The principal investigator will solve a number of problems in the area of isometric inequalities. These involve generalized convex hypersurfaces, inequalities for affine surface area, intersection and projection problems, and the dual Brunn- Minkowski theory. Generalized convex bodies are related to convex bodies in the way generalized functions (distributions) are related to ordinary functions. Their utilization will enable the principal investigator to establish a number of conjectured affine isometric inequalities. The classical Brunn-Minkowski theory is ideal for handling questions involving projections of convex shapes. The slow progress on a number of important questions is due to the fact that this theory is unsuited for dealing with intersection problems. The principal investigator will use a dual Brunn- Minkowski theory to overcome these difficulties. This work has applications outside mathematics to such fields as stereology.
