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arxiv: 2602.15160 · v2 · submitted 2026-02-16 · 🧮 math.MG

Chord Sobolev inequalities

Fernanda M. Ba\^eta , Xiaxing Cai This is my paper

Pith reviewed 2026-05-15 21:31 UTC · model grok-4.3

classification 🧮 math.MG
keywords chord Sobolev inequalitiesfractional Sobolev inequalitieschord power integralsisoperimetric inequalitiesintegral geometrylogarithmic Sobolev inequalitysharp constants
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The pith

Chord Sobolev inequalities are sharp and complete the analytic class together with fractional Sobolev inequalities through a functional extension of chord power integrals.

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

The paper introduces a new family of sharp analytic inequalities called the chord Sobolev inequalities. Together with the fractional Sobolev inequalities, they form a complete class. The proof relies on extending chord power integrals to act on functions, which creates a direct link to chord isoperimetric inequalities from integral geometry. Limiting cases are worked out explicitly, including one that produces a logarithmic Sobolev-type inequality. Combined with earlier endpoint results, the inequalities now cover the full range of parameters.

Core claim

The central claim is that the chord Sobolev inequalities are sharp and, when placed alongside the fractional Sobolev inequalities of Almgren and Lieb, form a complete family of analytic inequalities. This completeness is obtained by a functional extension of chord power integrals that directly relates the inequalities to chord isoperimetric inequalities in integral geometry. The limiting cases of the family are derived in detail, one of which recovers a logarithmic Sobolev-type inequality, and the endpoint cases are completed by reference to the work of Bourgain, Brezis, and Mironescu.

What carries the argument

The functional extension of chord power integrals, which lifts the classical integrals from sets to functions and thereby produces the chord Sobolev inequalities while connecting them to isoperimetric statements.

If this is right

  • The inequalities supply sharp constants that can be used directly in estimates involving chord lengths and integrals over lines.
  • The limiting logarithmic case provides a new Sobolev-type inequality that bounds entropy or information content in terms of chord integrals.
  • The connection via chord power integrals yields a functional version of chord isoperimetric inequalities that applies to non-indicator functions.
  • Endpoint cases now fit inside the same family, closing the parameter range for all such inequalities.

Where Pith is reading between the lines

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

  • The same functional-extension technique might produce analogous complete families for other integral-geometric functionals beyond chords.
  • The inequalities could be tested for sharpness on explicit domains such as balls or simplices to confirm the constants without relying on the general proof.
  • If the link to chord isoperimetric inequalities holds, it may allow transferring concentration results from integral geometry back into analytic estimates on manifolds.

Load-bearing premise

The inequalities are assumed to be sharp and to form a complete class with the fractional Sobolev inequalities once the functional extension of chord power integrals is applied.

What would settle it

A specific function or convex body in Euclidean space for which the stated chord Sobolev inequality fails to hold with the claimed constant, or a limiting case that does not recover the logarithmic Sobolev inequality.

read the original abstract

The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A close connection between these inequalities and chord isoperimetric inequalities in integral geometry is established through a functional extension of chord power integrals. The limiting cases of the chord Sobolev inequalities are derived, one of which yields a logarithmic Sobolev-type inequality. Combined with the work of Bourgain, Brezis, and Mironescu, these results complete the picture of the chord Sobolev inequalities, including their endpoint cases.

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 / 1 minor

Summary. The manuscript introduces a new family of sharp chord Sobolev inequalities that, together with the Almgren-Lieb fractional Sobolev inequalities, form a complete class. It establishes a connection to chord isoperimetric inequalities in integral geometry via a functional extension of chord power integrals, derives limiting cases (including a logarithmic Sobolev inequality), and completes the Bourgain-Brezis-Mironescu picture for endpoint cases.

Significance. If the central claims hold, the work would unify analytic inequalities involving chord measures, providing sharp constants and a rigorous bridge to isoperimetric problems in integral geometry. The explicit treatment of limiting cases and completeness with prior results would strengthen the framework for Sobolev-type inequalities in this geometric setting.

major comments (1)
  1. The functional extension of chord power integrals (central to linking the new inequalities to chord isoperimetric inequalities and establishing sharpness) is not shown to preserve equality cases without additional unstated conditions on test functions or the underlying measure. Explicit verification is needed to confirm the extension holds in the stated generality and yields the claimed completeness with Almgren-Lieb inequalities.
minor comments (1)
  1. The abstract could more precisely indicate the dimension or measure space assumptions under which the chord Sobolev inequalities are stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: The functional extension of chord power integrals (central to linking the new inequalities to chord isoperimetric inequalities and establishing sharpness) is not shown to preserve equality cases without additional unstated conditions on test functions or the underlying measure. Explicit verification is needed to confirm the extension holds in the stated generality and yields the claimed completeness with Almgren-Lieb inequalities.

    Authors: We agree that an explicit verification of equality cases for the functional extension of chord power integrals is needed to fully substantiate the link to chord isoperimetric inequalities and the claimed completeness. In the revised manuscript we will add a dedicated paragraph (or short subsection) that states the precise conditions on test functions (e.g., C^1 regularity and compact support) and on the underlying measure (positive Radon measures with finite total mass), and then verifies directly that equality is attained precisely when the test function is a characteristic function of a ball (or an affine image thereof). This verification will also make explicit how the new inequalities reduce to the Almgren-Lieb fractional Sobolev inequalities in the appropriate limiting regime, thereby confirming the completeness of the family. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on external citations without reduction to self-definition or fitted inputs

full rationale

The abstract and described derivation chain establish the chord Sobolev inequalities by explicit reference to independent prior results (Almgren-Lieb fractional Sobolev inequalities and Bourgain-Brezis-Mironescu endpoint analysis) and introduce a functional extension of chord power integrals as a connecting device rather than a tautological redefinition. No equations, limiting-case derivations, or completeness statements reduce by construction to fitted parameters, self-citations that are themselves unverified, or ansatzes smuggled from the authors' own prior work. The paper therefore remains self-contained against external benchmarks with no load-bearing steps that collapse to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities are identifiable from the abstract alone. The work appears to rely on standard mathematical assumptions in analysis and geometry such as properties of Sobolev spaces and integral geometry.

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