Geometric Aspects of Functional Analysis Israel Seminar (GAFA) 2017-2019 Volume II

• Main Authors: Klartag, Bo'az, Milman, Emanuel
• Format: eBook
• Language: English
• Published: Cham Springer International Publishing AG 2020; Springer International Publishing
• Edition: 1
• Series: Lecture Notes in Mathematics
• Subjects: Convex and Discrete Geometry; Functional Analysis; Functional analysis-Congresses; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes
• ISBN: 9783030467616; 3030467619
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/978-3-030-46762-3

Table of Contents:

• Intro -- Preface -- Jean Bourgain: In Memoriam -- Contents Overview for Volume I -- Contents -- 1 A Generalized Central Limit Conjecture for Convex Bodies -- 1.1 Introduction -- 1.2 Preliminaries -- 1.2.1 Notation and Definitions -- 1.2.2 Stochastic Calculus -- 1.2.3 Log-Concave Functions -- 1.2.4 Distance Between Probability Measures -- 1.2.5 Matrix Inequalities -- 1.2.6 From Generalized CLT to Third Moment Bound -- 1.3 Stochastic Localization -- 1.3.1 The Process and Its Basic Properties -- 1.3.2 Bounding the KLS Constant -- 1.3.3 Bounding the Potential -- 1.4 From Third Moment Bound to KLS -- 1.4.1 Tensor Inequalities -- 1.4.2 Derivatives of the Potential -- 1.4.3 Bounding the Potential -- 1.4.4 Proof of Theorem 1.9 -- 1.5 From KLS to Generalized CLT -- 1.5.1 Connection to Classical CLT for Convex Sets -- Appendix 1: Missing Proofs in Sect.1.2.4 -- Appendix 2: Missing Proofs in Sect.1.4.1 -- References -- 2 The Lower Bound for Koldobsky's Slicing Inequality via Random Rounding -- 2.1 Introduction -- 2.2 The Random Rounding and the Net Construction -- 2.3 The Key Estimate -- 2.3.1 Asymptotic Estimates -- 2.3.2 Union Bound -- 2.3.3 An Application of Random Rounding and Conclusion of the Proof of the Proposition 2.3.1 -- 2.4 Proof of Theorem 2.1.1 -- 2.5 Further Applications -- 2.5.1 Comparison via the Hilbert-Schmidt Norm for Arbitrary Matrices -- 2.5.2 Covering Spheres with Strips -- References -- 3 Two-Sided Estimates for Order Statistics of Log-Concave Random Vectors -- 3.1 Introduction and Main Results -- 3.2 Exponential Concentration -- 3.3 Sums of Largest Coordinates of Log-Concave Vectors -- 3.4 Vectors Satisfying Condition (3.3) -- 3.5 Lower Estimates for Order Statistics -- 3.6 Upper Estimates for Order Statistics -- References
• 6.4.2 Information Content and Intrinsic Volumes -- 6.4.3 A Bound for the mgf -- 6.4.4 Proof of Theorem 6.4.1 -- 6.4.5 Proof of Theorem 6.1.11 -- 6.5 Example: Rectangular Parallelotopes -- 6.5.1 Generating Functions and Intrinsic Volumes -- 6.5.2 Intrinsic Volumes of a Rectangular Parallelotope -- 6.5.3 Intrinsic Volumes of a Cube -- 6.6 Maximum-Entropy Distributions of Intrinsic Volumes -- 6.6.1 Ultra-Log-Concavity and Convex Bodies -- 6.6.2 Proof of Theorem 6.1.13 -- References -- 7 Two Remarks on Generalized Entropy Power Inequalities -- 7.1 Introduction -- 7.2 Failure of Schur-Concavity -- 7.3 Entropy Power Inequalities Under Symmetries -- References -- 8 On the Geometry of Random Polytopes -- 8.1 Introduction -- 8.2 Proof of Theorem 8.1.5 -- 8.3 Concluding Remarks -- References -- 9 Reciprocals and Flowers in Convexity -- 9.1 Introduction -- 9.2 Properties of Reciprocity and Flowers -- 9.3 The Spherical Inversion and a Proof of Theorem 9.9 -- 9.4 Structures on the Class of Flowers and Applications -- 9.5 Geometric Inequalities -- References -- 10 Moments of the Distance Between Independent Random Vectors -- 10.1 Introduction -- 10.1.1 Geometric Motivation -- 10.1.2 Probabilistic Discussion -- 10.1.3 Complex Interpolation -- 10.2 Proof of Theorem 10.1.1 -- 10.3 