Stochastic and Integral Geometry

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry - random sets, point processes, random mosaics - and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

List of contents

Foundations of Stochastic Geometry.- Prolog.- Random Closed Sets.- Point Processes.- Geometric Models.- Integral Geometry.- Averaging with Invariant Measures.- Extended Concepts of Integral Geometry.- Integral Geometric Transformations.- Selected Topics from Stochastic Geometry.- Some Geometric Probability Problems.- Mean Values for Random Sets.- Random Mosaics.- Non-stationary Models.- Facts from General Topology.- Invariant Measures.- Facts from Convex Geometry.

About the author

Rolf Schneider, 1932 in Chemnitz geboren, studierte Germanistik, Pädagogik, Anglistik und Romanistik und war Redakteur der kulturpolitischen Monatszeitschrift "Aufbau" in Berlin. Im Juni 1979 wurde er aus dem DDR-Schriftstellerverband ausgeschlossen, nachdem er mit elf anderen Autoren in einer Resolution gegen die Zwangsausbürgerung Wolf Biermanns protestiert hatte. Seine Romane und Theaterstücke wurden mit zahlreichen Preisen ausgezeichnet. Rolf Schneiderlebt in Schöneiche bei Berlin.

Summary

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry – random sets, point processes, random mosaics – and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

Additional text

From the reviews:
“In ‘Stochastic and Integral Geometry,’ R. Schneider and W. Weil give priority to the basic concepts in stochastic geometry … while keeping from integral geometry only what is relevant for applications in stochastic geometry. … Each chapter section is concluded by notes in which the main references are cited and numerous possible extensions are discussed. … Stochastic and Integral Geometry is a profound work by two eminent specialists which is essential reading for those willing to learn deep theory.” (Pierre Calka, Mathematical Geosciences, Vol. 45, 2013)
“This book … provides the systematic and exhaustive account of mathematical foundations of stochastic geometry with particular emphasis on tools from convex geometry. … 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 … . an absolutely indispensable part of all mathematical libraries. … also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects.” (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1175, 2010)
“The book presents a number of results that are otherwise scattered among an immense number of research papers and mostly provides full proofs for them. … The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. The book is also worth owning not only for those working in stochastic geometry and immediately related fields of theoretical and applied probability and spatial statistics. … This book … will be an essential part of every mathematical library.” (V. K. Oganyan, Mathematical Reviews, Issue 2010 g)

Report

From the reviews:
"In 'Stochastic and Integral Geometry,' R. Schneider and W. Weil give priority to the basic concepts in stochastic geometry ... while keeping from integral geometry only what is relevant for applications in stochastic geometry. ... Each chapter section is concluded by notes in which the main references are cited and numerous possible extensions are discussed. ... Stochastic and Integral Geometry is a profound work by two eminent specialists which is essential reading for those willing to learn deep theory." (Pierre Calka, Mathematical Geosciences, Vol. 45, 2013)
"This book ... provides the systematic and exhaustive account of mathematical foundations of stochastic geometry with particular emphasis on tools from convex geometry. ... 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 ... . an absolutely indispensable part of all mathematical libraries. ... also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1175, 2010)
"The book presents a number of results that are otherwise scattered among an immense number of research papers and mostly provides full proofs for them. ... The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. The book is also worth owning not only for those working in stochastic geometry and immediately related fields of theoretical and applied probability and spatial statistics. ... This book ... will be an essential part of every mathematical library." (V. K. Oganyan, Mathematical Reviews, Issue 2010 g)

Product details

• Authors: Rol Schneider, Rolf Schneider, Wolfgang Weil
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 12.10.2010
• EAN: 9783642097669
• ISBN: 978-3-642-09766-9
• No. of pages: 694
• Dimensions: 156 mm x 39 mm x 236 mm
• Weight: 1028 g
• Illustrations: XII, 694 p.
• Series: Probability and Its Applications; Probability and Its Applications
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; B, Mathematics and Statistics, Algebraic Geometry, Probability Theory and Stochastic Processes, Discrete Mathematics, Discrete geometry, Convex and Discrete Geometry, Convex geometry, Probabilities, Stochastics, Probability Theory