Integral geometry and geometric probability:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2004
|
| Edition: | 2. ed. |
| Series: | Cambridge mathematical library
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 404 S. graph. Darst. |
| ISBN: | 0521523443 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Editor s Statement xi
Foreword xiii
Preface xv
Part I: INTEGRAL GEOMETRY IN THE PLANE
Chapter 1. Convex Sets in the Plane 1
1. Introduction 1
2. Envelope of a Family of Lines 2
3. Mixed Areas of Minkowski 4
4. Some Special Convex Sets 6
5. Surface Area of the Unit Sphere and Volume of the Unit
Ball 9
6. Notes and Exercises 10
Chapter 2. Sets of Points and Poisson Processes in the Plane .... 12
1. Density for Sets of Points 12
2. First Integral Formulas 13
3. Sets of Triples of Points 16
4. Homogeneous Planar Poisson Point Processes 17
5. Notes 20
Chapter 3. Sets of Lines in the Plane 27
1. Density for Sets of Lines 27
2. Lines That Intersect a Convex Set or a Curve 30
3. Lines That Cut or Separate Two Convex Sets 32
4. Geometric Applications 34
5. Notes and Exercises 37
Chapter 4. Pairs of Points and Pairs of Lines 45
1. Density for Pairs of Points 45
V
vi Contents
2. Integrals for the Power of the Chords of a Convex Set. . 46
3. Density for Pairs of Lines 49
4. Division of the Plane by Random Lines 51
5. Notes 58
Chapter 5. Sets of Strips in the Plane 68
1. Density for Sets of Strips 68
2. Buffon s Needle Problem 71
3. Sets of Points, Lines, and Strips 72
4. Some Mean Values 75
5. Notes 77
Chapter 6. The Group of Motions in the Plane: Kinematic Density . . 80
1. The Group of Motions in the Plane 80
2. The Differential Forms on »! 82
3. The Kinematic Density 85
4. Sets of Segments 89
5. Convex Sets That Intersect Another Convex Set .... 93
6. Some Integral Formulas 95
7. A Mean Value; Coverage Problems 98
8. Notes and Exercises 100
Chapter 7. Fundamental Formulas of Poincare and Blaschke 109
1. A New Expression for the Kinematic Density 109
2. Poincare s Formula Ill
3. Total Curvature of a Closed Curve and of a Plane Domain 112
4. Fundamental Formula of Blaschke 113
5. The Isoperimetric Inequality 119
6. Hadwiger s Conditions for a Domain to Be Able to Contain
Another 121
7. Notes 123
Chapter 8. Lattices of Figures 128
1. Definitions and Fundamental Formula 128
2. Lattices of Domains 131
3. Lattices of Curves 134
4. Lattices of Points 135
5. Notes and Exercise 138
Contents v;j
Part II. GENERAL INTEGRAL GEOMETRY
Chapter 9. Differential Forms and Lie Groups 143
1. Differential Forms 143
2. Pfaffian Differential Systems 145
3. Mappings of Differentiable Manifolds 148
4. Lie Groups; Left and Right Translations 149
5. Left Invariant Differential Forms 150
6. Maurer Cartan Equations 152
7. Invariant Volume Elements of a Group: Unimodular
Groups 156
8. Notes and Exercises 160
Chapter 10. Density and Measure in Homogeneous Spaces 165
1. Introduction 165
2. Invariant Subgroups and Quotient Groups 169
3. Other Conditions for the Existence of a Density on Homo¬
geneous Spaces 170
4. Examples 171
5. Lie Transformation Groups 173
6. Notes and Exercises 175
Chapter II. The Affine Groups 179
1. The Groups of Affine Transformations 179
2. Densities for Linear Spaces with Respect to Special Homo¬
geneous Affinities 183
3. Densities for Linear Subspaces with Respect to the Special
Nonhomogeneous Affine Group 187
4. Notes and Exercises 189
Chapter 12. The Group of Motions in £„ 196
1. Introduction 196
2. Densities for Linear Spaces in £„ 199
3. A Differential Formula 200
4. Density for r Planes about a Fixed / Plane 201
5. Another Form of the Density for r Planes in £„ 204
6. Sets of Pairs of Linear Spaces 205
7. Notes 207
viii Contents
Part III. INTEGRAL GEOMETRY IN £„
Chapter 13. Convex Sets in £„ 215
1. Convex Sets and Quermassintegrale 215
2. Cauchy s Formula 217
3. Parallel Convex Sets; Steiner s Formula 220
4. Integral Formulas Relating to the Projections of a Convex
Set on Linear Subspaces 221
5. Integrals of Mean Curvature 222
6. Integrals of Mean Curvature and Quermassintegrale . . . 224
7. Integrals of Mean Curvature of a Flattened Convex Body 227
8. Notes 229
Chapter 14. Linear Subspaces, Convex Sets, and Compact Manifolds . 233
1. Sets of / Planes That Intersect a Convex Set 233
2. Geometric Probabilities 235
3. Crofton s Formulas in £„ 237
4. Some Relations between Densities of Linear Subspaces . . 240
5. Linear Subspaces That Intersect a Manifold 243
6. Hypersurfaces and Linear Spaces 247
7. Notes 248
Chapter 15. The Kinematic Density in £„ 256
1. Formulas on Densities 256
2. Integral of the Volume (Tr + ((_,1(Mrn M ) 258
3. A Differential Formula 260
4. The Kinematic Fundamental Formula 262
5. Fundamental Formula for Convex Sets 267
6. Mean Values for the Integrals of Mean Curvature .... 267
7. Fundamental Formula for Cylinders 270
8. Some Mean Values 272
9. Lattices in £„ 274
