  All-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills
theory can be formulated geometrically in terms of the "amplituhedron". We
study the mathematical structures of the one-loop amplituhedron, and present a
new formula for its canonical measure, or the one-loop Grassmannian measure
formula. Using the recently proposed momentum-twistor diagrams, we show that
there is a correspondence between the cells of one-loop amplituhedron, BCFW
terms or equivalently on-shell diagrams, and residues of the one-loop
Grassmannian formula. In particular, for the first non-trivial case of one-loop
NMHV, these structures are naturally associated with a nice geometric picture
as polygons in projective space, as we discuss in various illustrative
examples.
