The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms

In this paper we provide a formula for the canonical differential form of the hypersimplex ${\mathrm{\Delta }}_{k,n}$ for all $n$ and $k$. We also study the generalization of the momentum amplituhedron ${\mathcal{M}}_{n,k}$ to $m=2$, and we conclude that the existing definition does not possess the desired properties. Nevertheless, we find interesting momentum amplituhedron-like logarithmic differential forms in the $m=2$ version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms.