Probabilistic Approach in the Local Theory of Banach Spaces and Convex Geometry

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9706835 Rudelson Probabilistic methods constitute a very rapidly developing field of Convex Geometry and Local Theory of Banach spaces. Recent results in this area are connected in particular to the majorizing measure approach, introduced by Talagrand. This method enabled the solution of several deep problems in Combinatorics, Harmonic Analysis, and Convex Geometry. This work will continue investigation of constructions of majorizing measures as well as study possible applications of this method to concrete problems of Local Theory and Harmonic Analysis. Another direction is related to the study of convex bodies which are not necessary symmetric. This is a new project inspired by recent progress in generalization of the results of the Local Theory to star-shaped bodies. It turned out to be possible to prove the most important results of the Local Theory, including inverse Santalo and Brunn-Minkowski inequalities, for p-convex bodies. Unlike p-convex symmetric bodies, in the case of general convex bodies almost nothing is known. The basic technical tools, which work for convex and p-convex symmetric bodies, do not work any more if general convex bodies are considered. Moreover, the known facts about non-symmetric convex bodies suggest that there might be a significant difference between their properties and the properties of symmetric bodies.The aim of the project is to construct a theory of general convex bodies which is parallel to the existing theory of convex symmetric bodies. Convex Geometry is a field of Mathematics which studies the properties of convex bodies of high dimension. This includes finding sections of a given body with certain nice properties, computing the volume of a body, approximating a convex body by another one having a better structure, etc. While these problems cannot be solved in general by an explicit construction, it is often possible to show that a randomly chosen section or a random modification of a body has the desired prop erties. This approach enables the reduction of the analytic and geometric problems to the investigation of the behavior of a certain random process. The project will concentrate on developing a method to analyze such processes and on the investigation of the structure of general convex bodies. The results obtained in this direction are applicable to various problems in Harmonic Analysis and Banach Space theory as well as in Computer Science, like construction of effective optimization algorithms and fast computation of the volumes of convex bodies.