William Firey will carry out research into some of the classical problems of geometric convexity. His approach is very much in the modern vein and will involve extending results about spherical functions to functions on more complicated manifolds. Many of these problems involve inequalities which have significance in geometric probability and stochastic geometry. Firey will investigate higher order support functions of convex bodies. These are natural extensions of support functions to Grassmannian manifolds. He will study their relationship to various inequalities and investigate how their derivatives relate to the curvature of the bodies. This work holds out the prospect of contributing to the intermediate Christoffel-Minkowski problem concerning preassigned intermediate curvature functions. It may also shed light on the extremal conditions in the Alexandrov- Fenchel inequalities.
