Applications of Fourier analysis to convex geometry

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Abstract: The PI plans to study the geometry of convex bodies usingmethods of harmonic and functional analysis. One directionis to express different properties of convex bodies in termsof the Fourier transform and then use methods of Fourieranalysis to solve geometric problems. This approach will leadto new results on sections and projections of convex bodies,characterizations of different classes of bodies, new resultsof the Busemann-Petty type. Another direction of research isrelated to a new connection between convex geometry and thetheory of L_p-spaces, which has recently been found by the PIand allows to get new geometric results by extending differentfacts about L_p-spaces to negative values of p. Particularlyinteresting is the question of whether intersection and polarprojection bodies are isomorphically equivalent. A solution tothis problem will represent a big step in understanding theduality between sections and projections, which remains oneof the most intriguing mysteries of convex geometry. The PIalso plans to apply these methods to several open problems ofasymptotic geometric analysis, including the central limitproblem for convex bodies.The study of geometric properties of convex bodies based oninformation about sections and projections of these bodies hasimportant applications to many areas of mathematics and science,from the classical theory of x-rays and geometric tomography todifferent problems of engineering and medicine. A new approach tosections and projections of convex bodies, based on methods ofFourier analysis, has recently been developed by the PI. Thisapproach has already led to several results, including a Fourieranalytic solution of the Busemann-Petty problem on sections ofconvex bodies asking whether bodies with uniformly smaller centralsections necessarily have smaller volume. The proposed researchwill lead to better understanding of the geometry of convex bodiesand will further relate methods and results of convex geometry toharmonic analysis, functional analysis and probability. New techniquesfor computing the Fourier transform, developed in this project, willhave independent value and can be applied to signal processing andstatistics.