Polytropes and Tropical Linear Spaces.

A polytrope is a convex polytope that is expressed as the tropical convex hull of a finite number of points. Every bounded cell of a tropical linear space is a polytrope. It is a conjecture that conversely every polytrope arises as a bounded cell of a tropical linear space. We investigate vertices and edges of an arbitrary polytrope, develop general settings, and completely solve the conjecture by examining possible dual matroid tilings. This paper offers a new innovative but elementary approach to tropical convexity and tropical linearity, and studies their relationship. The paper also provides a computational base for the Dressian Dr(4,n).