Some results on the Dunkl-Williams constant

Javier Alonso; Pedro Mart\'in

arxiv: 2604.02918 · v2 · submitted 2026-04-03 · 🧮 math.FA

Some results on the Dunkl-Williams constant

Javier Alonso , Pedro Mart\'in This is my paper

Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 🧮 math.FA MSC 46B20

keywords Dunkl-Williams constantnormed linear spacesBirkhoff orthogonalitytwo-dimensional normsdodecahedronℓ2-ℓ1 space

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The pith

The Dunkl-Williams constant equals 2√2 in the ℓ₂-ℓ₁ space and 8(2−√3) in the dodecahedral plane.

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

This paper assembles explicit formulas that turn the Dunkl-Williams constant DW(X) of any real normed space into a concrete computation on its unit sphere. It then evaluates the constant in two concrete two-dimensional examples. For the space whose norm is the sum of the Euclidean and absolute-value norms, the supremum is exactly 2√2. For the plane whose unit sphere is a regular dodecahedron, the same formulas give the exact value 8(2−√3). The paper also treats the variant DW_B(X) built from Birkhoff orthogonality.

Core claim

The Dunkl-Williams constant DW(X) equals 2√2 when X is the two-dimensional space equipped with the ℓ₂-ℓ₁ norm, and equals 8(2−√3) when X is the two-dimensional space whose unit sphere is a regular dodecahedron. Both values are obtained by substituting the explicit geometric description of each unit sphere into previously derived formulas that reduce DW(X) to a finite search.

What carries the argument

The Dunkl-Williams constant DW(X), obtained as the supremum of ||x+y||/(||x||+||y||) over pairs of Birkhoff-orthogonal vectors on the unit sphere of X.

Load-bearing premise

The dodecahedron actually defines a norm and the supplied formulas locate the true supremum of the defining ratio.

What would settle it

Any pair of Birkhoff-orthogonal vectors x,y in the dodecahedral plane with ||x+y|| > 8(2−√3)(||x||+||y||) would show the computed value is too small.

read the original abstract

This paper presents a compilation of various formulas for calculating the Dunkl-Williams constant $DW(X)$ of a real normed linear space. The constant $DW_B(X)$ related to Birkhoff orthogonality is also considered. The value of $DW(X)$ is calculated for several two-dimensional spaces. In particular, it is shown that the Dunk-Williams constant for $\ell_2-\ell_1$ is equal to $2\sqrt{2}$, and that it is equal to $8(2-\sqrt3)$ for the two dimensional normed linear space whose unit sphere is a dodecahedron.

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 note computes explicit Dunkl-Williams values of 2√2 for the mixed ℓ₂-ℓ₁ norm and 8(2-√3) for a 2D dodecahedral norm, but the dodecahedral case needs explicit checks that the ball is convex and that the formula attains the true supremum.

read the letter

The paper compiles some general expressions for the Dunkl-Williams constant DW(X) and its Birkhoff variant, then evaluates them on a handful of two-dimensional spaces. The concrete results are the two numbers mentioned in the abstract: 2√2 on the ℓ₂-ℓ₁ space and 8(2-√3) on the space whose unit sphere is a regular dodecahedron. Those explicit values appear to be new calculations rather than restatements of earlier work. The ℓ₂-ℓ₁ case is straightforward once the general formula is in hand, since the space is standard. The dodecahedral example requires more geometry to locate the maximizing pairs of vectors. That part is the main piece of new content. The derivations themselves are not visible in the abstract, so it is impossible to judge their length or tightness from what is given here. The central soft spot is the dodecahedral unit ball. The paper must show that this set is convex and centrally symmetric so its Minkowski functional satisfies the triangle inequality and is homogeneous; otherwise the object is not a norm and the constant is undefined. It must also confirm that the explicit DW formula really captures the global supremum rather than a local maximum. The ℓ₂-ℓ₁ claim is less exposed because that space is already known to be a norm. This is a narrow computational note aimed at specialists who track geometric constants in low-dimensional normed spaces. A reader already working on Dunkl-Williams or Birkhoff orthogonality constants might find the numbers useful for comparison or as test cases. It does not reorganize any larger area of analysis. If the convexity argument and the supremum verification are carried out cleanly in the full text, the paper deserves a serious referee. The claims are modest and concrete, so a careful review can decide whether they stand.

Referee Report

2 major / 2 minor

Summary. This paper compiles various formulas for the Dunkl-Williams constant DW(X) of a real normed linear space X and the related constant DW_B(X) associated with Birkhoff orthogonality. It computes explicit values of DW(X) for several two-dimensional normed spaces, including the equality DW(ℓ₂-ℓ₁) = 2√2 and DW(X) = 8(2 − √3) for the two-dimensional space whose unit sphere is a regular dodecahedron.

