Applications of Fourier analysis to convex geometry

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Abstract: The PI plans to study the geometry of convex bodies using methods of harmonic and functional analysis. One direction is to express different properties of convex bodies in terms of the Fourier transform and then use methods of Fourier analysis to solve geometric problems. This approach will lead to new results on sections and projections of convex bodies, characterizations of different classes of bodies, new results of the Busemann-Petty type. Another direction of research is related to a new connection between convex geometry and the theory of L_p-spaces, which has recently been found by the PI and allows to get new geometric results by extending different facts about L_p-spaces to negative values of p. Particularly interesting is the question of whether intersection and polar projection bodies are isomorphically equivalent. A solution to this problem will represent a big step in understanding the duality between sections and projections, which remains one of the most intriguing mysteries of convex geometry. The PI also plans to apply these methods to several open problems of asymptotic geometric analysis, including the central limit problem for convex bodies. The study of geometric properties of convex bodies based on information about sections and projections of these bodies has important applications to many areas of mathematics and science, from the classical theory of x-rays and geometric tomography to different problems of engineering and medicine. A new approach to sections and projections of convex bodies, based on methods of Fourier analysis, has recently been developed by the PI. This approach has already led to several results, including a Fourier analytic solution of the Busemann-Petty problem on sections of convex bodies asking whether bodies with uniformly smaller central sections necessarily have smaller volume. The proposed research will lead to better understanding of the geometry of convex bodies and will further relate methods and results of convex geometry to harmonic analysis, functional analysis and probability. New techniques for computing the Fourier transform, developed in this project, will have independent value and can be applied to signal processing and statistics.