Initial steps in the classification of maximal mediated sets

Maximal mediated sets (MMS), introduced by Reznick, are distinguished subsets of lattice points in integral polytopes with even vertices. MMS Newton AGI-forms and nonnegative circuit polynomials determine whether these sums squares. In this article, we take initial steps classifying both theoretically practically. Theoretically, show that simplices isomorphic if only the generate same up to permutations. Furthermore, generalize a result Iliman third author. Practically, fully characterize for all sufficiently small dimensions maximal 1-norms. particular, experimentally prove conjecture Reznick 2 dimensional 1-norm 150 provide indications on distribution density MMS.