Harmonic Analysis in Convex Geometry

这个项目解决的问题是关于周围空间(比如人体的心脏)中形状的属性，这些属性可以从阴影或切片的信息中推断出来(比如在医学成像中)。这些问题的好处是，其中许多问题的解决方案不仅对研究生，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(经常)是违反直觉的，需要使用属于现代数学不同分支的最先进和复杂的工具。此外，许多问题不是源于“纯粹的数学”，而是源于医学成像和断层扫描。出于这个原因，这些解决方案可能会发现非常有趣的生物医学应用。目前的项目是两位主要研究人员长期合作的延续。他们将继续使用和发展调和分析的方法来解决凸几何和离散几何中出现的问题。这些问题包括关于一般测度的Brunn-Minkowski型不等式的问题，以及与不同版本的切片不等式有关的问题，包括用于一般测度的切片不等式和离散版本的切片不等式。首席调查人员还计划在提供有关投影和截面的大小(或某些其他性质)的信息的情况下，继续他们关于凸体的独特确定的工作。这将涉及拓扑学、概率学和傅立叶分析方法的混合，部分工作将集中在主要研究人员及其合作者先前在偶数维上解决的Bonnensen经典问题。在这项工作中开发的许多技术都是新的，人们对这些方法可以帮助解决几何断层扫描中的一些经典(但仍未解决)问题寄予厚望。

The problems addressed in this project ask about properties of shapes in an ambient space (like a human heart in the body) that can be inferred from information about their shadows or slices (as in medical imaging). The advantage of the problems is that the solutions to many of them are "intuitively clear" not only to graduate students, but also to undergraduate students, and in some cases, even to high-school pupils! On the other hand, the answers are (very often) counterintuitive, requiring the use of the most advanced and sophisticated tools belonging to the different branches of modern mathematics. Moreover, many of the problems originated not in "pure math," but in medical imaging and tomography. For this reason the solutions might find very interesting biomedical applications.The current project is a continuation of the long-time collaboration between the two principal investigators. They will continue to use and develop the methods of harmonic analysis to solve problems arising in convex and discrete geometry. These problems include ones about Brunn-Minkowski-type inequalities for general measures, as well as questions related to different versions of slicing inequalities, including a slicing inequality for general measures and a discrete version of the slicing inequality. The principal investigators also plan to continue their work on the unique determination of convex bodies given information on the size (or some other properties) of projections and sections. This will involve a mixture of topological, probabilistic, and Fourier-analytic methods, and part of the work will be concentrated around a classical problem of Bonnensen that the principal investigators and their collaborators have previously solved in even dimensions. Many of the techniques developed in that work are new, and there are high hopes that the methods could help to solve a number of classical (but still open) questions in geometric tomography.