Shelby
Cox
MPI MiS Leipzig
Weighted tropical Fermat-Weber points
Abstract.
Let $v_1,\ldots,v_m$ be points in a metric space $X$ with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $x$ which minimize $\sum w_i d(v_i, x)$. When $X$ is the tropical projective torus, and $d$ is the asymmetric tropical distance, Comaneci and Joswig proved that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $v_1,\ldots,v_m$. In a recent paper, Maize Curiel and I extend this result to the weighted setting using the combinatorics of tropical hyperplane arrangements and polyhedral geometry. Furthermore, we show that for any fixed data points $\bfv_1, \ldots, \bfv_m$, and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. In the context of phylogenetics, the (weighted) Fermat-Weber points furnish a method for computing consensus trees.