Schneider / Weil Stochastic and Integral Geometry

1. Auflage 2008
ISBN: 978-3-540-78859-1
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 694 Seiten

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.

Rolf Schneider: Born 1940, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1964, PhD 1967 (Frankfurt), Habilitation 1969 (Bochum), 1970 Professor TU Berlin, 1974 Professor Univ. Freiburg, 2003 Dr. h.c. Univ. Salzburg, 2005 EmeritusWolfgang Weil: Born 1945, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1968, PhD 1971 (Frankfurt), Habilitation 1976 (Freiburg), 1978 Akademischer Rat Univ. Freiburg, 1980 Professor Univ. Karlsruhe

Autoren/Hrsg.

Schneider, Rolf

Weil, Wolfgang

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;1 Prolog;13
3.1;1.1 Introduction;13
3.2;1.2 General Hints to the Literature;20
3.3;1.3 Notation and Conventions;22
4;Part I Foundations of Stochastic Geometry;27
4.1;2 Random Closed Sets;29
4.1.1;2.1 Random Closed Sets in Locally Compact Spaces;29
4.1.2;2.2 Characterization of Capacity Functionals;34
4.1.3;2.3 Some Consequences of Choquet’s Theorem;43
4.1.4;2.4 Random Closed Sets in Euclidean Space;49
4.2;3 Point Processes;59
4.2.1;3.1 Random Measures and Point Processes;60
4.2.2;3.2 Poisson Processes;70
4.2.3;3.3 Palm Distributions;82
4.2.4;3.4 Palm Distributions – General Approach;91
4.2.5;3.5 Marked Point Processes;94
4.2.6;3.6 Point Processes of Closed Sets;107
4.3;4 Geometric Models;111
4.3.1;4.1 Particle Processes;112
4.3.2;4.2 Germ-grain Processes;121
4.3.3;4.3 Germ-grain Models, Boolean Models;129
4.3.4;4.4 Processes of Flats;136
4.3.5;4.5 Surface Processes;152
4.3.6;4.6 Associated Convex Bodies;157
5;Part II Integral Geometry;177
5.1;5 Averaging with Invariant Measures;179
5.1.1;5.1 The Kinematic Formula for Additive Functionals;180
5.1.2;5.2 Translative Integral Formulas;192
5.1.3;5.3 The Principal Kinematic Formula for Curvature Measures;202
5.1.4;5.4 Intersection Formulas for Submanifolds;215
5.2;6 Extended Concepts of Integral Geometry;223
5.2.1;6.1 Rotation Means of Minkowski Sums;223
5.2.2;6.2 Projection Formulas;232
5.2.3;6.3 Cylinders and Thick Sections;235
5.2.4;6.4 Translative Integral Geometry, Continued;240
5.2.5;6.5 Spherical Integral Geometry;260
5.3;7 Integral Geometric Transformations;277
5.3.1;7.1 Flag Spaces;278
5.3.2;7.2 Blaschke–Petkantschin Formulas;282
5.3.3;7.3 Transformation Formulas Involving Spheres;299
6;Part III Selected Topics from Stochastic Geometry;303
6.1;8 Some Geometric Probability Problems;305
6.1.1;8.1 Historical Examples;305
6.1.2;8.2 Convex Hulls of Random Points;310
6.1.3;8.3 Random Projections of Polytopes;340
6.1.4;8.4 Randomly Moving Bodies and Flats;347
6.1.5;8.5 Touching Probabilities;361
6.1.6;8.6 Extremal Problems for Probabilities and Expectations;371
6.2;9 Mean Values for Random Sets;389
6.2.1;9.1 Formulas for Boolean Models;391
6.2.2;9.2 Densities of Additive Functionals;405
6.2.3;9.3 Ergodic Densities;416
6.2.4;9.4 Intersection Formulas and Unbiased Estimators;425
6.2.5;9.5 Further Estimation Problems;441
6.3;10 Random Mosaics;457
6.3.1;10.1 Mosaics as Particle Processes;458
6.3.2;10.2 Voronoi and Delaunay Mosaics;482
6.3.3;10.3 Hyperplane Mosaics;496
6.3.4;10.4 Zero Cells and Typical Cells;505
6.3.5;10.5 Mixing Properties;527
6.4;11 Non-stationary Models;533
6.4.1;11.1 Particle Processes and Boolean Models;534
6.4.2;11.2 Contact Distributions;546
6.4.3;11.3 Processes of Flats;555
6.4.4;11.4 Tessellations;562
7;Part IV Appendix;570
7.1;12 Facts from General Topology;571
7.1.1;12.1 General Topology and Borel Measures;571
7.1.2;12.2 The Space of Closed Sets;575
7.1.3;12.3 Euclidean Spaces and Hausdor. Metric;582
7.2;13 Invariant Measures;587
7.2.1;13.1 Group Operations and Invariant Measures;587
7.2.2;13.2 Homogeneous Spaces of Euclidean Geometry;593
7.2.3;13.3 A General Uniqueness Theorem;605
7.3;14 Facts from Convex Geometry;609
7.3.1;14.1 The Subspace Determinant;609
7.3.2;14.2 Intrinsic Volumes and Curvature Measures;611
7.3.3;14.3 Mixed Volumes and Inequalities;622
7.3.4;14.4 Additive Functionals;629
7.3.5;14.5 Hausdor. Measures and Recti.able Sets;645
8;References;649
9;Author Index;687
10;Subject Index;693
11;Notation Index;701