Topics in integral geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
Singapore u.a.
World Scientific
1994
|
| Series: | Series in pure mathematics
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 236 S. |
| ISBN: | 9810211015 9810211074 |
Staff View
MARC
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| 100 | 1 | |a Ren, De-lin |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Topics in integral geometry |c Ren De-lin |
| 264 | 1 | |a Singapore u.a. |b World Scientific |c 1994 | |
| 300 | |a XVIII, 236 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Series in pure mathematics |v 19 | |
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Record in the Search Index
| _version_ | 1804124393236856832 |
|---|---|
| adam_text | CONTENTS
Foreword
by S. S. Chern v
by C. C. Hstung vii
Chapter I. Basic Properties of Convex Sets 1
1.1 Basic Concepts 1
1.1.1 Convex sets and convex curves 1
1.1.2 Support lines and their existence 1
1.2 Support Functions and Width Functions 2
1.2.1 Generalized normal equations of lines 2
1.2.2 Support function and width function of a convex set 3
1.2.3 Convex curve as envelope of a family of lines 4
1.2.4 An elementary proof of the formula of perimeter 5
1.3 Some Special Convex Sets 7
1.3.1 Convex sets of constant widths 7
1.3.2 Parallel convex sets g
1.4 Mixed Areas of Minkowski 8
1.4.1 Mixed convex sets 8
1.4.2 Mixed areas of Minkowski 8
1.5 Surface Area of the Unit Sphere and Volume of the Unit Ball 10
Chapter II. Measure for Sets of Geometric Elements 11
2.1 Measure for Sets of Points 11
2.1.1 Measure for sets of points 11
2.1.2 Remarks 12
2.1.3 An integral formula 13
2.2 Measure for Sets of Lines 15
2.2.1 Measure for sets of lines 15
2.2.2 Two corollaries 16
2.2.3 Other forms of density for sets of lines 17
2.2.4 Proof of isoperimetric inequality 18
xiii
xiv Contents
2.3 Pairs of Points and Lines 19
2.3.1 Density of pairs of points 19
2.3.2 Integral for the power of chords of a convex set 20
2.3.3 Remarks on integrals for power of chords of a convex set 21
2.3.4 Inequalities for the integrals of the power of chord of a
convex set 22
2.3.5 Density for pairs of lines 22
2.3.6 Crofton s formula 22
2.4 Division of the Plane by Random Lines 23
2.4.1 Division of the convex set by random lines 23
2.4.2 Division of plane by random lines 26
2.4.3 Notes on random division 28
2.5 Sets of Strips in the Plane 28
2.5.1 Density for sets of strips 28
2.5.2 Generalized Buffon s needle problem 29
2.5.3 Further generalizations 29
Chapter III. Fundamental Formulas of Integral Geometry in the Plane 33
3.1 The Group of Motions in the Plane 33
3.1.1 The group of motions in the plane 33
3.1.2 Left and right translations 34
3.1.3 The differential forms on M 35
3.2 The Kinematic Density 36
3.2.1 Left and right invariant 1 forms 36
3.2.2 The kinematic density 37
3.2.3 Geometrical meaning of the kinematic measure 38
3.2.4 Other expressions for the kinematic measure 39
3.3 Poincare s Formula 40
3.3.1 A new expression for the kinematic density 40
3.3.2 Poincare s Formula 42
3.4 The Fundamental Kinematic Formula of Blaschke 43
3.4.1 Total curvature of a closed curve and of a plane domain 43
3.4.2 Fundamental kinematic formula of Blaschke 44
3.4.3 Some immediate consequences of Blaschke s formula 46
Chapter IV. Applications of Integral Geometry in the Plane 48
4.1 The Isoperimetric Inequality 48
4.1.1 The integral geometric proof of the isoperimetric inequality 48
4.1.2 Stronger isoperimetric inequalities 49
4.1.3 An upper limit for isoperimetric deficit 51
4.2 Conditions for One Domain to be Able to Contain Another 52
4.2.1 Sufficient conditions for one domain to contain another 53
Contents xv
4.2.2 Hadwiger s conditions 54
4.2.3 Some consequences 55
4.2.4 Pseudodiameter of a domain 55
4.3 Kinematic Measure of a Segment of Fixed Length within a
Convex Domain 56
4.3.1 Problem 56
4.3.2 A formula for kinematic measure of a segment within a
convex domain 56
4.3.3 Generalized support function and the restricted chord function 58
