Weak Optimal Transport: When is the Dual Potential Convex?

Filip Pramenkovi\'c

REVIEW 3 major objections 4 minor 1 cited by

Weak Optimal Transport: When is the Dual Potential Convex?

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One order condition decides when weak transport duals can be convex

desk verdict A genuinely unifying, sharp condition for convex dual potentials in weak optimal transport; the core proof is sound and it deserves a serious referee, with some extensions sketched too briefly. read the letter →

arxiv 2507.07200 v2 pith:R7VS3VFC submitted 2025-07-09 math.PR math.OC

Filip Pramenković This is my paper · ORCID

classification math.PRmath.OC MSC 49Q2260E15

keywords weakoptimaltransportconvexdualpotentialsorderincreasingStrassen'stheoremmartingaleBenamou-Breniermechanismdesignstablecones

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper identifies the exact condition under which the dual of a weak optimal transport problem can be maximized over convex potentials: the cost function must be nonincreasing in convex order in its second argument. The main theorem, Theorem 1.1, proves that this monotonicity is necessary and sufficient for the restricted duality to hold for all marginals, and a parallel theorem does the same for increasing convex potentials with the increasing convex order. The paper then shows that, under a boundedness condition and a mild continuity condition, the restricted supremum is attained by a $\nu$-integrable convex function. Applications recover several results that previously looked separate: Strassen's martingale theorem, barycentric transport, the martingale Benamou--Brenier problem, the multiple-good monopolist problem, and classical quadratic-cost Brenier duality. If the criterion is correct, checking one monotonicity property of the cost tells you whether the dual simplifies to convex potentials and whether an optimal convex potential exists.

What carries the argument

The machine that carries the argument is the convex-hull representation $\operatorname{conv}\psi(y)=\inf\{\rho(\psi):\delta_y\preceq_c\rho,\ \rho\in P_p(\mathbb{R}^d)\}$ from Lemma 2.1(i), together with its increasing-convex analogue and the monotone approximation $\operatorname{conv}(\psi+|\cdot|^p/n)\downarrow\operatorname{conv}\psi$. This identity converts convex-order monotonicity of the cost into an equality of $C$-conjugates, $\psi^C=(\operatorname{conv}\psi)^C$: if replacing $\rho$ by a measure above it in convex order only lowers the cost, then the cheapest measure for $\psi$ is no cheaper for its convex hull. That equality is precisely what lets the unrestricted dual over all test functions be restricted to convex ones, and the converse half of the theorem runs the same machinery backwards to force monotonicity of $C$. Section 6 generalizes the hull representation into a stability condition on cones, so the same argument works for any order generated by a stable cone.

What would settle it

Compute the primal weak transport value and the convex-restricted dual value for $C(x,\rho)=\int y^4\,d\rho(y)$ with $\mu=\delta_0$ and $\nu=(\delta_{-1}+\delta_1)/2$; they are $1$ and $0$, a gap that the converse direction demands for a cost that is not convex-order decreasing, so an analogous computation producing a gap for a cost that is convex-order decreasing would refute the forward direction.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that convexity of dual potentials in weak optimal transport is governed by convex-order monotonicity of the cost. For a lower semicontinuous cost $C:X\times P_p(\mathbb{R}^d)\to[0,\infty]$ that is convex in its second argument, the paper proves the identity $\inf_{\pi\in\Pi(\mu,\nu)}\int C(x,\pi_x)\,d\mu(x)=\sup_{\psi\in C_{b,p}(\mathbb{R}^d),\ \psi\text{ convex}}(\mu(\psi^C)-\nu(\psi))$ holds for every pair of marginals if and only if $C$ is $\preceq_c$-decreasing in its second argument. The same statement with increasing convex functions and the increasing convex order $\preceq_{icx}$ is Theorem 3.1. Under boundedness condition (B) and continuity condition (C), Theorem 4.1 upgrades the equality to attainment: the supremum is reached by a $\nu$-integrable convex function $\psi_{\mathrm{opt}}:\mathbb{R}^d\to(-\infty,\infty]$. Section 6 abstracts the mechanism to stable cones of functions, yielding the same restriction for any order whose cone satisfies a hull-representation condition.

Load-bearing premise

The argument relies on the classical representation of the convex hull of a test function as an infimum of expectations taken over probability measures that dominate the Dirac mass at that point in convex order, and on the monotone approximation that realizes that hull; if this representation failed for some admissible function, the equality of conjugates that powers the restriction theorem would break.

