Asymptotic Geometric Analysis, Part I
About this Title
Shiri Artstein-Avidan, Tel Aviv University, Tel Aviv, Israel, Apostolos Giannopoulos, University of Athens, Athens, Greece and Vitali D. Milman, Tel Aviv University, Tel Aviv, Israel
Publication: Mathematical Surveys and Monographs
Publication Year: 2015; Volume 202
ISBNs: 978-1-4704-2193-9 (print); 978-1-4704-2345-2 (online)
MathSciNet review: MR3331351
MSC: Primary 52A21; Secondary 28Axx, 46-02, 46Bxx, 52A23, 52A40
The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an “isomorphic” point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the “isomorphic isoperimetric inequalities” which led to the discovery of the “concentration phenomenon”, one of the most powerful tools of the theory, responsible for many counterintuitive results.
A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple “possibilities”, so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality.
The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.
Graduate students and research mathematicians interested in geometric functional analysis and applications.
Table of Contents
- Chapter 1. Convex bodies: Classical geometric inequalities
- Chapter 2. Classical positions of convex bodies
- Chapter 3. Isomorphic isoperimetric inequalities and concentration of measure
- Chapter 4. Metric entropy and covering numbers estimates
- Chapter 5. Almost Euclidean subspaces of finite dimensional normed spaces
- Chapter 6. The $\ell $-position and the Rademacher projection
- Chapter 7. Proportional theory
- Chapter 8. $M$-position and the reverse Brunn-Minkowski inequality
- Chapter 9. Gaussian approach
- Chapter 10. Volume distribution in convex bodies
- Appendix A. Elementary convexity
- Appendix B. Advanced convexity