On a solution to the Monge transport problem on the real line arising from the strictly concave case

Nicolas Juillet (IRMA)

arxiv: 1907.00681 · v1 · pith:4LYTW5R4new · submitted 2019-07-01 · 🧮 math.PR · math.OC

On a solution to the Monge transport problem on the real line arising from the strictly concave case

Nicolas Juillet (IRMA) This is my paper

Pith reviewed 2026-05-25 11:58 UTC · model grok-4.3

classification 🧮 math.PR math.OC

keywords optimal transportMonge problemreal lineexcursion couplingconcave costsdistance costuniquenesslimit of plans

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The pith

A unique solution to the Monge problem on the real line emerges as the limit of optimal plans for distance powers p<1 as p approaches 1.

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

The classical distance cost on the real line often admits multiple optimal transport plans between two measures. The paper recovers a distinguished solution by solving the problem first for the strictly concave costs |x-y|^p with 0

Core claim

The family of optimal plans for the costs |x-y|^p with p<1 converges, under the assumption of finite first moments, to a unique limit plan that remains optimal for the distance cost |x-y|. This limit, termed the excursion coupling, admits an explicit construction reminiscent of the convex case and is also recovered as the solution of secondary optimization problems; the routes it uses are described by a combinatoric and geometric characterization based on excursions of the cumulative distribution functions.

What carries the argument

The excursion coupling, defined as the limit of optimal plans for the family of strictly concave costs |x-y|^p as the exponent p tends to 1 from below.

If this is right

The excursion coupling selects a canonical plan among all optimal plans for the distance cost.
It satisfies optimality for certain secondary transport problems in addition to the primary distance cost.
The support of the coupling admits an explicit geometric description in terms of excursions of the distribution functions.
The construction parallels the monotone coupling obtained for convex costs and extends it to the concave regime.

Where Pith is reading between the lines

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

The same limiting procedure could be applied to other families of costs that approach the distance cost from the concave side.
On the line the excursion coupling may interact with the usual monotone rearrangement in a way that produces new identities for cumulative distributions.
The geometric route characterization might extend to discrete approximations or to measures with atoms.

Load-bearing premise

The optimal plans for the strictly concave costs converge in a suitable topology to a well-defined limit plan as the exponent tends to one.

What would settle it

For two measures on the line with finite first moments, check whether the sequence of optimal plans for |x-y|^p as p approaches 1 from below fails to converge or produces more than one limit point.

read the original abstract

It is well-known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance, and letting this parameter tend to one. A complete construction of this solution that we call excursion coupling is given. This is reminiscent to the one in the convex case. It is also characterized as the solution of secondary transport problems. Moreover, a combinatoric/geometric characterization of the routes used for this transport plan is provided.

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.

Desk Editor's Note private letter to a colleague

Juillet constructs the excursion coupling as the limit of unique p<1 plans for the |x-y| Monge problem on the line and gives an independent combinatorial characterization of its support.

read the letter

The paper's main contribution is an explicit construction of one particular Monge plan for the |x-y| cost on R, obtained by taking limits of the unique optima for the strictly concave costs |x-y|^p as p approaches 1 from below. Juillet also supplies a direct geometric/combinatorial description of the intervals that get transported and shows that the same plan solves certain secondary variational problems. This looks like a genuine addition rather than a restatement of earlier work on the line. The finite-first-moment assumption is standard and the starting point from known concave cases avoids obvious circularity. The combinatorial side gives an independent way to identify the plan, which helps. The potential soft spot is whether the limit is independent of the sequence p_n to 1. If two different sequences produced different weak limits, both optimal for the linear cost, the selection rule would not be canonical. The paper claims a complete construction, so it presumably proves the limit coincides with the combinatorial object for any sequence, but that step needs to be checked carefully in the full argument. This is a technical but self-contained piece in one-dimensional optimal transport. It is aimed at readers who work on explicit plans or selection mechanisms rather than general theory. The thinking is clear and the citations fit the topic. I would send it to peer review; a referee can verify the convergence details and the rigor of the combinatorial characterization.

Referee Report

2 major / 2 minor

Summary. The manuscript addresses non-uniqueness of optimal plans for the Monge problem on R with cost |x-y| by considering the family of problems with costs |x-y|^p for 0<p<1 (strictly concave), constructing the limit of the unique p-optimal plans as p->1^-, and identifying this limit with an explicitly constructed 'excursion coupling'. The paper supplies a combinatorial/geometric description of the support of this plan, shows it solves certain secondary variational problems, and claims the construction recovers uniqueness independently of the approximating sequence.

