Fourier Analytic Methods in the Study of Projections and Sections of Convex Bodies

It has been noticed long ago that many results on sections and projections are dual to each other, though methods used in the proofs are quite different and don't use the duality of underlying structures directly. In the paper [KRZ], the authors attempted to start a unified approach connecting sections and projections, which may eventually explain these mysterious connections. The idea is to use the recently developed Fourier analytic approach to sections of convex bodies (a short description of this approach can be found in [K7]) as a prototype of a new approach to projections. The first results seem to be quite promising. The crucial role in the Fourier approach to sections belongs to certain formulas connecting the volume of sections with the Fourier transform of powers of the Minkowski functional. An analog of these formula for the case of projections was found in [KRZ] and connects the volume of projections to the Fourier transform of the curvature function. This formula was applied in [KRZ] to give a new proof of the result of Barthe and Naor on the extremal projections of l p-balls with p > 2, which is similar to the proof of the result on the extremal sections of l p-balls with 0 < p < 2 in [K5]. Another application is to the Shephard problem, asking whether bodies with smaller hyperplane projections necessarily have smaller volume. The problem was solved independently by Petty and Schneider, and the answer is affirmative in the dimension two and negative in the dimensions three and higher. The paper [KRZ] gives a new Fourier analytic solution to this problem that essentially follows the Fourier analytic solution to the Busemann-Petty problem (the projection counterpart of Shephard's problem) from [K3]. The transition in the Busemann-Petty problem occurs between the dimensions four and five. In Section 4, we show that the transition in both problems has the same explanation based on similar Fourier analytic characterizations of intersection and projection bodies. The goal of this survey is to bring together certain aspects of the Fourier approaches to sections and projections, in order to emphasize the similarities between the results and the proofs. We do not include