Skew-Parameterized Geometric Constants in Banach Spaces

Qi Liu; Qing Du; Wenwen Zhang; Zhiyao Fang

REVIEW 4 major objections 3 minor 27 references

Skew-Parameterized Geometric Constants in Banach Spaces

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read One skew constant aims to identify Hilbert spaces

desk verdict The skew constant family is a plausible small extension, but the paper's claimed Hilbert values rest on a misoriented balance condition and are contradicted by admissible choices. read the letter →

arxiv 2608.00933 v1 pith:XBX2RSX5 submitted 2026-08-02 math.FA

Qing Du , Wenwen Zhang , Zhiyao Fang , Qi Liu This is my paper

classification math.FA MSC 46B2046C15

keywords geometricconstantsskewparameterizationHilbertspacecharacterizationuniformnon-squarenessnormalstructureinnerproductspacesBanachJamesconstant

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 introduces a skew-parameterized family of geometric constants for real normed spaces, formed by replacing the classical symmetric pair $ru+sv, ru-sv$ with the rotated pair $ru+sv, su-rv$. The authors claim that when the weight function $\$\lambda$$ satisfies a differential Hilbert-balance condition, the constant $C_{\$\lambda$,p}(H)$ on any real Hilbert space equals $$2^{{1/2-1/p}}$$ for $2\le p\le \infty$, and that a real normed space of dimension at least three attaining this value must itself be an inner product space. They also establish reduction formulas, sharp bounds, parameter-stability estimates, and sufficient conditions for uniform non-squareness and normal structure. If correct, the constant would unify classic invariants like the von Neumann–Jordan, James, and geometric mean constants under one numerical detector of Euclidean geometry.

What carries the argument

The mechanism is the skew coefficient matrix $M(r,s)=\begin{pmatrix} r & s \\ s & -r \end{pmatrix}$, which maps the unit-vector pair $(u,v)$ to $(ru+sv,\,su-rv)$. It satisfies $M(r,s)^T M(r,s)=($r^{2}$+$s^{2}$)I_2$, so on an inner product space the transformation is an orthogonal coordinate rotation combined with uniform scaling. The paper's technical engine is the differential Hilbert-balance condition (1) together with the reduction Lemma 1, which splits the two-parameter supremum into two one-parameter branches $A_{\$\lambda$,p}$ and $B_{\$\lambda$,p}$; the two-branch structure is the distinctive feature introduced by the skew parameterization.

What would settle it

In a real Hilbert space $H$ with $\dim H\ge 2$, take $\$\lambda$(a,b)=b$, $p=2$, $u=-v$ with $\|u\|=\|v\|=1$, and $r=s=$2^{{-1/2}}$$. Then the skew constant $C_{\$\lambda$,p}(H)$ admits the value $\$\lambda$(\|ru+sv\|,\|su-rv\|)=\|su-rv\|=\sqrt{2}$, whereas Theorem 14 asserts $C_{\$\lambda$,p}(H)=$2^{{1/2-1/2}}$=1$. Computing this admissible value for this $\$\lambda$$ and $p$ settles the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the skew constant $C_{\$\lambda$,p}(X)$ — the supremum of $\$\lambda$(\|ru+sv\|,\|su-rv\|)$ over unit vectors $u,v$ and nonnegative $r,s$ with $\|(r,s)\|_p=1$ — is exactly $$2^{{1/2-1/p}}$$ on Hilbert spaces for $p\ge 2$, under the Hilbert-balance condition $y\,\partial\$\lambda$/\partial x \le x\,\partial\$\lambda$/\partial y$ for $0<x\le y$. The value is obtained by applying Lemma 12, which asserts that this balance condition forces $\Phi(a)=\$\lambda$(\sqrt{R+a},\sqrt{R-a})$ to attain its maximum at $a=0$, so that in Hilbert space the identity $\|ru+sv\|^2+\|su-rv\|^2=2($r^{2}$+$s^{2}$)$ gives $\$\lambda$(\|ru+sv\|,\|su-rv\|)\le ($r^{2}$+$s^{2}$)^{1/2}$. Conversely,

Load-bearing premise

The paper assumes the Hilbert-balance condition (1), $y\,\partial\$\lambda$/\partial x \le x\,\partial\$\lambda$/\partial y$ for $0<x\le y$, makes $\Phi(a)=\$\lambda$(\sqrt{R+a},\sqrt{R-a})$ maximized at $a=0$, which is the step that yields the exact Hilbert value; if that maximization fails, the characterization collapses.

