In this monograph, the authors provide a comprehensive introduction to a central core of stochastic geometry called Poisson hyperplane tessellations. This is a quite modern concept and we can find some motivation behind this subject coming from random linear algebra. To give the reader a feel for the subject, we present an overview and then proceed to a description of the contents of the monograph. We consider the Euclidean space \(\mathbb{R}^{d}\) of dimension \(d\geq 2\), where the origin is denoted by \(o\). We denote by \(B(x,r)\) and \(\mathbb{S}^{d-1}(x,r)\), respectively, the ball and the sphere of center \(x\) and the radius \(r\). A convex body in \(\mathbb{R}^{d}\) is a compact and convex set with non-empty interior. We denote by \(\mathcal{K}\) the set of convex bodies. The set \(\mathcal{K}\) has an algebraic structure with the scale and translation actions, namely \[ tA := \{ ta : a \in A\}, \qquad A+x = x + A = \{ a + x : a \in A\} \] or any \(A \in \mathcal{K}\), \(t \in \mathbb{R} \setminus \{0\}\), and \(x \in \mathbb{R}^{d}\). The set \(\mathcal{K}\) is equipped with the standard Hausdorff metric. Let \(f : \mathcal{K} \rightarrow \mathbb{R}\) be a map. If there exists \(k\) such that \(f(tK) = t^{k}f(K)\) with \(K \in \mathcal{K}\) and \(t > 0\), then we say that \(f\) is homogeneous of degree \(k\). We say that \(f\) is scale invariant if \(f\) is homogeneous of degree \(k=0\). Moreover, if \(f(K+x) = f(x)\) for any \(K \in \mathcal{K}\) and \(x \in \mathbb{R}^{d}\), then we say that \(f\) is translation invariant. Finally, \(f\) is called a shape factor if \(f\) is scale and translation invariant. After this brief conceptual preparation, we proceed to define Poisson hyperplane mosaics. A hyperplane is a set of the form \[ H(u,t) = \{ x \in \mathbb{R}^{d} : \langle x, u \rangle = t \} \subset \mathbb{R}^{d-1} \] with \(u \in \mathbb{S}^{d-1}\) and \(t \in \mathbb{R}\). We denote by \(\mathcal{H}\) the space of hyperplanes. The surjection \(\mathbb{S}^{d-1} \times \mathbb{R} \rightarrow \mathcal{H}\) defined by \((u,t) \mapsto H(u,t)\) induces on \(\mathcal{H}\) a topology, so from now on we equip \(\mathcal{H}\) with this topology and the corresponding \(\sigma\)-algebra. Let \(\Theta\) be a homogeneous of degree \(r>0\) measure on \(\mathcal{H}\). We denote by \(\aleph\) the set of Borel probability measure on \(\mathbb{S}^{d-1}\) with support not contained in some closed hemisphere, by \(\aleph_{e}\) the set of measures \(\phi \in \aleph\) which are even, and by \(\aleph_{e,c}\) the set of measures \(\phi \in \aleph_{e}\) which are absolutely continuous with respect to the spherical Lebesgue measure \(\sigma\). Recall that a discrete real random variable \(Y\) is Poisson distributed with the parameter \(\theta \in (0,1)\) if \[ \mathbb{P}(Y=k) = \frac{\theta^{k}e^{-k}}{k!}, \,\, k \in \mathbb{N}. \] This definition extends to the case \(\theta = 0\), in which case \(Y=0\) almost surely. A Poisson hyperplane process \(\eta\) of intensity measure \(\Theta\) is a discrete random set in \(\mathcal{H}\) such that for any Borel set \(B \subset \mathcal{H}\) the random variable \(|\eta \cap B|\) is Poisson distributed with parameter \(\Theta(B)\), and for any pairwise disjoint Borel sets \(B_{1}, \dots B_{n} \subset \mathcal{H}\), the random variables \(|\eta \cap B_{1}|, \ldots, |\eta \cap B_{n}|\) are independent. We say that \(\eta\) is stationary when \(\Theta\) is stationary (and \(\Theta\) is stationary if and only if the associated Borel probability measure \(\phi\) on \(\mathbb{S}^{d-1}\) is homogeneous of degree \(1\) and \(\phi \in \aleph_{e}\)), and \(\eta\) is isotropic when \(\Theta\) is isotropic (which means that the \(\phi\) is the Haar measure). A polyhedron is a finite intersection of halfspaces. A polytope is a bounded polyhedron. A tessellation is a collection of polytopes which are covering the whole space and have pairwise disjoint interiors. The polytopes forming a tessellation \(X\) are called the cells of \(X\). Let \(\eta\) be a Poisson hyperplane process of intensity measure \(\Theta\). The closure of each of the connected components of the complement \(\mathbb{R}^{d} \setminus_{H \in \eta}H\) is almost surely a polytope. These are the cells of the so-called Poisson hyperplane tessellation \(X = X_{\eta}\) associated with \(\eta\). These Poisson hyperplane tessellations are the main object of study in this monograph.
