Fourier Analysis in Convex Geometry

这个项目的重点是研究固体的几何性质，基于其截面和投影的信息。凸几何的这一分支被称为几何层析成像。其中一个例子是x射线层析成像，它在科学、医学和工程中有许多应用。PI最近发现，几何层析的重要问题可以使用最流行的调和分析工具--傅里叶变换来解决。该工具允许人们将数据分解成具有周期性重复值的谐波函数的简单组合，通过这样做，人们可以将几何问题归结为与谐波相关的计算，其理论非常发达。这一方法解决了1956年凸体截面上的Busemann-Petty问题，1938年勋伯格关于正定函数的问题，以及任意函数的切片问题。PI计划进一步发展傅立叶方法，并将其应用于凸几何、泛函分析、概率论和代数之间的一系列问题。例如，你能找到一个代数方程，它的解是给定体的截面吗？一个人能从涉及该固体的某些截面或投影的面积的数据中估计该固体的体积吗？哪些随机变量是稳定的，也就是说，这些变量的几个副本的和总是复制相同的变量，直到一个常量？该项目的一个重要部分是研究生和博士后的参与和培训。所考虑的问题连接了几个数学领域--凸几何、泛函分析、概率和代数。然而，对于大多数结果，求解策略是共同的--问题被转化为傅里叶变换的语言，然后作为调和分析的问题来处理。PI计划通过考虑Arnold问题，进一步发展凸几何和代数之间的联系，回到牛顿《原理》中的引理28。问题是刻画那些截断面积是割平面参数的代数函数的凸域。经典的Bourain切片问题是问是否每一个体积为1的对称凸体都有一个面积大于绝对常数的超平面截面。PI计划研究Busemann-Petty问题的低维和非对称版本，询问具有均匀较小的平面截面面积的凸体是否一定具有较小的体积。另一个方向是研究与凸体体积结果相关的Radon变换的一般性质。泛函分析的一个关联是研究凸体之间的距离、嵌入和对偶问题。关于正定函数和嵌入勒贝格空间的问题与概率论中的一个老问题有关，该问题要求刻画稳定的随机向量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on the study of geometric properties of solids based on information about their sections and projections. This branch of convex geometry is called geometric tomography. One of the examples is x-ray tomography, which has numerous applications in science, medicine and engineering. The PI recently discovered that important problems of geometric tomography can be solved using the most popular tool of harmonic analysis, the Fourier transform. This tool allows one to decompose the data into a simple combination of harmonics, functions with periodically repeating values, and, by doing this, one can reduce geometric problems to computations related to the harmonics, the theory of which is well developed. This approach has led to solutions of the 1956 Busemann-Petty problem on sections of convex bodies, the 1938 Schoenberg's problem on positive definite functions, the slicing problem for arbitrary functions. The PI plans to further develop the Fourier approach and apply it to a range of problems at the interface between convex geometry, functional analysis, probability theory and algebra. For example, can one find an algebraic equation whose solutions are sections of a given solid? Can one estimate the volume of a solid from data involving areas of certain sets of sections or projections of this solid? Which random variables are stable, that is, have the property that the sums of several copies of these variables always reproduce the same variable up to a constant? An important part of the project is the involvement and training of graduate students and postdocs.The problems considered connect several areas of mathematics - convex geometry, functional analysis, probability and algebra. 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. The PI plans to develop further connections between convex geometry and algebra by considering the problem of Arnold, going back to Lemma 28 from Newton's "Principia." The problem is to characterize those convex domains whose cut-off area is an algebraic function of the parameters of the cutting plane. The classical slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. The PI plans to study the lower-dimensional and non-symmetric versions of the Busemann-Petty problem asking whether a convex body with uniformly smaller areas of plane sections necessarily has smaller volume. Another direction is to study the general properties of the Radon transform associated with volumetric results about convex bodies. A connection with functional analysis is the study of distances between convex bodies, embedding and duality problems. The question about positive definite functions and embeddings in Lebesgue spaces is related to an old problem in probability theory asking for a characterization of stable random vectors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.