The Fourier Transform and Convex Bodies

本文提出的研究目标是在傅里叶解析方法的发展上取得相当大的进展，以解决凸几何的几个问题，与截面和投影有关，以及凸体的唯一性、重构和对偶问题。最近发现的一个用曲率函数的傅里叶变换来表示投影体积的公式，已经导致了关于投影的几个结果的傅里叶解析证明，包括用极体的截面来表征投影体，谢泼德问题的傅里叶解析解（询问具有较小投影的对称凸体是否必然具有较小体积）与Busemann-Petty问题（谢泼德问题的部分对应）惊人地相似。这些结果的傅里叶解析证明的相似性特别表明，凸体的投影体积和凸体的截面之间一定存在深刻的对偶联系。为了在获得这种对偶现象的傅里叶解析描述方面取得进展，PI计划找到某些类物体的极值投影，对投影体和相交体进行进一步的研究，获得有关非中心截面的结果，并构造一个极也是高维投影体的非光滑投影体。凸性是一个非常古老的话题，至少可以追溯到阿基米德。由于它在线性规划、断层扫描、医学上的大量应用，它仍然受到青睐，令人惊讶的是，傅里叶分析方法直到最近才被应用于这一学科。这些方法可以作为思想的额外来源，来到这两个领域，凸几何和谐波分析，并将找到新的应用。同时，凸性是一个极其简单和自然的概念。它本身很有趣，也说明了一些关于数学的事实，这些事实或多或少是经典的，但总是很重要的，所以对本科生来说是一个完美的领域。首先，出现的问题或问题非常简单，易于表述和理解，因此学生不需要在接近材料之前上几节课。其次，直觉有时会误导“明显的问题”，本科生感受到学科的美。许多问题可以用相当基本的方法解决，但另一方面，许多问题的答案仍然未知或最近才被发现，通常使用与数学其他部分不同的技术。因此，对于更多的高年级学生和所有从事精确科学的人来说，这是一个完美的研究项目领域。

DMS 0400789D RyaboginKansas State UniversityThe Fourier Transform and Convex BodiesThe proposed research aims at achieving considerable progress towardsthe development of a Fourier analytic approach to the solution ofseveral problems of convex geometry, related to sections and projections, and also to the problems of uniqueness, reconstruction and duality of convex bodies. A recently discovered formulaexpressing the volume of projections in terms of the Fourier transformof the curvature function, has led to Fourier analytic proofs ofseveral results on projections, including the characterization of projection bodies in terms of sections of the polar body, and theFourier analytic solution to Shephard's problem (asking whether symmetric convex bodies with smaller projections necessarily have smaller volume) surprisingly similar to that of the Busemann-Petty problem (a section counterpart of the Shephard problem). The similarities in the Fourier analytic proofs of these resultsindicate in particular that there must exist deep dual connections between volumes of projections and sections of convex bodies.To achieve progress in obtaining the Fourier analytic description of this duality phenomena, the PI plans to find extremal projections of certain classes of bodies, to undertake a further study of the projection and intersection bodies, to obtain results concerning non-central sections, and to construct a nonsmooth projection bodywhose polar is also a projection body in higher dimensions.Convexity is a very old topic which can be traced at very least to Archimedes. It is still in favor due to its numerous applications to linear programming, tomography, medicine, and it is a surprise that Fourier analytic methods have been applied to the subject only very recently. These methods can serve as an additional source of ideas, coming to both fields, convex geometry and harmonic analysis, and will find new applications. At the same time, convexity is an extremely simple and natural notion. Interesting in itself, it also illustrates some facts about mathematics, facts that are more or less classical, but always important to realize, so it is a perfect field for undergraduates. First of all, questions or problems arise that are very simple to formulate and understand, so students do not need to take several classes before approaching the material.Secondly, intuition is sometimes misleading in ``obvious problems'', and the undergraduate feels the beauty of the subject. Many problems can be solved by fairly elementary means, but on the other hand, answers to many problems are still unknown or have been found recently,often using different techniques from other parts of mathematics.Therefore, it is a perfect field for research projects for more senior students and all people dealing with exact sciences.