Probabilistic Approach in Geometric Functional Analysis

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The PI plans to study convex bodies using the methods of probability.For many problems of convex geometry, such as finding sections of a convexbody with certain nice properties or approximating a convex body by anotherbody having a better structure, explicit constructions are unknown.In these cases random constructions turn out to be very effective.It is often possible to define a random section or approximation andto show that it has the desired property with high probability.This approach combined with advanced probabilistic tools, such as measureconcentration, have led to major discoveries in convex geometry and functionalanalysis.The PI also plans to study the combinatorial dimensionof a set of functions. This is a new notionwhich arises from questions in probability and computer science.The PI plans to investigate the connection between the entropy and thecombinatorial dimension and apply the results to study properties ofconvex bodies.The proposed research will provide new connections betweenfunctional analysis, convex geometry and probability.The study of combinatorial dimension is likely to have concretepractical applications in machine learning. The non-symmetric convex sets, which are one of the main objectsof the proposed research, arise naturally in a broad classof optimization problems. So, better understanding of the structureof such bodies will result in constructing more effective optimizationalgorithms.