Tropical and Ordinary Convexity Combined

TL;DR

This paper studies polytropes, which are polytopes that are both tropical and convex in the usual sense, classifies their types up to dimension three, and explores their algebraic properties related to polynomial rings.

Contribution

It characterizes polytropes as tropical simplices and classifies their combinatorial types up to dimension three, linking geometric and algebraic properties.

Findings

A $d$-dimensional polytrope is a tropical simplex.

Classification of polytrope types up to dimension three.

Connection between polytrope properties and Gorenstein condition of polynomial rings.

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.