We undertake a combinatorial study of the piecewise linear map g : R^{ 2m+2n} --> R^{ mn} which assigns to the four vectors a, A in R^m and b, B in R^n the m by n matrix given by g_{ ij} = min (a_i + b_j, A_i+B_j). This map arises naturally in Pachter and Sturmfels's work on the tropical geometry of statistical models. The image of g has been a subject of recent interest; it is the positive part of the tropical algebraic variety which parameterizes n-tuples of points on a tropical line in m-space. The domains of linearity of g are the regions of the real hyperplane arrangement A_{ m,n}, corresponding to the complete bipartite graph K_{ m,n}. We explain how the images of (some of) the regions provide two polyhedral subdivisions of the image of g, one of which is a refinement of the other. The finer subdivision is particularly nice enumeratively: it has 2 { m \choose 2} { n \choose 2} r_{ m-2,n-2} maximum-dimensional cells, where r_{ m-2,n-2} is the number of regions of the arrangement A_{ m-2,n-2}.
Categories: math.CO