Tropical and ordinary convexity combined

Michael Joswig 1  and Katja Kulas 2
  • 1 Fachbereich Mathematik, TU Darmstadt, 64289 Darmstadt, Germany. Email:
  • 2 Fachbereich Mathematik, TU Darmstadt, 64289 Darmstadt, Germany. Email:


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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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.