Max-plus convex sets and max-plus semispaces. II

Abstract

We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I.Optimization, 56, 171–205. We show that the theories of max-plus convexity in and -convexity in are equivalent, and we deduce some consequences. We show that max-plus convexity in Rⁿ is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unless ≤ is a total order onA, the results ofloc. cit. on semispaces cannot be generalized in a natural way to the framework ofAⁿ =(Aⁿ , ≤, ⊗), whereA:=M∪ {−∞}, withM=(M,≤,⊗) being a lattice ordered group and −∞ a “least element’ adjoined toM.

Keywords:

• Max-plus convex set
• Max-plus semispace
• Max-plus hemispace
• Max-plus segment
• Max-prod convexity
• B-Convexity
• Multi-order convexity

Keywords:

• Mathematics Subject Classifications 2000:
• Primary 52A01
• Secondary: 52A07
• 46A40

Acknowledgement

The authors would like to thank an anonymous referee for some useful remarks. V. Nitica was partially supported by NSF grant DMS-0500832. I. Singer was partially supported by grant CEx05-D11-23/2005.