Poisson Hyperplane Tessellations

Preface
Contents
1 Notation
	1.1 Basic Structures
	1.2 Measure and Integration
	1.3 Euclidean Space and Its Structures
2 Hyperplane and Particle Processes
	2.1 Point Processes
	Notes for Sect. 2.1
	2.2 Flat and Hyperplane Processes
	Notes for Sect. 2.2
	2.3 Particle Processes
	Notes for Sect. 2.3
	2.4 Random Tessellations
	Notes for Sect. 2.4
	2.5 Measurability
3 Distribution-Independent Density Relations
	3.1 Functionals and Their Densities
	Notes for Sect. 3.1
	3.2 Density Relations for Hyperplane Tessellations
	Notes for Sect. 3.2
4 Poisson Hyperplane Processes
	4.1 Poisson Point Processes
	Notes for Sect. 4.1
	4.2 Poisson Hyperplane Processes
	Notes for Sect. 4.2
	4.3 Poisson Hyperplane Tessellations
	Notes for Sect. 4.3
5 Auxiliary Functionals and Bodies
	5.1 The Hitting Functional and the Blaschke Body
	Notes for Sect. 5.1
	5.2 The Matheron Zonoid
	5.3 Intersection Densities
	Notes for Sect. 5.3
	5.4 Affine Intersection Densities
	5.5 Face Intensities and Functional Densities
	5.6 Section Processes
	Notes for Sect. 5.6
6 Zero Cell and Typical Cell
	6.1 The Zero Cell
	Notes for Sect. 6.1
	6.2 The Typical Cell
	Notes for Sect. 6.2
	6.3 K-Cells
7 Mixing and Ergodicity
	7.1 Mixing Properties
	7.2 Ergodicity and Applications
	Notes for Sect. 7.2
8 Observations Inside a Window
	8.1 The Number of k-Faces Hitting the Window
	Notes for Sect. 8.1
	8.2 Intersections: Expectations
	Notes for Sect. 8.2
	8.3 Intersections: Variances
	Notes for Sect. 8.3
	8.4 The Isotropic Case
	Notes for Sect. 8.4
9 Central Limit Theorems
	9.1 An Approach by Classical Methods
	Notes for Sect. 9.1
	9.2 Central Limit Theorems with Rates
	Notes for Sect. 9.2
	9.3 Stein\'s Method and Poisson Random Sums
	Notes for Sect. 9.3
10 Palm Distributions and Related Constructions
	10.1 Palm Distributions of Stationary Random Measures
	10.2 Palm Distributions of Hyperplane Processes
	Notes for Sect. 10.2
	10.3 Reduced Second Moment Measures
	10.4 The Isotropic Case
	10.5 Pair-Correlation Functions
	Notes for Sect. 10.5
11 Typical Faces and Weighted Faces
	11.1 Typical and Weighted k-Faces
	Notes for Sect. 11.1
	11.2 Faces with Given Directions
	11.3 More General Weights and Second Moments
	Notes for Sect. 11.3
12 Large Cells and Faces
	12.1 A General Version of Kendall\'s Problem
	12.2 Limit Shapes and Special Cases
	Notes for Sect. 12.2
	12.3 Proof of the Main Result
	Notes for Sect. 12.3
	12.4 A Survey of Additional Results
	12.5 Large Faces
13 Cells with a Given Number of Facets
	13.1 The Typical Cell, Conditioned with the Facet Number
	13.2 The Zero Cell, Conditioned with the Facet Number
	13.3 Some Preparatory Lemmas
	13.4 Cells with Many Facets
	Notes for Sect. 13.4
14 Small Cells
	14.1 Subspace-Free Directional Distributions
	Notes for Sect. 14.1
	14.2 Typical Cells with Small Circumradius
	14.3 Typical Cells with Small Inradius
15 The K-Cell Under Increasing Intensities
	15.1 The Possibility of Approximation
	15.2 Some Preparatory Lemmas
	15.3 The Rate of Convergence Under Increasing Intensities
	15.4 Estimation of the Hitting Functional
	15.5 The Mean Width Difference
	Notes for Sect. 15.5
	15.6 The Volume Difference
	Notes for Sect. 15.6
	15.7 The Facet Number
	15.8 The Selection Expectation
	Notes for Sect. 15.8
16 Isotropic Zero Cells
	16.1 Face Numbers of the Zero Cell
	Notes for Sect. 16.1
	16.2 Expected Face Numbers in High Dimensions
	Notes for Sect. 16.2
	16.3 Lower-Dimensional Sections of the Zero Cell
	Notes for Sect. 16.3
	16.4 The Volume of the Zero Cell
	16.5 Variance Inequalities for the Volume of the Zero Cell
	Notes for Sect. 16.5
	16.6 Asymptotic Comparison with Balls of Fixed Volume
	Notes for Sect. 16.6
	16.7 Second Moment of the Volume in the Stationary Case
	Notes for Sect. 16.7
17 Functionals of Poisson Processes and Applications
	17.1 Poisson U-Statistics
	17.2 General Poisson Functionals
	Notes for Sect. 17.2
	17.3 Normal Approximation of Poisson Functionals
	Notes for Sect. 17.3
	17.4 Application to Poisson Flats in Hyperbolic Space
	Notes for Sect. 17.4
18 Appendix: Some Auxiliary Results
	18.1 Mixed Volumes and Intrinsic Volumes
	18.2 Valuations
	18.3 The Volume Product
	18.4 Some Results on Zonoids
	18.5 Proof of Lemma 17.2.5
	18.6 The Characteristic Functional of a Poisson Process
References
Author Index
Subject Index
Notation Index