Operator capacity, the Brascamp--Lieb inequality and geometric programming

Anthony Gauvan; Hiroshi Tsuji; Neal Bez

arxiv: 2508.02118 · v2 · submitted 2025-08-04 · 🧮 math.FA · math.CA· math.OC

Operator capacity, the Brascamp--Lieb inequality and geometric programming

Neal Bez , Anthony Gauvan , Hiroshi Tsuji This is my paper

Pith reviewed 2026-05-19 01:20 UTC · model grok-4.3

classification 🧮 math.FA math.CAmath.OC

keywords operator capacityBrascamp-Lieb constantgeometric programmingHölder regularitynear-minimisersquiver datacompletely positive operators

0 comments

The pith

Geometric programming interprets operator capacity and the Brascamp-Lieb constant as programs with a group minimization step, producing new near-minimiser and Hölder regularity statements.

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

The paper recasts the capacity of completely positive operators and the Brascamp-Lieb constant as unconstrained geometric programs up to an extra minimization over a compact group. This shared viewpoint lets the authors import recent regularity results to prove that operator capacity admits near-minimisers and is locally Hölder continuous in the input data. The same reduction extends the statements to capacities of quiver data and supplies a new argument for the known algebraic characterisation of when the Brascamp-Lieb constant remains finite, provided Lieb's Gaussian saturation theorem is granted.

Core claim

By viewing both the operator capacity and the Brascamp-Lieb constant through the lens of geometric programming, the authors obtain near-minimisers together with local Hölder regularity for operator capacity, carry the conclusions over to quiver capacities, and recover the finiteness characterisation of the Brascamp-Lieb constant from Bennett-Carbery-Christ-Tao under the assumption that Lieb's theorem on Gaussian saturation holds.

What carries the argument

The reduction of operator capacity and the Brascamp-Lieb constant to unconstrained geometric programming problems that include an auxiliary minimization over a compact group.

If this is right

Near-minimisers of operator capacity exist and can be recovered from solutions of the corresponding geometric program.
Operator capacity varies in a locally Hölder continuous manner with respect to small changes in the completely positive operator.
The same near-minimiser and regularity conclusions hold for capacities defined from quiver data.
The Brascamp-Lieb constant is finite precisely when the associated geometric program satisfies the expected algebraic feasibility conditions, assuming Gaussian saturation.

Where Pith is reading between the lines

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

Numerical geometric-programming solvers could now be used to approximate operator capacities or to locate near-minimisers in concrete examples.
The formulation may open routes to semidefinite-programming relaxations or other convex-optimization techniques already developed for geometric programs.
Analogous reductions might be attempted for other multilinear inequalities that currently lack regularity statements.

Load-bearing premise

The recent regularity theorems for geometric programs apply directly to the operator-capacity setting without extra restrictions, and Lieb's theorem on Gaussian saturation remains valid in the relevant cases.

What would settle it

An explicit family of completely positive operators for which the associated geometric program is feasible yet operator capacity fails to be locally Hölder continuous at some point would disprove the regularity claim.

read the original abstract

The capacity of completely positive operators and the Brascamp--Lieb constant can both be interpreted in terms of unconstrained geometric programming up to an additional minimisation over a compact group. We shine light on this perspective and make use of it to make novel contributions in both directions. For example, by making use of recent work of Bennett--Bez--Buschenhenke--Cowling--Flock, we prove new results regarding near-minimisers and local H\"older regularity of operator capacity. In addition, we observe that these results may be extended to the more general notion of capacity of quiver data. Furthermore, the geometric programming viewpoint allows us to give a new proof of the finiteness characterisation of the Brascamp--Lieb constant due to Bennett--Carbery--Christ--Tao (assuming Lieb's theorem on gaussian saturation).

