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<p>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.</p>

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Tropical Fermat–Weber Polytropes

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年7月28日

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  • John Sabol,
  • David Barnhill,
  • Ruriko Yoshida,
  • Keiji Miura

摘要

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.