# Extremal behavior of large cells in the
Poisson hyperplane mosaic

Moritz Otto Department of Mathematics, Aarhus University, Ny Munkegade 118,
8000 Aarhus C, Denmark, email<EMAIL_ADDRESS>
###### Abstract

We study the asymptotic behavior of a size-marked point process of centers of
large cells in a stationary and isotropic Poisson hyperplane mosaic in
dimension $d\geq 2$. The sizes of the cells are measured by their inradius or
their $k$th intrinsic volume ($k\geq 2$), for example. We prove a Poisson
limit theorem for this process in Kantorovich-Rubinstein distance and thereby
generalize a result in Chenavier and Hemsley (2016) in various directions. Our
proof is based on a general Poisson process approximation result that extends
a theorem in Bobrowski, Schulte and Yogeshwaran (2021).

AMS 2020 Subject Classifications: Primary 60G55, 60F17 Secondary 60D05

Keywords: Blaschke-Petkantschin formula, Chen-Stein method, Kantorovich-
Rubinstein metric, Kendall’s problem, maximum cell, Palm distribution, point
process approximation, Poisson hyperplane process, random mosaic

##  1 Introduction

Random mosaics are important objects in both theory and practice of modern
probability. They are used in a wide range of applications to model random
spatial phenomena. Besides the Voronoi mosaic (and its dual, the Delaunay
mosaic), the hyperplane mosaic is an important model class. In this article we
study stationary and isotropic Poisson hyperplane mosaics. These are random
mosaics where the generating process is a stationary and isotropic Poisson
hyperplane process.

Different aspects of this mosaic have been investigated in the literature. For
example, in [8] it is shown that the shape of the zero cell (it is the cell
containing the origin $o\in{\mathbb{R}}^{d}$) is with high probability close
to the shape of a ball if its $k$th intrinsic volume (for some $k\geq 2$) is
large. This result is the answer to the Kendall problem (formulated for the
zero cell of a Poisson hyperplane mosaic) and is adapted to the typical cell
of the mosaic in [9]. In [3] cells with large (and small) inradius and with
center in a window are considered. For dimension $d=2$ it is shown that the
limit distribution of the largest and smallest order statistics for the
inradii converge to a Poisson distribution when the size of the window goes to
infinity.

In the present paper we considerably extend this result in various directions.
We consider marked point processes of centers of large (w.r.t. the inradius or
an intrinsic volume, e.g. volume, surface area) cells in a compact window in a
stationary, isotropic Poisson hyperplane mosaic, where the mark is a
transformation of the cell size. We study the asymptotics of a scaling of this
process as the volume of the window tends to infinity. Using Stein’s method
(via a coupling of the marked process with a Palm version of itself), we prove
convergence in the Kantorovich-Rubinstein distance to a marked Poisson
process. To this end, we give an extension of [2, Theorem 4.1]. In the proofs
of our main Theorems 1 and 2 we face two kinds of obstacles. The first one
concerns long range dependencies and comes from the fact that two cells that
are arbitrarily far apart from each other can share joint facet hyperplanes.
The second one deals with local dependencies in the mosaic and concerns the
sizes of the clusters in which large cells appear. The typical cluster size
heavily depends on the shape of large cells. For the size functions that we
consider (where all extremal bodies are balls) we use the answer to Kendall’s
problem and obtain clusters of size one.

Our article is structured as follows. In Section 2 we give precise definitions
of the objects that we study and present our main results. In Section 3 we
prove two important auxiliary lemmas that give a bound on the number of
hyperplanes in a stationary, isotropic Poisson hyperplane process that hit two
disjoint balls. Moreover, we present a spherical Blaschke-Petkantschin formula
that generalizes [16, Theorem 7.3.2] and might be of independent interest. In
Section 4 we give an extension of [2, Theorem 6.1]. In Section 5 we give the
proof of Theorem 1 (where we consider the process of centers of cells with a
large inradius). The remaining sections are devoted to the proof of Theorem 2
(which is a generalization of the proof of Theorem 1 to the process of centers
of cells that are large with respect to a more general size function). In
Section 6 we study the distribution of a size functional of the typical cell
in the Poisson hyperplane mosaic in more detail. In Section 7 we construct
stopping sets for large cells in the mosaic and demonstrate how they can be
used to derive a decorrelation inequality. Finally, we complete the proof of
Theorem 2 in Section 8.

##  2 Preliminaries and Main Results

We use the notation from [9] and [2]. For a locally compact second countable
Hausdorff (lcscH) space $(\mathbb{X},\mathcal{X})$ we write
$\mathbf{N}_{\mathbb{X}}$ for the space of all $\sigma$-finite counting
measures on $\mathbb{X}$ and $\widehat{\mathbf{N}}_{\mathbb{X}}$ for the space
of all finite counting measures on $\mathbb{X}$. We equip
$\mathbf{N}_{\mathbb{X}}$ and $\widehat{\mathbf{N}}_{\mathbb{X}}$ with their
corresponding $\sigma$-algebras $\mathcal{N}_{\mathbb{X}}$ and
$\widehat{\mathcal{N}}_{\mathbb{X}}$ that are induced by the mappings
$\omega\mapsto\omega(B)$ for all $B\in\mathcal{X}$. These are the
Borel-$\sigma$-algebras with respect to the Fell topologies on
$\mathbf{N}_{\mathbb{X}}$ and $\widehat{\mathbf{N}}_{\mathbb{X}}$,
respectively. For $x\in\mathbb{X}$ let $\delta_{x}\in\mathbf{N}_{\mathbb{X}}$
be the Dirac measure in $x$, i.e. $\delta_{x}(B)=\mathbf{1}_{B}(x)$ for
$B\in\mathcal{X}$. More generally, for
$\bm{x}=(x_{1},\dots,x_{m})\in\mathbb{X}^{m}$ we write
$\delta_{\bm{x}}=\delta_{x_{1}}+\cdots+\delta_{x_{m}}$.

The real Euclidean vector space ${\mathbb{R}}^{d}$ is equipped with the
standard scalar product $\langle\cdot,\cdot\rangle$ and induced norm
$\|\cdot\|$. We write $\kappa_{d}$ for the volume of the closed unit ball
$B^{d}:=\\{x\in{\mathbb{R}}^{d}:\,\|x\|\leq 1\\}$, $\omega_{d}=d\kappa_{d}$
for the surface area of the unit sphere $S^{d-1}:=\partial
B^{d}=\\{x\in{\mathbb{R}}^{d}:\,\|x\|=1\\}$ and
$B(z,r):=z+rB^{d}=\\{x\in{\mathbb{R}}^{d}:\,\|x-z\|\leq r\\}$ for the closed
ball with radius $r>0$ around $z\in{\mathbb{R}}^{d}$. The $\ell$-dimensional
($\ell\in\\{0,\dots,d\\}$) Lebesgue measure on an $\ell$-dimensional affine
subspace of ${\mathbb{R}}^{d}$ is denoted by $\lambda_{\ell}$ and $\sigma_{k}$
is the normalized Lebesgue measure on a $k$-dimensional
($k\in\\{0,\dots,d-1\\}$) great subsphere of $S^{d-1}$. Moreover, for a linear
subspace $L\subset\mathbb{R}^{d}$ we let $S_{L}:=\\{u\in L:\|u\|=1\\}$.

The lcscH space (with the standard topology) of hyperplanes in
${\mathbb{R}}^{d}$ is denoted by ${\mathbb{H}}$. Every element
$H\in\mathbb{H}$ can by represented as
$H=H(u,r)=\\{x\in{\mathbb{R}}^{d}:\,\langle x,u\rangle=r\\}$ for some $u\in
S^{d-1}$ and $r\in{\mathbb{R}}$. For $H\in{\mathbb{H}}$ and
$z\in{\mathbb{R}}^{d}\setminus H$ we write $H_{z}^{-}$ for the closed
halfspace bounded by $H$ that contains $z$ and we denote $H^{-}:=H_{o}^{-}$
for the closed halfspace bounded by $H$ (not passing through the origin
$o\in{\mathbb{R}}^{d}$) that contains $o$.

In this article all random objects are defined on some fixed probability space
$(\Omega,\mathbf{A},{\mathbb{P}})$ and $\eta$ is a stationary and isotropic
Poisson hyperplane process in ${\mathbb{R}}^{d}$. This is a random element in
the space $\mathbf{N}_{\mathbb{H}}$. As usual we write $\eta$ for the simple
(induced) counting measure and its support. The distribution of $\eta$ is
invariant under rotations and translations and its intensity measure
${\mathbb{E}}\eta$ is of the form ${\mathbb{E}}\eta=\gamma\mu_{d-1}$, where
$\gamma>0$ is the intensity of $\eta$ and $\mu_{d-1}$ is the motion invariant
measure on ${\mathbb{H}}$ given by

$\displaystyle\mu_{d-1}(\cdot)=2\int_{S^{d-1}}\int_{0}^{\infty}\mathbf{1}\\{H(u,r)\in\cdot\\}\,\mathrm{d}r\,\sigma_{d-1}(\mathrm{d}u).$
(2.1)

The space $\mathcal{K}^{d}$ of convex bodies (non-empty, compact, convex
subsets of ${\mathbb{R}}^{d}$) is endowed with the Hausdorff metric. For
$K\in\mathcal{K}^{d}$ we write ${\mathbb{H}}_{K}:=\\{H\in{\mathbb{H}}:\,H\cap
K\neq\emptyset\\}$ and ${\mathbb{H}}^{K}:=\\{H\in{\mathbb{H}}:\,H\cap
K=\emptyset\\}$. The random number $\eta({\mathbb{H}}_{K})$ of hyperplanes
passing through $K$ follows a Poisson distribution with parameter
$\gamma\Phi(K)$ with

$\displaystyle\Phi(K):=2\int_{S^{d-1}}h(K,u)\,\sigma_{d-1}(\mathrm{d}u),$
(2.2)

where $h(K,\cdot)$ is the support function of $K$.

In this article we study point processes related to large cells in Poisson
hyperplane mosaics. Let $\omega\in\mathbf{N}_{\mathbb{H}}$ be a locally finite
system of hyperplanes in $\mathbb{R}^{d}$ in general position. This means that
every $k$-dimensional plane of $\mathbb{R}^{d}$ is contained in at most $d-k$
hyperplanes of $\omega$. The closures of the connected components of the
complement of $\bigcup_{H\in\eta}H$ in ${\mathbb{R}}^{d}$ are called cells of
the mosaic generateed by $\omega$. For a tuple $\bm{H}=(H_{1},\dots,H_{d+1})$
of $d+1$ pairwise distinct hyperplanes in $\omega$ let $B(\bm{H})$ be the
closed inball of the unique simplex $\Delta(\bm{H})$ for which
$H_{1},\dots,H_{d+1}$ are the facet hyperplanes. Moreover, let $z(\bm{H})$ be
the center and $r(\bm{H})$ be the radius of $B(\bm{H})$. Since $\omega$ is in
general position, the inballs of the cells are unique and every inball is
touched by precisely $d+1$ hyperplanes in $\omega$. For every cell this allows
us to find unique (up to permutations) facet hyperplanes
$H_{1},\dots,H_{d+1}\in\omega$. We write

$\displaystyle
C(\bm{H},\omega):=\Delta(\bm{H})\cap\bigcap_{H\in\omega\cap{\mathbb{H}}^{B(\bm{H})}}H_{z(\bm{H})}^{-}.$
(2.3)

It is easy to see that almost surely any $d+1$ hyperplanes of $\eta$ are in
general position. The resulting Poisson hyperplane mosaic is stationary and
isotropic and has intensity

$\displaystyle\gamma^{(d)}:=\frac{(2\gamma)^{d}}{d+1}\int_{\mathsf{P}}\Delta_{d}(\bm{u})\sigma_{d-1}^{d+1}(\mathrm{d}\bm{u})$
(2.4)

(see [16, Section 10.3]), where $\mathsf{P}\subset(S^{d-1})^{d+1}$ is the set
of all $(d+1)$-tuples of unit vectors not lying in a closed hemisphere and
$\Delta_{d}(\bm{u})$ is the $d$-dimensional volume of the convex hull of
$u_{1},\dots,u_{d+1}$.

For $n\geq 1$ and $\omega\in\mathbf{N}_{{\mathbb{H}}}$ we consider the point
process

$\displaystyle\zeta_{n}[\omega]=\frac{1}{(d+1)!}\sum_{\bm{H}\in\omega^{(d+1)}}$
$\displaystyle\mathbf{1}\\{(\omega-\delta_{\bm{H}})\cap{\mathbb{H}}_{B(\bm{H})}=\emptyset\\}\,\delta_{(n^{-1/d}z(\bm{H}),2\gamma
r(\bm{H})-\log n)},$ (2.5)

where $\omega^{(d+1)}$ is the set of all $(d+1)$-tuples of hyperplanes of
$\omega$ with pairwise distinct entries. Hence,
$\zeta_{n}\equiv\zeta_{n}[\eta]$ is a scaling of the process of inball centers
of cells, marked by a transformation of their radius.

Let $\varphi$ be the Borel measure on ${\mathbb{R}}$ given by
$\varphi((c,\infty))=e^{-c}$ for all $c\in\mathbb{R}$. We compare the process
$\zeta_{n}$ with a Poisson process $\nu$ in the lcscH space
$\mathbb{Y}:={\mathbb{R}}^{d}\times\mathbb{R}$ restricted to
$W\times(c,\infty)$ for some compact $W\subset{\mathbb{R}}^{d}$. We consider
the Kantorovich-Rubinstein (KR) distance between the distributions of two
finite point processes $\zeta$ and $\nu$ that is given by

$\displaystyle\mathbf{d_{KR}}(\zeta,\nu):=\sup_{h\in\text{Lip}(\mathbb{Y})}|{\mathbb{E}}h(\zeta)-{\mathbb{E}}h(\nu)|,$

where $\text{Lip}(\mathbb{Y})$ is the class of all measurable 1-Lipschitz
functions $h:\widehat{\mathbf{N}}_{\mathbb{Y}}\to{\mathbb{R}}$ with respect to
the total variation between measures $\omega_{1},\omega_{2}$ on $\mathbb{Y}$
given by

$\displaystyle
d_{\text{TV}}(\omega_{1},\omega_{2}):=\sup|\omega_{1}(A)-\omega_{2}(A)|,$

where the supremum is taken over all Borel sets $A\subset\mathbb{Y}$ with
$\omega_{1}(A),\omega_{2}(A)<\infty$. The KR distance between two point
processes $\xi,\zeta$ dominates their total variation distance

$\displaystyle\mathbf{d_{TV}}(\xi,\zeta):=\sup_{A\in\mathcal{N}_{\mathbb{Y}}}|{\mathbb{P}}(\xi\in
A)-{\mathbb{P}}(\zeta\in A)|,$

where $\mathcal{N}_{\mathbb{Y}}$ is the standard $\sigma$-algebra on the space
of $\sigma$-finite counting measures on $\mathbb{Y}$ (see [4]).

######  Theorem 1.

Let $\zeta_{n}\equiv\zeta_{n}[\eta]$ be as above and let $\nu$ be a Poisson
process on ${\mathbb{R}}^{d}\times\mathbb{R}$ with intensity measure
$\gamma^{(d)}\lambda_{d}\otimes\varphi$. For all $c\in\mathbb{R}$ and all
compact $W\subset{\mathbb{R}}^{d}$ there exists a constant $C>0$ such that

$\displaystyle\mathbf{d_{KR}}(\zeta_{n}\cap(W\times(c,\infty)),\nu\cap(W\times(c,\infty)))\leq
Cn^{-\delta^{*}}(\log n)^{d+1}$

for $n$ large enough, where $\delta^{*}=\delta^{*}(d)\in(0,1/d)$ is the
solution of the fixed point equation

$\displaystyle\delta=\frac{\omega_{d-1}}{\omega_{d}}\int_{\frac{2+\delta}{3+\delta}}^{1}(1-x^{2})^{\frac{d-3}{2}}\,\mathrm{d}x.$
(2.6)

In the second part of this article we study point processes of centers of
cells that are large with respect to more general functions. Let
$\Sigma:\mathcal{K}^{d}\to{\mathbb{R}}$ be continuous (with respect to the
Hausdorff metric), homogeneous of degree $k$ for some $k>0$, not identically
zero and increasing under set inclusion. Such functions are called size
functions in [9]. Additionally, we assume that there is a constant $c_{0}>0$
such that $V_{d}(K)\leq c_{0}\Sigma(K)^{d/k}$ for all $K\in\mathcal{K}^{d}$,
where $V_{d}(K)$ is the volume of $K$. Note that $\Sigma$ and $\Phi$ satisfy
the sharp isoperimetric inequality

$\displaystyle\Phi(K)\geq\tau\Sigma(K)^{1/k},\quad K\in\mathcal{K}^{d},$ (2.7)

for some constant $\tau>0$ (see [8, Section 3]). That this inequality is sharp
means that there exists some $K\in\mathcal{K}^{d}$ such that equality holds in
(2.7). Every such body is called an extremal body (for given $\Sigma$ and
$\Phi$). For example, for $\Sigma=V_{d}$ we have $\tau=2\kappa_{d}^{1/d}$. We
assume that all extremal bodies of $\Sigma$ are Euclidean balls. For instance,
all intrinsic volumes $V_{k}$ ($k\in\\{2,\dots,d\\}$) have this property
($V_{d}$ is the volume and $2V_{d-1}$ is the surface area; see [16, Section
14.2]).

