A note on mediated simplices

Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem come from monomial substitutions into the arithmetic-geometric inequality. In 1989, the second author gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms (Math. Ann., (283), 431--464), involving the arrangement of lattice points in the simplex whose vertices were the $n$-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.