Minkowski’s classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.

中文翻译：

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锥凸集的Minkowski型定理

Minkowski的经典存在定理为欧氏空间的单位球面上的Borel度量作为凸体的表面积度量提供了充要条件。该解决方案在翻译之前是唯一的。我们处理无界凸集的相应问题，这些无穷凸集的无穷大行为由给定的封闭凸锥决定。我们提供了一个存在性定理和一个稳定性结果。