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arxiv: 2606.27503 · v1 · pith:WDJF47EUnew · submitted 2026-06-25 · 🧮 math.CO · math.OC

Tropical Fermat--Weber Problems over Non-Finite Data and their Inverse Formulations

John Sabol , Ruriko Yoshida This is my paper

Pith reviewed 2026-06-29 01:43 UTC · model grok-4.3

classification 🧮 math.CO math.OC
keywords tropical pseudonormFermat-Weber problemlinear programminginverse problemstropical geometrylocation problemsnon-finite data
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The pith

The tropical one-infinity pseudonorm enables linear programming formulations for Fermat-Weber problems on non-finite data and inverse variants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends tropical Fermat-Weber location problems to non-finite data sets by means of the one-infinity pseudonorm. The pseudonorm is a polyhedral hybrid gauge with tunable asymmetry whose translation invariance lets it descend to tropical projective spaces. The authors derive linear programming formulations for the location problem itself and for several inverse variants. A reader would care because the extension moves tropical methods from finite point sets to more general data.

Core claim

The tropical one-infinity pseudonorm permits linear programming formulations for Fermat-Weber problems over non-finite data and for several inverse variants.

What carries the argument

The tropical one-infinity pseudonorm, a polyhedral hybrid gauge with tunable asymmetry.

If this is right

  • Fermat-Weber problems over infinite data admit linear programming solutions.
  • Several inverse formulations of the problem also reduce to linear programs.
  • All formulations remain valid after passage to tropical projective space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same gauge might support formulations for other tropical optimization problems on infinite domains.
  • Discretization of infinite data could serve as a practical test of the linear programs.
  • The tunable asymmetry parameter may allow control over robustness in location problems.

Load-bearing premise

The pseudonorms are invariant under translation by a constant vector.

What would settle it

A concrete non-finite data set on which one of the proposed linear programs returns a point that is not a minimizer of the tropical one-infinity distance sum.

read the original abstract

The term tropical pseudonorm refers to a family of (not necessarily symmetric) gauge functions that arise in tropical or idempotent geometry. An important characteristic of these gauges is their invariance under translation by a constant vector, allowing them to descent naturally to tropical projective spaces. In this work, we explore the tropical one-infinity pseudonorm, a polyhedral hybrid gauge that allows for tunable asymmetry, in the context of a Fermat--Weber location problem. We extend previous formulations in considering non-finite data, and we investigate several variants of the inverse problem, providing linear programming formulations for their solution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces the tropical one-infinity pseudonorm, a polyhedral hybrid gauge with tunable asymmetry that is invariant under constant-vector translation and thus descends to tropical projective spaces. It extends prior tropical Fermat-Weber work to non-finite data sets and formulates several inverse variants, all reducible to linear programs.

Significance. If the claimed LP reductions hold, the work supplies a concrete algorithmic route for location problems over infinite tropical data and for inverse recovery of the pseudonorm parameters, which would enlarge the scope of tropical optimization beyond finite-point instances.

minor comments (1)
  1. Abstract, line 4: 'allowing them to descent naturally' contains a grammatical error; the intended verb is 'descend'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential algorithmic value of the LP reductions for non-finite tropical data. Since the report lists no specific major comments, we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and available context describe extending prior formulations to non-finite data and inverse variants via linear programming, with no equations, fitted parameters renamed as predictions, or self-citation chains presented as load-bearing. No self-definitional steps, ansatzes smuggled via citation, or uniqueness theorems imported from the authors appear. The derivation chain is self-contained against external benchmarks, with the invariance property stated as a characteristic rather than derived circularly from the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all such items are therefore recorded as empty.

pith-pipeline@v0.9.1-grok · 5624 in / 1039 out tokens · 19063 ms · 2026-06-29T01:43:02.103986+00:00 · methodology

discussion (0)

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Reference graph

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