凸几何分析是上世纪80年代在经典Brunn-Minkowski理论基础上发展起来的几何学与泛函分析相结合的一门交叉学科。 近年来凸几何分析的发展越来越与泛函分析的发展密不可分，两者相互渗透，相互促进。 仿射等周不等式的函数形式是一个具有代表性的、既与凸体的Brunn-Minkowski理论有关又涉及Sobolev函数理论的一个重要问题，它强于经典等周不等式的函数形式并以仿射等周不等式为其几何背景。 关于Lp空间的仿射等周不等式(如仿射Lp-Sobolev不等式、仿射Lp-Polya-Szego不等式等)已经被证明，并且凸几何分析已经发展到Orlicz-Brunn-Minkowski理论阶段， 其中Orlicz空间仿射等周不等式的存在性是一个重要的公开问题。 本项目主要研究仿射Orlicz-Polya-Szego不等式、仿射Orlicz-Sobolev不等式、以及它们的稳定形式。

Convex geometric analysis is an interdisciplinary subject of the geometry and functional analysis which has developed on the basis of the classical Brunn-Minkowski theory in 1980s. In recent years, the development of convex geometric analysis is becoming more and more closely related to the development of functional analysis theory. Both of them permeate each other and promote each other. The functional form of affine isoperimetric inequality is a representative problem, which is related to the theory of convex body Brunn-Minkowski and involves the theory of Sobolev functions, it is stronger than the classical functional isoperimetric inequality and the geometry behind it is the affine isoperimetric inequality. The affine isoperimetric inequality in Lp space (e.g., the affine Lp-Sobolev inequality, the affine Lp-Polya-Szego inequality, ect.) has been proved and recently the convex geometric analysis has been developed to Orlicz-Brunn-Minkowski theory, the existence of the affine isoperimetric ineuality in Orlicz space is an important open problem. The main research objective of this project is to prove that affine Polya-Szego inequalities； affine Orlicz-Sobolev inequalities, and their stability versions.