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."

The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject. The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex. With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes. For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.

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."

At the heart of this monograph is the Brunn-Minkowski theory, which can be used to great effect in studying such ideas as volume and surface area and their generalizations. In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems. The book provides hints and pointers to connections with other fields and an exhaustive reference list. This second edition has been considerably expanded to reflect the rapid developments of the past two decades. It includes new chapters on valuations on convex bodies, on extensions like the Lp Brunn-Minkowski theory, and on affine constructions and inequalities. There are also many supplements and updates to the original chapters, and a substantial expansion of chapter notes and references.

The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject. The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex. With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes. For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.

This book provides the foundations for geometric applications of convex cones and presents selected examples from a wide range of topics, including polytope theory, stochastic geometry, and Brunn-Minkowski theory. Giving an introduction to convex cones, it describes their most important geometric functionals, such as conic intrinsic volumes and Grassmann angles, and develops general versions of the relevant formulas, namely the Steiner formula and kinematic formula. In recent years questions related to convex cones have arisen in applied mathematics, involving, for example, properties of random cones and their non-trivial intersections. The prerequisites for this work, such as integral geometric formulas and results on conic intrinsic volumes, were previously scattered throughout the literature, but no coherent presentation was available. The present book closes this gap. It includes several pearls from the theory of convex cones, which should be better known.

Vom 1. bis 8. Oktober 1989 fand im Kloster Neresheim das DMV-Seminar "Stochastische Geometrie" statt. Das Ziel dieser Veranstaltung war es, die Stochastische Geometrie, die sich in den letzten Jahren lebhaft entwickelt hat und die auch fur Anwendungen in der Bildverarbeitung, der Stereologie und der Statistik von raumlichen Daten eine grundlegende Bedeutung bekommen hat, einem breiteren Kreis von Mathematikern nahe zu bringen. Dabei sollte auch das Zusammenwirken geometrischer Ideen und stochastischer Modelle exemplarisch aufgezeigt werden. Die Vortrage uber Integralgeometrie (R. Schneider), zufallige Mengen und geometrische Punktprozesse (W. Weil), zufallige Mosaike und Ebenenprozesse (J. Mecke), Kenngroessen geometrischer Strukturen und Statistik von Punktprozessen, zufalligen Mengen und Mosaiken (D. Stoyan) wurden erganzt durch speziellere Themen (zufallige Geraden, allgemeine Poissonprozesse, Boolesche Modelle, Punkt prozessmodelle ), Computer-Simulationen und Fallbeispiele. Der folgende Text enthalt die ausgearbeiteten Vortrage, wobei einige der Erganzungen eingearbeitet wurden. Eine Einfuhrung in die Theorie allgemeiner Poissonprozesse (J. Mecke) wurde als Anhang A aufgenommen. Es erschien uns nicht sinnvoll, die vollstandigen Programme zu den Simulationen abzudrucken. Wir haben aber fur einige Grundstrukturen der Stochastischen Geome trie Simulationsprogramme als Anhang B beigefugt. Bilder solcher Simulationen sowie Bilder von realen geometrischen Daten und zufalligen geometrischen Strukturen aus der Praxis sind in den Text aufgenommen worden. Die Programme stammen in der vorliegenden Form von Herrn Dipl.-Math. H. Fallert, der auch einen grossen Teil der Simulationen durchgefuhrt hat. Die Reinschrift der Manuskripte wurde von Frau U. Peters vorgenommen. Beiden 6 moechten wir an dieser Stelle fur ihre Mithilfe danken.

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.