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

Geometric programming recasts operator capacity to import new regularity results and an alternative finiteness proof, but the transfer of external hypotheses needs direct verification.

read the letter

The paper recasts capacities of completely positive operators and Brascamp-Lieb constants as unconstrained geometric programs plus a minimization over a compact group. This leads to new results on near-minimisers and local Hölder regularity for operator capacity, an extension to quiver data, and an alternative proof of the finiteness characterisation for the Brascamp-Lieb constant. The geometric programming lens is the useful part. It lets the authors apply recent regularity theorems from Bennett et al. directly to get concrete control on how the capacity behaves near its minimum. The quiver extension is a natural broadening. The new proof route for finiteness is straightforward once the setup is in place, assuming Lieb's theorem. The main concern is whether the structural conditions from the Bennett-Bez work carry over cleanly to the completely positive operator functional. The paper asserts the applicability, but without seeing the details it is hard to rule out extra singularities or missing lower bounds that could block the Hölder conclusion. The finiteness argument is conditional on Lieb's result, which is a standard assumption but keeps the contribution from being fully self-contained. Specialists in Brascamp-Lieb inequalities and related operator theory will find this worth reading. It adds specific new results rather than a broad overhaul, so a reader already familiar with the cited papers gets the most value. The work is grounded enough to deserve a serious referee. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper interprets the capacity of completely positive operators and the Brascamp-Lieb constant as unconstrained geometric programming problems up to an additional minimization over a compact group. It uses this viewpoint together with recent results of Bennett--Bez--Buschenhenke--Cowling--Flock to obtain new statements on near-minimisers and local Hölder regularity for operator capacity, extends the same conclusions to quiver data, and supplies a new proof of the finiteness characterisation of the Brascamp-Lieb constant (assuming Lieb's Gaussian saturation theorem).

Significance. If the regularity transfer is valid, the geometric-programming reformulation supplies a coherent analytic framework that yields concrete information on the stability of minimisers for operator capacity and its quiver generalisation. The alternative proof of the finiteness characterisation, while conditional on Lieb's theorem, organises existing ingredients in a potentially more transparent way. These contributions would be of interest to researchers working on Brascamp-Lieb-type inequalities and on optimisation over positive operators.

major comments (2)

[Section on regularity results (invoking Bennett--Bez et al.)] The local Hölder regularity and near-minimiser statements for operator capacity (and their quiver extension) rest on the direct applicability of the structural hypotheses in Bennett--Bez--Buschenhenke--Cowling--Flock (typically uniform convexity or smoothness on the relevant domain). The geometric-programming reformulation does not automatically guarantee these hypotheses; the manuscript does not contain an explicit verification that the completely-positive operator functional satisfies the required lower bounds or avoids extra singularities.
[Section containing the finiteness characterisation proof] The new proof of the finiteness characterisation of the Brascamp-Lieb constant is presented as following from the geometric-programming viewpoint. It remains unclear whether this viewpoint yields any reduction in the assumptions beyond the invocation of Lieb's theorem, or whether the argument is essentially a reorganisation of the Bennett--Carbery--Christ--Tao approach.

minor comments (2)

The notation for the auxiliary minimization over the compact group should be introduced once and used consistently; at present the group action appears with varying symbols in different sections.
Add a short paragraph recalling the precise statement of Lieb's theorem (including the domain on which Gaussian saturation is assumed) so that the dependence of the finiteness proof is fully explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, agreeing where clarification or verification is needed and explaining the contributions of the geometric programming viewpoint where appropriate. Revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Section on regularity results (invoking Bennett--Bez et al.)] The local Hölder regularity and near-minimiser statements for operator capacity (and their quiver extension) rest on the direct applicability of the structural hypotheses in Bennett--Bez--Buschenhenke--Cowling--Flock (typically uniform convexity or smoothness on the relevant domain). The geometric-programming reformulation does not automatically guarantee these hypotheses; the manuscript does not contain an explicit verification that the completely-positive operator functional satisfies the required lower bounds or avoids extra singularities.

