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

Abstract

Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ 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 Com\u{a}neci and Joswig, that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $\mathbf{v}_1,\ldots,\mathbf{v}_m$, to the weighted setting. In particular, we show that for any fixed data points $\mathbf{v}_1, \ldots, \mathbf{v}_m$, 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 Com\u{a}neci and Joswig for computing consensus trees in phylogenetics.