Theory of Convex Bodies

Title page......Page 1
Copyright page......Page 2
Table of Contents......Page 3
Translators' Preface......Page 7
Foreword......Page 8
Preliminary Remarks on n-dimensional Geometry......Page 11
1. Convex Sets, Bodies and Cones......Page 13
2. Bounding Planes and Support Planes of a Closed Set......Page 14
3. The Convex Hull of a Closed Set......Page 15
4. Support Properties of Convex Bodies......Page 16
5. Mass Distributions and their Centroids......Page 18
6. Representations of the Centroid of the Convex Hull......Page 19
7. Generation of the Convex Hull by Chords......Page 20
8. Centroids of Subbodies cut off by Hyperplanes and of Cross Sections......Page 21
9. Singular Boundary Points and Support Planes. Projections Cones and the Normal Cones. Vertices and Face Points......Page 25
10. Extreme Boundary Points and Support Planes......Page 26
11. Convex Polyhedra......Page 27
12. Cap Bodies and Tangential Bodies......Page 28
13. Convex Functions and their Directional Derivatives......Page 30
14. The Distance Function of a Convex Body......Page 33
15. The Support Function of a Convex Body......Page 36
16. Representation of the Boundary Points of a Convex Body by Support Functions......Page 37
17. The Determination of a Convex Body by the Support Function......Page 39
18. Polar Bodies......Page 41
20. Linear Combinations of Convex Bodies......Page 42
22. Behavior of Projections and Boundary Points under Linear Combinations......Page 44
24. Linear and Concave Families of Convex Bodies......Page 46
26. The Support Functions of Convergent Sequences of Bodies. The Function Space of Support Functions......Page 48
27. Approximation by Convex Polyhedra and Analytically Bounded Convex Bodies......Page 49
29. The Volume of a Body in a Linear Family. Mixed Volumes......Page 52
30. Cross Sectional Measure. Projection Bodies......Page 59
31. The Surface Area of a Convex Body......Page 61
32. Cauchy's Surface Area Formula. Cross Sectional Measure Integrals......Page 63
33. Breadth, Diameter, Width......Page 66
34. Centroids and Other Special Points of a Convex Body......Page 67
35. Circumscribed and Inscribed Balls. Minimal Annulus and other Related Figures......Page 69
36. Formulas in Terms of the Coordinates of Points......Page 71
37. Representation of Mixed Volumes in Terms of Support Functions......Page 73
38. Curvature Functions and Curvature Integrals. Relative Differential Geometry......Page 77
39. Special Formulas. Geometric Probability Theory for Convex Bodies......Page 81
40. Steiner Symmetrization and Annular Symmetrization......Page 86
41. Schwarz Rounding. Blaschke's Proof of the Brunn-Minkowski Theorem......Page 87
42. Central Symmetrization and Related Ideas......Page 89
43. Generalities Concerning Extremal Problems......Page 91
44. Inequalities between two Quantities......Page 92
45. Inequalities involving more than two Quantities for a Region of the Plane......Page 97
46. Inequalities among several Quantities of Convex Bodies......Page 100
47. Coverings......Page 102
48. The Brunn-Minkowski Theorem......Page 104
49. Minkowski's Inequalities......Page 107
50. Refinements of the Brunn-Minkowski Theorem and of the Minkowski Inequalities......Page 110
51. More about the Case of the Plane......Page 114
52. More about Space. Hubert's Proof of the Minkowski Inequalities......Page 116
53. The Volume of a Vector Body......Page 121
54. Estimates of Cross Sectional Measure Integrals in Terms of Width and Diameter......Page 122
55. The Surface Area of the Bodies of a Linear Family......Page 123
56. Special Cases of the Minkowski Inequalities......Page 125
57. The Isoperimetric Problem......Page 127
58. Continuously Curved Bodies......Page 131
59. Uniqueness Theorems......Page 132
60. Existence Theorems......Page 135
61. Characterizing Properties......Page 142
62. Convex Bodies with Center and Lattice Points......Page 143
63. Characterizing and Other Properties......Page 145
64. Complete Sets......Page 146
65. Orbiforms......Page 148
66. Extremal Problems for Orbiforms. . . t......Page 150
67. Spheroforms......Page 154
68. Related Classes of Convex Bodies......Page 157
69. Disk and Ball......Page 160
70. Ellipse and Ellipsoid......Page 161
71. Curvature Properties of Convex Curves. The Four Vertex Theorem and Related Matters......Page 163
72. Surfaces of Positive Gaussian Curvature. Bending Questions......Page 164
Bibliography......Page 168
Cover......Page 183