The PI plans to study finite dimensional normed spaces and convex bodies using methods of Fourier analysis. The idea is to express different properties of convex bodies in terms of the Fourier transform and then solve geometric problems using Fourier analytic methods. This approach has already led to several results including the Fourier transform formulae for the volume of sections of star bodies and a complete analytic solution to the Busemann-Petty problem on sections of convex bodies. The emphasis is on developing new connections between the Fourier transform and parallel section functions (X-rays) of convex bodies that will lead to new information about the structure of subspaces of Banach spaces, geometric properties of certain classes of convex bodies. The results will also be applied to the study of positive definite norm dependent functions (theorems of Schoenberg's type) and their connections with stable and isotropic stochastic processes.
This research will provide new connections between several areas of mathematics and engineering. Many of the results will have equivalent formulations in the languages of functional analysis, geometry and harmonic analysis. The study of subspaces of normed spaces is closely related to geometric tomography. The classical problem is to determine certain geometric properties of a solid by means of the areas of its slices obtained with differently placed sources of X-rays. The Fourier transform approach will contribute to this important problem in medical engineering. An application to statistics is the study of different parameters of stochastic processes under restricted data. An important example is the problem of determining the main frequences of a signal when only the maximal amplitude can be measured. New techniques for calculating the Fourier transform will serve as an important tool and will have other applications to signal processing.
