Tropical Fermat–Weber Points
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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 Rn/R1 with a fixed n ∈ N, motivated by its application to the space of equidistant phylogenetic trees with N leaves (in this case n = N ) realized 2 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.
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