Probabilistic Approach in Geometric Functional Analysis

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The PI plans to study convex bodies using the methods of probability. For many problems of convex geometry, such as finding sections of a convex body with certain nice properties or approximating a convex body by another body 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 and to show that it has the desired property with high probability. This approach combined with advanced probabilistic tools, such as measure concentration, have led to major discoveries in convex geometry and functional analysis. The PI also plans to study the combinatorial dimension of a set of functions. This is a new notion which arises from questions in probability and computer science. The PI plans to investigate the connection between the entropy and the combinatorial dimension and apply the results to study properties of convex bodies. The proposed research will provide new connections between functional analysis, convex geometry and probability. The study of combinatorial dimension is likely to have concrete practical applications in machine learning. 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.