Non-Linear Aspects of the Geometry of Convex Sets

摘要奖：DMS-9971202首席研究员：Paul Goodey和Marilyn BreenMarilyn Breen将继续她在欧几里得空间能见度问题上的工作。这些将集中在Helly-型定理上，旨在确定一个集合是否为星形的。一个主要目的是将Krasnosel‘skii定理推广到紧集以外。Paul Goodey的工作将集中在从低维数据中确定高维几何信息。这些数据通常来自高维对象的截面或投影。近年来，投影和截面之间出现了一些惊人的联系。他将试图找到更多的联系，并获得对这些现象的系统解释。预计将使用的技术将来自调和分析、群表示理论和经典凸几何。在这项拨款下要考虑的几何问题在相当不同的领域有应用。在纯粹的几何层面上，能见度问题的研究与机器人学有关。Helly-型定理已被证明对几何优化算法具有重要意义。事实上，Helly数决定了某些算法进行的速度界限。从低维数据重建物体是体视学和层析成像的核心。在这种情况下，对切片的研究相当于研究所考虑的物体的切片。从这些切片中获取有关原始物体的信息的能力在整个地球科学和医学研究中都是如此。物体的投影是X射线的同义词。一些有待解决的项目将对从X射线重建物体以及设计有效的算法来实现这种重建产生影响。

AbstractAward: DMS-9971202Principal Investigator: Paul Goodey and Marilyn BreenMarilyn Breen will continue her work on visibility problems inEuclidean space. These will focus on Helly-type theorems and areaimed at determining whether or not a set is starshaped. A majorobjective is to extend Krasnosel'skii's theorem beyond thesetting of compact sets. Paul Goodey's work will focus on thedetermination of high dimensional geometric information from lowdimensional data. This data usually arises from sections orprojections of the high dimensional object. In recent years somesurprising connections between projections and sections haveemerged. He will try to find further connections and to obtain asystematic explanation of these phenomena. It is anticipatedthat the techniques to be used will be drawn from harmonicanalysis, group representation theory and classical convexgeometry.The geometric questions to be considered under this grant haveapplications in quite diverse areas. At a purely geometriclevel, the study of visibility questions is connected withrobotics. Helly-type theorems have been shown to havesignificance for geometric optimization algorithms. In fact,Helly numbers determine bounds for the speed at which certainalgorithms proceed. Reconstruction of objects from lowdimensional data lies at the heart of stereology and tomography.The study of sections, in this context, amounts to studyingslices of the object under consideration. The ability to obtaininformation about the original object from these slices hasimplications throughout geosciences and in medical research, forexample. Projections of objects are synonymous with X-rays.Some of the projects to be addressed will have implications forthe reconstruction of objects from their X-rays and for thedesign of efficient algorithms to achieve such reconstruction.