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Asymptotic Geometric Analysis, Part I
 
Shiri Artstein-Avidan Tel Aviv University, Tel Aviv, Israel
Apostolos Giannopoulos University of Athens, Athens, Greece
Vitali D. Milman Tel Aviv University, Tel Aviv, Israel
Asymptotic Geometric Analysis, Part I
eBook ISBN:  978-1-4704-2345-2
Product Code:  SURV/202.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Asymptotic Geometric Analysis, Part I
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Asymptotic Geometric Analysis, Part I
Shiri Artstein-Avidan Tel Aviv University, Tel Aviv, Israel
Apostolos Giannopoulos University of Athens, Athens, Greece
Vitali D. Milman Tel Aviv University, Tel Aviv, Israel
eBook ISBN:  978-1-4704-2345-2
Product Code:  SURV/202.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 2022015; 451 pp
    MSC: 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 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.

    Readership

    Graduate 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 Brunn-Minkowski inequality
    • Chapter 9. Gaussian approach
    • Chapter 10. Volume distribution in convex bodies
    • Appendix A. Elementary convexity
    • Appendix B. Advanced convexity
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2022015; 451 pp
MSC: 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 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.

Readership

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 Brunn-Minkowski 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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