Tropical Fermat-Weber Polytropes
Combinatorics
2026-05-13 v5
Abstract
We study the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. We also provide a simple gradient descent algorithm that converges to the Fermat--Weber polytrope.
Cite
@article{arxiv.2402.14287,
title = {Tropical Fermat-Weber Polytropes},
author = {John Sabol and David Barnhill and Ruriko Yoshida and Keiji Miura},
journal= {arXiv preprint arXiv:2402.14287},
year = {2026}
}