Cells in the box and a hyperplane

On the cells in a stationary Poisson hyperplane mosaic

Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $\hat X$ in ${\mathbb R}^d$. Under some mild conditions on the spherical directional distribution of $\hat X$ (which are satisfied, for example, if the process is isotropic), we show that with probability one the set of cells ($d$-polytopes) of $X$ has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple $d$-polytopes is realized infinitely often by the cells of $X$. A further result concerns the distribution of the typical cell.

Small cells in a Poisson hyperplane tessellation

Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size functionals and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $φ$ and with respect to an arbitrary size functional $Σ$. More precisely, we investigate the distribution of the typical cell $Z$, conditioned on the event $\{Σ(Z)<a\}$, w…

Cells with many facets in a Poisson hyperplane tessellation

Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity $n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the o…

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent…

线面排列包络的复杂性

The complexity of the envelope of line and plane arrangements

超平面排列的一个小平面称为外部的，如果它恰好属于一个有界胞元。所有外部刻面的集合形成布置的包络。由维度$d$中的$n$超平面定义的简单布置的外部刻面的数量被假设为至少为$d{n-2 \choose d-1}$。在本文中，我们表明，对于4行或更多的简单安排，外部小平面的最小数目等于$2(n-1)$，对于5个平面或更多的简单安排，外部小平面的最小数目介于$\frac{n(n-2)+6}{3}$和$(n-4)(2n-3)+5$之间。