凸几何分析是20世纪80年代在经典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个交叉学科。 Mahler猜想是凸几何分析中的一个重要的未解决问题。 Mahler猜想可以表述为：（1）在中心对称的凸体中，立方体具有最小的Mahler体积；（2）在非中心对称的凸体中，单纯形具有最小的Mahler体积。本项目的主要研究内容包括：三维空间中心对称凸体的Mahler猜想；特殊凸体类（关于一个坐标平面对称的凸体、平行截面体、旋转体、最多十个顶点的多胞形）上的Mahler猜想；Mahler猜想的函数形式及相关分析不等式；凸体极体的极值性质。在研究过程中，我们将把几何方法和现代分析工具（Fourier变换、球面调和、局部渐进理论等）结合起来，力争创造一些新的研究方法，以加深我们对凸体和空间的对偶性认识，推动Mahler猜想的研究。

Convex geometric analysis is an interdisciplinary subject of the geometry and functional analysis which has developed on the basis of the classical Brunn-Minkowski theory in 1980s. Mahler conjecture is an important open problem in convex geometric analysis. Mahler conjecture can be expressed as following: (1) For the centrally symmetric convex bodies, cubes have the minimal Mahler volumes; (2) For the non-symmetric convex bodies, simplexes have the minimal Mahler volumes. This project mainly includes: Mahler conjecture on the three-dimensional centrally symmetric convex bodies; Mahler conjecture on certain special classes of convex bodies (e.g., convex bodies which are symmetric with respect to a coordinate plane, parallel sections homothety bodies, bodies of revolution, polytopes with at most ten vertices); the functional forms of Mahler conjecture and related analysis inequalities; the extremal properties of the polar body of a convex body. In the research process, we will combine the geometrical methods with modern analytical tools (Fourier transform, spherical harmonics, the local asymptotic theory, etc.), and strive to create some new research methods to enhance our awareness of the duality of convex bodies and spaces, promote the research on Mahler conjecture.