Harmonic Analysis in Convex Geometry

这项研究的目标是发展调和分析方法来解决凸几何问题，包括几何层析问题和有关对偶和体积的问题。许多猜想规定，凸体的投影和截面之间一定存在直接的对偶联系。为了获得对这些联系的理解，重要的是要尝试获得所涉及的二元性现象的描述。这就是傅立叶分析发挥作用的地方。这项建议的一个主要组成部分是研究调查人员最近的工作中自然出现的问题。他们将使用傅里叶变换和其他调和分析工具作为凸体体积与其极轴之间的联系。他们还将讨论和研究有关凸体和星体的投影和截面的其他对偶问题。凸几何在现实生活中有很多应用。它的目标与许多相关且往往实用的领域的目标相似。最著名的例子之一是经典计算机辅助层析成像，它的目标是通过物体的线积分来重建物体的密度。其他的例子还有结晶学、机器人学、体视学和电子显微镜。该方案的一个想法是通过几何层析和调和分析将凸几何的理论结果与实际领域联系起来。此外，这份提案所涉及的问题的好处是，其中大多数问题不仅对本科生，而且对高中甚至初中的孩子都是直观的。即使是几何上的二元性概念也可以解释给高中的孩子听。另一方面，答案(经常)是反直觉的，在平面和空间中是不同的，这激发了学生对这门学科的兴趣。研究人员认为，这些问题将有助于吸引更多的学生学习调和分析和凸性，以及数学和自然科学。

The goal of the proposed research is to develop methods of Harmonic Analysis to solve problems from Convex Geometry including problems from Geometric Tomography and questions concerning duality and volume. Many conjectures stipulate that there must exist direct duality connections between projections and sections of convex bodies. In order to gain an understanding of these connections, it is important to try to obtain a description of the duality phenomena involved. This is where the Fourier Analysis comes into play. A major component of this proposal is to study problems, which arise naturally from the recent work of the investigators. They will use Fourier Transform and other Harmonic Analysis tools as a link between volumes of a convex body and its polar. They will also address and study other duality problems about projections and sections of convex and star bodies. Convex Geometry has a lot of real life applications. It has goals similar to those of many related and often practical areas. One of the best-known examples is Classical Computer Aided Tomography, which aims to reconstruct the density of objects by means of their line integrals. Other examples are crystallography, robotics, stereology, and electron microscopy. One of the ideas of this proposal is to connect theoretical results from Convex Geometry to those practical areas via Geometric Tomography and Harmonic Analysis. In addition, the advantage of the problems addressed in this proposal is that most of them are "intuitively clear" not only to undergraduate students, but also to children in high school or even middle school. Even the geometric notion of duality can be explained to a child in the high school. On the other hand, the answers are (very often) counter-intuitive, different in the plane and in the space, and this stimulates an interest of students to the subject. The investigators believe that those problems will help to attract more students to Harmonic Analysis and Convexity as well as to Mathematics and Science in General.