Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2020-2022

Preface
Contents
Asymptotic Geometric Analysis: Achievements and Perspective
1 A Few Words About Asymptotic Geometric Analysis
2 Semyon Alesker, Valuations
2.1 Valuations on Convex Sets
2.2 Valuations on Functions
2.3 Comments by V.M.
3 Shiri Artstein-Avidan, A Playground of Dualities
3.1 Comments by V.M.
4 Ronen Eldan, Dimension-Free Concentration in High-Dimensional Distributions
4.1 Concentration Under a Convexity Condition
4.2 The Discrete Hypercube
4.3 Comments by V.M.
5 Dmitry Faifman, the Weyl Principle in Valuation Theory, and Projective Geometries
5.1 Intrinsic Volumes and the Weyl Principle
5.2 Volume in Funk and Hilbert Geometries
5.3 Comments by V.M.
6 Bo'az Klartag, Recent Developments Towards Understanding the Distribution of Volume in High Dimensions
6.1 Comments by V.M.
7 Alexander E. Litvak, Random Matrices and Related Topics
7.1 Random Matrices in Asymptotic Geometric Analysis
7.2 Approximation of Convex Bodies by Polytopes with a Small Number of Vertices or Faces
7.3 Comments by V.M.
8 Emanuel Milman, Isomorphic Version of the Log-Brunn–Minkowski Inequality
8.1 Comments by V.M.
8.1.1 Isomorphic Position of Convex Body
9 Yaron Ostrover, Convex Bodies in the Classical Phase Space
9.1 Comments by V.M.
10 Liran Rotem, Flowers and Convex Bodies
10.1 Comments by V.M.
Examples
References
On the Gaussian Surface Area of Spectrahedra
1 Introduction
2 Preliminaries
3 Proof of Main Theorem
References
Asymptotic Expansions and Two-Sided Bounds in Randomized Central Limit Theorems
1 Introduction
2 Typical Distributions
3 Upper Bound for the L2-Distance at Standard Rate
4 General Approximations for the L2-Distance with Error of Order at Most 1/n
5 Proof of Theorem 1.1 for the L2-Distance
6 General Lower Bounds for the L2-Distance: Proof of Theorem 1.2
7 Lipschitz Systems
8 Berry-Esseen-Type Bounds
9 Quantitative Forms of Sudakov's Theorem for the Kolmogorov Distance
10 Proof of Theorem 1.1 for the Kolmogorov Distance
11 Relations Between L1, L2 and Kolmogorov Distances
12 Lower Bounds: Proof of Theorem 1.3
13 Functional Examples
14 The Walsh System; Empirical Measures
15 Improved Rates for Lacunary Systems
16 Improved Rates for Independent and Log-Concave Summands
17 Improved Rates Under Correlation-Type Conditions
References
The Case of Equality in Geometric Instances of Barthe's Reverse Brascamp-Lieb Inequality
1 Introduction
2 The Structure Theory of the Geometric Brascamp-Lieb Data and Barthe's Determinantal Inequality
3 Splitting Smooth Extremizers Along Independent and Dependent Subspaces
4 Convolution and Product of Extremizers
5 hi0 is Gaussian in Proposition 15
6 Proof of Theorem 4
7 An Application: Equality in Liakopoulos' Dual Bollobas-Thomason Inequality
References
A Journey with the Integrated 2 Criterion and its Weak Forms
1 Introduction, Framework and Presentation of the Results
1.1 Framework (The Heart of Darkness Following 5:BaGLbook)
1.2 Presentation of the Main Results
2 Weak Integrated 2
3 The Log-Concave Case
4 Some Applications: Perturbation of Product Measures and Radial Measures
5 The Case of Compactly Supported Measures
6 Super 2 Condition
6.1 From (p-SPI) to (pSI-2)
6.2 From (pSI-2) to (p-SPI)
7 Appendix: About the Heart of Darkness
References
The Entropic Barrier Is n-Self-Concordant
1 Introduction
2 From the Entropic Barrier to the Dimensional Brascamp–Lieb Inequality
3 Proof of the Dimensional Brascamp–Lieb Inequality
3.1 Proof by Hörmander's L^2 Method
3.2 Proof by Convexity of the Entropy Along Bregman Divergence Couplings
4 A Tensorization Trick
References
Local Tail Bounds for Polynomials on the Discrete Cube
1 Introduction
2 Proofs
3 Remarks
References
Stable Recovery and the Coordinate Small-Ball Behaviour of Random Vectors
1 Introduction
1.1 Point Separation and Stable Point Separation
1.2 The Small-Ball Assumption
1.2.1 Example 1.3 Revisited
1.3 Small-Ball Estimates
1.4 Coordinate Small Ball
2 Proofs: Small Ball Estimates
2.1 Proof of Theorem 1.18
3 Proofs: Coordinate Small-Ball Estimates
3.1 Coordinate Small-Ball for General Operators
4 Proofs: Applications
4.1 Random Sub-Sampled Convolutions and Stable Point Separation
4.2 Sparse Recovery Under Malicious Noise
4.3 Small-Ball Estimates for the p-Norm
