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.
The main goal of this project is to study geometric parameters of solids based on information about their sections and projections. This direction of convex geometry is often called geometric tomography. The classical example is the x-ray tomography and the underlying mathematical methods. The problems in the project are concentrated around the famous Busemann-Petty problem, which was raised in 1956 and solved at the end of the 1990's as the result or work of many mathematicians including the PI. The formulation of the problem can easily be explained to a non-expert, while the solution appeares to be very difficult. Suppose we have two solids with the same center, and suppose that the areas of central plane sections of the first solid in all directions are smaller than the same for the second solid. Is it necessarily true that the volume of the first solid is smaller than the volume of the second solid? The answer is affirmative only in dimensions up to four, and it is negative in dimensions five and higher. The problem is closely related to the Hyperplane Conjecture, one of the most important open problems in this direction, asking whether solids of volume one necessarily have a big enough central plane section. One of the main results of this project establishes stability in the Busemann-Petty problem and other volume comparison problems, namely given that the areas of central plane sections of the first solid are smaller only up to a constant, one can show that the volume of the first solid is smaller up to a proportional constant. These results lead to hyperplane inequalities for certain classes of solids, which are more general than what is required in the Hyperplane Conjecture. The method of solution is based on the Fourier approach developed by the PI for the analytic solution of the Busemann-Petty problem, where the data is decomposed into the sum of periodic signals. The Fourier approach has many other applications in convex geometry, in particular in this project it was also applied to study the properties of intersection bodies, one of the main objects in convex geometry, for finding maximal and minimal plane sections of certain solids and for other problems. The Fourier approach has applications to other areas of mathematics, for example in this project it was applied to an old problem in probability theory. The normal law in probability has the self-reproduction property, namely the sum of two random quantities distributed according to the normal law is also distributed according to the normal law. Because of this property the normal law is crucial in statistics, it appears every time when one considers the sum of many random factors. In the 1930's Paul Levy found stable laws that also have the self-reproduction property and asked if there any other laws with the same property. A lot of work has been done on this problem, for example in 1991 the PI solved the 1938 Schoenberg's problem giving a partial answer to the question of Levy. One of the results of this project almost solves the problem of Levy by establishing that every other probabilistic law with the self-reproduction property must be extremely exotic. The Fourier approach employed in this project is based on new techniques for calculating the Fourier transform (decomposing data into periodic signals) that are of interest to harmonic analysts. These techniques have been applied by the PI to image processing in an earlier paper published in the journal Pattern Recognition. As a part of the work on the project, the PI has trained two graduate students and a postdoc. Each of them has produced interesting results in the directions of the project. One of the students, Daniel Fresen, has graduated with PhD and is currently a Gibbs Assistant Professor at Yale. The results of the project were disseminated in eleven publications in the leading mathematical journals, two prerpints in the Max Planck Institute Preprint Series, and in thirty one invited lectures at national and international universities and conferences. The PI has served as the organizer of three major conferences, including meetings at the American Institute of Mathematics in Palo Alto, CA and at the Banff International Research Station in Banff, Canada.
