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arXiv · math/0308166

Max-plus convex sets and functions

Guy Cohen·Stephane Gaubert·Jean-Pierre Quadrat·Ivan Singer
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Abstract

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 $\bar{K}$, 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.

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Guy Cohen, Stephane Gaubert, Jean-Pierre Quadrat, Ivan Singer. 2004-02-16. Max-plus convex sets and functions. https://arxiv.org/abs/math/0308166

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