Geometric Aspects of Functional Analysis

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.

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

Inhalt

• Asymptotic Geometric Analysis: Achievements and Perspective. - On the Gaussian Surface Area of Spectrahedra. - Asymptotic Expansions and Two-Sided Bounds in Randomized Central Limit Theorems. - The Case of Equality in Geometric Instances of Barthe's Reverse Brascamp-Lieb Inequality. - A Journey with the Integrated 2 Criterion and its Weak Forms. - The Entropic Barrier Is n-Self-Concordant. - Local Tail Bounds for Polynomials on the Discrete Cube. - Stable Recovery and the Coordinate Small-Ball Behaviour of Random Vectors. - On the Lipschitz Properties of Transportation Along Heat Flows. - A Short Direct Proof of the Ivanisvili-Volberg Inequality. - The Anisotropic Total Variation and Surface Area Measures. - Chasing Convex Bodies Optimally. - Shephard's Inequalities, Hodge-Riemann Relations, and a Conjecture of Fedotov. - The Local Logarithmic Brunn-Minkowski Inequality for Zonoids. - Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices.