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.

该项目着重于分析技术的应用，特别是傅立叶变换，用于研究某些类别的凸体的几何和凸性特性：Zonoids and交叉体。从几何到功能分析，这些物体已经在数学的几个领域中独立出现。在过去的二十年中，傅立叶分析技术的引入对这两类物体产生了重要的结果，并已用于统一其某些特性的研究。但是，仍然存在许多开放问题。拟议的研究指向三个主要方向：（a）更好地理解宗派与交叉体之间的关系，例如确保相交体是层状双重双重的条件，或研究其腐蚀力为ZONOIDS的差异行为的研究； （b）对交叉体操作员的凸性特性和固定点的研究； （c）从较低维信息（例如其平行截面的面积或周长）中重建非对称体的重建。凸体在许多数学领域的研究对象出现，并在物理，医学和计算机科学方面有应用。例如，从一个维度信息和二维信息中重建三维体的问题是医疗技术（例如X射线和CT扫描）的基础，并且还用于研究晶体的结构和物理特性。凸性在优化和线性编程问题中至关重要。该项目致力于发展数学理解凸体及其分析和几何特性之间的关系。 拟议的教育计划包括研究生班和研讨会，北达科他州立大学的本科项目以及与其他大学的游客交流，将有助于数学家准备在这些领域工作。进一步的外展活动，例如Sonia Kovalevsky Day的高中女生和NDSU数学俱乐部，将吸引代表性不足的学生进行数学研究。