Let $Z$ be the typical cell of the random mosaic generated by $\eta$. It can
be understood as a cell picked uniformly at random from all cells centered in
a large compact region and will be defined rigorously in Section 6. Let $F$
denote the distribution function of $\Sigma(Z)$ and set $G:=1/(1-F)$. Since
$F$ is continuous and strictly increasing on $[\gamma^{-1/k},\infty)$ (see
Lemma 10), $G$ is well-defined and invertible on $[\gamma^{-1/k},\infty)$. By
definition of $G$, we have

$\displaystyle{\mathbb{P}}(G(\Sigma(Z))>u)=u^{-1},\quad
u>G^{-1}(\gamma^{-1/k}),$ (2.8)

i.e. $G(\Sigma(Z))$ is Pareto(1)-distributed.

For $n\geq 1$ and $\omega\in\mathbf{N}_{{\mathbb{H}}}$ we consider the point
process

$\displaystyle\xi_{n}[\omega]:=\frac{1}{(d+1)!}\sum_{\bm{H}\in\eta^{(d+1)}}$
$\displaystyle\mathbf{1}\\{(\eta-\delta_{\bm{H}})\cap{\mathbb{H}}_{B(\bm{H})}=\emptyset\\}\,\delta_{(n^{-1/d}z(\bm{H}),n^{-1}G(\Sigma(C(\bm{H},\eta))))}.$
(2.9)

Let $\xi_{n}\equiv\xi_{n}[\eta]$ and $\psi$ be the Borel measure on
$(0,\infty)$ given by $\psi((a,b))=a^{-1}-b^{-1}$ for $0<a<b<\infty$.

######  Theorem 2.

Let $\xi_{n}\equiv\xi_{n}[\eta]$ be as above and let $\nu$ be a Poisson
process on ${\mathbb{R}}^{d}\times(0,\infty)$ with intensity measure
$\gamma^{(d)}\lambda_{d}\otimes\psi$. There is some $b\in(0,\delta^{*})$ such
that for all $c>0$ and all compact $W\subset{\mathbb{R}}^{d}$ there exists a
constant $C>0$ such that for all $n$ large enough we have

$\displaystyle\mathbf{d_{KR}}(\xi_{n}\cap(W\times(c,\infty)),\nu\cap(W\times(c,\infty)))\leq
Cn^{-b}.$

Here, $\delta^{*}=\delta^{*}(d)$ is the solution of the fixed point equation
(2.6).

In the proof of Theorem 2 we combine probabilistic bounds used in the proof of
Theorem 1 with geometric estimates. Intuitively spoken, we exploit that the
shape of a large typical cell in m is with high probability close to the shape
of a ball. This fact is known as Kendall’s conjecture that was answered in a
series of articles (see e.g. [8], [9]). This brings us in a position where we
can argue similarly to the proof of Theorem 1. The exponent $-b$ on the right-
hand side of the bound in Theorem 2 depends on the considered size function
$\Sigma$ and the stability function $s$ (see Section 6). Our proof method does
not allow us to make $b$ more explicit. We leave this as an open problem.

Outlook. We believe that many of the techniques developed in this article
might be of an independent interest and be useful to establish further
asymptotic results for (Poisson) hyperplane mosaics. One direction could be to
consider score sums defined by Poisson hyperplanes and to investigate whether
a central limit theorem holds for score functions depending on the cell sizes.
This would generalize existing normal approximation results in [5]. Another
line of further research could be to establish Poisson process approximation
for Poisson hyerplane mosaics defined in the hyperbolic space (see [6] and
[12] for recent results on Poisson and normal approximation in this space).

##  3 Some Integral Geometry

###  3.1 Geometry of random hyperplanes

In the proofs of Theorem 1 and Theorem 2 we will need to control the number of
hyperplanes passing through two fixed balls. The following lemma gives the
probability that a random hyperplane with uniformly distributed normal vector
and in distance $r>0$ from some point $z\in{\mathbb{R}}^{d}$ passes through a
fixed ball.

######  Lemma 3.

Let $w,z\in{\mathbb{R}}^{d}$ and $r,s>0$. Then we have

$\displaystyle\int_{S^{d-1}}\mathbf{1}\\{H(u,\langle
z,u\rangle+r)\in\mathbb{H}_{B(w,s)}\\}\,\sigma_{d-1}(\mathrm{d}u)=\frac{\omega_{d-1}}{\omega_{d}}\int_{\max(\frac{r-s}{\|w-z\|},-1)}^{\min(\frac{r+s}{\|w-z\|},1)}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x.$

###### Proof.

We use that the left-hand side only depends on $w$ and $z$ through $w-z$ and
write out the definition of $H(u,r)$. This gives

$\displaystyle\int_{S^{d-1}}\mathbf{1}\\{H(u,\langle
z,u\rangle+r)\in\mathbb{H}_{B(w,s)}\\}\,\sigma_{d-1}(\mathrm{d}u)$
$\displaystyle\quad=\int_{S^{d-1}}\mathbf{1}\\{H(u,r)\in\mathbb{H}_{B(w-z,s)}\\}\,\sigma_{d-1}(\mathrm{d}u)$
$\displaystyle\quad=\int_{S^{d-1}}\mathbf{1}\\{\langle
w-z,u\rangle\in[r-s,r+s]\\}\,\sigma_{d-1}(\mathrm{d}u).$

Choosing $S:=S^{d-1}\cap(\text{lin}\\{w-z\\})^{\perp}$ (where “lin” stands for
the linear hull) in [16, Lemma 6.5.1] and substituting $x=\cos(t)$ in a second
step, we find that the above is given by

$\displaystyle\frac{\omega_{d-1}}{\omega_{d}}\int_{S}\int_{0}^{\pi}\sin^{d-2}(t)\mathbf{1}\\{\cos(t)\|w-z\|\in[r-s,r+s]\\}\,\mathrm{d}t\,\sigma_{d-2}(\mathrm{d}v)$
$\displaystyle\quad=\frac{\omega_{d-1}}{\omega_{d}}\int_{-1}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathbf{1}\\{x\|w-z\|\in[r-s,r+s]\\}\,\mathrm{d}x,$

which implies the assertion. ∎

Fixing $r>0$, $\|w-z\|=r+s$ and letting $s\to\infty$ in Lemma 3 shows that

$\displaystyle
1=\frac{\omega_{d-1}}{\omega_{d}}\int_{-1}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x.$
(3.1)

In the next remark we give a bound on the expected number of hyperplanes
passing through two disjoint balls. Note that in dimension $d=2$, [3, Lemma
4.2] provides an upper bound on this number and that [14] gives the exact
number using an explicit geometric construction that does not seem to work in
higher dimension.

######  Remark 4.

Let $w,z\in{\mathbb{R}}^{d}$ and $r,s>0$ with $r+s\leq\|w-z\|$. Then we find
from the definition of the measure $\mu_{d-1}$ (see (2.1)) and Lemma 3 that

$\displaystyle\mu_{d-1}(\mathbb{H}_{B(z,r)}\cap\mathbb{H}_{B(w,s)})$
$\displaystyle=\frac{2\omega_{d-1}}{\omega_{d}}\int_{0}^{s}\int_{\frac{t-r}{\|w-z\|}}^{\frac{t+r}{\|w-z\|}}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\frac{2s\omega_{d-1}}{\omega_{d}}\int_{\frac{-r}{s+r}}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x.$
(3.2)

For $a\in(0,1)$ let
$L(a):=\frac{\omega_{d-1}}{\omega_{d}}\int_{\frac{-1}{1+a}}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x$
and note that $L(a)<1$ by (3.1). From (3.2) we find for $ar\leq s$ that

$\displaystyle\mu_{d-1}(\mathbb{H}_{B(z,r)}\cap\mathbb{H}_{B(w,s)})\leq
2L(a)s.$ (3.3)

######  Remark 5.

Note that for $d=3$ we instantly find from the equality in Remark 4 that

$\displaystyle\mu_{2}(\mathbb{H}_{B(z,r)}\cap\mathbb{H}_{B(w,s)})=\frac{2rs}{\|w-z\|},\quad
r,s>0,\,r+s\leq\|w-z\|.$

###  3.2 A spherical Blaschke-Petkantschin formula

The following lemma of spherical Blaschke-Petkantschin type is a
generalization of [16, Theorem 7.3.2] to the situation where $\ell\leq d$
hyperplanes are fixed and the integration over the remaining $d+1-\ell$
hyperplanes is carried out. To formulate the statement, we need to introduce
some notation. For $v_{1},\dots,v_{\ell}\in\mathbb{R}^{d}$ let
$\nabla_{\ell}(v_{1},\dots,v_{\ell})$ denote the $\ell$-dimensional volume of
the paralleliped spanned by $v_{1},\dots,v_{\ell}$. Moreover, we define for
$v_{1},\dots,v_{\ell+1}\in\mathbb{R}^{d+1}$ the number

$\displaystyle\Delta_{\ell}(v_{1},\dots,v_{\ell+1}):=\frac{1}{\ell!}\nabla_{\ell}(v_{1}-v_{\ell+1},\dots,v_{\ell}-v_{\ell+1}).$

Hence, $\Delta_{\ell}(v_{1},\dots,v_{\ell})$ is the $\ell$-dimensional volume
of the convex hull of $\\{v_{1},\dots,v_{\ell+1}\\}$. In the following we use
the abbreviations $u_{1:\ell}:=(u_{1},\dots,u_{\ell})$ and

$\displaystyle H(u_{\ell+1:d+1},\bm{\tau}):=(H(u_{\ell+1},\langle
z,u_{\ell+1}\rangle+r),\dots,H(u_{d+1},\langle z,u_{d+1}\rangle+r)),$

where $z$, $\ell$ and $r$ are always clear from the context. Recall the
definition of $\mathsf{P}$ from Section 2. Given $H\in\mathbb{H}$ with (unit)
normal vector $u$, we write $\sigma_{0}$ for the (discrete) uniform
distribution on $S_{H^{\perp}}=\\{-u,u\\}$. The underlying hyperplane $H$ will
always be clear from the context.

######  Lemma 6.

Let $1\leq\ell\leq d$ and $f:{\mathbb{H}}^{d+1}\to[0,\infty)$ be a measurable
function. We have

$\displaystyle\int_{{\mathbb{H}}^{d+1}}f\mathrm{d}\mu_{d-1}^{d+1}$
$\displaystyle=2^{d+1}d!\int_{{\mathbb{H}}^{\ell}}\int_{\bigtimes_{i=1}^{\ell}S_{H_{i}^{\perp}}}\int_{0}^{\infty}\int_{(S^{d-1})^{d+1-\ell}}\int_{\bigcap_{i=1}^{\ell}(H_{i}-ru_{i})}f(H_{1:\ell},H(u_{\ell+1:d+1},\bm{\tau}))$
$\displaystyle\times\frac{\Delta_{d}(u_{1:d+1})}{\nabla_{\ell}(u_{1:\ell})}\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\,\lambda_{d-\ell}(\mathrm{d}z)\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\,\mathrm{d}r\,\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})\,\mu_{d-1}^{\ell}(\mathrm{d}H_{1:\ell}).$

###### Proof.

We use [16, Theorem 7.3.2] (note the missing factor $2^{d+1}$ on the right-
hand side of the statement there) and obtain

$\displaystyle\int_{{\mathbb{H}}^{d+1}}f\,\mathrm{d}\mu_{d-1}^{d+1}=2^{d+1}d!\int_{{\mathbb{R}}^{d}}\int_{0}^{\infty}\int_{\mathsf{P}}f(H(\bm{u},\bm{\tau}))\Delta_{d}(\bm{u})\sigma_{d-1}^{d+1}(\mathrm{d}\bm{u})\,\mathrm{d}r\,\mathrm{d}z.$

We now replace the integration over the inner $\ell$ unit vectors
$u_{1},\dots,u_{\ell}$ by an integration over their orthogonal complements and
obtain for the above

$\displaystyle
2^{d+1}d!\int_{G(d,d-1)^{\ell}}\int_{\bigtimes\limits_{i=1}^{\ell}S_{G_{i}^{\perp}}}\int_{{\mathbb{R}}^{d}}\int_{0}^{\infty}\int_{S^{d+1-\ell}}f(z+G_{1:\ell}+ru_{1:\ell},H(u_{\ell+1:d+1},\bm{\tau}))$
$\displaystyle\qquad\times\,\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\,\mathrm{d}r\,\mathrm{d}z\,\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})\nu_{d-1}^{\ell}(\mathrm{d}G_{1:\ell}),$
(3.4)

where $z+G_{1:\ell}+ru_{1:\ell}:=(z+G_{1}+ru_{1},\dots,z+G_{\ell}+ru_{\ell})$
and $\nu_{q}$ ($q\in\\{0,\dots,d\\}$) is the unique Haar measure on the
Grassmannian $G(d,q)$ of $q$-dimensional linear subspaces, normalized by
$\nu_{q}(G(d,q))=1$ (see [16, Theorem 13.2.11]). For $k\in{\mathbb{N}}$ and
$q\in[k]$ we write
$b_{kq}:=\frac{\omega_{k-q+1}\cdots\omega_{k}}{\omega_{1}\cdots\omega_{q}}$
and $\overline{b}:=b_{d\ell}\left(\frac{b_{\ell 1}}{b_{d1}}\right)^{\ell}$.
Moreover, for $L\in G(d,q)$ we denote by $G(L,d-1)$ the space of all
$(d-1)$-dimensional linear subspaces containing $L$ with invariant measure
$\nu_{d-1}^{L}$ (see [16, Section 13.2]). Similarly, let $A(d,q)$ be the
affine Grassmannian of $q$-dimensional affine subspaces of ${\mathbb{R}}^{d}$
equipped with unique Haar measure $\mu_{q}$, normalized by $\mu_{q}(\\{E\in
A(d,q):\,E\cap B^{d}\neq\emptyset\\})=\kappa_{d-q}$ (see [16, Theorem
13.2.12]). Note that $A(d,d-1)=\mathbb{H}$. For an affine subspace $E\in
A(d,q)$ we write $A(E,d-1)$ for the space of affine subspaces containing $E$
with invariant measure $\mu_{d-1}^{E}$. Using Theorem [16, Theorem 7.2.5] we
find that (3.4) is given by

$\displaystyle
2^{d+1}d!\overline{b}\int_{G(d,d-\ell)}\int_{G(L,d-1)^{\ell}}\int_{\bigtimes\limits_{i=1}^{\ell}S_{G_{i}^{\perp}}}\int_{{\mathbb{R}}^{d}}\int_{0}^{\infty}\int_{S^{d+1-\ell}}f(z+G_{1:\ell}+ru_{1:\ell},H(u_{\ell+1:d+1},\bm{\tau}))$
$\displaystyle\quad\times\nabla_{\ell}(u_{1:\ell})^{d-\ell}\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\mathrm{d}r\mathrm{d}z\,\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})$
$\displaystyle\quad\times(\nu_{d-1}^{L})^{\ell}(\mathrm{d}G_{1:\ell})\nu_{d-\ell}(\mathrm{d}L).$

Now we use that ${\mathbb{R}}^{d}=L\oplus L^{\perp}$, write $\tau_{i}:=\langle
z+z^{\prime},u_{i}\rangle+r$ with $z\in L$ and $z^{\prime}\in L^{\perp}$ for
$i=\ell+1,\dots,d+1$ and find that the above is given by

$\displaystyle
2^{d+1}d!\overline{b}\int_{G(d,d-\ell)}\int_{G(L,d-1)^{\ell}}\int_{\bigtimes\limits\limits_{i=1}^{\ell}S_{G_{i}^{\perp}}}\int_{L^{\perp}}\int_{L}\int_{0}^{\infty}\int_{S^{d+1-\ell}}f(z+G_{1:\ell}+ru_{1:\ell},H(u_{\ell+1:d+1},\bm{\tau}))$
$\displaystyle\quad\times\nabla_{\ell}(u_{1:\ell})^{d-\ell}\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\,\mathrm{d}r\,\lambda_{d-\ell}(\mathrm{d}z)\,\lambda_{\ell}(\mathrm{d}z^{\prime})\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})$
$\displaystyle\quad\times(\nu_{d-1}^{L})^{\ell}(\mathrm{d}G_{1:\ell})\nu_{d-\ell}(\mathrm{d}L).$

From [16, (13.9)] we obtain for the above

$\displaystyle
2^{d+1}d!\overline{b}\int_{A(d,d-\ell)}\int_{A(E,d-1)^{\ell}}\int_{\bigtimes\limits_{i=1}^{\ell}S_{G_{i}^{\perp}}}\int_{E}\int_{0}^{\infty}\int_{S^{d+1-\ell}}f(H_{1:\ell}+ru_{1:\ell})\nabla_{\ell}(u_{1:\ell})^{d-\ell}\Delta_{d}(u_{1:d+1})$
$\displaystyle\quad\times\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\mathrm{d}r\lambda_{d-\ell}(\mathrm{d}z)\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})(\mu_{d-1}^{E})^{\ell}(\mathrm{d}H_{1:\ell})\mu_{d-\ell}(\mathrm{d}E),$

where
$H_{1:\ell}+ru_{1:\ell}:=(H_{1}+ru_{1},\dots,H_{\ell}+ru_{\ell},H(u_{\ell+1:d+1},\bm{\tau}))$.
Using [16, Theorem 7.2.8] and [16, Lemma 14.1.1], the last term is given by