10. Notes and Exercise 275
Chapter 16. Geometric and Statistical Applications; Stereology .... 282
1. Size Distribution of Particles Derived from the Size
Distribution of Their Sections 282
2. Intersection with Random Planes 286
3. Intersection with Random Lines 289
4. Notes 290
Contents ix
Part IV. INTEGRAL GEOMETRY IN SPACES OF
CONSTANT CURVATURE
Chapter 17. Noneuclidean Integral Geometry 299
1. The n Dimensional Noneuclidean Space 299
2. The Gauss Bonnet Formula for Noneuclidean Spaces . . 302
3. Kinematic Density and Density for r Planes 304
4. Sets of r Planes That Meet a Fixed Body 309
5. Notes 311
Chapter 18. Crofton s Formulas and the Kinematic Fundamental Formula
in Noneuclidean Spaces 316
1. Crofton s Formulas 316
2. Dual Formulas in Elliptic Space 318
3. The Kinematic Fundamental Formula in Noneuclidean
Spaces 320
4. Steiner s Formula in Noneuclidean Spaces 321
5. An Integral Formula for Convex Bodies in Elliptic Space 322
6. Notes 323
Chapter 19. Integral Geometry and Foliated Spaces; Trends in Integral
Geometry 330
1. Foliated Spaces 330
2. Sets of Geodesies in a Riemann Manifold 331
3. Measure of Two Dimensional Sets of Geodesies 334
4. Measure of (2n — 2) Dimensional Sets of Geodesies . . . 336
5. Sets of Geodesic Segments 338
6. Integral Geometry on Complex Spaces 338
7. Symplectic Integral Geometry 344
8. The Integral Geometry of Gelfand 345
9. Notes 349
Appendix. Differential Forms and Exterior Calculus 353
1. Differential Forms and Exterior Product 353
2. Two Applications of the Exterior Product 357
3. Exterior Differentiation 358
4. Stokes Formula 359
5. Comparison with Vector Calculus in Euclidean Three
Dimensional Space 360
6. Differential Forms over Manifolds 361
Bibliography and References 363
Author Index 395
Subject Index 399
|
| any_adam_object | 1 |
| author | Santaló Sors, Luis Antonio |
| author_facet | Santaló Sors, Luis Antonio |
| author_role | aut |
| author_sort | Santaló Sors, Luis Antonio |
| author_variant | s l a s sla slas |
| building | Verbundindex |
| bvnumber | BV017401672 |
| classification_rvk | SK 370 SK 800 |
| ctrlnum | (OCoLC)470401757 (DE-599)BVBBV017401672 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 0521523443 |
| language | English |
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| spelling | Santaló Sors, Luis Antonio Verfasser aut Integral geometry and geometric probability Luis A. Santaló 2. ed. Cambridge Cambridge Univ. Press 2004 XVII, 404 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical library Géométrie intégrale ram Probabilités géométriques ram connexion inriac densite inriac geometrie differentielle inriac geometrie integrale inriac probabilite geometrique inriac stereologie inriac theoreme limite inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Integralgeometrie (DE-588)4161911-0 gnd rswk-swf Geometrische Wahrscheinlichkeit (DE-588)4156727-4 s DE-604 Integralgeometrie (DE-588)4161911-0 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Geometrie (DE-588)4020236-7 s 2\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010485924&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Santaló Sors, Luis Antonio Integral geometry and geometric probability Géométrie intégrale ram Probabilités géométriques ram connexion inriac densite inriac geometrie differentielle inriac geometrie integrale inriac probabilite geometrique inriac stereologie inriac theoreme limite inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Geometrie (DE-588)4020236-7 gnd Integralgeometrie (DE-588)4161911-0 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4156727-4 (DE-588)4020236-7 (DE-588)4161911-0 |
| title | Integral geometry and geometric probability |
| title_auth | Integral geometry and geometric probability |
| title_exact_search | Integral geometry and geometric probability |
| title_full | Integral geometry and geometric probability Luis A. Santaló |
| title_fullStr | Integral geometry and geometric probability Luis A. Santaló |
| title_full_unstemmed | Integral geometry and geometric probability Luis A. Santaló |
| title_short | Integral geometry and geometric probability |
| title_sort | integral geometry and geometric probability |
| topic | Géométrie intégrale ram Probabilités géométriques ram connexion inriac densite inriac geometrie differentielle inriac geometrie integrale inriac probabilite geometrique inriac stereologie inriac theoreme limite inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Geometrie (DE-588)4020236-7 gnd Integralgeometrie (DE-588)4161911-0 gnd |
| topic_facet | Géométrie intégrale Probabilités géométriques connexion densite geometrie differentielle geometrie integrale probabilite geometrique stereologie theoreme limite Wahrscheinlichkeitsrechnung Geometrische Wahrscheinlichkeit Geometrie Integralgeometrie |
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