离散对数 Minkowski 问题的研究

Convex geometric analysis is an interdisciplinary subject of geometry and functional analysis which has developed on the basis of the classical Brunn-Minkowski theory at the end of the twentieth Century. The Minkowski problem is an important open problem in convex geometric analysis, and it is also one of the hottest issues studied by modern geometers. The logarithmic Minkowski problem is considered to be the most important and difficult part of the Minkowski problem, and it is not completely solved even in low dimensional space. In this project,we mainly concentrate on the following contents: the existence of discrete logarithmic Minkowski problem in three-dimensional space; the uniqueness of the logarithmic Minkowski problem for the special convex bodies in three-dimensional space. During the research, by combining partial differential equation, the local asymptotic theory and the Brunn-Minkowski theory together, we will strive to find a more reasonable mathematical tool to solve the problems of the existence and uniqueness for the discrete logarithmic Minkowski problem in three-dimensional space.

凸几何分析是20世纪末在经典Brunn-Minkowski理论的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。 Minkowski 问题是凸几何分析中一个重要的公开问题，也是现代几何学家们研究的热点问题之一。对数Minkowski问题被认为是Minkowski问题中最重要而且难度也最大的一部分，甚至在低维空间中该问题也没有得到完全解决。本项目的主要研究内容是：三维空间中离散对数Minkowski问题的存在性；三维空间中特殊凸体类上对数Minkowski问题的唯一性。在研究过程中，我们将偏微分方程、局部渐进理论和Brunn-Minkowski理论相结合，寻找更合理的数学工具，力争解决三维空间中离散对数Minkowski问题的存在性和唯一性。