Editorial extensions

If this is right

If the Hilbert-value theorem is right, a Banach space can be certified as Euclidean simply by computing the skew constant and checking whether it equals $2^{1/2-1/p}$.
The equality $C_{\lambda,p}(X)=2^{1-1/p}$ would diagnose the presence of asymptotic squares, giving a quantitative version of uniform non-squareness.
The strict bound $C_{\lambda,p}(X)<\gamma_p$, with $\gamma_p$ the unique root of the displayed equation, would imply uniform normal structure and hence the fixed point property for nonexpansive mappings.
The two-dimensional reduction formula means that infinite-dimensional questions about these constants reduce to checking all two-dimensional subspaces.
The parameter-stability estimate would let one replace complicated weight functions by simpler approximations without changing the constant by more than a controlled amount.

Reading between the lines

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

The paper's balance condition as written appears to have the reverse ordering of what the derivative computation in Lemma 12 requires; if that is so, the Hilbert-space value and the converse characterization hold only for a narrower subclass of $\lambda$, not the whole class $\Lambda_H$.
A natural test is whether asymmetric $\lambda$ such as $\lambda(a,b)=b$ recover the claimed Hilbert value; the paper's own Example 4 suggests that when the balance condition fails, the value can be maximized at non-orthogonal vectors, so the exact identity is sensitive to asymmetry in $\lambda$.
If the claimed Hilbert characterization collapses, the remaining content — global bounds, reduction formulas, and non-squareness criteria — may survive as an interesting numerical invariant even without the exact value identity.
The skew construction suggests a new family of renorming probes: by varying $\lambda$ and $p$ one might obtain a hierarchy of constants measuring deviation from Euclidean geometry, with the von Neumann–Jordan constant and James constant as special slices.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Quick take: this paper defines a skew analogue of the Amini-Harandi-Rahimi constants, and some of the scaffolding is fine—reduction to a single parameter, monotonicity bounds, stability estimates. But the central Hilbert result (Theorem 14) and the converse characterization (Theorem 15) are false, and the error is not subtle.

The load-bearing Lemma 12 asserts Φ(a)=λ(√(R+a),√(R−a)) is maximized at a=0. The balance condition (1) is imposed only for 02—the derivative is actually nonnegative, so Φ increases and the maximum is at a=R, not 0. The orientation is simply backwards. Then Theorem 14 fails even more directly: λ(a,b)=b is C^1, homogeneous, nondecreasing, and satisfies (1); in a Hilbert space with u=−v and r=s=2^{−1/p}, the admissible expression equals 2^{1−1/p}, which is larger than the claimed 2^{1/2−1/p} for every finite p. So the 'Hilbert-balance condition' does not constrain the antiparallel case the way the proof needs, and 'replace c by |c|' is invalid for order-sensitive λ.

What is actually useful here is the package around the false core: the skew family, the reduction formula, and the universal inequalities (Theorem 6) are reasonable. But two additional flaws undercut even that. Lemma 1 claims two 'independent extremal branches'; swapping (u,v) shows A_{λ,p}=B_{λ,p} for every λ, not only symmetric λ. And Theorem 19 feeds the wrong skew combination into the normal-structure argument: with r=R, s=1, the second slot is u−Rv, not v−Ru, so inequality (29) is not of the required form. Theorem 18 is also asserted rather than proved, with the key step deferred to a remark.

Net: this is a paper with a real family and some correct peripheral lemmas, but its advertised contributions (exact Hilbert values, converse characterization, two-branch behavior) do not survive. The reader's rejection is right, and the stress-test counterexample lands exactly where the proof breaks.

