Applications of Fourier analysis to convex geometry and geometric tomography

拟议研究项目的主要主题是利用傅里叶分析方法研究凸体和星体的截面、投影、体积。基于从各种测量获得的数据来检索凸体或星体的信息的区域称为几何层析成像。由R.J.Gardner所著的《几何层析》一书对这一领域的已知结果、技术和公开问题进行了极好的概述。该领域位于几个数学分支的交叉点：泛函分析、调和分析、凸几何、微分几何、积分几何等。这解释了几何层析成像所采用的方法的多样性。我们的方法是基于谐波分析技术的。上个世纪末，A.Koldobsky在这个方向上取得了根本性的发现，他将傅立叶变换应用于凸体截面的研究(参见《凸几何中的傅立叶分析》一书)。从那时起，这种方法已经成功地用于解决几何层析、凸几何、几何泛函分析等各种问题。提出的研究项目的目标是寻找这些技术的新应用。尽管这个项目涉及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。最广为人知的应用之一是计算机辅助断层成像，它允许从人类患者的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.