Harmonic Analysis in Convex Geometry

该项目中解决的问题涉及周围空间中形状的属性（如人体中的心脏），这些属性可以从有关其阴影或切片的信息（如医学成像）中推断出来。这些问题的美妙之处在于，其中许多问题的表述不仅对研究生，而且对本科生，在某些情况下，甚至对高中生来说都是“直观清晰的”。另一方面，答案往往是反直觉的，需要使用属于现代数学不同分支的最先进和最复杂的工具。许多问题的起源不是纯数学，而是医学成像和断层扫描，其解决方案可能会在生物医学中找到非常有趣的应用。该项目将通过培养研究生为美国的劳动力发展做出贡献。目前的项目是主要研究者之间长期合作的延续。PI和co-PI将继续使用和发展调和分析的方法来解决凸和离散几何中出现的问题。这些问题包括一组新的Bezout不等式的混合体积，起源于经典的代数几何。这些不等式涉及凸体投影的大小，并导致信息论和概率中非常意想不到的结果。主要研究人员还计划继续他们的工作的一套问题的几何断层摄影提出的Busemann和佩蒂在1956年。到目前为止，只有列表中的第一个问题得到了解决。PI和co-PI已经在一些与这个列表相关的问题上取得了进展，这些问题使用了交叉和投影体算子的迭代思想。他们还将继续研究与凸体的独特确定有关的问题，并提供有关投影和截面尺寸（或其他一些属性）的信息。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The problems addressed in this project concern the properties of shapes in an ambient space (like a human heart in the body) that can be inferred from the information about their shadows or slices (as in medical imaging). The beauty of the problems is that the formulations of many of them are "intuitively clear" not only to graduate students, but also to undergraduates, and in some cases, even to high-school pupils. On the other hand, the answers are very often counter-intuitive, requiring the use of the most advanced and sophisticated tools belonging to the different branches of modern mathematics. Many problems take their origin not in pure mathematics, but in medical imaging and tomography and the solutions might find very interesting biomedical applications. This project will contribute to US workforce development through the training of graduate students.The current project is a continuation of the long-time collaboration between the principal investigators. The PI and co-PI will continue to use and develop the methods of harmonic analysis to solve the problems arising in convex and discrete geometry. These problems include a new set of Bezout inequalities for mixed volumes, originating from classical Algebraic Geometry. The inequalities involve the size of projections of convex bodies and lead to very unexpected results in Information Theory and Probability. The principal investigators also plan to continue their work on the set of problems in Geometric Tomography proposed by Busemann and Petty in 1956. Only the first problem in the list has been solved so far. The PI and co-PI have made progress on a number of problems related to this list using the idea of the iteration of intersection and projection body operators. They will also continue to work on the questions related to the unique determination of convex bodies given the information on the size (or some other properties) of projections and sections.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.