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A note on non-commutative polytopes and polyhedra. (English) Zbl 07335424

Summary: 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.

MSC:

47L07 Convex sets and cones of operators
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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