The Fourier Transform and Convex Bodies

傅立叶变换和凸体

• 批准号: 0400789
• 负责人: Dmitry Ryabogin
• 金额: $ 3.2万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400789&HistoricalAwards=false
• 关键词: Fourier; Transform; Convex; Bodies

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.

DMS 0400789 D Ryabogin堪萨斯州立大学傅立叶变换和凸体拟议的研究旨在实现相当大的进展towards发展的傅立叶分析方法来解决几个问题的凸几何，有关的部分和投影，也问题的唯一性，重建和对偶的凸体。最近发现的一个用曲率函数的傅里叶变换表示投影体积的公式，导致了几个关于投影结果的傅里叶分析证明，包括用极体的截面表征投影体，Shephard问题的Fourier解析解（询问具有较小投影的对称凸体是否一定具有较小体积）与Busemann-Petty问题惊人地相似（Shephard问题的一个对应部分）。这些结果的傅立叶分析证明的相似性特别表明，在投影体积和凸体截面之间必须存在深刻的对偶联系。为了在获得这种对偶现象的傅立叶分析描述方面取得进展，PI计划找到某些类别的物体的极值投影，对投影和相交物体进行进一步研究，得到关于非中心截面的结果，构造一个非光滑的投影体，其极线也是高维的投影体。它仍然是有利的，由于其众多的应用，线性规划，断层扫描，医学，这是一个惊喜，傅立叶分析方法已被应用到这个问题只是最近。这些方法可以作为一个额外的来源的想法，来到这两个领域，凸几何和谐波分析，并会找到新的应用。同时，凸性是一个非常简单和自然的概念。有趣的是，它还说明了一些关于数学的事实，这些事实或多或少是经典的，但总是很重要的实现，所以它是一个完美的领域为本科生。首先，出现的问题或问题很容易表述和理解，所以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”，本科生会感受到学科的美。许多问题可以通过相当基本的方法解决，但另一方面，许多问题的答案仍然是未知的，或者最近才发现，通常使用与数学其他部分不同的技术。因此，它是更高年级学生和所有从事精确科学的人的研究项目的理想领域。