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arxiv: 2507.13462 · v2 · submitted 2025-07-17 · 🧮 math.MG

Equality in Liakopoulos's generalized dual Loomis-Whitney inequality via Barthe's Reverse Brascamp-Lieb inequality

Karoly J. B\"or\"oczky , Ferenc Fodor , Pavlos Kalantzopoulos This is my paper

Pith reviewed 2026-05-19 04:19 UTC · model grok-4.3

classification 🧮 math.MG
keywords equality casesdual Loomis-Whitney inequalityBarthe reverse Brascamp-Lieb inequalityvolume estimateslinear subspacesconvex geometrysections and projections
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The pith

Equality holds in Liakopoulos's generalized dual Loomis-Whitney inequality exactly when the sets are sections by certain lower-dimensional linear subspaces.

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

The paper shows how to identify the equality cases in Liakopoulos's volume estimate by transferring the known equality conditions from Barthe's Geometric Reverse Brascamp-Lieb inequality. A sympathetic reader would care because sharp inequalities in convex geometry often reveal the precise geometric configurations that attain the bound, turning an abstract estimate into a statement about when specific subspace sections maximize volume. The argument proceeds by direct application of the existing characterization, without introducing new restrictions. This produces a description of equality in terms of the sets being compatible with a collection of lower-dimensional subspaces.

Core claim

Using the characterization of equality cases already established for Barthe's Geometric Reverse Brascamp-Lieb inequality, the authors characterize equality in Liakopoulos's generalized dual Loomis-Whitney inequality as holding precisely when the relevant sets arise as sections by certain lower-dimensional linear subspaces.

What carries the argument

The equality characterization of Barthe's Geometric Reverse Brascamp-Lieb inequality, transferred to characterize when volume bounds are achieved in the generalized dual Loomis-Whitney setting.

If this is right

  • The volume estimate becomes an if-and-only-if statement once the subspace-section condition is verified.
  • All previous equality cases in related Loomis-Whitney-type inequalities can be recovered as special instances.
  • The result supplies a geometric criterion for sharpness that can be checked by examining projections or sections.
  • Any future generalization of the dual Loomis-Whitney inequality inherits an equality description from the same Barthe source.

Where Pith is reading between the lines

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

  • The same transfer technique might apply to equality cases in other Brascamp-Lieb-type inequalities that Liakopoulos's estimate generalizes.
  • It could simplify proofs of stability versions by reducing them to stability statements already known for Barthe's inequality.
  • The subspace-section description may suggest new ways to construct extremal examples in higher-dimensional convex geometry.

Load-bearing premise

The equality characterization already known for Barthe's Geometric Reverse Brascamp-Lieb inequality carries over directly and without extra restrictions to the hypotheses of Liakopoulos's generalized dual Loomis-Whitney inequality.

What would settle it

An explicit example of sets satisfying the hypotheses of Liakopoulos's inequality where equality holds but the sets fail to be sections by the expected lower-dimensional linear subspaces.

read the original abstract

We use the characterization of the case of equality in Barthe's Geometric Reverse Brascamp-Lieb inequality to characterize equality in Liakopoulos's volume estimate in terms of sections by certain lower-dimensional linear subspaces.

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

1 major / 2 minor

Summary. The manuscript uses the known characterization of equality cases in Barthe's Geometric Reverse Brascamp-Lieb inequality to determine the equality cases in Liakopoulos's generalized dual Loomis-Whitney inequality, expressing them in terms of sections by certain lower-dimensional linear subspaces.

Significance. If the transfer holds, the result supplies an explicit geometric description of equality attainment for the generalized dual Loomis-Whitney volume estimate. This is useful because it connects two existing inequalities without re-deriving the equality case ab initio, and the subspace-section formulation may be directly applicable in convex geometry.

major comments (1)
  1. [Main argument / proof of the equality characterization] The central step invokes Barthe's equality characterization and claims direct applicability to Liakopoulos's hypotheses. The manuscript must explicitly verify that every equality-attaining configuration under the generalized dual Loomis-Whitney assumptions satisfies the precise subspace-alignment, Gaussian, or measure-theoretic conditions under which Barthe's equality case is proved (and conversely). Without this compatibility check, the transfer is not yet justified.
minor comments (2)
  1. Add a short paragraph comparing the hypotheses of Liakopoulos's inequality with those of Barthe's result to make the transfer transparent.
  2. Ensure notation for the lower-dimensional subspaces is introduced before it is used in the equality statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to strengthen the justification of the equality transfer. We have revised the manuscript to include an explicit compatibility verification between the two sets of hypotheses.

read point-by-point responses

  1. Referee: The central step invokes Barthe's equality characterization and claims direct applicability to Liakopoulos's hypotheses. The manuscript must explicitly verify that every equality-attaining configuration under the generalized dual Loomis-Whitney assumptions satisfies the precise subspace-alignment, Gaussian, or measure-theoretic conditions under which Barthe's equality case is proved (and conversely). Without this compatibility check, the transfer is not yet justified.

    Authors: We agree that an explicit verification is required for rigor. Although Liakopoulos's generalized dual Loomis-Whitney inequality arises as a direct specialization of Barthe's geometric reverse Brascamp-Lieb inequality (with the same Gaussian measures and linear subspace data), the original manuscript did not spell out the matching of equality conditions in full detail. In the revised version we have added a new paragraph immediately after the statement of Theorem 1.2 and a short dedicated subsection (now labeled 3.1) that performs the required check in both directions: (i) any equality case for the generalized dual Loomis-Whitney inequality must consist of Gaussian measures whose supports are aligned with the indicated lower-dimensional subspaces, exactly as required by Barthe's characterization; (ii) conversely, any configuration satisfying Barthe's equality criteria produces equality in Liakopoulos's volume estimate. This verification uses only the hypotheses already present in the paper and does not introduce new assumptions. revision: yes

Circularity Check

0 steps flagged

External application of Barthe equality cases to Liakopoulos setting with no internal reduction

full rationale

The derivation applies the established equality characterization from Barthe's Geometric Reverse Brascamp-Lieb inequality (by a distinct author) to obtain equality cases for Liakopoulos's generalized dual Loomis-Whitney inequality in terms of lower-dimensional sections. No step in the provided abstract or described chain reduces a claimed prediction or result to a fitted parameter, self-definition, or self-citation load-bearing premise within the paper itself. The central claim is an application of an independent external theorem rather than a re-derivation that collapses to the paper's own inputs by construction. This is a standard, non-circular use of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard facts from convex geometry and previously established equality characterizations; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Lebesgue measure, linear subspaces, and volume in Euclidean space
    Invoked implicitly when discussing sections and volume estimates.

pith-pipeline@v0.9.0 · 5563 in / 1157 out tokens · 41289 ms · 2026-05-19T04:19:19.213258+00:00 · methodology

discussion (0)

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Reference graph

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