Geometric, convexity and regularity properties of certain classes of convex bodies

某些类凸体的几何、凸性和正则性性质

• 批准号: 1100657
• 负责人: Maria Alfonseca
• 金额: $ 6.24万
• 依托单位: North Dakota State University Fargo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100657&HistoricalAwards=false
• 关键词: Geometric; convexity; regularity; properties; certain

This project focuses on the application of Analysis techniques, in particular the Fourier transform, to the study of the geometric and convexity properties of certain classes of convex bodies: Zonoids and intersection bodies. These bodies have appeared independently in several areas of Mathematics, from Geometry to Functional Analysis. In the last two decades, the introduction of Fourier analytic techniques has yielded important results about these two classes of bodies, and has been used to unify the study of some of their properties. However, many open questions still remain. The proposed research is directed in three main directions: (a) a better understanding of the relation between zonoids and intersection bodies, such as conditions that ensure that an intersection body is the dual of a zonoid, or the study of the asymptotic behavior of zonoids whose polars are zonoids; (b) the study of the convexity properties and fixed points of the intersection body operator; (c) reconstruction of non-symmetric bodies from lower dimensional information, such as the area or the perimeter of their parallel sections. Convex bodies appear as objects of study in many areas of Mathematics, and have applications in Physics, Medicine, and Computer Science. For example, the problem of reconstructing a three dimensional body from one and two dimensional information is the basis of medical techniques such as X-rays and CT-scans, and it is also used in the study of the structure and physical properties of crystals. Convexity holds central importance in optimization and linear programming problems. This project works towards the advancement of the mathematical understanding convex bodies and the relation between their analytic and geometric properties. The proposed educational program, which includes graduate classes and seminars, undergraduate projects at North Dakota State University, and visitor exchanges with other Universities, will help prepare mathematicians to work in these areas. Further outreach activities, such as Sonia Kovalevsky Day for high school girls and the NDSU Math Club, will engage underrepresented students in the study of Mathematics.

该项目重点关注分析技术（特别是傅里叶变换）的应用，以研究某些类别的凸体（Zonoid 和交集体）的几何和凸性特性。这些机构独立出现在数学的多个领域，从几何到泛函分析。在过去的二十年中，傅里叶分析技术的引入已经取得了关于这两类物体的重要成果，并已被用来统一对其某些特性的研究。然而，许多悬而未决的问题仍然存在。拟议的研究主要针对三个方向：（a）更好地理解类流星体和交集体之间的关系，例如确保交集体是类流星体的对偶的条件，或者研究类流星体的渐近行为其极地是黄带； (b)交体算子的凸性和不动点的研究； (c) 从低维信息重建非对称物体，例如其平行截面的面积或周长。凸体作为许多数学领域的研究对象出现，并且在物理、医学和计算机科学中都有应用。例如，从一维和二维信息重建三维身体的问题是X射线和CT扫描等医学技术的基础，也用于晶体结构和物理性质的研究。凸性在优化和线性规划问题中具有核心重要性。该项目致力于增进对凸体及其解析和几何性质之间关系的数学理解。 拟议的教育计划包括北达科他州立大学的研究生课程和研讨会、本科项目以及与其他大学的访客交流，将有助于数学家为这些领域的工作做好准备。进一步的外展活动，例如高中女生索尼娅·科瓦列夫斯基日和 NDSU 数学俱乐部，将让代表性不足的学生参与数学学习。