Tropical Fermat-Weber Polytropes

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.