Editorial extensions

If this is right

Applying the theorem to the order-indicator cost $C(x,\rho)=0$ if $\delta_x\preceq_c\rho$ and $\infty$ otherwise gives Strassen's theorem: $\mu\preceq_c\nu$ exactly when a martingale coupling exists.
For barycentric costs $C(x,\rho)=\theta(x-\operatorname{mean}\rho)$ with convex $\theta$, the cost is automatically $\preceq_c$-decreasing, reproducing the known restriction of the dual to convex functions and, for quadratic $\theta$, the Brenier--Strassen mixture result.
The increasing-convex version recovers the multiple-good monopolist duality and the increasing-convex Kantorovich--Rubinstein formula, with dual attainment by increasing convex potentials under the regularity conditions.
For classical transport costs $C(x,\rho)=\int c(x,y)\,d\rho(y)$, convex-order monotonicity is equivalent to concavity of $c$ in its second argument, so the quadratic-cost duality in the paper is a direct instance of the same criterion.
Under conditions (B) and (C), the convex-restricted dual supremum is attained by a $\nu$-integrable convex function, meaning the simplification is not only an equality of values but an achieved optimum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A screening rule follows that the paper leaves implicit: one can certify the convex-potential simplification by checking $C(x,\cdot)$ against convex-order comparisons on a finite set of test measures, without solving the transport problem itself.
The stable-cone framework suggests the same reduction should hold for other orders, such as directionally convex or $k$-convex functions, whenever the associated hull admits an infimum-over-measures representation; the paper only instantiates convex and increasing convex.
In the martingale Benamou--Brenier example the boundedness condition (B) fails, so attainment is delicate; an extension to that setting would need a problem-specific replacement for (B) rather than a direct application of Theorem 4.1.
The converse direction could be used diagnostically: if a transport-type model provably has convex optimal dual potentials for all marginals, then its cost must be convex-order decreasing, which may reveal hidden monotonicity in models not currently formulated as weak transport.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a clean, genuinely unifying result. The main theorem — convex-order monotonicity of the cost is equivalent to restricting the WOT dual to convex potentials — is stated in that generality for the first time, and the proof is largely sound. The converse direction via delta marginals is a nice argument. I agree with the reader's ACCEPT and soundness 8.

What the paper does well: it takes a set of known dual restrictions (barycentric, martingale Benamou–Brenier, mechanism design, Strassen) and shows they all check the same simple condition. Lemma 2.3 is the heart: under monotonicity, psi^C = (conv psi)^C. The use of Lemma 2.1(i)/(vi) is legitimate; the stress-test concern about that imported identity does not land on reading the proof. I could not construct a counterexample either. The application section is a useful checklist for the condition.

Soft spots, in order of softness: Theorem 3.2 (allowing negative costs) is asserted as a modification of earlier proofs without being written out. Same for the stable-cone Theorem 6.1, which relies on semi-stable cone condition (5) — the paper itself calls it 'somewhat artificial'. These are not flaws in the central argument, but they mean the extensions are more sketch than proof. The attainment section is dense and relies on the external [7, Theorem 2.2] plus conditions (B)–(C); it is coherent but technical. A referee should ask for Theorem 3.2 to be proved, not just described.

I'd send this to a serious referee. The core result is important enough and the proof detailed enough. It will be a well-cited unification paper. My only wish is that the sketched parts were fuller, but that's a revision, not a rejection.

Referee Report

3 major / 4 minor

Summary. The paper studies the weak optimal transport duality and asks when the dual maximization can be restricted to convex (or increasing convex) potentials. The main result, Theorem 1.1, states that for a lower semicontinuous cost C that is convex in its second argument, equality between the primal weak transport problem and the dual restricted to convex potentials holds for all marginals if and only if C is nonincreasing in convex order in its second argument. The proof combines the general weak transport duality theorem with an approximation argument using the convex hull and a measurable selection. The paper also proves the increasing-convex analogue, establishes dual attainment under additional boundedness and continuity conditions, and derives several known results as applications: barycentric transport, Strassen's theorem, the martingale Benamou–Brenier formulation, the multiple-good monopolist problem, and classical quadratic transport. A general stable-cone framework is proposed in the final section.

