Stochastic and Integral Geometry (eBook)

Rolf Schneider, Wolfgang Weil (Autoren)

eBook Download: PDF

2008 | 2008
XII, 694 Seiten
Springer Berlin (Verlag)
978-3-540-78859-1 (ISBN)

Infos

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 Emeritus

Wolfgang 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

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 8.9.2008
• Reihe/Serie: Probability and Its Applications
• Reihe/Serie: Probability and Its Applications
• Zusatzinfo: XII, 694 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Geometric probability • Integral Geometry • Point Process • random set • stochastic geometry • Variance
• ISBN-10: 3-540-78859-X / 354078859X
• ISBN-13: 978-3-540-78859-1 / 9783540788591