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。该项目将重点研究某些非线性几何问题。长期目标是将研究者和其他人的各种已知结果扩展到更复杂的格拉斯曼模型。这些问题的动机是几何的，并且经常与积分几何和随机几何有关。Radon变换、球谐和分布理论的技术将用于这项新的研究。像球、立方体和金字塔这样的实体都是凸集的例子。这样的人从不向内屈服。两个这样的集合的交集总是凸的，而它们的并集却常常不是凸的。高维的类似集在业务应用程序中尤为重要。在这个项目中，研究者将从凸集理论扩展思想到更广泛的几何设置。

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.