Convex bodies are at once simple and amazingly rich in structure. While the classical results go back many decades, during the past ten years the integral geometry of convex bodies has undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This collection arises from an MSRI program held in the Spring of 1996, involving researchers in classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis. It is representative of the best research in a very active field that brings together ideas from several major strands in mathematics.
Author(s): Keith M. Ball, Vitali Milman
Series: MSRI
Publisher: CUP
Year: 1998
Language: English
Pages: 250
Contents......Page 1
The Convex Geometry And Geometric Analysis Program Msri, Spring 1996......Page 3
1. Introduction And Statement Of Main Results......Page 15
2. Preliminaries......Page 16
3. Proofs Of Theorems 1 And 3......Page 19
4. Polynomial Valuations......Page 22
1. Related De Nitions And Formulation Of The Gromov–milman Theorem......Page 31
2. Rohlin’s Theory......Page 32
3. Convex Restrictions Of Measures......Page 34
4. Convex Partitions......Page 36
5. Proof Of Theorem 1.1......Page 40
1. Introduction......Page 43
2. Notation And Basic Results......Page 45
3. Derivative Securities With Log-concave Payo Functions......Page 48
4. Extremal Properties Of Calls......Page 52
5. Extremal Properties Of Puts......Page 62
Random Points In Isotropic Convex Sets......Page 67
Threshold Intervals Under Group Symmetries......Page 73
1. Introduction......Page 79
2. Two-dimensional Sections......Page 82
3. High-dimensional Sections......Page 86
Isotropic Constants Of Schatten Class Spaces......Page 91
On The Stability Of The Volume Radius......Page 95
1. Introduction......Page 103
2. A Polytope Approximation Of The Unit Ball Of `n......Page 104
3. An Algorithm For Calculating The Norm......Page 106
4. Approximating `n......Page 107
5. Limits Of St -spaces......Page 115
6. Two Results About St -spaces......Page 118
7. Suggestions For Further Research......Page 121
1. Introduction......Page 125
2. Construction Of The Subspace And Operator Into X......Page 126
3. Further Questions......Page 128
Another Low-technology Estimate In Convex Geometry......Page 131
On The Equivalence Between Geometric And Arithmetic Means For Log-concave Measures......Page 137
On The Constant In The Reverse Brunn–minkowski Inequality For P-convex Balls......Page 143
The Extension Of The Finite-dimensional Version Of Krivine’s Theorem To Quasi-normed Spaces......Page 153
A Note On Gowers’ Dichotomy Theorem......Page 163
An “isomorphic” Version Of Dvoretzky’s Theorem, Ii......Page 173
1. Preliminaries......Page 179
2. Asymptotic Versions Of Operators......Page 183
3. Asymptotic Versions Of Operator Ideals......Page 187
1. Introduction And Notation......Page 195
2. Basic Inequalities......Page 196
3. Metric Entropy Of The Grassmann Manifold......Page 198
1. De Nition Of Curvature......Page 203
2. Curvature Of Graphs......Page 206
1. Introduction And Notation......Page 213
2. The Proof......Page 214
1. Introduction......Page 217
2. The Floating Body......Page 218
3. The Illumination Body......Page 232
1. Introduction......Page 245
2. The Case Where The Boundary Of K Is A C2......Page 246
