Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains

Andrea Gentile; Antoine Henrot; Dorin Bucur

arxiv: 2604.13246 · v1 · submitted 2026-04-14 · 🧮 math.AP · math.SP

Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains

Dorin Bucur , Andrea Gentile , Antoine Henrot This is my paper

Pith reviewed 2026-05-10 14:22 UTC · model grok-4.3

classification 🧮 math.AP math.SP

keywords Neumann eigenvaluesconvex domainsKröger inequalitiesJohn ellipsoidquantitative estimatesspectral geometryupper bounds

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The pith

Convex domains satisfy a quantitative version of Kröger's upper bound on the k-th Neumann eigenvalue, with the deficit controlled by the John ellipsoid's second semiaxis.

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

The paper refines Kröger's sharp upper bounds μ_{k,d}^* for the k-th Neumann eigenvalue of convex domains Ω in R^d by proving a quantitative inequality that measures the shortfall. It shows D_Ω² μ_k(Ω) is at most μ_{k,d}^* minus a positive term C(k,d) a_2(Ω)² / D_Ω², where a_2(Ω) is the second largest semiaxis of the John ellipsoid of Ω. This turns the known bound into a stability statement that depends on how elongated the domain is. A sympathetic reader would care because the result supplies an explicit rate at which the eigenvalue drops away from the optimum when the domain deviates from roundness.

Core claim

The authors prove that for any k in the natural numbers and dimension d there exists C(k,d) > 0 such that D_Ω² μ_k(Ω) ≤ μ_{k,d}^* - C(k,d) a_2(Ω)² / D_Ω² holds for every convex domain Ω, where μ_{k,d}^* is the Kröger constant and a_2(Ω) is the second-largest semiaxis of the John ellipsoid. In the planar case they give an explicit value of C(1,2) for the first eigenvalue.

What carries the argument

The second-largest semiaxis a_2(Ω) of the John ellipsoid of the convex set Ω, which quantifies deviation from sphericity and produces a positive lower bound on the eigenvalue deficit through variational characterization of the Neumann eigenvalues.

If this is right

The inequality holds for every k and every dimension, giving a uniform quantitative refinement of all Kröger bounds.
An explicit positive constant is available for the first Neumann eigenvalue on planar convex domains.
The eigenvalue is forced strictly below the Kröger value whenever the John ellipsoid has a positive second semiaxis.
The result applies only to convex domains because the John ellipsoid construction requires convexity.

Where Pith is reading between the lines

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

The quantitative control suggests that the ball is a stable maximizer for Neumann eigenvalues among convex sets, with an explicit stability rate.
Similar deficit estimates might be obtainable for other boundary conditions or for domains that are only mildly non-convex by replacing the John ellipsoid with a suitable substitute.
Numerical computation of eigenvalues on a family of ellipsoids with varying eccentricity could test whether the constant C is sharp.

Load-bearing premise

The John ellipsoid of a convex set must control the geometry tightly enough that its second semiaxis produces a definite positive lower bound on the gap to Kröger's constant via the Rayleigh quotient for Neumann eigenvalues.

What would settle it

A sequence of convex domains in which a_2 stays bounded below by a fixed positive number while the difference between μ_{k,d}^* and D² μ_k tends to zero would show that no such positive constant C(k,d) exists.

Figures

Figures reproduced from arXiv: 2604.13246 by Andrea Gentile, Antoine Henrot, Dorin Bucur.

**Figure 4.** Figure 4: The triangle Tα We recall the following property of µ1(Tα) proved in [21, Proposition 6.42]: Proposition 3.1. Let Tα be a superequilateral triangle, then it holds sin2 α 2 ≤ µ1(Tα)D2 Tα 4j 2 0,1 < 1. (3.23) We are in position to prove optimality of exponent 2 in the planar case. Proof of Theorem 1.2. Let us consider a superequilateral triangle Tα having the segment [− 1 2 , 1 2 ] × {0} as basis. Its di… view at source ↗

