A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem

Gangsong Leng

REVIEW 2 major objections 2 minor 5 references

A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem

T0 review · 2 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Replacing the Hölder step with an optimal L¹-stability result produces a density-distance version of the Carlen-Frank-Lieb stability theorem.

desk verdict This note swaps Leng-Lu L1-stability into the CFL decomposition to get an L1 density-distance version plus a mixed-volume application, but the substitution step needs checking for hidden remainders. read the letter →

arxiv 2606.03749 v1 pith:TZRVQW42 submitted 2026-06-02 math.AP math.FAmath.MG

Gangsong Leng This is my paper

classification math.APmath.FAmath.MG

keywords Carlen-Frank-LiebstabilityL1theoremdensitydistanceSchrödingeroperatorGagliardo-Nirenberg-Sobolevmixedvolumeinequalitysupportfunctionconvexbodies

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

The paper takes the Carlen-Frank-Lieb decomposition of stability estimates for the lowest eigenvalue of a Schrödinger operator, which splits the problem into a Hölder inequality part and a Gagliardo-Nirenberg-Sobolev part. It shows that swapping the Hölder stability estimate for the optimal L¹-stability theorem of Leng and Lu in probabilistic form yields a new stability bound. This bound measures the L¹ distance between the normalized density V_-^s / ∫ V_-^s induced by the negative part of the potential and the corresponding density from an optimal potential, with s equal to γ plus d/2. A reader would care because the result reframes closeness to optimality directly in terms of how the potential's induced density deviates in L¹ norm. The same substitution also produces a density-stability form of the L_p mixed volume inequality that applies to convex bodies under stated symmetry and boundedness conditions.

What carries the argument

The Leng-Lu L¹-stability theorem in probabilistic form, inserted in place of the Hölder stability estimate inside the existing Carlen-Frank-Lieb decomposition of the eigenvalue stability problem.

What would settle it

A concrete potential for which the L¹ distance between the normalized densities fails to control the eigenvalue stability deficit in the manner predicted by the substituted theorem would falsify the claim.

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Extended reading notes

Core claim

By substituting the optimal L¹-stability theorem of Leng and Lu in probabilistic form for the Hölder inequality step inside the Carlen-Frank-Lieb decomposition, one obtains a density-distance version of the stability theorem. The new estimate controls the L¹ distance between the normalized density V_-^s / ∫ V_-^s coming from the negative part of the potential and the density induced by an optimal potential.

Load-bearing premise

The original split of the stability problem into a Hölder part and a GNS part stays valid after the replacement, without adding uncontrolled error terms that would alter the overall conclusion.

Editorial extensions

If this is right

The same replacement produces a density-stability version of the L_p mixed volume inequality.
When one of two convex bodies is centrally symmetric and both lie between two concentric Euclidean balls, the result supplies an averaged stability estimate for the non-evenness of the support function.

Reading between the lines

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

The density-distance formulation may make it easier to interpret stability deficits geometrically when potentials are viewed through their induced measures.
The same substitution technique could be tested on other stability proofs that currently rely on Hölder decompositions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main new piece is the explicit density-distance form of the Carlen-Frank-Lieb stability estimate: it measures L1 distance between the normalized V_-^s density and the one coming from an optimal potential, obtained by replacing the Hölder step with the probabilistic Leng-Lu L1 result. The same move yields a density-stability version of the Lp mixed volume inequality, which under the concentric-ball trapping condition gives an averaged stability bound on how far the support function deviates from evenness.

The paper lays out the decomposition cleanly and shows that the GNS part carries over unchanged, so the reformulation is direct if the interface works. That is useful for anyone who prefers to work with densities rather than potentials.

The soft spot is the load-bearing substitution itself. The stress-test note is right to flag whether the change of measure from the Schrödinger ground state back to the normalized V density produces remainders controlled only by the claimed L1 distance; any leftover factors depending on the potential or higher moments would spoil the clean bound. The note also rests on the Leng-Lu theorem, which is the author's own earlier work, so the circularity risk is real until that result is independently checked or the current argument is written out in full detail.

This is a short, focused note for people already working on stability estimates in inequalities or convex geometry. It deserves a serious referee to verify the substitution step and confirm there are no uncontrolled terms.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a density-distance version of the Carlen--Frank--Lieb stability theorem for Schrödinger operators. By replacing the stability estimate for Hölder's inequality with the optimal L¹-stability theorem of Leng and Lu in probabilistic form, the stability deficit is expressed in terms of the L¹ distance between the normalized density V_-^s / ∫V_-^s and the density induced by an optimal potential, where s = γ + d/2. The paper also applies the same idea to obtain a density-stability version of the L_p mixed volume inequality, with a geometric application to averaged stability estimates for the non-evenness of support functions when one body is centrally symmetric and both are trapped between concentric balls.

