Positive geometry, local triangulations, and the dual of the Amplituhedron

Understand

We initiate the systematic study of \emph{local positive spaces} which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.

• We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms.
• In the maximal sign-flip case they are finite one-loop octagons.
• Particular combinations of sign-flip spaces can be glued into new local positive geometries.