L-Infinity Optimization in Tropical Geometry and Phylogenetics

We investigate uniqueness issues that arise in l∞-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of R based on the dimension of the set of l∞-nearest neighbors. This implies that the l∞-nearest neighbor in a linear space is unique if and only if the underlying matroid is uniform. For Bergman fans of matroids, we show that the set of l∞-nearest points is a tropical polytope and give an algorithm to compute its tropical vertices. A key ingredient here is a notion of topology that generalizes tree topology. These results have practical implications for distance-based phylogenetic reconstruction using the l∞-metric. We analyze the possible dimensions of the set of l∞-nearest equidistant tree metrics to an arbitrary dissimilarity map and the number of tree topologies represented in this set. For both 3 and 4-leaf trees, we decompose the space of dissimilarity maps relative to the tree topologies represented.