Exterior Cyclic Polytopes and Convexity of Amplituhedra

The amplituhedron is a semialgebraic set in the Grassmannian. We study convexity and duality of amplituhedra. We introduce a notion of convexity, called extendable convexity, for real semialgebraic sets in any embedded projective variety. We show that the k=m=2 amplituhedron is extendably convex in the Grassmannian of lines in projective three-space. In the process we introduce a new polytope called the exterior cyclic polytope, generalizing the cyclic polytope. It is equal to the convex hull of the amplituhedron in the Plücker embedding. We undertake a combinatorial analysis of the exterior cyclic polytope, its facets, and its dual. Finally, we introduce the (extendable) dual amplituhedron, which is closely related to the dual of the exterior cyclic polytope. We show that the dual amplituhedron for k=m=2 is again an amplituhedron, where the external matrix data is changed by the twist map.