Tropical convexity

By M. Develin, Bernd Sturmfels. Documenta Mathematica, 2003

321 citations recorded by the source corpus.

. The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications to phylogenetic trees are discussed.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9b383c1382b0f2dd3747d64ec2514c865d452e9c

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Citations in the corpus

Tropical Halfspaces (2003, methodology): “Like their ordinary counterparts tropical polytopes also have an exterior description.”
Tropical hyperplane arrangements and oriented matroids (2007, background): “2 Notice that this definition of type is the transpose of the definition in [7].”
Arboreal Singularities (2013, background): “This space arises in many places (all intimately related to symplectic topology): (1) a tropical hyperplane in n-dimensional tropical projective space ([1, 5, 16]),(1) (2) the universal planar tree over the (n−2)-dimensional associahedronKn−2 ([11, 15, 17]), (3) in geometric realizations of Waldhaus”
DRESSIANS, TROPICAL GRASSMANNIANS, AND THEIR RAYS (2011, background): “imensional polyhedral fan with f-vector (1;15;470; 642;677; 8;892;898; 57;394;505; 194;258;750; 353;149;650; 324;404;880; 117;594;645; 113;400): Modulo the natural Sym(8)-symmetry, the f-vector reads (1; 12; 155; 1;149; 5;013; 12;737; 18;802; 14;727; 4;788; 14): There are 116;962;265 maximal cones, ”
On visualization scaling, subeigenvectors and Kleene stars in max algebra (2008, background): “We note that the convex extremals ext(V∗(A)) correspond to the pseudovertices of tropical polytropes [32] (Kleene cones in themin-plus setting), and it is known that the number of thesemay be up to (2(n − 1))!/(n − 1)! [15,32], unlike the number of max extremals ext⊕(V∗(A)) which is not more than n.”
The Minkowski theorem for max-plus convex sets (2006, background): “al geometry (in this context, “tropical”is essentiallyused as a synonymof “max-plus”, orrather, ofthe dual term, “min-plus”). Tropicalanaloguesofpolytopes havebeen considered by Develin and Sturmfels [DS04], and also by Joswig [Jos05] (the tropical polytopes they consider are special ﬁnitely generat”
On the frontiers of polynomial computations in tropical geometry (2004, background): “ence, by the choice of r, ar + zr = as + zs. In other words, the minimum in fi is attained at least twice at the point z, i.e., z ∈ T (fi). Remark 4.13. Using the framework of tropical convexity from [4], Lemma 4.12 also follows from the fact that tropical hyperplanes are tropically convex [4, Propo”
Flag arrangements and triangulations of products of simplices (2006, background): “They also arise very naturally in connection with tropical geometry [10], transportation problems, and Segre embeddings [40].”
Hyperconvexity and tight-span theory for diversities (2010, background): “…metrics [4, 11, 12]; intriguing connections between the Buneman graph, the tight span and related constructions [9, 12–16]; links with tropical geometry and hyperdeterminants [17, 18]; classiﬁcation of ﬁnite metrics [4, 19]; and properties of the tight span for special classes of metrics [20, 21].”
Minimal half-spaces and external representation of tropical polyhedra (2009, methodology): “Lemma 10 of [16] shows that the sets XS are closed convex polyhedra (both in the max-plus and usual sense) which are given by XS = { x ∈ Rnmax ∣ ∣ xjv r i ≤ xiv j , ∀i, j ∈ {1, .”
Generators, extremals and bases of max cones (2006, background): “Note that in the tropical geometry [12,18] vertices of a polytope are defined to be the essentially unique generators determined in Corollary 20 and hence vertices correspond to our extremals (and to Wagneur’s irreducible elements).”