Ternary relations and their polytopes

Abstract: Graev introduced the construction of a convex polytope associated with a symmetric ternary relation. He showed that the number of left-invariant Einstein metrics on a homogeneous space under some conditions is no more than the normalized volume of certain polytope of such form. It happens that the construction of a cosmological polytope introduced by Arkani-Hamed, Benincasa and Postnikov for computation of the wave function of the Universe is the special case of the Graev construction. The paper is devoted to unification of these two theories from combinatorial perspective.