Proof of Theorem 10.1.6 and Its Consequences -- References -- 11 The Alon-Milman Theorem for Non-symmetric Bodies -- 11.1 Introduction -- 11.2 Proof of Theorem 11.1 -- References -- 12 An Interpolation Proof of Ehrhard's Inequality -- 12.1 Introduction -- 12.2 An ``Improved Jensen'' Inequality -- 12.3 A Short Proof of Prékopa-Leindler Inequality -- 12.4 Proof of Ehrhard's Inequality -- 12.4.1 An Example -- 12.4.2 Allowing J to Depend on t -- 12.4.3 The Hessian of J -- 12.4.4 Adding the Dependence on t -- References -- 13 Bounds on Dimension Reduction in the Nuclear Norm -- 13.1 Introduction
• 4 Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities -- 4.1 Introduction -- 4.2 Convergence Along the CLT for Rényi Entropies -- 4.3 Rényi EPIs of Order r(0,1) -- 4.3.1 Failure of a Generic Rényi EPI -- 4.3.2 Rényi EPIs for s-Concave Densities -- 4.3.2.1 Proof of Theorem 4.2 -- 4.3.2.2 Proof of Theorem 4.3 -- 4.4 An Entropic Characterization of s-Concave Densities -- References -- 5 Small Ball Probability for the Condition Number of Random Matrices -- 5.1 Introduction -- 5.2 Preliminaries -- 5.3 Proofs of Main Results -- 5.4 Small Ball Estimates for Singular Values -- 5.4.1 Proof of Theorem 5.2.1, the Case k≥lnn -- 5.4.2 Proof of Theorem 5.2.1, the Case k≤lnn -- References -- 6 Concentration of the Intrinsic Volumes of a Convex Body -- 6.1 Introduction and Main Results -- 6.1.1 Convex Bodies and Volume -- 6.1.2 The Intrinsic Volumes -- 6.1.2.1 Geometric Functionals -- 6.1.2.2 Properties -- 6.1.2.3 Hadwiger's Characterization Theorems -- 6.1.2.4 Quermassintegrals -- 6.1.3 The Intrinsic Volume Random Variable -- 6.1.4 Concentration of Intrinsic Volumes -- 6.1.5 Concentration of Conic Intrinsic Volumes -- 6.1.6 Maximum-Entropy Convex Bodies -- 6.1.7 Other Inequalities for Intrinsic Volumes -- 6.1.7.1 Ultra-Log-Concavity -- 6.1.7.2 Isoperimetric Ratios -- 6.2 Steiner's Formula and Distance Integrals -- 6.2.1 Steiner's Formula -- 6.2.2 Distance Integrals -- 6.2.3 Moments of the Intrinsic Volume Sequence -- 6.3 Variance of the Intrinsic Volume Random Variable -- 6.3.1 The Varentropy of a Log-Concave Distribution -- 6.3.2 A Log-Concave Density -- 6.3.3 Information Content and Intrinsic Volumes -- 6.3.4 Proof of Theorem 6.3.1 -- 6.4 Concentration of the Intrinsic Volume Random Variable -- 6.4.1 Moment Generating Function of the Information Content
• 13.2 Preliminaries -- 13.3 Certifying Projections -- 13.4 Certifying Anti-commutation -- 13.5 Replacing sigma with Identity -- 13.6 Dimension Bounds -- References -- 14 High-Dimensional Convex Sets Arising in Algebraic Geometry -- 14.1 Introduction -- 14.1.1 Organization -- 14.2 Asymptotic Log Positivity -- 14.2.1 Positivity -- 14.2.2 Asymptotic Log Positivity -- 14.3 The Body of Ample Angles -- 14.4 Asymptotically Log Fano/Canonically Polarized Varieties -- 14.4.1 Asymptotic Logarithmic Positivity Associated to (Anti)Canonical Divisors -- 14.4.2 Relation to the Body of Ample Angles -- 14.5 Convex Optimization and Classification in Algebraic Geometry -- 14.5.1 Tail Blow-Ups -- 14.5.2 Towards a Classification of Nested ``Tail'' Blow-Ups -- 14.5.3 The Set-Up -- 14.5.4 The Easy Direction and the Sub-critical Case -- 14.5.5 Dealing with the Quadratic Constraint and the Critical Case -- 14.5.6 Proof of Theorem 14.5.5 -- 14.5.7 Reduction of the Linear Intersection Constraints -- References -- 15 Polylog Dimensional Subspaces of ∞N -- 15.1 Introduction -- 15.2 Proofs -- 15.3 Remarks -- References -- 16 On a Formula for the Volume of Polytopes -- 16.1 Introduction -- 16.2 Formulation of the Result -- 16.3 Proof of Theorem 16.1 -- 16.4 Mixed Volumes -- References