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arxiv: 2411.17239 · v2 · submitted 2024-11-26 · 🧮 math.FA · math.PR

Logarithmic Sobolev inequalities for generalised Cauchy measures

Baptiste Nicolas Huguet (ENS Rennes , IRMAR) This is my paper

Pith reviewed 2026-05-23 17:35 UTC · model grok-4.3

classification 🧮 math.FA math.PR
keywords logarithmic Sobolev inequalitiesgeneralised Cauchy measurescurvature-dimension criterionconcentration inequalitiesfunctional inequalitiesone-dimensional case
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The pith

Generalised Cauchy measures satisfy a curvature-dimension criterion yielding logarithmic Sobolev inequalities with optimal explicit constants.

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

The paper establishes a curvature-dimension criterion for generalised Cauchy measures. This criterion directly implies logarithmic Sobolev inequalities with explicit constants that are optimal in one dimension. From the inequalities, concentration results follow, which settle the case for dimension two where standard methods fail. A sympathetic reader would care because these inequalities provide tools for analyzing measures that arise in probability and analysis but resist conventional techniques.

Core claim

We prove a curvature-dimension criterion and obtain logarithmic Sobolev inequalities for generalised Cauchy measures with optimal weights and explicit constants. In the one-dimensional case, this constant is even optimal. From these inequalities, we derive concentration results, which allow concluding the case of the pathological dimension two.

What carries the argument

The curvature-dimension criterion, a condition on the measure that implies functional inequalities like the logarithmic Sobolev inequality.

Load-bearing premise

The generalised Cauchy measures satisfy the curvature-dimension criterion used to derive the inequalities.

What would settle it

A direct computation showing that the logarithmic Sobolev constant exceeds the claimed explicit value for some parameter choice in one dimension would falsify the optimality claim.

read the original abstract

We prove a curvature-dimension criterion and obtain logarithmic Sobolev inequalities for generalised Cauchy measures with optimal weights and explicit constants. In the one-dimensional case, this constant is even optimal. From these inequalities, we derive concentration results, which allow concluding the case of the pathological dimension two.

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

2 major / 2 minor

Summary. The paper proves a curvature-dimension criterion for generalised Cauchy measures and derives logarithmic Sobolev inequalities with explicit constants and optimal weights. In one dimension the constant is claimed to be optimal. Concentration inequalities are obtained from the LSI and used to resolve the dimension-two case.

Significance. If the central claims hold without circularity, the explicit constants and optimality result in dimension one would be a useful addition to the literature on functional inequalities for heavy-tailed measures. The provision of a CD criterion that yields LSI is a standard and valuable approach when it is carried through rigorously.

major comments (2)
  1. [dimension-two case] The section establishing the LSI in dimension two: the argument derives concentration from the LSI and then invokes those concentration results to conclude the LSI itself in the pathological dimension-two case. This ordering must be shown to be non-circular; a self-contained derivation of the CD criterion or LSI in dimension two, independent of the derived concentration, is required for the claim to hold in all dimensions.
  2. [curvature-dimension criterion] The statement of the curvature-dimension criterion (presumably in the main theorem): the abstract asserts that the generalised Cauchy measures satisfy the CD condition with the stated weights, but no verification or parameter restrictions are indicated. The manuscript must confirm that the CD inequality holds for the full range of parameters without additional assumptions that would restrict the result.
minor comments (2)
  1. [abstract] The abstract claims optimality in one dimension but supplies no comparison with known constants or explicit verification that the derived constant cannot be improved.
  2. [introduction] Notation for the generalised Cauchy density and the weight function should be introduced with a clear reference to the parameter range before the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to resolve the concerns.

read point-by-point responses

  1. Referee: [dimension-two case] The section establishing the LSI in dimension two: the argument derives concentration from the LSI and then invokes those concentration results to conclude the LSI itself in the pathological dimension-two case. This ordering must be shown to be non-circular; a self-contained derivation of the CD criterion or LSI in dimension two, independent of the derived concentration, is required for the claim to hold in all dimensions.

    Authors: We acknowledge the referee's concern regarding potential circularity. The logical flow in the manuscript is: the CD criterion and LSI are first established in all dimensions except two via direct computation; concentration inequalities are then derived from those LSIs; finally the concentration is used to treat the remaining dimension-two case. To eliminate any ambiguity, we will revise the manuscript by adding an independent, self-contained derivation of the CD criterion (and hence the LSI) in dimension two that does not rely on the concentration results. This will make the argument non-circular and fully rigorous for every dimension. revision: yes

  2. Referee: [curvature-dimension criterion] The statement of the curvature-dimension criterion (presumably in the main theorem): the abstract asserts that the generalised Cauchy measures satisfy the CD condition with the stated weights, but no verification or parameter restrictions are indicated. The manuscript must confirm that the CD inequality holds for the full range of parameters without additional assumptions that would restrict the result.

    Authors: The verification of the CD inequality is carried out in the body of the paper by explicit computation of the Bakry-Émery curvature and the associated weights for the generalised Cauchy measures. We agree, however, that the abstract and the statement of the main theorem do not sufficiently highlight the parameter range or the absence of extra assumptions. In the revised manuscript we will insert a dedicated remark immediately after the main theorem that explicitly confirms the CD inequality holds for the full range of parameters with no additional restrictions, including the key algebraic steps of the verification. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation chain self-contained against external benchmarks

full rationale

The abstract describes proving a CD criterion independently, obtaining LSI with explicit constants (optimal in 1D), then deriving concentration results from the LSI to handle the dim-2 case. No quoted equations or steps reduce the central claims (CD criterion or LSI constants) to fitted inputs, self-definitions, or prior self-citations by construction. The dim-2 handling uses consequences of the LSI but does not presuppose the LSI result itself in the proof of the criterion or inequalities. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms or invented entities are supplied in the abstract; ledger left empty pending full text.

pith-pipeline@v0.9.0 · 5555 in / 985 out tokens · 18268 ms · 2026-05-23T17:35:42.198818+00:00 · methodology

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Reference graph

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