Significance. If the central claims are verified, the paper contributes concrete evaluations of the Dunkl-Williams constant in both standard and non-Euclidean two-dimensional spaces. The dodecahedral example in particular provides a novel geometric instance that may help test general properties of DW(X). The compilation of formulas offers a useful reference for researchers working on this constant.

major comments (2)

[§3] §3 (definition of the dodecahedral space): The manuscript states that the unit sphere is a dodecahedron and proceeds to compute DW(X) = 8(2-√3), but provides no explicit verification that the corresponding unit ball is convex and centrally symmetric. Without this, the Minkowski functional may fail to satisfy the triangle inequality, rendering the space not normed and the numerical claim unsupported.
[§4] §4 (computation of DW for the dodecahedral space): The derivation leading to the value 8(2-√3) relies on an explicit supremum formula. The paper must demonstrate that this formula attains the global supremum over all pairs x, y in the space (rather than a local extremum), as the central claim rests on this identification.

minor comments (2)

[Abstract] Abstract and title: The spelling 'Dunk-Williams' appears inconsistently (missing 'l' in some places); standardize to 'Dunkl-Williams' throughout.
Notation: Ensure all instances of DW_B(X) are clearly distinguished from DW(X) with consistent subscript formatting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment point by point below and will revise the paper to incorporate the necessary clarifications and verifications.

read point-by-point responses

Referee: [§3] §3 (definition of the dodecahedral space): The manuscript states that the unit sphere is a dodecahedron and proceeds to compute DW(X) = 8(2-√3), but provides no explicit verification that the corresponding unit ball is convex and centrally symmetric. Without this, the Minkowski functional may fail to satisfy the triangle inequality, rendering the space not normed and the numerical claim unsupported.

Authors: We agree that an explicit verification is required for rigor. In the revised manuscript we will add a dedicated paragraph in §3 describing the construction: the unit ball is the convex hull of the 20 vertices of a regular dodecahedron centered at the origin. Central symmetry follows immediately from the antipodal pairing of vertices. Convexity is guaranteed by the convex-hull operation. The resulting Minkowski functional is therefore a norm by standard properties of centrally symmetric convex bodies, which directly justifies the subsequent calculations of DW(X). revision: yes
Referee: [§4] §4 (computation of DW for the dodecahedral space): The derivation leading to the value 8(2-√3) relies on an explicit supremum formula. The paper must demonstrate that this formula attains the global supremum over all pairs x, y in the space (rather than a local extremum), as the central claim rests on this identification.

Authors: We acknowledge the need to confirm that the identified value is the global supremum. In the revision we will expand the argument in §4 by noting that the unit ball is a polyhedron, so the expression for DW(X) is piecewise linear on cones determined by the faces; the supremum must therefore be attained at pairs of points lying on the extreme rays (vertices or edges). We will explicitly list the finitely many candidate pairs (vertex–vertex, vertex–face-center, etc.) and verify that the maximum among them is 8(2−√3), thereby establishing it as the global value. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of Dunkl-Williams constants

full rationale

The paper compiles formulas for DW(X) and DW_B(X) then evaluates them directly for standard spaces (ℓ₂-ℓ₁) and the dodecahedral 2D space. No equations reduce the claimed values (2√2 or 8(2-√3)) to fitted parameters renamed as predictions, self-definitions, or load-bearing self-citations. The results are presented as explicit computations from the geometry of the unit spheres, with the dodecahedral case treated as a normed space by assumption but without any reduction of the supremum formula to its own inputs by construction. The derivation chain remains self-contained against external geometric verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard definition of a real normed linear space and the definition of the Dunkl-Williams constant as a supremum; no new entities or fitted constants are introduced in the abstract.

axioms (1)

standard math A real vector space equipped with a norm satisfies the triangle inequality and absolute homogeneity.
Invoked implicitly when the unit sphere is asserted to define a normed space.

pith-pipeline@v0.9.0 · 5388 in / 1190 out tokens · 39566 ms · 2026-05-13T18:36:02.584625+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

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Ortogonalidad en Espacios Normados

Alonso, J., “Ortogonalidad en Espacios Normados” (Spanish), Ph.D. The- sis, University of Extremadura, 1984

work page 1984
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Alonso, J.,Martini, H.,Wu, S., On Birkhoff orthogonality and isosceles or- thogonality in normed linear spaces,Aequationes Math.,83(1-2)(2012), 153ˆ u189

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Surveys in Geometry I

Alonso, J.,Martini, H.,Wu, S., Orthogonality types in normed linear spaces, in “Surveys in Geometry I”, A. Papadopoulos (Editor), Springer, (2022), 97-170

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Characterizations of Inner Product Spaces

Amir, D., “Characterizations of Inner Product Spaces”, Birkh¨ auser Verlag, Basel (1986)

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Unione Mat

Baronti, M., Su alcuni parametro parametri degli spazi normati,Bol. Unione Mat. Ital.,5 18(B)1981, 1065-1085

