This booklet is based on a series of ten lectures given by the first author at a NSF/CBMS conference in 2006. It provides a very nice introduction to the treatment of sections and projections of convex bodies by methods of Fourier analysis, which was inititated by the first author and was very successful in recent years. Its differences from the book by [A. Koldobsky, Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs 116. Providence) RI: American Mathematical Society (AMS) (2005; Zbl 1082.52002)] are characterized by the authors as follows. The current book exposes in a short form the main ideas of the Fourier approach to geometry so that interested researchers and students can quickly learn the subject and start working on related problems. Beyond that, included here are several interesting new results that have appeared after the first book, in particular the solution of the Busemann-Petty problem in non-Euclidean spaces, non-equivalence of several generalizations of intersection bodies, new methods of constructing non-intersection bodies, and a continuous path between intersection and projection bodies leading to some insights about the mysterious duality between sections and projections of convex bodies.
The chapter headings are: 1. Hyperplane sections of \(\ell_p\)-balls. 2. Volume and the Fourier transform. 3. Intersection bodies. 4. The Busemann-Petty problem. 5. Projections and the Fourier transform. 6. Intersection bodies and \(L_p\)-spaces. 7. On the road between polar projection bodies and intersection bodies.
The book concludes with the description of several open problems.