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REVIEW 3 major objections 4 minor 16 references

On the $L_{\mathrm{YJ}}(\xi, \eta, X)$ constant for the Bana\'s-Fr\k{a}czek space

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Exact LYJ constant of Banas-Fraczek space computed

desk verdict The paper's central formula is false: explicit unit vectors give a larger LYJ quotient than the claimed exact value, and Lemma 2.2 is numerically violated. read the letter →

arxiv 2411.13285 v1 pith:L2FHUK72 submitted 2024-11-20 math.FA

Yuxin Wang , Qi Liu , Linhui Chen , Xiewei Tan , Muhammad Sarfraz This is my paper · ORCID
classification math.FA MSC 46B20
keywords BanachspacesgeometricconstantsBanas-FraczekspaceLYJconstantvonNeumann-Jordansuper-reflexiveweaknormalstructure
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

This paper sets out to compute the exact value of the $L_{\mathrm{YJ}}(\xi,\eta,X)$ geometric constant for the Banas-Fraczek space $R_\lambda^2$, the normed plane with norm $\|(a,b)\|=\max\{\lambda|a|,\sqrt{a^2+b^2}\}$. Its central claim is that for every $\lambda\ge 1$ and positive $\xi,\eta$, the constant equals $1+\frac{2\xi\eta}{\xi^2+\eta^2}\left(1-\frac{1}{\lambda^2}\right)$. If true, the formula gives a direct geometric handle on the space: it interpolates from the Euclidean value $1$ at $\lambda=1$ toward $1+\frac{2\xi\eta}{\xi^2+\eta^2}$ as $\lambda\to\infty$, and it keeps the constant below $2$, which implies super-reflexivity. The paper also reads off the von Neumann-Jordan constant at $\xi=\eta$ and a weak-normal-structure condition.

What carries the argument

The central objects are the Banas-Fraczek norm and the quotient defining the constant: for nonzero $x,y$, the ratio is $\frac{\|\xi x+\eta y\|^2+\|\eta x-\xi y\|^2}{(\xi^2+\eta^2)(\|x\|^2+\|y\|^2)}$, and $L_{\mathrm{YJ}}$ is its supremum. The workhorse is a four-case split depending on whether $\lambda|a|$ or the Euclidean part dominates each vector, which reduces the quotient to expressions in $|x_1y_1|$ and $\sqrt{1-x_1^2}\sqrt{1-y_1^2}$. Lemma 2.1 bounds the cross term for $\lambda\ge\sqrt{2}$, and Lemma 2.2 claims that two auxiliary functions $f$ and $g$ on $[0,1/\lambda]^2$ peak at the corner $(1/\lambda,1/\lambda)$, supplying the upper bound in the range $1\le\lambda<\sqrt2$. The lower-bound direction uses $x=(1/\lambda,\sqrt{1-1/\lambda^2})$, $y=(1/\lambda,-\sqrt{1-1/\lambda^2})$, for which the quotient equals the claimed formula.

What would settle it

Check the Case(2) expression at $\lambda=6/5$, $\xi=1$, $\eta=3$, $t=1/2$, $x_1=5/6$, $y_1=7/10$: the auxiliary function $f$ gives $f(5/6,7/10)\approx 11.1308$, larger than the corner value $f(5/6,5/6)\approx 11.0833$ used in Lemma 2.2. Since that lemma is exactly what produces inequalities (5) and (6), the discrepancy invalidates the proof's $1\le\lambda<\sqrt2$ step; testing the corresponding full ratio against the claimed formula then settles whether Theorem 2.1 itself is correct.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 2.1: for $R_\lambda^2=(\mathbb{R}^2,\|\cdot\|_\lambda)$ with $\|(a,b)\|_\lambda=\max\{\lambda|a|,\sqrt{a^2+b^2}\}$ and $\lambda\ge 1$, the constant $L_{\mathrm{YJ}}(\xi,\eta,R_\lambda^2)$ equals $1+\frac{2\xi\eta}{\xi^2+\eta^2}\left(1-\frac{1}{\lambda^2}\right)$. The proof splits the unit sphere into cases according to which branch of the norm controls each of the two vectors $\xi x+\eta t y$ and $\eta x-\xi t y$. An upper bound is obtained by applying Lemma 2.1 for $\lambda\ge\sqrt{2}$ and Lemma 2.2 for $1\le\lambda<\sqrt{2}$, and the lower bound is attained by the two extreme points $\left(\frac{1}{\lambda},\pm\sqrt{1-\frac{1}{\lambda^2}}\right)$. Corollaries identify the $\xi=\eta$ case with the von Neumann-Jordan constant and conclude super-reflexivity.

