Affine Invariants in Geometric Analysis

鉴于仿射几何在图像处理、信息技术和随机几何中的应用爆炸式增长，需要对凸体进行新的仿射不变量研究。此外，有必要开发新技术来解决传统方法无法解决的长期问题。

Given the explosion of applications of affine geometry to image processing, information technology and stochastic geometry, new research is needed to investigate novel affine invariants for convex bodies. Additionally, there is a need to develop new techniques to tackle long-standing problems which resisted conventional methods. Our project addresses precisely these issues on the subject of affine invariants understood here at large as to include centro-affine and equi-affine invariants. While part of our study will employ new curvature flows to study affine invariants for convex bodies with sufficiently regular boundary, we will start an original study of new affine invariants for convex bodies which are lacking smooth positive Gauss curvature. The long-term goal is fitting the two classes of problems into a global coherent theory which includes all convex bodies disregarding a particular boundary structure. The direct consequences of the proposed project will be novel research results in the area of affine and isoperimetric inequalities, and solutions to generalized Minkowski problems arising in the Brunn-Minkowski-Firey theory of convex bodies. A particular attention will be given to open problems in the class of polytopes and connections to the theory of valuations. Adding to the significance of the project, our methods will range from convex, discrete and differential geometry to partial differential equations and curvature flows, while following the common thread of invariance under certain groups of transformations.