4.3.4 A formula for m(/) expressed by the generalized support
function 58
4.3.5 The measure m(l) for a.rectangle 63
4.4 Applications of m(l) to Geometric Probability 63
4.4.1 Laplace extension of Buffon problem 63
4.4.2 Applications of m(l) to generalized Buffon problem 64
4.4.3 Formulas of m(/) for equilateral triangle and regular hexagon 66
4.5 A Problem Related to the Statistical Estimating of n 67
4.5.1 Equidistant parallel lines 67
4.5.2 Grid of rectangles, independence 67
4.5.3 Efficiency analysis 68
4.6 Random Convex Set in a Lattice of Parallelograms 70
4.6.1 Width functions of convex sets 70
4.6.2 Distribution of the number of intersections 72
4.6.3 Hitting probabilities 76
4.6.4 Independence 77
Chapter V. Foundations of Integral Geometry in Homogeneous Spaces 80
5.1 Differentiable Manifolds 80
5.5.1 Topological space 80
5.5.2 Topologicl manifolds and differentiable manifolds 81
5.5.3 Differentiable functions and mappings 82
5.2 Vector Fields on a Manifold 82
5.2.1 Tangent spaces and vector fields 82
5.2.2 The differential of a mapping between manifolds 83
5.2.3 Local expressions of vector fields 83
5.3 Differential Forms and Exterior Differentiation 84
5.3.1 Covector fields 84
5.3.2 Tensor fields 85
5.3.3 The exterior algebra on a manifold 86
5.3.4 Exterior differentiation 89
5.3.5 Expression of exterior differential by the usual differential 90
xvi Contents
5.4 Integral Manifolds and Pfaffian Systems 91
5.4.1 Integral manifolds 91
5.4.2 Pfaffian system 92
5.5 Lie Groups and Kinematic Density of a Lie Group 95
5.5.1 Lie group 95
5.5.2 Left and right translations 95
5.5.3 Left invariant differential forms 97
5.5.4 Structure equations and structure constants for a Lie group 99
5.5.5 Kinematic density for Lie group 105
5.6 Density and Measure in Homogeneous Space 110
5.6.1 Actions of a Lie group on a manifold, homogeneous space 110
5.6.2 Conditions for the existence of invariant density on G/H 111
5.6.3 Weil s condition 113
5.6.4 Normal subgroups 115
5.6.5 Chern s conditions 115
5.6.6 Stable subgroups 116
5.7 A Brief Review of Integral Geometry in the Plane 117
Chapter VI. Integral Geometry in En 121
6.1 The Group of Motions in En 121
6.1.1 The group of motions and its structure equations 121
6.1.2 Invariant volume elements of the group of motions and
its subgroups 123
6.2 The Density of r Planes in En 125
6.2.1 The density of r planes 125
6.2.2 Density of r planes about a fixed g plane 126
6.2.3 The volume of the Grassmann manifold G ,n_r 127
6.2.4 Another form of the density of r planes in En 128
6.2.5 The density of the pairs of hyperplanes 129
6.2.6 The density of the pairs (Lr, LQ) 130
6.2.7 A density formula for points and flats 131
6.2.8 The density of the pairs of non intersecting flats 133
6.3 Convex Sets in En 134
6.3.1 Convex sets and mixed volumes 134
6.3.2 Quermassintegrale 136
6.3.3 Cauchy s formula and Steiner s formula 140
6.3.4 The mean value of W!(K n_r) 142
6.4 Integrals of Mean Curvature 143
6.4.1 Integrals of mean curvatures of hypersurfaces in En 143
6.4.2 Relations between integrals of mean curvature and
quermassintegrale 145
6.4.3 Some particular results 146
Contents xvii
6.4.4 Integrals of mean curvature of a flattened convex body 148
6.5 Sets of r Planes that Intersect a Convex Set 149
6.5.1 Measures of the sets of r planes that intersect a convex set 149
6.5.2 The integral of w£ (Lr ( 1 K) over the set
{Lr:LrnKjtH,} 150
6.5.3 Crofton and Hadwiger s formula 151
6.6 Chern s Formulas 152
6.6.1 A density formula 152
6.6.2 The integral of Ar+« 153
6.6.3 Chern s formula 154
6.7 Santalo s Formula 156
6.7.1 A density formula 156
6.7.2 Santalo s formula 157
6.8 Integral of the Volume of the Intersection of Two Manifolds 159
6.8.1 A density formula 159
6.8.2 Another density formula 160
6.8.3 Integral of the volume rr+,_n(Mr n A/«) 161
6.9 Chern Yen s Kinematic Fundamental Formula 162
6.9.1 An important density formula 162
6.9.2 Chern and Yen s kinematic fundamental formula 164
