几何分析系列报告之13【Deping Ye】

Abstract: The Minkowski type problem in convex geometry aims to solve the measure equation $\mu=\mathcal{M}(K, \cdot)$ for unknown convex bodies $K$, where $\mu$ is a pregiven Borel measure on the unit sphere $S^{n-1}$ and $\mathcal{M}(K, \cdot)$ is  a Borel measure on $S^{n-1}$ depending on $K$. Typical examples of the Minkowski type problems include the classical Minkowski problem and the dual Minkowski problem, etc. These problems have found their important applications in other areas, such as,  partial differential equations, computer science, etc. 

The Minkowski type problems for unbounded convex sets have attracted more attentions recently. These unbounded convex sets are closely related to log-concave functions and convex hypersurfaces, and play important roles in analysis, probability, algebraic geometry, singularity theory, etc. In this talk, I will discuss some recent progress on these problems. In particular, I will provide a detailed explanation on a special case: the dual Minkowski problem for unbounded closed convex sets. These include the setting of this problem and the existence of solutions to this problem.