Chen-Kou-Lyu theory: I. Convex cones in finite dimensions and their cross-dual matrix cones

In the July 2026 post, I explained how Yuxu Chen, Hui Kou and Zhenchao Lyu managed to show that the category of RB-domains is a proper subcategory of that of FS-domains, solving a long-standing open problem. At the end of that same post, I mentioned that they had also solved the question whether the probabilistic powerdomain of an RB-domain is an RB-domain. I am planning to explain how this works, and I will start this month by explaining the basics of the theory of cones in finite dimensions and their (cross-dual) matrix cones, which they use to this purpose. This is a well-known theory, which was developed by Barker and Loewy, or by Berman and Gaiha among others, in the 1970s. I should perhaps warn you: this post will be mostly about algebra, with little topology. If you have nothing against reading about algebra on a topology blog, then read the full post.