$\displaystyle
2^{d+1}d!\int_{\mathbb{H}^{\ell}}\int_{\bigtimes\limits_{i=1}^{\ell}S_{G_{i}^{\perp}}}\int_{0}^{\infty}\int_{S^{d+1-\ell}}\int_{\bigcap_{i=1}^{\ell}(H_{i}-ru_{i})}f(H_{1},\dots,H_{\ell},H(u_{\ell+1:d+1},\bm{\tau}))$
$\displaystyle\times\frac{\Delta_{d}(u_{1:d+1})}{\nabla_{\ell}(u_{1:\ell})}\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\,\lambda_{d-\ell}(\mathrm{d}z)\sigma_{d-1}^{d+1-\ell}(\mathrm{d}u_{\ell+1:d+1})\,\mathrm{d}r\,\sigma_{0}^{\ell}(\mathrm{d}u_{1:\ell})\,\mu_{d-1}^{\ell}(\mathrm{d}(H_{1},\dots,H_{\ell})).$

∎

##  4 Poisson process approximation

In this section we give an extension of [2, Theorem 6.1] and use its notation.
We begin with a brief repetition of the setup. Let $(\mathbb{X},\mathcal{X})$
and $(\mathbb{Y},\mathcal{Y})$ be locally compact second countable Hausdorff
(lcscH) spaces. Let
$g:\mathbb{X}^{k}\times\mathbf{N}_{\mathbb{X}}\to\\{0,1\\}$,
$f:\mathbb{X}^{k}\times\mathbf{N}_{\mathbb{X}}\to\mathbb{Y}$ be measurable
functions that are symmetric in the $\mathbf{x}$ coordinates and let
$\mathcal{F}$ be the space of closed subsets of $\mathbb{X}$ equipped with the
Fell topology. We assume that
$\mathcal{S}:\mathbb{X}^{k}\times\mathbf{N}_{\mathbb{X}}\to{\mathcal{F}}$ is
measurable and that $f,g$ are localized to $\mathcal{S}$, i.e., for all
$\omega\in\mathbf{N}_{\mathbb{X}}$ and for all
$S\supset\mathcal{S}(\mathbf{x},\omega)$ we have that

$\displaystyle g(\mathbf{x},\omega)=g(\mathbf{x},\omega\cap S),$
$\displaystyle f(\mathbf{x},\omega)=f(\mathbf{x},\omega\cap
S)\quad\text{if}\quad g(\mathbf{x},\omega)=1.$ (4.1)

Moreover, we assume for all $\mathbf{x}\in\mathbb{X}^{k}$ that
$\mathcal{S}(\mathbf{x},\cdot):\mathbf{N}_{\mathbb{X}}\to\mathcal{F}$ is a
stopping set. A mapping $\mathcal{S}:\mathbf{N}_{\mathbb{H}}\to\mathcal{F}$ is
called stopping set if

$\displaystyle\\{\omega\in\mathbf{N}_{\mathbb{H}}:\mathcal{S}(\omega)\subset
S\\}=\\{\omega\in\mathbf{N}_{\mathbb{H}}:\mathcal{S}(\omega\cap S)\subset
S\\}$ (4.2)

for all compact $S\subset\mathbb{H}$.

Define

$\displaystyle\xi[\omega]:=\frac{1}{k!}\sum_{\mathbf{x}\in\omega^{(k)}}g(\mathbf{x},\omega)\delta_{f(\mathbf{x},\omega)},\quad\omega\in\mathbf{N}_{\mathbb{X}},$

and let $\xi\equiv\xi[\eta]$, where $\eta$ is a Poisson process on
$\mathbb{X}$ with $\sigma$-finite intensity measure $\mathbf{K}$. Then we
obtain from the multivariate Mecke equation (see [10, Theorem 4.4]) that the
intensity measure $\mathbf{L}$ of $\xi$ is given by

$\displaystyle\mathbf{L}(\cdot)=\frac{1}{k!}\int_{\mathbb{X}^{k}}\mathbb{E}\mathbf{1}\\{f(\mathbf{x},\eta+\delta_{\mathbf{x}})\in\cdot\\}g(\mathbf{x},\eta+\delta_{\mathbf{x}})\,\mathbf{K}^{k}(\mathrm{d}\mathbf{x}).$

######  Theorem 7.

Let $\xi$ be the process defined above with $f,g$ satisfying (4.1) and
$\mathbf{L}(\mathbb{Y})<\infty$. Let $\zeta$ be a Poisson process with finite
intensity measure $\mathbf{M}$. Further, suppose that we are given a
measurable mapping $\mathbf{x}\mapsto S_{\mathbf{x}}$ from $\mathbb{X}^{k}$ to
$\mathcal{F}$ satisfying $\mathbf{x}\subset S_{\mathbf{x}}$. For
$\omega\in\mathbf{N}_{\mathbb{X}}$ let

$\displaystyle\tilde{g}(\mathbf{x},\omega):=g(\mathbf{x},\omega)\mathbf{1}\\{\mathcal{S}(\mathbf{x},\omega)\subset
S_{\mathbf{x}}\\}.$

Then

$\displaystyle d_{\mathbf{KR}}(\xi,\zeta)\leq
d_{\text{TV}}(\mathbf{L},\mathbf{M})+E_{1}+E_{2}+E_{3}+E_{4}+E_{5}+E_{6}$

with

$\displaystyle
E_{1}=\frac{2}{k!}\int_{\mathbb{X}^{k}}{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})\mathbf{1}\\{\mathcal{S}(\mathbf{x},\eta+\delta_{\mathbf{x}})\not\subset
S_{\mathbf{x}}\\}\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$ $\displaystyle
E_{2}=\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}){\mathbb{E}}\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\quad\quad\times\,\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$
$\displaystyle
E_{3}=\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}})$
$\displaystyle\quad\quad\quad\times\,{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$
$\displaystyle
E_{4}=\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\times\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$
$\displaystyle
E_{5}=\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\times\,{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$
$\displaystyle E_{6}=\frac{2}{k!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,k\\}}\frac{1}{(k-|I|)!}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k-|I|}}{\mathbb{E}}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\times\,\mathbf{K}^{k-|I|}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$

where for $I=\\{i_{1},\dots,i_{m}\\}$ we set
$\mathbf{x}_{I}=(x_{i_{1}},\dots,x_{i_{m}})$ and
$(\mathbf{x}_{I},\mathbf{z})=(x_{i_{1}},\dots,x_{i_{m}},z_{1:k-m})$.

######  Remark 8.

(a) By interchanging the roles of $\mathbf{x}$ and $\mathbf{z}$ in the term
$E_{3}$ in Theorem 8, one sees that $E_{3}\leq E_{2}$.
(b) Note that

$\displaystyle E_{2}+E_{3}$
$\displaystyle\leq\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}\mathbf{1}\\{S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}{\mathbb{E}}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}){\mathbb{E}}\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})=:E_{2}^{\prime},$
$\displaystyle E_{4}+E_{5}$
$\displaystyle\leq\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}\mathbf{1}\\{S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}{\mathbb{E}}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})$
$\displaystyle=:E_{3}^{\prime}.$

This shows that Theorem 7 is a refinement of [2, Theorem 4.1], where
$d_{\mathbf{KR}}(\xi,\zeta)$ is bounded by
$d_{TV}(\mathbf{L},\mathbf{M})+E_{1}+E_{2}^{\prime}+E_{3}^{\prime}+E_{6}$.

###### Proof of Theorem 7.

We proceed along the same lines as in the proof of [2, Theorem 4.1]. First
assume that for all $\mathbf{x}\in\mathbb{X}^{k}$ and
$\omega\in\mathbf{N}_{\mathbb{X}}$ we have that
$\mathcal{S}(\mathbf{x},\omega)\subset S_{\mathbf{x}}$. Then we have that
$\tilde{g}=g$, $E_{1}=0$ and

$\displaystyle g(\mathbf{x},\omega)=g(\mathbf{x},\omega\cap
S_{\mathbf{x}}),\quad f(\mathbf{x},\omega)=f(\mathbf{x},\omega\cap
S_{\mathbf{x}})\quad\text{if}\quad g(\mathbf{x},\omega)=1.$

Let $\eta^{\prime}$ be a Poisson process on $\mathbb{X}$ that is independent
of $\eta$ and has intensity measure $\mathbf{K}$. For fixed
$h\in\text{Lip}(\mathbb{Y})$ we need to bound the difference
${\mathbb{E}}h(\zeta)-{\mathbb{E}}h(\xi)$ which is by [2, (4.5)] given by

$\displaystyle{\mathbb{E}}h(\zeta)-{\mathbb{E}}h(\xi)=\frac{1}{k!}$
$\displaystyle\int_{0}^{\infty}\int_{\mathbb{X}^{k}}\left({\mathbb{E}}g(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})}[P_{s}h(\xi[\eta])]\right.$
$\displaystyle-\left.{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})}[P_{s}h(\xi[\eta+\delta_{\mathbf{x}}]-\delta_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})})]\right)\mathbf{K}^{k}(\mathrm{d}\mathbf{x})\mathrm{d}s,$
(4.3)

where $D_{x}h(\omega):=h(\omega+\delta_{x})-h(\omega)$ and $P_{s}$ is the
Markov semigroup corresponding to the generator $\mathcal{L}$ that is given by

$\displaystyle\mathcal{L}h(\omega):=\int_{\mathbb{X}}D_{x}h(\omega)\mathbf{M}(\mathrm{d}x)-\int_{\mathbb{X}}D_{x}h(\omega-\delta_{x})\omega(\mathrm{d}x).$

For $\mathbf{x}\in\mathbb{X}^{k}$ and $\omega\in\mathbf{N}_{\mathbb{X}}$ we
define

$\displaystyle\xi_{\mathbf{x}}[\omega]:$
$\displaystyle=\frac{1}{k!}\sum_{\mathbf{z}\in\omega^{(k)}}\mathbf{1}\\{(\omega+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap
S_{\mathbf{x}}=\emptyset\\}g(\mathbf{z},\omega)\delta_{f(\mathbf{z},\omega)}$
$\displaystyle=\frac{1}{k!}\sum_{\mathbf{z}\in\omega^{(k)}}\mathbf{1}\\{(\omega+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}=\emptyset\\}g(\mathbf{z},\omega\cap
S_{\mathbf{z}})\delta_{f(\mathbf{z},\omega\cap S_{\mathbf{z}})}$
$\displaystyle=\frac{1}{k!}\sum_{\mathbf{z}\in\omega^{(k)}}\mathbf{1}\\{(\omega+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}=\emptyset\\}g(\mathbf{z},\omega\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}+\omega\cap S_{\mathbf{z}}\cap
S_{\mathbf{x}}^{c})$
$\displaystyle\qquad\qquad\qquad\times\delta_{f(\mathbf{z},\omega\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}+\omega\cap S_{\mathbf{z}}\cap
S_{\mathbf{x}}^{c})}$
$\displaystyle=\frac{1}{k!}\sum_{\mathbf{z}\in\omega^{(k)}}\mathbf{1}\\{(\omega+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}=\emptyset\\}g(\mathbf{z},\omega\cap
S_{\mathbf{x}}^{c})\delta_{f(\mathbf{z},\omega\cap S_{\mathbf{x}}^{c})}.$

It follows from [2, (2.9)] and the multivariate Mecke equation (see [10,
Theorem 4.4] that

$\displaystyle|{\mathbb{E}}g(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})}[P_{s}h(\xi[\eta])]-{\mathbb{E}}g(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})}[P_{s}h(\xi_{\mathbf{x}}[\eta])]|$
$\displaystyle\quad\leq\frac{2}{k!}e^{-s}{\mathbb{E}}g(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})\sum_{\mathbf{z}\in\eta^{(k)}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}\neq\emptyset\\}g(\mathbf{z},\eta)$
$\displaystyle\quad\leq\frac{2}{k!}e^{-s}\int_{\mathbb{X}^{k}}{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}}){\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{z}}\cap
S_{\mathbf{x}}\neq\emptyset\\}g(\mathbf{z},\eta+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z}).$
(4.4)

By assumption (4.1), $f(\mathbf{x},\eta+\delta_{\mathbf{x}})$ and
$g(\mathbf{x},\eta+\delta_{\mathbf{x}})$ depend only on $\eta\cap
S_{\mathbf{x}}$. Since $\eta\stackrel{{\scriptstyle d}}{{=}}\eta^{\prime}$, we
obtain from the independence property of the Poisson process that

$\displaystyle{\mathbb{E}}g(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta^{\prime}+\delta_{\mathbf{x}})}[P_{s}h(\xi_{\mathbf{x}}[\eta])]={\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})}[P_{s}h(\xi_{\mathbf{x}}[\eta\cap
S_{\mathbf{x}}^{c}+\eta^{\prime}\cap S_{\mathbf{x}}])].$ (4.5)

Note that

$\displaystyle\xi_{\mathbf{x}}[\eta\cap S_{\mathbf{x}}^{c}+\eta^{\prime}\cap
S_{\mathbf{x}}]=\frac{1}{k!}\sum_{\mathbf{z}\in(\eta\cap
S_{\mathbf{x}}^{c})^{(k)}}\mathbf{1}\\{(\eta^{\prime}+\delta_{\mathbf{x}})\cap
S_{\mathbf{z}}\cap S_{\mathbf{x}}=\emptyset\\}g(\mathbf{z},\eta\cap
S_{\mathbf{x}}^{c})\delta_{f(\mathbf{z},\eta\cap S_{\mathbf{x}}^{c})}.$

Hence, we obtain that

$\displaystyle\mathbf{d_{TV}}\Big{(}\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}g(\mathbf{z},\eta+\delta_{\mathbf{x}})\delta_{f(\mathbf{z},\eta+\delta_{\mathbf{x}})},\xi_{\mathbf{x}}[\eta\cap
S_{\mathbf{x}}^{c}+\eta^{\prime}\cap S_{\mathbf{x}}]\Big{)}$
$\displaystyle\quad\leq\frac{1}{k!}\sum_{\mathbf{z}\in(\eta\cap
S_{\mathbf{x}}^{c})^{(k)}}\mathbf{1}\\{(\eta^{\prime}+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap S_{\mathbf{z}}=\emptyset,\,\eta\cap S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta\cap S_{\mathbf{x}}^{c})$
$\displaystyle\quad\quad+\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap S_{\mathbf{z}}=\emptyset,\,\eta^{\prime}\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta+\delta_{\mathbf{x}})$
$\displaystyle\quad\quad+\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta+\delta_{\mathbf{x}}).$ (4.6)

As in [2], we define the point process

$\displaystyle\hat{\xi}_{\mathbf{x}}:=\xi[\eta+\delta_{\mathbf{x}}]-g(\mathbf{x},\eta+\delta_{\mathbf{x}})\delta_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})}-\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}g(\mathbf{z},\eta+\delta_{\mathbf{x}})\delta_{f(\mathbf{z},\eta+\delta_{\mathbf{x}})}.$

From [2, (2.9)], (4.6) and the Mecke equation we obtain that

$\displaystyle|{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})}[P_{s}h(\xi[\eta+\delta_{\mathbf{x}}]-\delta_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})})]$
$\displaystyle\quad\quad-{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})D_{f(\mathbf{x},\eta+\delta_{\mathbf{x}})}[P_{s}h(\xi_{\mathbf{x}}[\eta\cap
S_{\mathbf{x}}^{c}+\eta^{\prime}\cap S_{\mathbf{x}}])]|$ $\displaystyle\leq
2e^{-s}{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})\Big{(}\frac{1}{k!}\sum_{\mathbf{z}\in(\eta\cap
S_{\mathbf{x}}^{c})^{(k)}}\mathbf{1}\\{(\eta^{\prime}+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap S_{\mathbf{z}}=\emptyset,\,\eta\cap S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta\cap S_{\mathbf{x}}^{c})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap S_{\mathbf{z}}=\emptyset,\,\eta^{\prime}\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta+\delta_{x})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{k!}\sum_{\mathbf{z}\in\eta^{(k)}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{z},\eta+\delta_{x})+\hat{\xi}_{\mathbf{x}}(\mathbb{Y})\Big{)}.$

Using that $\eta\cap S_{\mathbf{x}}$ and $\eta\cap S_{\mathbf{x}}^{c}$ are
independent point processes, we obtain from the Mecke equation that the above
is bounded by

$\displaystyle 2e^{-s}$
$\displaystyle\left(\frac{1}{k!}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset,\,(\eta^{\prime}+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}g(\mathbf{x},\eta+\delta_{\mathbf{x}})\right.$
$\displaystyle\quad\quad\quad\times\,g(\mathbf{z},\eta^{\prime}+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})$
$\displaystyle+\frac{1}{k!}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{\eta^{\prime}\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset,\,(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}g(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\times\,g(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})$
$\displaystyle+\frac{1}{k!}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}g(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})g(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\,\mathbf{K}^{k}(\mathrm{d}\mathbf{z})$
$\displaystyle+\left.{\mathbb{E}}g(\mathbf{x},\eta+\delta_{\mathbf{x}})\hat{\xi}_{\mathbf{x}}(\mathbb{Y})\right).$
(4.7)

Now we substitute (4.4) and (4.7) into (4.3) and obtain with the triangle
inequality

$\displaystyle|{\mathbb{E}}h(\zeta)-{\mathbb{E}}h(\xi)|$
$\displaystyle\,\leq\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}){\mathbb{E}}\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})$
$\displaystyle\,+\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\,{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{z}})\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})$
$\displaystyle\,+\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\,\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})$
$\displaystyle\,+\frac{2}{(k!)^{2}}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k}}{\mathbb{E}}\mathbf{1}\\{(\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}=\emptyset\\}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\,{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\mathbf{x}}\cap
S_{\mathbf{z}}\neq\emptyset\\}\mathbf{K}^{k}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x})$
$\displaystyle\,+\frac{2}{k!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,k\\}}\frac{1}{(k-|I|)!}\int_{\mathbb{X}^{k}}\int_{\mathbb{X}^{k-|I|}}{\mathbb{E}}\tilde{g}(\mathbf{x},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})\tilde{g}(\mathbf{z},\eta+\delta_{\mathbf{x}}+\delta_{\mathbf{z}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\,\mathbf{K}^{k-|I|}(\mathrm{d}\mathbf{z})\mathbf{K}^{k}(\mathrm{d}\mathbf{x}),$

where the terms on the right-hand side are $E_{2}$, $E_{3},E_{4},E_{5}$ and
$E_{6}$. The rest of the proof goes along the lines of Step 2 in the proof of
[2, Theorem 4.1]. ∎

##  5 Proof of Theorem 1

To prepare the proof of Theorem 1, we determine the intensity measure
${\mathbb{E}}\zeta_{n}$ of $\zeta_{n}$. Let $A\in\mathcal{B}^{d}$ and $y>0$.
From [16, Theorem 4.1.1] we find that

$\displaystyle{\mathbb{E}}\zeta_{n}(A\times(y,\infty))$
$\displaystyle=n\gamma^{(d)}\lambda_{d}(A)\mathbb{Q}_{0}(\\{K\in\mathcal{K}^{d}:\,2\gamma
r(K)>y+\log n\\}),$ (5.1)

where $\mathbb{Q}_{0}$ is the distribution of the typical cell of a Poisson
hyperplane distribution generated from a stationary, isotropic Poisson
hyperplane process of intensity $\gamma$. From [16, Theorem 10.4.6] and using
(2.4) we find that for all $R>0$,

$\displaystyle\mathbb{Q}_{0}(\\{K\in\mathcal{K}^{d}:\,r(K)>R\\})=\mathrm{e}^{-2\gamma
R}.$ (5.2)

This gives for $n$ large enough

$\displaystyle{\mathbb{E}}\zeta_{n}(A\times(y,\infty))$
$\displaystyle=\gamma^{(d)}\lambda_{d}(A)\mathrm{e}^{-y}.$ (5.3)

The following lemma gives upper bounds for the expected number of pairs of
cells with centers in some compact set $W\subset\mathbb{R}^{d}$ in the
following two scenarios. In the first one the inradius of both cells is larger
than some $R>0$. In the second scenario the distance of their centers is
larger than some $D>0$ and both inradii are in the interval $(aR,R]$ for some
$a\in(0,1)$.

