Geometric Aspects of Functional Analysis

1. Front Matter

   Pages i-xiii

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2. Gromov’s Waist of Non-radial Gaussian Measures and Radial Non-Gaussian Measures

   • Arseniy Akopyan, Roman Karasev

   Pages 1-27

3. Zhang’s Inequality for Log-Concave Functions

   • David Alonso-Gutiérrez, Julio Bernués, Bernardo González Merino

   Pages 29-48

4. Bobkov’s Inequality via Optimal Control Theory

   • Franck Barthe, Paata Ivanisvili

   Pages 49-61

5. Arithmetic Progressions in the Trace of Brownian Motion in Space

   • Itai Benjamini, Gady Kozma

   Pages 63-69

6. Edgeworth Corrections in Randomized Central Limit Theorems

   • Sergey G. Bobkov

   Pages 71-97

7. Three Applications of the Siegel Mass Formula

   • Jean Bourgain, Ciprian Demeter

   Pages 99-111

8. Decouplings for Real Analytic Surfaces of Revolution

   • Jean Bourgain, Ciprian Demeter, Dominique Kemp

   Pages 113-125

9. On Discrete Hardy–Littlewood Maximal Functions over the Balls in \({\boldsymbol {\mathbb {Z}^d}}\): Dimension-Free Estimates

   • Jean Bourgain, Mariusz Mirek, Elias M. Stein, Błażej Wróbel

   Pages 127-169

10. On the Poincaré Constant of Log-Concave Measures

    • Patrick Cattiaux, Arnaud Guillin

    Pages 171-217

11. On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures

    • Djalil Chafaï, Joseph Lehec

    Pages 219-246

12. Several Results Regarding the (B)-Conjecture

    • Dario Cordero-Erausquin, Liran Rotem

    Pages 247-262

13. A Dimension-Free Reverse Logarithmic Sobolev Inequality for Low-Complexity Functions in Gaussian Space

    • Ronen Eldan, Michel Ledoux

    Pages 263-271

14. Information and Dimensionality of Anisotropic Random Geometric Graphs

    • Ronen Eldan, Dan Mikulincer

    Pages 273-324

15. On the Ekeland-Hofer-Zehnder Capacity of Difference Body

    • Efim Gluskin

    Pages 325-340

16. Back Matter

    Pages 341-342

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Continuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressedis the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.

• Asymptotic Geometric Analysis
• Convex Geometry
• Functional Analysis
• Functional Inequalities
• Geometric Probability

• School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel, Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

  Bo'az Klartag

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

  Emanuel Milman

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