论文标题
离散的高斯图像问题
The Discrete Gauss Image Problem
论文作者
论文摘要
我们研究了离散的高斯图像问题，这是对Aleksandrov的经典问题的概括，该问题是存在具有规定积分曲率的凸体的存在。我们引入了一个称为分配问题的组合问题，并显示了与离散高斯图像问题的等效性。我们在解决这两个问题的措施上建立了足够（几乎必要的）几何条件。此外，我们还提供了与Aleksandrov的整体介绍有关的某些经典概念的新离散解释，例如，将Aleksandrov与Hall的婚姻定理联系起来。
We study the Discrete Gauss Image Problem, a generalization of Aleksandrov's classical question on the existence of convex bodies with prescribed integral curvature. We introduce a combinatorial problem called the Assignment Problem and show its equivalence to the Discrete Gauss Image Problem. We establish sufficient (and nearly necessary) geometric conditions on measures that solve both problems. Additionally, we provide new discrete interpretations of some classical concepts related to Aleksandrov's integral curtvature, such as, for example, connecting Aleksandrov relation to Hall's Marriage Theorem.
