The proposed research lies in the analytic part of geometric convexity. More specifically, it focuses on the Brunn-Minkowski theory, also known as the theory of mixed volumes. The principal investigators plan to generalize the theory to objects more general than convex bodies and extend the theory in the same way that the theory of Lp spaces extends the theory of L1 and L2 spaces. They will also continue work on the creation of a dual of the Brunn-Minkowski theory, where intersections replace projections. Following on this, they intend to study the partial differential equations which arise in this extended Brunn-Minkowski theory, specifically generalizations of the classical Minkowski problem. A central part of this work is establishing new isoperimetric inequalities within the extended Brunn-Minkowski theory and its dual. The theory of mixed volumes has already found a wide variety of applications ranging from statistics to geometric tomography. (Geometric tomography is the subject which attempts to describe three-dimensional objects from lower dimensional information such as X-rays, projections, or sections of the objects in question.) The principal investigators intend to extend and build new analogs of this important mathematical theory. A central part of this work will involve the invention (or discovery) of new isoperimetric inequalities. These inequalities, in particular affine isoperimetric inequalities, have proven to be of importance in subjects ranging from partial differential equations to geometric tomography to image analysis. They are very useful because they provide estimates about how large one quantity can be, say the volume of a body, when all that is known is some other quantity, say the surface area of the body.
