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 Δ k,n for all n and k. We also study the generalization of the momentum amplituhedron Mn,k to m = 2, which has been conjectured to share many properties with the hypersimplex, and we provide counterexamples for these conjectures. 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.