Fourier analysis in geometric tomography

几何层析成像是一个数学领域，其中一个调查的几何性质的基础上的信息，这些固体的部分和投影。X射线断层扫描就是其中的一个例子，它在科学、医学和工程领域有着广泛的应用。在数学中，几何层析成像与凸几何和泛函分析重叠。主要研究人员最近发现，几何层析成像领域的重要问题可以使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成谐波的简单组合（即，转换成具有周期性重复值的函数），并且通过这样做，将几何问题减少到与谐波相关的计算，谐波理论已经得到了很好的发展。这种方法导致了几个长期存在的问题在该地区的分析解决方案，包括Busemann-Petty问题，谢泼德问题，切片问题的比例尺寸的部分。在这个项目中，首席研究员计划将傅立叶方法应用于几种类型的问题。体积比较问题询问关于截面或投影的什么样的数据是必要的，以得出一个物体的体积大于另一个物体的体积的结论。体积差问题询问与固体的平面截面或投影相关的数据中的误差如何影响该固体的体积的计算。切片问题是关于一个实体的体积的边界，它是以通过中心的切片的面积来表示的。主要研究者期望在这个项目中开发的方法也将适用于函数分析和概率中的几个问题，特别是稳定随机过程的研究（即，随机定律继承了正常定律的自复制特性）。统计学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。该项目中考虑的问题属于数学的几个领域：几何层析成像，凸几何，泛函分析和概率。然而，大多数结果的解决方案的策略是共同的：问题被翻译成傅立叶变换的语言，然后作为一个问题，从谐波分析。在几何层析成像的主要研究者计划应用傅立叶方法的稳定性和分离的体积比较问题，即体积差异和切片不等式的部分，投影和曲率函数。特别是，他计划研究Busemann-Petty问题的定量版本，该问题询问在所有方向上具有均匀较小的中心平面截面面积的凸体是否一定具有较小的体积。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和泛函分析之间的联系。主要研究者最近发现，相交体，凸几何中的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问题的对偶性之间的交叉和投影机构，以及问题的估计距离从任意凸体类的交叉和投影机构。然而另一个研究方向与概率论中的一个问题有关，可以追溯到利维。这个问题是随机向量的特点，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，唯一具有这种性质的向量是稳定随机向量的混合。主要研究者已经在这个方向上做出了一些贡献，包括1938年勋伯格问题的解决方案，但主要问题仍然悬而未决。

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.