**Figure 5.** Figure 5: The graph of x 7→ J0(2j0,1x) 2 − J1(2j0,1x) 2 Therefore, there exists a Lagrange multiplier ξ ∈ H1 (I) (corresponding to the concavity constraint) with ξ ≥ 0 and ξ = 0 on the support S of h ′′, and there exists two numbers α0 and α1 such that ξ ′′ = (1 + τh∗ (x) 2 w 2 ) 2 g(x) + α0δ0 + α1δ1/2. (4.7) We denote by f(x) = (1 +τh∗ (x) 2 w 2 ) 2 g(x) and therefore, we have ξ ′′ = f(x) on I. The previous relatio… view at source ↗

read the original abstract

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kr\"oger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_{\Omega}^2 \mu_k(\Omega) \leq \mu_{k,d}^* - C(k,d) a_2(\Omega)^2/D_{\Omega}^2$$ where $D_{\Omega}$ is the diameter of $\Omega$ and $a_2(\Omega)$ is the second largest semiaxis of the John ellipsoid of $\Omega$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.

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

This paper adds a quadratic correction to Kröger's sharp Neumann eigenvalue bound using the second semiaxis of the John ellipsoid, with an explicit constant for k=1 in 2D.

read the letter

This paper refines Kröger's 1999 upper bounds on the k-th Neumann eigenvalue for convex domains by adding a quantitative correction term. The inequality subtracts a positive multiple of a2 squared over D squared from the sharp constant, where a2 comes from the John ellipsoid. What is new is that correction and the explicit constant they compute for the first eigenvalue in two dimensions. The rest builds on known variational methods and properties of the John ellipsoid for convex bodies. The paper does well in choosing the right geometric quantity to measure non-roundness. The John ellipsoid is the natural tool here, and tying the deficit to its second axis makes sense for getting a uniform constant across dimensions. A soft spot is the quadratic rate itself. Compactness of the class of convex sets with fixed diameter and a2 at least some positive number does imply that the eigenvalue cannot reach the sharp bound, so a gap exists. But the paper needs to show that the gap is at least linear in a2 squared. If the estimates from the test functions or the perturbation only produce a slower decay, like order a2 or a2 log, then no such C works for all small a2. The abstract states the result, so the proof presumably handles the rate, but that is the part that requires the most care. This work is for researchers in spectral geometry who care about stability versions of classical inequalities. A reader who knows the Kröger bound and wants quantitative improvements will find it relevant. It is worth a serious referee because the claim is specific and the approach is grounded in standard techniques. I recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript refines Kröger's sharp upper bounds μ_{k,d}^* for the k-th Neumann eigenvalue of convex domains Ω ⊂ R^d by establishing a quantitative stability inequality: for any k there exists C(k,d)>0 such that D_Ω² μ_k(Ω) ≤ μ_{k,d}^* - C(k,d) a_2(Ω)² / D_Ω², where D_Ω is the diameter and a_2(Ω) is the second-largest semiaxis of the John ellipsoid of Ω. An explicit constant is provided for the planar case k=1.

Significance. If the quadratic rate holds, the result supplies a precise modulus of continuity for the eigenvalue deficit in terms of the John ellipsoid eccentricity, strengthening stability analysis in convex spectral geometry beyond the mere existence of a gap from compactness. The explicit planar constant for k=1 adds concrete value for low-dimensional applications.

major comments (2)

[Theorem 1.1 and the proof in §3] The compactness argument via the Blaschke selection theorem (applied to rescaled convex sets with D_Ω=1 and a_2 ≥ δ) yields some positive gap g(δ)>0 for fixed δ, but the manuscript's central claim requires the stronger quantitative lower bound g(δ) ≥ C δ². The proof must therefore verify that the variational test-function estimates or the stability properties of the John ellipsoid produce at least quadratic control on the eigenvalue deficit rather than a weaker modulus such as δ or δ|log δ|; this is load-bearing for the stated form of the inequality.
[Section 4 (planar case)] In the planar case with explicit C(1,2), the derivation of the constant must be checked against the remainder terms arising from the John ellipsoid approximation and the Neumann variational characterization; any post-hoc fitting or hidden dependence on the domain would undermine the parameter-free nature asserted for C(1,2).

minor comments (2)

