Tropical FermatWeber points
Abstract
In a metric space, the FermatWeber points of a sample are statistics to measure the central tendency of the sample and it is wellknown that the FermatWeber point of a sample is not necessarily unique in the metric space. We investigate the computation of FermatWeber 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 FermatWeber 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 FermatWeber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical FermatWeber points.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1604.04674
 Bibcode:
 2016arXiv160404674L
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 52B11;
 13P25;
 92B05
 EPrint:
 20 Pages, 2 figures. To appear in SIAM Journal on Discrete Mathematics