凸体几何学中的几何测度及其 Minkowski 问题研究

本项目以凸体几何学为基础， 综合利用变分学、泛函分析和 Brunn-Minkowski 理论，重点研究凸体几何学中的几何测度及其 Minkowski 问题。具体包括： Lp （p>1）电容测度及其 Minkowski 问题； Lp (p<1) 表面积测度及其 Minkowski 问题；混合锥体积测度及其 Minkowski 问题等。此外，本项目继续攻关 Lutwak 仿射均质积分等周问题猜想和混合锥体积泛函的仿射等周问题。. 本项目的研究课题是近五年来国际凸体几何学界研究的热点和主流。也是几何分析、调和分析和椭圆偏微分方程等领域关注的重要问题。具有较大的难度和挑战性, 亟待解决。. 开展本项目研究势必促进课题组成员迈向凸体几何研究的高点。 研究过程中所发展的概念和技巧必定会得到实质性有科学意义的成果, 丰富凸体几何的内涵。

With the basis of convex geometry, this project will intensively focus on investigating geometric measures and their associated Minkowski problems, by comprehensively using methods from calculus of variations, functional analysis and Brunn-Minkowski theory. To be specific, it includes the following: Lp electrostatic capacitary measure and its Minkowski problem for p>1; Lp surface area measure and its Minkowski problem for p<1; Mixed cone volume measure and its Minkowski problem, etc. Furthermore, this project will continue to tackle the conjecture for Lutwak affine quermassintegrals, and affine isoperimetric problem for mixed cone volume functional...The above mentioned topics are closely traced the research hotspot and lie at the mainstream of convex geometry since 2012. They are also important problems absorbed by experts from geometric analysis, harmonic analysis and elliptic PDEs and urgently awaited to be solved , with great difficulty and challenge...The investigators would be strongly promoted to the highlighted position if the project is granted. We believe that the concepts and techniques developed in this project would have to yield some essential and significant achivements, which will greatly enrich the connotations of convex geometry.