Stochastic and Integral Geometry

Since this book is about relations between stochastic geometry and integral geometry, we begin with an imaginary experiment that demonstrates the need for and use of integral geometry for certain geometric probability questions and at the same time leads in a natural way to a basic model of stochastic geometry.

A random set in a space E is defined, in agreement with the usual approach of axiomatic probability, as a set-valued random variable, that is, as a measurable map from some abstract probability space into a system of subsets of E, endowed with a suitable σ-algebra. It has turned out to be particularly tractable to assume that E is a locally compact space with a countable base and to consider the system F of its closed subsets, equipped with the topology of closed convergence and the induced σ-algebra of Borel sets. This approach is described in Section 2.1.

The notion of a random closed set, as developed so far, is still very general. To obtain tractable models for applications, one has to restrict the admissible set classes suitably. One possibility consists in considering sets which are generated as the union set of a countable family of simpler sets, such as compact sets, convex bodies, curves, lines, or flats. The appropriate notion for randomizing such families is that of a point process in a space of geometric objects. Point processes are, besides random sets, the second basic object of stochastic geometry. In many applications, the ‘points’ of the process are ordinary points of R ^d , but in others, like those employing random closed sets, the ‘points’ may themselves be sets. For that reason, we study point processes in a general locally compact space E.

Having laid the general foundations in the previous chapters, we now study geometric processes in R ^d and the random sets derived from them. By geometric processes we understand point processes of closed sets which are concentrated on geometrically distinguished subclasses of F′. In particular, we consider particle processes and flat processes. Particle processes are point processes in the subset C′ of nonempty compact sets. Special processes, in general more tractable, are obtained if only particles from the convex ring R or even the class K of convex bodies are admitted. A k-flat process is a point process in F′ whose intensity measure is concentrated on the space A(d, k) of k-dimensional flats (planes, affine subspaces) of R ^d .

As soon as stochastic geometry deals with structures satisfying invariance properties with respect to some group, such as stationarity or isotropy in Euclidean spaces, there arises the need for a theory allowing averaging with respect to invariant measures. Integral geometry in the sense of Blaschke and Santaló is perfectly made for obtaining such averaging formulas. In this chapter we develop the basic tools, namely intersection formulas for fixed and moving geometric objects, where suitable geometric quantities of the inter-sections are integrated with respect to invariant measures. Basic facts about invariant measures on locally compact topological groups and homogeneous spaces, as far as they are needed for our purposes, are collected in the Appendix in Chapter 13.

In this chapter, we derive further integral geometric formulas for convex bodies. They are related to the principal kinematic formula, either directly or indirectly. As in the latter formula, we have a fixed and a moving set, but in the two subsequent sections we do not consider intersections of both; we form sums of convex bodies or projections of convex bodies to subspaces. First we treat rotation means of Minkowski sums, which will later (Section 8.5) be applied to touching probabilities. The global version is an immediate consequence of the principal kinematic formula; the local version will be proved by techniques similar to those in Sections 5.2 and 5.3. From the formulas for rotation means of sums we deduce projection formulas.

In Section 6.3, we admit (infinite) convex cylinders as moving sets. For these, we derive a local kinematic formula, and we also obtain a formula that combines sections with projections.

Mean value formulas with respect to invariant measures, as treated in the preceding two chapters, are a central topic of integral geometry. Another one is transformation formulas for integrals over various spaces of geometric objects. The need for such results in stochastic geometry can be demonstrated by simple examples. Consider, for instance, two independent, identically distributed random hyperplanes in R ^d . Suppose the distribution is such that the intersection of the two hyperplanes is almost surely a (d − 2)-flat. What is the distribution of this random (d − 2)-flat? Or, consider k ≤ d independent, identically distributed random points in R ^d , and suppose their distribution is such that they almost surely span a (k − 1)-flat. What is its distribution? In the cases where the original distributions are derived from invariant measures (by restriction, for example), the answers can be obtained from simple cases of the transformation formulas of this chapter. Generally, these transformation formulas relate integrations over tuples of flats, with respect to invariant measures, to integrations over other sets of flats (or other geometric objects) that are obtained by geometric operations, such as intersection or span. As an example, consider the integral of a function depending on d points. It may happen that the function depends only on the hyperplane spanned (almost everywhere) by the points. Then it may have a simplifying effect to integrate first over the d-tuples of points lying in a fixed hyperplane, and then over all hyperplanes. In principle, the required transformation formulas are just versions of the transformation rule for multiple integrals under differentiable mappings. However, since the mappings are defined by geometric operations, the Jacobians have geometric interpretations, and therefore direct geometric arguments are often simpler and more perspicuous than the use of special parametrizations.

Geometric probability deals with randomly generated objects and configurations of elementary geometry. Whereas stochastic geometry employs more sophisticated models, such as random sets and particle processes, the typical questions of geometric probability involve only finitely many random geometric objects of a simple nature, like points, lines, planes, convex bodies, and simple operations performed with them, for example, taking convex hulls or intersections. The geometric events or random variables under investigation are in general of an elementary nature. Many related questions are easily formulated, but the answers may differ widely in their levels of difficulty.

Geometric probability problems and the early development of integral geometry have been closely related. From the wealth of questions on geometric probabilities that can be considered, we present here a selection, guided by the criteria of intrinsic interest (a personal choice, of course) and of applicability of integral geometry. After a glimpse of the early examples of geometric probability problems in Section 8.1, we devote Section 8.2 to convex hulls of random points, and Section 8.6 to various inequalities for geometric probabilities and expectations of geometric random variables related to convex bodies. In both sections, Blaschke–Petkantschin type transformations are a useful tool. Section 8.3 applies spherical integral geometry to random projections of polytopes, and Section 8.4 treats randomly moving convex bodies and flats by means of Euclidean integral geometry. Section 8.5 develops a theory of randomly touching convex bodies.

