Applications of Fourier analysis to convex geometry and geometric tomography

拟议研究项目的主题是围绕使用傅立叶分析方法研究凸体和星星体的截面、投影、体积。根据从各种测量中获得的数据处理有关凸体或星星体的信息的检索的领域被称为几何层析成像。R.加德纳给出了一个很好的概述已知的结果，技术，和开放的问题，在这一领域。该地区位于交叉的几个分支的数学：功能分析，调和分析，凸几何，微分几何，积分几何等，这解释了多样性的方法采用几何层析成像。我们的方法是基于谐波分析技术。这方面的基本发现是由A. Koldobsky在上个世纪末，谁应用傅立叶变换的研究部分凸体（见书“傅立叶分析凸几何”）。从那时起，这样的方法已被成功地用于解决各种问题的几何层析成像，凸几何，几何功能分析等建议的研究项目的目标是找到新的应用这些技术。尽管这个项目涉及的是几何层析成像的纯理论方面，但这一领域的结果和方法在许多科学领域都有应用。最著名的应用之一是计算机辅助断层扫描，其允许从人类患者的X射线生成图像。

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.