Asymptotic Geometric Analysis, Part II

ISBN-13: 9781470463601

Shiri Artstein-Avidan;Apostolos Giannopoulos;Vitali D. Milman

Opis

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

Spis treści

• Functional inequalities and concentration of measure
• Isoperimetric constants of log-concave measures and related problems
• Inequalities for Guassian measures
• Volume inequalities
• Local theory of finite dimensional normed spaces: Type and cotype
• Geometry of the Banach-Mazur compactum
• Asymptotic convex geometry and classical symmetrizations
• Restricted invertibility and the Kadison-Singer problem
• Functionalization of geometry
• Bibliography
• Subject index
• Author index