Positivity of tree-level amplituhedra for general parameters

Determine whether, for all integer triples (n,k,m) with 0 < k ≤ k + m ≤ n, the tree-level amplituhedron A_{n,k,m}(Z) inside the complex Grassmannian Gr(k,k+m) is a positive geometry in the sense of Arkani-Hamed, Bai, and Lam (2017), beyond the known cases k = 1 and k + m = n.

Background

Tree-level amplituhedra are defined as images of the positive Grassmannian Gr(k,n){≥0} under the linear map induced by a totally positive matrix Z ∈ ℝ^{n×(k+m)}. When k = 1 they recover cyclic polytopes, and when k + m = n they are isomorphic to Gr(k,n){≥0}, both of which are known positive geometries. The general case is central to the program of positive geometry because the associated canonical forms encode scattering amplitudes in N=4 Super Yang–Mills theory. Establishing positivity in the remaining parameter regimes would place amplituhedra firmly within the axiomatic framework of positive geometries.