积分几何与凸几何分析不等式

Integral geometry is closely related with convex geometric analysis and probability. Crofton investigated invariant measures on set of geometric subjects (points, lines, surfaces, subspaces, groups of transformations and etc.). Czube, Poincare and Blaschke investigated the global differential geometry via probability idea. S. S. Chern and Weil pioneered integral geometry on homogeneous space by introducing invariant measure on local compact group. The relations among invariant measures on set of geometric subjects are geometric equalities and inequalities. Those geometric equalities and inequalities describe some natural and important physical phenominas. In the past decades, the intervince among theory of Lie group, algebraic geometry, functional analysis and etc. greatly help the development of geometric inequalities. Integral and convex geometric inequalities has been a very important branch of global differential geometry since last century. Integral geometry is a fundamental branch of mathematics and has a wide range of applications and development and overlaps with physics, analysis, algbra, mining, stereologists, medical science (geometric tomography), material science and information science. In the past years, we spent times and ernergy on this beautiful and fasnating branch of mathematics and achieved some exciting results and prograss that are recognized by scholars and experts in this field. We are going to continue our research. We also hope to cultivate young researchers, keep and inhense our strong teaching and research characteristcs and style. We hope to work on some great and well-known mathematical works, such as Alexandrov-Fenchel inequality in homogeneous space, Bonnesen-style Alexandrov-Fenchel inequalities, the Bonnesen-Fenchel conjecture.

积分几何与凸几何分析及概率论密切相关。Crofton 研究几何元素集的测度，Poincare-Blaschke用概率思想研究整体微分几何问题。陈省身，Weil 将局部紧群上不变测度引入积分几何，形成齐性空间积分几何学。几何元素（如空间中的点、线、面、子空间、变换群等）集上的几何测度之间的关系表现为一些几何等式和不等式，这些等式和不等式刻划了最自然的事实及物理现象。过去几十年来，李群论、代数几何、泛函分析等领域的交叉对几何不等式的发展起了巨大推动作用，积分几何不等式已发展成整体微分几何的重要分支。积分几何与凸几何分析不等式的应用遍及数学，物理很多分支,如分析、代数、采矿学（探针搜索）、医学（肿瘤切片），信息工程等。过去几年我们已经重视和加强了这一领域的研究并取得了一些得到国际同行认可的成果。我们将继续研究，同时继续培养优秀年轻人，保持和巩固自己的研究特色，争取作出有影响的世界水平的工作。

积分几何与凸几何分析及概率论密切相关，在过去几十年中，积分几何不等式已发展成整体微分几何的重要分支。积分几何与凸几何分析不等式的应用遍及数学，物理很多分支,如分析、代数、采矿学（探针搜索）、医学（肿瘤切片），信息工程等。过去十多年来我们已经重视和加强了这一领域的研究并取得了一些得到国际同行认可的成果，同时培养了一批优秀的年轻学者，我们将保持和巩固自己的研究特色，争取作出有影响的世界水平的工作。.古典的等周问题是确定固定周长的最大面积平面图形，这一问题起源于古希腊。但第一个严格的数学证明直到19世纪才由Weierstrass给出，他的工作是在Bernoulli, Euler和Lagrange的基础上完成的并且假设了图形边界具有光滑性. Hurwitz后来发表的一篇简短证明中用到了Fourier 级数并去掉了光滑性条件。Schmidt（1938）把区域与圆盘比较给出了一个非常优美的证明，Schmidt的证明中用到了Green定理和Caushy - Schwarz不等式 表示的的弧长和面积公式。自那以后就有了许多证明而且很多非常简明。等周问题已经被推广到曲面上以及（难度高的）高维空间区域。另一个非常重要的问题是Minkowski problem，即给点单位球面S^{n-1}上的一个连续函数g, “”存在一个光滑闭曲面M使单位外法向量u处的Gauss曲率为g的充分必要条件”是什么？.我们在这些问题上作了教深入研究并取得一些进展。