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Geometric Aspects of Functional Analysis

Israel Seminar (GAFA) 2020-2022

  • Book
  • © 2023

Overview

Editors:
  1. Ronen Eldan

    1. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

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  2. Bo'az Klartag

    1. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

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  3. Alexander Litvak

    1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

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  4. Emanuel Milman

    1. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

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  • Features an interdisciplinary mix of harmonic analysis, computational geometry, optimization & learning algorithms
  • Includes a unique combination of papers on convex geometry and high-dimensional analysis
  • Presents the state-of-the-art in asymptotic geometric analysis

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2327)

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About this book

This book reflects general trends in the study of geometric aspects of functional analysis, understood in a broad sense. A classical theme in the local theory of Banach spaces is the study of probability measures in high dimension and the concentration of measure phenomenon. Here this phenomenon is approached from different angles, including through analysis on the Hamming cube, and via quantitative estimates in the Central Limit Theorem under thin-shell and related assumptions. Classical convexity theory plays a central role in this volume, as well as the study of geometric inequalities. These inequalities, which are somewhat in spirit of the Brunn-Minkowski inequality, in turn shed light on convexity and on the geometry of Euclidean space. Probability measures with convexity or curvature properties, such as log-concave distributions, occupy an equally central role and arise in the study of Gaussian measures and non-trivial properties of the heat flow in Euclidean spaces. Also discussed are interactions of this circle of ideas with linear programming and sampling algorithms, including the solution of a question in online learning algorithms using a classical convexity construction from the 19th century.

Keywords

Table of contents (15 chapters)

  1. Front Matter

    Pages i-viii
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  2. Asymptotic Geometric Analysis: Achievements and Perspective

    • Vitali Milman
    Pages 1-55

  3. On the Gaussian Surface Area of Spectrahedra

    • Srinivasan Arunachalam, Oded Regev, Penghui Yao
    Pages 57-65

  4. Asymptotic Expansions and Two-Sided Bounds in Randomized Central Limit Theorems

    • Sergey G. Bobkov, Gennadiy P. Chistyakov, Friedrich Götze
    Pages 67-128

  5. The Case of Equality in Geometric Instances of Barthe’s Reverse Brascamp-Lieb Inequality

    • Karoly J. Boroczky, Pavlos Kalantzopoulos, Dongmeng Xi
    Pages 129-165

  6. A Journey with the Integrated \(\Gamma 2\) Criterion and its Weak Forms

    • Patrick Cattiaux, Arnaud Guillin
    Pages 167-208

  7. The Entropic Barrier Is n-Self-Concordant

    • Sinho Chewi
    Pages 209-222

  8. Local Tail Bounds for Polynomials on the Discrete Cube

    • Bo’az Klartag, Sasha Sodin
    Pages 223-230

  9. Stable Recovery and the Coordinate Small-Ball Behaviour of Random Vectors

    • Shahar Mendelson, Grigoris Paouris
    Pages 231-267

  10. On the Lipschitz Properties of Transportation Along Heat Flows

    • Dan Mikulincer, Yair Shenfeld
    Pages 269-290

  11. A Short Direct Proof of the Ivanisvili-Volberg Inequality

    • Piotr Nayar, Jacek Rutkowski
    Pages 291-296

  12. The Anisotropic Total Variation and Surface Area Measures

    • Liran Rotem
    Pages 297-312

  13. Chasing Convex Bodies Optimally

    • Mark Sellke
    Pages 313-335

  14. Shephard’s Inequalities, Hodge-Riemann Relations, and a Conjecture of Fedotov

    • Ramon van Handel
    Pages 337-354

  15. The Local Logarithmic Brunn-Minkowski Inequality for Zonoids

    • Ramon van Handel
    Pages 355-379

  16. Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

    • Santosh S. Vempala, Andre Wibisono
    Pages 381-438

  17. Back Matter

    Pages 439-440
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Editors and Affiliations

  • Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Ronen Eldan, Bo'az Klartag

  • Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

    Alexander Litvak

  • Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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