6.9.3 Kinematic fundamental formula for convex sets 170
6.9.4 Integral formulas for the integrals of mean curvature 171
Chapter VII. Applications of Integral Geometry 173
7.1 Introduction to Integral Geometry in ffi3 173
7.1.1 Group of the motions in K3 173
7.1.2 Densities for lines and planes in M3 175
7.1.3 Some fundamental formulas 177
7.1.4 Moving cylinders 179
7.2 Elements of Stereology 181
7.2.1 Objects of study in stereology 181
7.2.2 General discussion 181
7.2.3 Intersection with random planes 183
7.2.4 Spherical particles 186
7.2.5 Nearly spherical particles 187
7.2.6 Intersection with random lines 188
7.2.7 Estimation for the number of crystals 189
7.3 Sufficient Conditions for One Domain to Contain Another 190
7.3.1 A density formula 191
7.3.2 A sufficient condition for one convex body to contain another 192
7.3.3 Sufficient conditions for one domain to contain another in M3 195
7.3.4 Analogues of Hadwiger s theorem in Kn (n 4) 200
xviii Contents
7.4 Kinematic Measure of a Segment of Fixed Length Within a
Convex Body 201
7.4.1 A general formula for the kinematic measure of a segment
of fixed length within a convex body 201
7.4.2 Transformation of formula 203
7.4.3 Formulas of m(/) for a cylinder 206
7.4.4 Kinematic measure m(l) for a right parallelepiped in K3
and Buffon problem 208
7.4.5 Kinematic measure m(/) for a right parallelepiped in K
and Buffon problem 212
7.5 Unified Inequalities Relating to Integrals for the Power of Chords 213
7.5.1 Inequalities relating to integrals for the power of
chords in I3 214
7.5.2 Applications to geometric probability 217
7.5.3 Inequalities of integrals for the power of chords in ]Rn 218
7.5.4 Integral geometric inequalities for moments 223
7.5.5 Pairs of non intersecting random flats meeting two
convex bodies 227
7.6 Inequalities Characterizing Simplices 228
7.6.1 Lemmas 228
7.6.2 Inequalities characterizing simplices 230
Index 234
|
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| author | Ren, De-lin |
| author_facet | Ren, De-lin |
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| author_sort | Ren, De-lin |
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| discipline | Mathematik |
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| id | DE-604.BV010012948 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:44:57Z |
| institution | BVB |
| isbn | 9810211015 9810211074 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006639546 |
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| owner_facet | DE-91G DE-BY-TUM DE-12 DE-703 |
| physical | XVIII, 236 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | World Scientific |
| record_format | marc |
| series | Series in pure mathematics |
| series2 | Series in pure mathematics |
| spelling | Ren, De-lin Verfasser aut Topics in integral geometry Ren De-lin Singapore u.a. World Scientific 1994 XVIII, 236 S. txt rdacontent n rdamedia nc rdacarrier Series in pure mathematics 19 Integralgeometrie (DE-588)4161911-0 gnd rswk-swf Integralgeometrie (DE-588)4161911-0 s DE-604 Series in pure mathematics 19 (DE-604)BV000016845 19 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006639546&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ren, De-lin Topics in integral geometry Series in pure mathematics Integralgeometrie (DE-588)4161911-0 gnd |
| subject_GND | (DE-588)4161911-0 |
| title | Topics in integral geometry |
| title_auth | Topics in integral geometry |
| title_exact_search | Topics in integral geometry |
| title_full | Topics in integral geometry Ren De-lin |
| title_fullStr | Topics in integral geometry Ren De-lin |
| title_full_unstemmed | Topics in integral geometry Ren De-lin |
| title_short | Topics in integral geometry |
| title_sort | topics in integral geometry |
| topic | Integralgeometrie (DE-588)4161911-0 gnd |
| topic_facet | Integralgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006639546&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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