Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santal? and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems.
Author(s): Daniel A. Klain, Gian-Carlo Rota
Series: Lezioni Lincee
Publisher: CUP
Year: 1997
Language: English
Pages: 191
Cover......Page 1
About......Page 2
Series: Lezioni Lincee......Page 4
Goto 4 /FitH 555521596548......Page 5
Contents......Page 6
Preface......Page 10
Using this book......Page 13
1.1 The classical problem......Page 14
1.2 The space of lines......Page 16
1.3 Notes......Page 18
2.1 Valuations......Page 19
2.2 Groemer's integral theorem......Page 21
2.3 Notes......Page 24
3.1 Subsets of a finite set......Page 26
3.2 Valuations on a simplicial complex......Page 34
3.3 A discrete analogue of Helly's theorem......Page 41
3.4 Notes......Page 42
4.1 The lattice of parallelotopes......Page 43
4.2 Invariant valuations on parallelotopes......Page 48
4.3 Notes......Page 54
5.1 Polyconvex sets......Page 55
5.2 The Euler characteristic......Page 59
5.3 Helly's theorem......Page 63
5.4 Lutwak's containment theorem......Page 67
5.5 Cauchy's surface area formula......Page 68
5.6 Notes......Page 71
6.1 The lattice of subspaces......Page 73
6.2 Computing the flag coefficients......Page 76
6.3 Properties of the flag coefficients......Page 83
6.4 A continuous analogue of Sperner's theorem......Page 86
6.5 A continuous analogue of Meshalkin's theorem......Page 90
6.6 Helly's theorem for subspaces......Page 94
6.7 Notes......Page 96
7.1 The affine Grassmannian......Page 99
7.2 The intrinsic volumes and Hadwiger's formula......Page 100
7.3 An Euler relation for the intrinsic volumes......Page 106
7.4 The mean projection formula......Page 107
7.5 Notes......Page 108
8.1 Simple valuations on polyconvex sets......Page 111
8.2 Even and odd valuations......Page 119
8.3 The volume theorem......Page 122
8.4 The normalization of the intrinsic volumes......Page 124
8.5 Lattice points and volume......Page 125
8.6 Remarks on Hilbert's third problem......Page 128
8.7 Notes......Page 130
9.1 A proof of Hadwiger's characterization theorem......Page 131
9.2 The intrinsic volumes of the unit ball......Page 133
9.3 Crofton's formula......Page 136
9.4 The mean projection formula revisited......Page 138
9.5 Mean cross-sectional volume......Page 141
9.6 The Buffon needle problem revisited......Page 142
9.7 Intrinsic volumes on products......Page 143
9.8 Computing the intrinsic volumes......Page 148
9.9 Notes......Page 153
10.1 The principal kinematic formula......Page 159
10.2 Hadwiger's containment theorem......Page 163
10.3 Higher kinematic formulas......Page 165
10.4 Notes......Page 166
11.1 Convexity in the sphere......Page 167
11.2 A characterization for spherical area......Page 169
11.3 Invariant valuations on spherical polytopes......Page 172
11.4 Spherical kinematic formulas......Page 175
11.5 Remarks on higher dimensional spheres......Page 177
11.6 Notes......Page 179
Bibliography......Page 181
Index of symbols......Page 187
Index......Page 189
