Applications of Fourier analysis to convex geometry and geometric tomography

点击翻译按钮获取中文摘要

The main theme of the proposed research project is centered around the study of sections, projections, volumes of convex and star bodies using methods of Fourier analysis. The area dealing with the retrieval of information about convex or star bodies based on data obtained from various measurements is known as geometric tomography. The book "Geometric Tomography" by R. J. Gardner gives an excellent overview of known results, techniques, and open problems in this area. The area lies at the intersection of several branches of mathematics: functional analysis, harmonic analysis, convex geometry, differential geometry, integral geometry etc. This explains the diversity of methods employed in geometric tomography. Our approach is based on techniques of harmonic analysis. Fundamental discoveries in this direction were made by A. Koldobsky at the end of the last century, who applied the Fourier transform to the study of sections of convex bodies (see the book "Fourier Analysis in Convex Geometry"). Since then such methods have been successfully used to solve a variety of problems in geometric tomography, convex geometry, geometric functional analysis etc. The goal of the proposed research project is to find new applications of such techniques. Even though this project concerns purely theoretical aspects of geometric tomography, results and methods from this area find applications in many scientific fields. One of the most well-known applications is computer assisted tomography, which allows to generate images from X-rays of human patients.