Significance. If the results are correct, Theorem 1.1 provides a sharp and unifying explanation for a number of disparate dual-restriction results in weak optimal transport, martingale transport, mechanism design, and classical Brenier duality. The proof of the main equivalence is clean and self-contained apart from two imported ingredients: the general weak transport duality theorem of Backhoff-Veraguas, Beiglböck, and Pammer, and standard convex-hull representations. The paper is honest about the conditional nature of the stable-cone framework and about the delicacy of dual attainment in unbounded settings, for example in Remark 5.1. No free parameters are introduced and the converse direction of Theorem 1.1 gives a falsifiable characterization, which strengthens confidence in the result.

major comments (3)

[§3, Theorem 3.2] Theorem 3.2 is stated without proof: the text says only that the proof of Theorem 1.1 'can be modified' to accommodate costs bounded below by -(a_l(x) + rho(b_l)). This theorem is then used in §5.1.2 and §5.1.3 to recover the martingale Benamou-Brenier duality and classical quadratic transport duality, so the missing argument is load-bearing for the applications claimed in the abstract. In particular, when the cost is not nonnegative and the potential class is enlarged to Cb,p + b_l, the C-conjugate may take the value -infinity and the finiteness arguments in Lemma 2.3 need to be reworked. Please supply a full proof or a precise reference for this extension.
[§6, Theorem 6.1] Theorem 6.1 is stated as a theorem but its proof is not given; the text says only that the proof of Theorem 1.1 'naturally extends to the setting of stable cones.' The result depends crucially on the dual-representation assumption (5), and Theorem 6.4 relies on Theorem 6.1. Since the authors themselves note that the validity of such representations for semi-stable cones on noncompact spaces is subtle (with [11] flawed and [10] giving a counterexample), the proof should be written out to make clear exactly which properties (1)-(5) are used and where compactness or additional assumptions are needed.
[§4, Eq. (14)] In Step 1 of the proof of Theorem 4.1, the representation conv psi(y) = inf{xi(psi) : delta_y <=_c xi, |supp xi| <= d+1} is asserted without proof. This identity is used to control conv_R psi and to prove its nu-integrability, and is therefore central to the attainment result. Please add a proof or a precise reference for this Caratheodory-type hull formula.

minor comments (4)

[§2, Lemma 2.1(i)] Please add a reference or proof for the convex-hull representation conv psi(y) = inf{rho(psi) : delta_y <=_c rho}; the current proof is only written for the increasing convex version (ii).
[§4, Step 2] Egorov's theorem is applied to conv_R psi, which is only known to be nu-integrable and may take the value +infinity; please clarify that one first restricts to a set of full rho-measure on which these functions are finite, so that the uniform-convergence argument is justified.
[§5.1.2] The lower bound used for the martingale Benamou-Brenier example appears to be off by a dimension constant: for rho = gamma one has MCov(rho,gamma) = 1 in d = 1 while (1/2)rho(|y|^2) = 1/2. The bound should read -MCov(rho,gamma) >= -1/2 rho(|y|^2) - d/2, and b_l should be chosen accordingly.
[Throughout] There are several typos and typesetting issues: the running title contains 'TRANSPOR T' with a stray space, the notation chi_{x=mean(pi_x)} in Eq. (3) should be defined, and the measure e_rho is introduced with the unusual notation 'de_rho' in the proof of Lemma 2.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 follows from external WOT duality and standard convex-analysis identities; applications are recovered after the fact.

full rationale

The central derivation chain is not circular. Theorem 1.1 is proved by combining Theorem 2.4, the general weak-transport duality theorem of Backhoff-Veraguas, Beiglböck and Pammer, with Lemma 2.3, which uses the classical convex-hull representation conv psi(y) = inf{rho(psi): delta_y ⪯_c rho} (Lemma 2.1(i)) and the monotone approximation conv(psi + |.|^p/n) down to conv psi (Lemma 2.1(vi)). These are standard convex-analysis facts; they do not encode the target inequality (5) and are not derived from the applications. The converse direction assumes the restricted duality for all marginals and derives ⪯_c-decreasingness from it, which is the logically correct structure for an equivalence proof. Theorem 4.1 imports [7, Theorem 2.2] for existence of an optimal dual pair in the full admissible class; that theorem is an external published result, and the remaining argument verifies convexity of an optimizer. No fitted parameters, benchmark-tuned normalizations, or author-overlapping uniqueness theorems are used. The known applications, including Strassen's theorem, barycentric transport, martingale Benamou-Brenier, and the multiple-good monopolist problem, are consequences of Theorem 1.1, not premises. The paper honestly flags unproved modifications, such as Theorem 3.2 and the stable-cone discussion, and explicitly labels assumption (5) in the stable-cone framework as 'somewhat artificial'; these are limitations or omitted proofs, not circular dependencies. The derivation is self-contained relative to its cited external duality and convex-analysis inputs, and no load-bearing step reduces to its own conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters and no invented entities. The paper's contribution is structural: all objects, costs, potentials, hulls, and orders already exist in the cited literature. The main external axioms are the general WOT duality theorem and the convex-hull dual representation, with measurable selection and regularity assumptions in support.