Authors: We agree that the manuscript would benefit from an explicit verification that the completely positive operator functional satisfies the structural hypotheses (uniform convexity, smoothness, and suitable lower bounds) of Bennett--Bez--Buschenhenke--Cowling--Flock. The geometric programming reformulation recasts the capacity as an unconstrained problem, but this does not automatically transfer the required analytic properties; a direct check is necessary to rule out additional singularities. In the revised version we will add a dedicated subsection performing this verification for the operator capacity case and confirming that it extends to the quiver setting without introducing new issues. revision: yes
Referee: [Section containing the finiteness characterisation proof] The new proof of the finiteness characterisation of the Brascamp-Lieb constant is presented as following from the geometric-programming viewpoint. It remains unclear whether this viewpoint yields any reduction in the assumptions beyond the invocation of Lieb's theorem, or whether the argument is essentially a reorganisation of the Bennett--Carbery--Christ--Tao approach.

Authors: The geometric programming viewpoint does not reduce the assumptions beyond those already present in Lieb's Gaussian saturation theorem; the proof remains conditional on that result. Its contribution is a reorganisation that interprets the finiteness characterisation through the unconstrained geometric program together with the auxiliary minimisation over the compact group. This makes the link to operator capacity transparent and organises the ingredients from Bennett--Carbery--Christ--Tao in a manner that may be more accessible for future extensions. We will revise the relevant section to state this explicitly and to compare the new argument with the original approach. revision: partial

Circularity Check

0 steps flagged

Geometric programming reformulation is independent; new regularity results transfer from prior overlapping-author work without reducing claims to self-definition or fitted inputs.

full rationale

The paper's central contribution is an unconstrained geometric programming interpretation of operator capacity and the Brascamp-Lieb constant (up to group minimisation), which is used to derive new near-minimiser and Hölder regularity statements by direct application of the external Bennett--Bez--Buschenhenke--Cowling--Flock results, plus a new proof of the Bennett--Carbery--Christ--Tao finiteness characterisation that explicitly assumes Lieb's Gaussian saturation theorem. No equation or step equates a claimed output to an input by construction, renames a known pattern, or loads the main argument on an unverified self-citation chain. The cited prior work supplies independent structural hypotheses that the present reformulation invokes rather than derives, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard functional-analysis background plus two external theorems whose proofs are not reproduced here.

axioms (2)

domain assumption Lieb's theorem on Gaussian saturation for the Brascamp-Lieb constant
Invoked explicitly for the new proof of the finiteness characterisation.
domain assumption Applicability of the Bennett--Bez--Buschenhenke--Cowling--Flock near-minimiser results to operator capacity
Used to obtain the new regularity statements.

pith-pipeline@v0.9.0 · 5677 in / 1443 out tokens · 40667 ms · 2026-05-19T01:20:27.447951+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 (J-cost uniqueness) unclear

?

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

cap0(T)^m = inf_y ∑_{j∈J_{n,m}} d_j(T) exp(<y,u_j>) with u_j = j - (m/n)1 (Prop 4.1); local Hölder via Bennett-Bez-Buschenhenke-Cowling-Flock Thm 4.5 on Ψ(d)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

BL(L,θ)^{-2} = inf_R Ψ(d(L,θ,R)) after Gaussian saturation (Lieb) and diagonalization (eq 2.1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

R. B. Bapat,Mixed discriminants of positive semidefinite matrices, Linear Algebra Appl.126(1989), 107–124

work page 1989
[2]

Barthe,On a reverse form of the Brascamp–Lieb inequality, Invent

F. Barthe,On a reverse form of the Brascamp–Lieb inequality, Invent. Math.134 (1998), 335–361

work page 1998
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Bennett, N

J. Bennett, N. Bez, S. Buschenhenke, M. G. Cowling, T. C. Flock,On the nonlinear Brascamp–Lieb inequality, Duke Math. J.169(2020), 3291–3338

work page 2020
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Bennett, N

J. Bennett, N. Bez, M. G. Cowling, T. C. Flock,Behaviour of the Brascamp–Lieb constant, Bull. Lond. Math. Soc.49(2017), 512–518

work page 2017
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N. Bez, A. Gauvan, H. Tsuji,A note on ubiquity of geometric Brascamp–Lieb data, Bull. Lond. Math. Soc.57(2025), 302–314

work page 2025
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Franks,Operator scaling with specified marginals, STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 190–203

