  Over a non-archimedean local place, the height of a projective variety with
respect to a very ample line bundle equipped with a Fubini-Study metric is
related to the naive height of its Chow form. Using a non-Archimedean
Kempf-Ness criteria, we characterize Fubini-Study metrics that minimize the
height under the special linear action in terms of their Monge-Amp\`ere
polytopes. This polytope can be constructed either as its non-Archimedean
Bergman functional or as the weight polytope for the residual action on its
Chow form; it is associated with a polymatroid.
