Tropical Fermat-Weber Polytropes (2402.14287v4)

Abstract: We study the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well-known that a tropical Fermat-Weber point of a given sample is not unique and we show that the set of all possible Fermat-Weber points forms a polytrope. To prove this, we show that the tropical Fermat-Weber is the dual of a minimum-cost flow problem, and that its polytrope is a bounded cell of a tropical hyperplane arrangement given by both max- and min-tropical hyperplanes with apices given by the sample. We also define tropical Fermat-Weber gradients and provide a gradient descent algorithm that converges to the Fermat-Weber polytrope.