The principal investigator will continue his work on the geometry of convex sets and on some nonlinear problems associated with the geometry. A major motivation for the research is a need for the development of formulas of translative integral geometry. The research will involve the study of certain newly discovered functionals which are related to mixed volumes. This award will support research in the general area of integral geometry. The geometry of a set may often be analyzed and described by a decomposition of the set into slices. The geometry of the slices is often fairly simple and easy to describe. The main difficulty in the analysis is usually the reconstruction of the underlying geometric features of the entire set from the geometry of the slices. One well know application of this area of mathematics is the CAT scan.
