A note on non-commutative polytopes and polyhedra

TL;DR

This paper explores the differences between commutative and non-commutative polyhedral cones, providing a direct proof and revealing that these differences appear at the matrix size of 2, regardless of the cone's complexity.

Contribution

It offers a direct, constructive proof of the non-equivalence of non-commutative cones and polyhedra, highlighting the fundamental role of 2x2 matrices in this distinction.

Findings

Non-commutative cones differ from polyhedra almost always.

Differences appear at the matrix size of 2, regardless of cone complexity.

A constructive proof clarifies the relationship between non-commutative cones and polyhedra.

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. 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.