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arxiv: 2411.05678 · v3 · submitted 2024-11-08 · 🧮 math.MG · math.AT· math.FA· math.OC· math.PR

Relative Optimal Transport

Peter Bubenik , Alex Elchesen This is my paper

Pith reviewed 2026-05-23 17:22 UTC · model grok-4.3

classification 🧮 math.MG math.ATmath.FAmath.OCmath.PR
keywords relative optimal transportKantorovich-Rubinstein normRadon measuresLipschitz functionsRiesz conesPolish spacesdualityWasserstein distance
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The pith

Relative optimal transport to a distinguished subset makes relative 1-finite Radon measures the sequentially order continuous dual of relative Lipschitz functions under the relative Kantorovich-Rubinstein norm on boundedly compact Polish sp

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

The paper builds optimal transport around a fixed distinguished subset that serves as an unrestricted reservoir of mass. This setup lets one compare measures whose total masses differ, without forcing normalization or other adjustments. Relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, and both Kantorovich-Rubinstein and Monge-Kantorovich dualities are obtained, together with a relative form of the Riesz-Markov-Kakutani representation theorem. The central result states that, on any boundedly compact Polish space, the space of relative 1-finite real-valued Radon measures equipped with the relative Kantorovich-Rubinstein norm is precisely the sequentially order continuous dual of the space of relative Lipschitz functions equipped with the operator norm. A supporting theory of Riesz cones is developed along the way.

Core claim

For a boundedly compact Polish space, the space of relative 1-finite real-valued Radon measures with the relative Kantorovich-Rubinstein norm coincides with the sequentially order continuous dual of the space of relative Lipschitz functions with the operator norm. The relative transportation problem is shown to admit an optimal solution, and relative forms of Kantorovich-Rubinstein duality, Monge-Kantorovich duality, and the Riesz-Markov-Kakutani theorem are established.

What carries the argument

The relative transportation problem with a distinguished subset acting as an unrestricted reservoir of mass, which induces the relative Kantorovich-Rubinstein norm and the associated dualities between measures and functions.

If this is right

  • The relative transport problem always possesses an optimal solution.
  • Relative Kantorovich-Rubinstein duality and relative Monge-Kantorovich duality both hold.
  • A relative Riesz-Markov-Kakutani theorem identifies the measure spaces arising from the construction with spaces of Lipschitz functions.
  • The new theory of Riesz cones supplies an ordered-cone framework that supports the duality statements.

Where Pith is reading between the lines

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

  • The construction supplies a direct way to compare measures of unequal total mass without artificial normalization steps.
  • The same reservoir device may extend to other metric-measure settings where total mass is not conserved.
  • Riesz cones could be applied independently to other ordered function spaces that lack a global unit.
  • The coincidence result suggests that numerical schemes for relative transport can be validated by checking duality gaps on the function side.

Load-bearing premise

The distinguished subset functions purely as a reservoir of mass and adds no further constraints on the underlying metric or on the definition of relative Lipschitz functions.

What would settle it

An explicit boundedly compact Polish space together with a concrete relative 1-finite Radon measure whose pairing with relative Lipschitz functions fails to match the value given by the relative Kantorovich-Rubinstein norm.

read the original abstract

We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest.

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 / 3 minor

Summary. The paper develops a theory of optimal transport relative to a distinguished subset that acts as a mass reservoir, permitting comparison of measures with differing total variations. It establishes existence of optimal plans, defines relative Kantorovich-Rubinstein and Wasserstein distances, proves relative versions of Kantorovich-Rubinstein and Monge-Kantorovich duality, and obtains a relative Riesz-Markov-Kakutani theorem. The central result states that, on a boundedly compact Polish space, the space of relative 1-finite real-valued Radon measures equipped with the relative Kantorovich-Rubinstein norm coincides with the sequentially order continuous dual of the space of relative Lipschitz functions equipped with the operator norm. The work also introduces a supporting theory of Riesz cones.

Significance. If the stated identifications hold, the framework extends classical optimal transport and duality results to settings without mass conservation, which is relevant for applications involving sources or sinks. The relative Riesz-Markov-Kakutani theorem provides a concrete functional-analytic representation on standard Polish spaces, and the Riesz-cone theory is flagged as potentially reusable. The results rest on the usual boundedly-compact Polish assumptions and introduce no free parameters or circular derivations.

minor comments (3)
  1. [Introduction / §1] The abstract and introduction should explicitly state the precise definition of 'relative 1-finite' measures and the precise norm on the space of relative Lipschitz functions before the duality theorem is announced.
  2. [§2] Notation for the distinguished subset (reservoir) and the relative measures should be introduced once and used consistently; several passages appear to switch between different symbols for the same objects.
  3. [§3] The paper should include a short comparison table or paragraph contrasting the relative KR norm with the classical KR norm when the reservoir is empty or the total masses coincide.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of the paper, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces new definitions for relative optimal transport, relative 1-finite Radon measures, relative Lipschitz functions, and relative Kantorovich-Rubinstein norm on boundedly compact Polish spaces, then derives relative versions of standard duality and representation theorems from those definitions. The central result equates the space of relative measures (with the new norm) to the sequentially order continuous dual of relative Lipschitz functions (with operator norm) via a relative Riesz-Markov-Kakutani theorem. This construction is self-contained against external benchmarks; it does not reduce any prediction or theorem to a fitted input, self-citation chain, or definitional tautology. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The theory rests on standard properties of Polish spaces and introduces new objects (relative measures, Riesz cones) whose definitions are not supplied in the abstract.

axioms (1)
  • domain assumption The underlying space is a boundedly compact Polish space.
    Invoked for the identification of relative measures with the dual of relative Lipschitz functions.
invented entities (2)
  • Relative measures no independent evidence
    purpose: Measures allowed to exchange mass with the distinguished reservoir subset.
    Core new object enabling comparison of unequal total variation measures.
  • Riesz cones no independent evidence
    purpose: Ordered cones linking relative measures to relative Lipschitz functions.
    Introduced to support the relative Riesz-Markov-Kakutani theorem.

pith-pipeline@v0.9.0 · 5680 in / 1311 out tokens · 34113 ms · 2026-05-23T17:22:03.384690+00:00 · methodology

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