Geometric Aspects of Functional Analysis

1. Front Matter

   Pages i-xii

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2. A Generalized Central Limit Conjecture for Convex Bodies

   • Haotian Jiang, Yin Tat Lee, Santosh S. Vempala

   Pages 1-41

3. The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding

   • Bo’az Klartag, Galyna V. Livshyts

   Pages 43-63

4. Two-Sided Estimates for Order Statistics of Log-Concave Random Vectors

   • Rafał Latała, Marta Strzelecka

   Pages 65-94

5. Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

   • Jiange Li, Arnaud Marsiglietti, James Melbourne

   Pages 95-123

6. Small Ball Probability for the Condition Number of Random Matrices

   • Alexander E. Litvak, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann

   Pages 125-137

7. Concentration of the Intrinsic Volumes of a Convex Body

   • Martin Lotz, Michael B. McCoy, Ivan Nourdin, Giovanni Peccati, Joel A. Tropp

   Pages 139-167

8. Two Remarks on Generalized Entropy Power Inequalities

   • Mokshay Madiman, Piotr Nayar, Tomasz Tkocz

   Pages 169-185

9. On the Geometry of Random Polytopes

   • Shahar Mendelson

   Pages 187-198

10. Reciprocals and Flowers in Convexity

    • Emanuel Milman, Vitali Milman, Liran Rotem

    Pages 199-227

11. Moments of the Distance Between Independent Random Vectors

    • Assaf Naor, Krzysztof Oleszkiewicz

    Pages 229-256

12. The Alon–Milman Theorem for Non-symmetric Bodies

    • Márton Naszódi

    Pages 257-261

13. An Interpolation Proof of Ehrhard’s Inequality

    • Joe Neeman, Grigoris Paouris

    Pages 263-278

14. Bounds on Dimension Reduction in the Nuclear Norm

    • Oded Regev, Thomas Vidick

    Pages 279-299

15. High-Dimensional Convex Sets Arising in Algebraic Geometry

    • Yanir A. Rubinstein

    Pages 301-323

16. Polylog Dimensional Subspaces of \(\ell _\infty ^N\)

    • Gideon Schechtman, Nicole Tomczak-Jaegermann

    Pages 325-334

17. On a Formula for the Volume of Polytopes

    • Rolf Schneider

    Pages 335-345

18. Back Matter

    Pages 347-348

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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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