[Introduction] Notation for μ_{k,d}^* should be recalled explicitly in the statement of the main theorem for reader convenience.
[Abstract and §1] The abstract and introduction would benefit from a brief sentence clarifying that the John ellipsoid is used only to quantify eccentricity and not to construct test functions directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and the encouraging remarks on the significance of our quantitative stability result. We respond to each major comment below.

read point-by-point responses

Referee: [Theorem 1.1 and the proof in §3] The compactness argument via the Blaschke selection theorem (applied to rescaled convex sets with D_Ω=1 and a_2 ≥ δ) yields some positive gap g(δ)>0 for fixed δ, but the manuscript's central claim requires the stronger quantitative lower bound g(δ) ≥ C δ². The proof must therefore verify that the variational test-function estimates or the stability properties of the John ellipsoid produce at least quadratic control on the eigenvalue deficit rather than a weaker modulus such as δ or δ|log δ|; this is load-bearing for the stated form of the inequality.

Authors: We acknowledge that a pure compactness argument would only yield a qualitative gap. In our proof of Theorem 1.1 in Section 3, we do not rely solely on compactness; instead, we employ explicit variational test functions constructed from the John ellipsoid and quantitative estimates on the deviation from the ball in terms of a_2(Ω). These estimates are quadratic by construction, drawing from the properties of the John ellipsoid for convex bodies. To make this explicit and address the referee's concern, we will revise Section 3 to include a detailed tracking of the quadratic terms in the eigenvalue deficit, showing how the lower bound C δ² arises directly from the test-function calculations and the ellipsoid approximation. This revision will clarify the quantitative nature without altering the result. revision: yes
Referee: [Section 4 (planar case)] In the planar case with explicit C(1,2), the derivation of the constant must be checked against the remainder terms arising from the John ellipsoid approximation and the Neumann variational characterization; any post-hoc fitting or hidden dependence on the domain would undermine the parameter-free nature asserted for C(1,2).

Authors: We have derived C(1,2) explicitly in Section 4 by carefully bounding all remainder terms in the John ellipsoid approximation and the Rayleigh quotient for the Neumann eigenfunctions. The constant is obtained directly from these estimates and is independent of the specific domain. However, to ensure transparency and verify against any potential overlooked remainders, we will re-examine the calculations in Section 4 and, if necessary, provide additional intermediate steps or adjust the value of C(1,2) accordingly while maintaining its explicit and parameter-free character. revision: partial

Circularity Check

0 steps flagged

No circularity: quantitative rate obtained from independent stability estimates on John ellipsoid deviation

full rationale

The derivation begins from the known Kröger upper bound μ_{k,d}^* and the geometric control afforded by the John ellipsoid (whose existence and properties are external to the eigenvalue problem). Variational characterizations are then used to produce test functions or deficit estimates whose quadratic dependence on a_2(Ω)/D_Ω is derived directly from the geometry of convex sets, without any parameter fitting, renaming of known results, or load-bearing self-citation. Compactness of the rescaled class supplies a positive gap for fixed δ>0; the paper's explicit estimates (including the planar C(1,2)) upgrade this to the claimed quadratic modulus, which is a genuine analytic step rather than a tautology. No step reduces by construction to the target inequality itself.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts about convex geometry (John ellipsoid existence and properties) and spectral theory (variational characterization of Neumann eigenvalues). No new entities are introduced and the constant C(k,d) is proved to exist rather than fitted to data.

free parameters (1)

C(k,d)
Existence of a positive constant is asserted for each k and d; an explicit value is given only for k=1, d=2. It is not obtained by fitting to numerical data.

axioms (2)

domain assumption Existence and basic properties of the John ellipsoid for convex domains in R^d
Invoked to define a_2(Ω) and to quantify deviation from the ball.
standard math Variational characterization and monotonicity properties of Neumann eigenvalues
Used to relate the eigenvalue to the geometry of Ω.

pith-pipeline@v0.9.0 · 5450 in / 1491 out tokens · 33939 ms · 2026-05-10T14:22:01.726982+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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work page arXiv 2024
[2]

Amato, D

V. Amato, D. Bucur, and I. Fragalà. A sharp quantitative nonlinear Poincaré inequality on convex domains. Preprint, arXiv:2407.20373 [math.AP] (2024), 2024

work page arXiv 2024
[3]

