Asymptotic Geometric Analysis, Part II

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. Cover Title page Preface to Part II Preface to Part I Notation and background from asymptotic geometric analysis 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 Index Back Cover "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 ini 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"--Back cover "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." Provided by publisher