Klain, Daniel A.; Rota, Gian-Carlo Introduction to geometric probability. (English) Zbl 0896.60004 Lezioni Lincee. Cambridge: Cambridge University Press. Rome: Accademia Nazionale dei Lincei, xiv, 178 p. (1997). In the mathematical literature there are a lot of books on geometric probability and integral geometry. Just to cite a few, we can mention L. A. Santaló’s “Integral geometry and geometric probability” (1976; Zbl 0342.53049), and R. Schneider’s “Convex bodies: The Brunn-Minkowski theory” (1993; Zbl 0798.52001)]. The book under review develops a theory of geometric probability using the notion of “invariant valuations” as the basic concept. It is interesting to compare it with Santaló’s book which is based on the theory of differentiable manifolds and homogeneous spaces.As a motivation to the topic, in Chapter I, a solution, due to Barbier [J. Math. Pures Appl., II. Sér. 5, 273-286 (1860)], to the Buffon needle problem is presented, and the concept of geometric probability is extended to the space of straight lines in the Euclidean space. In Chapter 2, the authors introduce the concept of “valuation” and they state and prove “Groemer’s integral theorem” for valuations. This theorem is used systematically and it is applied, in subsequent chapters, to find the set of generators that is most appropriate in each case. Chapter 3 is devoted to the study of combinatorial properties of the lattice of subsets of a finite set, which will be used in later chapters. A particularly interesting result is the characterization of those valuations invariant under the permutation group. One notes that the idea of characterizing the valuations invariant under the action of a group is crucial in the study of geometric probability and is present throughout the book. It is also very interesting the application of the discrete kinematic formula in order to get combinatorial-like identities.In Chapter 4 a theory is developed about invariant valuations for the lattice \(\text{Par} (n)\) of finite unions of orthogonal parallelotopes having edges parallel to a fixed frame. This chapter can be considered as a model to develop the (more difficult) task of constructing the theory of lattices of finite unions of compact convex sets in the Euclidean space. For example, it is proved that the invariant valuations on \(\text{Par} (n)\) are normalized independently of the dimension of the ambient space; that is, they are intrinsic. In Chapter 5 the authors analyze, in the Euclidean space, the integral geometry of polyconvex sets, \(\text{Polycon} (n)\), finite unions of compact convex sets. After proving some classical theorems (Euler-Schläfli-Poincaré, Klee, Carathéodory and Helly), the chapter ends with a proof of Cauchy’s boundary area formula and some interesting and useful comments about the normalization constants of this formula. In Chapter 6, starting from the rotation invariant normalized measure on the Grassmann manifold \(\text{Gr} (n,1)\), they define the corresponding measure on \(\text{Gr} (n,k)\) in terms of the flag coefficients; that is, in terms of generalized combinatorial numbers, with which they develop an interesting theory. Helly’s theorem for subspaces is proved at the end of this chapter. It is noteworthy how the values of the normalized invariant measures here differ from those used by other authors (see, for instance, L. A. Santaló’s book).Chapter 7 begins with the analysis of the invariance of the measure of the affine Grassmann manifold under the group of motions. Further, they study its relation with the intrinsic volumes defined in Chapter 4. In fact, the main result asserts that the intrinsic volume \(\mu_{n-k}\) (which for parallelotopes coincides with the symmetric function of the side lengths) also evaluates the “measure” of the set of \(k\)-dimensional planes meeting a given parallelotope. This interpretation extends to convex sets via Groemer’s theorem. The mean projection formula puts an end to this chapter, along with, as in all the others, some interesting “Notes”. In Chapter 8, the authors state and prove the fundamental theorem of geometric probability; namely the characterization of the volume of polyconvex sets as a continuous and simple valuation on \(\text{Polycon} (n)\) invariant under rigid motions. They also prove the universal normalization theorem for the intrinsic volumes \(\mu_i\). Further, working on the lattice points of the Euclidean space \(L\), they prove a nice result about the expectation of \(gK\cap L\) \((g\) being a random Euclidean motion and \(K\) a convex set) as the quotient \(\mu_n(K)/ \mu_n(C)\), where \(C\) is the fundamental domain of the lattice \(L\). They finish the chapter with some remarks on Hilbert’s third problem. Chapter 9 is devoted to one of the most important theorems of the theory of geometric probability: “Hadwiger’s characterization theorem”. The authors use it to derive simple proofs of a good number of results both in integral geometry and geometric probability and show how the chosen normalization for the intrinsic measures on the Grassmann manifolds appear to be the best. It is interesting to compare these results on integral geometry with the corresponding ones in other books (L. A. Santaló, R. Schneider, \(\dots)\). The paragraph “Notes” at the end of this Chapter is of particular interest.Using Hadwiger’s characterization theorem, the authors prove in Chapter 10 the kinematic formula for the volumes \(\mu_i\) in the Euclidean space and make some remarks about Hadwiger’s containment theorem. Finally, in Chapter 11, the authors extend some of the properties of invariant valuations to the spheres that were previously studied in the Euclidean space. They start by considering the convexity in the sphere \(\Sigma^n\) and prove the spherical area theorem and the characterization theorem for continuous invariant valuations on the sphere \(\Sigma^2\). They also prove, but only in dimension 2, the fundamental formula of kinematics and make some remarks on higher dimension spheres.The reviewer considers this book of great interest for all post-graduates in mathematics, especially for those interested in the theory of geometric probability (integral geometry), since it allows to find relations, in a simple way, among areas which, a priori, look very far apart, such as geometry and probability. The book is written in a rigorous and simple style, which makes it understandable for most of the post-graduates in mathematics. Its contents is exposed in a very suggestive way, but perhaps one of the most interesting aspects of this book are the “Notes” at the end of each chapter, since they show a historical view of the development of many results of geometric probability, and there a lot of interesting and attractive open problems are posed. Reviewer: A.M.Naveira (Valencia) Cited in 2 ReviewsCited in 172 Documents MathOverflow Questions: Does this formula for caliper diameter hold for concave polyhedra? MSC: 60D05 Geometric probability and stochastic geometry 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory 53C65 Integral geometry Keywords:geometric probability; integral geometry; invariant measures; kinematic formula Citations:Zbl 0342.53049; Zbl 0798.52001 PDFBibTeX XMLCite \textit{D. A. Klain} and \textit{G.-C. Rota}, Introduction to geometric probability. Cambridge: Cambridge University Press; Rome: Accademia Nazionale dei Lincei (1997; Zbl 0896.60004)