TITLE: The amplituhedron minus the physics
QUESTION [72 upvotes]: Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume familiarity with quantum field theory,
but perhaps there exist more purely geometric explications...?
If so, I would appreciate a reference—Thanks!

(Image from Quanta Magazine)
(Added). Here is a snapshot from p.64 of the paper which jc cited,
"Scattering Amplitudes and the Positive Grassmannian."
(I am guessing that what they called the "positroid" in this paper
is either closely related or perhaps identical to what
was later named the "amplitudedron"?)

REPLY [4 votes]: Alexander Postnikov gave a detailed series of lectures on the positive Grassmanian at the Hebrew University of Jerusalem (see links below). He also briefly referred to the amplituhedron.
Folowwing Postinkov, I can briefly explain the situation as follows: 
A) The stratification of the positive Grassmanian:
Regard the Grassmanian as represented by equivalence classes of 
$m$ by $n$ matrices ($m \le n$) under row operators. 
The positive grassmanian is the set of totally-non negative matrices namely those where all $m$ by $m$ minors are nonnegative. It has an important cell-like structure (the open cells are known to be homeomorphic to open balls, their closures are conjectured to be homehomorphic to closed balls) that can be defined in two equivalent ways as follows: 
Given an ordering of the $n$ columns the cell of the Shubert stratification indexed by a subset $S$ (of size $m$) are matrices whose lexicographically first non-zero minor (w.r.t. the ordering) is $S$.The matroidal stratification is the common refinement of these Shubert decomposition with respect to all permutations.   
The cell structure studied by Postnikov can be seen
a) as the matroidal stratification reduced to this part of the grassmanian; 
b) Start with the common refinement of Shubert cells with respect to a cyclic family of permutations (for the whole Grassmanian this lies between the Shubert stratification and the matroid one). Then restrict it to the positive Grassmanian.
This cell structure has beautiful combinatorial description in terms of certain planar graphs and statistics of permutations.
B) The amplituhedron:
Recall that every polytope is a projection of a simplex and projections with respect to totally positive matrices give precisely the cyclic polytope.
Now, replace the simplex by the positive Grassmanian: The amplitutahedron is a projection of the positive Grassmanian based on a totally positive matrix. So the amplituhedron  is a common generalization of the positive grassmanian and the cyclic polytope.

Video1, Video2, Video3, Video4 
Postnikov's ten minutes explanation of the amplituhedron starts here.
Update: Some more details and links can be found in the blog post The simplex, the cyclic polytope, the positroidron, the amplituhedron and beyond.