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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.

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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