Stochastic and integral geometry.

*(English)*Zbl 1175.60003
Probability and its Applications. Berlin: Springer (ISBN 978-3-540-78858-4/hbk; 978-3-642-09766-9/pbk; 978-3-540-78859-1/ebook). xi, 693 p. (2008).

Stochastic geometry studies probability distributions of geometric objects, for instance polytopes, lines, tessellations, and random sets. This book, by two eminent specialists in this subject, provides the systematic and exhaustive account of mathematical foundations of stochastic geometry with particular emphasis on tools from convex geometry.

The book is split into four parts. The first part describes the main objects studied in stochastic geometry: random closed sets, point processes and various geometric models, e.g. processes of flats and surfaces, the germ-grain model and its special case, the Boolean model.

The second part lays out the necessary background from integral and convex geometry, which is recognised as the most important mathematical arsenal which (alongside with the probability theory) is widely used in stochastic geometry. The topics covered there include the kinematic formula, integral geometry for cylinders and thick sections, translative integral geometry and spherical integral geometry. It also discusses integral geometric transformation, in particular the Blaschke-Petkantschin formula and its generalisations.

The third part elaborates further on probabilistic properties of geometric objects, in particular, random polytopes, extremal problems, mean values for functionals of germ-grain models with statistical applications, and random tessellations (mosaics). A special chapter deals with non-stationary germ-grain models and non-stationary tessellations.

The fourth part is composed of several appendices that collect facts from general topology, invariant measures and convex geometry.

Many results presented in this book can only be found in journal papers. The notes to each chapter cover also the most recent journal and preprint literature.

In comparison with the classical treatise by [D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications. 2nd ed. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons Ltd. (1995; Zbl 0838.60002)] the book under review is written in theorem-proofs style and so provides a solid foundation for those who would like to learn mathematical methods used in stochastic geometry.

Some topics of this book can be complemented by further texts, [e.g. D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I: Elementary theory and methods. 2nd ed. Probability and Its Applications. (New York), NY: Springer. (2008; Zbl 1026.60061) and An introduction to the theory of point processes. Vol. II: General theory and structure. 2nd revised and extended ed. Probability and Its Applications. (New York), NY: Springer (2008; Zbl 1159.60003)], which develops the probability theory of point processes, and [I. Molchanov, Theory of random sets. Probability and Its Applications. (London): Springer. (2005; Zbl 1109.60001)] specially devoted to random sets.

The material presented in the whole book constitutes an encyclopedic treatment of the subject rather than a text that is immediately suitable for a lecture course of a reasonable length. The thorough and up-to-date presentation in this text makes it an invaluable source for researchers pursuing studies not only in stochastic geometry, but also in convex geometry and various applications, e.g. in material science and spatial statistics. This book is an absolutely indispensable part of all mathematical libraries. Such unique reference text would be also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects.

The book is split into four parts. The first part describes the main objects studied in stochastic geometry: random closed sets, point processes and various geometric models, e.g. processes of flats and surfaces, the germ-grain model and its special case, the Boolean model.

The second part lays out the necessary background from integral and convex geometry, which is recognised as the most important mathematical arsenal which (alongside with the probability theory) is widely used in stochastic geometry. The topics covered there include the kinematic formula, integral geometry for cylinders and thick sections, translative integral geometry and spherical integral geometry. It also discusses integral geometric transformation, in particular the Blaschke-Petkantschin formula and its generalisations.

The third part elaborates further on probabilistic properties of geometric objects, in particular, random polytopes, extremal problems, mean values for functionals of germ-grain models with statistical applications, and random tessellations (mosaics). A special chapter deals with non-stationary germ-grain models and non-stationary tessellations.

The fourth part is composed of several appendices that collect facts from general topology, invariant measures and convex geometry.

Many results presented in this book can only be found in journal papers. The notes to each chapter cover also the most recent journal and preprint literature.

In comparison with the classical treatise by [D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications. 2nd ed. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons Ltd. (1995; Zbl 0838.60002)] the book under review is written in theorem-proofs style and so provides a solid foundation for those who would like to learn mathematical methods used in stochastic geometry.

Some topics of this book can be complemented by further texts, [e.g. D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I: Elementary theory and methods. 2nd ed. Probability and Its Applications. (New York), NY: Springer. (2008; Zbl 1026.60061) and An introduction to the theory of point processes. Vol. II: General theory and structure. 2nd revised and extended ed. Probability and Its Applications. (New York), NY: Springer (2008; Zbl 1159.60003)], which develops the probability theory of point processes, and [I. Molchanov, Theory of random sets. Probability and Its Applications. (London): Springer. (2005; Zbl 1109.60001)] specially devoted to random sets.

The material presented in the whole book constitutes an encyclopedic treatment of the subject rather than a text that is immediately suitable for a lecture course of a reasonable length. The thorough and up-to-date presentation in this text makes it an invaluable source for researchers pursuing studies not only in stochastic geometry, but also in convex geometry and various applications, e.g. in material science and spatial statistics. This book is an absolutely indispensable part of all mathematical libraries. Such unique reference text would be also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects.

Reviewer: Ilya S. Molchanov (Bern)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60D05 | Geometric probability and stochastic geometry |

52A22 | Random convex sets and integral geometry (aspects of convex geometry) |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

62M30 | Inference from spatial processes |