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.
@article{arxiv.1510.03553,
title = {The Amplituhedron and the One-loop Grassmannian Measure},
author = {Yuntao Bai and Song He and Thomas Lam},
journal= {arXiv preprint arXiv:1510.03553},
year = {2016}
}