Amato, A

V. Amato, A. L. Masiello, G. Paoli, and R. Sannipoli. Sharp and quantitative estimates for the p-torsion of convex sets. NoDEA Nonlinear Differential Equations Appl., 30(1):Paper No. 12, 22, 2023

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Bogosel, A

B. Bogosel, A. Henrot, and M. Michetti. Optimization of Neumann eigenvalues under convexity and geometric constraints.SIAM J. Math. Anal., 56(6):7327–7349, 2024

work page 2024
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Brasco, G

L. Brasco, G. De Philippis, and B. Velichkov. Faber-Krahn inequalities in sharp quantitative form. Duke Math. J., 164(9):1777–1831, 2015

work page 2015
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Brasco, C

L. Brasco, C. Nitsch, and C. Trombetti. An inequality à la Szegö-Weinberger for thep-Laplacian on convex sets.Commun. Contemp. Math., 18(6):1550086, 23, 2016

work page 2016
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Bucur and I

D. Bucur and I. Fragalà. On poincaré constants related to isoperimetric problems in convex bodies. Preprint, arXiv:2504.06900, 2026

work page arXiv 2026
[9]

Bucur and A

D. Bucur and A. Henrot. Maximization of the second non-trivial Neumann eigenvalue.Acta Math., 222(2):337–361, 2019

work page 2019
[10]

Bucur, E

D. Bucur, E. Martinet, and E. Oudet. Maximization of Neumann eigenvalues.Arch. Ration. Mech. Anal., 247(2):Paper No. 19, 36, 2023

work page 2023
[11]

S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications.Math.Z., 143:289–297, 1975

work page 1975
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de Guzmán.Differentiationof integrals in Rn, volume Vol

M. de Guzmán.Differentiationof integrals in Rn, volume Vol. 481 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Feffer- man, and two by Roberto Moriyón

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Fusco, F

N. Fusco, F. Maggi, and A. Pratelli. The sharp quantitative isoperimetric inequality.Ann. of Math. (2), 168(3):941–980, 2008

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R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.), 39(3):355–405, 2002

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Girouard, N

A. Girouard, N. Nadirashvili, and I. Polterovich. Maximization of the second positive Neumann eigenvalue for planar domains.J. Differential Geom., 83(3):637–661, 2009

work page 2009
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Henrot, editor.Shape optimization and spectral theory

A. Henrot, editor.Shape optimization and spectral theory. De Gruyter Open, Warsaw, 2017

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Henrot and M

A. Henrot and M. Michetti. Optimal bounds for Neumann eigenvalues in terms of the diameter. Annales mathématiques du Québec, 2023

work page 2023
[18]

Henrot and M

A. Henrot and M. Pierre. Shape variation and optimization, volume 28 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2018. A geometrical analysis, English version of the French publication [ MR2512810] with additions and updates. REFERENCES 29

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P. Kröger. On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space. Proc. Am. Math. Soc., 127(6):1665–1669, 1999

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Lamboley and A

J. Lamboley and A. Novruzi. Polygons as optimal shapes with convexity constraint.SIAM J. Control Optim., 48(5):3003–3025, 2010

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R. S. Laugesen and B. A. Siudeja. Triangles and other special domains. InShape optimization and spectral theory, pages 149–200. De Gruyter Open, Warsaw, 2017

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Schneider

R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014

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[23]

G. Szegö. Inequalities for certain eigenvalues of a membrane of given area.J. Rational Mech. Anal., 3:343–356, 1954

work page 1954
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H. F. Weinberger. An isoperimetric inequality for theN-dimensional free membrane problem. J. Rational Mech. Anal., 5:633–636, 1956

work page 1956

[1] [1]

Amato, D

V. Amato, D. Bucur, and I. Fragalà. The geometric size of the fundamental gap. Preprint, arXiv:2407.01341 [math.SP] (2024), 2024

work page arXiv 2024

[2] [2]

Amato, D

V. Amato, D. Bucur, and I. Fragalà. A sharp quantitative nonlinear Poincaré inequality on convex domains. Preprint, arXiv:2407.20373 [math.AP] (2024), 2024

work page arXiv 2024

[3] [3]

