev, E. (2001). On the rose of intersections of stationary flat processes. Adv. Appl. Prob. 33, 584–599.CrossRefGoogle Scholar
Spodarev, E. (2003). Isoperimetric problems and roses of neighborhood for stationary flat processes. To appear in Math. Nachr.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, New York.Google Scholar
Weil, W. (1979). Kinematic integral formulas for convex bodies. In Contributions to Geometry, Proc. Geometry Symp. (Siegen 1978), eds Tölke, J. and Wills, J. M., Birkhäuser, Basel, pp. 60–76.CrossRefGoogle Scholar
Weil, W. (1981). Zufällige Berührung konvexer Körper durch q-dimensionale Ebenen. Resultate Math. 4, 84–101.CrossRefGoogle Scholar