Convexity of tropical polytopes

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    We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of Rn. We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovič [16] and show that row spaces of tropical Kleene star matrices are exactly the “polytropes” studied by Joswig and Kulas [13].
    Original languageEnglish
    Pages (from-to)531-544
    Number of pages13
    JournalLinear Algebra and its Applications
    Publication statusPublished - 15 Nov 2015


    • Tropical convexity; Min-plus convex hull; Polytopes


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