Methods of harmonic analysis in convex geometry

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The research project outlined below is concerned with the study of convex bodies and belongs to the area known as geometric tomography. One of the typical questions here is the extent to which knowledge of the size of sections or projections of a convex body yields information about the body itself. The main theme of this proposal is the study of applications of harmonic analysis to convex geometry. One of the earliest such applications goes back to A. Hurwitz's 1901 note on the use of Fourier series to prove the isoperimetric inequality. More recently, A. Koldobsky pioneered a Fourier transform approach to sections of convex bodies, which led, in particular, to an analytic solution to the Busemann-Petty problem.