Fourier Analysis in Convex Geometry

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This project is focused on the study of geometric properties of solids based on information about their sections and projections. This branch of convex geometry is called geometric tomography. One of the examples is x-ray tomography, which has numerous applications in science, medicine and engineering. The PI recently discovered that important problems of geometric tomography can be solved using the most popular tool of harmonic analysis, the Fourier transform. This tool allows one to decompose the data into a simple combination of harmonics, functions with periodically repeating values, and, by doing this, one can reduce geometric problems to computations related to the harmonics, the theory of which is well developed. This approach has led to solutions of the 1956 Busemann-Petty problem on sections of convex bodies, the 1938 Schoenberg's problem on positive definite functions, the slicing problem for arbitrary functions. The PI plans to further develop the Fourier approach and apply it to a range of problems at the interface between convex geometry, functional analysis, probability theory and algebra. For example, can one find an algebraic equation whose solutions are sections of a given solid? Can one estimate the volume of a solid from data involving areas of certain sets of sections or projections of this solid? Which random variables are stable, that is, have the property that the sums of several copies of these variables always reproduce the same variable up to a constant? An important part of the project is the involvement and training of graduate students and postdocs.The problems considered connect several areas of mathematics - convex geometry, functional analysis, probability and algebra. 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. The PI plans to develop further connections between convex geometry and algebra by considering the problem of Arnold, going back to Lemma 28 from Newton's "Principia." The problem is to characterize those convex domains whose cut-off area is an algebraic function of the parameters of the cutting plane. The classical slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. The PI plans to study the lower-dimensional and non-symmetric versions of the Busemann-Petty problem asking whether a convex body with uniformly smaller areas of plane sections necessarily has smaller volume. Another direction is to study the general properties of the Radon transform associated with volumetric results about convex bodies. A connection with functional analysis is the study of distances between convex bodies, embedding and duality problems. The question about positive definite functions and embeddings in Lebesgue spaces is related to an old problem in probability theory asking for a characterization of stable random vectors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.