######  Lemma 9.

Let $\gamma>0$, $W\subset\mathbb{R}^{d}$ be compact and
$I\subsetneq\\{1,\dots,d+1\\}$.

  1. $(a)$

For all $R>0$ we have

$\displaystyle\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in
W\\}\,\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{H}_{I},\bm{G})\\}>R\\}$
$\displaystyle\qquad\times\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
$\displaystyle\quad\leq\begin{cases}c_{0}\lambda_{d}(W)^{2}R\mathrm{e}^{-2\gamma
R},&\quad I=\emptyset,\\\
c_{1}\lambda_{d}(W)(\mathrm{diam}(W)+R)^{d-1}\mathrm{e}^{-2\gamma R},&\quad
1\leq|I|\leq d.\end{cases}$

  2. $(b)$

Let $D>0$, $R>0$, $a\in(0,1)$ and let
$L(a)=\frac{\omega_{d-1}}{\omega_{d}}\int_{\frac{-1}{1+a}}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x$
as in Remark 4. We have

$\displaystyle\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\leq
D\\}\ $
$\displaystyle\qquad\times\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})}\mathbf{1}\\{r(\bm{H})\in(aR,R],r(\bm{H}_{I},\bm{G})\in(aR,R]\\}$
$\displaystyle\qquad\times\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
$\displaystyle\quad\leq\begin{cases}c_{2}\frac{D^{d}\lambda_{d}(W)}{1-L(a)}\mathrm{e}^{-2\gamma
aR(2-L(a))},&\quad I=\emptyset,\\\
c_{3}\frac{(D+R)^{d-1}\lambda_{d}(W)}{1-L(a)}\mathrm{e}^{-2\gamma
aR(2-L(a))},&\quad 1\leq|I|\leq d.\end{cases}$

Here, $\bm{H}_{I}=(H_{i_{1}},\dots,H_{i_{m}})$ for $I=\\{i_{1},\dots,i_{m}\\}$
and the constants $c_{0},c_{1},c_{2},c_{3}>0$ depend on the dimension $d$ and
on $\gamma$.

###### Proof.

(a) By symmetry we can assume (at the cost of a factor 2) that $r(\bm{H})\geq
r(\bm{H}_{I},\bm{G})$. Using the estimate
$\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})\geq\mu_{d-1}({\mathbb{H}}_{B(\bm{H})})=2r(\bm{H})$
gives that the left-hand side of the statement is bounded by

$\displaystyle
2\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in
W\\}\,\mathbf{1}\\{r(\bm{H})>R\\}\mathbf{1}\\{r(\bm{H}_{I},\bm{G})\leq
r(\bm{H})\\}$ $\displaystyle\quad\times\,\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
(5.4)

Now we consider the three cases $I=\emptyset$, $|I|=1$ and $2\leq|I|\leq d$
separately. If $|I|=0$ we obtain from [16, Theorem 7.3.2] and (2.4) that (5.4)
is given by

$\displaystyle\frac{4((d+1)!)^{2}(\gamma^{(d)})^{2}}{\gamma^{2d+1}}\lambda_{d}(W)^{2}\int_{R}^{\infty}r\mathrm{e}^{-2\gamma
r}\mathrm{d}r=\frac{2((d+1)!)^{2}(\gamma^{(d)})^{2}(2\gamma
R+1)}{\gamma^{2d+2}}\lambda_{d}(W)^{2}\mathrm{e}^{-2\gamma R},$

where we recall that $\gamma^{(d)}$ is the expected number of cells centered
in a Borel set with Lebesgue measure one in the mosaic generated by a Poisson
hyperplane process with intensity $\gamma$.

Now let $|I|=1$ and assume that $I=\\{1\\}$. Then we get from Lemma 6 (note
that $\nabla_{1}(u)=1$ for all $u\in S^{d-1}$) that (5.4) is given by

$\displaystyle 2^{d+2}d!$
$\displaystyle\int_{\mathbb{H}^{d+1}}\int_{S_{H_{1}^{\perp}}}\int_{0}^{r(\bm{H})}\int_{(S^{d-1})^{d}}\int_{(H_{1}-su_{1})\cap
W}\mathbf{1}\\{z(\bm{H})\in
W\\}\mathbf{1}\left\\{r(\bm{H})>R\right\\}\mathrm{e}^{-2\gamma r(\bm{H})}$
$\displaystyle\quad\times\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\lambda_{d-1}(\mathrm{d}w)\,\sigma_{d-1}^{d}(\mathrm{d}u_{2:d+1})\,\mathrm{d}s\,\sigma_{0}(\mathrm{d}u_{1})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.5)

Note that since the measure $\sigma_{d-1}$ is isotropic and $\Delta_{d}$ is
rotation-invariant, we have for all $u\in S^{d-1}$

$\displaystyle\int_{(S^{d-1})^{d}}\Delta_{d}(u,u_{2:d+1})\mathbf{1}_{\mathsf{P}}(u,u_{2:d+1})\,\sigma_{d-1}^{d}(\mathrm{d}u_{2:d+1})$
$\displaystyle=\int_{\mathsf{P}}\Delta_{d}(u_{1:d+1})\,\sigma_{d-1}^{d+1}(\mathrm{d}u_{1:d+1})$
$\displaystyle=\frac{(d+1)1^{(d)}}{2^{d}},$ (5.6)

where the second equality holds by (2.4). Since $\lambda((H_{1}-su_{1})\cap
W)\leq\mathrm{diam}(W)^{d-1}$ for all $H_{1}\in\mathbb{H}$, $s>0$ and
$u_{1}\in S^{d-1}$, we find that (5.5) is bounded by

$\displaystyle
4(d+1)!1^{(d)}\mathrm{diam}(W)^{d-1}\int_{\mathbb{H}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W\\}\mathbf{1}\left\\{r(\bm{H})>R\right\\}r(\bm{H})\mathrm{e}^{-2\gamma
r(\bm{H})}\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$

and the claim now follows after an application of [16, Theorem 7.3.2].

If $2\leq|I|\leq d$ we use that the triangle inequality gives

$\displaystyle d(G_{i},z(\bm{H}))\leq
r(\bm{H}_{I},\bm{G})+\|z(\bm{H}_{I},\bm{G})-z(\bm{H})\|,\quad
i\in\\{1,\dots,d+1\\}\setminus I.$ (5.7)

This yields for (5.4) the bound

$\displaystyle
2\int_{{\mathbb{H}}^{d+1}}\mu_{d-1}(\mathbb{H}_{B(o,r(\bm{H})+\text{diam}(W))})^{d+1-|I|}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\,\mathbf{1}\\{r(\bm{H})>R\\}\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Since $\mu_{d-1}(\mathbb{H}_{B(z,r)})=2r$ for all $z\in\mathbb{R}^{d}$ and all
$r>0$, the above is given by

$\displaystyle
2^{d+2-|I|}\int_{{\mathbb{H}}^{d+1}}(r(\bm{H})+\text{diam}(W))^{d+1-|I|}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{r(\bm{H})>R\\}\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$

and the claim again follows after an application of [16, Theorem 7.3.2].

(b) For $r(\bm{H})\leq R$ and $aR<r(\bm{H}_{I},\bm{G})$ we have
$ar(\bm{H})\leq r(\bm{H}_{I},\bm{G})$. Hence, we obtain from Remark 4 that
$\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})\leq
2L(a)r(\bm{H}_{I},\bm{G})$. Consequently, by the inclsuion-exclusion
principle,

$\displaystyle\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})\geq
2r(\bm{H})+2(1-L(a))r(\bm{H}_{I},\bm{G}).$

Therefore, the left-hand side in the statement of part (b) of the lemma is
bounded by

$\displaystyle\int_{{\mathbb{H}}^{d+1}}$
$\displaystyle\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\leq D\\}\
\mathrm{e}^{-2\gamma r(\bm{H})-2\gamma(1-L(a))r(\bm{H}_{I},\bm{G})}$
$\displaystyle\,\times\,\mathbf{1}\left\\{r(\bm{H})\in(aR,R],r(\bm{H}_{I},\bm{G})\in(aR,R]\right\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.8)

If $I=\emptyset$ we apply [16, Theorem 7.3.2] twice and find that (5.8) is
bounded by

$\displaystyle\frac{\kappa_{d}((d+1)!)^{2}(\gamma^{(d)})^{2}}{\gamma^{2d+2}(1-L(a))}D^{d}\lambda_{d}(W)\mathrm{e}^{2\gamma
aR(2-L(a))}.$

For $|I|=1$ we obtain from Lemma 6 and (5.6) for (5.8) the bound

$\displaystyle\frac{2(d+1)!1^{(d)}}{\gamma(1-L(a))}D^{d-1}\int_{\mathbb{H}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W\\}\mathbf{1}\left\\{r(\bm{H})>aR\right\\}r(\bm{H})\mathrm{e}^{-2\gamma
r(\bm{H})}\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

From here the assertion follows by [16, Theorem 7.3.2]

If $2\leq|I|\leq d$ we find using (5.7) that (5.8) is bounded by

$\displaystyle\mathrm{e}^{-2\gamma aR(1-L(a))}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}\mu_{d-1}(\mathbb{H}_{B(z(\bm{H}),R+D)})^{d+1-|I|}\mathbf{1}\\{z(\bm{H})\in
W\\}\mathbf{1}\left\\{r(\bm{H})>aR\right\\}\,\mathrm{e}^{-2\gamma r(\bm{H})}$
$\displaystyle\times\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

The claim now follows from the fact that $\mu_{d-1}(\mathbb{H}_{B(z,r)})=2r$
for all $z\in\mathbb{R}^{d}$ and all $r>0$ and from [16, Theorem 7.3.2]. ∎

###### Proof of Theorem 1.

Let $c\in\mathbb{R}$, $W\subset{\mathbb{R}}^{d}$ be compact and let $\nu$ be a
Poisson process on ${\mathbb{R}}^{d}\times\mathbb{R}$ with intensity measure
$\gamma^{(d)}\lambda_{d}\otimes\varphi$, where $\varphi$ is given by
$\varphi((y,\infty))=e^{-y}$ for all $y\in\mathbb{R}$. Since by (5.3) the
intensity measures of $\zeta_{n}\cap(W\times(c,\infty))$ and
$\nu\cap(W\times(c,\infty))$ coincide, their total variation is zero for $n$
large enoguh. We apply Theorem 7 with

$\displaystyle g(\bm{H},\omega)=\mathbf{1}\\{n^{-1/d}z(\bm{H})\in
W\\}\mathbf{1}\\{(\omega-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset\\}\mathbf{1}\\{2\gamma
r(\bm{H})-\log n>c\\},$ $\displaystyle
f(\bm{H},\omega)=(n^{-1/d}z(\bm{H}),2\gamma r(\bm{H})-\log
n),\quad\omega\in\mathbf{N}_{\mathbb{H}},\,\bm{H}\in\omega^{(d+1)},$

and the (deterministic) stopping set $\mathcal{S}$ given by
$\mathcal{S}(\bm{H},\omega)=\mathbb{H}_{B(\bm{H})}$ for
$\bm{H}\in\mathbb{H}^{d+1}$ in general position and all
$\omega\in\mathbf{N}_{\mathbb{H}}$. Letting
$S_{\bm{H}}=\mathbb{H}_{B(\bm{H})}$ gives $g=\tilde{g}$ and, therefore,
$E_{1}=0$. Moreover, since $g(\bm{G},\omega+\delta_{\bm{G}})=1$ implies that
$\omega\cap\mathbb{H}_{B(\bm{G})}=\emptyset$, we find that $E_{3}=E_{4}=0$.
This yields the bound

$\displaystyle\mathbf{d_{KR}}(\zeta_{n}\cap(W\times(c,\infty)),\nu\cap(W\times(c,\infty)))\leq
E_{2}+E_{5}+E_{6}$ (5.9)

with

$\displaystyle E_{2}$
$\displaystyle=\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{\delta_{(\bm{H},\bm{G})}\cap{\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})}\neq\emptyset\\}\mathbf{1}\\{z(\bm{H})\in
W_{n},\,z(\bm{G})\in W_{n}\\}$
$\displaystyle\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-2\gamma r(\bm{H})}\,\mathrm{e}^{-2\gamma
r(\bm{G})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
$\displaystyle E_{5}$
$\displaystyle=\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}{\mathbb{P}}(\eta\cap{\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})}\neq\emptyset)\mathbf{1}\\{z(\bm{H})\in
W_{n},\,z(\bm{G})\in W_{n}\\}$
$\displaystyle\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})}$
$\displaystyle\times\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
$\displaystyle E_{6}$
$\displaystyle=\frac{2\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{|I|}}{(d+1-|I|)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W_{n},\,z(\bm{H}_{I},\bm{G})\in W_{n}\\}$
$\displaystyle\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{H}_{I},\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})}$
$\displaystyle\times\mathbf{1}\\{r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$

where $W_{n}:=n^{1/d}W=\\{x\in{\mathbb{R}}^{d}:\,x\in n^{1/d}W\\}$
$\bm{H}_{I}=(H_{i_{1}},\dots,H_{i_{m}})$ for $I=\\{i_{1},\dots,i_{m}\\}$.
Moreover, we have used that
$\delta_{(\bm{H}_{|I|+1:d+1},\bm{G})}\cap{\mathbb{H}}_{B(\bm{H})\cup
B(\bm{H}_{I},\bm{G})}=\emptyset$ implies that
$r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|$ for
$I\subsetneq\\{1,\dots,d+1\\}$.

Next we consider the terms $E_{2},E_{5},E_{6}$ separately. In the following,
$c_{i}>0$ $(i\in\mathbb{N})$ are positive constants. Their exact values are
not important for the argument.

