Fourier analysis in geometric tomography

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Geometric tomography is the area of mathematics in which one investigates geometric properties of solids based on the information about sections and projections of these solids. One of the examples, x-ray tomography, has numerous applications in science, medicine, and engineering. Within mathematics, geometric tomography overlaps with convex geometry and functional analysis. The principal investigator has recently discovered that important problems in the area of geometric tomography can be solved using the most popular tool of harmonic analysis, the Fourier transform. This tool allows one to decompose data into a simple combination of harmonics (i.e., into functions with periodically repeating values) and, by doing this, to reduce geometric problems to computations related to the harmonics, the theory of which is well developed. This approach has led to analytic solutions of several longstanding problems in the area, including the Busemann-Petty problem, the Shephard problem, and the slicing problem for sections of proportional dimensions. In this project, the principal investigator plans to apply the Fourier approach to several types of problems. Volume comparison problems ask what kind of data about sections or projections is necessary to conclude that the volume of one body is greater than the volume of another body. Volume difference problems ask how errors in the data related to plane sections or projections of a solid affect the computation of the volume of this solid. Slicing problems are concerned with bounds for the volume of a solid in terms of the areas of its slices through the center. The principal investigator expects that methods developed in this project will also be applicable to several problems in functional analysis and probability, in particular to the study of stable random processes (i.e., random laws inheriting the self-reproducing property of the normal law). Stable laws are frequently used in statistics. An important part of the project is the involvement and training of graduate students and postdocs. The problems considered in this project belong to several areas of mathematics: geometric tomography, convex geometry, functional analysis, and probability. However, the strategy of solution is common for most of the results: the question is translated into the language of the Fourier transform and then treated as a problem from harmonic analysis. In geometric tomography the principal investigator plans to apply Fourier methods to stability and separation in volume comparison problems, namely, volume difference and slicing inequalities for the section, projection, and curvature functions. In particular, he plans to study quantitative versions of the Busemann-Petty problem, which asks whether convex bodies with uniformly smaller areas of central plane sections in all directions necessarily have smaller volumes. He also plans to continue his work on the slicing problem of Bourgain. Another direction of research in the project is a connection between convex geometry and functional analysis. The principal investigator has recently found that intersection bodies, one of the main objects of interest in convex geometry, can be considered as the unit balls of certain Lebesgue spaces. He plans to apply this connection to the problem of duality between intersection and projection bodies, as well as to the problem of estimating the distance from an arbitrary convex body to the classes of intersection and projection bodies. Yet another direction of research is related to a problem in probability theory going back to Levy. The problem is to characterize random vectors having the property that all linear combinations of their coordinates have the same distribution, up to a constant. A conjecture is that the only vectors with this property are mixtures of stable random vectors. The principal investigator has already made several contributions in this direction, including a solution of a 1938 problem of Schoenberg, but the main question remains open.