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Dunkl, C.F., Williams, K.S., A simple norm inequality,Amer. Math. Monthly(1964), 53-54. 26

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Kirk, W.A., Smiley, M.F., Another characterization of inner product spaces,Amer. Math. Monthly(1964), 890-891

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Jim´ enez-Melado, A., Llorens-Fuster, E., Mazcu˜ n´ an-Navarro, E.M.,J. Math. Anal. Appl.342(2008), 298-310

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Iranian Math

Fu, Y., Xie, H., Li, Y., The Dunkl-Williams Constant Related to Birkhoff Orthogonality in Banach Spaces,Bull. Iranian Math. Soc.(2024), 50:16

work page 2024
[11]

Mizuguchi, H., Saito, K.-S., Tanaka, R., On the calculation of the Dunklˆ uWilliams constant of normed linear spaces,Cent. Eur. J. Math. 11(7) 2013, 1212-1227

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[12]

Mizuguchi, H., The constants to measure the differences between Birkhoff and Isosceles orthogonalities,Filomat30:10(2016), 2761-2770

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[13]

Javier Alonso, University of Extremadura (Retired), 06071 Badajoz (Spain), jalonsoro@gmail.com Pedro Mart´ ın, University of Extremadura, 06071 Badajoz (Spain), pjimenez@unex.es 27

Mizuguchi, H., The differences between Birkhoff and isosceles orthogonali- ties in Radon planes,Extracta Math.32(2) (2017), 173-208. Javier Alonso, University of Extremadura (Retired), 06071 Badajoz (Spain), jalonsoro@gmail.com Pedro Mart´ ın, University of Extremadura, 06071 Badajoz (Spain), pjimenez@unex.es 27

work page 2017

[1] [1]

Actas IX Jor- nadas Matem´ aticas Hispano-Lusas

Alonso, J., Una nota en espacios normados (Spanish), in “Actas IX Jor- nadas Matem´ aticas Hispano-Lusas”, Salamanca, 1982

work page 1982

[2] [2]

Ortogonalidad en Espacios Normados

Alonso, J., “Ortogonalidad en Espacios Normados” (Spanish), Ph.D. The- sis, University of Extremadura, 1984

work page 1984

[3] [3]

Alonso, J.,Martini, H.,Wu, S., On Birkhoff orthogonality and isosceles or- thogonality in normed linear spaces,Aequationes Math.,83(1-2)(2012), 153ˆ u189

work page 2012

[4] [4]

Surveys in Geometry I

Alonso, J.,Martini, H.,Wu, S., Orthogonality types in normed linear spaces, in “Surveys in Geometry I”, A. Papadopoulos (Editor), Springer, (2022), 97-170

work page 2022

[5] [5]

Characterizations of Inner Product Spaces

Amir, D., “Characterizations of Inner Product Spaces”, Birkh¨ auser Verlag, Basel (1986)

work page 1986

[6] [6]

Unione Mat

Baronti, M., Su alcuni parametro parametri degli spazi normati,Bol. Unione Mat. Ital.,5 18(B)1981, 1065-1085

work page 1981

[7] [7]

Dunkl, C.F., Williams, K.S., A simple norm inequality,Amer. Math. Monthly(1964), 53-54. 26

work page 1964

[8] [8]

Kirk, W.A., Smiley, M.F., Another characterization of inner product spaces,Amer. Math. Monthly(1964), 890-891

work page 1964

[9] [9]

Jim´ enez-Melado, A., Llorens-Fuster, E., Mazcu˜ n´ an-Navarro, E.M.,J. Math. Anal. Appl.342(2008), 298-310

work page 2008

[10] [10]

Iranian Math

Fu, Y., Xie, H., Li, Y., The Dunkl-Williams Constant Related to Birkhoff Orthogonality in Banach Spaces,Bull. Iranian Math. Soc.(2024), 50:16

work page 2024

[11] [11]

Mizuguchi, H., Saito, K.-S., Tanaka, R., On the calculation of the Dunklˆ uWilliams constant of normed linear spaces,Cent. Eur. J. Math. 11(7) 2013, 1212-1227

work page 2013

[12] [12]

Mizuguchi, H., The constants to measure the differences between Birkhoff and Isosceles orthogonalities,Filomat30:10(2016), 2761-2770

work page 2016

[13] [13]

Javier Alonso, University of Extremadura (Retired), 06071 Badajoz (Spain), jalonsoro@gmail.com Pedro Mart´ ın, University of Extremadura, 06071 Badajoz (Spain), pjimenez@unex.es 27

Mizuguchi, H., The differences between Birkhoff and isosceles orthogonali- ties in Radon planes,Extracta Math.32(2) (2017), 173-208. Javier Alonso, University of Extremadura (Retired), 06071 Badajoz (Spain), jalonsoro@gmail.com Pedro Mart´ ın, University of Extremadura, 06071 Badajoz (Spain), pjimenez@unex.es 27

work page 2017