Load-bearing premise

The proof for $1\le\lambda<\sqrt2$ rests on Lemma 2.2, which asserts that two auxiliary functions on $[0,1/\lambda]^2$ reach their maximum at the corner $(1/\lambda,1/\lambda)$; if that corner-maximum claim fails, the upper-bound inequalities (5) and (6) do not follow.

Editorial extensions

If this is right

  • For $\xi=\eta$, the theorem gives $L_{\mathrm{YJ}}(1,1,R_\lambda^2)=2-\frac{1}{\lambda^2}$, recovering the known von Neumann-Jordan constant of the Banas-Fraczek space.
  • Because the claimed value is always below $2$ for $\lambda\ge 1$, the space $R_\lambda^2$ is super-reflexive under the criterion supplied by the $L_{\mathrm{YJ}}$ constant.
  • Taking $\lambda=1$ makes the norm Euclidean and the formula collapses to $L_{\mathrm{YJ}}(\xi,\eta,R_1^2)=1$, the minimal possible value of the constant.
  • For $\eta\le\xi<\frac32\eta$ and $1\le\lambda<\sqrt{4\xi\eta/(6\xi\eta-3\eta^2)}$, the claimed value falls below the threshold in Lemma 2.3, so the space has weak normal structure.

Reading between the lines

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

  • A direct numerical stress-test targets the proof's bottleneck: at $\lambda=6/5$, $\xi=1$, $\eta=3$, $t=1/2$, the auxiliary function $f$ in Lemma 2.2 takes value about $11.1308$ at $(x_1,y_1)=(5/6,7/10)$, exceeding the corner value $11.0833$ asserted by the lemma.
  • The same case-splitting strategy could be tried on the generalized spaces $X_{\lambda,p}$ with norm $\max\{\lambda|a|,(|a|^p+|b|^p)^{1/p}\}$, replacing the Euclidean cross-term identities by $\ell^p$ analogues.
  • Because the formula depends on $\xi,\eta$ only through $2\xi\eta/(\xi^2+\eta^2)$, it suggests that for any two-dimensional norm whose unit ball is cut by vertical strips, the $L_{\mathrm{YJ}}$ constant may obey a one-parameter interpolation between the Euclidean value and the anisotropic limit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the LYJ constant L_YJ(ξ,η,X) for the Banas-Fraczek space R²_λ, defined by the norm ‖(a,b)‖ = max{λ|a|, √(a²+b²)}. The main result, Theorem 2.1, asserts that for every λ ≥ 1, L_YJ(ξ,η,R²_λ) equals 1 + (2ξη/(ξ²+η²))(1 − 1/λ²). The proof proceeds by a case analysis and relies on Lemma 2.2, which claims that two auxiliary functions f and g on [0,1/λ]² attain their maxima at the corner (1/λ,1/λ). The paper also derives corollaries on the von Neumann–Jordan constant, super-reflexivity, and weak normal structure.

Significance. An exact formula for a geometric constant of a concrete family of norms would be a useful contribution to the literature on geometric constants in Banach spaces, and the lower-bound construction in equations (8)–(9) is valid and does establish the claimed lower bound. However, the upper-bound proof depends entirely on Lemma 2.2, which is false, and the claimed formula is directly contradicted by an explicit numerical example using the paper's own definitions. Since the central result is incorrect, the paper cannot be accepted in its current form.