Who is this for? Specialists in Banach-space geometric constants might cite the family definition, but not the false theorems. A serious referee would be needed to catch this, and the paper should not be accepted without major repair. My recommendation: do not publish in current form; if the authors reverse the balance condition or restrict the class so Lemma 12 works, the Hilbert computation might be salvageable for a narrower family, but the present claims are not supported.

Referee Report

4 major / 3 minor

Summary. The paper defines a skew-parameterized family of geometric constants C_{λ,p}(X) = sup λ(‖ru+sv‖,‖su−rv‖) over u,v∈S_X and r,s≥0 with ‖(r,s)‖_p=1, together with the primed version C′_{λ,p}(X). It claims reduction formulas to a single ratio parameter, global comparison inequalities, exact Hilbert-space values under a differential 'Hilbert-balance condition' (Theorem 14: C=C′=2^{1/2−1/p} for p≥2 and =1 for p≤2), a converse characterization of Hilbert spaces in dimension at least three (Theorem 15), sufficient conditions for uniform non-squareness and normal structure, and explicit computations in classical spaces. The central Hilbert-space claims are false: the balance condition is applied outside its stated range, and admissible choices with λ(a,b)=b produce values larger than the claimed ones.

Significance. If the Hilbert-space computation and characterization were correct, the paper would provide a unified one-parameter family of geometric constants detecting Hilbert spaces and uniformly non-square spaces. The reduction lemmas and global inequalities (Lemmas 1–11, Theorem 6, Propositions 7–9) are elementary and appear likely to survive; the skew-pair construction is a natural idea. However, the load-bearing results of Section 4 are contradicted by explicit admissible substitutions. The advertised Hilbertian characterization is therefore not established. The paper does not provide machine-checked proofs or reproducible code, and the claimed exact values are falsified by simple examples.

major comments (4)

[Definition 1(iv); Lemma 12] Lemma 12 applies condition (1) at (x,y)=(√(R+a),√(R−a)), where x>y, but (1) is stated only for 0<x≤y. The proof uses the inequality outside its stated domain. This is not a harmless gap: for λ_q(a,b)=2^{−1/q}(a^q+b^q)^{1/q} with q>2, which is C^1, nondecreasing, and satisfies (1) on 0<x≤y, direct differentiation gives Φ′(a)=2^{−1/q}(x^q+y^q)^{1/q−1}(x^{q−2}−y^{q−2})/2>0 for x>y. Thus Φ is increasing and its maximum is at a=R, not a=0, contradicting (17). Lemma 12 is false for symmetric elements of Λ_H.
[Theorem 14] The step 'replacing c by |c| if necessary' is invalid for non-symmetric λ. When c=⟨u,v⟩<0, the pair is (√(R−a),√(R+a)) with a=2rs|c|, not the pair controlled by Lemma 12. For λ(a,b)=b, which lies in Λ_H, take H=ℝ^2, u=(1,0), v=(−1,0), p=2, r=s=2^{−1/2}. Then ‖ru+sv‖=0 and ‖su−rv‖=√2, so the admissible expression equals √2, giving C_{λ,2}(H)≥√2>1=2^{1/2−1/p}. For general p≥2, u=−v and r=s=2^{−1/p} give C_{λ,p}(H)≥2^{1−1/p}>2^{1/2−1/p}. Hence (18) is false and Theorem 14 must be rejected.
[Theorems 15 and 25] Theorem 15's final equivalence relies on Theorem 14 for the forward implication that a Hilbert space attains C_{λ,p}=2^{1/2−1/p}. Since Theorem 14 is false for admissible λ∈Λ_H, the bi-implication as stated is false: for λ(a,b)=b and H Hilbert, C_{b,2}(H)≥√2, not 1. Theorem 25 repeats the same Hilbert-space computation with a weighted gauge and therefore inherits the same defect. The conditional part of Theorem 15 may be salvageable, but the advertised equivalence is unsupported.
[Definition 1(iv), paragraph after (1)] The text says that min{a,b} and √(ab) satisfy the balance condition. For min, on the region 0<x<y one has λ_x=1 and λ_y=0, so yλ_x≤xλ_y is false. For √(ab), yλ_x−xλ_y = (1/2)(y^{3/2}/x^{1/2}−x^{3/2}/y^{1/2})>0 when x<y, so the inequality fails. Thus the examples given do not satisfy (1) as written; the qualifying 'limiting form' comment does not repair the sign. This indicates that the class Λ_H and the statement of Lemma 12 are internally inconsistent, and it explains why the counterexample λ(a,b)=b is admissible.

