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arxiv: 2606.01068 · v1 · pith:YO532PZ5new · submitted 2026-05-31 · 🧮 math.FA

A weighted Birkhoff orthogonal James-type constant

Junxiang Qi , Zhouping Yin , Qi Liu , Yongjin Li This is my paper

Pith reviewed 2026-06-28 16:32 UTC · model grok-4.3

classification 🧮 math.FA
keywords Birkhoff orthogonalityJames constantuniformly nonsquare spacesBanach space geometryweighted constantstrict convexity
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The pith

A weighted Birkhoff orthogonal James constant equals 1 for some λ in (0,1) if and only if the Banach space is not uniformly nonsquare.

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

The paper defines a family of constants J_λ^⊥(X) that take the supremum of the smaller norm after weighting two Birkhoff-orthogonal unit vectors by λ and 1-λ. It establishes that these constants are 2-Lipschitz continuous in λ, satisfy basic bounds, and reduce to the ordinary orthogonal James constant in limiting cases. The central result is a characterization of uniform nonsquareness: J_λ^⊥(X) reaches the value 1 for at least one weight λ between 0 and 1 exactly when the space is not uniformly nonsquare. The work also links the new constants to strict convexity, uniform convexity, the modulus of smoothness, and the von Neumann-Jordan constant.

Core claim

The weighted Birkhoff orthogonal James-type constant J_λ^⊥(X) satisfies J_λ^⊥(X)=1 for some λ ∈ (0,1) if and only if X is not uniformly nonsquare. The authors prove its 2-Lipschitz continuity with respect to λ and its relations to the orthogonal James constant J_⊥(X), along with connections to strict convexity, uniform convexity, the modulus of smoothness, and the von Neumann-Jordan constant.

What carries the argument

The weighted Birkhoff orthogonal James-type constant J_λ^⊥(X), defined as the supremum over Birkhoff-orthogonal unit vectors x, y of the minimum of ||λx + (1-λ)y|| and ||λx - (1-λ)y||.

If this is right

  • J_λ^⊥(X) supplies new characterizations of uniformly nonsquare spaces.
  • The constant is 2-Lipschitz continuous with respect to the weight λ.
  • Relations hold between J_λ^⊥(X) and the orthogonal James constant J_⊥(X).
  • Links exist to strict convexity, uniform convexity, the modulus of smoothness, and the von Neumann-Jordan constant.

Where Pith is reading between the lines

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

  • The family of constants could be used to quantify how strongly a space deviates from uniform nonsquareness by tracking the interval of λ where the value reaches 1.
  • Analogous weighted constructions might be applied to other orthogonality-based constants to obtain finer geometric distinctions.

Load-bearing premise

Birkhoff orthogonality x ⊥_B y and the unit sphere behave as standard in any real Banach space.

What would settle it

A real Banach space that is uniformly nonsquare yet satisfies J_λ^⊥(X)=1 for some λ in (0,1), or a space that is not uniformly nonsquare yet has J_λ^⊥(X) less than 1 for every λ in (0,1).

Figures

Figures reproduced from arXiv: 2606.01068 by Junxiang Qi, Qi Liu, Yongjin Li, Zhouping Yin.

Figure 1
Figure 1. Figure 1: Geometric construction of weighted orthogonal geometric constant [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Let $X$ be a real Banach space and $\lambda \in[0,1]$. Motivated by orthogonal versions of the James constant, we introduce the weighted Birkhoff orthogonal James-type constant $$J_\lambda^{\perp}(X)=\sup \left\{\min \{\|\lambda x+(1-\lambda) y\|,\|\lambda x-(1-\lambda) y\|\}: x, y \in S_X, x \perp_B y\right\},$$ where \(\lambda\in[0,1]\) and $x \perp_B y$ stands for Birkhoff orthogonality. We establish its basic bounds, stability properties, and reduction principles, and clarify its relations with the orthogonal James constant $J_{\perp}(X)$. The 2 -Lipschitz continuity of $J_\lambda^{\perp}(X)$ with respect to $\lambda$ is proved. New characterizations of uniformly nonsquare spaces are obtained; in particular, $J_\lambda^{\perp}(X)=1$ for some $\lambda \in(0,1)$ if and only if $X$ is not uniformly nonsquare. We also discuss connections with strict convexity, uniform convexity, modulus of smoothness, and the von Neumann-Jordan constant.

