The Minkowski theorem for max-plus convex sets
By Stephane Gaubert, Ricardo D. Katz. 2006
112 citations recorded by the source corpus.
Abstract We establish the following max-plus analogue of Minkowski’s theorem. Any point of a compact max-plus convex subset of ( R ∪ { - ∞ } ) 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 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.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/efa0ea46cdce35413907da8da34129ad6b35d565
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In Computational Theory and Mathematics
- Stephane Gaubert
- Marianne Akian
- Zur Izhakian
- Alexander E. Guterman
- Louis H. Rowen
- Manfred Knebusch
- Peter Butkovic
- Ricardo D. Katz
In Geometry and Topology
- Zur Izhakian
- Louis H. Rowen
- Stephane Gaubert
- Marianne Akian
- Manfred Knebusch
- Alexander E. Guterman
- Peter Butkovic
- Ricardo D. Katz
Citations in the corpus
- Supertropical linear algebra (2010, background): “Critical elements correspond to “extreme points” over the max-plus algebra in [4], who show that every point in R is a linear combination of at most n+1 extreme points.”
- Minimal half-spaces and external representation of tropical polyhedra (2009, background): “The proof is based on the idea of [18] (Theorem 1 below), which is combined with the results of [20].”
- On visualization scaling, subeigenvectors and Kleene stars in max algebra (2008, background): “7 [11, 24] If K ⊆ R+ is a closed max cone, then the set ext⊕(K) is non-empty and it is the unique scaled basis for K.”
- Linear independence over tropical semirings and beyond (2008, result): “This result was made more precise in further works by Butkovič and Cuninghame-Green [CGB04], Gaubert and Katz [GK07], Butkovič, Sergeev and Schneider [BSS07].”