minor comments (3)

[Lemma 2] The expression 'yx −x y' in (6) is garbled; the intended normalized difference should be written with clear vector notation, e.g. ‖‖y‖ x/‖x‖ − ‖x‖ y/‖y‖‖.
[Theorem 18] The proof is abbreviated: the construction of the normalized pair and the inequality '21/pC′λ,p(X)≥λ(a+t−1,b+t−1)/t' are asserted without full derivation, and the limiting arguments are only sketched. This would need a complete proof in any revision.
[References and typography] There are numerous formatting/OCR artifacts, including garbled names in references [23]–[24] and in the main text, and inconsistent notation for C_{λ,p} in the introduction. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the skew constants are defined independently and all cited inputs are external; the substantive flaws are mathematical, not circular.

full rationale

The paper introduces C_{λ,p}(X) as a new family of constants via Definition 2, with no parameter fitted to any target value. The reduction formulas (Lemma 1), inequalities (Theorems 6 and 8), and Hilbert-space computations (Lemmas 12–13 and Theorem 14) proceed from the definition and stated hypotheses. The Hilbert-balance condition (1) is imported from Amini-Harandi–Rahimi [1], an external prior work with no overlapping authors; it is an assumption, not a renamed conclusion. No uniqueness theorem from the authors' own prior work is invoked, and no ansatz is smuggled in via self-citation. The converse characterization (Theorem 15) uses a classical Aronszajn–Jordan–von Neumann criterion, which is independent of the new constant. Although the balance condition appears to be misoriented and Theorem 14 has counterexamples (e.g., λ(a,b)=b with antiparallel u,v in R^2), those are mathematical errors in the proof, not circularity: the claimed values do not reduce by construction to the input, and no fitted constant is relabeled as a prediction. The paper is not circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on a mis-oriented balance condition (1) and on several constructions quoted from the fixed-point literature whose exact constants are not re-derived. The class Λ_H as defined admits functions for which the Hilbert computation fails, and the proof of Theorem 19 appears to mismatch the admissible skew combination (with coefficients (R,1), the second argument is ||u - Rv||, not ||v - Ru||).

assumptions (7)

ad hoc to paper Hilbert-balance condition (1): y·∂λ/∂x ≤ x·∂λ/∂y for 0 < x ≤ y, defining the class Λ_H and assumed to imply Lemma 12.
The ordering in (1) is opposite to what Lemma 12's derivative computation requires. min and √(ab) fail (1) as written; λ(a,b) = b satisfies (1) and refutes Theorem 14; the ℓ_q means (q > 2) satisfy (1) and contradict Lemma 12. This is the load-bearing assumption that fails.
standard math Aronszajn-Jordan-von Neumann isosceles-pair criterion: if every isosceles unit pair has sum-norm √2, then X is an inner product space (dim ≥ 3).
Invoked without citation in the proof of Theorem 15 to pass from 'all isosceles unit pairs have norm √2' to Hilbertian structure.
standard math James constant bound J(X) ≥ √2; uniformly non-square spaces are reflexive.
Used in Theorem 6 and Corollary 17; classical results from the Banach space geometry literature.
standard math Goebel-Kirk construction with modulus R(1,X) giving inequalities (27)-(28).
Theorem 19 depends on these exact lower bounds; they are not re-derived, and the application to the skew pair appears to require ||u - Rv|| while (28) bounds ||v - Ru||.
domain assumption Saejung-Llorens-Fuster scheme constants in Theorem 20.
The proof asserts asymptotic lower estimates on skew combinations; the details are sketched and the constants are taken from the cited scheme.
ad hoc to paper Translation condition (22): λ(a,b) + t ≤ λ(a+t, b+t) for -min{a,b,1} ≤ t ≤ 0, used in Theorem 18.
The key comparison estimate is asserted, not derived, and Remark 4 defers the supporting lemma to a future version. Verification for the listed examples is only asserted.
standard math Ultrapower characterization of uniform normal structure.
Used at the end of Theorem 19 to upgrade normal structure to uniform normal structure.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Skew-Parameterized Geometric Constants in Banach Spaces." pith.science (2026). https://pith.science/paper/XBX2RSX5

