Poisson超平面镶嵌中的小单元
Small cells in a Poisson hyperplane tessellation
摘要：
到目前为止，很少有人知道的泊松超平面镶嵌中的小细胞的性质。现有的几个结果是启发式的或仅适用于二维的情况下，非常具体的大小泛函和方向分布。本文填补了这一空白，提供了一个系统的研究小细胞在泊松超平面镶嵌的任意尺寸，任意方向分布$φ$和相对于一个任意大小的功能$Σ$。更确切地说，我们调查的典型细胞$Z$的分布，条件的事件$\{Σ(Z)<a\}$，其中$a\to0$和$Σ$是一个大小功能，即功能的凸体是连续的，不全同零，齐次$k>0$，并增加相对于集包含。我们专注于这些小细胞的小平面数量和形状。我们在各种一般设置中表明，小细胞往往会使小平面的数量最小化，并且它们具有非退化的极限形状分布，该分布取决于尺寸$Σ$和方向分布。我们还展示了一类的方向性分布的细胞与小内径不倾向于最小化的方面的数量。
英文摘要：
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 $\{Σ(Z)<a\}$, where $a\to0$ and $Σ$ is a size functional, i.e. a functional on the set of convex bodies which is continuous, not identically zero, homogeneous of degree $k>0$, and increasing with respect to set inclusion. We focus on the number of facets and the shape of such small cells. We show in various general settings that small cells tend to minimize the number of facets and that they have a non degenerated limit shape distribution which depends on the size $Σ$ and the directional distribution. We also exhibit a class of directional distribution for which cells with small inradius do not tend to minimize the number of facets.
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