Mathematical Sciences: Non-linear Aspects of the Geometry ofConvex Sets

首席研究员将继续他对凸集几何的研究。该项目将专注于某些非线性几何问题。一个长期目标是将研究人员和其他人的各种已知结果推广到更复杂的格拉斯曼尼亚人身上。这些问题的动机是几何的，往往与积分几何和随机几何有关。这项新的研究将使用Radon变换、球谐函数和分布理论中的技术。球体、立方体和棱锥体等实体都是凸集的例子。这样的布景永远不会向内低头。两个这样的集合的交集总是凸集，而它们的并集通常不是凸集。高维相似集在商业应用中尤为重要。在这个项目中，研究人员将把凸集理论的思想扩展到更广泛的几何环境。

The principal investigator will continue his research into the geometry of convex sets. The project will focus on certain non-linear geometric problems. A long-term goal is to extend various known results by the investigator and others to more complex Grassmannians. The motivation for these problems is geometrical and frequently has connections with integral geometry and stochastic geometry. Techniques from the theory of Radon transforms, spherical harmonics, and distributions will be used in this new study. Solids such as the ball, cube, and pyramid are examples of convex sets. Such sets never bow inward. The intersection of two such sets is always convex, while their union, often not. Analogous sets in higher dimensions are particularly important in business applications. In this project the investigator will extend ideas from the theory of convex sets to a wider range of geometric settings.