The estimate of $E_{2}$. Note that

$\mathbf{1}\\{\delta_{(\bm{H},\bm{G})}\cap{\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})}\neq\emptyset\\}\leq\sum_{i=1}^{d+1}\mathbf{1}\\{H_{i}\in\mathbb{H}_{B(\bm{G})}\\}+\sum_{i=1}^{d+1}\mathbf{1}\\{G_{i}\in\mathbb{H}_{B(\bm{H})}\\}.$

From symmetry in $\bm{H}$ and $\bm{G}$ we find that $E_{2}$ is bounded by

$\displaystyle\frac{4\gamma^{2d+2}}{(d+1)!d!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{G_{1}\in{\mathbb{H}}_{B(\bm{H})}\\}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\mathbf{1}\\{z(\bm{G})\in
W_{n}\\}\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log t}{2\gamma}\right\\}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\mathrm{e}^{-2\gamma r(\bm{H})}\,\mathrm{e}^{-2\gamma
r(\bm{G})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$

where $\bm{G}=(G_{1},\dots,G_{d+1})$. Next we apply Lemma 6 with $\ell=1$ to
the inner integral (note that $\nabla_{1}(v)=1$ for all $v\in S^{d-1}$) and
obtain

$\displaystyle\frac{2^{d+3}\gamma^{2d+2}}{(d+1)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}_{B(\bm{H})}}\int_{G_{1}^{\perp}}\int_{(S^{d-1})^{d}}\int_{\frac{c+\log
n}{2\gamma}}^{\infty}\int_{W_{n}\cap(G_{1}-ru_{1})}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}$ $\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-2\gamma r(\bm{H})}\mathrm{e}^{-2\gamma
s}\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})$
$\displaystyle\quad\times\,\lambda_{d-1}(\mathrm{d}w)\,\mathrm{d}s\,\sigma_{d-1}^{d}(\mathrm{d}u_{2:d+1})\,\sigma_{1}(\mathrm{d}u_{1})\,\mu_{d-1}(\mathrm{d}G_{1})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Since $\lambda_{d-1}(W_{n}\cap(G_{1}-rv_{1}))\leq
n^{(d-1)/d}\text{diam}(W)^{d-1}$, (2.4) and
$\mu_{d-1}(\mathbb{H}_{B(\bm{H})})=2r(\bm{H})$ (see (2.1)), the above is
bounded by

$\displaystyle\frac{4\gamma^{d+1}\gamma^{(d)}\mathrm{diam}(W)^{d-1}\mathrm{e}^{-c}}{n^{1/d}d!}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}r(\bm{H})\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathrm{e}^{-2\gamma r(\bm{H})}$
$\displaystyle\quad\times\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

From [16, Theorem 7.3.2] we conclude that $E_{2}$ is bounded by
$c_{1}n^{-1/d}\log n.$

The estimate of $E_{5}$. Given $\delta>0$ (to be specified later), we split
$E_{5}$ into the sum

$\displaystyle\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}$
$\displaystyle\quad\times\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{G})\\}>\frac{(2+\delta)(c+\log
n)}{2\gamma}\text{ or }\|z(\bm{H})-z(\bm{G})\|\leq(\log n)^{(d+1)/d}\\}\,$
$\displaystyle\quad\times\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})}\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
(5.10)
$\displaystyle+\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}{\mathbb{P}}(\eta\cap{\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})}\neq\emptyset)\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}$
$\displaystyle\quad\times\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{G})\\}\leq\frac{(2+\delta)(c+\log
n)}{2\gamma}\\}\,\mathbf{1}\\{\|z(\bm{H})-z(\bm{G})\|>(\log n)^{(d+1)/d}\\}$
$\displaystyle\quad\times\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})}\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.11)

Note that (5.10) is bounded by

$\displaystyle\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{G})\in
W_{n}\\}\,\mathbf{1}\left\\{\max\\{r(\bm{H}),r(\bm{G})\\}>\frac{(2+\delta)(c+\log
n)}{2\gamma}\right\\}$
$\displaystyle\quad\times\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
(5.12)
$\displaystyle\,+\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{\|z(\bm{H})-z(\bm{G})\|\leq(\log
n)^{(d+1)/d}\\}\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{\frac{(2+\delta)(c+\log
n)}{2\gamma}\geq r(\bm{G})>\frac{c+\log n}{2\gamma},\,\frac{(2+\delta)(c+\log
n)}{2\gamma}\geq r(\bm{H})>\frac{c+\log n}{2\gamma}\right\\}$
$\displaystyle\quad\times\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.13)

From Lemma 9(a) with $I=\emptyset$ and $R:=\frac{(2+\delta)(c+\log
n)}{2\gamma}$ we get for (5.12) the bound $c_{2}n^{-\delta}\log n$. By part
(b) of the same lemma with $a:=(2+\delta)^{-1}$ and $D:=(\log n)^{(d+1)/d}$ we
conclude that (5.13) is bounded by $c_{3}n^{-(1-L(\frac{1}{2+\delta}))}(\log
n)^{d+1}$. Hence, letting $\delta^{*}$ be the solution of the fixed point
equation

$\displaystyle\delta=1-L\left(\frac{1}{2+\delta}\right):=\frac{\omega_{d-1}}{\omega_{d}}\int_{\frac{2+\delta}{3+\delta}}^{1}(1-x^{2})^{\frac{d-3}{2}}\,\mathrm{d}x,$
(5.14)

we find that (5.10) is bounded by $c_{3}n^{-\delta^{*}}(\log n)^{d+1}$.

Now we discuss (5.11). If $r(\bm{G})\leq r(\bm{H})$, (3.2) gives

$\displaystyle\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})})=\frac{2\omega_{d-1}}{\omega_{d}}\int_{0}^{r(\bm{G})}\int_{\frac{s-r(\bm{H})}{\|z(\bm{H})-z(\bm{G})\|}}^{\frac{s+r(\bm{H})}{\|z(\bm{H})-z(\bm{G})\|}}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x\,\mathrm{d}s\leq\frac{4\omega_{d-1}r(\bm{H})r(\bm{G})}{\omega_{d}\|z(\bm{H})-z(\bm{G})\|},$
(5.15)

where the inequality is obtained by bounding the integrand by 1. Hence, for
$\max\\{r(\bm{H}),r(\bm{G})\\}\leq\frac{(2+\delta)(c+\log n)}{2\gamma}$ and
$\|z(\bm{H})-z(\bm{G})\|>(\log n)^{(d+1)/d}$, we obtain from the inclusion-
exclusion principle the bound

$\displaystyle\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{G})})\geq
2\gamma r(\bm{H})+2\gamma r(\bm{G})-c_{3}.$ (5.16)

This helps us as follows to bound (5.11). Assuming that $r(\bm{G})\leq
r(\bm{H})$ (at the cost of a factor 2), we use the bound
${\mathbb{P}}(\eta\cap{\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})}\neq\emptyset)\leq\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cap{\mathbb{H}}_{B(\bm{G})})$
together with (5.15). To bound the exponential function in the integrand of
(5.11) we use (5.16). This gives for (5.11) the bound

$\displaystyle\frac{16\omega_{d-1}\gamma^{2d+3}e^{c_{3}}}{\omega_{d}((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\frac{r(\bm{H})^{2}}{\|z(\bm{H})-z(\bm{G})\|}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{H})\geq
r(\bm{G})>\frac{c+\log n}{2\gamma}\right\\}\,\mathrm{e}^{-2\gamma
r(\bm{H})-2\gamma
r(\bm{G})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.17)

From [16, Theorem 7.3.2] and (5.2) we obtain that the above is given by

$\displaystyle\frac{16\omega_{d-1}\gamma^{(d)}\gamma^{d+2}}{n\omega_{d}(d+1)!e^{-c_{3}}}\int_{{\mathbb{H}}^{d+1}}$
$\displaystyle\int_{W_{n}}\frac{r(\bm{H})^{2}}{\|z(\bm{H})-z\|}\mathbf{1}\Big{\\{}z(\bm{H})\in
W_{n}\Big{\\}}\mathbf{1}\Big{\\{}r(\bm{H})>\frac{c+\log
n}{2\gamma}\Big{\\}}\,\mathrm{e}^{-2\gamma r(\bm{H})}$
$\displaystyle\quad\times\mathrm{d}z\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

We introduce spherical coordinates around $z(\bm{H})$ in the inner
integration, let $w:=\mathrm{diam}(W)$ and obtain the bound

$\displaystyle\frac{16\omega_{d-1}\gamma^{(d)}\gamma^{d+2}}{n(d+1)!e^{-c_{3}}}\int_{{\mathbb{H}}^{d+1}}\
\int_{0}^{n^{1/d}w}t^{d-2}r(\bm{H})^{2}\mathbf{1}\Big{\\{}z(\bm{H})\in
W_{n}\Big{\\}}\mathbf{1}\Big{\\{}r(\bm{H})>\frac{c+\log n}{2\gamma}\Big{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\mathrm{e}^{-2\gamma
r(\bm{H})}\mathrm{d}t\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
$\displaystyle\quad=\frac{16\omega_{d-1}\gamma^{(d)}\gamma^{d+2}w^{d-1}}{n^{1/d}(d-1)(d+1)!e^{-c_{3}}}\int_{{\mathbb{H}}^{d+1}}r(\bm{H})^{2}\mathbf{1}\Big{\\{}z(\bm{H})\in
W_{n}\Big{\\}}\mathbf{1}\Big{\\{}r(\bm{H})>\frac{c+\log n}{2\gamma}\Big{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\mathrm{e}^{-2\gamma
r(\bm{H})}\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

From [16, Theorem 7.3.2] and (5.2) we conclude that (5.11) is bounded by
$c_{4}n^{-1/d}(\log n)^{2}$.

The estimate of $E_{6}$. Analogously to $E_{5}$, we split $E_{6}$ into the sum

$\displaystyle\frac{2\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{|I|}}{(d+1-|I|)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{H}_{I},\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})}$
$\displaystyle\quad\times\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{H}_{I},\bm{G})\\}>\frac{(2+\delta)(c+\log
n)}{2\gamma}\text{ or }\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\leq(\log n)^{2}\\}$
$\displaystyle\quad\times\mathbf{1}\\{r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
(5.18) $\displaystyle\,+\frac{2\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{|I|}}{(d+1-|I|)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\,\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\mathbf{1}\left\\{r(\bm{H}_{I},\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-\gamma\mu_{d-1}({\mathbb{H}}_{B(\bm{H})}\cup{\mathbb{H}}_{B(\bm{H}_{I},\bm{G})})}$
$\displaystyle\quad\times\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{H}_{I},\bm{G})\\}\leq\frac{(2+\delta)(c+\log
n)}{2\gamma},\,\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|>(\log n)^{2}\\}$
$\displaystyle\quad\times\mathbf{1}\\{r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.19)

Argueing analogously to the bound of (5.10), we obtain that (5.18) is bounded
by $c_{5}n^{-\delta^{*}}\log n$.

Now we consider (5.19). Using (5.16) we obtain that (5.19) is bounded by

$\displaystyle\frac{4\gamma^{d+1}e^{c_{3}}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{d+1-|I|}}{(d+1-|I|)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\,\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\mathbf{1}\left\\{r(\bm{H})>r(\bm{H}_{I},\bm{G})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-2\gamma r(\bm{H})-2\gamma
r(\bm{H}_{I},\bm{G})}\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(5.20)

Now we distinguish the cases $|I|=1$ and $2\leq|I|\leq d$. For every summand
in (5.20) with $|I|=1$ we obtain from Lemma 6 with $\ell=1$

$\displaystyle
2^{d+1}\gamma^{d}\int_{\mathbb{H}^{d+1}}\int_{H_{1}^{\perp}}\int_{\frac{c+\log
n}{2\gamma}}^{r(\bm{H})}\int_{(S^{d-1})^{d}}\int_{(H_{1}-su_{1})\cap
W_{n}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log n}{2\gamma}\right\\}$
$\displaystyle\qquad\times\mathrm{e}^{-2\gamma r(\bm{H})-2\gamma
s}\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\lambda_{d-1}(\mathrm{d}w)\,\sigma_{d-1}^{d}(\mathrm{d}u_{2:d+1})\,\mathrm{d}s\,\sigma_{1}(\mathrm{d}u_{1})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
$\displaystyle\,\leq(d+1)1^{(d)}w^{d-1}\gamma^{d-1}n^{-1/d}\int_{\mathbb{H}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}r(\bm{H})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\mathrm{e}^{-2\gamma
r(\bm{H})}\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$ (5.21)

where $w:=\textrm{diam}(W)$ and we have used the bound
$\lambda_{d-1}((H_{1}-su_{1})\cap W_{n})\leq\textrm{diam}(W)n^{\frac{d-1}{d}}$
and (2.4). An application of [16, Theorem 7.3.2] gives that (5.21) is bounded
by $c_{6}n^{-1/d}\log n$.

For $2\leq|I|\leq d$ we exploit the bound

$\displaystyle d(G_{i},o)\leq
d(G_{i},z(\bm{H}_{I},\bm{G}))+d(z(\bm{H}_{I},\bm{G}),o)\leq
r(\bm{H})+\textrm{diam}(W_{n}),\quad i\in\\{1,\dots,d+1\\}\setminus I.$

This yields for every summand in (5.21) with $2\leq|I|\leq d$ the bound

$\displaystyle\frac{\gamma^{d+1-|I|}}{(d+1-|I|)!}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}\mu_{d-1}(\mathbb{H}_{B(o,r(\bm{H})+\textrm{diam}(W_{n}))})^{d+1-|I|}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}$ $\displaystyle\times\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log
n}{2\gamma}\right\\}\,\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Using that $\mu_{d-1}(\mathbb{H}_{B(o,r)})=2r$ for $r>0$ and letting
$w:=\textrm{diam}(W)$ we obtain the bound

$\displaystyle\frac{(2\gamma)^{d+1-|I|}}{(d+1-|I|)!}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}\Big{(}r(\bm{H})+\frac{w}{n^{1/d}}\Big{)}^{d+1-|I|}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\mathbf{1}\left\\{r(\bm{H})>\frac{c+\log n}{2\gamma}\right\\}$
$\displaystyle\quad\times\,\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

By another application of [16, Theorem 7.3.2] we arrive at the bound
$c_{7}n^{-1/d}\log n$.

It remains to show that the solution $\delta^{*}=\delta^{*}(d)$ of (5.14) is
less than or equal to $1/d$ for all $d\geq 2$. For $d=2$ this can be checked
easily. Since $1-L(\frac{1}{2+\delta})$ is decreasing in $\delta$, it suffices
for $d\geq 3$ to show that $1-L(\frac{d}{2d+1})\leq\frac{1}{d}$. To establish
this, we first bound $\frac{\omega_{d-1}}{\omega_{d}}$. Since
$\omega_{d}:=\frac{2\pi^{d/2}}{\Gamma(d/2)}$ for $d\in\mathbb{N}$,
$\Gamma(n+\frac{1}{2})=\frac{(2n)!\sqrt{\pi}}{4^{n}n!}$ for $n\in\mathbb{N}$
and using Stirling’s formula ([15])

$\displaystyle\sqrt{2\pi n}\,e^{1/(12n+1)}(n/\mathrm{e})^{n}\leq
n!\leq\sqrt{2\pi n}\,e^{1/(12n)}(n/\mathrm{e})^{n},$

we find that $\frac{\omega_{d-1}}{\omega_{d}}\leq\sqrt{\frac{d-1}{2\pi}}$.
Using that $x\mapsto(1-x^{2})^{\frac{d-3}{2}}$ is for $d\geq 3$ decreasing on
the interval $[0,1]$, we thus obtain

$\displaystyle
1-L\Big{(}\frac{1}{2+\delta}\Big{)}=\frac{\omega_{d-1}}{\omega_{d}}\int_{\frac{2d+1}{3d+1}}^{1}(1-x^{2})^{\frac{d-3}{2}}\mathrm{d}x\leq\sqrt{\frac{d-1}{2\pi}}\frac{d}{3d+1}\Big{(}1-\Big{(}\frac{2d+1}{3d+1}\Big{)}^{2}\Big{)}^{\frac{d-3}{2}}.$

Since $\frac{d}{3d+1}\leq\frac{1}{3}$ and
$1-(\frac{2d+1}{3d+1})^{2}\leq\frac{5}{9}$ for $d\in\mathbb{N}$, the above is
bounded by

$\displaystyle\frac{9\sqrt{d-1}}{5\sqrt{10\pi}}\Big{(}\frac{\sqrt{5}}{3}\Big{)}^{d}.$

Since $\frac{9d\sqrt{d-1}}{5\sqrt{10\pi}}(\frac{\sqrt{5}}{3})^{d}<1$ for
$d\in\mathbb{N}$, we deduce that $\delta^{*}(d)\in(0,1/d)$.

We conclude that for $n$ large enough

$\displaystyle\mathbf{d_{KR}}(\zeta_{n}\cap(W\times(c,\infty)),\nu\cap(W\times(c,\infty)))\leq
Cn^{-\delta^{*}}(\log n)^{d+1},$

where $\delta^{*}>0$ is the solution of the fixed point equation (5.14) and
the constant $C>0$ depends on $d$, $\gamma$, $W$ and $c$. ∎

##  6 Asymptotic shape of the typical cell

For the proof of Theorem 2 we need the notion of the typical cell $Z$ of the
random mosaic generated by $\eta$. That is any random polytope with
distribution $\mathbb{Q}_{0}$ on $\mathcal{K}^{d}$ given by

$\displaystyle\gamma^{(d)}\mathbb{Q}_{0}(\cdot)=\sum_{\bm{H}\in\eta^{(d+1)}}\mathbf{1}\\{(\eta-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset\\}\mathbf{1}\\{C(\bm{H},\eta)-z(\bm{H})\in\cdot\\}\mathbf{1}\\{z(\bm{H})\in[0,1]^{d}\\}.$
(6.1)

For an explicit integral representation of $\mathbb{Q}_{0}$ we refer to [16,
Theorem 10.4.6]. The following lemma ensures that the distribution function
$F$ of $\Sigma(Z)$ is strictly increasing and continuous on
$[\gamma^{-1/k},\infty)$. This implies that the function $G:=\frac{1}{1-F}$
introduced in Section 2 is well-defined and invertible on
$[\gamma^{-1/k},\infty)$.

######  Lemma 10.

Let $Z$ be the typical cell in the random mosaic generated by a stationary
Poisson hyperplane process with intensity $\gamma>0$ and let $\Sigma$ be a
$k$-homogeneous size function. On the interval $[\gamma^{-1/k},\infty)$, the
distribution $\mathbb{P}_{\Sigma(Z)}$ of $\Sigma(Z)$ and the Lebesgue measure
$\lambda_{1}$ are equivalent.