assumptions (6)

domain assumption General weak optimal transport duality: inf over plans of integral C dmu equals sup over psi in Cb,p of mu(psi^C) - nu(psi) for lsc convex-in-second-argument C (Theorem 2.4, from Backhoff-Veraguas, Beiglbock, Pammer).
Imported black box used as the first equality in the proof of Theorem 1.1 and throughout; the paper's result inherits its hypotheses.
standard math Dual representation of the convex hull: conv psi(y) = inf{rho(psi): delta_y <=_c rho, rho in P_p}, and the analogous increasing convex hull identity (Lemma 2.1(i),(ii)).
This is the bridge between convex-order monotonicity of C and equality of psi^C with (conv psi)^C in Lemma 2.3.
standard math Monotone approximation conv(psi + |.|^p/n) decreases to conv psi as n goes to infinity (Lemma 2.1(vi)).
Used in Lemma 2.3 to obtain finite epsilon estimates; the paper states it as classical without proof.
standard math Von Neumann uniformization and measurable selection of the family xi_y used in Lemma 2.3 and Theorem 4.1.
Needed to select, in a measurable way, measures witnessing the convex hull representation and then to average them into e rho.
domain assumption Existence of an optimal dual pair and weak transport duality from Beiglbock, Pammer, Riess, Schrott [7, Theorem 2.2], together with regularity conditions (B) and (C).
Theorem 4.1 is built directly on this external result; the paper adds convexity and monotonicity but does not reprove the underlying dual attainment.
domain assumption In the compact setting, a semi-stable cone of functions is a stable cone via Meyer [16, Corollary XI, T46].
Used in Section 6 to obtain the dilation theorem and the two-step transport identity for general stable cones.

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Cite this review

Pith. "Pith review of Weak Optimal Transport: When is the Dual Potential Convex?." pith.science (2026). https://pith.science/paper/R7VS3VFC

@misc{pith2026250707200,
  author       = {Pith},
  title        = {Pith review of: Weak Optimal Transport: When is the Dual Potential Convex?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7VS3VFC}},
  note         = {Machine review of arXiv:2507.07200}
}

read the original abstract

Weak optimal transport generalizes the classical theory of optimal transportation to nonlinear cost functions and covers a range of problems that lie beyond the traditional theory - including entropic transport, martingale transport, and applications in mechanism design. As in the classical case, the weak transport problem can also be written as a dual maximization problem over a pair of conjugate potentials. We identify sharp monotonicity conditions on the cost under which the dual problem can be restricted to convex potentials. This framework unifies several known results from the literature, including barycentric transport, martingale Benamou-Brenier, the multiple-good monopolist problem, Strassen's theorem, stochastic order projections and the classical Brenier theorem.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Brenier-Strassen Theorem on CAT(kappa) Spaces
math.FA 2026-07 accept novelty 7.0 of 10

On CAT(0) spaces, every probability measure has a unique Wasserstein projection onto the set of measures dominated by ν in convex order, and the optimal transport is a 1-Lipschitz map.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [11]

Gogus, Tony L

Nihat G. Gogus, Tony L. Perkins, and Evgeny A. Poletsky. Non-compact versions of Edwards’ Theorem. Positivity, 17(3):459–473, Sep 2013

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Ibrahim K. Djire and Jan Wiegerinck. On a question of sadullaev concerning boundary relative extremal functions. MATHEMATICA SCANDINA VICA, 124(1):51–60, Jan. 2019

work page 2019

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J-J Alibert, Guy Bouchitt´ e, and Thierry Champion. A new class of costs for optimal transport planning. European Journal of Applied Mathematics , 30(6):1229–1263, 2019

work page 2019

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J. Backhoff-Veraguas, M. Beiglb¨ ock, and G. Pammer. Existence, duality, and cyclical mono- tonicity for weak transport costs. Calculus of Variations and Partial Differential Equations , 58(6):203, Nov 2019

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J. Backhoff-Veraguas and G. Pammer. Stability of martingale optimal transport and weak optimal transport. Ann. Appl. Probab., 32(1):721–752, 2022

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