The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces a remarkable new way for computations in quantum field theory based on volume computations of certain polyhedra. This and related works may also have profound implications for the foundation of particle physics. It is related to various beautiful mathematics, and, in particular, to the combinatorics of certain stratifications of the Grassmanians.(See also the Quanta Magazine article, A Jewel at the Heart of Quantum Physics, by Natalie Wolchover, and Nima Arkani-Hamed’s on-line SUSY 2013 video lecture The Amplituhedron.)
The amplituhedron is remarkable new subsequent geometric object which comes in this study extending the notion of "positive grassmanian." (I did not see it explicitly defined in the above paper, at least not by this name.) It is very briefly described in Arkani-Hamed's lecture.
My question is quite simple: what is the mathematical definition of the amplituhedron?