  The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian
$Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$
which lies in $\mathbb{R}^n$. Meanwhile, the amplituhedron
$\mathcal{A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian
$Gr^{\geq 0}_{k,n}$ into the Grassmannian $Gr_{k,k+2}$ under the amplituhedron
map $\tilde{Z}$. It is not a polytope, and has full dimension $2k$ inside
$Gr_{k,k+2}$. Nevertheless, as was first discovered by Lukowski-Parisi-Williams
[LPW], these two objects appear to be closely related via T-duality. In this
paper we use the geometry of the hypersimplex and positroid polytopes to obtain
a deeper understanding of the amplituhedron. We show that the inequalities
cutting out positroid polytopes - images of positroid cells of $Gr^{\geq
0}_{k+1,n}$ under the moment map - translate into sign conditions
characterizing the T-dual Grasstopes - images of positroid cells of $Gr^{\geq
0}_{k,n}$ under the amplituhedron map. Moreover, we subdivide the amplituhedron
into chambers enumerated by the Eulerian numbers, just as the hypersimplex can
be subdivided into simplices enumerated by the Eulerian numbers. We use this
property to prove one direction of the conjecture of [LPW]: whenever a
collection of positroid polytopes gives a triangulation of the hypersimplex,
the T-dual Grasstopes give a triangulation of the amplituhedron. Along the way,
we prove: Arkani-Hamed-Thomas-Trnka's conjecture that $\mathcal{A}_{n,k,2}(Z)$
can be characterized using sign conditions, and
Lukowski-Parisi-Spradlin-Volovich's conjectures about characterizing
generalized triangles (the $2k$-dimensional positroid cells which map
injectively into the amplituhedron $\mathcal{A}_{n,k,2}(Z)$), and $m=2$ cluster
adjacency. Finally, we discuss new cluster structures in the amplituhedron.
