: Rolf Schneider, Wolfgang Weil
: Stochastic and Integral Geometry
: Springer-Verlag
: 9783540788591
: 1
: CHF 124.10
:
: Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik
: English
: 694
: Wasserzeichen/DRM
: PC/MAC/eReader/Tablet
: PDF

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

Preface6
Contents10
1 Prolog13
1.1 Introduction13
1.2 General Hints to the Literature20
1.3 Notation and Conventions22
Part I Foundations of Stochastic Geometry27
2 Random Closed Sets29
2.1 Random Closed Sets in Locally Compact Spaces29
2.2 Characterization of Capacity Functionals34
2.3 Some Consequences of Choquet’s Theorem43
2.4 Random Closed Sets in Euclidean Space49
3 Point Processes59
3.1 Random Measures and Point Processes60
3.2 Poisson Processes70
3.3 Palm Distributions82
3.4 Palm Distributions – General Approach91
3.5 Marked Point Processes94
3.6 Point Processes of Closed Sets107
4 Geometric Models111
4.1 Particle Processes112
4.2 Germ-grain Processes121
4.3 Germ-grain Models, Boolean Models129
4.4 Processes of Flats136
4.5 Surface Processes152
4.6 Associated Convex Bodies157
Part II Integral Geometry177
5 Averaging with Invariant Measures179
5.1 The Kinematic Formula for Additive Functionals180
5.2 Translative Integral Formulas192
5.3 The Principal Kinematic Formula for Curvature Measures202
5.4 Intersection Formulas for Submanifolds215
6 Extended Concepts of Integral Geometry223
6.1 Rotation Means of Minkowski Sums223
6.2 Projection Formulas232
6.3 Cylinders and Thick Sections235
6.4 Translative Integral Geometry, Continued240
6.5 Spherical Integral Geometry260
7 Integral Geometric Transformations277
7.1 Flag Spaces278
7.2 Blaschke–Petkantschin Formulas282
7.3 Transformation Formulas Involving Spheres299
Part III Selected Topics from Stochastic Geometry303
8 Some Geometric Probability Problems305
8.1 Historical Examples305
8.2 Convex Hulls of Random Points310
8.3 Random Projections of Polytopes340
8.4 Randomly Moving Bodies and Flats347
8.5 Touching Probabilities361
8.6 Extremal Problems for Probabilities and Expectations371
9 Mean Values for Random Sets389
9.1 Formulas for Boolean Models391
9.2 Densities of Additive Functionals405
9.3 Ergodic Densities416
9.4 Intersection Formulas and Unbiased Estimators425
9.5 Further Estimation Problems441
10 Random Mosaics457
10.1 Mosaics as Particle Processes458
10.2 Voronoi and Delaunay Mosaics482
10.3 Hyperplane Mosaics496
10.4 Zero Cells and Typical Cells505
10.5 Mixing Properties527
11 Non-stationary Models533
11.1 Particle Processes and Boolean Models534
11.2 Contact Distributions546
11.3 Processes of Flats555
11.4 Tessellations562
Part IV Appendix570
12 Facts from General Topology571
12.1 General Topology and Borel Measures571
12.2 The Space of Closed Sets575
12.3 Euclidean Spaces and Hausdor. Metric582
13 Invariant Measures587
13.1 Group Operations and Invariant Measures587
13.2 Homogeneous Spaces of Euclidean Geometry593
13.3 A General Uniqueness Theorem605
14 Facts from Convex Geometry609
14.1 The Subspace Determinant609
14.2 Intrinsic Volumes and Curvature Measures611
14.3 Mixed Volumes and Inequalities622
14.4 Additive Functionals629
14.5 Hausdor. Measures and Recti.able Sets645
References649
Author Index687
Subject Index693
Notation Index701