Applications of Fourier analysis to convex geometry

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The principal investigator plans to apply methods of Fourier analysis to convex geometry, functional analysis and probability. The study of geometric properties of convex bodies based on information about sections and projections of these bodies has important applications to many areas of mathematics and science. A new approach to sections of convex bodies, based on methods of Fourier analysis, has recently been developed by the investigator. The idea of this approach is to express different cross-sectional characteristics of a body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. The investigator plans to apply this approach to characterizations of different classes of bodies, determination of convex bodies from data about their sections and projections, geometric inequalities of the Busemann-Petty type. A connection between intersection bodies, one of the main objects in convex geometry, and functional spaces has recently been found by the investigator. The investigator plans to use this connection to establish new results about intersection bodies using methods of functional analysis. An old problem in probability is to characterize all random vectors having the property that all linear combinations of coordinates have the same distribution, up to a constant. The classical examples are stable random vectors. The investigator plans to characterize all random vectors with this property. The problems considered in this proposal belong to three areas of mathematics: convex geometry, functional analysis and probability. However, the strategy of solution is common for most of the results - the question is translated into the language of the Fourier transform and then treated as a problem from harmonic analysis. In convex geometry, the principal investigator studies properties of convex bodies based on data related to sections and projections of these bodies. This direction, often named geometric tomography, has numerous applications to engineering and medicine. The problems in functional analysis are related to the theory of embedding of metric spaces that has many applications to computer science. In probability, the investigator plans to study generalizations of stable processes that play important role in several areas of mathematics and statistics. The Fourier transform of distributions will serve as the main technical tool, and a significant part of the work will depend on new techniques for calculating the Fourier transform. These techniques have independent value and have already been applied to other areas of mathematics, signal processing and statistics. An important part of the project is the involvement and training of graduate students.