The limit shape of the zero cell in a stationary Poisson hyperplane tessellation.
The authors generalize to higher dimensions, and to not necessarily isotropic stationary Poisson hyperplane process, a proof by \textit{I. N. Kovalenko} [Cybern. Syst. Anal. 33, No. 4, 461--467 (1997); translation from Kibern. Sist. Anal. 1997, No. 4, 3--10 (1997; Zbl 0922.60022)] of D. G. Kendall's conjecture concerning the shape of the zero cell of the random tessellation generated by the process. In the case of a stationary and isotropic Poisson line process in the plane, Kovalenko proved that, given that the area of the zero cell tends to infinity, its shape tends to circularity. Let \(X\) be a stationary Poisson hyperplane process in \(R^d\), \(d\geq 2\), with intensity \(\lambda>0\). An even measure \(\varphi\) on the unit sphere describes the distribution of the unit normals of the hyperplane of \(X\). By Minkowski's existence and uniqueness theorem from the geometry of convex bodies, there exists a unique centrally symmetric convex body \(B\), called the direction body of \(X\), for which \(\varphi\) is the area measure. Let \(Z_0\) be the zero cell of the tessellation induced by \(X\). Then it is shown that there exists a constant \(c_0\), depending only on \(B\), such that the following is true. If \(\varepsilon\in (0,1)\) and \(I=[a,b)\) is any interval with \(a^{1/d}\lambda\geq\sigma_0>0\), then \[ P(r_B(Z_0)\geq\varepsilon\mid V(Z_0)\in I)\leq c \exp\{-c_0 \varepsilon^{d+1} a^{1/d}\lambda\}, \] where \(P\) denotes probability, \[ r_B(Z_0):=\inf \{s/r-1: rB\subset Z_0\subset sB+z,\;z\in R^d,\;r,s>0\}, \] and \(c\) is a constant depending on \(B,\varepsilon,\sigma_0\). As a consequence, the interesting result \[ \lim_{a\to\infty}P(r_B(Z_0)\geq \varepsilon\mid V(Z_0)\geq a)=0 \] for every \(\varepsilon>0\) is obtained, which shows that the conditional law for the shape of \(Z_0\), given a lower bound for \(V(Z_0)\), converges weakly, as that lower bound tends to infinity, to the law concentrated at the shape of the direction body \(B\) associated to the given process
scientific article; zbMATH DE number 2100744
scientific article