  Given a distance matrix consisting of pairwise distances between species, a
distance-based phylogenetic reconstruction method returns a tree metric or
equidistant tree metric (ultrametric) that best fits the data. We investigate
distance-based phylogenetic reconstruction using the $l^\infty$-metric. In
particular, we analyze the set of $l^\infty$-closest ultrametrics and tree
metrics to an arbitrary dissimilarity map to determine its dimension and the
tree topologies it represents. In the case of ultrametrics, we decompose the
space of dissimilarity maps on 3 elements and on 4 elements relative to the
tree topologies represented.
  Our approach is to first address uniqueness issues arising in
$l^\infty$-optimization to linear spaces. We show that the $l^\infty$-closest
point in a linear space is unique if and only if the underlying matroid of the
linear space is uniform. We also give a polyhedral decomposition of $\rr^m$
based on the dimension of the set of $l^\infty$-closest points in a linear
space.
