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 \textit{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 \emph{exterior cyclic polytope}, generalizing the cyclic polytope. It is equal to the convex hull of the amplituhedron in the Pl\"ucker embedding. We undertake a combinatorial analysis of the exterior cyclic polytope, its facets, and its dual. Finally, we introduce the \textit{(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.