@misc{pith2026260800933,
  author       = {Pith},
  title        = {Pith review of: Skew-Parameterized Geometric Constants in Banach Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBX2RSX5}},
  note         = {Machine review of arXiv:2608.00933}
}

read the original abstract

Building on the family of geometric constants introduced by Amini-Harandi and Rahimi, we define a skew-parameterized family on real normed spaces by replacing the classical symmetric pair with a rotated coefficient pair. This modification reveals new extremal behavior, particularly for asymmetric homogeneous weight functions. We establish reduction formulas, comparison inequalities, and parameter-stability estimates. Under an appropriate differential balance condition, we determine the exact values of these constants on Hilbert spaces and obtain a converse characterization in dimensions at least three. We further derive sufficient conditions for uniform non-squareness and normal structure, together with explicit computations in classical normed spaces. These results provide a unified framework for detecting Hilbertian and fixed-point-related geometric properties of Banach spaces.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Amini-Harandi and M

A. Amini-Harandi and M. Rahimi, On some geometric constants in Banach spaces, Mediterr. J. Math., 16 (2019), 120

work page 2019
[2]

Dinarvand, On a general class of geometric constants and normal structure in Banach spaces, J

M. Dinarvand, On a general class of geometric constants and normal structure in Banach spaces, J. Fixed Point Theory Appl., 27 (2025), 12

work page 2025
[3]

Alonso and E

J. Alonso and E. Llorens-Fuster, Geometric mean and triangles inscribed in a semicircle in Banach spaces,J. Math. Anal. Appl.340(2008), 1271–1283

work page 2008
[4]

Alonso, P

J. Alonso, P. Martin and P. L. Papini, Wheeling around von Neumann-Jordan constant in Banach spaces,Studia Math.188(2008), 135–150

work page 2008
[5]

Alonso, H

J. Alonso, H. Martini and S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces,Aequationes Math.83(2012), 153–189. 19

work page 2012
[6]

Baronti and P

M. Baronti and P. L. Papini, Triangles, parameters, modulus of smoothness in normed spaces,Math. Inequal. Appl.19(2016), 197–207

work page 2016
[7]

J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces,Ann. of Math.38(1937), 114–115

work page 1937
[8]

Y. Cui, W. Huang, H. Hudzik and R. Kaczmarek, Generalized von Neumann- Jordan constant and its relationship to the fixed point property,Fixed Point Theory Appl.2015(2015), Article 40

work page 2015

Show all 27 references

[9]

Gao and K

J. Gao and K. S. Lau, On the geometry of spheres in normed linear spaces,J. Aust. Math. Soc. Ser. A48(1990), 101–112

1990
[10]

Garcia-Falset, E

J. Garcia-Falset, E. Llorens-Fuster and E. M. Mazcunan-Navarro, Uniformly non- square Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal.233(2006), 494–514

2006
[11]

Goebel and W

K. Goebel and W. A. Kirk,Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990

1990
[12]

R. C. James, Uniformly non-square Banach spaces,Ann. of Math.80(1964), 542–550

1964
[13]

Jimenez-Melado, E

A. Jimenez-Melado, E. Llorens-Fuster and S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces,Proc. Amer. Math. Soc.134(2006), 355–364

2006
[14]

M. Kato, L. Maligranda and Y. Takahashi, On James and Jordan-von Neumann constants and normal structure coefficient of Banach spaces,Studia Math.144 (2001), 275–295

2001
[15]

Kato and Y

M. Kato and Y. Takahashi, On the von Neumann-Jordan constant for Banach spaces,Proc. Amer. Math. Soc.125(1997), 1055–1062

1997
[16]

