{"paper":{"title":"Tropical Fermat-Weber Polytropes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The set of all Fermat-Weber points under the tropical metric forms a polytrope for any finite sample.","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Barnhill, John Sabol, Keiji Miura, Ruriko Yoshida","submitted_at":"2024-02-22T05:00:00Z","abstract_excerpt":"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 d"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The location problem is dual to a particular minimum-cost flow problem whose optimal solutions exactly parametrize the Fermat-Weber set (abstract, paragraph 3). If this duality does not hold for the chosen tropical dissimilarity measure, the polytrope claim collapses.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The set of tropical Fermat-Weber points forms a polytrope via duality to min-cost flow, with a gradient-descent algorithm to locate it.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The set of all Fermat-Weber points under the tropical metric forms a polytrope for any finite sample.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d2fcfb30a8d9f7b9e2a4efa83eafb73cdd0f0644aa297e8aa24927518fc1ca42"},"source":{"id":"2402.14287","kind":"arxiv","version":6},"verdict":{"id":"057149c1-11a8-4de4-b066-5961c043ad16","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T04:07:35.617356Z","strongest_claim":"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.","one_line_summary":"The set of tropical Fermat-Weber points forms a polytrope via duality to min-cost flow, with a gradient-descent algorithm to locate it.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The location problem is dual to a particular minimum-cost flow problem whose optimal solutions exactly parametrize the Fermat-Weber set (abstract, paragraph 3). If this duality does not hold for the chosen tropical dissimilarity measure, the polytrope claim collapses.","pith_extraction_headline":"The set of all Fermat-Weber points under the tropical metric forms a polytrope for any finite sample."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2402.14287/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}