The Minkowski Theorem for Max-plus Convex Sets

We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set can be decomposed as a max-plus sum of its recession cone and of the max-plus convex hull of its extreme points.