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

• 成果类型: Article
• 署名作者: Hug, D; Reitzner, M; Schneider, R
• 署名单位: University of Freiburg; Technische Universitat Wien
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 1140-1167
• 关键词: large random polygons; conjecture; kendall; d.g.; PROOF

摘要

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 infinity. 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.