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Asymptotic Geometric Analysis Part II 1st Edition by Shiri Artstein Avidan, Apostolos Giannopoulos, Vitali D Milman ISBN 9781470463601 1470463601

Name: Asymptotic Geometric Analysis Part II 1st Edition by Shiri Artstein Avidan, Apostolos Giannopoulos, Vitali D Milman ISBN 9781470463601 1470463601
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Asymptotic Geometric Analysis Part II 1st Edition Shiri Artstein-Avidan Digital Instant Download

Author(s): Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

ISBN(s): 9781470463601, 1470463601

Edition: 1

File Details: PDF, 5.93 MB

Year: 2021

Language: English

SKU: EB-37419142 Category: Mathematics - Analysis Tags: Apostolos Giannopoulos, Shiri Artstein-Avidan, Vitali D. Milman

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Asymptotic Geometric Analysis Part II 1st Edition by Shiri Artstein Avidan, Apostolos Giannopoulos, Vitali D Milman – Ebook PDF Instant Download/Delivery: 9781470463601 ,1470463601
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ISBN 10: 1470463601
ISBN 13: 9781470463601
Author: Shiri Artstein Avidan, Apostolos Giannopoulos, Vitali D Milman

Subject categories: • General convexity • Research exposition (monographs, survey articles) pertaining to functional analysis • Normed linear spaces and Banach spaces; Banach lattices • Geometric probability and stochastic geometry • Probabilistic methods in Banach space theory • Geometry and structure of normed linear spaces • Convex sets in n dimensions (including convex hypersurfaces) • Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) • Asymptotic theory of convex bodies • Inequalities and extremum problems involving convexity in convex geometryThis 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.

Asymptotic Geometric Analysis Part II 1st Edition Table of contents:

Chapter 1. Functional inequalities and concentration of measure
1.1. The Poincaré inequality
1.2. Cost induced transforms and concentration
1.3. Logarithmic Sobolev inequality
1.4. Further reading
1.5. Notes and remarks
Chapter 2. Isoperimetric constants of log-concave measures and related problems
2.1. Isotropic log-concave probability measures
2.2. Kannan-Lovász-Simonovits conjecture
2.3. Isoperimetric constants of log-concave probability measures
2.4. Thin-shell estimates and the central limit theorem
2.5. Variance problem and the slicing problem
2.6. Stochastic localization and the KLS conjecture
2.7. Further reading
2.8. Notes and remarks
Chapter 3. Inequalities for Gaussian measures
3.1. Gaussian isoperimetric inequality
3.2. Ehrhard’s inequality
3.3. Gaussian measure of dilates of centrally symmetric convex bodies
3.4. Gaussian correlation inequality
3.5. The B-theorem
3.6. Applications to discrepancy
3.7. Some technical results
3.8. Notes and remarks
Chapter 4. Volume inequalities
4.1. Rearrangement of functions
4.2. Brascamp-Lieb-Luttinger inequality
4.3. The original proof of the Brascamp-Lieb inequality
4.4. Multidimensional versions
4.5. Applications to convex geometry
4.6. Vaaler’s inequality and related results
4.7. Stochastic dominance and geometric inequalities
4.8. Blaschke-Petkantschin formulas
4.9. Further reading
4.10. Notes and remarks
Chapter 5. Local theory of finite dimensional normed spaces: type and cotype
5.1. Type and cotype
5.2. Operator norms
5.3. Maurey’s lemma and duality of entropy
5.4. Spaces with bounded cotype
5.5. Grothendieck’s inequality
5.6. Factorization through a Hilbert space and Kwapien’s theorem
5.7. The complemented subspace problem
5.8. Krivine’s theorem
5.9. Maurey-Pisier theorem
5.10. Stable type p and the dimension of l_{p}^{m} subspaces
5.11. Further reading
5.12. Notes and remarks
Chapter 6. Geometry of the Banach-Mazur compactum
6.1. Diameter of the Banach-Mazur compactum
6.2. Random orthogonal factorizations
6.3. Diameter of the compactum in the non-symmetric case
6.4. Banach-Mazur distance to the cube
6.5. Elton’s theorem
6.6. Spaces with maximal distance to Euclidean space
6.7. Alon-Milman theorem
6.8. Dvoretzky theorem: dependence on ε
6.9. Further reading
6.10. Notes and remarks
Chapter 7. Asymptotic convex geometry and classical symmetrizations
7.1. Random Minkowski symmetrizations
7.2. Minkowski symmetrizations
7.3. Steiner symmetrizations
7.4. Spherical harmonics
7.5. Almost isometric symmetrization
7.6. Notes and remarks
Chapter 8. Restricted invertibility and the Kadison-Singer problem
8.1. Sparse approximations of graphs
8.2. Interlacing polynomials
8.3. Restricted invertibility
8.4. Proportional Dvoretzky-Rogers factorization
8.5. The Kadison-Singer problem
8.6. Further reading
8.7. Notes and remarks
Chapter 9. Functionalization of Geometry
9.1. Extending convex geometry to the log-concave realm
9.2. Functional Duality
9.3. Functional forms of geometric inequalities
9.4. Notes and remarks
Bibliography
Index

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Tags: Shiri Artstein Avidan, Apostolos Giannopoulos, Vitali D Milman, Asymptotic Geometric Analysis