What are the equations and constraints defining an/the Amplituhedron?

Question

No doubt we've all read quite a bit about amplituhedra in recent years, the best thing since sliced bread apparently for obtaining scattering amplitudes. But finding on the web formal definitions of it have seemed to me surprisingly elusive, and in fact I don't recall ever seeing any.

So can it be readily defined explicitly by a set of equations and/or constraints which a mathematician might be able to study, perhaps finding parametric solutions for example, without reference to physics?

(On the same topic, although this may not be the right place to mention it, and possibly nowhere is, I can't help thinking the name "amplituhedron" is rather clunky and it would have been better called "amplihedron"! But it's probably too late to change now, even if this was wanted.)

Answer 1

You could try the original paper: Arkani-Hamed, Nima; Trnka, Jaroslav (2014). "The Amplituhedron". Journal of High Energy Physics. 2014 (10): 30. arXiv preprint. See especially Section 9 for the formal definition (which defeats my Mathjax).

By many definitions it is not a kind of polyhedron anyway but a higher-dimensional polytope, the number of dimensions being dependent on the interaction under analysis. "Amplitope" is the obvious choice, but the authors based their maths on a formalism with a particular usage of "polyhedron" and decided to stick with it.