积分几何与凸体几何分析中的等周问题及Minkowski问题研究

积分几何与凸几何中的等周问题、Minkowski问题是数学中的重要问题.陈省身、苏步青、吴大任、任德麟等早期的不变运动公式和等周问题等作出的先驱工作和贡献影响着积分几何与凸体几何分析的发展.. . 等周问题对应的等周不等式是数学中最重要的几何不等式，与等周不等式等价的分析不等式是著名的Sobolev不等式. 等周不等式的自然和重要推广是Bonnesen型不等式、Minkowski不等式、Alexandrov-Fenchel 不等式等，它们对Minkowski问题的解决至关重要.Minkowski问题是几何和分析中的重要问题，包括青年数学家黄勇在内的中国几何学家最近在积分几何与凸几何分析中的（Lp，对偶, log）Minkowski问题上取得令人振奋的新进展.. . 我们将在过去所取得的一些重要成果和突破的基础上，继续研究等周问题、Minkowski问题极其应用并期望取得新的突破.

Ioperimetric problem and Minkowski problem, in integral and convex geometry, are among the most important problems in Mathematics. Modern integral and convex geometric analysis on set of convex bodies have developed to a new stage by pioneer works and contributions in invariant kinematic formula and isoperimetric problem of S. S. Chern, B. Su, D. Wu and D. Ren...Associate isoperimetric inequality of the isoperimetric problem is the most important and profound geometric inequality in Mathematics. It is known that the isoperimetric inequality is equivalent to Sobolev inequality, the dual functional inequality of isoperimetric inequality. The Bonnesen-style inequalities are strengthens and extensions of isoperimetric inequality. More natural and important extensions include Minkowski inequality, Alexandrov- Fenchel inquality, Gaussian isoperimetric inequality (on probabilistic distribution on convex domain). These inequalities are crucial for Minkowski problem. In the past decades, we investigated these Bonnesen-style inequalities and accumulated experiences and achieved progress. The Minkowski problem is one of the most important problems in global differential geometry and analysis. Recently, mathematicians achieved astonished progress in Minkowski problem in convex geometry, such as Lp Minkowski problem, dual Minkowski problem (Huang-Lutwak-Yang-Zhang, Acta Math. 2016), and log-Minkowski problem...Based on our experiences and achievements in past years, we will focus our investigations on isoperimetric problem (Bonnesen-style Minkowski inequalities, Bonnesen-style Alexandrov-Fenchel inequalities and their dual analytic inequalities) and Minkowski problem. We are faithful to achieve more results and breakthrough.