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arXiv · 2402.14287

Tropical Fermat-Weber Polytropes

John Sabol·David Barnhill·Ruriko Yoshida·Keiji Miura
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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.

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John Sabol, David Barnhill, Ruriko Yoshida, Keiji Miura. 2026-06-25. Tropical Fermat-Weber Polytropes. https://arxiv.org/abs/2402.14287

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