The Minkowski Theorem for Max-plus Convex Sets

Abstract

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.

13 pages, 4 figures (5 eps files)

References in corpus (1)

• Duality and separation theorems in idempotent semimodules

Cited by in corpus (3)

• The tropical analogue of polar cones
• Cyclic projectors and separation theorems in idempotent convex geometry
• Idempotent and tropical mathematics and problems of mathematical physics (Volume II)