**Mathematical Surveys and Monographs**

Volume: 202;
2015;
451 pp;
Hardcover

MSC: Primary 52; 46; 60; 28; 68;

Print ISBN: 978-1-4704-2193-9

Product Code: SURV/202

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-2345-2
Product Code: SURV/202.E**

List Price: $110.00

Individual Member Price: $88.00

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#### Supplemental Materials

# Asymptotic Geometric Analysis, Part I

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*Shiri Artstein-Avidan; Apostolos Giannopoulos; Vitali D. Milman*

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.

#### Table of Contents

# Table of Contents

## Asymptotic Geometric Analysis, Part I

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Chapter 1. Convex bodies: Classical geometric inequalities 122
- Chapter 2. Classical positions of convex bodies 4768
- Chapter 3. Isomorphic isoperimetric inequalities and concentration of measure 79100
- Chapter 4. Metric entropy and covering numbers estimates 131152
- Chapter 5. Almost Euclidean subspaces of finite dimensional normed spaces 161182
- Chapter 6. The ℓ-position and the Rademacher projection 203224
- Chapter 7. Proportional theory 233254
- Chapter 8. 𝑀-position and the reverse Brunn-Minkowski inequality 257278
- Chapter 9. Gaussian approach 287308
- Chapter 10. Volume distribution in convex bodies 315336
- Chapter 11. Elementary convexity 369390
- Chapter 12. Advanced convexity 389410
- Bibliography 415436
- Subject index 439460
- Author index 447468
- Back Cover Back Cover1473

#### Readership

Graduate students and research mathematicians interested in geometric functional analysis and applications.

#### 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