###### Proof.

The proof follows the strategy of Section 9 in [8]. We first show that
$\mathbb{P}_{\Sigma(Z)}$ is absolutely continuous with respect to
$\lambda_{1}$ on $[\gamma^{-1/k},\infty)$. For $K\in\mathcal{K}^{d}$ let
$D(K)$ be the diameter of $K$ and
$\Delta(K):=\frac{D(K)}{c_{1}\Sigma(K)^{1/k}}$ be the relative diameter, where
$c_{1}$ is chosen such that $\Delta(K)\geq 1$ for all $K\in\mathcal{K}^{d}$.
For $a>0$, $h>0$ and $m\in\mathbb{N}$ let

$\displaystyle\mathcal{K}_{a,h}(m):=\\{K\in\mathcal{K}^{d}:\,\Sigma(K)\in
a(1,1+h),\,\Delta(K)\in[m,m+1)\\}$

and $q_{a,h}(m):=\mathbb{P}(\Sigma(Z)\in\mathcal{K}_{a,h}(m))$. Fix
$\kappa>0$. Analogously to the argumentation before $(41)$ in [8] (with [8,
Lemma 5] replaced by [9, Lemma 4.5]), we find some $\nu\in\mathbb{N}$ such
that

$\displaystyle q_{a,1}(m)\leq c(\kappa)m^{d\nu}\exp(-(1-\kappa/4)\tau
d^{1/k}\gamma).$

From [9, Lemma 4.3] we obtain $q_{a,1}(m)\leq c_{2}\exp(-c_{3}ma^{1/k}\gamma)$
for some $c_{2},c_{3}>0$. By [9, Lemma 4.8] we have for some $c_{4}>0$

$\displaystyle\mathbb{P}(\Sigma(Z)\in a(1,1+h))$
$\displaystyle=\sum_{m\in\mathbb{N}}q_{a,h}(m)=c_{4}ha^{1/k}\gamma\Big{(}\sum_{m\leq
m_{0}}q_{a,1}(m)+\sum_{m>m_{0}}q_{a,1}(m)\Big{)},$

where $m_{0}\in\mathbb{N}$ is chosen such that

$\displaystyle c_{3}m\geq 2(1-\kappa/4)\tau,\quad m>m_{0}.$ (6.2)

Now we can argue analogously to the proof of Proposition 7.1 in [7] (where [7,
(24)] is replaced by (6.2)) to arrive that

$\displaystyle\mathbb{P}(\Sigma(Z)\in a(1,1+h))\leq
c_{5}(\kappa)h\exp(-(1-\kappa/2)\tau a^{1/k}\gamma).$ (6.3)

As in [8, Section 9] we can now conclude from (6.3) that if a set
$M\subset[\gamma^{-1/k},\infty)$ is covered by countably many intervals of
total length $\varepsilon$, then the $\mathbb{P}_{\Sigma(Z)}$-measure of $M$
is at most $c_{6}\varepsilon$, where $c_{6}$ does not depend on $\varepsilon$.
This proves that $\mathbb{P}_{\Sigma(Z)}$ is absolutely continuous with
respect to $\lambda_{1}$ on $[\gamma^{-1/k},\infty)$.

Next we show that the Radon-Nikodým density of $\mathbb{P}_{\Sigma(Z)}$ with
respect to the Lebesgue measure $\lambda_{1}$ is positive on
$[\gamma^{-1/k},\infty)$. From [9, Lemma 4.1] we obtain for
$a\geq\gamma^{-1/k}$

$\displaystyle\frac{\mathrm{d}\mathbb{P}_{\Sigma(Z)}}{\mathrm{d}\lambda_{1}}(a)=\lim_{h\downarrow
0}\frac{\mathbb{P}(\Sigma(Z)\in
a(1,1+h))}{ah}\geq\frac{c}{a}\exp(-(1+\beta)\tau a^{1/k}\gamma)>0.$

This gives that $\mathbb{P}_{\Sigma(Z)}$ and $\lambda_{1}$ are equivalent
measures on $[\gamma^{-1/k},\infty)$. ∎

In order to measure the deviation of the shape of a convex body $K$ from the
shape of a Euclidean ball, we use the deviation function

$\displaystyle\vartheta(K):=\min\left\\{\frac{R-r}{R+r}:rB^{d}+z\subset
K\subset RB^{d}+z,\,r,R>0,\,z\in{\mathbb{R}}^{d}\right\\}$ (6.4)

from [9] and note that $\vartheta(K)=0$ if and only if $K$ is a Euclidean
ball. Let $\Sigma$ be a size function as defined in Section 2. By [9,
Theorem], there exists a continuous function
$s:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with $s(\varepsilon)>0$ for all
$\varepsilon>0$ and $s(0)=0$ and some constant $c_{0}>0$ (depending only on
$\tau$) such that

$\displaystyle{\mathbb{P}}(\vartheta(Z)\geq\varepsilon\mid\Sigma(Z)>u)\leq
c_{1}\exp(-c_{0}s(\varepsilon)u^{1/k}\gamma),\quad u>0,$ (6.5)

where $c_{1}>0$ depends only on $\Sigma,s,\varepsilon$. In [9] the function
$s$ is called a stability function for $\Sigma$ and $\vartheta$.

The distributions of the typical cell $Z$ and the zero cell $Z_{0}$ (this is
the (a.s. unique) cell containing the origin $o\in{\mathbb{R}}^{d}$) are
linked via

$\displaystyle{\mathbb{P}}(\Sigma(Z)\geq u)\leq{\mathbb{P}}(\Sigma(Z_{0})\geq
u),\quad u>0,$ (6.6)

which is a direct consequence of [9, Lemma 3.1]. Nevertheless, the
distributions of $\Sigma(Z)$ and $\Sigma(Z_{0})$ show the same asymptotic
behavior on a logarithmic scale, i.e.

$\displaystyle\lim_{u\to\infty}u^{-1/k}\log{\mathbb{P}}(\Sigma(Z)\geq
u)=\lim_{u\to\infty}u^{-1/k}\log{\mathbb{P}}(\Sigma(Z_{0})\geq
u)=-\tau\gamma,$ (6.7)

where $\tau$ is the constant from (2.7). This result is a direct consequence
of [8, Theorem 2] and [9, Lemma 4.1].

Next we determine the asymptotic behavior of $G^{-1}$. Let $y>0$ and note that
the fact that $F$ has unbounded support implies that
$G^{-1}(ny)^{-1/k}\to\infty$ as $n\to\infty$. Hence, we obtain from (6.7) and
the definition of $G$ that

$\displaystyle\lim_{n\to\infty}\frac{\log
n}{G^{-1}(ny)^{1/k}}=\lim_{n\to\infty}G^{-1}(ny)^{-1/k}\log(1-F(G^{-1}(ny)))\frac{\log(ny)}{\log(1-F(G^{-1}(ny)))}=\tau\gamma.$
(6.8)

##  7 Stopping sets and decay of correlation

In the proof of Theorem 2 we need to control the circumradius of the typical
cell $Z$. Following [13, Section 6.3], for $\alpha\in(0,\frac{\pi}{6})$ and
$z\in{\mathbb{R}}^{d}$ we let $K^{\prime}_{i}(z),\,i\in\mathcal{I},$ be a
finite collection of infinite open cones with apex $z$, angular radius
$\frac{\alpha}{2}$ and union ${\mathbb{R}}^{d}$. Let
$\omega\in\mathbf{N}_{\mathbb{H}}$ be locally finite and let $R_{i}(z,\omega)$
be the minimal $r>0$ such that there is $u\in S^{d-1}\cap K_{i}(z)$ with
$H(u,\langle z,u\rangle+r)\in\omega$ if such $r>0$ exists and set
$R_{i}(z,\omega):=\infty$, otherwise.

For $\bm{H}\in\omega^{(d+1)}$ in general position let
$R(\bm{H},\omega):=(\cos\alpha)^{-1}\max_{1\leq i\leq
I}R_{i}(z(\bm{H}),\omega\cap{\mathbb{H}}^{B(\bm{H})})$. If
$(\omega-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset$, then
$R(\bm{H},\omega)$ is an upper bound for the circumradius of
$C(\bm{H},\omega)$ and we have

$\displaystyle
C(\bm{H},\omega)=C(\bm{H},\omega_{B(z(\bm{H}),R(\bm{H},\omega))}).$ (7.1)

Next we determine the distribution of $R(\bm{H},\eta)$. For all
$u>(\cos\alpha)^{-1}r(\bm{H})$ we obtain

$\displaystyle{\mathbb{P}}(R(\bm{H},\eta)>u)$
$\displaystyle=\prod_{i\in\mathcal{I}}{\mathbb{P}}(R_{i}(z(\bm{H}),\eta\cap{\mathbb{H}}^{B(\bm{H})})>u\cos\alpha)$
$\displaystyle=1-(1-\mathrm{e}^{-2\gamma(u\cos\alpha-r(\bm{H}))/|\mathcal{I}|})^{|\mathcal{I}|}.$
(7.2)

In the proof of Theorem 2 we will use the stopping set
$\omega\mapsto\mathbb{H}_{B(z(\bm{H}),R(\bm{H},\omega))}$ for
$\bm{H}\in\mathbb{H}^{d+1}$ in general position. In the following lemma we
prove that this is indeed a stopping set.

######  Lemma 11.

For all $\bm{H}\in{\mathbb{H}}^{d+1}$ in general position the mapping
$\mathcal{S}(\bm{H},\cdot):\mathbf{N}_{\mathbb{H}}\to{\mathcal{F}}$ given by

$\displaystyle\mathcal{S}(\bm{H},\omega):=\mathbb{H}_{B(z(\bm{H}),R(\bm{H},\omega))}$
(7.3)

is a stopping set.

###### Proof.

We need to verify (4.2). Let $\bm{H}\in{\mathbb{H}}^{d+1}$ be in general
position, $\omega\in\mathbf{N}_{\mathbb{H}}$ and $S\subset\mathbb{H}$ be
compact. Assume that $\mathcal{S}(\bm{H},\omega)\subset S$ and let
$z:=z(\bm{H})$. Then we have

$\displaystyle\mathcal{S}(\bm{H},\omega)$
$\displaystyle=\mathbb{H}_{B(z(\bm{H}),(\cos\alpha)^{-1}\max_{i\in\mathcal{I}}\inf\\{r>0:\,\exists
u\in K_{i}(z)\cap S^{d-1}:\,H(u,\langle z,u\rangle+r)\in\omega\\})}\cap S$
$\displaystyle=\mathbb{H}_{B(z(\bm{H}),(\cos\alpha)^{-1}\max_{i\in\mathcal{I}}\inf\\{r>0:\,\exists
u\in K_{i}(z)\cap S^{d-1}:\,H(u,\langle z,u\rangle+r)\in\omega\cap S\\})}$
$\displaystyle=\mathcal{S}(\bm{H},\omega\cap S).$

Since $\mathcal{S}(\bm{H},\omega_{1})\subset\mathcal{S}(\bm{H},\omega_{2})$
for $\omega_{1}\geq\omega_{2}$, $\mathcal{S}(\bm{H},\omega\cap S)\subset S$
implies that $\mathcal{S}(\bm{H},\omega)\subset S$. Hence, the proof is
complete. ∎

In the following we consider the events that two cells (whose circumradii are
not too large) are large with respect to $\Sigma$. We will prove that the
correlation of these two events decays when the distance of the centers of the
two cells becomes large. This will be an important step towards the proof of
Theorem 2. For $z\in\mathbb{R}^{d}$, $\alpha\in(0,\frac{\pi}{6})$ and
$i\in\mathcal{I}$ let $N(i):=\\{j\in\mathcal{I}:\,K_{j}(z)\cap
K_{i}(z)\neq\emptyset\\}$ be the set of indices $j\in\mathcal{I}$ for which
the cone $K_{j}(z)$ intersects $K_{i}(z)$. Let
$R_{i}^{\prime}(z,\omega):=\min\\{R_{j}(z,\omega):\,j\in N(i)\\}$ (see Figure
1). For $\bm{H}\in\mathbb{H}^{d+1}$ in general position let
$R^{\prime}(\bm{H},\omega):=(\cos 3\alpha)^{-1}\max_{i\in
I}R_{i}^{\prime}(z(\bm{H}),\omega)$ .

Figure 1: Here we have $N(1)=\\{1,2,3\\}$. Since $\min\limits_{j\in
N(1)}R_{j}(z,\omega)=R_{3}(z,\omega)$, we find that
$R_{3}(z,\omega)=R_{1}^{\prime}(z,\omega)$.

For $r>0$ let $r_{\alpha}:=\frac{r}{(\cos\alpha)(\cos 3\alpha)}$. Let
$I\subsetneq\\{1,\dots,d+1\\}$, $\bm{H}\in\mathbb{H}^{d+1}$,
$\bm{G}\in\mathbb{H}^{d+1-|I|}$ such that $\bm{H}$ and $(\bm{H}_{I},\bm{G})$
are in general position and define

$\mathbb{H}(\bm{H},\bm{G},\alpha):=\mathbb{H}_{B(z(\bm{H}),r(\bm{H})_{\alpha})}\cap\mathbb{H}_{B(z(\bm{H}_{I},\bm{G}),r(\bm{H}_{I},\bm{G})_{\alpha})}.$

Note that if
$\frac{\max(r(\bm{H}),r(\bm{H}_{I},\bm{G}))}{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|}$
is small enough, then for every $i\in\mathcal{I}$ we find some $j\in N(i)$
such that

$\displaystyle
R_{j}(z(\bm{H}),\omega\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})$
$\displaystyle=R_{j}(z(\bm{H}),\omega),\quad\omega\in\mathbf{N}_{\mathbb{H}}.$

This implies that
$R_{i}^{\prime}(z(\bm{H}),\omega\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\leq\max_{j\in
N(i)}R_{j}(z(\bm{H}),\omega),\,i\in\mathcal{I},$ and, hence,

$\displaystyle
R^{\prime}(\bm{H},\omega\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\leq\frac{\cos\alpha}{\cos
3\alpha}R(\bm{H},\omega)$ (7.4)

and the statement also holds if $\bm{H}$ is replaced by $(\bm{H}_{I},\bm{G})$.

Fix $\alpha\in(0,\frac{\pi}{6})$ and $u>0$. For $\bm{H}\in\omega^{(d+1)}$
define

$\displaystyle h(\bm{H},\omega)$
$\displaystyle=\mathbf{1}\\{(\omega-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset\\}\mathbf{1}\\{\Sigma(C(\bm{H},\omega))>u\\}\,\mathbf{1}\Big{\\{}R(\bm{H},\omega)\leq\frac{r(\bm{H})}{(\cos\alpha)^{2}}\Big{\\}},$
$\displaystyle h^{\prime}(\bm{H},\omega)$
$\displaystyle=\mathbf{1}\\{(\omega-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset\\}\mathbf{1}\\{\Sigma(C(\bm{H},\omega))>u\\}\,\mathbf{1}\Big{\\{}R^{\prime}(\bm{H},\omega)\leq\frac{r(\bm{H})}{(\cos\alpha)(\cos
3\alpha)}\Big{\\}}.$

Using (7.4) together with the inequality
$\Sigma(C(\bm{H},\omega))\leq\Sigma(C(\bm{H},\omega\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c}))$,
we obtain

$\displaystyle h(\bm{H},\omega)\leq
h^{\prime}(\bm{H},\omega\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c}).$ (7.5)

In the next lemma we exploit (7.5) to derive an approximative decorrelation
inequality for $h$.

######  Lemma 12.

Let $I\subsetneq\\{1,\dots,d+1\\}$, $\bm{H}\in\mathbb{H}^{d+1}$,
$\bm{G}\in\mathbb{H}^{d+1-|I|}$ such that $\bm{H}$ and $(\bm{H}_{I},\bm{G})$
are in general position. Let
$H\in\mathbb{H}_{B(\bm{H})}\cap\mathbb{H}_{B(\bm{H}_{I},\bm{G})}$ and
$\mu\in\\{0,\delta_{H}\\}$. If
$\frac{\max(r(\bm{H}),r(\bm{H}_{I},\bm{G}))}{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|}$
is small enough, we have

$\displaystyle\mathbb{E}h(\bm{H},\eta_{\bm{H},\bm{G}}+\mu)\,h((\bm{H}_{I},\bm{G}),\eta_{\bm{H},\bm{G}}+\mu)\leq
2\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}})\,\mathbb{E}h^{\prime}((\bm{H}_{I},\bm{G}),\eta_{\bm{H}_{I},\bm{G}}).$

###### Proof.

We apply (7.5) with $\omega:=\eta_{\bm{H},\bm{G}}+\mu$ and obtain

$\displaystyle\mathbb{E}h(\bm{H},\eta_{\bm{H},\bm{G}}+\mu)\,h((\bm{H}_{I},\bm{G}),\eta_{\bm{H},\bm{G}}+\mu)$
$\displaystyle\quad\leq\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\,h^{\prime}((\bm{H}_{I},\bm{G}),\eta_{\bm{H}_{I},\bm{G}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c}).$
(7.6)

Note that
$h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})$ is
measurable with respect to
$\eta\cap\mathbb{H}_{B(z(\bm{H}),r(\bm{H})_{\alpha})}$ and that
$h^{\prime}((\bm{H}_{I},\bm{G}),\eta_{\bm{H}_{I},\bm{G}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})$
is measurable with respect to
$\eta\cap\mathbb{H}^{B(z(\bm{H}),r(\bm{H})_{\alpha})}$. Hence, by independence
of the processes $\eta\cap\mathbb{H}_{B(z(\bm{H}),r(\bm{H})_{\alpha})}$ and
$\eta\cap\mathbb{H}^{B(z(\bm{H}),r(\bm{H})_{\alpha})}$, (7.6) factorizes into

$\displaystyle\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\,\mathbb{E}h^{\prime}(\bm{H}_{I},\bm{G},\eta_{\bm{H}_{I},\bm{G}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c}).$

Thus it remains to show that
$\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\leq\sqrt{2}\,\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}})$
to conclude the proof. Let
$p:=\mathbb{P}(\eta\cap\mathbb{H}(\bm{H},\bm{G},\alpha)=\emptyset)=\exp(-\gamma\mu_{d-1}(\mathbb{H}(\bm{H},\bm{G},\alpha)))$.
Since the processes $\eta\cap\mathbb{H}(\bm{H},\bm{G},\alpha)$ and
$\eta\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c}$ are independent, we obtain

$\displaystyle\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})$
$\displaystyle=p^{-1}\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}\cap\mathbb{H}(\bm{H},\bm{G},\alpha)^{c})\mathbf{1}\\{\eta\cap\mathbb{H}(\bm{H},\bm{G},\alpha)=\emptyset\\}$
$\displaystyle=p^{-1}\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}})\mathbf{1}\\{\eta\cap\mathbb{H}(\bm{H},\bm{G},\alpha)=\emptyset\\}$
$\displaystyle\leq p^{-1}\mathbb{E}h^{\prime}(\bm{H},\eta_{\bm{H}}).$

Since $p^{-1}\leq\sqrt{2}$ if
$\frac{\max(r(\bm{H}),r(\bm{H}_{I},\bm{G}))}{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|}$
is small enough, the proof of the lemma is complete. ∎

##  8 Proof of Theorem 2

As a last intermediate result before the proof of Theorem 2, we show in the
following lemma how to bound the expected number of pairs of large cells whose
centers are within a distance less than or equal to some $D>0$. The lemma will
be used together with (6.5) and Lemma 9 to exclude local clustering of large
cells. It can be understood as a counterpart to Lemma 12 that excludes
asymptotic long-range dependencies.

