Develin and Sturmfels showed that regular triangulations of $\Delta_{ n-1} \times \Delta_{ d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{ n-1} \times \Delta_{ d-1}$. In this paper, we show that any triangulation of $\Delta_{ n-1} \times \Delta_{ d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.
Categories: math.CO