  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.