######  Lemma 13.

Let $D>0$, $W\subset\mathbb{R}^{d}$ be compact,
$I\subsetneq\\{1,\dots,d+1\\}$, $a\in(0,1)$, $R>0$ and $y>0$. We have

$\displaystyle\frac{\gamma^{d+1}}{(d+1)!}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W\\}\,\mathbf{1}\\{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\leq D\\}$
$\displaystyle\,\,\times\mathbb{P}(\eta\cap{\mathbb{H}}_{B(\bm{H})\cup
B(\bm{H}_{I},\bm{G})}=\emptyset,\,\Sigma(C(\bm{H},\eta_{\bm{H},\bm{G}}))>y,\,\Sigma(C((\bm{H}_{I},\bm{G}),\eta_{\bm{H},\bm{G}}))>y)$
$\displaystyle\,\,\times\mathbf{1}\\{\min\\{r(\bm{H}),r(\bm{H}_{I},\bm{G})\\}\leq
aR\\}\mathbf{1}\\{\max\\{r(\bm{H}),r(\bm{H}_{I},\bm{G})\\}\leq
R\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
$\displaystyle\leq
2^{d+2-|I|}\gamma^{(d)}(D+R)^{d+1-|I|}\lambda_{d}(W)\mathbb{P}\Big{(}\Sigma(Z)>y,\vartheta(Z)\geq\frac{\tau
y^{1/k}-2aR}{\tau y^{1/k}+2aR}\Big{)},$

where for $I=\\{i_{1},\dots,i_{m}\\}$ we set
$\bm{H}_{I}=(H_{i_{1}},\dots,H_{i_{m}})$ and $\tau$ is the constant from
(2.7).

###### Proof.

We assume that $r(\bm{H}_{I},\bm{G})\geq r(\bm{H})$ (at the cost of a factor
2). By the triangle inequality, $\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\leq D$ and
$r(\bm{H}_{I},\bm{G})\leq R$ imply that $d(G_{i},z(\bm{H}))\leq D+R$ for
$i\in\\{1,\dots,d+1\\}\setminus I$. Using that
$\mu_{d-1}(\mathbb{H}_{B(z(\bm{H}),D+R)})\leq 2(D+R)$ and that
$\Sigma(C(\bm{H},\eta_{\bm{H},\bm{G}}))\leq\Sigma(C(\bm{H},\eta_{\bm{H}}))$,
we obtain for the left-hand side in the statement of the lemma

$\displaystyle\frac{2^{d+2-|I|}(D+R)^{d+1-|I|}\gamma^{d+1}}{(d+1)!}\int_{{\mathbb{H}}^{d+1}}$
$\displaystyle\mathbf{1}\\{z(\bm{H})\in
W\\}\mathbb{P}(\eta\cap{\mathbb{H}}_{B(\bm{H})}=\emptyset,\,\Sigma(C(\bm{H},\eta_{\bm{H}}))>y)$
$\displaystyle\times\,\mathbf{1}\\{r(\bm{H})\leq
aR\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Now we use that the conditions $r(\bm{H})\leq aR$ and $\Sigma(\bm{H},\eta)>y$
imply by the definition of $\Phi$ (see (2.2)) and the isoperimetric inequality
(2.7) that

$\displaystyle\min\left\\{\frac{R}{r}:\,rB^{d}+z\subset K\subset
RB^{d}+z,\,r,R>0,\,z\in\mathbb{R}^{d}\right\\}\geq\frac{\Phi(K)}{2r(K)}\geq\frac{\tau\Sigma(K)^{1/k}}{2r(K)},\quad
K\in\mathcal{K}^{d},$

where $r(K)$ is the inradius of $K$. Hence, for
$\Sigma(C(\bm{H},\eta_{\bm{H}}))>y$ and $r(\bm{H})\leq aR$ we have that
$\vartheta(C(\bm{H},\eta_{\bm{H}}))\geq\frac{\tau y^{1/k}-2aR}{\tau
y^{1/k}+2aR}$, where $\vartheta$ is the deviation function given at (6.4).
This yields the bound

$\displaystyle\frac{2^{d+2-|I|}(D+R)^{d+1-|I|}\gamma^{d+1}}{(d+1)!}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{z(\bm{H})\in
W\\}$
$\displaystyle\qquad\quad\times\,\mathbb{P}\Big{(}\eta\cap{\mathbb{H}}_{B(\bm{H})}=\emptyset,\,\Sigma(C(\bm{H},\eta_{\bm{H}}))>y,\vartheta(C(\bm{H},\eta_{\bm{H}}))\geq\frac{\tau
y^{1/k}-2aR}{\tau y^{1/k}+2aR}\Big{)}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

From here the assertion follows from the definition of the typical cell $Z$
(see (6.1)). ∎

###### Proof of Theorem 2.

Let $W\subset\mathbb{R}^{d}$ be compact and $c>0$. Using (2.8) we find for all
Borel sets $A\subset W$, $y\geq c$ and $nc\geq G^{-1}(\gamma^{-1/k})$ that

$\displaystyle{\mathbb{E}}\xi_{n}(A\times(y,\infty))=n\gamma^{(d)}\lambda_{d}(A){\mathbb{P}}(n^{-1}G(\Sigma(Z))>y)=\gamma^{(d)}\lambda_{d}(A)y^{-1},$
(8.1)

where $Z$ is the typical cell in m. Hence, for $nc\geq G^{-1}(\gamma^{-1/k})$
the intensity measure of the restricted process
$\xi_{n}\cap(W\times(c,\infty))$ is given by
$\gamma^{(d)}\lambda_{d}(\cdot\cap W)\otimes\psi(\cdot\cap(c,\infty))$, where
$\psi((a,b))=a^{-1}-b^{-1}$ for $0<a<b<\infty$ as defined in Section 2.

We apply Theorem 7 with

$\displaystyle g(\bm{H},\omega)=\mathbf{1}\\{n^{-1/d}z(\bm{H})\in
W\\}\mathbf{1}\\{(\omega-\delta_{\bm{H}})\cap\mathbb{H}_{B(\bm{H})}=\emptyset\\}\mathbf{1}\\{n^{-1}G(\Sigma(C(\bm{H},\omega)))>c\\},$
$\displaystyle
f(\bm{H},\omega)=(n^{-1/d}z(\bm{H}),n^{-1}G(\Sigma(C(\bm{H},\omega)))),\quad\omega\in\mathbf{N}_{\mathbb{H}},\,\bm{H}\in\omega^{(d+1)},$

the stopping set $\mathcal{S}$ from (7.3) and
$S_{{\bm{H}}}:=\mathbb{H}_{B(z(\bm{H}),(\cos\alpha)^{-2}r(\bm{H}))}$, where
the value of $\alpha\in(0,\frac{\pi}{6})$ will be specified later. Note that
(4.1) holds by (7.1). Let $\nu$ be a Poisson process on
${\mathbb{R}}^{d}\times(0,\infty)$ with intensity measure
$\gamma^{(d)}\lambda_{d}\otimes\psi$. Using Remark 8(a), we find

$\displaystyle\mathbf{d_{KR}}(\xi_{n}\cap(W\times(c,\infty)),\nu\cap(W\times(c,\infty)))\leq
E_{1}+2E_{2}+E_{4}+E_{5}+E_{6}$

with

$\displaystyle E_{1}$
$\displaystyle=\frac{2\gamma^{d+1}}{(d+1)!}\int_{\mathbb{H}^{d+1}}{\mathbb{E}}g(\bm{H},\eta_{\bm{H}})\mathbf{1}\\{\mathcal{S}(\bm{H},\eta_{\bm{H}})\not\subset
S_{\bm{H}}\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$ $\displaystyle E_{2}$
$\displaystyle=\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1}}{\mathbb{E}}\mathbf{1}\\{\eta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{\bm{H}}){\mathbb{E}}\tilde{g}(\bm{G},\eta_{\bm{G}})$
$\displaystyle\quad\quad\quad\quad\times\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
$\displaystyle E_{4}$
$\displaystyle=\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1}}{\mathbb{E}}\mathbf{1}\\{\eta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{(\bm{H},\bm{G})})\tilde{g}(\bm{G},\eta_{(\bm{H},\bm{G})})$
$\displaystyle\quad\quad\quad\quad\times\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
$\displaystyle E_{5}$
$\displaystyle=\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1}}{\mathbb{E}}\mathbf{1}\\{\eta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}=\emptyset\\}\tilde{g}(\bm{H},\eta_{(\bm{H},\bm{G})})\tilde{g}(\bm{G},\eta_{(\bm{H},\bm{G})})$
$\displaystyle\quad\quad\quad\quad\times\,{\mathbb{P}}(\eta\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset)\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
$\displaystyle E_{6}$
$\displaystyle=\frac{2\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{d+1-|I|}}{(d+1-|I|)!}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1-|I|}}{\mathbb{E}}\tilde{g}(\bm{H},\eta_{(\bm{H},\bm{G})})\tilde{g}(\bm{G},\eta_{(\bm{H},\bm{G})})$
$\displaystyle\quad\quad\quad\quad\times\mathbf{1}\\{r(\bm{H})+r(\bm{H}_{I},\bm{G})\leq\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$

where for $I=\\{i_{1},\dots,i_{m}\\}$ we set
$\bm{H}_{I}=(H_{i_{1}},\dots,H_{i_{m}})$,
$(\bm{H}_{I},\bm{G})=(H_{i_{1}},\dots,H_{i_{m}},\bm{G})$ and write
$\tilde{g}(\bm{H},\omega):=g(\bm{H},\omega)\mathbf{1}\\{\mathcal{S}(\bm{H},\omega)\subset
S_{\bm{H}}\\}$. Now we bound $E_{1},E_{2},E_{4},E_{5},E_{6}$ separately.

The estimate of $E_{1}$. Since
$\mathcal{S}(\bm{H},\omega)=\mathcal{S}(\bm{H},\omega\cap{\mathbb{H}}^{B(\bm{H})})$,
we find from independence of the processes $\eta\cap{\mathbb{H}}_{B(\bm{H})}$
and $\eta\cap{\mathbb{H}}^{B(\bm{H})}$ that $E_{1}$ is given by

$\displaystyle\frac{2\gamma^{d+1}}{(d+1)!}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}{\mathbb{P}}(\mathcal{S}(\bm{H},\eta)\not\subset
S_{\bm{H}},n^{-1}G(\Sigma(C(\bm{H},\eta_{\bm{H}})))>c)\,\mathrm{e}^{-2\gamma
r(\bm{H})}\,\mathbf{1}\\{z(\bm{H})\in W_{n}\\}$
$\displaystyle\quad\times\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Note that $\mathcal{S}(\bm{H},\eta)\not\subset S_{\bm{H}}$ implies by the
definition of the stopping set $\mathcal{S}$ (see (7.3)) that $\max_{i\in
I}R_{i}(z(\bm{H}),\eta\cap{\mathbb{H}}^{B(\bm{H})})>(\cos\alpha)^{-1}r(\bm{H})$.
This yields that
$\vartheta(C(\bm{H},\eta_{\bm{H}}))>\frac{1-\cos\alpha}{1+\cos\alpha}$, where
$\vartheta$ is given in (6.4). Hence, we obtain that $E_{1}$ is bounded from
above by

$\displaystyle
n\gamma^{(d)}\lambda_{d}(W){\mathbb{P}}(n^{-1}G(\Sigma(Z))>c){\mathbb{P}}\Big{(}\vartheta(Z)>\frac{1-\cos\alpha}{1+\cos\alpha}\Big{|}\,n^{-1}G(\Sigma(Z))>c\Big{)}$
$\displaystyle\quad\leq
c_{1}\exp\left\\{-c_{0}s\Big{(}\frac{1-\cos\alpha}{1+\cos\alpha}\Big{)}G^{-1}(nc)^{1/k}\gamma\right\\},$
(8.2)

where $Z$ is the typical cell with distribution given in (6.5) and the
inequality follows from (2.8) and (6.5). Since $\tau\gamma
G^{-1}(nc)^{1/k}\sim\log n$ by (6.8) and $s(\varepsilon)>0$ for
$\varepsilon>0$, $E_{1}$ is bounded by $c_{1}n^{-b}$ for some $b>0$.

The estimate of $E_{2}$. We split the event $\\{\eta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap S_{\bm{G}}\neq\emptyset\\}$ from the first indicator in $E_{2}$
into $\\{\delta_{(\bm{H},\bm{G})}\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}$ and $\\{\eta\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}$. This allows us to bound $E_{2}$ by

$\displaystyle\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}\mathbf{1}\\{\delta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}{\mathbb{E}}\tilde{g}(\bm{H},\eta_{\bm{H}}){\mathbb{E}}\tilde{g}(\bm{G},\eta_{\bm{G}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H})$
(8.3)
$\displaystyle\,+\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}^{d+1}}{\mathbb{E}}\mathbf{1}\\{\eta\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{\bm{H}})\,{\mathbb{E}}\tilde{g}(\bm{G},\eta_{\bm{G}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(8.4)

Analogously to the bound of $E_{2}$ in the proof of Theorem 1, we find from
Lemma 6 with $\ell=1$ (note that $\nabla_{1}(v)=1$ for all $v\in S^{d-1}$)
that (8.3) is bounded by

$\displaystyle\frac{2^{d+2}\gamma^{2d+2}}{(d+1)!(d+1)}\int_{{\mathbb{H}}^{d+1}}\int_{{\mathbb{H}}_{B(\bm{H})}}\int_{S_{G_{1}^{\perp}}}\int_{(S^{d-1})^{d}}\int_{0}^{\infty}\int_{W_{n}\cap(G_{1}-su_{1})}{\mathbb{E}}\tilde{g}(\bm{H},\eta_{\bm{H}}){\mathbb{E}}\tilde{g}(\bm{G},\eta_{\bm{G}})$
$\displaystyle\quad\times\,\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\lambda_{d-1}(\mathrm{d}w)\,\mathrm{d}s\,\sigma_{d-1}^{d}(\mathrm{d}u_{2:d+1})\,\sigma_{0}(\mathrm{d}u_{1})\,\mu_{d-1}(\mathrm{d}G_{1})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
(8.5)

where $\bm{G}:=(G_{1},H(u_{2},\langle
u_{2},w\rangle+s),\dots,H(u_{d+1},\langle u_{d+1},w\rangle+s))$. Since
$\mu_{d-1}(\mathbb{H}_{B(\bm{H})})=2r(\bm{H})$ (see (2.1)), we have

$\displaystyle\int_{\mathbb{H}_{B(\bm{H})}}\int_{S_{G_{1}^{\perp}}}\mathbf{1}\\{u\in\cdot\\}\,\sigma_{0}(\mathrm{d}u)\,\mu_{d-1}(\mathrm{d}G_{1})=2r(\bm{H})\sigma_{d-1}(\cdot).$

Hence, (8.5) is given by

$\displaystyle\frac{2^{d+3}\gamma^{2d+2}}{(d+1)!(d+1)}$
$\displaystyle\int_{{\mathbb{H}}^{d+1}}\int_{(S^{d-1})^{d+1}}\int_{0}^{\infty}\int_{W_{n}\cap(G_{1}-su_{1})}r(\bm{H})\,{\mathbb{E}}\tilde{g}(\bm{H},\eta_{\bm{H}}){\mathbb{E}}\tilde{g}(\bm{G},\eta_{\bm{G}})$
$\displaystyle\quad\times\,\Delta_{d}(u_{1:d+1})\mathbf{1}_{\mathsf{P}}(u_{1:d+1})\lambda_{d-1}(\mathrm{d}w)\,\mathrm{d}s\,\sigma_{d-1}^{d+1}(\mathrm{d}u_{1:d+1})\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$

Using that $\lambda_{d-1}(W_{n}\cap(G_{1}-su_{1}))\leq
n^{(d-1)/d}\text{diam}(W)^{d-1}$, we obtain by definition of the typical cell
$Z$ the bound

$\displaystyle
c_{2}n^{\frac{d-1}{d}}{\mathbb{P}}(n^{-1}G(\Sigma(Z))>c)\int_{{\mathbb{H}}^{d+1}}r(\bm{H}){\mathbb{E}}\tilde{g}(\bm{H},\eta_{\bm{H}})\,\mathrm{e}^{-2\gamma
r(\bm{H})}\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$ (8.6)

Next we distinguish by the value of $r(\bm{H})$. If $r(\bm{H})>\frac{\log
n}{\gamma}$, the integral in (8.6) is by [16, Theorem 7.3.2] of order
$n^{-1}$. If $r(\bm{H})\leq\frac{\log n}{\gamma}$, it is by (8.1) of order
$\log n$. Hence, (8.3) is bounded by $c_{3}n^{-1/d}\log n$.

