Some Variants of the Kakeya Problem

挂屋问题的一些变体

• 批准号: 0552041
• 负责人: Daniel Oberlin
• 金额: $ 8.45万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0552041&HistoricalAwards=false
• 关键词: Some; Variants; Kakeya; Problem

(1) This project will focus on certain geometric questions from classical analysis which have their origins in the well-known Kakeya problem: find the minimum dimension of a subset of d-dimensional space which contains a line segment in each direction. Of particular interest to the p.i. are estimates from below for the Hausdorff dimension of sets which have large intersections with certain collections of lines, planes, or spheres (or, in higher dimensions, with certain collections of k-planes or hyperspheres). A theme of this work will be to explore the scope and limitations of a particular method, due to the p.i., for treating the averaging operators which arise in the study of these problems. (2) This project is concerned with the structure of certain complicated subsets of higher dimensional spaces. The study of these sets, while of compelling intrinsic interest, should properly be viewed as a test of our current understanding of certain objects called averaging operators. The relevant averaging operators here are of two kinds: spherical averaging operators and Radon transforms. Spherical averaging operators are basic to our understanding of the propagation of waves through fluid media such as the ocean or the atmosphere. The Radon transform and its relatives are essential, for example, in problems of tomography such as those which arise in the field of medical imaging.

(1) 该项目将重点关注经典分析中的某些几何问题，这些问题起源于著名的挂谷问题：找到 d 维空间子集的最小维数，该子集在每个方向上都包含一条线段。 p.i 特别感兴趣。是与某些直线、平面或球体集合（或者，在更高维度中，与某些 k 平面或超球面集合）有较大交集的集合的 Hausdorff 维数的下面估计。这项工作的主题是探索特定方法的范围和局限性，由于 p.i.，用于处理这些问题研究中出现的平均算子。 (2) 该项目涉及高维空间的某些复杂子集的结构。对这些集合的研究虽然具有引人注目的内在兴趣，但应该正确地视为对我们当前对称为平均算子的某些对象的理解的测试。这里相关的平均算子有两种：球面平均算子和Radon变换。球面平均算子是我们理解波在海洋或大气等流体介质中传播的基础。例如，在诸如医学成像领域中出现的断层摄影问题中，氡变换及其相关变换是必不可少的。