Max-plus convex sets and functions
Guy Cohen, Stéphane Gaubert, Jean-Pierre Quadrat, Ivan Singer · Contemporary mathematics - American Mathematical Society · 2005
We 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 K, a convex function K n → 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.