In order to carry out further investigations on Poisson hyperplane tessellations and to translate this setting using the language of polytopes, the authors introduce an interesting functional called the hitting functional and they relate it geometrically to the Blaschke bodies. Then they introduce the Matheron zonoids, which provide interesting connections. Since the origin is almost certainly not contained in any hyperplane \(\eta\), the cell of \(X\) containing \(o\) is called the zero cell and is denoted by \(Z_{o}\). In the stationary case, i.e. \(r=1\) and \(\phi\) is even, one can define the typical cell \(Z_{typ}\). These cells are studied in Chapter 6. It is worth noting that cells play a fundamental role in the theory of tessellations, and that’s why the authors pay a lot of attention to them.
Then the authors study mixing and the ergodicity of stationary Poisson hyperplane processes and tessellations. In Chapter 8 the authors deal with expectations and variances of geometric random variables obtained from stationary Poisson hyperplane processes and tessellations inside a bounded observation window.
As a next step, the authors explore the asymptotic behavior of a class of geometric random variables, which are defined in a bounded window, as the window expands. Then they develop the classical method of U-statistics in order to use them to prove central limit theorems. Then they provide a self-contained introduction to Stein’s method and probabilistic coupling arguments to obtain quantitative results also for a bounded observation window. It turns out that an alternative and general introduction of typical objects of stationary random tessellations can be based on Palm measures. Since this approach is used to introduce weighted \(k\)-faces, the authors present a brief introduction to Palm distributions.
In Chapter 11, the authors explore different approaches, including Palm measures, to typical \(k\)-faces, with or without weights. The \(k\)-faces with \(k \in \{1,\dots ,d - 1\}\) have a direction, and this fact offers new aspects. The results on weighted \(k\)-faces yield some information about mixed second moments.
In Chapter 12, the authors study a very general versions of D. G. Kendall’s problem. This class of problems deals with limit shapes and asymptotic shapes of, for example, the zero cell under the condition that it is large under some measurement. The obtained results include deviation bounds for general classes of geometric functionals of the cells and information about their asymptotic distributions. Prescribing the number of facets leads to different phenomena, as set out in Chapter \(13\). This chapter ends with estimates for the probability that the typical cell of a stationary Poisson hyperplane tessellation has a large number of facets.
Chapter 15 is devoted to a detailed study of the \(K\)-cell. Given a convex body \(K\), the \(K\)-cell of a stationary Poisson hyperplane process is the intersection of all closed halfspaces bounded by hyperplanes of the process and containing \(K\). Thus, the \(K\)-cell is a random polytope generalizing the zero polytope. The chapter deals with various aspects of the question of how well the convex body \(K\) is approximated by the \(K\)-cell if the intensity of the hyperplane process tends to infinity. Chapter 16 gives a closer look at zero cells of isotropic Poisson hyperplane processes without stationarity. The distance of the hyperplanes from the origin is governed by a parameter, and the expected face numbers and the volume of the zero cell are studied in dependence on this parameter. In addition to mean values, moments and variances of the volume of the zero cell and the asymptotic behavior of the intersection volume of the zero cell and a lower dimensional ball are explored. In Chapter 17, the authors provide another perspective for the study of variance formulas and asymptotic normality of Poisson U-statistics.
Obviously, our outline of the content of the monograph is not exhaustive, since the authors deliver a plethora of topics that are in the field of interest of stochastic geometry, and it is almost impossible to cover them in a concise way here. The present monograph is nicely written and can be a good introduction to this subject also for beginners (as the reviewer of this monograph is), especially that the topics are presented in detail with many instructive figures. This increases readability (even if some of the considerations are more technical). To sum up, I believe that this is a very nicely written monograph that unites in one piece different areas of contemporary mathematics.