For a stationary random closed set Z in R ^d , the volume density or specific volume was defined in Section 2.4 by , where \(B \subset {\rm{R}}^d\) can be an arbitrary Borel set with 0 < λ(B) < ∞. This important parameter describes the mean volume of the random set per unit volume of the space. It is obtained by a double averaging, stochastic and spatial. The straightforward definition (9.1) has the advantage that it immediately exhibits λ(Z ∩ B)/λ(B) as an unbiased estimator for the specific volume. The situation becomes less simple if one wants to take other quantitative aspects of point sets into account. For example, in several applications one is interested in the mean surface area (the mean perimeter in the plane) per unit volume. Clearly, one cannot just proceed as in the case of (9.1), since the surface area of Z ∩ B is in general not defined. Evidently, we must restrict the realizations of the random set Z as well as the ‘observation window’ B. For that reason, we shall assume in the following that the realizations of the closed random set Z belong to the extended convex ring S, the sets of which have the property that the intersection with any convex body is a finite union of convex bodies. Moreover, the observation window will be a compact convex set W with positive volume. In that case, Z ∩ W has a well-defined surface area. However, part of it generally comes from Z ∩ bd W and not from the boundary of Z. To overcome boundary effects caused by the window W, the definition of densities for functionals other than the volume will require additional devices, for example, limit procedures.

By a mosaic we understand a system of convex polytopes in R ^d that cover the whole space and have pairwise no common interior points. A random mosaic can alternatively be described as a special random closed set (formed by the boundaries of the cells of the mosaic), or as a special point process of convex polytopes. The k-dimensional faces of these polytopes themselves generate point processes of k-dimensional sets. Thus, a random mosaic is in a natural way associated with d + 1 particle processes (the processes of vertices, edges, …, cells). This special structure and the copious relations between the intensities and spccific intrinsic volumes of the various face processes make random mosaics a rich topic for mathematical studies. The planar case has been investigated most thoroughly, but also for three-dimensional random mosaics there are many particular results. In this chapter, we maintain the general frame as before and study random mosaics in R ^d , though on some occasions we restrict ourselves to the two- or three-dimensional case, when general results are not known or would be too complicated.

Although the main theme of this book is random geometric structures with invariance properties, such as stationarity or isotropy, we conclude with an outlook to some of the extensions that are possible without such assumptions. The invariance properties in previous chapters allowed us to employ integral geometric formulas for obtaining results on geometric mean values. Our set-up followed also the historical development of the field, where from the beginning stationarity and isotropy seemed to be natural and convenient conditions to get simple and applicable formulas. Their counterparts for non-isotropic random sets and particle processes are necessarily more complicated, as we have seen in some of the previous sections. However, once the step from isotropic to non-isotropic structures is made, the question arises whether a similar generalization from stationary to non-stationary structures is possible. Although random sets and point processes without any invariance properties have been studied by many authors under different aspects, one might get the impression that, for example, the mean value formulas for Boolean models, which are at the heart of stochastic geometry, rely on the invariance of the model. Surprisingly, this is not the case. As the dissertation of Fallert [222] showed (see also [223]), specific intrinsic volumes for Boolean models with convex or polyconvex grains can be introduced without any invariance requirements, and the formulas obtained in Section 9.1 transfer to this situation in a suitably generalized form. Even more astonishing is the fact that these local mean value formulas for non-stationary Boolean models (and Poisson particle processes) make heavy use of the iterated formulas of translative integral geometry, as we have discussed in Section 6.4. Thus, although we do not require that the distributions of our random structures are invariant with respect to the translation group, the corresponding integral geometric setting still plays an essential role.

Throughout this book, we have to use various facts from general topology, the theory of invariant measures, and the geometry of convex sets. In order not to delay the access to stochastic geometry by lengthy preparations, we have collected them here in the Appendix, so that we can refer to the results when they are needed.

The present chapter deals with topological notions and results, including some basic facts on Borel measures. Invariant measures are the topic of Chapter 13, and the necessary material from convex geometry is presented in Chapter 14.

Integral geometry, as it is used in this book for the treatment of random geometric structures with stationarity or isotropy properties, is based on the notion of an invariant measure. Here invariance refers to a group operation and thus to a homogeneous space. Invariant measures on topological groups and homogeneous spaces are known as Haar measures. The general theory of such measures can be found, for example, in Hewitt and Ross [342] and Nachbin [571]. However, we do not presuppose here any knowledge of the theory of Haar measure (with the exception of Section 13.3, which is rarely used in this book and could be dispensed with). For the topological groups and homogeneous spaces that are relevant for integral geometry in Euclidean spaces, the existence and uniqueness of invariant measures will be proved in Section 13.2 in a direct and elementary way, starting from Lebesgue measure and assuming only basic facts from measure theory.

In this book, the more concrete examples of random sets are generated as unions of random systems of convex bodies. The quantitative description of such random sets is based on functionals of convex bodies which are particularly adapted to taking unions: they are additive. In Section 14.2 we collect the basic facts about the most important of these functionals, the rigid motion invariant intrinsic volumes, and their local counterparts, the curvature measures. In Section 14.4 we provide general information about additive functionals, as far as needed.