**Mathematical Surveys and Monographs**

Volume: 261;
2021;
645 pp;
Softcover

MSC: Primary 52; 46; 60;

**Print ISBN: 978-1-4704-6360-1
Product Code: SURV/261**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

**Electronic ISBN: 978-1-4704-6777-7
Product Code: SURV/261.E**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

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#### Supplemental Materials

# Asymptotic Geometric Analysis, Part II

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

This book is a continuation of Asymptotic
Geometric Analysis, Part I, which was published as volume 202 in
this series.

Asymptotic geometric analysis studies properties of geometric
objects, such as normed spaces, convex bodies, or convex functions,
when the dimensions of these objects increase to infinity. The
asymptotic approach reveals many very novel phenomena which influence
other fields in mathematics, especially where a large data set is of
main concern, or a number of parameters which becomes uncontrollably
large. One of the important features of this new theory is in
developing tools which allow studying high parametric families.

Among the topics covered in the book are measure concentration,
isoperimetric constants of log-concave measures, thin-shell estimates,
stochastic localization, the geometry of Gaussian measures, volume
inequalities for convex bodies, local theory of Banach spaces, type
and cotype, the Banach-Mazur compactum, symmetrizations, restricted
invertibility, and functional versions of geometric notions and
inequalities.

#### Readership

Graduate students and researchers interested in analysis and geometry of high dimensional spaces.

#### Table of Contents

# Table of Contents

## Asymptotic Geometric Analysis, Part II

- Cover Cover11
- Title page iii4
- Preface to Part II ix10
- Preface to Part I xix20
- Notation and background from asymptotic geometric analysis xxxiii34
- Chapter 1. Functional inequalities and concentration of measure 140
- Chapter 2. Isoperimetric constants of log-concave measures and related problems 67106
- 2.1. Isotropic log-concave probability measures 69108
- 2.2. Kannan-Lovász-Simonovits conjecture 71110
- 2.3. Isoperimetric constants of log-concave probability measures 80119
- 2.4. Thin-shell estimates and the central limit theorem 91130
- 2.5. Variance problem and the slicing problem 97136
- 2.6. Stochastic localization and the KLS conjecture 102141
- 2.7. Further reading 111150
- 2.8. Notes and remarks 115154

- Chapter 3. Inequalities for Gaussian measures 121160
- 3.1. Gaussian isoperimetric inequality 123162
- 3.2. Ehrhard’s inequality 133172
- 3.3. Gaussian measure of dilates of centrally symmetric convex bodies 139178
- 3.4. Gaussian correlation inequality 142181
- 3.5. The 𝐵-theorem 151190
- 3.6. Applications to discrepancy 157196
- 3.7. Some technical results 163202
- 3.8. Notes and remarks 178217

- Chapter 4. Volume inequalities 185224
- 4.1. Rearrangement of functions 187226
- 4.2. Brascamp-Lieb-Luttinger inequality 189228
- 4.3. The original proof of the Brascamp-Lieb inequality 194233
- 4.4. Multidimensional versions 197236
- 4.5. Applications to convex geometry 203242
- 4.6. Vaaler’s inequality and related results 216255
- 4.7. Stochastic dominance and geometric inequalities 227266
- 4.8. Blaschke-Petkantschin formulas 231270
- 4.9. Further reading 236275
- 4.10. Notes and remarks 246285

- Chapter 5. Local theory of finite dimensional normed spaces: type and cotype 257296
- 5.1. Type and cotype 259298
- 5.2. Operator norms 265304
- 5.3. Maurey’s lemma and duality of entropy 281320
- 5.4. Spaces with bounded cotype 286325
- 5.5. Grothendieck’s inequality 291330
- 5.6. Factorization through a Hilbert space and Kwapien’s theorem 295334
- 5.7. The complemented subspace problem 297336
- 5.8. Krivine’s theorem 302341
- 5.9. Maurey-Pisier theorem 323362
- 5.10. Stable type 𝑝 and the dimension of ℓ_{𝑝}^{𝑚} subspaces 333372
- 5.11. Further reading 344383
- 5.12. Notes and remarks 357396

- Chapter 6. Geometry of the Banach-Mazur compactum 365404
- 6.1. Diameter of the Banach-Mazur compactum 367406
- 6.2. Random orthogonal factorizations 373412
- 6.3. Diameter of the compactum in the non-symmetric case 380419
- 6.4. Banach-Mazur distance to the cube 383422
- 6.5. Elton’s theorem 396435
- 6.6. Spaces with maximal distance to Euclidean space 400439
- 6.7. Alon-Milman theorem 401440
- 6.8. Dvoretzky theorem: dependence on 𝜖 407446
- 6.9. Further reading 412451
- 6.10. Notes and remarks 422461

- Chapter 7. Asymptotic convex geometry and classical symmetrizations 429468
- Chapter 8. Restricted invertibility and the Kadison-Singer problem 481520
- Chapter 9. Functionalization of Geometry 539578
- Bibliography 595634
- Index 629668
- Index 639678
- Back Cover Back Cover1686