Fourier analytic methods in convex geometry and geometric tomography

所提出的研究方案属于凸几何、几何泛函分析、几何层析和离散层析等领域。这些都是非常丰富和活跃的领域。这些领域的结果和方法在数学的许多其他分支中得到应用，以及其他科学，如材料科学、医学、地球物理等。例如，几何层析成像为CT扫描仪、研究地震活动、寻找石油等提供了数学基础。这项提议的总体目标是通过观察凸体(和其他物体)的截面或投影的大小来研究它们。我计划用来解决各种几何问题的主要工具是调和分析。几个长期存在的问题，包括关于凸体中心截面的Busemann-Petty问题和关于最大截面的Klee问题，已经用傅立叶分析得到了解决。这项提议的目标之一是进一步发展和改进这种技术，以及寻找傅立叶分析在几何问题中的新应用。这些方法将与其他方法结合使用，特别是那些来自几何泛函分析、离散几何和代数的方法。这项提案有三个主要主题。第一部分是关于代数可积域的Arnold问题。第二部分是Gruenbaum型不等式的推广和应用。在最后一部分，我将描述几何层析成像和离散层析成像中的一些唯一性问题。

The proposed research program belongs to the areas of convex geometry, geometric functional analysis, geometric tomography, and discrete tomography. These are very rich and active areas. Results and methods from these areas find applications in many other branches of mathematics, as well as other sciences, such as materials science, medicine, geophysics, etc. For example, geometric tomography provides a mathematical foundation for CT-scanners, the study of seismic activity, the search for oil, to name a few. The general objective of this proposal is to study convex bodies (and other objects) by looking at the sizes of their sections or projections. The main tools that I plan to use to attack various geometric problems are those of harmonic analysis. Several longstanding problems, including the Busemann-Petty problem about central sections of convex bodies and Klee's problem about maximal sections, have been solved with the help of Fourier analysis. One of the goals of this proposal is to further develop and refine such techniques, as well as to find new applications of Fourier analysis to geometric problems. These methods will be used in combination with other methods, especially those coming from geometric functional analysis, discrete geometry, and algebra. There are three main topics in this proposal. The first part is dedicated to Arnold's problem about algebraically integrable domains. The second part is about generalizations and applications of Gruenbaum-type inequalities. In the last part I will describe some uniqueness problems in geometric tomography and discrete tomography.