Geometric Aspects of Functional Analysis

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 BrunnMinkowski 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.

Features a unique mixture of papers on convex geometry and high-dimensional analysis Describes state-of-the-art progress in asymptotic geometric analysis Written from an interdisciplinary perspective, relations to differential geometry, information theory and computer science are included

Inhalt

• Jean Bourgain: In Memoriam. - A Generalized Central Limit Conjecture for Convex Bodies. - The Lower Bound for Koldobsky's Slicing Inequality via Random Rounding. - Two-Sided Estimates for Order Statistics of Log-Concave Random Vectors. - Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities. - Small Ball Probability for the Condition Number of Random Matrices. - Concentration of the Intrinsic Volumes of a Convex Body. - Two Remarks on Generalized Entropy Power Inequalities. - On the Geometry of Random Polytopes. - Reciprocals and Flowers in Convexity. - Moments of the Distance Between Independent Random Vectors. - The AlonMilman Theorem for Non-symmetric Bodies. - An Interpolation Proof of Ehrhard's Inequality. - Bounds on Dimension Reduction in the Nuclear Norm. - High-Dimensional Convex Sets Arising in Algebraic Geometry. - Polylog Dimensional Subspaces of lN/. - On a Formula for the Volume of Polytopes.