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."

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn-Minkowski theory, with an exposition of mixed volumes, the Brunn-Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn-Minkowski theory, with an exposition of mixed volumes, the Brunn-Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

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."

Die von Blaschke begriindete Integralgeometrie handelt von beweglichen Fi- guren im Raum und von invarianten Integralen, die sich bei ihnen bilden lassen. Dieses Zitat aus Hadwiger [1957] (S. 225) beschreibt recht gut die wesentlichen Elemente der Integralgeometrie: Es geht urn bewegte Figuren, also der Operation einer Gruppe unterworfene geometrische Objekte, und urn invariante Mittelwerte im Zusammenhang mit solchen bewegten Figuren. Integralgeometrie ist also ein Teilgebiet der Geometrie, das sich mit der Bestimmung und Anwendung von Mittelwerten geometrisch definierter Funk- tionen beziiglich invarianter Maf3e befaBt. Zu den Grundlagen der Integral- geometrie gehoren daher einerseits Teile der Theorie invarianter Maf3e auf topologischen Gruppen und homogenen Raumen, andererseits gewisse Ge- biete aus der Geometrie der Punktmengen, wie etwa der Polyeder, konvexen Mengen oder differenzierbaren Untermannigfaltigkeiten. Urspriinglich aus Fragestellungen iiber geometrische Wahrscheinlichkei- ten entstanden und von Blaschke, Chern, Hadwiger, Santal6 und anderen ab 1935 entwickelt, hat sich die Integralgeometrie in jiingerer Zeit als wichtiges Hilfsmittel in der Stochastischen Geometrie und deren Anwendungsgebieten (Stereologie, Bildanalyse, raumliche Statistik) erwiesen. Dies hat zu neuen Resultaten gefiihrt, zu Verallgemeinerungen klassischer integralgeometrischer Formeln, aber auch zu andersartigen Zugangen und zu neuen Gesichtspunk- ten. Das vorliegende Buch ist sowohl klassischen Ergebnissen der Integralgeo- metrie gewidmet als auch neueren Entwicklungen. Es unterscheidet sich in mehrfacher Hinsicht wesentlich von den vorhandenen Monographien.