To bound (8.4), we consider the first probability in the integrand and find by
the Mecke equation (see [10, Theorem 4.1]) that for fixed
$\bm{H}\in\mathbb{H}^{d+1}$,

$\displaystyle{\mathbb{E}}\mathbf{1}\\{\eta\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{\bm{H}})$
$\displaystyle\leq{\mathbb{E}}\sum_{H\in\eta}\mathbf{1}\\{H\in S_{\bm{H}}\cap
S_{\bm{G}}\\}\tilde{g}(\bm{H},\eta_{\bm{H}})$
$\displaystyle=\gamma\int_{S_{\bm{H}}\cap
S_{\bm{G}}}{\mathbb{E}}\tilde{g}(\bm{H},\eta_{\bm{H}}+\delta_{H})\,\mu_{d-1}(\mathrm{d}H).$
(8.7)

Since $g(\bm{H},\omega_{1})\geq
g(\bm{H},\omega_{2})\geq\tilde{g}(\bm{H},\omega_{2})$ for
$\omega_{1}\subset\omega_{2}$, we have
$g(\bm{H},\eta_{\bm{H}})\geq\tilde{g}(\bm{H},\eta_{\bm{H}}+\delta_{H})$ a.s.
Hence, (8.7) is bounded by

$\displaystyle\gamma\mu_{d-1}(S_{\bm{H}}\cap
S_{\bm{G}})\,{\mathbb{E}}g(\bm{H},\eta_{\bm{H}}).$

From (5.15) we find that $\mu_{d-1}(S_{\bm{H}}\cap
S_{\bm{G}})\leq\frac{4(\cos\alpha)^{-4}r(\bm{H})r(\bm{G})\omega_{d-1}}{\omega_{d}\|z(\bm{H})-z(\bm{G})\|}$.
Similarly to the bound of $E_{5}$ in the proof of Theorem 1, we conclude that
(8.4) (and hence also $E_{2}$) is bounded by $c_{4}n^{-1/d}(\log n)^{2}$.

The estimate of $E_{4}$. Fix $\delta>0$ and let $r_{n}:=\frac{\log
n}{2\gamma}$. We partition the integration area of $E_{4}$ into the Borel sets

$\displaystyle I_{1}$
$\displaystyle=\\{(\bm{H},\bm{G})\in\mathbb{H}^{2d+2}:r(\bm{H})\vee
r(\bm{G})>(2+\delta)r_{n}\\},$ $\displaystyle I_{2}$
$\displaystyle=\\{(\bm{H},\bm{G})\in\mathbb{H}^{2d+2}:r(\bm{H})\wedge
r(\bm{G})>\frac{r_{n}}{2},r(\bm{H})\vee r(\bm{G})\leq(2+\delta)r_{n},$
$\displaystyle\qquad\|z(\bm{H})-z(\bm{G})\|\leq(\log n)^{2}\\},$
$\displaystyle I_{3}$
$\displaystyle=\\{(\bm{H},\bm{G})\in\mathbb{H}^{2d+2}:r(\bm{H})\wedge
r(\bm{G})\leq\frac{r_{n}}{2},r(\bm{H})\vee r(\bm{G})\leq(2+\delta)r_{n},$
$\displaystyle\qquad\|z(\bm{H})-z(\bm{G})\|\leq(\log n)^{2}\\},$
$\displaystyle I_{4}$
$\displaystyle=\\{(\bm{H},\bm{G})\in\mathbb{H}^{2d+2}:r(\bm{H})\vee
r(\bm{G})\leq(2+\delta)r_{n},\|z(\bm{H})-z(\bm{G})\|>(\log n)^{2}\\},$

where $a\vee b:=\max(a,b)$ and $a\wedge b:=\min(a,b)$ for $a,b\in\mathbb{R}$.
This yields for $E_{4}$ the bound

$\displaystyle\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\Big{(}\int_{I_{1}}\mathbf{1}\\{z(\bm{H}),z(\bm{G})\in
W_{n}\\}\,e^{-\gamma\mu_{d-1}(\mathbb{H}_{B(\bm{H})}\cup\mathbb{H}_{B(\bm{G})})}\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{2}}\mathbf{1}\\{z(\bm{H}),z(\bm{G})\in
W_{n}\\}\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mathrm{e}^{-\gamma\mu_{d-1}(\mathbb{H}_{B(\bm{H})}\cup\mathbb{H}_{B(\bm{G})})}$
$\displaystyle\qquad\times\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{3}}\mathbb{E}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{4}}\mathbb{E}\mathbf{1}\\{\eta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))\Big{)}.$

From Lemma 9(a) (with $R:=\frac{(2+\delta)\log n}{2\gamma}$) we find that the
integral over $I_{1}$ is bounded by $c_{4}n^{-\delta}\log n$. By Lemma 9(b)
(with $a:=(2+\delta)^{-1}(2-L(\frac{1}{4+2\delta}))^{-1/2}$ and $D:=(\log
n)^{2}$) we have that the integral over $I_{2}$ is bounded by
$c_{5}n^{1-(2+\delta)a(2-L(a))}(\log n)^{2d}$. To see that the exponent of $n$
is negative, observe that the fact that $L(\cdot)\in(0,1)$ yields
$a>\frac{1}{4+2\delta}$. Since $L$ is decreasing, this implies that

$\displaystyle(2+\delta)a(2-L(a))=\frac{2-L(a)}{\sqrt{2-L(\frac{1}{4+2\delta})}}\geq\sqrt{2-L(a)}>1.$

The integral over $I_{3}$ is by Lemma 13 (with the same choices of $a$, $D$
and $R$ as above) bounded by

$\displaystyle c_{6}n(\log
n)^{2d+2}\mathbb{P}\Bigg{(}n^{-1}G(\Sigma(Z))>c,\,\vartheta(Z)>\frac{\sqrt{2-L(\frac{1}{4+2\delta})}\tau\gamma
G^{-1}(nc)^{1/k}-\log n}{\sqrt{2-L(\frac{1}{4+2\delta})}\tau\gamma
G^{-1}(nc)^{1/k}+\log n}\Bigg{)}$ $\displaystyle\quad\leq c_{7}(\log
n)^{2d+2}\mathbb{P}\Bigg{(}\vartheta(Z)>\frac{\sqrt{2-L(\frac{1}{4+2\delta})}\tau\gamma
G^{-1}(nc)^{1/k}-\log n}{\sqrt{2-L(\frac{1}{4+2\delta})}\tau\gamma
G^{-1}(nc)^{1/k}+\log n}\,\Bigg{|}\,n^{-1}G(\Sigma(Z))>c\Bigg{)}.$

Since $\sqrt{2-L(\frac{1}{4+2\delta})}>1$ and $\tau\gamma
G^{-1}(nc)^{1/k}\sim\log n$ by (4.1), it follows from (2.8) and (6.5) that the
integral over $I_{3}$ is bounded by $c_{7}n^{-b}$ for some $b>0$. We split the
integral over $I_{4}$ into

$\displaystyle\int_{I_{4}}\mathbf{1}\\{\delta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\mathbb{E}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
(8.8) $\displaystyle\quad+\int_{I_{4}}\mathbb{E}\mathbf{1}\\{\eta\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G})).$
(8.9)

For (8.8) we obtain from Lemma 12 with $\mu=0$ (where we use that
$\frac{\max(r(\bm{H}),r(\bm{H}_{I},\bm{G}))}{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|}$
becomes on $I_{4}$ arbitrarily small for $n$ large enough) the bound

$\displaystyle\int_{I_{4}}\mathbf{1}\\{\delta_{(\bm{H},\bm{G})}\cap
S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset\\}\mathbb{E}{g}(\bm{H},\eta_{\bm{H}})\,\mathbb{E}{g}(\bm{G},\eta_{\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G})).$

From here we can proceed as in the bound of (8.3) and find that (8.8) is
bounded by $c_{8}n^{-1/d}\log n$.

Now we discuss (8.9). First notice that (8.9) is by (8.7) (with
$\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})$
instead of $\tilde{g}(\bm{H},\eta_{\bm{H}})$) bounded by

$\displaystyle\gamma\int_{I_{4}}\int_{S_{\bm{H}}\cap
S_{\bm{G}}}\mathbb{E}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}}+\delta_{H})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}}+\delta_{H})\,\mu_{d-1}(\mathrm{d}H)\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G})).$

Here we use Lemma 12 with $\mu:=\delta_{H}$ and obtain the bound

$\displaystyle 2\gamma\int_{I_{4}}\mu_{d-1}(S_{\bm{H}}\cap
S_{\bm{G}})\mathbb{E}g(\bm{H},\eta_{\bm{H}})\mathbb{E}g(\bm{G},\eta_{\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G})).$
(8.10)

Now we invoke the bound $\mu_{d-1}(S_{\bm{H}}\cap
S_{\bm{G}})\leq\frac{4(\cos\alpha)^{-4}r(\bm{H})r(\bm{G})\omega_{d-1}}{\omega_{d}\|z(\bm{H})-z(\bm{G})\|}$
from (5.15). Introducing spherical coordinates, we find analogously to the
bound of (8.4) that (8.9) is bounded by $c_{9}n^{-1/d}(\log n)^{2}$.

The estimate of $E_{5}$. We partition the integration area of $E_{5}$ into the
same sets as for $E_{4}$. This gives the bound

$\displaystyle\frac{2\gamma^{2d+2}}{((d+1)!)^{2}}\Big{(}\int_{I_{1}}\mathbf{1}\\{z(\bm{H}),z(\bm{G})\in
W_{n}\\}\,e^{-\gamma\mu_{d-1}(\mathbb{H}_{B(\bm{H})}\cup\mathbb{H}_{B(\bm{G})})}\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{2}}\mathbf{1}\\{z(\bm{H}),z(\bm{G})\in
W_{n}\\}\mathbf{1}\\{r(\bm{H})+r(\bm{G})\leq\|z(\bm{H})-z(\bm{G})\|\\}\,\mathrm{e}^{-\gamma\mu_{d-1}(\mathbb{H}_{B(\bm{H})}\cup\mathbb{H}_{B(\bm{G})})}$
$\displaystyle\qquad\times\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{3}}\mathbb{E}\tilde{g}(\bm{H},\eta_{\bm{H},\bm{G}})\,\tilde{g}(\bm{G},\eta_{\bm{H},\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))$
$\displaystyle+\int_{I_{4}}{\mathbb{P}}(\eta\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset)\mathbb{E}\tilde{g}(\bm{H},\eta_{\bm{H}})\,\tilde{g}(\bm{G},\eta_{\bm{G}})\,\mu_{d-1}^{2d+2}(\mathrm{d}(\bm{H},\bm{G}))\Big{)}.$

The integrals over $I_{1}$, $I_{2}$ and $I_{3}$ can be bounded as the
analogous terms in the estimate of $E_{4}$. For the integral over $I_{4}$ we
note that ${\mathbb{P}}(\eta\cap S_{\bm{H}}\cap
S_{\bm{G}}\neq\emptyset)\leq\gamma\mu_{d-1}(S_{\bm{H}}\cap S_{\bm{G}})$ and
can thus argue as for (8.10).

The estimate of $E_{6}$. Analogously to the bounds of $E_{4}$ and $E_{5}$, we
split the integration area of $E_{6}$ into the sets $I_{1}$, $I_{2}$, $I_{3}$
and $I_{4}$ (with $\bm{G}$ replaced by $(\bm{H}_{I},\bm{G})$). As above, we
apply Lemma 9 and Lemma 13 to bound the integrals over $I_{1}$, $I_{2}$ and
$I_{3}$. For the integral over $I_{4}$ we use Lemma 12 and find for $n$ large
enough (by which we mean that
$\frac{\max(r(\bm{H}),r(\bm{H}_{I},\bm{G}))}{\|z(\bm{H})-z(\bm{H}_{I},\bm{G})\|}\leq\frac{(2+\delta)\log
n}{2\gamma(\log n)^{2}}$ is small enough to apply Lemma 12) the bound

$\displaystyle\frac{4\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{d+1-|I|}}{(d+1-|I|)!}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1-|I|}}\mathbb{E}{g}(\bm{H},\eta_{\bm{H}})\mathbf{1}\\{R^{\prime}(\bm{H},\eta)\leq
r(\bm{H})_{\alpha}\\}$
$\displaystyle\quad\times\,\mathbb{E}g((\bm{H}_{I},\bm{G}),\eta_{\bm{H}_{I},\bm{G}})\mathbf{1}\\{R^{\prime}((\bm{H}_{I},\bm{G}),\eta)\leq
r(\bm{H}_{I},\bm{G})_{\alpha}\\}\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}),$
(8.11)

where we recall the notation $r_{\alpha}:=\frac{r}{(\cos\alpha)(\cos
3\alpha)}$ for $r>0$ and $\alpha\in(0,\frac{\pi}{6})$ from Section 7. By
homogeneity of $\Sigma$ and monotonicity under set inclusion,
$\Sigma(C(\bm{H},\eta))>G^{-1}(nc)$ implies that
$\Sigma(B^{d})\,R^{\prime}(\bm{H},\eta)^{k}>G^{-1}(nc)$. Since
$r(\bm{H})_{\alpha}\geq R^{\prime}(\bm{H},\eta)$ and
$\tau=2\Sigma(B^{d})^{-1/k}$, this yields that
$r(\bm{H})>\frac{\tau((\cos\alpha)(\cos 3\alpha))^{1/k}}{2}G^{-1}(ny)^{1/k}$.
We apply the same argument to the terms with $(\bm{H}_{I},\bm{G})$ instead of
$\bm{H}$ and assume (at the cost of a factor 2) that $r(\bm{H})\geq
r(\bm{H}_{I},\bm{G})$. Thus we arrive for (8.11) at the bound

$\displaystyle\frac{8\gamma^{d+1}}{(d+1)!}\sum_{\emptyset\subsetneq
I\subsetneq\\{1,\dots,d+1\\}}\frac{\gamma^{d+1-|I|}}{(d+1-|I|)!}\int_{\mathbb{H}^{d+1}}\int_{\mathbb{H}^{d+1-|I|}}\mathbf{1}\\{z(\bm{H})\in
W_{n}\\}\mathbf{1}\\{z(\bm{H}_{I},\bm{G})\in W_{n}\\}$
$\displaystyle\quad\times\,\mathbf{1}\Big{\\{}r(\bm{H})\geq
r(\bm{H}_{I},\bm{G})>\frac{\tau((\cos\alpha)(\cos
3\alpha))^{1/k}}{2}G^{-1}(nc)^{1/k}\Big{\\}}\,\mathrm{e}^{-2\gamma
r(\bm{H})-2\gamma r(\bm{H}_{I},\bm{G})}$
$\displaystyle\quad\times\,\mu_{d-1}^{d+1-|I|}(\mathrm{d}\bm{G})\,\mu_{d-1}^{d+1}(\mathrm{d}\bm{H}).$
(8.12)

From (6.8) we have that $\tau\gamma G^{-1}(nc)^{1/k}\sim\log n$ as
$n\to\infty$ for all $c>0$. Hence, we can bound (8.12) analogously to (5.20).
If $\alpha>0$ is small enough, we find that (8.12) (and hence $E_{6}$) is
bounded by $c_{11}n^{-b}$ for some $b>0$. This finishes the proof of Theorem
2. ∎

## References

  * [1] Barbour, A. and Brown, T. (1992). Stein’s method and point process approximation. Stoch. Proc. Appl. 43, 9–31.
  * [2] Bobrowski, O., Schulte, M. and Yogeshwaran, D. (2021). Poisson process approximation under stabilization and Palm coupling. arXiv:2104.13261.
  * [3] Chenavier, N. and Hemsley, R. (2016). Extremes for the inradius in the Poisson line tessellation. Adv. Appl. Prob. 48, 544–573.
  * [4] Decreusefond, L., Schulte, M. and Thäle, C. (2016). Functional Poisson approximation in Kan-torovich–Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Appl. Prob. 44, 2147–2197.
  * [5] Heinrich, L., Schmidt, H. and Schmidt, V. (2006). Central limit theorems for Poisson hyperplane tessellations. Ann. Prob. 16, 919–950.
  * [6] Herold, F., Hug, D. and Thäle, C. (2021). Does a central limit theorem hold for the $k$-skeleton of Poisson hyperplanes in hyperbolic space? Prob. Th. Rel. Fields. 179, 889–968.
  * [7] Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Probab. 32, 1140–1167.
  * [8] Hug, D. and Schneider, R. (2007). Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191.
  * [9] Hug, D. and Schneider, R. (2007). Typical cells in Poisson hyperplane tessellations. Discrete Comput. Geom. 38, 305–319.
  * [10] Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press, Cambridge.
  * [11] Otto, M. (2020). Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry. arXiv:2005.10116.
  * [12] Otto, M. and Thäle, C. (2022). Large nearest neighbour balls in hyperbolic stochastic geometry. arXiv:2209.12730.
  * [13] Penrose, M. (2007). Gaussian Limits for Random Geometric Measures. Electron. J. Probab. 35, 989–1035.
  * [14] Santaló, L. (2004). Integral Geometry and Geometric Probability. Cambridge University Press, Cambridge.
  * [15] Robbins, H. (1955). A Remark on Stirling’s formula. Amer. Math. Monthly. 62, 26–29.
  * [16] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.