极小几何表面积是凸几何分析与积分几何中重要的仿射不变量，它与Lutwak推广的Lp极小几何表面积把仿射微分几何、相对微分几何及Minkowski几何紧密联系在一起。本项目拟运用凸几何分析中Brunn-Minkowski理论主攻以下问题：1.应用p-Petty体研究Lp极小几何表面积的连续性；2.利用Steiner对称化、影子系统等方法研究Orlicz空间中的函数，结合分析中Fourier变换、Radon变换、余弦变换及卷积公式等研究Orlicz极小几何表面积，进而研究它在赋值理论中的应用；3.研究“对偶”的浮游体、照明体和表面体，并借助它们给出极小几何表面积的几何解释。我们还将研究关于极小几何表面积的等周不等式，同时研究极小几何表面积的应用，及其与仿射表面积的关系。这些结论对发展和完善凸几何分析中的Brunn-Minkowski理论都非常重要，在理论和应用上具有重要的意义。

Geominimal surface ara is an important affine invariant in convex geometric analysis and integral geometry. Geominimal surface area and its Lp extensions introduced by Lutwak are the bridges connecting affine differential geometry, relative differential geometry and Minkowski geometry. We will consider the following problems in this research: 1.To study the continuity of Lp geominimal surface area by p-Petty body; 2.By using the steiner symmetrization、shadow system and other methods to study the functions in Orlicz space, combining the Fourier transforms、Radon transforms、Cosine transforms and convolutions transforms to research the Orlicz geominimal surface area, furthermore study its application in valuation theory；3.To study the “dual” floating body、illumination body、surface body，and study the geometric interpretation of Lp geominimal surface area by them. We will also research the affine isoperimetric inequalities for Lp geominnimal surface area, and the connection between geominnimal surface area and affine surface area. These results are important for the development to Brunn-Minkowski Theory.