TROPICAL AND ORDINARY CONVEXITY COMBINED

Abstract

Abstract 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.