A note on non-commutative polytopes and polyhedra

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. This was obtained as a byproduct in an earlier paper. In this note we give a direct and constructive proo...

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Bibliographic Details
Published in:arXiv.org 2019-02
Main Authors: Huber, Beatrix, Netzer, Tim
Format: Article
Language:English
Subjects:
Cones
Polytopes
Online Access:Get full text
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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. This was obtained as a byproduct in an earlier paper. In this note we give a direct and constructive proof of the statement. Our proof also yields a 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.
ISSN:2331-8422