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

0 major / 2 minor

Summary. The manuscript defines the weighted Birkhoff orthogonal James-type constant J_λ^⊥(X) for a real Banach space X and λ ∈ [0,1] as the supremum of min{‖λx + (1-λ)y‖, ‖λx − (1-λ)y‖} taken over all x, y on the unit sphere S_X satisfying Birkhoff orthogonality x ⊥_B y. It establishes the trivial upper bound J_λ^⊥(X) ≤ 1, proves 2-Lipschitz continuity of the map λ ↦ J_λ^⊥(X), derives stability and reduction principles relating the weighted constant to the unweighted orthogonal James constant J_⊥(X), and obtains new characterizations of uniformly nonsquare spaces, the principal one being that J_λ^⊥(X) = 1 for some λ ∈ (0,1) if and only if X is not uniformly nonsquare. Additional relations to strict convexity, uniform convexity, the modulus of smoothness, and the von Neumann-Jordan constant are discussed.

Significance. If the stated proofs hold, the work supplies a parameterized family of constants that refines the detection of uniform nonsquareness and furnishes a concrete 2-Lipschitz modulus of continuity in the weight parameter; these technical features are useful for stability arguments in Banach-space geometry and for relating the new constant to classical moduli.

minor comments (2)
  1. [Abstract] Abstract and §1: the range statement λ ∈ [0,1] for the definition is correct, but the characterization theorem is stated only for λ ∈ (0,1); a single sentence reconciling the two ranges would prevent minor reader confusion.
  2. [§3] The proof of 2-Lipschitz continuity (presumably in §3) is credited in the abstract; adding an explicit remark on whether the constant 2 is sharp (e.g., via a concrete pair of spaces) would strengthen the presentation without altering the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. No major comments were provided in the report, so there are no specific points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the new constant J_λ^⊥(X) directly from the standard Birkhoff orthogonality relation on the unit sphere and derives its properties (universal upper bound ≤1 via triangle inequality, 2-Lipschitz continuity in λ, relations to the unweighted J_⊥(X)) using only the axioms of real Banach spaces. The central iff characterization (J_λ^⊥(X)=1 for some λ∈(0,1) ⇔ X not uniformly nonsquare) is obtained by showing that equality to the universal bound detects failure of uniform nonsquareness; this is a non-trivial equivalence between the new quantity and a pre-existing geometric property, with no self-definitional reductions, fitted-input predictions, or load-bearing self-citations visible in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger entries are the minimal domain assumptions stated or implied.

axioms (1)
  • domain assumption X is a real Banach space with the standard norm and Birkhoff orthogonality relation
    Explicitly stated in the opening sentence of the abstract.

pith-pipeline@v0.9.1-grok · 5746 in / 1125 out tokens · 24181 ms · 2026-06-28T16:32:49.110081+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references

  1. [1]

    On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces[J]

    KATO M, MALIGRANDA L, TAKAHASHI Y. On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces[J]. Studia Mathematica, 2001, 144(3): 275-295

    2001

  2. [2]

    An Amir - Cambern theorem for quasi-isometries of spaces[J]

    GALEGO E, SILVA A L D. An Amir - Cambern theorem for quasi-isometries of spaces[J]. Pacific Journal of Mathematics, 2018, 297(1): 87-100

    2018

  3. [3]

    Norm inequalities inand a geometric constant[J]

    BHUNIA P, MAL A. Norm inequalities inand a geometric constant[J]. Banach Journal of Mathematical Analysis, 2024: 31

    2024

  4. [4]

    Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings[J]

    GARC´IA-FALSET J, LLORENS-FUSTER E, MAZCU ˜NAN-NAVARRO E M. Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings[J]. Journal of Functional Analysis, 2006, 233(2): 494-514

    2006

  5. [5]

    Two Geometric Constants Measuring Differences Between Orthogonality Types in Banach Spaces[J]

    BI J, LIU Q, et al. Two Geometric Constants Measuring Differences Between Orthogonality Types in Banach Spaces[J]. Mediterranean Journal of Mathematics, 2026

