A Note on Convex Realization of Halohedron

Abstract

In the recent works [arXiv:1803.05809],[arXiv: 1806.01842], Halohedron emerged as amplituhedron for 1-loop planar diagrams in bi-adjoint massless $\phi^3$ theory. Halohedron is a specific case of graph cubeahedron where the considered graph is a cycle-graph. In [arXiv: 1906.06861],[arXiv: 1501.07152], the authors provide construction of any graph cubeahedron and we use this construction to find the polytopal realization of Halohedron. We show that the Halohedron we obtain is equivalent to the proposed realization of Halohedron in `Big Kinematic Space'[arXiv: 1806.01842].