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A. Garg, L. Gurvits, R. Oliveira, A. Wigderson,Algorithmic and optimization aspects of Brascamp–Lieb inequalities, via Operator Scaling, Geom. Funct. Anal.28(2018) 100–145

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A. Garg, L. Gurvits, R. Oliveira, A. Wigderson,Operator scaling: theory and appli- cations, Found. Comput. Math.20(2020) 223–290

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Straszak, N

D. Straszak, N. K. Vishnoi,Maximum entropy distributions: Bit complexity and stability, Proceedings of the Thirty-Second Conference on Learning Theory, volume 99 of Proceedings of Machine Learning Research, pages 2861–2891. 2Alternatively, one may also check that log Φd(y) = supp∈P(J) (⟨y, P j∈J pj uj ⟩ −KL(p∥d)). 10 NEAL BEZ, ANTHONY GAUV AN, AND HIROSHI...

work page

[1] [1]

R. B. Bapat,Mixed discriminants of positive semidefinite matrices, Linear Algebra Appl.126(1989), 107–124

work page 1989

[2] [2]

Barthe,On a reverse form of the Brascamp–Lieb inequality, Invent

F. Barthe,On a reverse form of the Brascamp–Lieb inequality, Invent. Math.134 (1998), 335–361

work page 1998

[3] [3]

Bennett, N

J. Bennett, N. Bez, S. Buschenhenke, M. G. Cowling, T. C. Flock,On the nonlinear Brascamp–Lieb inequality, Duke Math. J.169(2020), 3291–3338

work page 2020

[4] [4]

Bennett, N

J. Bennett, N. Bez, M. G. Cowling, T. C. Flock,Behaviour of the Brascamp–Lieb constant, Bull. Lond. Math. Soc.49(2017), 512–518

work page 2017

[5] [5]

N. Bez, A. Gauvan, H. Tsuji,A note on ubiquity of geometric Brascamp–Lieb data, Bull. Lond. Math. Soc.57(2025), 302–314

work page 2025

[6] [6]

Franks,Operator scaling with specified marginals, STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 190–203

C. Franks,Operator scaling with specified marginals, STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 190–203

work page

[7] [7]

A. Garg, L. Gurvits, R. Oliveira, A. Wigderson,Algorithmic and optimization aspects of Brascamp–Lieb inequalities, via Operator Scaling, Geom. Funct. Anal.28(2018) 100–145

work page 2018

[8] [8]

A. Garg, L. Gurvits, R. Oliveira, A. Wigderson,Operator scaling: theory and appli- cations, Found. Comput. Math.20(2020) 223–290

work page 2020

[9] [9]

Gurvits,Classical complexity and quantum entanglement, J

L. Gurvits,Classical complexity and quantum entanglement, J. Comput. System Sci. 69(2004), 448–484

work page 2004

[10] [10]

T. C. Kwok, L. C. Lau, A. Ramachandran,Spectral analysis of matrix scaling and operator scaling, SIAM J. Comput.50(2021), 1034–1102

work page 2021

[11] [11]

E. H. Lieb,Gaussian kernels have only Gaussian maximizers, Invent. Math.102 (1990), 179–208

work page 1990

[12] [12]

Straszak, N

D. Straszak, N. K. Vishnoi,Maximum entropy distributions: Bit complexity and stability, Proceedings of the Thirty-Second Conference on Learning Theory, volume 99 of Proceedings of Machine Learning Research, pages 2861–2891. 2Alternatively, one may also check that log Φd(y) = supp∈P(J) (⟨y, P j∈J pj uj ⟩ −KL(p∥d)). 10 NEAL BEZ, ANTHONY GAUV AN, AND HIROSHI...

work page