本项目将在新近建立的凸体的 Orlicz Brunn-Minkowski 理论框架下， 综合利用凸体几何、泛函分析和变分的方法， 重点研究投影体的仿射极值问题，具体涉及： 投影体的均质积分、投影极体的均质积分、凸体的仿射均质积分、多重投影体的均质积分、仿射均质积分比。本项目的研究课题是凸体几何在最近几年的发展中自然出现的问题，它们紧密追随了凸体几何的国际主流和研究热点，亟待解决且具有一定的挑战性。研究过程中所发展的概念和技巧必定会得到一些实质性的有意义的成果，从而丰富 Brunn-Minkowski 理论的内涵。

In the framework of the newly established Orlicz Brunn-Minkowski theory of convex bodies, this project will focus on investigating some affine extremum problems of projection bodies, by using the methods of convex geometry, functional analysis and calculus of variations. To be specific, it will include the following: the quermassintegrals of projection bodies, the quermassintegrals of polar projection bodies, the affine quermassintegrals of convex bodies, the quermassintegrals of multi-projection bodies, and the affine-quermassintegral-ratios of convex bodies. The above mentioned topics are challenging and urgently awaited to be solved ones in convex geometry, which are naturally evolved from the developments of convex geometry since 2010 and are closely traced the hot topics and mainstream. We believe that the concepts and techniques developed in this project would have to yield some essential and significant achievements, which will enrich the connotations of the Brunn-Minkowski theory.