Tropical Fermat-Weber points

Bo Lin, Ruriko Yoshida

Published 2016-04-16Version 1

We investigate the computation of Fermat-Weber points under the tropical metric, motivated by its application to the space of equidistant phylogenetic trees realized as the tropical linear space of all ultrametrics. While the Fr\'echet mean with the ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman has been studied by many authors, the Fermat-Weber point under tropical metric in tree spaces is not well understood. In this paper we investigate the Fermat-Weber point under the tropical metric and we show that the set of tropical Fermat-Weber points is a classical convex polytope. We identify conditions under which this set is a singleton.