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.
