Max-plus convex sets and functions

Details Max-plus convex sets and functions Max-plus convex sets and functions

2003-08-18

arxiv.org/abs/math/0308166v2

Appears in "Idempotent Mathematics and Mathematical Physics", G.L. Litvinov and V.P. Maslov, Eds, vol. 377 of Contemporary Mat

Mathematics

Functional Analysis

25 pages, 4 Postscript figures, v2 (minor revision)

Scientific paper

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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