Mathematical Sciences: Lower Order Support Functions of Convex Bodies

威廉·费里将对几何凸性的一些经典问题进行研究。他的方法非常符合现代的脉络，将涉及将关于球函数的结果推广到更复杂的流形上的函数。这些问题中的许多都涉及在几何概率和随机几何中具有重要意义的不等式。Firey将研究凸体的高阶支撑函数。这些都是支持函数在Grassman流形上的自然推广。他将研究它们与各种不平等的关系，并研究它们的导数如何与物体的曲率相关。这项工作为解决关于预先分配的中间曲率函数的中间Christoffel-Minkowski问题提供了前景。它还可能揭示亚历山德罗夫-芬切尔不平等中的极端条件。

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.