  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.
