The Annals of Probability

The limit shape of the zero cell in a stationary Poisson hyperplane tessellation

Abstract

In the early 1940s, D. G. Kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic Poisson line process in the plane tends to circularity given that the area of the zero cell tends to $\infty$. A proof was given by I. N. Kovalenko in 1997. This paper generalizes Kovalenko's result in two directions: to higher dimensions and to not necessarily isotropic stationary Poisson hyperplane processes. In the anisotropic case, the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of Minkowski's existence theorem for convex bodies with given area measures.

Article information

Source
Ann. Probab., Volume 32, Number 1B (2004), 1140-1167.

Dates
First available in Project Euclid: 11 March 2004

https://projecteuclid.org/euclid.aop/1079021474

Digital Object Identifier
doi:10.1214/aop/1079021474

Mathematical Reviews number (MathSciNet)
MR2044676

Zentralblatt MATH identifier
1050.60010

Citation

Hug, Daniel; Reitzner, Matthias; Schneider, Rolf. The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Probab. 32 (2004), no. 1B, 1140--1167. doi:10.1214/aop/1079021474. https://projecteuclid.org/euclid.aop/1079021474

References

• Bronshtein, E. M. and Ivanov, L. D. (1975). The approximation of convex sets by polyhedra. Siberian Math. J. 16 852--853.
• Goldman, A. (1998). Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Probab. 26 1727--1750.
• Groemer, H. (1990). On an inequality of Minkowski for mixed volumes. Geom. Dedicata 33 117--122.
• Kovalenko, I. N. (1997). A proof of a conjecture of David Kendall on the shape of random polygons of large area. Kibernet. Sistem. Anal. 1997 3--10, 187. (English translation in Cybernet. Systems Anal. 33 461--467.)
• Kovalenko, I. N. (1999). A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons. J. Appl. Math. Stochastic Anal. 12 301--310.
• Mecke, J. (1999). On the relationship between the $0$-cell and the typical cell of a stationary random tessellation. Pattern Recognition 32 1645--1648.
• Miles, R. E. (1995). A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons. Adv. in Appl. Probab. 27 397--417.
• Schneider, R. (1982). Random hyperplanes meeting a convex body. Z. Wahrsch. Verw. Gebiete 61 379--387.
• Schneider, R. (1993). Convex Bodies: The Brunn--Minkowski Theory. In Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
• Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.
• Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.