5 Concluding Remarks
5.1 The Wrong Way
A Examples of Vectors That Satisfy The SBA
B Proof of Remark 1.15
References
On the Lipschitz Properties of Transportation Along Heat Flows
1 Introduction and Main Results
1.1 Applications
1.1.1 Eigenvalues Comparisons
1.1.2 Dimensional Functional Inequalities
1.1.3 Majorization
2 Proofs
2.1 Preliminaries
2.2 Lipschitz Properties of Transportation Along Heat Flows
2.2.1 Transportation from the Gaussian
2.2.2 Transportation to the Gaussian
References
A Short Direct Proof of the Ivanisvili-Volberg Inequality
1 Introduction
2 Proof of the Main Inequality
3 Proof of Lemma 2
References
The Anisotropic Total Variation and Surface Area Measures
1 Introduction
2 Anisotropic Total Variations
3 Completing the Proof
References
Chasing Convex Bodies Optimally
1 Introduction
2 Problem Setup
2.1 Notations and Conventions
2.2 Continuous Time Formulation
3 Functional Steiner Point and Work Function
3.1 The Work Function
4 Linear Competitive Ratio
5 Competitive Ratio O(dlogN) in Euclidean Space
6 Steiner Points of Level Sets
6.1 A Simplification for Chasing Convex Bodies
6.2 Steiner Points of Level Sets
Appendix: Proof of Lemma 3.7
References
Shephard's Inequalities, Hodge-Riemann Relations, and a Conjecture of Fedotov
1 Introduction
2 Linear Algebra
2.1 Hyperbolic Matrices
2.2 Shephard's Inequalities
2.3 A Sylvester Criterion
3 Hodge-Riemann Relations
4 Proof of Theorem 1.5
5 An Explicit Example
6 Hodge-Riemann Relations Fail for General Convex Bodies
References
The Local Logarithmic Brunn-Minkowski Inequality for Zonoids
1 Introduction
2 Preliminaries
2.1 Projections and Zonoids
2.2 Smooth Bodies and the Hilbert Operator
2.3 The Bochner Method
3 Proof of Theorem 1.4
3.1 The Induction Step
3.2 The Induction Base
4 Proof of Theorem 1.5
4.1 The Equality Condition
4.2 The Bochner Method Revisited
4.3 Characterization of Equality
5 Implications
References
Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices
1 Introduction
2 Review of KL Divergence Along Langevin Dynamics
2.1 KL Divergence
2.2 Log-Sobolev Inequality
2.2.1 Talagrand Inequality
2.3 Langevin Dynamics
2.3.1 Exponential Convergence of KL Divergence Along Langevin Dynamics Under LSI
3 Unadjusted Langevin Algorithm
3.1 Convergence of KL Divergence Along ULA Under LSI
4 Review of Rényi Divergence Along Langevin Dynamics
4.1 Rényi Divergence
4.1.1 Log-Sobolev Inequality
4.2 Langevin Dynamics
4.2.1 Convergence of Rényi Divergence Along Langevin Dynamics Under LSI
5 Rényi Divergence Along ULA
5.1 Decomposition of Rényi Divergence
5.2 Rapid Convergence of Rényi Divergence to Biased Limit Under LSI
5.3 Convergence of Rényi Divergence Along ULA Under LSI
6 Poincaré Inequality
6.1 Convergence of Rényi Divergence Along Langevin Dynamics Under Poincaré
6.2 Convergence of Rényi Divergence to Biased Limit Under Poincaré
6.3 Convergence of Rényi Divergence Along ULA Under Poincaré
7 Properties of Biased Limit
7.1 Bound on Bias Under Third-Order Smoothness
7.2 Isoperimetry of Biased Limit Under Strong Log-Concavity and Smoothness
8 Proofs and Details
8.1 Proofs for Sect. 2: KL Divergence Along Langevin Dynamics
8.1.1 Proof of Lemma 1
8.1.2 Proof of Lemma 2
8.1.3 Proof of Theorem 4
8.2 Proofs for Sect. 3: Unadjusted Langevin Algorithm
8.2.1 Proof of Lemma 3
8.2.2 Proof of Theorem 1
8.3 Details for Sect. 4: Rényi Divergence Along Langevin Dynamics
8.3.1 Properties of Rényi Divergence
8.3.2 Proof of Lemma 4
8.3.3 Proof of Lemma 5
8.3.4 Proof of Lemma 6
8.3.5 Proof of Theorem 2
8.3.6 Hypercontractivity
8.4 Proofs for Sect. 5: Rényi Divergence Along ULA
8.4.1 Proof of Lemma 7
8.4.2 Proof of Lemma 8
8.4.3 Proof of Theorem 5
8.5 Details for Sect. 6: Poincaré Inequality
8.5.1 Proof of Lemma 9
8.5.2 Proof of Theorem 3
8.5.3 Proof of Lemma 10
8.5.4 Proof of Theorem 6
8.6 Proofs for Sect. 7: Properties of Biased Limit
8.6.1 Bounding Relative Fisher Information
8.6.2 Bounding the Expected Value
8.6.3 Bounding the Fisher Information
8.6.4 Proof of Upper Bound in Theorem 7
8.6.5 Proof of Lower Bound in Theorem 7
8.6.6 Proof of Theorem 8
9 Discussion
Appendix
Review on Notation and Basic Properties
Derivation of the Fokker-Planck Equation
Remaining Proofs
Proof of Lemma 16
Proof of Lemma 17
Proof of Lemma 19
Proof of Lemma 20
References