Significance. If the key substitution is justified, this note provides an alternative formulation of stability results that measures deviations in density space rather than in potential space, which may facilitate connections to geometric inequalities. The geometric application offers an averaged stability estimate under symmetry assumptions. The approach leverages an existing L¹-stability result to refine the CFL decomposition.

major comments (2)

[Main argument (around the substitution of Leng-Lu theorem)] The central claim relies on substituting the probabilistic L¹-stability of Leng-Lu into the CFL decomposition. It is necessary to verify explicitly that the re-embedding from the probability measure setting back to the deterministic eigenvalue problem controls all remainder terms solely by the claimed L¹ density distance, without introducing factors depending on the potential or higher moments. The manuscript should provide the detailed estimate at this interface to confirm no uncontrolled errors arise.
[Geometric application section] For the L_p mixed volume inequality stability, the derivation of the averaged stability estimate for non-evenness should specify the constants and the precise averaging measure used, to ensure the bound is quantitative and not merely existential.

minor comments (2)

[Abstract] The abstract mentions 'this note we point out', but the full text should clarify if this is a new observation or includes new proofs.
[Notation] Ensure consistent use of s = γ + d/2 throughout, and define all terms like V_- upon first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. The comments highlight two places where additional explicit verification and quantification would strengthen the presentation. We address each point below and will incorporate the requested details in a revised version.

read point-by-point responses

Referee: The central claim relies on substituting the probabilistic L¹-stability of Leng-Lu into the CFL decomposition. It is necessary to verify explicitly that the re-embedding from the probability measure setting back to the deterministic eigenvalue problem controls all remainder terms solely by the claimed L¹ density distance, without introducing factors depending on the potential or higher moments. The manuscript should provide the detailed estimate at this interface to confirm no uncontrolled errors arise.

Authors: We agree that an explicit interface estimate is needed. The Leng–Lu theorem is applied to the normalized probability densities V_-^s / ∫V_-^s and the corresponding optimal density. Because the Schrödinger eigenvalue problem is homogeneous of degree -2s in the potential and the GNS stability term is already controlled in the CFL decomposition, the re-embedding step produces remainder terms that are bounded by a universal multiple of the L¹ distance alone; no additional factors involving ||V||_∞ or higher moments appear after normalization. We will insert a short lemma (new Lemma 2.3) that records this calculation in full detail. revision: yes
Referee: For the L_p mixed volume inequality stability, the derivation of the averaged stability estimate for non-evenness should specify the constants and the precise averaging measure used, to ensure the bound is quantitative and not merely existential.

Authors: We accept the point. The averaging is performed with respect to the normalized Haar measure on SO(d) (equivalently, the uniform probability measure on the sphere after fixing one body), and the constants depend only on the inner and outer radii R and r of the concentric balls that trap both bodies. We will state the precise constants (which are explicit functions of R/r and p) and the measure in the revised geometric section. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in the derivation chain

full rationale

The paper's main contribution is observing that substituting the Leng-Lu L¹-stability theorem into the Carlen-Frank-Lieb decomposition yields a density-distance stability estimate. This substitution is presented as a direct replacement without introducing new fitted parameters or self-referential definitions. The cited Leng-Lu result is treated as an established prior theorem, providing independent support for the Hölder step. No evidence of the new result reducing to its inputs by construction or relying on a self-citation chain that lacks external verification. The derivation remains self-contained as an application of existing results.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper builds on existing theorems; no new free parameters or entities introduced based on the abstract. Standard mathematical axioms from analysis are used.

assumptions (1)

standard math Standard properties of Schrödinger operators and inequalities like Gagliardo-Nirenberg-Sobolev hold.
Invoked in the decomposition mentioned in the abstract.

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Cite this review

Pith. "Pith review of A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem." pith.science (2026). https://pith.science/paper/TZRVQW42

@misc{pith2026260603749,
  author       = {Pith},
  title        = {Pith review of: A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZRVQW42}},
  note         = {Machine review of arXiv:2606.03749}
}

abstract

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schr\"odinger operator by decomposing the problem into a stability estimate for H\"older's inequality and a stability estimate for a Gagliardo--Nirenberg--Sobolev inequality. In this note we point out that, if the H\"older step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen--Frank--Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=\gamma+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.

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Works this paper leans on

5 extracted references · 1 canonical work pages

[1]

J. M. Aldaz, A stability version of Hölder’s inequality,J. Math. Anal. Appl.343 (2008), 842–852

2008
[2]

E. A. Carlen, R. L. Frank and E. H. Lieb, Stability estimates for the lowest eigenvalue of a Schrödinger operator,Geom. Funct. Anal.24 (2014), 63–84

2014
[3]

Lutwak, The Brunn–Minkowski–Firey theory

E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem,J. Differential Geom.38 (1993), 131–150

1993
[4]

Schneider,Convex Bodies: The Brunn–Minkowski Theory, 2nd expanded ed., Cambridge University Press, Cambridge, 2014

R. Schneider,Convex Bodies: The Brunn–Minkowski Theory, 2nd expanded ed., Cambridge University Press, Cambridge, 2014

2014
[5]

An Optimal Stability Theorem for H\"older's Inequality

G. Leng and Y. Lu, An optimal stability theorem for Hölder’s inequality. arXiv:2605.31179. 10

work page Pith review arXiv

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

J. M. Aldaz, A stability version of Hölder’s inequality,J. Math. Anal. Appl.343 (2008), 842–852

2008

[2] [2]

E. A. Carlen, R. L. Frank and E. H. Lieb, Stability estimates for the lowest eigenvalue of a Schrödinger operator,Geom. Funct. Anal.24 (2014), 63–84

2014

[3] [3]

Lutwak, The Brunn–Minkowski–Firey theory

E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem,J. Differential Geom.38 (1993), 131–150

1993

[4] [4]

Schneider,Convex Bodies: The Brunn–Minkowski Theory, 2nd expanded ed., Cambridge University Press, Cambridge, 2014

R. Schneider,Convex Bodies: The Brunn–Minkowski Theory, 2nd expanded ed., Cambridge University Press, Cambridge, 2014

2014

[5] [5]

An Optimal Stability Theorem for H\"older's Inequality

G. Leng and Y. Lu, An optimal stability theorem for Hölder’s inequality. arXiv:2605.31179. 10

work page Pith review arXiv