We introduce a new polytope called the exterior cyclic polytope. Our motivation comes from particle physics, and in particular from a semialgebraic set called the amplituhedron which lives in a Grassmannian Gr(k, r) and appears in calculations of particle scattering.

The exterior cyclic polytope is defined as the convex hull of the amplituhedron in the ambient Plücker space of Gr(k, r). We describe its face structure and facets, which in the case k=2 are controlled by a matroid called the hyperconnectivity matroid. Furthermore, we describe the dual of the k=2 and r=4 polytope in terms of the twist map of Marsh and Scott, and use this to define a notion of dual amplituhedron. This is joint work with Elia Mazzucchelli.