eBook ISBN:  9781470423452 
Product Code:  SURV/202.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470423452 
Product Code:  SURV/202.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsMathematical Surveys and MonographsVolume: 202; 2015; 451 ppMSC: Primary 52; 46; 60; 28; 68
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 nontrivial 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, Dvoretzkytype theorems, volume distribution in convex bodies, and more.
ReadershipGraduate students and research mathematicians interested in geometric functional analysis and applications.

Table of Contents

Chapters

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 BrunnMinkowski inequality

Chapter 9. Gaussian approach

Chapter 10. Volume distribution in convex bodies

Appendix A. Elementary convexity

Appendix B. Advanced convexity


Additional Material

Reviews

[T]he book makes much more accessible many essential recent and classical results from modern asymptotic geometric analysis and convexity and is an outstanding source for a course on this subject.
Artem Zvavitch, Mathematical Reviews 
This book (and its second volume) has become the essential reference and main source for the theory of asymptotic geometric analysis.
Maria A Hernández Cifre, Zentralblatt MATH


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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 nontrivial 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, Dvoretzkytype theorems, volume distribution in convex bodies, and more.
Graduate students and research mathematicians interested in geometric functional analysis and applications.

Chapters

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 BrunnMinkowski inequality

Chapter 9. Gaussian approach

Chapter 10. Volume distribution in convex bodies

Appendix A. Elementary convexity

Appendix B. Advanced convexity

[T]he book makes much more accessible many essential recent and classical results from modern asymptotic geometric analysis and convexity and is an outstanding source for a course on this subject.
Artem Zvavitch, Mathematical Reviews 
This book (and its second volume) has become the essential reference and main source for the theory of asymptotic geometric analysis.
Maria A Hernández Cifre, Zentralblatt MATH