在几何中，最优传输问题，闵可夫斯基问题和蒙奇-安培方程是三个表面上不同，但是本质上紧密相关的问题。它们不仅有丰富的结构和漂亮的理论，而且在很多工程领域有广泛的应用。但是传统的算法，无论是解最优传输问题的线性规划，还是解蒙奇-安培方程的有限元方法，都不能揭示这个本质的联系。本项目基于凸多面体几何，提出一个统一的框架，它可以同时解上述三个问题，从而能够揭示深层的联系和丰富相关的理论。同时该框架能给出更精确的解。我们的关键观察是1）它们共享一个具有几何意义的变分原理，2它们与计算几何中经典的Voronoi图有紧密联系。这使得我们能够发展有效算法，从而解决实际工程问题。本项目将从理论，算法，应用三个层次来开展研究,并将理论框架和算法从欧氏空间推广到一般的黎曼流形。

In geometry, optimal transport problem (OTP), Minkowski problem and Monge-Ampere equation (MAE) are three seemingly differet but closed related problems. They not only have rich structures and beautiful theories, but also have many applications in various areas of engineerings and biomedecine. However, the conventional methods, either linear programing for OTP or finite element method for MAE, do not recognize the fundamental relation among these problems. In this project, we propose a framework based on the convex geometry that can solve these three problems altogether and discovers the intrinsic relations among them. Our key insights are 1) there is a variational principle shared by these problems;2)they are closely connected to power diagram, a generalization of the classic Voronoi diangram in computational geometry. This enables us to develop efficient and robust algorithms and thus to solve the engineering problems in real world. We will carry out our research in three levels: developing theory, designing algorithms, and exploring applications in computer vision and computer graphics, geometric modeling and medical imaging.