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arxiv: 2605.19295 · v1 · pith:ZM4J7M3Inew · submitted 2026-05-19 · 🧮 math.FA

A von Neumann-Jordan Constant of Non-Normable Metrics

Doan Huu Hieu , Nguyen Duy Cuong This is my paper

Pith reviewed 2026-05-20 03:03 UTC · model grok-4.3

classification 🧮 math.FA
keywords von Neumann-Jordan constantnon-normable metricsvector spacesgeneralized constantp-metricsClarkson inequalitymetric-type inequality
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The pith

The von Neumann-Jordan constant generalizes to non-normable metrics when reasonable conditions are met.

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

The paper extends the von Neumann-Jordan constant, originally defined for norms, to non-normable metrics on vector spaces. It identifies conditions under which the known results for norms continue to hold for these more general metrics. A sympathetic reader would care because this broadens the scope of the constant beyond traditional normed spaces, allowing its use in settings where norms may not exist or be natural. Examples, counterexamples, and computations for p-metrics are used to validate the extension.

Core claim

The authors define a generalized version of the von Neumann-Jordan constant for non-normable metrics and prove that under certain reasonable conditions on the metrics, the properties and inequalities established for norms remain valid without additional restrictions. They support this with several examples and counterexamples, and derive precise formulas for the constant in the case of p-metrics on product spaces that satisfy a metric-type Clarkson inequality.

What carries the argument

The generalized von Neumann-Jordan constant for non-normable metrics, which transfers the norm results to this setting by satisfying analogous properties under the identified conditions.

Load-bearing premise

The reasonable conditions on the non-normable metrics are strong enough that no counterexamples to the norm results appear.

What would settle it

A specific non-normable metric that satisfies the reasonable conditions but for which one of the known norm results on the von Neumann-Jordan constant fails to hold would disprove the claim.

read the original abstract

The paper studies a generalized von Neumann-Jordan constant of non-normable metrics on vector spaces. To the best of our knowledge, all existing results of the von Neumann-Jordan constant and its generalizations have been established only in the normed setting. We identify reasonable conditions on non-normable metrics under which results known for norms remain valid. Several examples and counterexamples are provided to justify the established results. The computation for a class of non-normable metrics on product spaces is also investigated. In particular, we give precise formulas for the generalized von Neumann-Jordan constant of p-metrics under a metric-type Clarkson inequality. Comparisons with existing results are discussed throughout the paper whenever applicable.

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 / 3 minor

Summary. The paper introduces and studies a generalized von Neumann-Jordan constant defined for non-normable metrics on vector spaces. It identifies explicit conditions (including a metric-type Clarkson inequality) under which known results from the normed-space setting carry over, supplies supporting examples and counterexamples that delineate the boundary of those conditions, and derives closed-form expressions for the constant in the special case of p-metrics on product spaces.

Significance. If the central claims hold, the work is a modest but useful extension of geometric-functional-analytic constants beyond the normed setting. The explicit formulas for p-metrics and the systematic use of examples/counterexamples to mark the limits of the transfer are concrete strengths that make the results falsifiable and potentially applicable to other non-normable structures.

minor comments (3)
  1. [Transfer theorems section] §3 (or the section stating the transfer theorems): the precise statement of the 'reasonable conditions' on the metric should be isolated in a numbered definition or hypothesis so that the subsequent theorems can cite it directly.
  2. [p-metrics subsection] The formulas for the p-metric case are a positive feature, but the notation for the product metric and the range of p should be fixed once at the beginning of that subsection to avoid repeated re-definition.
  3. [Comparisons with existing results] A short remark comparing the obtained constant with the classical von Neumann-Jordan constant in the norm limit (when the metric comes from a norm) would help readers see continuity with the existing literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no point-by-point rebuttals to provide. We will incorporate any minor editorial or presentational improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends results on the von Neumann-Jordan constant from normed spaces to non-normable metrics by stating explicit conditions (including a metric-type Clarkson inequality) that allow known properties to transfer. It supplies independent definitions, derives closed-form expressions for p-metrics on product spaces, and provides supporting examples plus counterexamples that bound the conditions. No step reduces a claimed prediction or result to a quantity defined by the paper's own fitted parameters, self-citations, or ansatz smuggled from prior author work; the derivation chain remains self-contained against external normed-space benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work relies on standard background results from functional analysis and prior normed-space literature.

pith-pipeline@v0.9.0 · 5638 in / 955 out tokens · 53139 ms · 2026-05-20T03:03:05.266863+00:00 · methodology

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Reference graph

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