For k=m=2, the amplituhedron equals the intersection of Gr(2,4) with the exterior cyclic polytope C_{2,2,n}(Z), and its extendable convex dual is the twisted amplituhedron.
Parametrizations of Canonical Bases and Totally Positive Matrices
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Exterior Cyclic Polytopes and Convexity of Amplituhedra
For k=m=2, the amplituhedron equals the intersection of Gr(2,4) with the exterior cyclic polytope C_{2,2,n}(Z), and its extendable convex dual is the twisted amplituhedron.