Tropical secant varieties of linear spaces
  In this paper, we investigate tropical secant varieties of ordinary linear spaces. These correspond to the log-limit sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural subcomplex of the complex of regular subdivisions of a corresponding point set, and we display the range of behavior of this object. We also use this characterization to reformulate the question of determining Barvinok rank into a question regarding regular subdivisions of products of simplices. 
Mike Develin