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Tropical and ordinary convexity combined. (English) Zbl 1198.14060
This paper is concerned with polytropes, i.e. tropical polytopes which are also convex in the ordinary sense. Properties of polytropes are studied and illustrated with interesting examples. The main result is a method to construct the tropical vertices of a polytrope (given by inequalities) and the conclusion (since there are always $$d+1$$ vertices) that each polytrope is a tropical simplex. The authors also present an efficient method how to enumerate all combinatorial types of polytropes. There is an interesting connection to commutative algebra via the Stanley-Reisner ring of a product of simplices. (Polytropes are dual to regular subdivisions of a product of simplices, see [M. Develin, B. Sturmfels, Doc. Math., J. DMV 9, 1–27, erratum 205–206 (2004; Zbl 1054.52004)]). The authors show that the known theorem that the Segre product of two polynomial rings is Gorenstein if and only if the numbers of variables coincide is equivalent to their main result. Polytropes also occur as bounded cells of deformations of the Coxeter hyperplane arrangement of type $$A_d$$ [T. Lam, A. Postnikov, Discrete Comput. Geom. 38, No. 3, 453–478 (2007; Zbl 1134.52019)].

##### MSC:
 14T05 Tropical geometry (MSC2010) 52A01 Axiomatic and generalized convexity
##### Keywords:
Polytopes; tropical polytopes
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##### References:
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