The tropical polytope is the set of all weighted tropical Fermat-Weber points

Let ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_m$ be points in a metric space with distance $d$, and let $w_1,\dots ,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_{i}d({\mathbf{v}}_i,\mathbf{x})$. We extend a result of Comăneci and Joswig, that the set of unweighted Fermat-Weber points agrees with the “central” covector cell of the tropical convex hull of ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_m$, to the weighted setting. In particular, we show that for any fixed data points ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_m$, 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. We similarly extend the method of Comăneci and Joswig for computing consensus trees in phylogenetics.