Llorens-Fuster, Zbaganu constant and normal structure,Fixed Point Theory 9(2008), 159–172

E. Llorens-Fuster, Zbaganu constant and normal structure,Fixed Point Theory 9(2008), 159–172

2008
[17]

Llorens-Fuster, E

E. Llorens-Fuster, E. M. Mazcunan-Navarro and S. Reich, The Ptolemy and Zbaganu constants of normed spaces,Nonlinear Anal.72(2010), 3984–3993

2010
[18]

E. M. Mazcunan-Navarro, Banach space properties sufficient for normal structure, J. Math. Anal. Appl.337(2008), 197–218

2008
[19]

Saejung, On James and von Neumann-Jordan constants and sufficient condi- tions for the fixed point property,J

S. Saejung, On James and von Neumann-Jordan constants and sufficient condi- tions for the fixed point property,J. Math. Anal. Appl.323(2006), 1018–1024

2006
[20]

Takahashi and M

Y. Takahashi and M. Kato, On a new geometric constant related to the modulus of smoothness of a Banach space,Acta Math. Sinica30(2014), 1526–1538. 20

2014
[21]

Yang and F

C. Yang and F. Wang, An extension of an inequality between von Neumann- Jordan and James constants in Banach spaces,Acta Math. Sin. Engl. Ser.33 (2017), 1287–1296

2017
[22]

Zbaganu, An inequality of M

G. Zbaganu, An inequality of M. Radulescu and S. Radulescu which characterizes inner product spaces,Rev. Roumaine Math. Pures Appl.47(2001), 253–257

2001
[23]

Garc ´ ıa-Falset, J., Llorens-Fuster, E., Mazc?˜ n´ an Navarro, E.M.: Uniformly non- square Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal.,233, 494–514 (2006)

2006
[24]

Jim´ enez-Melado and E

A. Jim´ enez-Melado and E. Llorens-Fuster, The fixed point property for some uniformly nonsquare Banach spaces, Boll. Unione Mat. Ital-A., 10, 3 (1996), 587– 595

1996
[25]

Gao, A Pythagorean approach in Banach spaces,J

J. Gao, A Pythagorean approach in Banach spaces,J. Inequal. Appl.2006(2006), Article ID 94982

2006
[26]

Takahashi and M

Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly non- square Banach spaces,Nihonkai Math. J.9(1998), 155–169

1998
[27]

Baronti, E

M. Baronti, E. Casini and P. L. Papini, Triangles inscribed in a semicircle, in Minkowski plane and in normed spaces,J. Math. Anal. Appl.252(2000), 124– 146. 21

2000

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Amini-Harandi and M

A. Amini-Harandi and M. Rahimi, On some geometric constants in Banach spaces, Mediterr. J. Math., 16 (2019), 120

work page 2019

[2] [2]

Dinarvand, On a general class of geometric constants and normal structure in Banach spaces, J

M. Dinarvand, On a general class of geometric constants and normal structure in Banach spaces, J. Fixed Point Theory Appl., 27 (2025), 12

work page 2025

[3] [3]

Alonso and E

J. Alonso and E. Llorens-Fuster, Geometric mean and triangles inscribed in a semicircle in Banach spaces,J. Math. Anal. Appl.340(2008), 1271–1283

work page 2008

[4] [4]

Alonso, P

J. Alonso, P. Martin and P. L. Papini, Wheeling around von Neumann-Jordan constant in Banach spaces,Studia Math.188(2008), 135–150

work page 2008

[5] [5]

Alonso, H

J. Alonso, H. Martini and S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces,Aequationes Math.83(2012), 153–189. 19

work page 2012

[6] [6]

Baronti and P

M. Baronti and P. L. Papini, Triangles, parameters, modulus of smoothness in normed spaces,Math. Inequal. Appl.19(2016), 197–207

work page 2016

[7] [7]

J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces,Ann. of Math.38(1937), 114–115

work page 1937

[8] [8]

Y. Cui, W. Huang, H. Hudzik and R. Kaczmarek, Generalized von Neumann- Jordan constant and its relationship to the fixed point property,Fixed Point Theory Appl.2015(2015), Article 40

work page 2015