# The Minkowski Theorem for Max-plus Convex Sets

1 MAXPLUS - Max-plus algebras and mathematics of decision Inria Paris-Rocquencourt

Abstract : We establish the following max-plus analogue of Minkowski-s theorem. Any point of a compact max-plus convex subset of $\mathbbR\cup{-\infty}^n$ 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.

Keywords : TROPICAL ALGEBRA EXTREME POINTS POLYHEDRA POLYTOPES CONVEX CONES CONVEX SETS ABSTRACT CONVEXITY KREIN-MILMAN THEOREM CARATHÉODORY THEOREM MAX-PLUS ALGEBRA

Author: Stéphane Gaubert - Ricardo David Katz -

Source: https://hal.archives-ouvertes.fr/