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Stochastic and Integral Geometry

:	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

	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