Abstract
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 $\mathbb{R}^{n}/ \mathbb{R}{1}$ 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 with 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.
