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

该项目侧重于分析技术的应用，特别是傅里叶变换，以研究某些类凸体的几何和凸性性质：zonoid和交体。这些体已经独立地出现在数学的几个领域，从几何到泛函分析。在过去的二十年里，傅里叶分析技术的引入已经产生了关于这两类物体的重要结果，并已被用于统一对它们的一些性质的研究。然而，许多悬而未决的问题仍然存在。建议的研究方向主要有三个：(a)更好地理解分形体与相交体之间的关系，例如确保相交体是分形体的对偶条件，或研究极点为分形体的分形体的渐近行为；(b)交体算子的凸性和不动点的研究；(c)从较低维信息重建非对称体，例如其平行截面的面积或周长。凸体作为研究对象出现在数学的许多领域，并在物理学、医学和计算机科学中有应用。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，它也用于研究晶体的结构和物理性质。凸性在最优化和线性规划问题中占有中心地位。该项目致力于提高对凸体的数学理解及其解析和几何性质之间的关系。拟议的教育计划，包括研究生课程和研讨会，北达科他州立大学的本科生项目，以及与其他大学的访问交流，将有助于数学家在这些领域的工作。进一步的推广活动，如高中女生索尼娅·科瓦列夫斯基日和NDSU数学俱乐部，将吸引代表性不足的学生学习数学。

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.