2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68335We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if $K$ is a conditionally complete idempotent semifield, with completion $\bar{K}$, a convex function $K^n\to\bar{K}$ which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of $K^n$, which extends earlier results of Zimmermann, Samborski, and Shpiz.25 pages, 4 Postscript figures, v2 (minor revision)Functional Analysis26B25 (Primary) 06F20, 06F30 (Secondary)text