    2026

  6. [6]

    Symmetric form geometric constant related to isosceles orthogonality in Banach spaces[J]

    NI Q C, LIU Q, WANG Y X, XIA J Y, WANG R R. Symmetric form geometric constant related to isosceles orthogonality in Banach spaces[J]. Filomat, 2025

    2025

  7. [7]

    Generalized rectangular modulus related to isosceles orthogonality in Banach spaces[J]

    XIE H Y, LIU Q, LI Y J. Generalized rectangular modulus related to isosceles orthogonality in Banach spaces[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2025

    2025

  8. [8]

    Birkhoff

    G. Birkhoff. Orthogonality in linear metric spaces. (1935), 169-172

    1935

  9. [10]

    ”Parameters in Banach spaces and orthogonality.” Constructive Mathematical Analysis 5.1 (2022): 37-45

    Papini, Pier Luigi, and Marco Barontı. ”Parameters in Banach spaces and orthogonality.” Constructive Mathematical Analysis 5.1 (2022): 37-45

    2022

  10. [11]

    R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), no. 3, 542–550

    1964

  11. [12]

    Gao and K.S

    J. Gao and K.S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc. Ser. A 48 (1), 101-112, 1990

    1990

  12. [13]

    He and Y

    C. He and Y. Cui, Some properties concerning Milman’s moduli, J. Math. Anal. Appl. 329 (2007), 1260–1272

    2007

  13. [14]

    Demonstr

    Liu, Qi., Sarfraz, M., Li, Y.: Some aspects of generalized Zb˘ aganu and James constant in Banach spaces. Demonstr. Math. 54, 299–310 (2021)

    2021

  14. [15]

    Garc´ ıa-Falset, J., Llorens-Fuster, E., Mazcu˜ nan-Navarro, E.M.: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233, 494-514 (2006) 16

    2006

  15. [16]

    Day, M.M.: Some characterizations of inner-product spaces. Trans. Am. Math. Soc. 62, 320–337 (1947)

    1947

  16. [17]

    Lindenstrauss

    J. Lindenstrauss. On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J. 10(1963), 241-252

    1963

  17. [18]

    Casini: About some parameters of normed linear spaces, Atti Accad

    E. Casini: About some parameters of normed linear spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 80 (1986), 11–15

    1986

  18. [19]

    ”Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality.” Journal of mathematical analysis and applications 323.1 (2006): 1-7

    Ji, Donghai, and Senlin Wu. ”Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality.” Journal of mathematical analysis and applications 323.1 (2006): 1-7

    2006

  19. [20]

    Measurements of differences between orthogonality types.Journal of Mathematical Analysis and Applications 397.1 (2013): 285-291

    Papini, Pier Luigi, and Senlin Wu. Measurements of differences between orthogonality types.Journal of Mathematical Analysis and Applications 397.1 (2013): 285-291

    2013

  20. [21]

    Some moduli of convexity and smoothness related to Birkhoff orthogonality in Banach spaces: D

    Du, Dandan, Ruihui Liang, and Yongjin Li. Some moduli of convexity and smoothness related to Birkhoff orthogonality in Banach spaces: D. Du et al. Revista de la Real Academia de Ciencias Exactas, F´ ısicas y Naturales. Serie A. Matem´ aticas 119.4 (2025): 110

    2025

  21. [22]

    Some Geometric Constants Related to the Sine Function and Cosine Function in Banach Spaces: D

    Du, Dandan, and Yongjin Li. Some Geometric Constants Related to the Sine Function and Cosine Function in Banach Spaces: D. Du and Y. Li. Vietnam Journal of Mathematics 54.1 (2026): 1-17

    2026

  22. [23]

    Banach space theory: The basis for linear and nonlinear analysis

    Fabian, Mari´ an, et al. Banach space theory: The basis for linear and nonlinear analysis. Vol. 1. New York: Springer, 2011

    2011

  23. [24]

    Moduli in normed linear spaces and characterization of inner product spaces

    Alonso, Javier, and Antonio Ull´ an. Moduli in normed linear spaces and characterization of inner product spaces. Archiv der Mathematik 59.5 (1992): 487-495. 17

    1992