  Given a dissimilarity map $\delta$ on finite set $X$, the set of ultrametrics
(equidistant tree metrics) which are $l^\infty$-nearest to $\delta$ is a
tropical polytope. We give an internal description of this tropical polytope
which we use to derive a polynomial-time checkable test for the condition that
all ultrametrics $l^\infty$-nearest to $\delta$ have the same tree structure.
It was shown by Ardila and Klivans \cite{ardila-klivans2006} that the set of
all ultrametrics on a finite set of size $n$ is the Bergman fan associated to
the matroid underlying the complete graph on $n$ vertices. Therefore, we derive
our results in the more general context of Bergman fans of matroids. This added
generality allows our results to be used on dissimilarity maps where only a
subset of the entries are known.
