渐进泛函分析研究欧氏空间中凸体的几何性质随空间维数趋向无穷时的不变特征,它旨在应用分析的角度和工具研究凸几何问题.近二十年来渐进几何分析和凸几何领域一个主要的研究分支叫做“几何的函数化”.凸体几何量的函数化和与几何不等式相对应的泛函不等式是主要的研究内容.本项目在已有结果的基础上,继续研究函数的仿射不变量和在极变换下的泛函Blaschke-Santalo不等式.具体的,我们将建立凸函数(对数凹函数)的仿射不变点(映射),为函数的仿射等周不等式的建立奠定基础;我们还将找出极变换下使得泛函Blaschke-Santalo不等式等式成立的极值函数,并且进一步探讨建立一般对数凹测度下紧致的极变换泛函Blaschke-Santalo不等式的可能性,为解决Cordero–Erausquin问题提供新的角度.

Asymptotic functional analysis studies the properties of convex bodies with emphasis on the dependence of geometric and analytic invariants on the dimension. In other words, we study the convex bodies in the analytic and geometric point of view. In recent decades one of the main streams of research in this field and convex geometry is called “functionalization”, including the functionalization of geometric notions and geometric functional inequalities. In this proposal, we continue to put forward the study of functional affine invariants and the research on functional Blaschke-Santalo type inequalities with respect to polarity transform. Specifically, we aim to establish the affine invariant point (and mapping) for convex (and log-concave) functions, paving the way for the future study of affine isoperimetric inequality; meanwhile, we will devote efforts on the characterization of the functional Blaschke-Santalo inequalities with respect to polarity transform, and furthermore establishing its analog with general log-concave measures in order to shed light on the Cordero–Erausquin conjecture.