Positive geometry of the general amplituhedron

Prove that the tree-level amplituhedron \(\mathcal{A}_{k,n,m}\) is a positive geometry in the sense of Definition 1 for arbitrary \(k,n,m\).

Background

The tree-level amplituhedron is one of the principal motivating examples for the theory of positive geometries and is realized in a Grassmannian. The notes state that its positive-geometry status is currently established only for k=1k=1, where it is a cyclic polytope, and for k=m=2k=m=2. The general case remains conjectural, so establishing the recursive canonical-form structure for arbitrary parameters is an explicit unresolved problem.

References

While the amplituhedron is one of the main motivations for the development of positive geometry, to my knowledge it is only known that ${\cal A}_{k,n,m}$ is a positive geometry in the sense of Definition \ref{def:posgeomABL} for $k = 1$, in which case it is a cyclic polytope, and for $k = m = 2$ . It is conjectured to be a positive geometry for arbitrary $k,n,m$.

— Positive Geometry of Polytopes and Polypols  (2506.05510 - Telen, 5 Jun 2025) in Remark following Definition 1, Section 1