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We initiate the systematic study of \emph{local positive spaces} which arisein the context of the Amplituhedron construction for scattering amplitudes inplanar maximally supersymmetric Yang-Mills theory. We show that all localpositive spaces relevant for one-loop MHV amplitudes are characterized bycertain sign-flip conditions and are associated with surprisingly simplelogarithmic forms. In the maximal sign-flip case they are finite one-loopoctagons. Particular combinations of sign-flip spaces can be glued into newlocal positive geometries. These correspond to local pentagon integrands thatappear in the local expansion of the MHV one-loop amplitude. We show that,geometrically, these pentagons do \emph{not} triangulate the originalAmplituhedron space but rather its twin "Amplituhedron-Prime." This newgeometry has the same boundary structure as the Amplituhedron (and thereforethe same logarithmic form) but differs in the bulk as a geometric space. Oncertain two-dimensional boundaries, where the Amplituhedron geometry reduces toa polygon, we check that both spaces map to the same dual polygon.Interestingly, we find that the pentagons internally triangulate that dualspace. This gives a direct evidence that the chiral pentagons are naturalbuilding blocks for a yet-to-be discovered dual Amplituhedron.

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