The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

Poisson-Laguerre tessellations的极限

Limits of Poisson-Laguerre tessellations

研究Poisson-Laguerre tessellations的极限性质，证明其收敛到经典Poisson-Voronoi或Poisson-Delaunay tessellation，并讨论典型单元格的收敛性。

Weighted Poisson-Delaunay Mosaics

Slicing a Voronoi tessellation in $\mathbb{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in $\mathbb{R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of…

Exponential moments for planar tessellations

In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical such tessellations like the Poisson-Voronoi, Poisson-Delaunay and Poisson line tessellation, we also treat the Johnson-Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

β-和β'型德劳内图中的度数

Degrees in the $β$- and $β'$-Delaunay graphs

研究β-和β'-型维布恩图的典型单元，推导出互补定理，揭示典型单元的Φ内容分布及与形状无关性，并推导出分布的积分表示，进而得出面数分布的界，利用对偶性得出β-和β'-型德劳内三角剖分的典型度数界。

球面空间中的几何不等式、稳定性结果与Kendall问题

Geometric inequalities, stability results and Kendall's problem in spherical space

本文研究球面空间中泊松随机镶嵌胞腔的Kendall形状问题，通过建立球面凸体尺寸泛函与命中泛函的新几何不等式及稳定性结果，在高强度渐近机制下获得了尺寸泛函的偏差不等式与渐近分布。

Cluster size distributions of extreme values for the Poisson-Voronoi tessellation

We consider the Voronoi tessellation based on a homogeneous Poisson point process in $\mathbf{R}^{d}$. For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us…

关于平面中高阶 Brillouin 镶嵌及相关铺砌的角度

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

研究平面中局部有限集合的高阶 Brillouin 镶嵌及相关铺砌的角度单调性，证明平稳 Poisson 点过程下各类镶嵌的角度分布独立于阶数。

锚定展开、速度和双曲Poisson Voronoi曲面细分

Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation

我们证明了具有正锚定扩张和指数体积增长的平稳随机图上的随机游动具有正速度。我们还表明，两个家庭的随机三角剖分的双曲平面，双曲Poisson Voronoi tessellation和双曲Poisson Delaunay三角剖分，有1骨架正锚定的扩展。因此，我们证明了这些图上的简单随机游动具有正速度。我们包括一个开放的问题和命题上的主题平稳的几何随机图和双曲泊松Voronoi镶嵌的一节。

Poisson-Delaunay Mosaics of Order $k$

The order-$k$ Voronoi tessellation of a locally finite set $X \subseteq \mathbb{R}^n$ decomposes $\mathbb{R}^n$ into convex domains whose points have the same $k$ nearest neighbors in $X$. Assuming $X$ is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the $k$ nearest points in $X$ are within a given distance threshold.

Poisson超平面镶嵌中的大面

Large faces in Poisson hyperplane mosaics

一个著名的问题D. G. Kendall指出，在${\mathbb{R}}^d$中，一个静态Poisson超平面镶嵌的零胞，在它具有大体积的条件下，以很高的概率近似于某个确定的形状，这个形状是由底层超平面过程的方向分布决定的。该结果在这里被扩展到用于$k\in\{2,...,d-1\}$的镶嵌的典型$k$面。这就需要一个额外的条件，即面的方向是在给定方向的一个足够小的邻域内。

平稳曲面的极值及其应用

A general study of extremes of stationary tessellations with applications

设$\mathfrak{m}$是在有界Borel子集$W$中观察到的$\mathbf{R}^d$中的随机镶嵌，并且$f(\cdot)$是定义在凸体集合上的可测函数。对于$\mathfrak{m}$的每个细胞$C$，我们将点$z(C)$与$C$的核相关联。将$f(\cdot)$应用于$\mathfrak{m}$的所有细胞，当$ρ$趋于无穷大时，我们研究了$f(C)$在所有细胞$C\in\mathfrak{m}$上的顺序统计量，其中细胞核在$\mathbf{W}_ρ=ρ^{1/d}W$中。在强混合性质和$\mathfrak{m}$和$f(\cdot)$上的局部条件下，我们给出了一个一般定理，该定理将顺序统计量的研究归结为随机变量$f(\mathscr{C})$，其中$\mathscr{C}$是$\mathfrak{m}$的典型胞元.证明推导出一个泊松近似依赖图通过陈斯坦方法。本文证明了点过程$\left\{(ρ^{-1/d}z(C), a_ρ^{-1}(f(C)-b_ρ)), C\in\mathfrak{m}, z(C)\in \mathbf{W}_ρ\right\}$（其中$a_ρ>0$和$b_ρ$是依赖于$ρ$的两个合适的函数）收敛于非齐次Poisson点过程.在Poisson-Voronoi和Poisson-Delaunay镶嵌的特定设置和不同的功能$f(\cdot)$，如内径，…

双曲空间中水平球内的随机双曲多面体

Random hyperbolic polyhedra in horoballs

本文研究双曲空间中水平球内泊松点过程的测地凸包，将其边界投影为对偶泊松-拉盖尔镶嵌，证明临界状态下收敛于泊松-德劳内镶嵌，并给出体积、表面积及顶点强度的精确渐近结果。

Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry

We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall's problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Vorono…