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Tropical convexity. (English) Zbl 1054.52004
Tropical convexity studies subsets of the Euclidean space that satisfy a certain convexity property w.r.t. tropical addition and multiplication of real numbers: Tropical addition of reals returns the ordinary maximum, tropical multiplication returns the ordinary sum. A set is tropically convex if with any two points it contains all tropical linear combinations. The tropical convex hull of a set of points is the smallest tropically convex set containing all the points. The tropical convex hull of a finite set of points is a tropical polytope. As one of their main results, the authors prove that the combinatorial types of tropical polytopes with given number of vertices $$r$$ in a given dimension $$n$$ are in bijection with the regular polyhedral subdivisions of the product of two simplices, one with $$r$$ and one with $$n$$ vertices. On their way to this result, a collection of properties are proved about tropically convex sets and tropical polytopes that resemble analogous statements in ordinary convexity. Finally, it is shown that the injective hull of a metric [J. R. Isbell, Comment. Math. Helv. 39, 65–76 (1964; Zbl 0151.30205)] can be regarded as a tropical polytope induced by the distance matrix of the metric. These objects have been proposed by Dress et al. [A. Dress, V. Moulton and W. Terhalle, Eur. J. Comb. 17, No. 2–3, 161–175 (1996; Zbl 0853.54027)] for the analysis of phylogenetic trees. In particular, a metric is a tree metric if and only if the tropical polytope associated to it has dimension one.

##### MSC:
 52A30 Variants of convex sets (star-shaped, ($$m, n$$)-convex, etc.) 52B11 $$n$$-dimensional polytopes 52B10 Three-dimensional polytopes 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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