Introduction to Geometric Probability
by
Table of Contents
Preface iv
Using this book vi
1 The Buffon needle problem 1
1.1 The classical problem 1
1.2 The space of lines 2
1.3 Notes 4
2 Valuation and integral 5
2.1 Valuations 5
2.2 Groemer's integral theorem 7
2.3 Notes 10
3 A discrete lattice 11
3.1 Subsets of a finite set 11
3.2 Valuations on a simplicial complex 18
3.3 A discrete analogue of Helly's theorem 25
3.4 Notes 25
4 The intrinsic volumes for parallelotopes 27
4.1 The lattice of parallelotopes 27
4.2 Invariant valuations on parallelotopes 31
4.3 Notes 37
5 The lattice of polyconvex sets 38
5.1 Polyconvex sets 38
5.2 The Euler characteristic 41
5.3 Helly's theorem 46
5.4 Lutwak's containment theorem 48
5.5 Cauchy's surface area formula 50
5.6 Notes 53
6 Invariant measures on Grassmannians 54
6.1 The lattice of subspaces 54
6.2 Computing the flag coefficients 56
6.3 Properties of the flag coefficients 62
6.4 A continuous analogue of Sperner's theorem 66
6.5 A continuous analogue of Meshalkin's theorem 69
6.6 Helly's theorem for subspaces 74
6.7 Notes 75
7 The intrinsic volumes for polyconvex sets 78
7.1 The affine Grassmannian 78
7.2 The intrinsic volumes and Hadwiger's formula 79
7.3 An Euler relation for the intrinsic volumes 84
7.4 The mean projection formula 85
7.5 Notes 86
8 A characterization theorem for volume 88
8.1 Simple valuations on polyconvex sets 88
8.2 Even and odd valuations 95
8.3 The volume theorem 98
8.4 The normalization of the intrinsic volumes 100
8.5 Lattice points and volume 101
8.6 Remarks on Hilbert's third problem 103
8.7 Notes 105
9 Hadwiger's characterization theorem 107
9.1 A proof of Hadwiger's characterization theorem 107
9.2 The intrinsic volumes of the unit ball 108
9.3 Crofton's formula 111
9.4 The mean projection formula revisited 113
9.5 Mean cross-sectional volume 116
9.6 The Buffon needle problem revisited 117
9.7 Intrinsic volumes on products 118
9.8 Computing the intrinsic volumes 122
9.9 Notes 127
10 Kinematic formulas for polyconvex sets 132
10.1 The principal kinematic formula 132
10.2 Hadwiger's containment theorem 135
10.3 Higher kinematic formulas 137
10.4 Notes 138
11 Polyconvex sets in the sphere 140
11.1 Convexity in the sphere 140
11.2 A characterization for spherical area 142
11.3 Invariant valuations on spherical polytopes 145
11.4 Spherical kinematic formulas 147
11.5 Remarks on higher dimensional spheres 150
11.6 Notes 151
References 153
Index of symbols 160
Index 162
Return to the Summary
Return to my Publications Page
Return to my Home Page