Abstract It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones, as was recently proved by different authors. In this note we give a direct and constructive proof of the statement. Our proof yields a new and surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size 2, independent of dimension and complexity of the initial convex cone.

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关于非交换多面体和多面体的注释

摘要 众所周知，每个多面体锥体都是有限生成的（即多面体），反之亦然。令人惊讶的是，正如最近不同作者所证明的那样，对于此类锥体的非交换版本，这两个概念几乎总是不同。在本笔记中，我们给出了该陈述的直接和建设性证明。我们的证明产生了一个新的令人惊讶的定量结果：两个概念的差异总是可以在非交换性的第一级看到，即对于大小为 2 的矩阵，与初始凸锥的维数和复杂性无关。