A polytrope is a tropical polytope which is convex in the ordinary sense . Since their facet normals (as an ordinary polytope) are vectors in the root system of type A, they are also known as "alcoved polytopes" (of type A) . Polytropes arise as tropical eigenspaces  and are closely related to the Kleene stars , which occur in combinatorial optimization (e.g., in the Floyd-Warshall algorithm for computing all shortest paths ).
The combinatorial study of tropical polytopes was started by Develin and Sturmfels . Joswig and Kulas investigated polytropes as geometric objects in their own right . Lam and Postnikov studied polytropes under the name "alcoved polytopes" .
A tropical polytope in the line R2/R1 is an interval and hence a polytrope.
Up to combinatorial equivalence there are precisely five distinct types of polytropes in the plane R3/R1.
In  it was erroneously claimed that there are only five combinatorial types of polytropes in the tropical projective 3-torus R4/R1 with exactly 20 ordinary vertices (the maximum number).
This was corrected by Jiménez and de la Puente , who found the sixth type and showed that this is the complete list; see also .
The full classification of polytropes in the tropical projective 3-torus, not only of the maximal ones, was obtained by Tran ; there are 1013 combinatorial types.
Here is the detailed statistics per number of ordinary vertices:
4:1 5:1 6:5 7:6 8:34 9:38 10:81 11:101 12:151 13:144 14:154 15:116 16:92 17:46 18:28 19:9 20:6
The collection is based on the data from Tran's classification. For each maximal cone in the fan computed there (and described in ) we determine the barycenter of the rays (scaled to coprime integer coordinates) as a representative. These yield lifting functions for certain regular subdivisions of Δ3xΔ3.
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Last modified: Die Dez 20:32:31 UTC 2016 by mic