  In a metric space, the Fermat-Weber points of a sample are statistics to
measure the central tendency of the sample and it is well-known that the
Fermat-Weber point of a sample is not necessarily unique in the metric space.
We investigate the computation of Fermat-Weber points under the tropical metric
on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {\bf 1}$ with a fixed $n
\in \mathbb{N}$, motivated by its application to the space of equidistant
phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as
the tropical linear space of all ultrametrics. We show that the set of all
tropical Fermat-Weber points of a finite sample is always a classical convex
polytope, and we present a combinatorial formula for a key value associated to
this set. We identify conditions under which this set is a singleton. We apply
numerical experiments to analyze the set of the tropical Fermat-Weber points
within a space of phylogenetic trees. We discuss the issues in the computation
of the tropical Fermat-Weber points.
