The Fourier Transform and Convex Bodies

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DMS 0400789D RyaboginKansas State UniversityThe Fourier Transform and Convex BodiesThe proposed research aims at achieving considerable progress towardsthe development of a Fourier analytic approach to the solution ofseveral problems of convex geometry, related to sections and projections, and also to the problems of uniqueness, reconstruction and duality of convex bodies. A recently discovered formulaexpressing the volume of projections in terms of the Fourier transformof the curvature function, has led to Fourier analytic proofs ofseveral results on projections, including the characterization of projection bodies in terms of sections of the polar body, and theFourier analytic solution to Shephard's problem (asking whether symmetric convex bodies with smaller projections necessarily have smaller volume) surprisingly similar to that of the Busemann-Petty problem (a section counterpart of the Shephard problem). The similarities in the Fourier analytic proofs of these resultsindicate in particular that there must exist deep dual connections between volumes of projections and sections of convex bodies.To achieve progress in obtaining the Fourier analytic description of this duality phenomena, the PI plans to find extremal projections of certain classes of bodies, to undertake a further study of the projection and intersection bodies, to obtain results concerning non-central sections, and to construct a nonsmooth projection bodywhose polar is also a projection body in higher dimensions.Convexity is a very old topic which can be traced at very least to Archimedes. It is still in favor due to its numerous applications to linear programming, tomography, medicine, and it is a surprise that Fourier analytic methods have been applied to the subject only very recently. These methods can serve as an additional source of ideas, coming to both fields, convex geometry and harmonic analysis, and will find new applications. At the same time, convexity is an extremely simple and natural notion. Interesting in itself, it also illustrates some facts about mathematics, facts that are more or less classical, but always important to realize, so it is a perfect field for undergraduates. First of all, questions or problems arise that are very simple to formulate and understand, so students do not need to take several classes before approaching the material.Secondly, intuition is sometimes misleading in ``obvious problems'', and the undergraduate feels the beauty of the subject. Many problems can be solved by fairly elementary means, but on the other hand, answers to many problems are still unknown or have been found recently,often using different techniques from other parts of mathematics.Therefore, it is a perfect field for research projects for more senior students and all people dealing with exact sciences.