Probabilistic Approach in Geometric Functional Analysis

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The PI plans to study metric spaces and convex bodies using the methods of Probability. For many problems of Convex Geometry, like finding sections of a convex body with certain nice properties or approximating a convex body by another body, having a better structure, the explicit constructions are unknown. In these cases random constructions were proved to be very effective. It is often possible to define a random section or approximation and to show that it has the desired property with high probability. This approach combined with advanced probabilistic tools, like measure concentration, led to major discoveries in Convex geometry and Functional Analysis, including Dvoretzky's Theorem and inverse Santalo Inequality. The PI intends to apply the probabilistic method to study embeddings of different metric spaces, such as groups, graphs etc., into Banach spaces. These metric spaces are often equipped with a probability measure, which has strong concentration properties. Then it is possible to obtain significant information about the embedding of such metric space into a Banach space by studying the distribution of the image of a random point. Another direction of the proposed research is related to the study of convex bodies, which are not necessary symmetric. The results obtained in the last 3 years by several researchers, including the PI, show that many properties, previously known only for convex symmetric bodies, hold without the assumption of symmetry. The PI plans to continue his work in this area with the aim of constructing a theory of general convex bodies, which would be parallel to the existing theory of convex symmetric bodies.The proposed research will provide new connections between Functional Analysis, Convex Geometry and Probability. The results on embedding graphs into Banach spaces will be useful for finding small separators of graphs. This can lead to construction of more effective algorithms in several Computer Science problems, in particular in numerical solution of partial differential equations. The non-symmetric convex sets, which are one of the main objects of the proposed research, arise naturally in a broad class of optimization problems. So, better understanding of the structure of such bodies will result in constructing more effective optimization algorithms. The stochastic processes related to a convex body are also of considerable interest for the Control Theory.