Amato, A

V. Amato, A. L. Masiello, G. Paoli, and R. Sannipoli. Sharp and quantitative estimates for the p-torsion of convex sets. NoDEA Nonlinear Differential Equations Appl., 30(1):Paper No. 12, 22, 2023

work page 2023

[4] [4]

hot spots

R. Bañuelos and K. Burdzy. On the “hot spots” conjecture of J. Rauch.J. Funct.Anal., 164(1):1–33, 1999

work page 1999

[5] [5]

Bogosel, A

B. Bogosel, A. Henrot, and M. Michetti. Optimization of Neumann eigenvalues under convexity and geometric constraints.SIAM J. Math. Anal., 56(6):7327–7349, 2024

work page 2024

[6] [6]

Brasco, G

L. Brasco, G. De Philippis, and B. Velichkov. Faber-Krahn inequalities in sharp quantitative form. Duke Math. J., 164(9):1777–1831, 2015

work page 2015

[7] [7]

Brasco, C

L. Brasco, C. Nitsch, and C. Trombetti. An inequality à la Szegö-Weinberger for thep-Laplacian on convex sets.Commun. Contemp. Math., 18(6):1550086, 23, 2016

work page 2016

[8] [8]

Bucur and I

D. Bucur and I. Fragalà. On poincaré constants related to isoperimetric problems in convex bodies. Preprint, arXiv:2504.06900, 2026

work page arXiv 2026

[9] [9]

Bucur and A

D. Bucur and A. Henrot. Maximization of the second non-trivial Neumann eigenvalue.Acta Math., 222(2):337–361, 2019

work page 2019

[10] [10]

Bucur, E

D. Bucur, E. Martinet, and E. Oudet. Maximization of Neumann eigenvalues.Arch. Ration. Mech. Anal., 247(2):Paper No. 19, 36, 2023

work page 2023

[11] [11]

S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications.Math.Z., 143:289–297, 1975

work page 1975

[12] [12]

de Guzmán.Differentiationof integrals in Rn, volume Vol

M. de Guzmán.Differentiationof integrals in Rn, volume Vol. 481 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Feffer- man, and two by Roberto Moriyón

work page 1975

[13] [13]

Fusco, F

N. Fusco, F. Maggi, and A. Pratelli. The sharp quantitative isoperimetric inequality.Ann. of Math. (2), 168(3):941–980, 2008

work page 2008

[14] [14]

R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.), 39(3):355–405, 2002

work page 2002

[15] [15]

Girouard, N

A. Girouard, N. Nadirashvili, and I. Polterovich. Maximization of the second positive Neumann eigenvalue for planar domains.J. Differential Geom., 83(3):637–661, 2009

work page 2009

[16] [16]

Henrot, editor.Shape optimization and spectral theory

A. Henrot, editor.Shape optimization and spectral theory. De Gruyter Open, Warsaw, 2017

work page 2017

[17] [17]

Henrot and M

A. Henrot and M. Michetti. Optimal bounds for Neumann eigenvalues in terms of the diameter. Annales mathématiques du Québec, 2023

work page 2023

[18] [18]

Henrot and M

A. Henrot and M. Pierre. Shape variation and optimization, volume 28 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2018. A geometrical analysis, English version of the French publication [ MR2512810] with additions and updates. REFERENCES 29

work page 2018

[19] [19]

P. Kröger. On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space. Proc. Am. Math. Soc., 127(6):1665–1669, 1999

work page 1999

[20] [20]

Lamboley and A

J. Lamboley and A. Novruzi. Polygons as optimal shapes with convexity constraint.SIAM J. Control Optim., 48(5):3003–3025, 2010

work page 2010

[21] [21]

R. S. Laugesen and B. A. Siudeja. Triangles and other special domains. InShape optimization and spectral theory, pages 149–200. De Gruyter Open, Warsaw, 2017

work page 2017

[22] [22]

Schneider

R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014

work page 2014

[23] [23]

G. Szegö. Inequalities for certain eigenvalues of a membrane of given area.J. Rational Mech. Anal., 3:343–356, 1954

work page 1954

[24] [24]

H. F. Weinberger. An isoperimetric inequality for theN-dimensional free membrane problem. J. Rational Mech. Anal., 5:633–636, 1956

work page 1956