Work

Tropical Fermat–Weber Polytropes

Year: 2026

Type: article

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.

Source: Discrete & Computational Geometry

DOI: 10.1007/s00454-026-00864-4

Authors: John Sabol, David Barnhill, Ruriko Yoshida, Keiji Miura

Institutions: Naval Postgraduate School, United States Naval Academy, Aalborg University, Kwansei Gakuin University

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090