Significance. If the identification of the excursion coupling with the p->1 limit holds for every sequence and in a topology preserving optimality, the result supplies a canonical, explicitly constructible selection among the (possibly non-unique) |x-y|-optimal plans. The independent combinatorial construction and secondary characterizations are positive features; they could be useful in 1D transport applications requiring a distinguished plan.

major comments (2)

[Section 4] The load-bearing claim is that the excursion coupling coincides with the weak limit of p-optimal plans for every sequence p_n ->1 (not merely for some sequences). Section 4 (or the convergence argument following the construction in Section 3) must explicitly rule out the possibility that distinct sequences produce distinct weak limits, both of which remain optimal for the limiting cost |x-y|; the current outline does not contain a quantitative estimate or tightness argument showing the limit is sequence-independent.
[Section 3.2] Assumption on finite first moments is stated, but the topology in which convergence of plans is proved (weak, narrow, or Wasserstein) is not specified with sufficient precision to guarantee that optimality for |x-y|^p passes to the limit; a counter-example sequence of plans converging in a weaker topology but losing optimality should be ruled out or addressed.

minor comments (2)

[Section 2] Notation for the excursion coupling (e.g., how the 'excursions' are indexed) should be introduced earlier and used consistently in the combinatorial characterization.
The abstract states that the construction is 'reminiscent to the one in the convex case'; a brief comparison paragraph citing the relevant convex-case reference would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for your thorough review of our manuscript. We respond to each major comment below, indicating revisions where appropriate.

read point-by-point responses

Referee: [Section 4] The load-bearing claim is that the excursion coupling coincides with the weak limit of p-optimal plans for every sequence p_n ->1 (not merely for some sequences). Section 4 (or the convergence argument following the construction in Section 3) must explicitly rule out the possibility that distinct sequences produce distinct weak limits, both of which remain optimal for the limiting cost |x-y|; the current outline does not contain a quantitative estimate or tightness argument showing the limit is sequence-independent.

Authors: We thank the referee for highlighting this point. The excursion coupling is defined combinatorially and independently of the sequence, and we prove it is the unique |x-y|-optimal plan satisfying the geometric support condition. Since any weak limit of p-optimal plans for p<1 must satisfy this support condition (as the condition is closed under weak limits), the limit is necessarily the excursion coupling regardless of the sequence chosen. We will add an explicit paragraph in Section 4 to emphasize this uniqueness argument and why it implies sequence-independence without needing additional quantitative estimates. revision: partial
Referee: [Section 3.2] Assumption on finite first moments is stated, but the topology in which convergence of plans is proved (weak, narrow, or Wasserstein) is not specified with sufficient precision to guarantee that optimality for |x-y|^p passes to the limit; a counter-example sequence of plans converging in a weaker topology but losing optimality should be ruled out or addressed.

Authors: The convergence is intended in the narrow topology. Given the finite first moment assumption, this topology ensures the plans are tight, and since the cost functions |x-y|^p converge uniformly on compacts to |x-y|, the optimality passes to the limit by standard lower semicontinuity arguments. We will revise Section 3.2 to explicitly state the topology as narrow convergence and include a brief justification that no counterexamples arise under the moment condition, as any sequence with finite moments converging narrowly will have the limit optimal for the limit cost. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent combinatorial construction of excursion coupling

full rationale

The paper begins from the known unique optima for strictly concave costs |x-y|^p (p<1) and takes the limit p→1, but supplies a direct, self-contained construction of the excursion coupling via combinatorial/geometric characterization of routes and secondary transport problems. This construction does not reduce by definition or fitting to the p-limits themselves, nor does it rely on load-bearing self-citations or imported uniqueness theorems. The verification that the constructed plan coincides with the limit is a separate proof step and does not create circularity. No steps match the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The construction rests on standard background results in optimal transport on the line and the existence of the limit; no free parameters or new entities with independent evidence are introduced in the abstract.

axioms (1)

standard math Optimal transport plans exist and can be characterized for costs |x-y|^α with α<1 on the real line.
Invoked implicitly when the paper considers the family of problems and takes the limit.

invented entities (1)

excursion coupling no independent evidence
purpose: Name for the limiting transport plan that restores uniqueness.
Defined as the limit of the family; no independent falsifiable prediction is stated in the abstract.

pith-pipeline@v0.9.0 · 5618 in / 1292 out tokens · 38686 ms · 2026-05-25T11:58:36.637809+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance, and letting this parameter tend to one... excursion coupling
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

arches do not cross, do not connect and have the same orientation when they are nested

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.