Fourier analysis in geometric tomography

几何层析成像是一个数学领域，人们根据这些固体的剖面和投影信息来研究固体的几何特性。其中一个例子是x射线断层扫描，它在科学、医学和工程领域有着广泛的应用。在数学中，几何断层扫描与凸几何和功能分析重叠。首席研究员最近发现几何层析成像领域的重要问题可以用最流行的谐波分析工具傅立叶变换来解决。该工具允许将数据分解为简单的谐波组合（即，分解为具有周期性重复值的函数），并通过这样做，将几何问题减少为与谐波相关的计算，其理论得到了很好的发展。这种方法已经导致了该领域几个长期存在的问题的分析解决方案，包括Busemann-Petty问题、Shephard问题和比例维截面的切片问题。在这个项目中，首席研究员计划将傅里叶方法应用于几种类型的问题。体积比较问题问的是需要什么样的截面或投影数据才能得出一个物体的体积大于另一个物体的体积的结论。体积差问题是关于一个实体的平面截面或投影的数据误差如何影响这个实体的体积计算。切片问题关注的是固体的体积边界，它是由穿过中心的切片的面积决定的。主要研究者期望本项目开发的方法也适用于泛函分析和概率中的几个问题，特别是稳定随机过程的研究（即继承正常规律的自我复制特性的随机规律）。统计中经常使用稳定定律。该项目的一个重要组成部分是研究生和博士后的参与和培训。在这个项目中考虑的问题属于数学的几个领域：几何断层扫描，凸几何，功能分析和概率。然而，大多数结果的解决策略是通用的：问题被转换成傅里叶变换的语言，然后作为谐波分析的问题来处理。在几何断层扫描中，首席研究员计划将傅里叶方法应用于体积比较问题中的稳定性和分离，即截面、投影和曲率函数的体积差和切片不等式。特别是，他计划研究Busemann-Petty问题的定量版本，该问题是关于在所有方向上中心平面截面面积均匀较小的凸体是否必然具有较小的体积。他还计划继续研究布尔甘的切片问题。该项目的另一个研究方向是凸几何与功能分析之间的联系。首席研究员最近发现，交叉体是凸几何中主要感兴趣的对象之一，可以被认为是某些勒贝格空间的单位球。他计划将这种联系应用于相交体和投影体之间的对偶问题，以及估计从任意凸体到相交体和投影体类的距离的问题。然而，另一个研究方向与一个可以追溯到列维的概率论问题有关。问题是描述随机向量的特征，使其坐标的所有线性组合具有相同的分布，直到一个常数。一个猜想是只有稳定随机向量的混合物才具有这种性质。首席研究员已经在这个方向上做出了一些贡献，包括解决了1938年勋伯格的一个问题，但主要问题仍然没有解决。

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.