Advances in Geometry
Managing Editor: Grundhöfer, Theo / Joswig, Michael
Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard
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Tropical and ordinary convexity combined
A polytrope is a tropical polytope which at the same time is convex in the ordinary sense. A d-dimensional polytrope turns out to be a tropical simplex, that is, it is the tropical convex hull of d + 1 points. This statement is equivalent to the known fact that the Segre product of two full polynomial rings (over some field K) has the Gorenstein property if and only if the factors are generated by the same number of indeterminates. The combinatorial types of polytropes up to dimension three are classified.
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