The m=1 amplituhedron and cyclic hyperplane arrangements

The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. When k+m=n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k=1, the amplituhedron is a cyclic polytope. While the case m=4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m=1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B(n,k,m), which we show is isomorphic to A(n,k,m). We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron A(n,k,1) using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that A(n,k,1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A(n,k,1) is homeomorphic to a ball.