Fourier analytic methods in convex geometry and geometric tomography

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The proposed research program belongs to the areas of convex geometry, geometric functional analysis, geometric tomography, and discrete tomography. These are very rich and active areas. Results and methods from these areas find applications in many other branches of mathematics, as well as other sciences, such as materials science, medicine, geophysics, etc. For example, geometric tomography provides a mathematical foundation for CT-scanners, the study of seismic activity, the search for oil, to name a few.***The general objective of this proposal is to study convex bodies (and other objects) by looking at the sizes of their sections or projections. The main tools that I plan to use to attack various geometric problems are those of harmonic analysis. Several longstanding problems, including the Busemann-Petty problem about central sections of convex bodies and Klee's problem about maximal sections, have been solved with the help of Fourier analysis. One of the goals of this proposal is to further develop and refine such techniques, as well as to find new applications of Fourier analysis to geometric problems. These methods will be used in combination with other methods, especially those coming from geometric functional analysis, discrete geometry, and algebra. There are three main topics in this proposal. The first part is dedicated to Arnold's problem about algebraically integrable domains. The second part is about generalizations and applications of Gruenbaum-type inequalities. In the last part I will describe some uniqueness problems in geometric tomography and discrete tomography.*****