major comments (3)
  1. [Section 2, Lemma 2.2] Lemma 2.2 is false. For λ = 6/5, t = 1/2, ξ = 1, η = 3, the function f defined in part (a) satisfies f(5/6, 7/10) ≈ 11.1308 > f(5/6, 5/6) ≈ 11.0833, contradicting the claimed corner maximum. The proof's inference that the absence of an interior critical point forces the maximum to occur at one of the four corners is invalid; the maximum can occur at a non-corner boundary point, as this example shows. Consequently, inequalities (5) and (6) in the proof of Theorem 2.1 are not justified.
  2. [Section 2, proof of Lemma 2.2] The algebraic step after multiplying equations (i) and (ii) is erroneous: from the displayed equality involving tξ²η² and positive terms, the manuscript claims equivalence to 0 = 2tξηλ² + ξ²λ² x/y + t²λ²η² y/x. The right-hand side is a sum of positive terms, so it cannot be zero. The claimed contradiction is therefore not established, and the exclusion of interior maxima is unsupported.
  3. [Section 2, Theorem 2.1] Theorem 2.1 is false. For λ = 6/5, ξ = 1, η = 3, the unit vectors x = (5/6, √11/6) and y = (−4/5, 3/5) in R²_λ satisfy ‖x‖ = ‖y‖ = 1, and the LYJ quotient (‖ξx + ηy‖² + ‖ηx − ξy‖²)/((ξ² + η²)(‖x‖² + ‖y‖²)) equals (13551 + 375√11)/12500 ≈ 1.18358, which strictly exceeds the claimed value 1 + (2ξη/(ξ²+η²))(1 − 1/λ²) = 71/60 ≈ 1.18333. This is a direct contradiction to the main formula using the paper's own definition of the constant.
minor comments (4)
  1. [Lemma 2.2] The displayed value of f(1/λ,1/λ) is inconsistent between the statement and the proof: the statement gives 4tξη·(λ²−1)/λ² + t²ξ² + η², while the proof's comparison uses 4tξη·(λ²−1)/λ + t²ξ² + η².
  2. [Theorem 2.1, Case(3)] The expression for ‖ηx − ξty‖² contains a typesetting error: (ηx₁ − ξty₁)²(ηx₂ − ξty₂)² should be (ηx₁ − ξty₁)² + (ηx₂ − ξty₂)².
  3. [Theorem 2.1, equations (5) and (6)] The term '2tξηsin √(1−x²)√(1−y²)' should be '2tξη√(1−x²)√(1−y²)'.
  4. [Abstract] The abstract contains the typo 'meticilous'; the phrase 'Bana´s-Fr ˛ aczek' is also hyphenated inconsistently throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning: the claimed exact value is derived by direct case analysis from the definition of the LYJ constant, not by assuming the target formula.

full rationale

The paper's main derivation proceeds from the definition of L_YJ(ξ, η, X) as a supremum of a ratio over unit vectors in R_λ^2. The lower bound in Theorem 2.1 is obtained by explicitly choosing the pair x = (1/λ, sqrt(1 - 1/λ^2)) and y = (1/λ, -sqrt(1 - 1/λ^2)) and computing the ratio; the upper bound is attempted by a four-case analysis using the auxiliary inequalities Lemma 2.1 and Lemma 2.2. Neither bound is obtained by substituting the claimed formula into itself, and no parameter is fitted to the target constant. The cited results from prior work, including Lemma 2.1 from [8] and Lemma 2.3 from [13], are auxiliary inequalities and criteria used in corollaries; they are not restatements of the main theorem, and the main exact-value proof does not reduce to them. The reader's reported counterexample, if correct, would show that Lemma 2.2 is false and therefore that Theorem 2.1 is false; but a false auxiliary lemma and a false theorem are matters of mathematical correctness, not circularity. Self-citations appear in the introduction and in the corollaries, but they are not load-bearing for the paper's central claimed derivation. Accordingly, no circular step satisfying the quoted-evidence standard is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central derivation imports an external inequality (Lemma 2.1), makes an unstated extreme-point reduction, and relies on Lemma 2.2. No parameters are fitted and no new entities are introduced. The decisive deficiency is that Lemma 2.2 is false, so the proof has no valid load-bearing support for 1 ≤ λ < √2.

assumptions (3)
  • standard math Extreme-point reduction for convex maxima over B_X × B_X
    The proof of inequality (1) considers only x, y in ext(B_X) and t in [0, 1]. The reduction from all unit vectors to extreme points and from general t to t ≤ 1 is not stated or justified.
  • domain assumption Lemma 2.1 imported from [8]
    Used for λ ≥ √2 in Case(2) and Case(3); accepted without proof from Yang and Xu. The lemma may be correct, but the paper does not verify it.
  • ad hoc to paper Lemma 2.2 maximization at the corner
    The proof of Theorem 2.1 for λ < √2 depends on this lemma, but the lemma is false; see the explicit counterexample in the red flags.

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Pith. "Pith review of On the $L_{\mathrm{YJ}}(\xi, \eta, X)$ constant for the Bana\'s-Fr\k{a}czek space." pith.science (2026). https://pith.science/paper/L2FHUK72

@misc{pith2026241113285,
  author       = {Pith},
  title        = {Pith review of: On the $L_\mathrmYJ(\xi, \eta, X)$ constant for the Bana\'s-Fr\kaczek space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L2FHUK72}},
  note         = {Machine review of arXiv:2411.13285}
}
abstract

In this paper, for any $\lambda \geq 1, R_\lambda^2$ is the Bana\'s-Fr\k{a}czek space. The exact value of $L_{\mathrm{YJ}}(\xi, \eta, X)$ for this space will be calculated. Specifically, $L_{\mathrm{YJ}}\left(\xi, \eta, R_\lambda^2\right)=1+\frac{2 \xi \eta}{\xi^2+\eta^2}\left(1-\frac{1}{\lambda^2}\right)$ is the result thereafter through meticilous computation.

Discussion (0). Continue with ORCID to comment.

Reference graph

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