凸几何学与积分几何学都是现代凸几何分析的一个重要研究领域, 以凸体或星体为主要研究对象的几何学科. 探索及研究凸体或星体的几何测度(面积, 体积等)之间的联系及性质是凸几何学与积分几何学的主要工作, 其表现形式为几何不变量之间的等式和不等式关系, 这些等式和不等式对发现和研究凸体的性质意义重大, 刻画了最自然最本质的几何事实. 本课题中我们主要研究：1, 凸体的 Orlicz Brunn-Minkowski 理论以及 Orlicz Sobolev型的不等式和仿射的 Orlicz Sobolev型不等式以及与等周型不等式之间的内在联系. 2, 研究Orlicz 迷向凸体, Orlicz 迷向测度及表面积之间的联系及其性质. 3, 研究凸体的 M-sum及其相关的几何性质. 4, 研究投影体, 中心体, 相交体等在复空间理论以及Orlicz混合体积理论下的几何性质.

Convex geometry and integral geometry are the most important areas in modern convex geometric analysis, and with convex (or star) bodies as their main research objects. The main task is to explore and study their relationships between the geometry measures (area, volume, etc.), and it’s relationship always in form of the equality and inequality, which characterized as the most nature of the geometric facts. In this project, firstly, we mainly investigate the Orlicz Brunn-Minkowski theory, and the relationships between the Orlicz Sobolev type inequality, affine Orlicz Sobolev type inequality and the Orlicz isoperimetric inequality. Secondly, the Orlicz isotropic convex bodies, Orlicz isotropic measure and surface area will be researched. Third, the property of M-sum of convex bodies will be researched. Last, the property of projection body, centroid body and intersection body in the theory of complex space and the theory of orlicz mixed volume will be investigated.